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Wiley Encyclopedia of Electrical and Electronics Engineering Electro-Optical Filters Standard Article Hajime Sakata1 1Canon, Inc., Atsugi, Kanagawa 243-01, Japan Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W4402 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (264K)

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Abstract The sections in this article are Directional Coupler Filters Spectral Selectivity Wavelength Tunability Weighted Coupling Dependence on the State of Polarization Survey of Applications About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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ELECTRO-OPTICAL FILTERS

693

ELECTRO-OPTICAL FILTERS Lightwaves hold a great deal of potential in the fields of communication and information processing due to their enormous wavelength resources. The wavelength division multiplexing (WDM) system, in which a data stream is divided into a set of substreams each having a different wavelength, enhances transmission capacity by many orders of magnitude over single-channel transmission lines. Moreover, the WDM concept makes photonic networks flexible and expandable, regardless of the physical topology (1). Such WDM systems require optical filters to extract a wavelength channel from a stream of multiplexed wavelength data. An optical filter that can be electrically tuned to a desired wavelength is known as an electro-optical filter. There are various types of electro-optical filters that have been reported so far. Typical filters are based on directional couplers, distributed Bragg reflectors (DBR), Fabry–Perot resonators, Mach–Zehnder interferometers, and polarization mode converters. Among them, the directionalcoupler-type filter has a single passband in transmission and can send the selected wavelength to the other port without inherent light reflection. This makes it possible to integrate other components into the coupler structure and to prevent optical noise from entering the system. The directional coupler filters thus contribute to important applications as wavelength-selective elements for a variety of photonic or optoelectronic devices including tunable lasers, optical amplifiers, wavelength-selective photodetectors, and photonic switches. DIRECTIONAL COUPLER FILTERS Directional Couplers Made of Asymmetric Waveguides Directional couplers can be made wavelength-selective by dissimilating a pair of optical waveguides. This is because complete power transfer between the two waveguide modes can occur only for one wavelength. The coupling wavelength can be varied through the induced refractive index change of the waveguide. In the early stage of development, coupler filters were mostly made of LiNbO3 crystal. Later, III-V compound semiconductors were used as a waveguide material due to their attractive characteristics, which include a wide refractive index variation, high optical gain, and compatibility with semiconductor devices. We can fabricate coupler filters with AlGaAs/GaAs systems for 0.8 애m operation, while for the 1.3 애m to 1.5 애m range, InGaAsP/InP systems are used. A pair of waveguides can be arranged in either the lateral or vertical direction, as shown in Fig. 1. Laterally arranged couplers are convenient for separating output ports and defining the tuning region. Nevertheless, vertically stacked couplers have further advantages based on epitaxial growth technology, which enables wide-ranging controllability of layer composition and thickness. Characteristics that can be controlled include device compactness, flexibility of the filter bandwidth, and the tunable wavelength range. Strong mode confinement is also attainable by using a large index difference, which is useful for providing one of the waveguide modes predominantly with effective index change, optical gain, and detector absorption. A schematic model of the vertically stacked asymmetric directional coupler is shown in Fig. 2. The coupler is composed of two waveguides, one of which has a high-contrast core with J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

694

ELECTRO-OPTICAL FILTERS

a narrow cross section and the other has a low-contrast core with a wide cross section. The waveguide parameters are determined for each waveguide to support a single mode. Figure 2 also illustrates the effective refractive indexes of the two waveguides as a function of wavelength. The synchronous wavelength (i.e., the filter center wavelength) shown in this figure is the one at which the dispersion curves intersect. The filter bandwidth is determined by the angle of intersection. Note that the shift in the crossover point is usually accompanied by a change in the angle of intersection, resulting in a change in the filter bandwidth. Letting N1 and N2 be the effective refractive indexes for the two waveguide modes, the filter bandwidth (FWHM: full width at half maximum) is approximately given by

(a)

λp =

(b) Figure 1. Waveguide allocation in asymmetric directional couplers: (a) a lateral coupler and (b) a vertical coupler.

0.8λc    ∂N2 ∂N1  − L ∂λ ∂λ λ=λ

(1) c

where L is the coupling length (2). To narrow the filter bandwidth, we fabricate the coupled waveguides so that the index dispersion difference is enhanced while the effective indexes meet at the filter center wavelength. Broberg et al. (3) first attained filter performance showing a bandwidth of 22 nm at the center wavelength of 1.12 애m in an InGaAsP/InP system. With a highly asymmetric coupler structure, a much narrower bandwidth of 1.5 nm has been attained with a 5 mm device length (4) Grating-Assisted Vertical Coupler Filters

n0 n1

n2

Waveguide 2

λ = λC

λ ≠ λC

Waveguide 1

Effective refractive index

n2 Waveguide 2 n1 Waveguide 1 n0

0

λC

Wavelength Figure 2. Synchronous vertical directional coupler filter: the filter structure with its refractive index profile (upper diagram) and wavelength dispersion curves for the two waveguide modes (lower diagram). The synchronous wavelength corresponds to the filter center wavelength, ␭c.

We have seen that the design of the phase-match wavelength in a synchronous asymmetric coupler results in stringent requirements concerning the material used and the geometry of the waveguides forming the coupler. A directional coupler with periodic variation of the coupling coefficient allows the phase-match position to be shifted by setting an appropriate grating period, and thus lessens the requirements for the specific waveguide configuration. Additionally, we can provide further dissimilarity between the two waveguides without respect to natural synchronous conditions; this makes the filter bandwidth narrower. Alferness et al. (5) was the first to realize filter performance in a grating-assisted vertical codirectional coupler. Figure 3 shows a schematic coupler structure in which each waveguide supports each guided mode with a different parity. We should design one waveguide to have a higher effective refractive index than the other waveguide to achieve phase matching via periodic coupling. Thus, the fundamental mode is strongly confined to the high-index waveguide, and we thus call it the high-index mode. In this way, grating-induced coupling with the low-index mode effectively occurs around the high-index waveguide. In a codirectional coupler, the grating period ⌳ is in fact much greater than the grating height, g. As a result the transition distance between the crest and the trough of the grating is negligible compared to the period. The rectangular grating illustrated in Fig. 3 is thus considered to represent a realistic model. Light launched into one waveguide is evanescently coupled into the other waveguide at ␭ ⫽ ␭c, while light detuned from ␭c is uncoupled, remaining in the launching waveguide. Codirectional coupling between the two waveguide modes takes place under

ELECTRO-OPTICAL FILTERS

High-index waveguide

Coupling-assisted grating Λ w g

waveguide. Placing the grating on top has the additional advantages of providing reliable thickness control of vertically coupled layers due to one-time epitaxial growth and allowing for smart integration with various photonic devices because the same material can be used for both the active layer and the high-index waveguide. In the following chapters, we will restrict ourselves to the grating-assisted vertical codirectional coupler for electro-optical filters.

n0 n1 n2

t2 t0

Low-index waveguide

SPECTRAL SELECTIVITY

t1

Bandwidth Narrowing Factors

z

A narrower bandwidth enables more channels to be stacked within a finite wavelength range. We will here consider the factors involved in narrowing bandwidth in grating-assisted codirectional couplers. According to the coupled mode theory (6), the filter bandwidth (FWHM) for the center wavelength ␭c is given by

x

Effective refractive index

High-index mode, nH

λp = Low-index mode, nL

+

λc

0.8λ2c L ng

where ⌬ng is equal to     ∂n ∂n

ng = nH − λc H − nL − λc L ∂λ ∂λ λ=λ c

Λ

λc

Wavelength Figure 3. Grating-assisted vertical coupler filter. The upper diagram shows the geometry and its refractive index profile. The lower diagram illustrates wavelength dispersion of the high-index mode (bold curve) and the diffracted wave of the low-index mode (solid curve).

the following first-order Bragg condition (6):

2π 2π n (λc ) − nL (λc ) = λc H 

695

(2)

where nH and nL are the effective refractive indexes for the high-index and low-index waveguide modes, respectively. A full crossover can be achieved if the coupler length represents L ⫽ 앟/2␬, where ␬ is the coupling coefficient between the two modes. Figure 3 also illustrates the wavelength dispersion curves of the effective refractive indexes for the two waveguide modes. The dispersion curve of the low-index mode intersects with that of the high-index mode at the filter center ␭c with the help of grating diffraction. As the grating period varies, the intersection point shifts according to Eq. (2), resulting in a filter center wavelength change. We can evaluate the position of the phase-match grating either above or below the high-index waveguide. If the grating is located below, the coupling strength rapidly decreases under certain conditions, resulting in no coupling. This is attributable to the zero crossing of the low-index mode found around the lower boundary of the high-index core (see Fig. 3). If, on the other hand, the grating is located above the waveguide, we have access to substantial coupling strength. Thus we place the phase-match grating on top of the high-index

(3)

(4)

Thus the effective group index mismatch of the two coupled filter modes, ⌬ng, and the coupling length L, are the essential factors in narrowing filter bandwidth. Figure 4 shows the dependency of filter characteristics on the refractive index difference between the two waveguides. As the refractive index increases in the high-index waveguide, the coupling length decreases monotonously in line with the stronger mode confinement in the grating region, and the filter bandwidth reveals a local minimum on balance between the coupling length and the effective index difference. Differentiating the refractive index between the two waveguides has an excellent effect on narrowing the bandwidth as well as shortening the coupler length. Additional bandwidth narrowing due to the dispersion difference is shown in Fig. 4(b). As the relative dispersion of the high-index waveguide increases, the filter bandwidth narrows by almost an order of magnitude compared to cases in which there are no dispersion differences. According to the experimental results of the filter center wavelengths and their changes as mentioned later, the group index mismatch is estimated to be ⌬ng ⫽ 0.09 ⫹ 0.43␭ at ␭ ⫽ 0.83 애m for the transverse electric (TE) mode. Note that the dispersion difference rather than the index difference contributes to the ⌬ng factor. The magnitude of dispersion difference is further emphasized by the compound semiconductor properties over dielectric waveguide considerations, resulting in a relatively narrow filter bandwidth. Figure 5 clarifies the correlation between the filter bandwidth and the coupling length by adjusting the structural parameters, which include the layers’ thickness, the height, the period, and the duty-ratio of the grating. Since the filter bandwidth is proportional to ␭c2 /L, it innately requires a longer device length for 1.5 애m operation to obtain a bandwidth equivalent to that in the 0.8 애m region. The bandwidth can be varied from something on the order of 10 nm down to the

ELECTRO-OPTICAL FILTERS

30

20

10

0.5

Coupling length (mm)

Filter bandwidth (nm)

1.0

0 3.5 3.35 3.4 3.45 Refractive index of high-index waveguide (a) 8

Filter bandwidth (nm)

L = 660 µm

guide was removed in 90 애m in order to couple the incident light into the lower waveguide only. A typical bandpass filter response as a function of incident wavelength is shown in Fig. 7(a) for TE polarized light. There is a well-defined peak in the response at 828.5 nm with a bandwidth FWHM of 2.5 nm, which is in good agreement with the calculated value. The theoretical curve is a fit in which only the refractive index of the high-index waveguide core has been adjusted so that the calculated center wavelength coincides with the measured wavelength. The inter-waveguide coupling efficiency at the filter center wavelength is 90%, corresponding to coupling length mismatch of 앑20%. The cyclic baseline structure is an artifact due to the sidelobes of the directional coupler. Figure 7(b) shows the transmission response escaping through the lower waveguide, which reveals the alternate notch at the filter center. Thus, we can pick up other wavelength channels as well with a series of couplers. Equivalent results have been obtained in the 1.5 애m wavelength region with InGaAsP/InP waveguides, for example, a 1.7 nm bandwidth for a device length of 3 mm (8).

6

10 4

8

2

0 1 0.4 0.6 0.8 0 0.2 Dispersion difference, ∂ (n2-n1)/ ∂ λ ( µ m–1)

Filter bandwidth (nm)

696

6

4

2

(b) Figure 4. Narrowing effects in the filter bandwidth. The parameters are: n1 ⫽ 3.39, n0 ⫽ 3.29, t2 ⫽ 0.4 애m, t1 ⫽ 0.1 애m, t0 ⫽ 1 애m, g ⫽ 0.1 애m, and w/⌳ ⫽ 0.6. (a) The filter bandwidth (bold curve) and coupling length (solid curve) versus the refractive index of the highindex waveguide for a 0.83-애m center wavelength. The wavelength dispersion is not included in the calculation. (b) The filter bandwidth as a function of the dispersion difference between the two waveguides; n2 ⫽ 3.44 and ⭸n1 /⭸␭ ⫽ ⭸n0 /⭸␭ ⫽ ⫺0.7 애m⫺1.

Filter Spectral Responses Figure 6 shows an example of a fabricated vertical coupler filter, in which the two buried cores consist of 55 periods of Al0.4Ga0.6As/GaAs (6 nm/3 nm) multiple quantum well (MQW) and 9 periods of Al0.5Ga0.5As/GaAs (8 nm/3 nm) MQW. The grating height is typically 0.1 애m at a grating period of 8.8 애m. The total device length is 600 애m, and the grating section length is 510 애m, that is, the entrance of the upper wave-

0.5 1.0 1.5 Coupling length (mm) (a)

2.0

1.0 2.0 3.0 4.0 Coupling length (mm)

5.0

20

Filter bandwidth (nm)

order of 0.5 nm, with a coupling length shorter than several millimeters. This bandwidth reduction may be achieved at the expense of increasing the coupling length. Such a wide range of available bandwidths is applicable to various types of WDM systems. One may obtain an even narrower bandwidth by the contradirectional coupling mechanism, in which mode coupling occurs in opposite directions via the submicron periodic structure (7).

0 0

15

10

5

0 0

(b) Figure 5. Calculated correlation between the filter bandwidth and the coupling length. (a) AlGaAs/GaAs system; the layer materials are assumed to be 765 nm and 755 nm band-gap AlGaAs for the highand low-index waveguides, respectively. (b) InGaAsP/InP system; the layer materials are assumed to be 1.3 애m and 1.1 애m band-gap InGaAsP for the high- and low-index waveguides, respectively.

ELECTRO-OPTICAL FILTERS

697

Top view Bandpass filter output Stacked channel waveguides Notch filter output Coupling-assisted grating

10 ␮m Cross-sectional view

High-index waveguide Incoming light Low-index waveguide 1 ␮m Figure 6. Schematic view of the grating-assisted vertical coupler filter with AlGaAs/GaAs buried heterostructure waveguides. Photographs of the top and of the cross section are shown in the insets.

Figure 8 shows the output field patterns of the gratingassisted coupler filter for the transverse magnetic (TM) polarized light. A cross-sectional view of the two asymmetric ridge waveguides is shown as an inset of Fig. 8, in which the ridge width is made as narrow as 2.3 애m for single transverse mode propagation in each waveguide (9). The light, which passes through the lower waveguide at a wavelength of 825 nm, moves to the upper waveguide as the wavelength shifts to become longer. Finally, perfect mode conversion is observed at the filter center of 828.1 nm. As the wavelength deviates from the filter center, the output light returns to the lower waveguide again. Figure 9 shows the spectral characteristics of the filters for different grating periods. These filters were made in lateral waveguides and the results are superimposed in Fig. 9. The filter center wavelength shifts by 앑4 nm almost linearly with a grating period change of 0.2 애m. An interwaveguide coupling efficiency of 앑95% and a filter bandwidth of 앑3 nm are achieved. Cross-talk at a wavelength spacing of 4 nm is found to be between 10 and 20 dB. This cross-talk is due to the sidelobes that are inherent to a uniform coupling structure. By varying the temperature of the filters mentioned above, wavelength changes at a rate of 앑0.13 nm/⬚C were measured over a temperature range of 20 to 90⬚C. The thermal wavelength change was fairly small compared to the filter bandwidth. This result corresponds to the relative index change of ⭸(nH ⫺ nL)/⭸T 앒 1.5 ⫻ 10⫺5 ⬚C⫺1. In contrast to single waveguide filters, one may control the temperature coefficient of the filter wavelength through the mismatch between the thermal coefficients of the two waveguides. Several loss factors can be assumed when the light is propagated through the vertical coupler. These include the power coupling efficiency between the entrance and the filter waveguide, inter-waveguide coupling efficiency, absorption loss,

and scattering loss in the two waveguides. The first two items are design-dependent factors that can be fully improved. When an operation wavelength is set to be far from the bandgap wavelength, the residual absorption in the waveguides will be negligible. However, the scattering loss, including an inherent radiation loss due to grating-induced higher-order coupling, is inevitable for the vertical couplers with the etched grating. The calculation by means of the power coupling to the radiation modes estimates the grating-induced radiation loss to be less than 1 dB/mm for a grating height of 0.1 애m (10,11). Systematic evaluations of the loss factors would be needed to assess the potential of grating-assisted coupler filters. WAVELENGTH TUNABILITY Tunable Wavelength Range Wavelength tuning can be realized by varying the refractive index of either of the two waveguides. Figure 10 shows a schematic diagram of asymmetric index change in a vertical coupler filter. If we define the induced index change in a tuning layer as 웃n, the change in filter center wavelength can be derived from the detuning of the phase-match condition of Eq. (2) as follows: δλ =

(H − L )δnλc

ng

(5)

where ⌫H and ⌫L are the confinement factors of the high- and low-index modes in the tuning layer, respectively. The wavelength change 웃␭ can be enhanced by suppressing the initial difference between the group indexes of the two waveguides. We can understand this dependency by Fig. 3, in which the

698

ELECTRO-OPTICAL FILTERS

Experimental Theoretical

Coupling efficiency

1

Nc = 0.5

810

820

830 Wavelength (nm)

1 (H − L )δnL md 0.8λc

(7)

The factor md depends on the filter response shape, which indicates the ratio between the channel spacing and the filter bandwidth. In Eq. (7), the group index mismatch ⌬ng is canceled out between the wavelength change and the filter bandwidth, so that ⌬ng cannot increase the number of accessible channels. Thus, increasing the coupling length with a largebiased index change is the key to accommodating many channels. A high-index waveguide should be used for the tuning layer to enhance the biased index change, (⌫H ⫺ ⌫L)웃n. This is due to two factors. First, the high-index waveguide reveals a larger mode confinement factor than the low-index waveguide. Second, the induced index change is greater in the high-index material because its band-gap wavelength ␭g is closer to the operation wavelength than is the low-index material. Figure 11 shows the calculated wavelength change and filter bandwidth dependence on the thickness of the highindex waveguide, that is, the tuning layer. As the layer thickens, the filter bandwidth narrows owing to the increase in ⌬ng and the decrescent overlapping of the two modes in the grating region, and meanwhile the wavelength change increases. It comes from the enhancement in ⌫H ⫺ ⌫L compared to that in ⌬ng. In such a way, we can determine optimal layer formations for enhancing the tunable channel number.

0 840

(a)

Coupling efficiency (dB)

is a design-adjustable parameter depending on the composition and thickness of the two waveguides (12). For WDM receiver applications, the number of accessible channels is important as opposed to the variable wavelength range itself. Assuming that the laser spectral bandwidth is narrow enough relative to the filter bandwidth, we can estimate the tunable channel number Nc by comparing the wavelength change with the filter bandwidth as follows:

–5

–10

Tunable Grating-Assisted Coupler Filters

810

820

830 Wavelength (nm)

840

(b) Figure 7. Measured filter responses of the interwaveguide coupling efficiency. The two output ports reveal (a) a bandpass filter response (upper waveguide) and (b) a notch filter response (lower waveguide).

crossover wavelength moves faster as the angle of interaction becomes narrower. This relationship can be contrasted with that found in mono-mode filters, such as Fabry–Perot, DBR, and distributed feedback (DFB) filters. The tuning range in a mono-mode device is restricted directly by the effective group index ng of the basic waveguide itself as follows: δλ =

δnλc ng

(6)

where ⌫ is the mode confinement factor. It is advantageous for the vertical coupler filter that the group index mismatch

Refractive index change can typically be attained by carrier injection or through the electric-field electrooptic effect. The maximum refractive index change that can be realized by carrier injection is on the order of 웃n/n ⫽ 0.01. Electric field effects are an order of magnitude smaller; they, however, require low power consumption that is suitable for integration with, for example, a photodetector. An increase in carrier density will result in free carrier absorption, while electrorefraction in semiconductors will accompany an absorption edge shift. Nevertheless, we can compensate for such induced absorption loss through the incorporation of optical amplifiers into the filters. Buhl et al. (13) demonstrated broad-band tunability by a current injection in an InP-based grating-assisted coupler filter. The device consists of two buried rib waveguides in which the upper waveguide is 1.15 애m thick, ␭g ⫽ 1.3 애m quaternary material, and the lower waveguide is composed of ␭g ⫽ 1.1 애m material with a thickness of 0.2 애m. The phase-match grating located above the upper waveguide is 50 nm thick with a period of 16 애m. The filter bandwidth (FWHM) is 5.5 nm at the center wavelength of 앑1.5 애m. The wavelength shift occurs toward shorter wavelengths, which is consistent with the reduction in the refractive index of the upper wave-

ELECTRO-OPTICAL FILTERS

␭ = 825 nm

826 nm

827 nm

699

828.1 nm

1␮m

High-index waveguide (Upper) Low-index waveguide (Lower)

Figure 8. Near-field patterns of the filter output. The inset shows a scanning electron microscope view of a ridge waveguide filter that has a grating period of 10 애m.

guide due to current injection. Figure 12 shows the wavelength change against tuning current and the estimated current density for a 1-mm-long device. Center wavelength shifts as wide as 37 nm have been achieved with injected currents up to 200 mA. The nonlinear shape of the tuning curve corresponds to the index change, which is proportional to the square root of the injection current density. The electric-field tunable structure was examined with a tuning high-index waveguide formed by an AlGaAs/GaAs MQW (14). The whole device forms a p-i-n junction and applies reverse bias voltage to the intrinsic tuning waveguide. The filter response had a passband bandwidth of 1.7 nm at ␭c ⫽ 826.6 nm. Figure 13 shows the measured dependence of the filter center wavelength on the applied voltage. As seen

WEIGHTED COUPLING

1 Λ ( µ m) 9.8

Coupling efficiency

in the figure, the center wavelength shifts to longer wavelengths in proportion to the approximate square of the applied voltage. Since TM polarization is used in this evaluation, the quadratic term in the electric field strongly contributes to the electrooptic effect, while the linear effect is negligible. Wavelength tuning by quantum well electrorefraction has also been achieved in the 1.5 애m region by using an InAlGaAs/InGaAsP MQW structure (15). In this device, the tuning layer consists of ten 1.3-애m-gap InGaAsP wells measuring 10 nm in thickness, with 1.17-애m-gap InAlGaAs for the barriers. Assisted by a grating with a 20 애m period, the filter exhibited a bandwidth of 3.2 nm for a center wavelength of 앑1.57 애m. A tuning experiment conducted under a reverse bias of 4 V yielded a 16 nm shift to a longer wavelength.

0.8

10.0

0.6

10.2 10.4 10.6

Sidelobe Suppression in Vertical Coupler Filters If the mode coupling is constant between the two waveguides, the coupling spectrum is similar to a sinc2-like function and

High-index mode

δn 0.4

Tuning layer ΓL

0.2

0 820

825

830

835 840 Wavelength (nm)

845

ΓH

850

Figure 9. Measured bandpass filter responses of the AlGaAs/GaAs MQW filters, where ⌳ is a grating period. The grating section length (i.e., the coupling length) is 430 애m long with a grating height of 85 nm.

Low-index mode Figure 10. Schematic illustration of tuning principle in a vertically stacked asymmetric coupler. The effective index change is predominantly induced in the high-index mode through the tuning layer.

ELECTRO-OPTICAL FILTERS

40

20

δ n/n2 = 0.5% 20

10

0

0 0.2 0.3 0.4 Thickness of high-index waveguide (µ m)

Figure 11. The wavelength change and the filter bandwidth as a function of the high-index waveguide thickness; ⭸n2 /⭸␭ ⫽ ⫺1.2 애m⫺1 and the other parameters are identical to those in Fig. 4. The parameter of 웃n/n2 indicates the fractional index change in the high-index waveguide, i.e., the tuning layer. The filter center wavelength is designed initially as ␭c ⫽ 0.83 애m.

has large sidelobes. Such sidelobes are undesirable since they require large interchannel wavelength spacing for cross-talk to be avoided. It is well-known that the filter response is approximately proportional to the Fourier transform of the coupling strength distribution (16). Thus, the sidelobes can be greatly reduced by tapering the separation between the two waveguides. This procedure is, however, only accurate for lateral couplers and requires vertical space control which may be difficult to realize in practice. For vertically grating-coupled filters, we can tailor the horizontal dimensions of the phase-match grating to shape the filter response (17). Owing to a relatively coarse grating period, the grating parameters

0

Tuning current (mA) 100 150 50

Wavelength change (nm)

828

827

826 0

20

10

1.25

2.50

3.75

5.00

Tuning current density (kA/cm2) Figure 12. Measured change in the filter center wavelength as a function of injected tuning current and estimated current density in the InGaAsP/InP vertical coupler. The filter response of interwaveguide coupling has an FWHM of 5.5 nm at ␭c 앒 1.5 애m. After L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. A. Burrus, and G. Raybon, Grating-assisted vertical coupler/filter for extended tuning range, Electron. Lett., 29 (1): 81–82, 1993.

3

that is, the duty ratio, the period, and the phase shift, can be formed easily by combining photomask exposure with uniform etching. Reduced sidelobes are also potentially useful for photonic switch applications to improve the extinction ratio and to reduce the sensitivity of the switch state to voltage variations. Since the mode coupling occurs mainly by grating-induced diffraction, the local coupling between the two waveguides is described by a coupling coefficient ␬ as follows (6):
ω0 g κ= E n2s EL dx (8) 8 0 H where ⌬ns2 is the Fourier coefficient in the squared refractive index of the grating, 웆 is the angular frequency, ⑀0 is the electric permittivity of the vacuum, and EH and EL are the electric fields for high- and low-index modes, respectively. The fundamental component of the Fourier series expansion is given by

n2s =

30

1 2 Applied voltage (V)

Figure 13. Measured change in the filter center wavelength as a function of applied reverse bias voltage, Vt, in the AlGaAs/GaAs MQW waveguide coupler. The solid curve is a fit of the form 웃␭ ⫽ q ⭈ V2t with q ⫽ 0.28 (nm/V2).

200

40

0 0.00

Filter center wavelength (nm)

δ n/n2 = 1% Filter bandwidth (nm)

829

60

30

Wavelength change (nm)

700

−

 w (n22 − n20 ) : 0<x n2 ≥ n3 ). Light is confined in the x direction and propagates in the z direction. We first use a zig-zag wave model

Because n2 ≥ n3 , it follows that θs ≥ θc . A guided wave can be understood as a light wave that undergoes total internal reflections at both cover-film and substrate-film interfaces and propagates along the film in a zig-zag manner, as shown in Fig. 3(a), where θ denotes the angle of incidence at the interfaces. The condition π/2 > θ > θs must therefore be satisfied for a guided wave to exist. In the case that θs > θ > θc , light escapes to the substrate, whereas in the case that θc > θ, light escapes to both the cover and the substrate. The light waves in these two cases are called radiation modes, which are not confined in the film. The condition θ = θs is called the cut-off condition. Consider a monochromatic wave with complex time√and z dependences in the form ex p[ j(ωτ − βz)], where j ≡ −1, τ is time, ω is the radian optical frequency, and β is the propagation constant or phase constant. The phase velocity

Waveguides, Optical

3

tion (4) is known as the transverse resonance condition. It is obvious that Eq. (4) admits only discrete incidence angles. The guided waves in a slab waveguide can have two orthogonal polarizations. The wave that is linearly polarized in the y direction is called the TE (transverse electric) mode, because it has a single electric-field component in the y (transverse) direction. The wave that is linearly polarized in the direction that is perpendicular to both the y direction and the wave vector is called the TM (transverse magnetic) mode, because it has a single magnetic-field component in the y direction. It should be noted that the TE mode has magnetic-field components in both the x and z directions, while the TM mode has electric-field components in both the x and z directions. It is known that the phase shifts −22 and −23 depend on the polarization of the wave (see, for example, chapter 2 of reference 4). By using the mode index instead of the angle of incidence as the parameter, the transverse resonance condition Eq. (4) can be written as Figure 3. (a) A guided wave in a step-index slab waveguide can be modeled by a zig-zag wave bouncing up and down in the film under total internal reflections. (b) The z component of the wave vector of the zig-zag wave gives the propagation constant β of the guided wave, while the x component gives the phase constant α in the x direction.

of the wave is given by vp = ω/β. As shown in Fig. 3(b), the propagation constant is simply the z component of the wave vector, i.e., β = n1 k sin θ, where k = 2π/λ is the freespace wavenumber (or free-space propagation constant of the plane wave) and λ is the free-space wavelength. From the range of θ for a guided wave, it is easy to show that n1 >

β > n2 . k

(3)

The dimensionless parameter β/k can be understood as the effective refractive index seen by the wave, and is often referred to as the effective index or mode index. When the mode is at cut-off, β/k = n2 . Not every value of β/k that satisfies Eq. (3) represents a physical guided wave. Suppose we travel in the z direction along with the wave at its phase velocity. We should always see the same phase of the guided wave, i.e., the same field distribution in the transverse direction. The guided wave must therefore have a field distribution in the x direction that is invariant in the z direction. However, according to the zig-zag wave model shown in Fig. 3(a), what we see also are plane waves that bounce up and down within the film. To obtain a z-invariant field distribution in the x direction, such plane waves must form a standing wave in the x direction. To form a standing wave, the phase acquired by the plane waves after traveling a round trip in the x direction must be equal to an integral number of 2π: 4tα − 22 − 23 = 2mπ,

m = 0, 1, 2, . . . ,

(4)

where 4tα represents the phase acquired in the film with α = n1 k cos θ being the phase constant in the x direction, as shown in Fig. 3(b), and −22 and −23 represent the phase shifts acquired by the wave under total internal reflections at the substrate and cover interfaces, respectively. Equa-

2tk[n21 − (β/k)2 ]1/2 = mπ + 2 + 3 ,

m = 0, 1, 2, . . . , (5)

with i = tan−1 {ri [

(β/k)2 − n2i 1/2 ] }, n21 − (β/k)2

i = 2 or 3,

(6)

where the factor ri distinguishes between the TE and TM modes, with ri = 1 for the TE mode and ri = (n1 /ni )2 for the TM mode. For a given free-space wavenumber k (or wavelength λ), the effective indices β/k for all the guided modes can be found from Eq. (5) by a root-searching technique. The integer m in Eq. (5) is called the mode order and the modes are labeled as TEm and TMm modes. In many practical applications, the refractive index of the thin film is only slightly larger than that of the substrate (the difference is of the order of one percent), i.e., n21 − n22  n21 − n23 . The phase shifts 2 and 3 (3 ∼ = π/2) then become insensitive to the polarization of the wave and the effective indices of the TE and TM modes of the same order are nearly equal. In that case, the angle θ shown in Fig. 3(a) is close to π/2 and, as a result, the TM mode is approximately linearly polarized in the x direction with a negligible electric-field component in the z direction. Putting β/k = n2 into Eq. (5) results in an expression for the cut-off wavelength λc : λc =

4πt(n21 − n22 )1/2 √ , mπ + tan−1 (r3 a)

(7)

n22 − n23 n21 − n22

(8)

where a=

is a factor that measures the degree of asymmetry of the waveguide structure (with a = 0 for the symmetric case n2 = n3 ). As an example, with n1 = 1.55, n2 = 1.50, n3 = 1.0 (air), and a film thickness of 2t = 1 µm, the cutoff wavelengths for the TE0 , TM0 , TE1 , TM1 modes are 1.987 µm, 1.720 µm, 0.561 µm, and 0.537 µm, respectively. If the operating wavelength is longer than 1.987 µm, the waveguide does not support any guided mode. If the wavelength lies between 1.720 µm and 1.987 µm, only the TE0 mode

4

Waveguides, Optical

is supported and a single-mode waveguide results. As the wavelength becomes shorter and shorter, more and more modes are allowed to propagate. The number of guided modes supported by the waveguide can be determined from mc , the value of m evaluated at β/k = n2 : mc =

√ 4t 2 1 (n − n22 )1/2 − tan−1 (r3 a). λ 1 π

(9)

The next integer larger than mc gives the number of modes. With the same refractive indices used in the previous example, at a wavelength of 1.3 µm, a film thickness of 1 µm gives mc = 0.21 and for the TE and TM modes, respectively. The waveguide therefore supports only the TE0 and TM0 modes. If the film thickness is increased to 10 µm, we find mc ∼ = 5.6 for both the TE and TM modes, and the waveguide supports six TE modes and six TM modes, a total of twelve modes. On the other hand, we can maintain single-mode operation with a thinner film by increasing the refractiveindex difference between the thin film and the substrate. The use of a high index contrast can therefore reduce the size of the waveguide and hence increase the level of device integration. Electromagnetic theory. While the zig-zag wave model gives the correct equation, Eq. (5), for finding the effective indices of the guided modes, it does not lead to the field distributions. A rigorous analysis of the waveguide should be based on the electromagnetic theory. In what follows, the electric- and magnetic-field components in the p (p = x, y, or z) direction are denoted by Ep and Hp , respectively. By applying Maxwell’s source-free equations to a planar waveguide with an arbitrary refractive-index profile n(x), the following wave equations can be derived (see, for example, reference 14): d 2 Ey + [n2 (x)k2 − β2 ]Ey = 0 dx2

(10)

where ε0 is the free-space permittivity. The problem is to solve for β and the field from the wave equation. There can be many sets of solutions, each set corresponding to a guided mode. For a three-layer step-index slab waveguide, Eqs. (10) and (11) can be solved analytically. With reference to the coordinate system shown in Fig. 2, the field distribution that satisfies Eq. (10) or (11) can be written as ψ(x) = {

U = kt[n21 − (β/k)2 ]1/2 ,

(17)

W2 = kt[(β/k)2 − n22 ]1/2 ,

(18)

W3 = kt[(β/k)2 − n23 ]1/2 .

(19)

and

It is clear from Eq. (16) that the spatial distribution of the field is sinusoidal within the film, while it decays exponentially in the cover and the substrate. The portions of the field that are localized in the cover and the substrate are referred to as the evanescent fields, which can carry a significant amount of power, especially when the mode is close to cut-off. It is known from the electromagnetic theory that tangential electric and magnetic fields must be continuous at an interface between two different dielectric media. Equation (16) ensures that one of the tangential fields is continuous everywhere. For the TE mode, the continuity of the other tangential field Hz requires dψ/dx be continuous, while for the TM mode, the continuity of the tangential field Ez requires [1/n2 (x)]dψ/dx be continuous. By enforcing these boundary conditions at x = 0 and x = 2t, we obtain B = r3

d 1 dHy β2 [ 2 ] + [k2 − 2 ]Hy = 0 dx n (x) dx n (x)

(11)

for the TM mode. For the TE mode, Ex = Ez = Hy = 0 and Hx and Hz are related to Ey by β Hx = − Ey ωµ0

(12)

and Hz = −

1 dEy , jωµ0 dx

(13)

where µ0 is the free-space permeability. Similarly, for the TM mode, Ey = Hx = Hz = 0 and Ex and Ez are related to Hy by β Hy ωε0 n2 (x)

(14)

1 dHy , jωε0 n2 (x) dx

(15)

Ex = and

(16)

where ψ = Ey for the TE mode, or ψ = Hy for the TM mode. In Eq. (16), the normalized parameters U, W2 , and W3 are defined by

for the TE mode, and

Ez =

A exp(W3 x/t), −∞ < x < 0, A cos(Ux/t) + B sin(Ux/t), 0 ≤ x ≤ 2t, (A cos 2U + B sin 2U)exp[−W2 (x − 2t)/t], 2t < x < + ∞,

W3 A U

(20)

and tan 2U =

U(r2 W2 + r3 W3 ) , U 2 − r2 r3 W2 W3

(21)

where r2 and r3 are the same factors that appear in Eq. (6). With Eq. (20), the constant B can be eliminated from Eq. (16), which is then left with an arbitrary amplitude A. The propagation constants of the guided modes can be determined from Eq. (21), which is, in fact, identical to Eq. (5) obtained from the zig-zag wave model. With the knowledge of the propagation constants, the corresponding field distributions can be calculated from Eqs. (12)–(16). Figure 4 shows qualitatively the distributions of the electric-field components for several low-order TE and TM modes. The number of nulls (in the x direction) in the field distribution is simply equal to the mode order m. It should be noted that Ex , the dominant electric-field component for the TM mode, is discontinuous at the interfaces; it is n2 (x)Ex that is continuous - see Eq. (14). A guided mode carries optical power in the z direction. The power density (i.e., power per unit length in the y di-

Waveguides, Optical

5

Figure 5. Universal dispersion curves for several low-order TE modes of a three-layer step-index slab waveguide.

Figure 4. Field distributions for several low-order TE and TM modes of a step-index slab waveguide.

rection) can be calculated by integrating the z component of the Poynting vector in the x direction. The results are P=

β 2ωµ0

for the TE mode, and β P= 2ωε0




+∞

|Ey |2 dx

(22)

|Hy |2 dx n2 (x)

(23)

−∞




+∞

−∞

for the TM mode. By substituting Eq. (16) into the above expressions, the amplitude A in Eq. (16) can be expressed in terms of the power density P. Dispersion curves. Equation (5) or (21), which is often called the dispersion relation or eigenvalue equation, allows us to calculate the dispersion characteristics of the guided mode, i.e., how the effective index β/k varies with the free-space wavenumber k. By scaling Eq. (5) with the normalized frequency V: V = kt(n21 − n22 )1/2 ,

(24)

and the normalized propagation constant b: b=

(β/k)2 − n22 , n21 − n22

(25)

we obtain 2V (1 − b)1/2 = mπ + tan−1 [r2 (

b 1/2 b + a 1/2 ) ] + tan−1 [r3 ( ) ], 1−b 1−b (26)

where a is the asymmetry factor defined by Eq. (8). It can be seen that the normalized parameters are related by V 2 = U 2 + W22 and b = W22 /V 2 (0 < b < 1). By using the normalized parameters, the physical parameters of the waveguide no longer appear explicitly in the equation. The solutions can then be expressed as b − V relations, which are universal in the sense that they are applicable to the entire class of waveguides. Figure 5 shows the universal dispersion curves for several low-order TE modes. By substituting b = 0 into Eq. (26), the cut-off value of V , denoted by Vc , can be determined: √ 1 Vc = [mπ + tan−1 (r3 a)]. (27) 2 In the case of a symmetric waveguide (a = 0), the TEm and TMm modes have the same cut-off V value m(/2, and there is no cut-off wavelength for the fundamental mode (m = 0).

Multilayer Waveguides The approach outlined in the previous section can be applied to a waveguide that consists of more than three layers. The fields in the individual layers are first written down separately and the coefficients in the fields are then determined by the successive application of the boundary conditions at the interfaces. This procedure can in general lead to the dispersion relation. The form and the complexity of the solutions depend on how the fields are represented. A straightforward extension of the three-layer solutions has led to relatively simple dispersion relations for several four-layer and five-layer slab waveguides (4, 11). As the number of layers in the waveguide increases, the dispersion relation becomes more complicated. Nevertheless, compact dispersion relations in a recurrent form have been derived for general multilayer structures (18) and slab waveguide arrays (19). In the special case of a waveguide array with alternating high-index and low-index layers, the guided mode can be treated as a set of coupled zig-zag waves, which satisfy the transverse resonance conditions simultaneously (19).

6

Waveguides, Optical

Another approach to analyzing multilayer slab waveguides is based on modeling a multilayer waveguide with a stack of thin films, each of which is characterized by a twoby-two matrix that relates the reflected and transmitted fields (20). The characteristic matrix of the entire stack is given by the product of the individual matrices. By enforcing the condition that the fields decay exponentially in the first and the last layer, a dispersion relation in the form of a matrix equation results, from which the effective indices can be found by a root-searching technique.

Anisotropic Film Waveguides Many active waveguide devices, such as electro-optic modulators and mode converters, rely on phase modulation of optical waves with applied electric fields. Such devices are usually built upon crystals with large electro-optic coefficients (such as lithium niobate and lithium tantalate), which possess anisotropic dielectric properties. An optically anisotropic crystal can be characterized by three principal refractive indices, which are associated with the principal axes of the crystal. When the three refractive indices have different values, it is called a biaxial crystal. When two of them are equal, it is called a uniaxial crystal. The actual refractive indices experienced by an optical wave propagating in the crystal depend on the direction in which the wave is launched into the crystal with respect to the principal axes of the crystal. In general, different electricfield components of the incident wave experience different refractive indices. This can give rise to the phenomenon of birefringence. In that case, the incident wave breaks up into two waves, which propagate independently in the crystal at different phase velocities. When the principal axes of the materials that form the waveguide are arbitrarily oriented, TE and TM modes do not normally exist. The modes are, in general, hybrid in nature and it is a tedious exercise to find them (21). Fortunately, in most practical applications, the principal axes of the materials are aligned with the waveguide axes. In such special cases, the guided modes can still be classified as TE and TM modes and the simple zig-zag wave model described earlier can be extended to obtain the dispersion relation (22). We refer to Fig. 2(a) for the geometry of the waveguide and, for generality, assume different anisotropic materials in different regions of the waveguide. The principal refractive indices are denoted by nix , niy , and niz , where i = 1 for the film, i = 2 for the substrate, and i = 3 for the cover. Because the TE mode possesses only an electric-field component in the y direction, it experiences only the refractive indices niy . The analysis for the TE mode is therefore the same as that in the isotropic case. The condition for the TE mode to exist is n1y > n2y , n3y , and the dispersion relation is given by Eqs. (5) and (6) with n1 and ni replaced by n1y and niy , respectively. The analysis for the TM mode is slightly more complicated, owing to the fact that there exist two electric-field components. It can be shown (22) that the condition for the TM mode to exist is given by n1x > n2x , n3x , and the dispersion relation for the TM mode obtained from the zig-zag

wave model can be expressed as 2tkK1 [n21x − (β/k)2 ]1/2 = mπ + 2 + 3 ,

m = 0, 1, 2, . . . ,(28)

with i = tan−1 {(

n1z 2 Ki (β/k)2 − n2ix 1/2 ) [ ] }, niz K1 n21x − (β/k)2

i = 2 or 3, (29)

where Ki = niz /nix (i = 1, 2, 3) is a measure of the degree of anisotropy. For most anisotropic materials, Ki has a value between 0.8 and 1.2. Because of the material anisotropy, the effective indices for the TE and TM modes can be very different, and devices based on anisotropic waveguides can be highly sensitive to the polarization state of light. In fact, it is possible to design an anisotropic waveguide to support only one class of modes, and thus function as a waveguide polarizer or a mode filter (23). Metal-Clad Waveguides The three-layer structure shown in Fig. 2(a) becomes a metal-clad waveguide, when metal is employed as the cover material. At the optical frequency, the refractive index of metal is a complex number with a predominant negative imaginary part. For example, the refractive index of silver at the wavelength 0.633 µm is 0.0646 − j4.04. Because of the complex nature of the refractive index, optical waves are attenuated when penetrating into the metal cover. The propagation constant for the guided mode is therefore also a complex number, where the real part is the phase constant and the (negative) imaginary part is the attenuation constant. The dispersion relation and the mode fields can be found by applying the electromagnetic theory to the waveguide in the same way as that described for the stepindex slab waveguide (24, 25). In fact, the dispersion relation for the metal-clad waveguide is identical in form to that for the step-index slab waveguide (24); it is also given by Eqs. (5) and (6), provided that β and n3 in these equations assume complex values. An important feature of the metal-clad waveguide is that it attenuates the TM0 mode much more severely than the other modes. The attenuation constant for the TM0 mode can be larger than that for the TE0 mode by several orders of magnitude (25). This property has been explored for forming polarizers and mode filters. Antiresonant Reflecting Optical Waveguides In the conventional design of an optical waveguide, the refractive index of the guiding layer should be larger than that of the substrate to ensure light confinement without loss. For practical reasons, the substrate material may have a large refractive index. For example, the common substrate material silicon (Si) has a refractive index of 3.4–3.8, which is larger than the indices of most other waveguide materials. To form a low-loss waveguide on a high-index substrate, we may insert a sufficiently thick low-index buffer layer between the guiding layer and the substrate. In the case of using a Si substrate, a layer of SiO2 (with a refractive index of (1.45) can be grown directly on Si by thermal oxidation. It is expensive, however, to produce a thick enough (> 4 µm) SiO2 layer on Si. To overcome this problem, an interference cladding, which consists of

Waveguides, Optical

7

a properly designed high-index film deposited on a SiO2 buffer layer, can be inserted between the guiding layer and the Si substrate to provide a large reflectivity (close to 100 %) at the operating wavelength of the waveguide. In this way, the SiO2 buffer layer required can be relatively thin (∼2 µm). Such a multilayer waveguide is known as an antiresonant reflecting optical waveguide (ARROW) (26). The dispersion relation of an ARROW can be cast in the same form as Eq. (5), except that the phase shift acquired at the interface between the guiding layer and the interference cladding, i.e., 2 in Eq. (5), is obtained from the analysis of a multilayer structure (27). Graded-Index Planar Waveguides Several fabrication techniques, such as ion-diffusion and ion-exchange, can produce waveguides with refractiveindex profiles that vary gradually in the transverse direction. Exact analytical solutions are available, however, only for a few symmetric refractive-index profiles, such as parabolic, sech-squared, and exponential profiles (4). More general profiles must be analyzed with either a numerical method or an approximate method. For a review of some numerical methods, see reference 11. In most practical cases, the refractive-index variation is smooth and slow, and the problem can be solved accurately with the WKB method (Wentzel-Kramers-Brillouin method), which is an approximate analytical method originally developed in quantum mechanics (28). The WKB method. Figure 6 shows a graded-index profile n(x), which decreases monotonically from the surface into the substrate with a peak index n1 at x = 0. The index at infinity n(+∞) is equal to the substrate index n2 and the index for 0 < x is equal to the cover (usually air) index n3 . Because the effective index of a guided mode must lie between n1 and n2 , there must exist a location (turning point) in the x direction at which the effective index is equal to the refractive index of the profile, as shown by the point x = xt in Fig. 6. The idea of the WKB method is to assume an oscillatory field in the region 0 < x < xt , and an exponentially decaying field in the region xt < x < + ∞. This assumption is accurate provided that the index variation is slow, compared with the optical wavelength. To be specific, the field for a guided mode is expressed as

ψ(x) = {

C3 exp(|q|x),
√ (C1 / q)cos( (C2 /



and  = tan−1 {r3 [

q dx − ),




0 ≤ x < xt ,

q2 (x) = a02 (x − xt ) where point:

a02

(33)

(34) 2

can be found from the slope of n (x) at the turning

a02 = k2

dn2 (x) , dx

at x = xt

(35)

With Eq. (34), the wave equation Eq. (10) or (11) has an exact solution: ψ(x) = C0 Ai (r)

(36)

where Ai (r) is the Airy function (29) evaluated at r(x) with −4/3 r(x) = −a0 q2 (x). The solution given by Eq. (36) can be connected to those given by Eq. (30) in the regions x > xt and x < xt with the help of the asymptotic expressions for the Airy function in the respective regions (29). We thus find 1 C1 , 2 √ π C0 = 1/3 C1 , a0 C2 =

(30)

x

|q|exp(−

(β/k)2 − n23 1/2 ] }, n21 − (β/k)2

where r3 = 1 for the TE mode and r3 = (n1 /n3 )2 for the TM mode. As q(xt ) = 0, the approximation given by Eq. (30) fails at the turning point. To find the solution at x = xt , the function q2 (x) in the neighborhood of the turning point is approximated by a linear function:

−∞ < x < 0,

x

0

Figure 6. Refractive-index profile of a graded-index planar waveguide.

|q| dx)), xt < x < + ∞, xt

(37) (38)

and the dispersion relation

where q(x) = [n2 (x)k2 − β2 ]1/2 .




(31)

For the TE mode, ψ and dψ/dx are continuous at x = 0, while for the TM mode, ψ and [1/n2 (x)]dψ/dx are continuous at x = 0. By enforcing these boundary conditions, we obtain r3 cos  C1 C3 =  n21 k2 − β2

(32)

xt

[n2 (x) − (β/k)2 ]1/2 dx = mπ +

k 0

π + , 4

m = 0, 1, 2, · · · .(39)

After the effective index has been solved from Eq. (39), the corresponding mode field can be calculated from Eqs. (30)–(38). Equation (39) has the same form as Eq. (5) and therefore can be interpreted by the zig-zag wave model. The left-hand

8

Waveguides, Optical

side of Eq. (39) represents the phase acquired by the zigzag wave in the guiding region (0 < x < xt ). The right-hand side implies that the wave acquires a (half) phase shift of −π/4 at the turning point, where total internal reflection takes place, and a (half) phase shift of − at the surface. The phase shift  in Eq. (39) is in fact identical to the phase shift 3 given by Eq. (6). Under the condition n21 − n22  n21 − n23 , it follows that 3 ∼ = π2, regardless of the polarization of the wave. If the profile is of the buried type, i.e., n(x) decreases monotonically in both the x and (x directions, the dispersion relation becomes




x2

[n2 (x) − (β/k)2 ]1/2 dx = (m +

k x1

1 )π, 2

m = 0, 1, 2, · · · ,(40)

where x1 and x2 are the turning points in the regions x < 0 and x > 0, respectively. The accuracy of the WKB method has been investigated for a number of profiles (30). The method is more accurate for multimode waveguides. Nevertheless, it gives an adequate accuracy for many single-mode waveguides. Some dispersion formulae. The refractive-index profile n(x) shown in Fig. 6 can be written as n2 (x) = n22 [1 + 2 f (x/d)]

where




(43)

Xt

F=

f (X)1/2 dX

(44)

0

with X = x/d. The number of modes supported by the waveguide is simply given by the next integer larger than mc - see Eq. (9) for the case of a step-index waveguide. We can define a normalized mode order M (0 ≤ M ≤ 1) by (31) M=

m+

3 4 mc + 43

.

Xt

[ f (X) − b(M)]1/2 dX = FM

∇t2 Et + [n2 (x, y)k2 − β2 ]Et = −[Et ·

(46)

0

with f (Xt ) = b(M). For a given profile shape, the normalized propagation constant b depends only on the normalized mode order M. The b − M relation applies to all guided

∇t n2 (x, y) ], n2 (x, y)

(47)

where ∇t is the transverse part of ∇. In general, the waveguide supports hybrid modes or vector modes, which contain all six field components. Once the transverse electric-field is found, the z component Ez and the magnetic field H can be calculated from Ez =

1 [∇t · Et + Et · ∇t lnn2 (x, y)], jβ

(48)

1 ∇ × E. jωµ0

(49)

and H =−

Alternatively, one can first solve the transverse magnetic field from a wave equation similar to Eq. (47) and then find the other field components from Maxwell’s equations. Equation (47) actually consists of two equations coupling the x and y components of the electric field. When the index difference between the guiding region and the substrate is small, as in most practical cases, one of the two transverse field components becomes small (32) and the coupling terms in Eq. (47) can be ignored. This results in two independent quasi-vector wave equations (32): ∇t2 Ex + [n2 (x, y)k2 − β2 ]Ex = −

∂ Ex ∂n2 [ ], ∂x n2 ∂x

(50)

∇t2 Ey + [n2 (x, y)k2 − β2 ]Ey = −

∂ Ey ∂n2 [ ], ∂y n2 ∂y

(51)

(45)

Putting Eqs. (41) and (42) into Eq. (39) (with  = π/2) and using Eqs. (43) and (45), we obtain the normalized WKB equation:




In a planar waveguide, light is confined only in one transverse dimension and can diffract away in the other transverse dimension. To achieve a better control of the light wave and form devices that are compatible with optical fibers, light confinement in both transverse dimensions is necessary. All the waveguides shown in Fig. 1, except for the slab waveguide, can provide two-dimensional light confinement. With the assumption that the wave propagates in the z direction with a propagation constantβ, the transverse electric field Et in a two-dimensional waveguide with refractive-index distribution n(x, y) satisfies the vector wave equation (32), which can be derived from Maxwell’s source-free equations:

(42)

where b is the normalized propagation constant defined by Eq. (25). The value of m at b = 0, denoted by mc , can be calculated by substituting Eqs. (41) and (42) into Eq. (39) (with  = π/2) and setting b = 0: 3 kdn2 (2)1/2 F mc = − π 4

TWO-DIMENSIONAL WAVEGUIDES

(41)

where d is a parameter characterizing the physical depth of the profile,  is the relative difference between the peak index and the substrate index, i.e.,  = (n21 − n22 )/2n22 , and f (x/d) is a normalized function 0 ≤ f ≤ 1 characterizing the shape of the profile. The effective index can also be written in a similar form: (β/k)2 = n22 (1 + 2b)

modes. Table 1 shows the explicit dispersion formulae for some common profile shapes (31), which are obtained either by solving Eq. (46) analytically or by least-squares fitting numerical data with polynomials.

which represent two different classes of modes. The modes governed by Eqs. (50) and (51) are linearly polarized and commonly referred to, respectively, as the Exmn and Eymn modes, where m and n are mode orders (integers ≥ 1), which specify the numbers of peaks in the field in the x and y directions, respectively. If the differentiation between the two polarizations is not important, the right-hand side of Eq. (47) can be dropped completely. This leads to the scalar

Waveguides, Optical

9

Table 1. Explicit dispersion formulae obtained from the WKB approximation (26) for some common profile shapes Profile shape, f (X) Power-law: 1 − Xα , 0 ≤ X ≤ 1 0, 1 ≤ X< + ∞ Sech-squared: {

sech2 (X), 0 ≤ X < + ∞ Tan-squared: π X), 4 Gaussian: 1 − tan2 (

exp(−X2 ),

0 ≤ X< + ∞

0 ≤ X< + ∞

Profile volume, F

1 0

(1 − Xα )1/2 dX

0 ≤ X< + ∞

(1 − M)2

√ 2( 2 − 1)

√ 2 − [1 + ( 2 − 1)M]2



π/2

0≤X+∞

1.0000 − 1.6227M + 0.6817M 2 −0.3837M 3 + 0.3247M 4

0.921915

1.0000 − 0.5877M 1/2 − 1.0819M +0.1446M 3/2 + 0.5250M 2

2

1.0000 − 0.8238M 1/2 − 2.0870M +2.6716M 3/2 − 0.7608M 2

Exponential:

exp(−X),

1 − M α/(2+α)

π/2

Error-function:

erfc(X),

Dispersion formula, b(M)

wave equation: ∇t2 ψ + [n2 (x, y)k2 − β2 ]ψ = 0,

(52)

where ψ can represent any transverse field component. Under the scalar approximation, the Exmn and Eymn modes have the same propagation constant and field distribution. They are often called the scalar modes and can be denoted simply by the Emn modes. The problem of analyzing a two-dimensional waveguide is to solve Eq. (47), or Eqs. (50) and (51), or Eq. (52), depending on the level of accuracy required. It is in general more difficult to solve the full vector wave equation Eq. (47) than the other equations. Conventional numerical methods for boundary value problems, such as the finite element method and the finite difference method, can be employed to solve these equations. These methods are very versatile, as they can handle arbitrary geometries and refractive-index profiles, as well as anisotropic materials. Complications in applying these methods arise, however, in modeling the infinite open space with a finite number of elements or grid points, and in eliminating unphysical spurious solutions that may appear in solving the vector wave equation. There exists another class of numerical methods which are based on series expansion of the mode field. The point-matching method, the mode-matching method, the spectral-index method, and the Fourier-series expansion method are some of the popular ones. These methods are relatively easy to implement and particularly suitable for isotropic step-index waveguides. In addition to numerical methods, a number of approximate semi-analytical methods, such as Marcatili’s method, the effective-index method, the perturbation method, and the variational method, are also available. They can give good results for certain waveg-

uide structures. A survey of various numerical and approximate methods available for analyzing two-dimensional waveguides can be found in reference 32. Rectangular-Core Waveguides Rectangular-core waveguides are the most commonly used waveguide structures in integrated optics. Figure 7(a) shows the cross-section of a rectangular-core waveguide, where 2t and 2w are the thickness and the width of the core, respectively, and n1 , n2 , n3 , and n4 are the refractive indices of the core and the surrounding claddings with n1 > n2 ≥ n3 , n4 . In most practical cases, the index difference between the core and the substrate is small, i.e., (n1 − n2 )/n1  1. The structure shown in Fig. 7(a) represents several important classes of waveguides, including fully buried channel waveguides (n2 = n3 = n4 ), embedded channel waveguides (n2 = n4 < n3 ), and strip waveguides (n2 < n3 = n4 ). Figure 7(b) shows qualitatively the field distributions of some low-order modes in a buried channel waveguide. Here we describe two popular approximate methods, namely, Marcatili’s method (33) and the effectiveindex method (34), for the analysis of the waveguide shown in Fig. 7(a). Both methods solve approximately Eqs. (50) and (51) for the vector modes, or Eq. (52) for the scalar modes. Marcatili’s method. In the method proposed by Marcatili (33), the field in the rectangular-core waveguide is approximated by the product of the fields in two slab waveguides, which are obtained by extending the width and the height, respectively, of the rectangular-core waveguide to infinity, as shown in Fig. 8(a). To be specific, the field for the Eymn (Exmn ) mode of the rectangular-core waveguide, ψmn , is ap-

10

Waveguides, Optical

Figure 7. (a) Cross-sectional geometry of a rectangular-core waveguide, and (b) field patterns for several low-order modes, where the arrows represent the directions of the electric fields.

proximated by ψmn (x, y) ∼ = ψm (x)ψn (y).

(53)

where ψm is the field for the TEm−1 (TEm−1 ) mode of the slab waveguide with thickness 2w, and ψn−1 is the field for the TMn−1 (TEn−1 ) mode of the slab waveguide with thickness 2t. Here all the fields are transverse electric fields. In the case that the polarization effect is not taken into account, i.e., for the scalar modes, the TM mode can be replaced by the TE mode of the same order. The normalized propagation constant for the mode of the rectangular-core waveguide is simply given by b = bm + bn − 1

(54)

where bm = [(βm /k)2 − n22 ]/(n21 − n22 ) and bn = [(βn /k)2 − n22 ]/(n21 − n22 ) are the normalized propagation constants for the modes of the corresponding slab waveguides (with βm and βn the propagation constants). With this method, we only need to analyze two slab waveguides, or use the universal dispersion curves shown in Fig. 5 twice, to obtain an approximate solution for the rectangular-core waveguide. It has been found that Marcatili’s method actually solves a separable rectangular structure, as shown in Fig. 8(b) (35). This structure differs from the original waveguide shown in Fig. 7(a) in having lower refractive indices in the corner regions. Marcatili’s method therefore always underestimates the propagation constant, and gives a poor accuracy in the near-to-cut-off regime, where the fields in the corner regions are significant. A correction factor for the propagation constant can be derived by treating the structure shown in Fig. 8(b) as a perturbation of the original structure shown in Fig. 7(a) (35). The effective-index method. The idea of the effectiveindex method is to replace the rectangular-core waveguide

Figure 8. (a) In Marcatili’s method, the rectangular-core waveguide in Fig. 7(a) is replaced by two slab waveguides. (b) The structure that is actually analyzed by the method differs from the original structure in having lower indices in the four corner regions.

by an equivalent slab waveguide with an effective refractive index obtained from another slab waveguide (34, 36). The effective-index method only requires solutions for two slab waveguides and is as efficient as Marcatili’s method. The effective-index method is, however, more general, as it can be applied to a wide range of geometries and composite structures (36). There are two ways of applying the effective-index method, depending on how the effective refractive index is calculated. In one method, as illustrated in Fig. 9(a), the mode index of the TMm−1 (TMm−1 ) mode in the slab waveguide with thickness of 2t is used as the effective refractive index nx of a second slab waveguide with thickness 2w. The propagation constant of the TEn−1 (TMn−1 ) mode of the second slab waveguide is regarded as the propagation constant of the Eymn (Exmn ) mode of the rectangular-core waveguide. In the other method, as illustrated in Fig. 9(b), the mode index of the TEn−1 (TMn−1 ) mode in the slab waveguide with thickness of 2w is used as the effective refractive index ny of a second slab waveguide with thickness 2t. The propagation constant of the TMm−1 (TEm−1 ) mode of the second slab waveguide is the propagation constant of the Eymn (Exmn ) mode of the rectangular-core waveguide. In both methods, the mode field in the rectangular-core waveguide is the product of the fields in the two slab waveguides. In the scalar analysis, the TM mode is replaced by the TE mode of the same order. The solutions obtained from the two methods for the same mode are in general different. The method that starts with a thinner slab usually gives a

Waveguides, Optical

11

Figure 10. Same as in Fig. 9(a) except that the refractive index in the claddings of the equivalent slab waveguide is represented in a more general way.

Figure 9. In the effective-index method, the rectangular-core waveguide in Fig. 7(a) is replaced by an equivalent slab waveguide, which has an effective index calculated from another slab waveguide. Two ways of applying the effective-index method are possible, depending on whether the slab waveguide with thickness (a) 2t or (b) 2w is used to determine the effective index.

better accuracy. By properly combining the solutions from the two methods, however, it is possible to derive a much more accurate solution (36). The approximations involved in the effective-index method have been investigated for both the scalar (37) and vector (38) modes. Expressions for the errors in the method have been derived (37, 38), which show explicitly under what conditions the method is accurate. It can be shown that the method always overestimates the propagation constants for buried and embedded channel waveguides, but may underestimate the propagation constants for strip waveguides. The method shown in Fig. 9(a) is particularly accurate for strip waveguides. Marcatili’s method and the effective-index method are, in fact, special cases of a more general method (39). A generalization of the method shown in Fig. 9(a) is shown in Fig. 10, where a parameter γ is introduced in the claddings of the equivalent slab waveguide to differentiate between various methods. The conventional effective-index method corresponds to γ = 0, whereas Marcatili’s method corresponds to γ = 1. It can be proved that a much better accuracy for the propagation constant can be obtained, if the following expression for γ is employed (40): γ =1−

(4 /3 )W2 + (4 /2 )W3 2W2 W3 + W2 + W3

(55)

where i = (n21 − n2i )/2n21 (i = 2, 3, 4), and W2 and W3 are the normalized parameters for the slab waveguide with thickness 2t. The method with γ given by Eq. (55) is called the effective-index method with built-in perturbation correction (39, 40).

Dispersion curves. Dispersion curves calculated by various methods for several low-order modes of a fully buried channel waveguide are shown in Fig. 11(a) for the case w/t = 2, and Fig. 11(b) for the case w/t = 1, where the normalized parameters are defined by b = [(β/k)2 − n22 ]/(n21 − n22 ) and V = kt(n21 − n22 )1/2 . The results obtained from Marcatili’s method, the conventional effective-index method (starting with a thinner slab), Marcatili’s method with perturbation correction (35), and the effective index with builtin perturbation correction (39) are compared. All these methods require the solutions of only two slab waveguides and, therefore, have the same degree of efficiency. Accurate numerical data calculated from a finite element method (41) are used as the references. Figure 11 shows clearly that the effective-index method with built-in perturbation correction is significantly more accurate than Marcatili’s method and the conventional effective-index method, and slightly more accurate than Marcatili’s method with perturbation correction. More numerical data for rectangularcore waveguides can be found in references 39 and 40. A rectangular-core waveguide can be designed to support only a number of Em1 modes by using a wide thin core. The propagation constants of these modes have approximately regular spacing and their interference gives rise to the effect of self-imaging, which forms the basis of a class of optical couplers, known as multimode interference (MMI) couplers (42). Diffused Channel Waveguides Diffused channel waveguides refer specifically to channel waveguides that are fabricated with the ion-diffusion technique. The refractive-index profile of a diffused channel waveguide can usually be described by a function that decreases monotonically in both the x and y directions in the substrate with a peak value at the surface of the waveguide. Many general numerical methods and approximate methods are available for analyzing such waveguides (32). Here we describe only the effective-index method (36, 43), which is particularly simple to implement. Figure 12 shows how the effective-index method is applied to a diffused channel waveguide. The idea is to approximate the diffused channel waveguide by an equivalent planar waveguide with an effective index profile ne (x). As shown in Fig. 12, the effective index at a particular point x = xi , i.e., ne (xi ), is given by the mode index calculated from the refractive-index profile of the diffused channel waveguide at x = xi , i.e., n(xi , y), which varies only in the y direction. The propagation constant calculated for the profile ne (x) is treated as the approximate solution for

12

Waveguides, Optical

Figure 12. A diffused channel waveguide is replaced by an equivalent graded-index planar waveguide with refractive-index profile n(xi , y).

Figure 13. Cross-sectional geometry of a rib waveguide.

Rib Waveguides

Figure 11. Universal dispersion curves for a buried channel waveguide calculated by various methods for (a) w/t = 2 and (b) w/t = 1: finite-element method (solid circles); conventional effectiveindex method (dashed curves); Marcatili’s method (dot-dashed curves); Marcatili’s method with perturbation correction (dotted curves); effective-index method with built-in perturbation correction (solid curves). (After Chiang (39). Reproduced by permission of the Institution of Electrical Engineers.)

the diffused channel waveguide. This method requires only the solutions for graded-index planar waveguides and is much more efficient than general two-dimensional numerical methods.

Rib waveguide and its variations are common waveguide structures used in semiconductor-based optoelectronic devices. Figure 13 shows the geometry of a rib waveguide with width 2w, rib-wall height h, and outer slab depth d. The thickness of the guiding region is 2t = h + d. Technically, the effective-index method developed for rectangularcore waveguides can be applied to rib waveguides. With the effective-index method, the rib waveguide shown in Fig. 13 can be approximated by a symmetric three-layer slab waveguide with effective indices for the guiding layer and the claddings determined by the slab waveguides with thickness 2t and d, respectively. It is found, however, that the effective-index method is accurate only when the relative outer slab depth of the waveguide, d/2t, is either large or small (44). A preferred method for the analysis of rib waveguides is the spectral-index method (44). The spectral-index method involves expanding the fields in the waveguide in terms of the local modes (modes of slab waveguides) and matching the fields along the base of the rib. The field matching, which is facilitated with a partial Fourier transform, can lead to a compact implicit dispersion relation. The spectral-index method can give good accuracy and requires much less computation time than general numerical methods. The details of the method as well

Waveguides, Optical

as its applications to more general rib structures are given in reference 44. In the special case of a strip waveguide, i.e., d = 0, a perturbation analysis leads to an accurate explicit dispersion relation for the d = 0 mode ( p = x, y) (45): ª 2

β2p = βp −

m2 π2 2t 2 1/2 [1 − ( ) (1 − 23 S p )], 4w2 wV 3

(56)

where Sx = 1, Sy = 0, i = (n21 − n2i )/2n21 (i = 2, 3), V is the normalized frequency defined by Eq. (24), and ªβ p is the propagation constant of the TEn−1 ( p = x) or TMn−1 ( p = y) mode of the three-layer slab waveguide shown in Fig. 2. For the scalar (Emn ) mode, S p = 0 and ªβx should be used. Equation (56) requires only the solution for a three-layer slab waveguide. It can be proved from Eq. (56) that the propagation constants of the Exmn and Eymn modes can be made equal at a certain value of V by controlling the aspect ratio of the core, w/t (45). A single-mode waveguide with a polarization-independent propagation constant, namely, a zero-birefringence waveguide, is the building block of many polarization-insensitive devices where phase matching must be satisfied for both polarized modes. Zerobirefringence single-mode rib and strip waveguides have been demonstrated experimentally (46). PHOTONIC BANDGAP WAVEGUIDES Photonic bandgap (PBG) structures, also known as photonic crystals, are periodic structures that can take many different forms, such as multilayer stacks, arrays of holes in a single-material dielectric film or fiber, arrays of dielectric pillars, etc. (47). The most important feature of a PBG structure is the existence of forbidden frequency bands that prohibit light waves within these bands from propagating in the structure. Many of the applications with PBG structures are based on the destruction of their perfect periodic configurations by introduction of defects in the structures. The defects can be in the form of a layer of low-index film in the middle of a periodic multilayer stack, one or more missing or additional holes in a film or fiber filled with otherwise strictly regular holes, etc. While light within the forbidden bands cannot propagate in the perfect PBG structure, it can propagate through the defects with a low loss. Defect control in a PBG structure thus provides a highly effective means for the manipulation of light. A number of miniaturized optical components, wavelength filters, and lasers have been demonstrated with various kinds of PBG structures fabricated in different materials. The physical principles of different kinds of PBG structures are detailed in reference 47, while a comprehensive review of the technologies for the fabrication of PBG devices is given in reference 48. As PBG waveguides are the most basic structures used in PBG devices, we discuss a simple one-dimensional PBG waveguide as an example to illustrate the general dispersion properties of this class of waveguides. Zig-Zag Wave Model Figure 14 shows a one-dimensional PBG waveguide, where the guiding layer has refractive index ng and thickness

13

2t, and the claddings on both sides of the guiding layer are semi-infinite PBG structures made up of multilayer stacks of alternating refractive indices n1 and n2 with corresponding thicknesses t1 and t2 . The structure is invariant in the y direction and light propagates in the z direction. The main feature of the PBG waveguide is that the guiding layer assumes a lower refractive index than its surrounding cladding (i.e., ng ≤ n2 < n1 ). While the principle of total internal reflection does not apply to the PBG waveguide, light guidance in such a waveguide can be understood from the fact that, within the forbidden bands of the PBG structures, light undergoes complete reflection and can be trapped in the guiding layer, regardless of its low index. The guiding layer acts as a defect in the infinite periodic structure. Therefore, light propagation in the PBG waveguide can be described as plane waves bouncing back and forth along the waveguide as in the case of a conventional slab waveguide. With the zig-zag wave model, the dispersion relation of the PBG waveguide can be expressed in the form of the transverse resonance condition as 2tk[n2g − (β/k)2 ]1/2 = mπ + 2,

m = 0, 1, 2, . . . ,

(57)

where k is the free-space wavenumber, β is the propagation constant, and −2 is the phase shift acquired when the wave is incident upon the boundary between the guiding layer and the infinite PBG structure. Equation (57) has the same form as Eq. (5). For a conventional three-layer slab waveguide, the phase shift at the boundary is derived from the condition of total internal reflection, as given by Eq. (6), and the range of the effective index β/k (and hence the range of the incidence angle θ) is small. For the PBG waveguide, however, the phase shift −2 is a characteristic of the PBG structure, which is derivable from a transfer-matrix method (49). The range of the effective index is given by 0 ≤ β/k ≤ ng , which is large. The upper bound (β/k = ng ) corresponds to grazing incidence (θ = π/2), while the lower bound (β/k = 0) corresponds to normal incidence (θ = 0), which suggests that the effective index can be much smaller than the refractive index of the material. Dispersion Curves To facilitate discussion, we define the normalized frequency V as V = k(n21 − n22 )1/2 ,

(58)

and the normalized propagation constant b as b=

(β/k)2 − n22 , n21 − n22

(59)

where  = t1 + t2 is the pitch of the periodic structure. The dispersion characteristics of the waveguide are expressed as the relationship between b and V (the b − V curves). Here, V is defined in terms of the parameters of only the PBG structure, so that the dispersion curves for all the waveguides, regardless of their thicknesses and refractive indices, can be compared with reference to the same band diagrams of the PBG structure. For the sake of simplicity, we assume ng = n2 . For a guided mode, the condition 0 ≤ β/k ≤ n2 is translated into 1/(1 − r12 ) ≤ b ≤ 0, where r12 =

14

Waveguides, Optical

Figure 14. A one-dimensional photonic bandgap (PBG) waveguide formed by sandwiching a low-index layer (film) between two identical semi-infinite periodic slab structures.

(n1 /n2 )2 . The dispersion curves of the TE modes are actually independent of r12 , so there is no need to specify the value of r12 for the calculation of their dispersion curves. The dispersion curves of the TM modes, however, depend on the value of r12 . As an example, we set t1 = t2 , t = , and r12 = 2.25. The dispersion curves for the TE and TM modes are shown in Fig. 15(a) and Fig. 15(b), respectively, where the shaded areas are the forbidden bands of the PBG structure. Each forbidden band is labeled by a band order n (= 1, 2, 3,. . . ). Within each band, the mode is labeled by a mode order m, the integer that appears in Eq. (57). A comparison of Fig. 15 and Fig. 5 shows that the dispersion characteristics of the PBG waveguide are distinctly different from those of a conventional slab waveguide. Apart from the difference in the range of the effective index, the dispersion curves of the PBG waveguide are fragmentary. As shown in Fig. 15, many dispersion curves have the same mode order m, but they are not connected because of the presence of band gaps. These curves terminate at the band edges at which the modes are at cut-off. It is therefore possible to operate the waveguide at a suitable value of V, so that the higher-order modes instead of the lower-order modes are supported. The dispersion curves of the TM modes show some additional features. As shown in Fig. 15(b), there exist two special lines, the B-line and the L-line. The B-line specifies Brewster incidence at which the effective index is given 2 by β/k = n1 n2 /(n21 + n22 )1/2 or b = 1/(1 − r12 ) = −0.246. Under this condition, light can pass through the PBG structure without decaying (49). The forbidden bands shrink to points along the B-line. The L-line specifies the minimum value of b that is allowed, i.e., b = 1/(1 − r12 ) = −0.8, which corresponds to β/k = 0. A comparison of Fig. 15(a) and Fig. 15(b) shows that the TM bands fall completely within the TE bands. In the analysis of omnidirectional reflection of the PBG structure, only the TM forbidden bands need to be considered (50). A careful inspection of the labels of the curves in Fig. 15 shows some missing mode orders (for

Figure 15. Universal dispersion curves for a one-dimensional PBG waveguide with ng = n2 , t1 = t2 , t = , and r12 = 2.25 for (a) the TE modes and (b) the TM modes.

Waveguides, Optical

example, m = 3 and 7 for the TE modes). These missing modes do not exist because their dispersion curves shrink into points together with the forbidden bands (not due to Brewster incidence). As discussed earlier, each of the guided modes can be labeled uniquely with the mode order m and the band order n. In fact, these mode orders allow the field patterns of the guided modes to be visualized intuitively. The electric field of the TE mode, i.e., Ey (x), contains m + 1 peaks in the guiding layer and n zero crossings in each period of the PBG structure. The magnetic field of the TM mode, i.e., Hy (x), contains m + 1 peaks in the guiding layer above the B-line while m zero crossings in the guiding layer below the B-line, and n zero crossings in each period of the PBG structure. It should be mentioned that the guiding layer of the PBG waveguide can have a refractive index larger than that of the surrounding PBG structure. In that case, forbidden bands still exist but the band gaps can become much narrower. Two-dimensional PBG waveguides such as one that has a low-index rectangular core surrounded by periodic cladding structures can be converted approximately into one-dimensional PBG waveguides by the effective-index method in the same way as depicted in Fig. 8 and Fig. 9. Analysis of more general PBG structures that involve index variations along the z direction (e.g., structures that contain arrays of air holes in the wave propagation direction) usually requires a purely numerical approach based on a propagation method, such as the finite-difference timedomain method (51). DIRECTIONAL COUPLERS When two parallel waveguides are placed in close proximity to each other, a directional coupler is formed. Under appropriate conditions, light launched into one of the waveguides can couple completely into the neighbor waveguide. Directional couplers are used widely in optical communication systems and integrated optical circuits as power dividers, wavelength multiplexers/demultiplexers, and polarization splitters. In this section, we describe two commonly used methods for analyzing directional couplers, with a coupler consisting of two parallel identical stepindex slab waveguides as an example. A more rigorous treatment of directional couplers can be found in reference 12. Normal-Mode Analysis Figure 16(a) shows two parallel identical three-layer stepindex slab waveguides, where 2t and n1 are the thickness and the index of the guiding slabs, respectively, n2 is the index of the surrounding medium, and 2d is the separation between the two waveguides. In the normal-mode analysis, the directional coupler is treated as a composite waveguide, i.e., a five-layer step-index slab waveguide in the present example. The composite waveguide supports two classes of modes: the even modes, which have symmetric field distributions, and the odd modes, which have antisymmetric field distributions. The field distributions for the lowest-

15

order even and odd TE modes are also shown in Fig. 16(a). For the sake of simplicity, we assume that the composite waveguide supports only the lowest-order even and odd modes. When light is launched into only one of the guiding slabs, these two modes are simultaneously excited and each carries approximately half of the input power. This can be seen by superposition of the two mode fields shown in Fig. 16(a), which results in a total field localized in only one guiding slab. Because the two modes propagate at different phase velocities, they acquire a phase difference as they propagate along the coupler. The phase difference acquired increases with the distance and is given by (β+ − β− )z, where β+ and β− are the propagation constants for the even and odd modes, respectively. We can assume that the two modes are in phase when they are excited at z = 0. After propagating a distance z = Lc given by Lc =

π , β+ − β −

(60)

the two modes become 180 degree out of phase. A superposition of the two mode fields then produces a total field localized in the opposite guiding slab, i.e., the optical power is transferred to the opposite guiding slab. In fact, the optical power is transferred back and forth between the two guiding slabs at every Lc . The behavior of a directional coupler is therefore characterized by Lc or β+ − β− ≡ 2C. Lc and C are commonly referred to as the coupling length and the coupling coefficient, respectively. To calculate Lc or C, we must find β+ and β− . Using the method outlined in the previous section for analyzing a three-layer slab waveguide, we obtain the following dispersion relations for the five-layer slab waveguide: tan 2U =

r2 UW2 [1 + tanh(W2 d/t)] U 2 − r22 W22 tanh(W2 d/t)

(61)

for the even modes, and tan 2U =

r2 UW2 [1 + coth(W2 d/t)] U 2 − r22 W22 coth(W2 d/t)

(62)

for the odd modes, where r2 = 1 for the TE mode and r2 = (n1 /n2 )2 for the TM mode, and U and W2 are given by Eqs. (17) and (18), respectively. The propagation constants β+ and β− can be solved exactly from Eqs. (61) and (62), respectively, by a root-searching technique. Coupled-Mode Analysis The normal-mode analysis just described relies on the knowledge of the modes of the composite waveguide. Except for very simple structures, exact solutions for such modes are difficult to find. A more practical approach to analyzing directional couplers, especially when complicated waveguide geometries are involved, is based on the coupled-mode theory (14), which requires only the solutions of the modes of the individual waveguides that form the coupler. In the coupled-mode theory, the modes of the composite waveguide are approximated by the modes of the individual waveguides in isolation. For the coupler shown in Fig. 16(a), the fields of the lowest-order even and odd modes, ψ+ (x) and ψ− (x), can be expressed, respectively, as

16

Waveguides, Optical

Figure 16. (a) Geometry of a directional coupler consisting of two parallel identical three-layer slab waveguides together with the field distributions for the two lowest-order TE modes of the composite five-layer waveguide structure. (b) The normalized coupling coefficient of a directional coupler decreases rapidly with increasing the normalized frequency and the relative waveguide separation. Numerical results are obtained for the coupler in (a) from the normal-mode theory (solid curves) and the coupled-mode theory (dotted curves).

the sum and the difference of the fields in the three-layer slab waveguides that form the coupler: ψ± (x) ∼ = ψ(x − d − t) ± ψ(x + d + t),

(63)

where ψ(x) is the field of the fundamental mode of the three-layer slab waveguide. For the sake of simplicity, we consider only the TE mode. By substituting Eq. (63) into the TE wave equation, Eq. (10), and integrating the resultant equation over the entire x domain, we find

 d+2t 2 ∼ 2 β± = β ± k2 (n21 − n22 )

d+t

ψ(x − d − t)ψ(x + d + t)dx

 +∞ −∞

ψ(x)2 dx

fields in the two waveguides over the guiding region of one of the waveguides. Equation (64) involves only the solutions for the fundamental mode of a single waveguide. Using the solutions for the three-layer slab waveguide, we obtain: 2 B 2 2 ∼ 2V β+ − β− = 2 t N

with B=

, (64)

where β is the propagation constant for the TE0 mode of the three-layer slab waveguide. The overlap integral in Eq. (64) measures the degree of overlapping between the mode

(65)

2d 2W2 exp(−W2 ) V2 t

(66)

V2 W2 1 + 2 + , U2 U W2

(67)

and N=

Waveguides, Optical

where the normalized parameters are defined for the threelayer slab waveguide – Eqs. (17), (18) and (24). It should be noted that the accuracy of the assumption Eq. (63), and hence, the coupled-mode solution Eq. (64), increases with the normalized waveguide separation d/t and the normalized frequency V . Figure 16(b) shows the normalized coupling coefficient C as a function of V for several values of relative waveguide separation d/t, where C ≡ 0.25V [(β+ /k)2 − (β− /k)2 ]/(n21 − n22 ). Results obtained from the normal-mode analysis and the coupled-mode analysis are hardly distinguishable, except when both d/t and V are small. As shown in Fig. 16(b), the normalized coupling coefficient decreases rapidly as the normalized frequency and the relative waveguide separation increase. As an example, with 2t = 2 µm, n1 = 2.20, n21 − n22 = 0.02n21 , we have V = 1.5 at the wavelength 1.3 µm. When the waveguide separation is equal to the thickness of the guiding slabs (2d = 2 µm), the coupling length is approximately 0.7 mm. If we double the waveguide separation (2d = 4 µm), the coupling length will increase by roughly ten times, to approximately 8 mm. The coupled-mode theory also leads to simple expressions to describe power evolution in the individual waveguides. When the two waveguides are identical and lossless and optical power is launched into only one of the waveguides at z = 0, the powers in the two waveguides, denoted by P1 (z) and P2 (z), respectively, are given by P1 (z) = Pi cos2 (Cz)

(68)

P2 (z) = Pi sin2 (Cz),

(69)

and

where Pi is the input power. These expressions are consistent with our earlier discussion with the normal-mode analysis that a complete transfer of optical power from one waveguide to the other takes place over a coupling length, i.e., at z = Lc = π/2C. When the length of the coupler is chosen to be equal to an odd multiple of Lc /2. The coupler functions as a power divider with a 50/50 splitting ratio (i.e., a 3-dB coupler), which distributes optical power equally between the two waveguides. In principle, any splitting ratio can be achieved by choosing a suitable length. It is also possible to design a coupler that functions as a wavelength multiplexer or demultiplexer, by making use of the fact that the coupling coefficient C depends on the wavelength. When the length of the coupler is equal to an even number of coupling lengths at one wavelength, and an odd number of coupling lengths at the other wavelength, the two wavelengths of light, both launched into one of the waveguides, can evolve separately from the two waveguides. Similarly, a polarization splitter can also be realized with a directional coupler, if the coupling coefficient is sensitive to the polarization state of light. A directional coupler can also be formed with two dissimilar waveguides. In general, a complete transfer of optical power cannot take place in such an asymmetric coupler at all wavelengths. However, it is possible to design an asymmetric coupler that allows a complete power transfer at a specific wavelength, and hence, serves as a wavelength filter.

17

Although we detail only the analysis of two parallel slab waveguides, the underlined principles apply to general waveguide geometries. In fact, the results obtained for parallel slab waveguides can be readily extended to directional couplers consisting of parallel rectangular-core waveguides by means of the effective-index method (52). Directional couplers with polarization-insensitive splitting ratios, which are formed with rectangular-core waveguides, have been demonstrated experimentally (53). As shown by the results in Fig. 16(b), evanescent-field coupling between two waveguide cores is effective only when the two cores are sufficiently close to each other. For widely separated cores where evanescent-field coupling is negligible, we can still achieve strong light coupling between the cores by introduction of properly designed periodic structures (i.e., gratings) along the cores (54). FABRICATION TECHNIQUES Techniques commonly used in fabricating optical waveguides can be divided into three categories, according to how the guiding layer of the waveguide is formed. The three categories are (i) deposition, in which material is deposited in the form of thin film on a lower-index substrate; (ii) ionmigration, in which ions are introduced into a substrate to raise the refractive index of the substrate near the surface; and (iii) epitaxial growth, in which a layer of crystallized material is grown epitaxially on a similar substrate material. In general, deposition and epitaxial techniques lead to step-index waveguides, while ion-migration techniques give rise to graded-index waveguides. A fabrication technique could be applied to many different material systems, while waveguides based on the same material system could be fabricated with more than one technique. Table 2 shows some typical waveguide materials together with the more commonly used fabrication techniques. More detailed discussions on waveguide fabrication can be found in the relevant chapters in the reference books (3,7,9,10,13,55). Deposition Techniques Sputtering is the most versatile deposition method and allows deposition of glass or crystalline films on glass substrates as well as on crystalline substrates. In the sputtering process, atoms from a solid target material are made to eject in a vacuum by the bombardment with atoms or ions, and the ejected particles are collected on a nearby substrate to form a thin film. In the glow-discharge sputtering method, the ions used for bombardment are generated from a plasma, which is formed by ionizing an inert gas (e.g. argon) with a high voltage (several kilovolts) applied to the target (which serves as the cathode). For a metal target, a dc voltage is applied (dc sputtering), while for a dielectric target, a radio-frequency (RF) voltage is used (RF sputtering) to prevent accumulation of positive ions at the target, which could counteract the applied voltage and stop the process. Alternatively, high-speed ions can be generated with an ion gun instead of a glow-discharge plasma (ionbeam sputtering). Ion-beam sputtering can allow a better control of the substrate temperature and the physical parameters in the deposition process. It is also possible to al-

18

Waveguides, Optical Table 2. Some common waveguide materials and fabrication techniques

Material Glasses

Refractive index 1.45 − 1.55

Polymers

1.45 − 1.65

LiNbO3 ZnO Silicon on insulator (SOI) SiOx Ny on silica Chalcogenide Ga1−x Alx As Ga1−x Inx As1−y Py

no = 2.29, ne = 2.20 (at λ = 0.633 µm) 2.0 (at λ = 0.633 µm) 3.47/1.46 (at λ = 1.55 µm) 1.46 − 2.0 2–3 3.4 – 3.6 3.2 – 3.6

Type of waveguide Step-index Graded-index Step-index Graded-index Step-index Step-index Step-Index Graded-index Step-index Step-index Step-index

low oxygen to react with a metal target during deposition to form a metal oxide film at the substrate surface (reactive sputtering). In general, sputtering provides pure, durable, and low-loss films, but the deposition rate is usually low (of the order of 0.01 – 0.1 µm per minute). For the deposition of polymers, the spin-coating and dipcoating methods can be used. A substrate is first covered with, or dipped into, a liquid of polymer solution. The thickness of the liquid layer is then controlled by either spinning the substrate at an appropriate rate (spin-coating) or by lifting it up vertically to allow excess liquid to run off (dip-coating). The film is subsequently dried in air and baked in an oven. This method is simple and capable of producing low-loss waveguides, but it is difficult to control the film thickness and uniformity to a high accuracy. A more elaborate method for producing polymer thin films is called plasma polymerization, which involves the use of an electrical discharge to convert a monomer (low molecular weight organic compound) chemically into a smooth polymer film. This method provides a much better control of the film thickness. Other deposition techniques include thermal vapor deposition and chemical vapor deposition. In the former, a thin film is formed by depositing a material vaporized in a vacuum chamber on a substrate, while in the latter, thinfilm deposition is the result of chemical reactions of gases. Chemical vapor deposition is a common method for making silica-based waveguides. In fact, it is a standard method for making optical fiber preforms. The thin-film deposition process can be enhanced at lower temperatures by using a vapor that contains electrically charged particles (plasma). This technique is known as plasma enhanced chemical vapor deposition (PECVD). It is widely employed for the fabrication of high-index-contrast waveguides based on siliconon-insulator (SOI) or silicon oxynitride (SiOx Ny ) on silica (56).

Fabrication technique RF sputtering Ion-exchange Spin-coating Polymerization Metal in-diffusion Proton-exchange RF sputtering PECVD PECVD Thermal vapor deposition Epitaxial growth Epitaxial growth

field assisted diffusion). The electric field can also drive the diffused ions well below the surface of the substrate to produce a buried waveguide and eliminate surface scattering. Titanium-diffused lithium niobate (Ti:LiNbO) waveguides and copper-diffused lithium tantalate (Cu:LiTaO3 ) waveguides are well-known examples of waveguides fabricated with thermal diffusion. In the ion-exchange method, a substrate is immersed in a molten salt at a suitable temperature (200 – 350 ◦ C) for a period of time. The ions in the substrate are then exchanged with the ions in the molten salt. This process in general results in a gradual change in the refractive index near the surface of the substrate. In some cases, the ionexchange process can be accelerated by applying an electric field (electric-field assisted ion-exchange). For example, glass waveguides can be made by exchanging sodium ions in glass with silver, potassium, or thallium ions. A detailed description of the ion-exchange method for a wide range of glass waveguides can be found in reference 55. In the special case that metal ions in a crystalline substrate are exchanged with hydrogen ions (protons) from an acid, the method is called the proton-exchange method. The proton-exchange method can usually lead to a relatively large refractive-index change and a step-like profile. For example, a lithium niobate waveguide can be made by exchanging lithium ions with protons in a solution of benzoic acid at around 250 ◦ C. The process may take a few minutes for a single-mode waveguide to many hours for a heavily moded waveguide. Graded-index waveguides can also be fabricated with the ion-implantation technique, in which a stream of accelerated ions are made to penetrate into a substrate material and, consequently, cause lattice disorder, and hence, index change in the material. This technique is expensive to implement, but it can be applied to a wide range of materials and provide an accurate control of the waveguide parameters.

Ion-Migration Techniques In the thermal diffusion method, a metal film is deposited on a substrate and heated at a high temperature (around 1000 ◦ C) for a period of several hours. The metal is then diffused into the substrate and a smooth graded-index profile near the surface of the substrate is formed. An electric field may be applied during the diffusion process to accelerate the process and reduce the temperature required (electric-

Epitaxial Techniques Epitaxial techniques are widely used for the fabrication of semiconductor waveguides. The principle of epitaxial growth is that a material in its molten state (liquid-phase epitaxy) or gaseous state (vapor-phase epitaxy) can be crystallized into thin films on the surface of a substrate material, when the two materials have similar crystal struc-

Waveguides, Optical

tures and lattice constants. In vapor-phase epitaxy, chemical reactions of gases take place at a high temperature. In general, epitaxial techniques can produce high-quality crystalline films with well-controlled thicknesses. It is also possible to grow crystalline thin films by allowing beams of atoms or molecules from different sources to react with a crystalline substrate under ultrahigh vacuum conditions (molecular-beam epitaxy). Molecular-beam epitaxy offers good flexibility and an extremely precise control of film thickness (layers as thin as 1 nm can be grown). Patterning The fabrication techniques described above only lead to planar waveguides. To make a two-dimensional waveguide, it is necessary to carry out patterning, either on a planar waveguides or on a substrate. Figure 17(a) shows the main steps involved in fabricating a two-dimensional waveguide on a planar waveguide. The planar waveguide is first coated with a layer of (negative) photoresist followed by a metal mask with a window with the desired dimension. The strip of photoresist in the window, after exposure to ultraviolet light, is developed chemically, and remains to serve as a protective shadow on the planar waveguide. The guiding layers of the waveguide that are not covered by the photoresist mask are then removed with an etching technique. If the etching is complete (down to the substrate), a strip waveguide results. If the etching is incomplete, a rib waveguide results. It is also possible to make a two-dimensional waveguide from a substrate material, as shown in Fig. 17(b). The substrate material is coated directly with a layer of positive photoresist and a metal mask. After exposure and development, in contrast with the previous case, the photoresist that has been exposed to ultraviolet light is removed chemically. This leaves a window of photoresist on the substrate. A layer of higher-index material is then sputtered on the patterned structure. In this way, only a strip of higherindex material is in touch of the substrate. With the rest of the photoresist removed, a strip waveguide results. A diffused channel waveguide is formed instead, if a layer of metal for thermal diffusion is deposited on the patterned structure, so that diffusion takes place only in the window area. When a lower-index layer is further deposited on a strip or embedded waveguide, a fully buried channel waveguide results. Two patterning techniques are available: photolithography, in which photoresists are used, as described in the examples, and electron-beam lithography, in which electronbeam resists are used. With the electron beam technique, patterns are written directly on the resists by the electron beam and no mask is required. A typical resolution of 0.1 µm can be achieved. There are also two kinds of etching techniques: “wet” etching techniques, which rely on liquid chemicals as etching agents, and “dry” etching techniques, which rely on gaseous chemicals. Dry etching usually results in better etched surfaces. Reactive ion etching (RIE) is a particularly versatile dry-etching technqiue, where highenergy ions generated under low pressure or in vacuum by an electromagnetic field are made to attack the substrate and react with it to remove the material.

19

There are two other general techniques that are particularly suitable for the fabrication of polymer waveguide devices: laser direct writing and imprinting/molding (57). Laser writing is based on changing the refractive index of photosensitive polymer by exposing it to a laser beam of a suitable wavelength. Waveguide patterns can be produced directly on a polymer substrate with a computer-controlled moving laser beam or an expanded stationery laser beam through a mask. When a polymer that is insensitive to the writing wavelength is used as the cladding material, it is possible to form buried waveguides directly (58). The laser writing technique can also be applied to the fabrication of embedded waveguides in photosensitive glass (59). The imprinting/molding technique is most suitable for mass production of polymer waveguides. This technique requires the preparation of a mold or stamp that contains the device pattern. There are several versions of the technique. In the case of imprinting, the mold or stamp is made in conformal contact with the polymer film coated on a substrate to transfer the pattern. In the case of hot embossing, the mold is pressed against a polymer film at an elevated temperature and peeled off from the film by cooling. It is also common to fill the mold with the desired polymer in the liquid form followed by ultra-violet curing. The advantage of the imprinting/molding technique is that the mold can be used repeatedly to produce the same device with a simple process and thus lower the manufacturing cost. The mold can be produced directly by photolithography on silicon or glass. More often the mold is made of a mechanically flexible elastomer poly(dimethysiloxane) (PDMS) (see, for example, reference 60), which is produced by casting from a master. The master to be replicated can be made of commonly used resist, glass, or semiconductor. CHARACTERIZATION AND MEASUREMENTS After a waveguide has been fabricated, the next step is to measure its properties. The results from the measurements can provide valuable information about how good the fabrication technique is, and how well the fabricated waveguide meets the design specifications. It is through a number of iterations of fabrication and measurement that a reproducible fabrication method can be established for a specific type of waveguide. The properties of a waveguide that are of particular interest are optical attenuation, refractive-index profile, and, in the case of a single-mode waveguide, mode-field distribution. More detailed discussions of various measurement methods can be found in the relevant chapters of the reference books (3,7,10,13,55). Prism-Coupling Technique Essentially all optical measurements involve launching light into a waveguide. Here the prism-coupling technique (61) is described briefly, which is widely used in waveguide characterization. Figure 18 shows a prism with base angle αp and refractive index np , placed against the surface of a planar waveguide with a small air gap. A collimated light beam is incident upon the prism at an angle θ and deflected onto the base of the prism at an angle θ to the normal of the base.

20

Waveguides, Optical

Figure 17. Procedures in the fabrication of a strip waveguide starting with (a) a step-index slab waveguide or (b) a substrate.

It can be shown that, only when the phase velocity of the incident beam along the base of the prism is equal to the phase velocity of the guided mode of the waveguide, light can be coupled into that mode with a high efficiency (61). This condition leads to β θ = np sin θ = np sin[sin−1 (sin ) + αp ] k np

(70)

where β/k is the effective index of the mode. For the method to work properly, the refractive index of the prism must be sufficiently larger than that of the guiding layer of the waveguide. It is clear from Eq. (70) that, by varying the incident angle θ, any guided mode can be selectively excited. This method is capable of providing a high coupling efficiency (better than 80 %). To obtain a stable and efficient coupling condition, however, the adjustment of the gap separation and beam position is critical. Exactly the same method can be used to couple light out of the waveguide. The other commonly used method for launching light into a waveguide is the end-coupling (end-fire) method, in which light is launched into the waveguide through the end face of the waveguide. This method requires the end face of the waveguide be flat and perpendicular to the input light beam. Loss Measurements A high-quality waveguide should have a low optical loss (typically less than 1 dB/cm). There are two kinds of losses: absorption loss caused by light absorption in the waveguide materials, and scattering loss caused by the imperfections in the waveguide structure. Absorption loss is an important concern in semiconductor waveguides, as there are many mechanisms in semiconductors that can cause absorption (interband absorption, impurity absorption, and carrier absorption). On the other hand, scattering loss is the major

Figure 18. Illustration of the prism-coupling technique. A highindex prism is placed against the surface of a planar waveguide and light is launched into the waveguide through the prism. Only when the phase velocity of the incident light along the base of the prism is equal to that of a guided mode of the waveguide, the mode can be excited with a high efficiency.

loss in many dielectric waveguides. In general, waveguides fabricated from diffusion techniques have a lower scattering loss. For curved waveguides, there also exists bendinduced radiation loss. In general, the optical loss in a waveguide, denoted by α (in dB/cm), can be calculated from optical power levels P1 and P2 measured respectively at two different positions z1 and z2 in the waveguide: α=

10 log(P1 /P2 ) . z1 − z2

(71)

To measure power levels at two different positions, the cutback method can be used. Light is launched into the waveguide by the end-coupling technique and the output power at the other end of the waveguide is measured. The waveguide is then cut by a known length and the same measurement is repeated to obtain another power reading. This method is accurate, but destructive. An alternative method is based

Waveguides, Optical

on the use of two prisms, one for launching light into the waveguide and the other for coupling light out of the waveguide. The output prism is allowed to slide along the waveguide and couple light out of the waveguide at different locations. In this way, output power levels at different positions can be measured. The prism-sliding technique is non-destructive and capable of measuring loss for individual mode. However, care must be taken to ensure that the coupling efficiency is optimized, and hence, remains constant during the measurement. This can be facilitated by filling the air gap between the prism and the waveguide with an index-matching liquid. The use of index-matching liquid can improve the coupling efficiency and reduce the friction. Measurements of losses down to ∼0.1 dB/cm with errors of ±0.01 dB/cm are possible with this method. The scattering loss in a waveguide can also be measured by detecting light scattered from the waveguide with an optical fiber probe or a camera placed near the surface of the waveguide. Refractive-Index Profile Measurements In the case of using prisms to couple light into and out of a waveguide, if the light incident upon the input prism spans a range of angles, all the guided modes of the waveguide can be excited and leave from the output prism at different angles. The output pattern displayed on a screen shows a number of parallel bright lines (“m-lines”), each of which represents a guided mode (2, 61). By measuring the exit angles of the modes, the effective indices can be calculated from Eq. (70). The prism-coupling method thus provides a convenient way for measuring mode indices. With the knowledge of the effective indices, the thickness and the refractive index of the thin film in a step-index waveguide can be determined from Eq. (5). As there are two unknowns, at least two effective indices are required. In the case of a graded-index waveguide, the refractiveindex profile can be determined from the inverse WKB method. A common way of applying the inverse WKB method is to treat the effective index in the WKB equation, Eq. (39), as a continuous function of m (the effective-index function) (62), i.e., N(m) ≡ β/k, where m is a real number instead of an integer. According to Eq. (39), there exists a one-to-one correspondence between the refractive-index profile n(x) and the effective-index function N(m). The idea is therefore to find approximately the function N(m) from a set of discrete effective indices (m = 0, 1, 2, . . .) measured by the prism-coupling method. This can be done by plotting the measured effective indices against the mode orders and then least-squares fitting the data to generate a continuous function of m (with a low-order polynomial). From Eq. (39), by extrapolating N(m) to m = −0.75, the peak index at the waveguide surface can be obtained, i.e., n1 = N(−0.75). To facilitate the calculation, the function N(m) is replaced by a large number of samples Ni (i = 0, 1, 2, . . .) with N0 > N1 > N2 > . . . and N0 = N(−0.75) = n1 , each of which corresponds to a turning point xi at which n(xi ) = Ni . The turning points can then be calculated from the following algorithm (62): xi =

(m + 0.25)π + (Ni ) − k

i−1

2

for i = 2, 3, · · · with x0 = 0 and x1 = [(m1 + 0.25)π + 2 (N1 )]/[k(N 1 − N12 )1/2 ], and N i = (Ni + Ni−1 )/2. Once the turning points are found, the refractive-index profile n(x) can be constructed. In general, the accuracy of the method increases with the number of effective indices available for forming the effective-index function. In the original version of the method (62), at least three effective indices from the same mode type (TE or TM) are required, which means that the waveguide must support at least three modes of the same type. By combining the measurements for both the TE and TM modes and/or at different wavelengths, the method can be applied to single-mode and two-mode waveguides (63). However, the technique of combining the measurements for both mode types cannot be applied to waveguides that do not support both mode types or contain unknown material birefringence, while the technique of combining the measurements at different wavelengths requires several laser sources and an accurate knowledge of the dispersion properties of the waveguide material. These problems can be overcome with the technique that combines the effective indices of the same mode measured with different index-matching liquids applied to the surface of the waveguide (64). Mode-Field Distribution To optimize the coupling efficiency between a single-mode fiber and a single-mode waveguide, the mode-field distributions in the fiber and the waveguide must match. The mode-field distribution in a waveguide, or the near-field intensity pattern, can be measured conveniently with a video camera system. In fact, the measured near-field intensity pattern can be used to determine the refractive-index profile of the waveguide. From Eq. (52), the refractive-index profile n(x, y) can be calculated from (65) β 1 1 n2 (x, y) = ( )2 − 2 [∇t2 I − (∇t I)2 ] k 2k I 2I

− Ni2 )1/2

(73)

where I(x, y) ∝ ψ2 (x, y) is the near-field intensity pattern. Because numerical calculation of first and second derivatives on the measured data is involved, very accurate experimental data are required and data smoothing is necessary. This method is capable of determining refractiveindex profiles in two-dimensional waveguides. CONCLUDING REMARKS With continuous research endeavors for almost four decades, numerous integrated optical devices have been demonstrated, varying from simple ones, which may function merely as passive signal distributors, to complicated ones, which may involve a complicated matrix of waveguides and a large number of thermal controls. Indeed, the advancement in waveguide fabrication over the years has greatly improved the control of waveguide dimensions and made possible the launch of a range of products at reasonable prices. For example, silica-waveguide star couplers and high-speed LiNbO3 modulators are among the most

2

{x j [(N j − Ni2 )1/2 − (N j+1 − Ni2 )1/2 ]}

j=1 i k(N 2

21

(72)

22

Waveguides, Optical

common integrated optic waveguide devices in the market. Commercial software packages are also available for the design of a wide range of waveguide devices. The continuing pursue for higher transmission capacity with optical fiber has brought about the concept of dense wavelength-division multiplexing (DWDM), which demands the transmission of a large number of closely packed wavelength channels along a single-mode fiber. The most important enabler of the DWDM technology is the integrated optic device known as the arrayed-waveguide grating (AWG). An AWG consists of a large array of waveguides with a fixed length difference between two neighboring waveguides, where the light outputs from all the waveguides are made to interfere to provide spatial separations for a large number of DWDM channels. An AWG thus fulfils the key function in the DWDM technology as a wavelength multiplexer/demultiplexer. A comprehensive review of the AWG technology can be found in chapter 9 of reference 17. At the same time, the rapid advances in the waveguide fabrication techniques allow further miniaturization of optical waveguides by using high-index-contrast structures, in particular, silicon-based structures (56). The recent developments in photonic crystals, which are also high-index-contrast structures, have opened up many new opportunities in the realization of compact devices (48) and the application of nonlinear optical phenomenon (66). On the other hand, polymer waveguides hold promise for mass production of low-cost and high-performance devices for the next generation of optical communication systems (57– 68). More exciting developments along these fronts are expected to come with time.

BIBLIOGRAPHY 1. S. E. Miller, Integrated optics - an introduction, Bell Syst. Tech. J., 48, 2059–2068, 1969. 2. P. K. Tien, Integrated optics and new wave phenomena in optical waveguides, Reviews of Modern Physics, 49, 361–420, 1977. 3. H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits, New York: McGraw-Hill, 1989. 4. T. Tamir (ed.), Guided-Wave Optoelectronics, 2nd ed., Berlin: Springer-Verlag, 1990. 5. R. Syms and J. Cozens, Optical Guided Waves and Devices, London: McGraw-Hill, 1992. 6. K. J. Ebeling, Integrated Optoelectronics, Berlin: SpringerVerlag, 1993. 7. S. Martellucci, A. N. Chester, andM. Bertolotti (ed.), Advances in Integrated Optics, New York: Plenum Press, 1994. 8. T. P. Lee (ed.), Current Trends in Integrated Optoelectronics, Singapore: World Scientific, 1994. 9. E. J. Murphy (ed.), Integrated Optical Circuits and Components: Design and Applications, New York: Marcel Dekker, 1999. 10. R. G. Hunsperger, Integrated Optics: Theory and Technology, 5th ed., Berlin: Springer-Verlag, 2002. 11. M. J. Adams, An Introduction to Optical Waveguides, New York: Wiley, 1981. 12. A. W. Snyder and J. D. Love, Optical Waveguide Theory, London: Chapman and Hall, 1983.

13. D. L. Lee, Electromagnetic Principles of Integrated Optics, New York: Wiley, 1986. 14. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed., San Diego: Academic Press, 1991. 15. K. Kawano, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations and the Schr¨odinger Equation, New York: John Wiley & Sons, 2001. 16. G. Lifante, Integrated Photonics: Fundamentals, New York: John Wiley & Sons, 2003. 17. K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed., San Diego: Academic Press, 2006. 18. Y.-F. Li and J. W. Y. Lit, General formulas for the guiding properties of a multilayer slab waveguide, J. Opt. Soc. Amer. A, 4, 671–677, 1987. 19. K. S. Chiang, Coupled-zigzag-wave theory for guided waves in slab waveguide arrays, J. Lightwave Technol., 10, 1380–1387, 1992. 20. L. M. Walpita, Solutions for planar waveguide equations by selecting zero elements in a characteristic matrix, J. Opt. Soc. Amer. A, 2, 595–602, 1985. 21. T. A. Maldonado and T. K. Gaylord, Hybrid guided modes in biaxial planar waveguides, J. Lightwave Technol., 14, 486–499, 1996. 22. V. Ramaswamy, Propagation in asymmetrical anisotropic film waveguides, Appl. Opt., 13, 1363–1371, 1974. 23. P. Sosnowski, Polarization mode filters for integrated optics, Optics Commun., 4, 408–412, 1972. 24. S. C. Rashleigh, Four-layer metal-clad thin film optical waveguides, Opt. Quantum Electron., 8, 49–60, 1976. 25. I. P. Kaminow, W. L. Mammel, and H. P. Weber, Metal-clad optical waveguides: analytical and experimental study, Appl. Opt., 13, 396–405, 1974. 26. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, Antiresonant reflecting optical waveguides in SiO2 -Si multi-layer structures, Appl. Phys. Lett., 49, 13–15, 1986. 27. T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration, J. Lightwave Technol., 6, 1440–1445, 1988. 28. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, New York: McGraw-Hill, Ch.9, 1953. 29. M. Abramowitz andI. A. Stegun (ed.), Handbook of Mathematical Functions, New York: Dover, 446–454, 1970. ˇ 30. J. Janta and J. Ctyrok y, ´ On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides, Optics Commun., 25, 49–52, 1978. 31. K. S. Chiang, Simplified universal dispersion curves for graded-index planar waveguides based on the WKB method, J. Lightwave Technol., 13, 158–162, 1995. 32. K. S. Chiang, Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides, Opt. Quantum Electron., 26, S113–134, 1994. 33. E. A. J. Marcatili, Dielectric rectangular waveguide and directional coupler for integrated optics, Bell Syst. Tech. J., 48, 2071–2102, 1969. 34. R. M. Knox and P. P. Toulios, Integrated circuits for the millimeter through optical frequency range, in Proc. Symp. Submillimeter Waves, Brooklyn: Polytechnic Press, 497–516, 1970. 35. A. Kumar, K. Thyagarajan, and A. K. Ghatak, Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach, Opt. Lett., 8, 63–65, 1983.

Waveguides, Optical 36. K. S. Chiang, Effective-index analysis of optical waveguides, in SPIE Proc. Physics and Simulation of Optoelectronic Devices III, 2399, 2–12, 1995. 37. K. S. Chiang, Performance of the effective-index method for the analysis of dielectric waveguides, Opt. Lett., 16, 714–716, 1991. 38. K. S. Chiang, Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides’ IEEE Trans. Microwave Theory Tech., 44, 692–700, 1996. 39. K. S. Chiang, Analysis of rectangular dielectric waveguides: effective-index method with built-in perturbation correction, Electron. Lett., 28, 388–389, 1992. 40. K. S. Chiang, C. H. Kwan, and K. M. Lo, Effective-index method with built-in perturbation correction for the vector modes of rectangular-core optical waveguides, J. Lightwave Technol., 17, 716–722, 1999. 41. K. S. Chiang, Finite element analysis of weakly guiding fibers with arbitrary refractive-index distribution, J. Lightwave Technol., LT-4, 980–990, 1986. 42. L. B. Soldano and E. C. M. Pennings, Optical multi-mode interference devices based on self-imaging: principles and applications, J. Lightwave Technol., 13, 615–627, 1995. 43. G. B. Hocker and W. K. Burns, Mode dispersion in diffused channel waveguides by the effective index method, Appl. Opt., 16, 113–118, 1977. 44. P. N. Robson andP. C. Kendall (ed.), Rib Waveguide Theory by the Spectral Index Method, New York: Research Studies Press, 1990. 45. K. S. Chiang, Dispersion characteristics of strip dielectric waveguides, IEEE Trans. Microwave Theory Tech., 39, 349–352, 1991. 46. S. Y. Cheng, K. S. Chiang, and H. P. Chan, Birefringence characteristics of benzocyclobutene rib optical waveguides, Electron. Lett., 40, 372–373, 2004. 47. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of Light, Princeton: Princeton University Press, 1995. 48. D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, and B. Miao, Photonic crystal structures and applications: perspective, overview, and development, IEEE J. Sel. Top. Quantum. Electron., 12,pp. 1416–1437, 2006. 49. P. Yeh, A. Yariv, and C. S. Hong, Electromagnetic propagation in periodic stratified media. I. General theory, J. Opt. Soc. Am., 67, 423–438, 1977. 50. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, A dielectric omnidirectional Reflector, Science, 282, 1679–1682, 1998. 51. A Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Boston: Artech House, 2005. 52. C. H. Kwan and K. S. Chiang, Effective-index method with built-in perturbation correction for the design of polarizationinsensitive optical waveguide directional couplers, J. Lightwave Technol., 20, 1018–1026, 2002. 53. S. Y. Cheng, K. S. Chiang, and H. P. Chan, Polarization dependence in polymer waveguide directional couplers, IEEE Photon. Technol. Lett., 17, 1465–1467, 2005. 54. Y. Bai, Q. Liu, K. P. Lor, and K. S. Chiang, Widely tunable long-period waveguide grating couplers, Optics Express, 14, 12644–12654, 2006. 55. S. I. Najafi (ed.), Introduction of Glass Integrated Optics, Norwood: Artech House, 1992.

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56. S. Janz, A. Bogdanov, P. Cheben, A. Delage, ˆ B. Lamontagne, M.-J. Picard, D.-X. Xu, K. P. Yap, and W. N. Ye, Silicon-based integrated optics: waveguide technology to microphotonics, in Proc. Mater. Res. Soc. Symp. 832, F1.1.1–F1.1.12, 2005. 57. L. Eldada and L. W. Shackelette, Advances in polymer integrated optics, IEEE J. Sel. Top. Quantum. Electron., 6, 54–68, 2000. 58. K. S. Chiang, K. P. Lor, Q. Liu, and H. P. Chan, UV-written buried waveguides in benzocyclobutene, in SPIE Proc. AsiaPacific Optical and Wireless Communications Conference and Exhibition, 6351,63511J( 1–8), 2006. 59. P. Yang, J. Guo, G. R. Burns, and T. S. Luk, Direct-write embedded waveguides and integrated optics in bulk glass by femosecond laser pulses, Opt. Eng., 44,051104( 1–6), 2005. 60. J. Narasimhan and I. Papautsky, Polymer embossing tools for rapid prototyping of plastic microfluidic devices, J. Micromech. Microeng., 14, 96–103, 2004. 61. P. K. Tien and R. Ulrich, Theory of prism-coupling and thin film light guides, J. Opt. Soc. Amer., 60, 1325–1337, 1970. 62. K. S. Chiang, Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes, J. Lightwave Technol., LT-3, 385–391, 1985. 63. K. S. Chiang, Refractive-index profiling of graded-index planar waveguides from effective indexes measured for both mode types and at different wavelengths, J. Lightwave Technol., 14, 827–832, 1996. 64. K. S. Chiang, C. L. Wong, S. Y. Cheng, and H. P. Chan, Refractive-index profiling of graded-index planar waveguides from effective indexes measured with different external refractive indexes, J. Lightwave Technol., 18, 1412–1417, 2000. 65. K. Morishita, Index profiling of three-dimensional optical waveguides by the propagation-mode near-field method, J. Lightwave Technol., LT-4, 1120–1124, 1986. 66. R. E. Slusher andB. J. Eggleton (ed.), Nonlinear Photonic Crystals, Berlin: Springer-Verlag, 2003. 67. A. Yeniay, R. Y. Gao, K. Takayama, R. F. Gao, and A. F. Garito, Ultra-low-loss polymer waveguides, J. Lightwave Technol. 22, 154–158, 2004. 68. L. W. Shacklette, Polymers in the light path: optical polymer technology for communications, Optics & Photonics News, 15, 22–27,Nov. 2004.

KIN SENG CHIANG Department of Electronic Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

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Wiley Encyclopedia of Electrical and Electronics Engineering Nonlinear Optics Standard Article Jeffrey H. Hunt1 1Boeing North American, Canoga Park, CA Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W4409 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (231K)

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Abstract The sections in this article are Stimulated Raman Scattering Two-Photon Absorption Four-Wave Mixing Spectroscopy High-Resolution Nonlinear Spectroscopy Detection of Rare Atoms and Molecules Multiphoton Spectroscopy Surface Nonlinear Optical Characterization About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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NONLINEAR OPTICS

601

with increasing sophistication, the applications are identical in their basic form. Nonlinear optics is useful because of the extreme versatility of the technique. When performing linear optical characterization, the resonance in the material parameters must match the wavelength of the light impinging on the sample. There is no other option. However, in nonlinear optical characterization, there are a large number of possible combinations to make the light waves resonant with the material. Certainly, any of the input waves can be matched to a material parameter. But the difference between input wavelengths can be resonated with a material. Also, the sum of the input waves could provide a resonance. Any imagined combination of frequencies can, under the appropriate circumstances, be used as a means of probing the resonances of a material. An article of this brevity can by no means contain a complete survey of all techniques for nonlinear optical characterization. As examples of the most visible and most universally applicable techniques, volumes could be written on the subject. There will also be no attempt here to include theoretical derivations of the equations, since excellent discourses are available in the literature (1). Specifically, this article will discuss stimulated Raman scattering, two-photon absorption, four-wave mixing, highresolution spectroscopy, detection of rare atoms, multiphoton spectroscopy, and surface nonlinear optical characterization.

STIMULATED RAMAN SCATTERING

NONLINEAR OPTICS Linear electromagnetic phenomena are attractive because of their simplicity, elegance, and ease of interpretation. The basic experimental implementation has not changed since the nineteenth century. Although more work continues to be done

Spontaneous Raman scattering is one of the oldest nonlinear optical effects observed. In a vibrational transition that is Raman active, the vibrational excitation contains an anisotropy. This anisotropy allows the vibration to couple to the local electronic excitation. Consequently, when an optical pulse of sufficient intensity travels through a sample, the output will contain frequency sidebands separated from the input frequency by the frequency of the vibrational excitation. In the stimulated case, the input wave, the material excitation,and the output wave travel as electromagnetically coupled phenomena. Consequently, the signal generated can experience gain analogous to that which occurs in an optical amplifier. Phase-matching considerations force the stimulated Raman scattering (SRS) signal to be generated in a well-defined collimated output. The signal is easier to detect, making it possible to see Raman excitations not seen in the spontaneous case. The stimulated Raman effect in gases was first reported by Minck, Terhune, and Rado (2). At room temperature, the Q1 vibrational transition is dominant in the stimulated Raman effect in H2. A variation in the vibrational frequency with pressure was observed in the spontaneous Raman spectrum. Since the stimulated emission spectrum can display much narrower and more intense lines, this shift was more accurately determined by utilizing Raman laser oscillators. The Stokes and anti-Stokes frequencies (above and below the pump beam) of two H2 Raman lasers are compared. One laser was operated at a variable frequency and the other has a fixed frequency constant within 0.001 cm⫺1. A schematic experimental setup for SRS is shown in Fig. 1 (3). Here the input lasers passes through the sample. PM1 provides calibration, while PM2 measures the forward-gener-

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

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Figure 1. The experimental system for SRS. Although only a schematic, the experimental system here depicts the main elements necessary for performing a measurement of this type. Note that the positioning of PM2 and PM3 permits both forward and backward mesurements to be performed simultaneously. Of importance here is the fact that nothing experimentally exotic is necessary, and the interaction itself provides the information.

ated signal and PM3 measures the backward-generated signal. Data produced from such an experiment are shown in Fig. 2. Although theoretical prediction had shown symmetry in the forward- and backward-generated signals, the data here show a large asymmetry. Work conducted subsequent to 100

Forward Stokes Backward Stokes P1 > P2 > P3

P1

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Stokes power (kW)

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Figure 3. The experimental set-up for the gas phase SRS measurements. Note the relative placement of each of the Stokes and Raman elements, particularly relative to the detector positions.

this showed that the discrepancy disappeared once self-focusing was taken into account. (These data were performed in liquids that were Kerr active and, consequently, had large degrees of self-focusing.) To eliminate the effect of self-focusing, the experiments can be performed in gas, which does not exhibit self-focusing, as liquids do. Experiments were performed to measure gain at the H2 vibrational resonance in the forward direction (4). The experimental setup is shown in Fig. 3. The gain of the stable backward amplifier is measured with the setup shown. An important feature is that the geometry of the laser beam in the amplifier cell in both cases is identical to that of the beam emerging from the ruby laser. The laser pulse of about 150 mJ had a 20 ns duration and a diameter of 4 mm. This brings the power flux density to approximately 60 MW/cm2. Since the geometry of the laser beam is modified in the oscillator cell, this requires the beam splitter arrangement to measure gain in the forward direction. The experimental gain for the forward and the backward direction as a function of the density is given by the points in Fig. 4. The theoretical curves are included for comparison.

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Figure 2. Here is the data collected from each of the detectors shown in Fig. 1. Several important features can be extracted from the data here. There is clearly a threshold intensity needed to initiate the interaction. There is nonlinear dependence of the output power with cell length. Also, the strong asymmetry between the forward and backward signals provides the first indication of the first experimental demonstration of self-focusing, leading to lower thresholds and higher powers in the forward direction.

1

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Figure 4. Included with the theoretical curves are the experimental results for forward and backward gain measurements. Since these data were taken in a gas sample, they represented the first measurements devoid of self-focusing. This allowed verification of the SRS theory.

The experimental points X for the forward gain should be compared with the dashed theoretical curve. The experimental points ⫹ for the backward gain should be compared with the drawn theoretical curve. Without the effect of self-focusing, there is good agreement between measurement and prediction. Although it is assumed that nonlinear optical investigations can only be realized with pulsed lasers, it has been demonstrated that SRS can be implemented using continuouswave (cw) laser sources. This work examines and compares a variety of forms of SRS where cw laser sources may be applied (5). More specifically, direct stimulated Raman gain spectra will be compared with those obtained using optically heterodyned schemes, such as the polarization interference techniques. The use of cw lasers with lock-in detection produces a technique that is extremely sensitive. Since this work was a demonstration of technique, the experiment was performed on the well-characterized Raman lines in common solvents and mixtures of common solvents. The isotopic mode of benzene at 983.1 cm⫺1 is clearly seen along with the 991.6 cm⫺1 mode of benzene and the 1003 cm⫺1 mode in toluene. As a demonstration of the sensitivity and resolution of this type of spectroscopy, it was applied to the acquisition highresolution Raman spectra of a monolayer of chemisorbed p-nitrobenzoic acid on aluminum oxide (6). The technique has broad application since it does not require special surface materials or morphology. The spectra are taken in a region near 1610 cm⫺1, where a strong mode has been observed for p-nitrobenzoic acid (PNBA) monolayers in SRS. The technique is introduced as a means of overcoming the sensitivity limitation and luminescence background problems of conventional Raman spectroscopy. A straightforward approach for employing surface picosecond Raman gain (SPRG) is realized for molecular monolayers on dielectric surfaces. In this case, substrate absorption is not important and a simple transmission geometry may be employed. In this article, the measurement explicitly demonstrates the detection of the 1610 cm⫺1 region of the ring quadrant stretch mode of a p-nitrobenzoic acid monolayer chemisorbed on an oxidized aluminum surface. In the experimental setup, two dye lasers are tuned so that their difference frequency coincides with the material excitation at 1610 cm⫺1. The gain in the anti-Stokes line is monitored as a function of the detuning. The data are shown in Fig. 5. The peaks at 1610, 1600, and 1580 are assigned as Raman peaks since their location did not shift when the stokes laser was shifted by 35 cm⫺1 and repeated spectra were obtained by scanning the pump laser wavelength. The peak at 1610 cm⫺1 is due to the PNBA ring quadrant stretch observed in infrared and enhanced Raman spectra reported previously; the other two peaks are unique to this measurement and seem to imply changes in the condition of the monolayer during the experiments. Evidence suggests that the peaks at 1580 and 1600 are new species formed by photochemical and or thermal reaction induced by the focused beams unique to the measurement. Stimulated Raman Scattering: Temporal Measurements Since there are several pulses that enter a sample during SRS, information can be gleaned from an experiment in which one pulse is delayed from the other. This provides a very pow-

603

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Figure 5. An early optical diagnostic. The existence of three lines in the surface SRS spectra showed that optically induced chemistry could take place at an interface. The data here were anticipated to contain only one peak, so the spectra provided an independent check of their validity.

erful tool when making temporal measurements on Raman systems. Ordinarily, one would have to measure Raman linewidth and infer the lifetime from the inverse of the bandwidth. A different method for the determination of ␶ is the use of time-dependent SRS (7). Here, molecular vibrations are coherently excited by a short light pulse and the rise and decay of the vibration amplitude are measured with a second light pulse properly delayed with respect to the first one. For materials with t ⬎ ␶, that material parameter can be determined. On the other hand, the shape of picosecond light pulses is conserved using samples with t ⬍ ␶. From this direct measurement, the dephasing time in various materials can be determined. The experimental system is depicted schematically in Fig. 6. It consists of a modelocked Nd : glass laser that is followed by a single pulse selector. The pulse passes an optical amplifier and a potassium dihydrogen phosphate crystal for conversion to the second harmonic frequency and into the measuring cell. The incident pulse traverses the medium, generating molecular vibration, and Stokes shifts the light. The laser and first Stokes pulses are measured with the fast photocells P1 and P4. The conversion efficiency of a couple percent was kept small to keep the experiment within the range of validity of our theoretical projections. Two beamsplitters provide two

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inferences from spectral data gave numbers of 0.26 ps, so the direct measurement is a considerable improvement.

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Figure 6. The experimental set-up for time dependent measurements is shown here. The essential element for input is a single short pulse of sufficient energy for the nonlinear interaction to proceed. Then a means of inducing the excitation and measuring its delay is needed. The prism combination is used here for the probe pulse delay.

light pulses of small intensity, approximately 10⫺3 of the pumping pulse. One pulse with variable time delay served as a probe pulse for the vibrational field in the medium, while a second probe pulse with fixed time delay was used as a reference signal to improve the accuracy of our measurements. Both probe pulses are coherently scattered by the molecular vibrations. In this system the anti-Stokes wave generates the phonon and the probe beam interacts with the same phonon to the observed anti-Stokes wave. The system is adjusted for close phase matching with the phonon wave vector collinear with the Stokes and anti-Stokes wave vectors. The experimental data are shown in Fig. 7. The antiStokes line of the probe beam is plotted versus the delay time for carbon tetrachloride and ethyl alcohol. From the data, the material response time ␶ is measured to be 8.3 ps. Previous

Anti-Stokes signal S(tD)

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Figure 7. Here are the data from the time delay measurement. For comparison, the open circles show the autocorrelation between pump and probe pulses. The material temporal response is shown in filled circles. Since that decay contains only a single term, the decay is linear on the log scale. Plotting the data on a semi-log scale, as shown, allows one to infer the material dephasing time.

The selection rules and, consequently, the spectroscopy of one- and two-photon absorption processes are completely different. This makes them complementary techniques, providing different types of spectroscopic information. In a two-photon absorption process, two photons are simultaneously absorbed with effective photon energy equal to two times the single photon energy. This means that an input laser at frequency 웆 will be absorbed as if it were light at 2 웆. As a higher-order process, the effective cross section is considerably lower, but with the appropriate choice of intensity, the absorption values can be strong enough to be easily detectable. One example of this technique is the two-photon absorption measurements performed in single crystal KBr and RbBr (8). The input laser frequency was selected to coincide with the vicinity of the fundamental absorption edge. This electronic absorption is transparent at single photon energy, so that the process has to be two photon in nature. In this experiment a cw source of ultraviolet light is used, which allows a broad spectrum to be evaluated at once. The basic experimental design is shown in Fig. 8. There are several optical filters that have been omitted form the schematic and that are designed to reject stray laser light, which would cause coloration of the sample. To demonstrate that the absorption was actually two photon in nature, the intensity of the laser at the sample was varied. Nonlinear absorption would lead to a decrease in the transmission, which would be proportional to the laser intensity. This was observed. The absorption spectra are show in Fig. 9. Notice that the linear and two-photon absorption spectra are very different in shape. The spectra presented here were sufficiently different that the measurements resulted in a new assignment for the bandgap in the bromide series. This was extremely important because an experimentally determined bandgap is the most important fitting feature in theoretical calculations. Two-photon spectroscopy can also be performed in gas (9). High-resolution measurements in sodium vapor were performed that were able to eliminate the Doppler background. In this experiment, the sodium atoms are excited to the 4D level by absorbing two photons of 5778 A light, and the resonance is detected by collecting light at the ultraviolet (UV) wavelengths emitted from the 4P level. Important to the application of the technique is that the two-input lasers are counterpropagating beams. Their coincidence in time and space causes the broadening due to the finite velocity of the atoms to be eliminated. The generated spectra clearly show the effect of the hyperfine sublevels in transition between the 3S and 4D states. These lines would be completely obscured by Doppler broadening in a single UV photon absorption measurement. FOUR-WAVE MIXING SPECTROSCOPY Technically, this technique is the generalized form of stimulated Raman scattering. The theoretical formalism is the same, only the material resonances do not have to be Raman

NONLINEAR OPTICS

Nitrogen temperature heat shield

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Figure 9. The two-photon and single-photon spectra have selection rules that are very different. The resulting spectra bare little resemblance. The different measurements resulted in a new assignment in the band gap of the bromide series. Linear spectroscopy was sufficiently ambiguous that the assignment would not have been possible without the comparative spectra.

Ac and dc signals to oscilloscope

Figure 8. The experimental system demonstrates that two photon mixing need not be derived strictly from a coherent radiation source. In the two photon process, one photon from the ruby laser is mixed with a broadband UV source from a Xenon arc source. The laser energy is near but below the band edge so that a single photon absorption is not possible.

active. Consequently, there will not be gain in the generated signal, but spectroscopic resonances can still be obtained. One of the most straightforward applications is the socalled induced-grating technique. Two pulses arriving from the same laser that coincide in time but arrive at different angles create an interference pattern on the sample. This pattern will be reflected in the material excitation by producing some photoexcited phenomena with a modulation present on it. A time-delayed third pulse can then be diffracted off the grating. By varying the arrival time of the third pulse and measuring the scattering efficiency as a function of the delay, one can measure the decay time of the excitation in the material. A typical experimental setup is shown in Fig. 10 (10). The design here is specifically for coherent anti-Stokes Raman spectroscopy (CARS) but is generically equivalent to any fourwave mixing experiment. Two laser inputs have to be provided to the sample. Appropriate control of the inputs and acquisition of the outputs is provided. As the difference between the two input frequencies is made to coincide with a material parameter, the anti-Stokes generated wave will be enhanced. Experimental measurements are shown in Fig. 11. Here the sample was calcite. The difference between the lasers was tuned to the 1088 cm⫺1 excitation in calcite. Note that the spectrum is very different from linear spectroscopy. In the linear case, one would see the classical Lorentzian shape. This arises from the imaginary part of the dielectric function in the vicinity of an absorption resonance. However, the presence of a large nonresonant, nonlinear background causes mixing to occur between the real and imaginary components of the dielectric tensor. The resulting spectrum will then become an admixture of the two components. However, different components of the tensor contribute only to different polarizations in the output signal. A detailed consideration of these contributions enable one to select which parts of the dielectric function will dominate the output spectrum (11). Results from this type of experiments are

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NONLINEAR OPTICS

Chart recorder

Stepping motor

Dual boxcar Log A/B B input A input

Reference sample Dye laser 2 Filters Delay line Double monochromator

1.7 MW N2 laser

Figure 10. The experimental schematic contains all the basic features necessary for a four wave mixing experiment. Although it is not necessary to have a single pump for the two lasers, a means of synchronizing the inputs is still required. Beyond that, careful references are needed to null any fluctuations, since the nonlinearity of the response could be completely washed out otherwise.

PMT PMT Double monochromator

Stepping motor Delay line

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shown in Fig. 12. Here the sample is dilute mixtures of benzene in carbon tetrachloride. By varying the appropriate polarization component of the anti-Stokes output, the desired spectral features can be ascertained as shown in the data. Graph (a) demonstrates how the choice of polarization can eliminate the nonresonant background. Graphs (b) and (c) show how the imaginary or real part of the dielectric function can be brought out from the sample.

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_

1

.25 mm calcite 2

3

_

10

0

10 20 30 (cm–1) (ω 1 – ω 2 ) – ω R

40

50

2πC

Figure 11. A typical four wave mixing resonance contains not just the peak when the frequency mixing is on resonance with the material excitation. There will also be a dip below the nonresonant background due to the interference between the real and imaginary parts of the dielectric function.

Although lasers themselves have linewidths that are intrinsically narrow, most spectroscopic lines in nature are inhomogeneously broadened. Consequently, the fine (narrow) features wash out when using standard linear absorption techniques. As an example, the Doppler width of the sodium D lines are over 1 GHz, while the hyperfine splittings are subGHz. Nonlinear optical spectroscopic techniques provide several means that reduce the effects of the inhomogeneous broadening in a sample. Several examples of these techniques are discussed here. As an example, high-resolution nonlinear optics was performed using quantum beats in Cs vapor (12). If an atom is prepared in a coherent superposition of states by a pulse excitation, modulations in the fluorescence signal at frequencies corresponding to the energy splittings may be observed. By carefully optimizing the experimental procedures, the optically excited hyperfine splittings were able to be observed. This was difficult because the small splittings result in very fast modulations that are difficult to observe. Specifically, the light emitted by the 72P3/2 level of Cs133 was observed after excitation by a short laser pulse. The energy-level diagram of Cs with the hyperfine splittings is shown in Fig. 13. The physical process is quantum mechanical in origin. Once the excitation process by laser absorption occurs, the hyperfine coherence starts evolving at different eigenfrequencies that correspond to the energy splittings between the ex-

NONLINEAR OPTICS

cited levels. Since the fluorescence from these levels depends on the state at the time of emission, the intensity of the fluorescence light must exhibit the same temporal behavior as the temporal behavior of the atoms. In the experimental setup, a 2 ns pulse of 1 GHz bandwidth impinges on the sample. The fluorescence signal is monitored by repetitive measurement of the fluorescence. (At the time this work was conducted, real-time signal processing did not exist and the signal temporal shape had to be measured by sampling. This would not be necessary with present technology.) Figure 14 shows the beat signals corresponding to the resonances in the Cs. The theoretical plots are shown below each picture. As can be seen, the predicted behavior matches the measured signal to a high degree of precision. Performing a Fourier transform on the signal, one finds the three peaks at the frequencies corresponding to each beat signal. The transform is shown at the bottom of Fig. 14.

607

F= 5

ν 54 = 82.9 MHz

4

ν 43 = 66.5 ν 32 = 49.9

3 2

Beats at 116.4, 66.5, 49.9 MHz b 4555Å a

149.4, 82.9, 66.5 MHz

4 ∆ g = 9193 MHz 3

24

Figure 13. The theoretical energy level diagram shows the multitude of hyperfine structure lines which exist within the Doppler broadened resonance.

(c) Re (XR)

20

Another example is the so-called Doppler free spectroscopy (13). Doppler broadening is caused by the large range of velocities within an atomic sample. Cooling the sample can narrow the spectrum, but only by a limited amount. A more effective means would be to create a situation in which only a limited velocity span could interact with the atoms. In Doppler free spectroscopy, two counterpropagating lasers interact with a sodium beam. The experimental setup is shown in Fig. 15. Here, the counterpropagating lasers are generated by retroreflecting the input laser. As a two-photon absorption, the laser frequency is tuned to half the absorption frequency. Since the absorption is weak, the return beam is almost the same strength as the input beam. This simple scheme allows quick resolution of the hyperfine levels in Na, as shown at the bottom of the figure.

θ 0 ≈ 8° n0 λ /4 0

16 5 (b) Im (XR)

103 × I /I⊥

4

3

θ 0 ≈ 2.7° n0 λ /4

2 0.6 θ0 = 0

(a) Im (XR)2

DETECTION OF RARE ATOMS AND MOLECULES

0.4

0.2

0 985

990 995 Stokes shift (cm–1)

1000

Figure 12. Using the polarization properties of four wave mixing, the signal can be changed to emphasize the real or imaginary parts of the signal. Part (a) of the figure is the magnitude of the Stokes resonance, but (b) and (c) contain dominantly imaginary and real parts, respectively.

A specialized but important application nonlinear optics is in the detection of a small number of atoms and molecules. Florescence techniques are known to be very sensitive due to the optical labeling of certain species. Careful optimization of the experimental apparatus can make it possible to detect single atoms. An example is presented here. Resonance fluorescence can detect the presence of a small amount of atoms because of the relatively large number of photons per second that can be scattered from the atoms in question (14). The first step here is the use of fluorescence to make absolute measurements of sodium vapor densities. The experimental apparatus is shown in Fig. 16. A laser input is tuned to one of the resonant sodium lines. The florescence is collected in sidelight and analyzed. The sodium oven temperature is varied as a means of controlling the density of vapor in the cell. Phase-sensitive (lock-in) electronics

608

NONLINEAR OPTICS Excitation from the F = 3 ground state Iσ Iπ

0 10

t (ns)

Iσ Iπ

Sa(ν )

Excitation from the F = 3 ground state

ν

0

100

t (ns)

0

32 ν 43

50

=ν

42 + ν 32

ν (MHz)

Figure 14. Observing the beat frequencies from the time dependent output spectra, one can see the interference in the terms of the signal. Fourier analysis of the signal shows the multiple components that are present. Their frequency and relative strength can be determined to a high degree of resolution and accuracy.

enables detection sensitivity to less than 100 atoms per cubic centimeter. The results of these measurements are shown in Fig. 17. As can be seen, the measurements cover many orders of magnitude, from 102 atoms/cm3 to 1011 atoms/cm3. Furthermore, coupling the spectroscopy data with thermodynamic measurements allows the curve to be absolutely calibrated without any variable parameters in the fit to the theoretical curve (solid line).

MULTIPHOTON SPECTROSCOPY One of the most exciting and surprising techniques in nonlinear optical characterization is multiphoton spectroscopy. Not only can one access states that cannot be probed by singlephoton methods, but transitions between excited states can be examined as well. One can think of multiphoton spectroscopy as a generalization of the two-photon and stimulated Raman spectroscopies discussed in this article. The formalism remains the same, but the applications here are more exotic.

In one experiment, multiple-photon excitation was used to prepare Sr in autoionizing Rydberg states (15). Separate excitations were used in this case. The process excited the first electron to a Rydberg state and then the excitation of the remaining 5s core electrons to the 5p1/2nd state. This produces large amounts of population in the latter state,which is autoionizing and can therefore be detected with nearly unit efficiency. Because the stepwise excitations are all dipole allowed, the required laser powers are low, resulting in no background because of photoionization. Using two synchronous pumped lasers at 460 m and 4194 nm, Sr molecules in an atomic beam are excited to the 5S2 states, to the 5s, 5p, and to the 5s, 20s bound Rydberg state. After excitation to this state, a laser at 4218 nm excites the remaining 5s core electron, producing an autoionizing state. A small electric field is applied, and this sweeps the electron out of the 5p1/220s state, which is counted. By measuring the electron’s number as a function of the third laser pulse, the spectra of the 5p1/2,nd state and 5p1/2,ns states can be attained. The spectra obtained are shown in Fig. 18. In these spectra, the quantum defect is strongly represented. The upper curve shows two peaks because of the strong coupling to the defect. The lower curve has only a single peak due to its relatively weak coupling. Another example is the use of multiphoton excitation of Na to study the diamagnetic structure of Na Rydberg states (16). The goal in this experiment was to study the one-electron atom in a field so intense that the Lorentz force on the electron dominates the Coulomb interaction. Although the field required cannot be achieved in the laboratory, the experimental measurements approaching the strength would help to explain some of the interactions that are not well understood. In the experiment shown in Fig. 19, an atomic beam is excited to a Rydberg state by a two-photon excitation. After this, an electric field is applied to ionize the Rydberg state and collected. The experiment then measured the collected energy as a function of applied magnetic field. The results are shown in Fig. 20. The solid lines represent the calculated shifts in each of the Rydberg states as a function of magnetic field. This shows very good agreement. Note that the use of the Rydberg state, not possible without the multiphoton excitation, makes it possible to create an atom that can be well characterized by a single electron model, so that it is possible to eliminate the multibody interaction and concentrate on the interaction with the magnetic field.

SURFACE NONLINEAR OPTICAL CHARACTERIZATION One of the most recent applications of nonlinear optics to characterization is the use of second-order nonlinear optics to study surfaces and interfaces. It is well known that symmetry considerations prevent second-order nonlinear optical processes from proceeding in centrosymmetric media. (To be completely precise, this is only the case in the electric dipole approximation. However, the magnetic dipole and electric quadrupole contributions are so weak that they are usually ignored on the surface.) Since the symmetry is broken at a surface or interface, second-order nonlinear optical processes become surface specific. Consequently, a whole host of surface physics and chemistry problems, which previously could only be performed in ultra-high vacuum (UHV), now become trac-

NONLINEAR OPTICS

Photomultiplier

To recorder

UV filter

Optical isolator Art laser

Al light pipe

c.w. dye laser Lens F = 10cm

No. cell

To monitor

Mirror R = 10cm

Oven (a)

a

b c d

0

1

Laser fequency (GHz)

(b)

σ+

(c) Figure 15. The upper half (a) of the figure shows the preparation of the sodium cell and the manner in which the saturation configuration is effected. Note that there is not a separate return beam. The retroreflection acts as its own second pass. The extremely narrow linewidths resulting from the Doppler free signal are shown in (b) and (c).

609

610

NONLINEAR OPTICS

Coincidence counter

PM1 Vo

Laser beam

5s18d

5p1/2,nd

5s19s

5p1/2,ns

d

ω

fl

ω les

PM2

Ion signal

Figure 16. The schematic representation is shown to illustrate how the coincidence counts from the fluorescence signal are created and detected.

1011

1010

109

108

Density (atoms / cm3)

4230

107

4220 4210 Wavelength (Å)

4200

Figure 18. The Rydberg spectroscopy allowed the first observation of quantum mechanical effects in high lying energy states. The strong defect coupling distorts the upper curve while not affecting the lower one. It is not possible to determine this type of interaction by other means.

106

105

table experimental problems. Presented here are the experimental techniques and some examples of the sorts of measurements that can be performed. The experimental setup for second harmonic generation is quite simple. The input laser is focused to the desired spot

104

Charged particle detector

103

Laser beam Electric field plates

102

Interaction region –20

0

20

40

60 80 T(°C)

100

120 140 160

Figure 17. The fluorescence signal from the rare atom detection. Note that the signal can detect the presence of fewer than 100 atoms per cubic centimer. Additionally, the signal range extends over 9 orders of magnitude, an extremely large range over which to produce data.

B

Atomic beam

Figure 19. The experimental configuration for the high magnetic field strength spectroscopy. The particular configuration is such that the atomic beam magnetic field and electric field (for ionization) are all collinear, while the laser propagates in a plane normal to that axis.

NONLINEAR OPTICS

611

10 N = 29

Rhodamine 110 Rhodamine 6 G

N = 28 140

N = 27 150 01

2

3

4 5 Magnetic field (T)

6

Figure 20. The spectroscopy of the high N states in the Rydberg atom shows the strong effect that the magnetic field can have on the spectroscopy. Note that the field predominates at 4 to 5 T such that the states of differing N begin to overlap. The theory does, however, still provide good agreement with experiment in spite of the large shifts.

Normalized SH intensity (arbitrary units)

Energy (cm–1)

130

5

0

size on a surface or interface. Due to the surface phase-matching constraints, the second harmonic generated at the surface will propagate in the same direction as the reflected fundamental. Optical filtering is necessary to eliminate the fundamental, but the second harmonic signal is strong enough to detect with standard detection schemes. An example spectroscopic sample is shown in Fig. 21 (17). The two rhodamine molecules have excitations that are near each other but do not exactly coincide. The shift is due to the

S2

30,000 cm–1

S1

320 340 SH wavelength (nm)

360

Figure 22. The surface second harmonic spectra of a molecular monolayer of the dye molecule adsorbed to a glass substrate. This represents one of the first examples of the use of second-order nonlinear optics as a means for providing surface sensitive optical characterization. This demonstrates the potential for performing surface species identification in an in-situ fashion. Odd order optical techniques could also access this information, but they would have large contributions from the bulk substrate.

slight difference in their molecular structures as shown. After preparing two samples of glass substrates with molecular coatings, the second harmonic spectra is as shown in Fig. 22. A slight shift is seen between the two samples. Note that these signals, with excellent signal to background, are generated from single molecular monolayers.

28,800 cm–1

ω 19,600 cm–1

300

18,900 cm–1

H 2N

O

CI– N +H 2

CI– H5C2NH

ω

X´

2H 5

CH3 CO2C2H5

H 3C Z´

CO2H

Z´

O

+NHC

X´

Surface coverage Ns (arbitrary units)

2ω

0

S0

0 Rhodamine 110

Rhodamine 6G

Figure 21. The second harmonic excitation spectra of the rhodamine 110 and 6G molecules differ slightly in the position of their resonant peaks. The slight change in the molecular structure, as shown, causes the shift.

0.2

0.4

0.6

2 4 Concentration ρ (mM)

6

Figure 23. Other than surface second-order nonlinear optics, there is no manner in which to perform surface sensitive spectroscopy at a solid liquid interface. The measurement demonstrates how the surface molecular adsorption from solution can be measured in-situ.

612

NONLINEAR SYSTEMS

The technique can also be used as an in situ probe of effects at a solid liquid interface. Figure 23 shows the second harmonic strength of a signal generated at a glass ethanol interface as a function of p-nitrobenzoic acid concentration (18). As the PNBA adsorbs to the glass, the surface coverage increase and the resulting second harmonic signal increases. There is no other technique that can monitor, to this level of sensitivity, a chemical process at a liquid/solid interface. BIBLIOGRAPHY 1. Y. R. Shen, The Principles of Nonlinear Optics, New York: Wiley, 1984. 2. R. W. Minck, R. W. Terhune, and W. G. Rado, Laser-stimulated Raman effect and resonant four-photon interactions in gases H2, D2, and CH4, Appl. Phys. Lett., 3: 181–184, 1963. 3. Y. R. Shen and Y. J. Shaham, Self-focusing and stimulated Raman and Brillouin scattering in liquids, Phys. Rev., 163: 224– 231, 1967. 4. N. Bloembergen et al., IEEE J. Quant. Elect., QE-3: 197, 1967. 5. A. Owyoung, IEEE J. Quant. Electr., QE-14: 192, 1978. 6. J. Heritage and D. L. Allara, Chem. Phys. Lett., 74: 507, 1980. 7. D. Von der Linde, A. Laubereau, and W. Kaiser, Phys. Rev. Lett., 26: 954, 1971. 8. D. Frolich and B. Staginnus, Phys. Rev. Lett., 19: 496, 1967. 9. J. J. Hopfield, J. M. Worlock, and K. J. Park, Phys. Rev. Lett., 11: 414, 1963. 10. M. D. Levenson, IEEE J. Quant. Electr., QE-10: 110, 1974. 11. J-L. Oudar, R. W. Smith, and Y. R. Shen, Appl. Phys. Lett., 34: 758, 1979. 12. S. Haroche, J. A. Paisner, and A. L. Schawlow, Phys. Rev. Lett., 30: 948, 1973. 13. T. W. Hansch, in N. Bloembergen (ed.), Nonlinear Spectroscopy, New York: North-Holland, 1977, p. 17. 14. W. M. Fairbank, T. W. Hansch, and A. L. Schawlow, J. Opt. Soc. Am., 65: 199, 1975. 15. W. E. Cooke et al., Phys. Rev. Lett., 40: 178, 1978. 16. M. L. Zimmerman, J. C. Castro, and D. Kleppner, Phys. Rev. Lett., 40: 1083, 1978. 17. T. F. Heinz et al., Spectroscopy of molecular monolayers by resonant second-harmonic generation, Phys. Rev. Lett., 48: 478–481, 1982. 18. T. F. Heinz, H. W. K. Tom, and Y. R. Shen, Detection of molecular orientation of monolayer adsorbates by optical second-harmonic generation, Phys. Rev., A28: 1883–1885, 1983.

JEFFREY H. HUNT Boeing North American

NONLINEAR OSCILLATORS. See RELAXATION OSCILLATORS AND NETWORKS.

NONLINEAR POLYNOMIAL EQUATIONS. See POLYNOMIALS.

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Wiley Encyclopedia of Electrical and Electronics Engineering Optical Communication Standard Article Casimer DeCusatis1 1IBM Corporation, Poughkeepsie, NY Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W4405 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (347K)

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Abstract The sections in this article are Introduction Fundamentals of Optical Fiber Link Budget Optical Link Standards Advanced Topics Conclusion About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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236

OPTICAL COMMUNICATION

OPTICAL COMMUNICATION The use of light as a signaling medium is not a recent development; communicating using light is a very basic concept. Visual communication systems have been used for centuries, including smoke signals, beacon fires, and semaphores. Even more modern concepts, such as optical communications for telephone systems, are far from being new ideas; a century ago, in 1880, Alexander Graham Bell developed the Photophone, capable of transmitting speech over several hundred meters using visible light beams; by comparison, Marconi demonstrated wireless radio for the first time in 1895. Although Bell’s early system was crude by today’s standards and proved to be impractical, it nevertheless set the stage for exploration of optical frequencies for communications (1,2). Over the years, progress in fiber optics has been the result of an interdisciplinary collaboration involving electrical engineers, physicists, materials scientists, and others (this method of development has also characterized semiconductor electronics). As a result, many of the key enabling technologies such as semiconductor lasers, low loss optical fibers, and integrated electronics were developed approximately at the same time. We can trace the development of modern optical fiber communications to a combination of incremental innovations in the existing art and scientific breakthroughs (such as the invention of the laser), which led to entirely new technologies. As communication engineers sought to investigate higher and higher frequencies for transmission, eventually leading to microwave systems in the early 1940s, speculation on the use of optical communications began in the years following World War II. This background of theory was put to use when the laser was first described by Townes and Schawlow in 1958, and subsequently demonstrated for the first time by J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

OPTICAL COMMUNICATION

Maiman in 1960. Thus began a prolific period of development in the technology that would make optical communications practical. Following a proposal in 1966 by Kao (a British engineer with the Standard Telecommunications Laboratory) that loss in glass fibers could be significantly reduced to very low levels, Corning Glass Works proceeded to develop the first practical optical fiber (with loss below 20 dB/km) in 1970. Advances in the following years led to losses of well below 1 dB/ km, at about the same time that semiconductor lasers became available that were capable of continuous operation at room temperature. By the mid-1970s demonstrations of optical fiber communications systems was well under way, which ultimately led to the proliferation of fiber systems in the telecommunications infrastructure. Initially, the predominant use of optical fiber has been as a replacement for point-to-point electrical wiring, except at much higher data rates and longer distances. While this represents an important advance in communication technology, it is far from realizing the inherent potential of optical communication systems. Just as electrical communications has evolved from the inception of point-to-point telegraph systems to modern networks which combine voice, data, video and other types of multimedia, optical communication has been growing from a simple drop-in replacement for copper wire to the development of new types of communication designed specifically to take advantage of optical interconnects. INTRODUCTION It is important to realize that while optical communication may appear to be simply an extension of electronic communications using copper wire, the two are fundamentally very different phenomena. While it can be useful to think of light traveling in a fiber in the same way that electricity does in a wire, this analogy can be misleading. Light is an electromagnetic wave and the optical fiber is a waveguide; the fundamental principles of optical communications are all based on guided waves, rather than voltage and current potentials. Even fairly simple concepts such as joining two fibers together (coupling) is done very differently from splicing a pair of electrical wires. While electronic and optical communications are closely related, they employ different principles and apply them in different ways. The basic components of an optical communication system are shown in the block diagram of Fig. 1; they consist of the following: An electrical data stream to be transmitted from the sender to the receiver.

Transmitter

Receiver Connector

Connector Glass fiber Amplifier

Modulator Light sensor

Light source Electricity

Light

Detector

Electricity

Figure 1. Block diagram, basic optical communication system.

237

An optical source (such as a laser or light emitting diode) that is modulated by the electrical data and couples the modulated light into an optical fiber. The optical fiber cable, which can be of several different types. Signals experience dispersion and lose their strength as they travel along the fiber. The fiber typically includes at least one optical connection at either end to couple light into and out of the fiber, possibly using additional lenses or optical elements, as well as couplers, splices, splitters, or other components along the path. An optical receiver (such as a photodiode), which converts the optical signal back into electrical form. This step may be followed by additional electronics that amplify the signal, decode the transitions of individual data bits and their relative timing, and reconstructs an approximation of the original data stream presented to the sender. There are many applications for optical communication links in telecommunications, data communications, and analog systems. Telecommunication applications consist primarily of voice traffic, although increasingly multimedia applications are running over the voice network. Datacom systems provide interconnections between computer equipment and have somewhat different requirements; they typically use shorter distances than telecom links, operate at lower bit error rates because the consequences of a single bit error are more severe, must maintain international class 1 (inherently safe) optical power levels at all times, and must be robust enough to withstand frequent handling. Telecom links, by contrast, are handled only by trained professionals, but because they are much longer they also encounter concerns with outdoor cable (either buried underground or suspended from towers) or undersea cable. Analog communication systems have been reserved for military applications, although recent developments in cable television distribution may take advantage of this technology. While digital systems often use direct (incoherent) detection schemes, some analog systems employ coherent detection. In this case, the received optical signal is combined with a local oscillator in the form of an unmodulated beam of light derived from the same source; mixing these two signals allows the detection of the modulated information in the same way that analog radio frequency (RF) communication systems operate. Because of the difficulty in deriving a suitable reference light beam, this technique is typically restricted to military applications and related areas. Optical sources, including semiconductor lasers, light emitting diodes (LEDs), and vertical cavity surface emitting lasers (VCSELs), as well as semiconductor optical detectors, including positive–intrinsic–negative (p–i–n) and avalanche photodiodes are treated in detail elsewhere in the encyclopedia. The manufacturing of optical fiber and cables is treated in detail elsewhere, and industry standards are available for fiber optic testing, installation of fiber, fire safety requirements for optical cables, and laser safety for optical transmitters (1– 10). The fundamentals of optical fiber, optical coupling, and link design will be treated in more detail later. Optical communication has been widely adopted because it offers many well-known advantages such as the following: 1. Information Capacity. Optical fibers offer inherently higher potential data transmission rates than electrical

238

OPTICAL COMMUNICATION

wires. The theoretical bandwidth of a single-mode optical fiber is approximately 25 THz or enough capacity to carry all of the world’s telephone traffic at peak usage on one pair of fibers. While it is not possible to exploit this full potential today, systems operating in the hundreds-of-gigabits range have been demonstrated in laboratory experiments. Existing electronics cannot be operated at such high speeds; however, fibers offer the ability to multiplex different optical wavelengths on a single fiber without interference. In this way, many high-speed data streams can be combined on a single fiber to approach data rates far exceeding conventional copper wiring. Furthermore, optical fibers offer a larger bandwidth-distance product, which is an important measure of the usefulness of a communication channel. A single copper wire may be able to carry gigabit data rates, but only over distances of a few meters; in contrast, optical fiber offers bandwidth-distance products in the range of terabits over hundreds of kilometers. Fiber optic technology is still in its infancy in this regard, and future systems are expected to take fuller advantage of this capacity. A related advantage of optical fiber is the ability to enhance the capacity of existing fiber links as new technology becomes available simply by changing the equipment at either end of the link; it is not required to change the fiber itself. 2. Distance Between Signal Regenerators. As noted above, the bandwidth-distance product of optical fiber enables high-fidelity transmission over long distances. As a signal travels along a communication link, it decreases in strength (attenuation) and is corrupted by various types of noise. Data pulses also tend to spread out as they propagate (dispersion) and begin to interfere with adjacent pulses. After some distance, the signal must be strengthened and its noise removed by using a repeater or amplifier; timing information may also be recovered by retiming or regenerating the signal. Because optical fiber offers a higher bandwidth-distance product, signals at a given data rate will travel farther in optical fiber before requiring regeneration. For example, commercial phone lines require repeaters spaced about every 12 km; optical links have increased this distance to around 40 km, and some recently installed systems (1997) extend the unrepeated distance up to 120 km or more. 3. Immunity to Electromagnetic Interference. It is an obvious but very important point that there are no electrical components along a fiber link, except for the electronics at either end or at the repeaters. This means that the transmission medium (glass fibers) can neither pick up nor create electrical interference. There are very few things which can interfere with or distort an optical signal traveling along a fiber; this is one reason why optical communication offers such high-fidelity transmission. This means that optical fiber can be used in locations where electrical cable would not be able to function, such as near a noise source such as an electrical motor or in a cable duct next to heavy-duty power lines. Since there is no danger of electrocution from an optical cable, it may be routed near water or other hazards; the optical fiber is very safe, because it isolates

the transmission medium from high voltages at either end of the link. Outdoor optical cable does not require special lightning protection. In a network environment, there is much greater flexibility in selecting a route for optical fiber. Electrical communication systems are also subject to ground loops, or small voltage potentials between points in the link which should be a common ground; this problem is particularly acute if electrical cables are grounded at both ends of a long link. Of course, this problem is not present when using optical fiber, which acts to electrically isolate equipment on both ends of the link. 4. Improved Security. Because optical fiber is nonconductive, it does not radiate electromagnetic fields; metallic cables require additional shielding to achieve similar effects. It is possible to eavesdrop by tapping a fiber optic cable, but it is very difficult to do and the additional loss caused by the tap is relatively easy to detect and locate. Tapping an optical fiber will generally require interrupting the link while the tap is inserted, and there are few access points where an intruder can gain the intimate access to a fiber cable necessary to insert a tap. Placement of active taps that insert false signals onto the optical link is even more difficult. Thus, optical fiber offers greater security for data transmission. 5. Weight, Size, and Material Cost. For the same transmission capacity, the material cost for optical fiber is significantly less than for copper cable, and offers a significant reduction in both cable diameter and cable weight. Optical cable is much more flexible due to its small size, and can be routed around tight bends or through small holes more easily. This also means that the cost of installing fiber cable is significantly reduced. We must also point out that any communication technology has limitations and weaknesses; some of the disadvantages of optical fiber include the following: 1. More Complex Terminations. Multimode fibers are about the same diameter as a human hair; single mode fibers are even smaller. Because optical fiber is a waveguide, two fibers cannot simply be spliced in the same way as electrical wires. Instead, the fiber ends must be aligned with each other, then joined by melting the glass (fusion splicing). Although splicing technology is fairly well developed, making durable low-loss splices remains a skilled task that requires precision equipment. It is particularly difficult to do under extremes of temperature or in tight physical locations. Pluggable optical connectors have been developed with good reliability and low loss, although single-mode connections remain more difficult than multimode connections and contamination from even small amounts of dirt can obscure the fiber endface if connectors are not properly cleaned before use. The difficulty of aligning fibers with optical sources or detectors remains one of the most important factors contributing to the high cost of optical transceivers as compared with electrical transceivers. 2. Bending Loss. As light travels along a fiber waveguide, the fiber’s composition is designed to bend light back toward the fiber core and away from the outer cladding.

OPTICAL COMMUNICATION

This guiding mechanism allows light to remain confined within the fiber when the fiber is bent; however, if the bending radius becomes too great, light will escape from the fiber core and the signal strength will greatly diminish. For this reason, optical fibers cannot bend too sharply; the allowed bend radius varies with specific cables because it depends on the difference in refractive index between the core and cladding. As we will see, there is a tradeoff involved; for many reasons we would like to keep this difference as small as possible, yet the smaller the index difference, the larger the bend radius limit. 3. Susceptibility to Different Noise Sources. Some types of optical fibers function as a waveguide with many different modes supported at the same time. If light is not coupled into the fiber in the proper way, or if there are flaws in the fiber, then some modes will propagate at greater velocities than others. This intermodal delay can cause severe limits in the performance of optical communication systems, and it is one reason why optical transceiver designs are different for single-mode and multimode fiber. The glass used in optical fiber is typically doped with small concentrations of impurities to improve the optical transmission properties; this makes optical fiber susceptible to ionizing (gamma) radiation. Similarly to high-energy X rays, large amounts of this type of radiation can cause glass to become opaque to the optical signal wavelengths. In some types of glass, gamma radiation can also cause spontaneous emission of light, which interferes with the desired signal. Optical cable placed in environments such as nuclear power plants, space satellites, or medical centers must allow for this effect; when optical fiber is buried or placed undersea, the long-term cumulative effects of background radiation may limit the fiber’s effective lifetime. Very high voltage electrical fields (above 30,000 V) can affect glass in the same way as gamma radiation; this presents a concern for routing optical cable along high-voltage power lines. Finally, although optical cables are surprisingly rugged, they must be carefully shielded in outdoor environments where insects and rodents (particularly gophers) can attack the cable. There is actually a standardized test for outdoor cables, conducted by approved wildlife organizations, which involves placing cables in a gopher enclosure for a specified length of time and observing the resulting damage.

Frequency (Hz)

239

Wavelength (m) Gamma rays

1020

X rays Ultraviolet Visible light Infrared

1010 Microwaves 106

10–9

1 nm

10–6

1 µm

10–3

1 mm

1

TV Radio

102

1m

103

1 km

106

1000 km

Figure 2. The electromagnetic spectrum.

face of a glass fiber (Fig. 3). We know from Snell’s Law (1) that incoming light rays will be refracted when they strike the glass core at an angle; once the rays enter the fiber, they undergo refraction again at the interface between the fiber core and the outside environment. To make the fiber function as a waveguide, the core is surrounded by another layer of material called the cladding. The glass fiber acts as a waveguide if there is a difference in refractive index between the fiber core and cladding materials. If the core has a higher refractive index than the cladding, light rays entering the fiber core at the proper angle are continuously refracted away from the core-cladding interface and continue to propagate along the fiber. This occurs only for light entering the fiber core with an acceptance cone defined by the angle ␪ in Fig. 3. It is conventional to describe the amount of light that can be coupled into a fiber in terms of the numerical aperture (NA), which is easily derived from Snell’s Law as NA = n1 sin θ

(1)

where n1 is the refractive index of the media outside the fiber. For air, n1 ⫽ 1 and the NA is simply defined as the sin of the largest angle contained within the acceptance cone. If the refractive index of the fiber core is n2, then we can also redefine NA in terms of the dimensionless parameter ⌬, which is given by  = n1 −

FUNDAMENTALS OF OPTICAL FIBER

n2 n2

(2)

so that we have To understand how light propagates in an optical fiber, we must first introduce the mathematical expressions for light wave propagation. Light is most accurately described as a vector electromagnetic wave (1–10), in the same manner as any other component of the electromagnetic spectrum (Fig. 2). This is the only satisfactory way to describe light propagation in a singlemode fiber; however, this level of complexity is often not required for describing many important properties of fiber optics. A good approximation to describing optical coupling can be obtained from geometric optics by considering light rays emanating from an optical source and impinging on the end-

NA = n1

√ 2

(3)

NA = sin θ

θ

Cladding Core

Figure 3. Definition of numerical aperture. Reprinted from Ref. 3, courtesy of Academic Press.

240

OPTICAL COMMUNICATION

or in phasor form, where the real part of the right-hand side is assumed,

Multimode step index 125

E = E0 (x, y)e j

50/62 Multimode graded index

(ωt −β z )

(5)

where E0 is the peak amplitude, 웆 is the radian frequency, which is equal to 2앟f, f is the frequency in hertz, and 웁 is the propagation constant defined by

125 50/62 Single mode

β = 2π/λ = ω/v

125 9 Figure 4. Definition of modes in an optical fiber. Ray paths are straight in step-index fiber and curved in graded index fiber due to the focusing properties of the graded index fiber core.

Consider many rays entering the fiber at various angles within the NA; all the rays propagate along the fiber, but at different angles (Fig. 4). Each ray path may be thought of as a mode of propagation; if the fiber core is large, many ray paths, or modes, are possible. This is known as a multimode optical fiber. Conceptually, if we reduce the size of the fiber core enough we reach a point where only one ray path, or mode, is bound within the fiber and all others are not guided. This is known as a single-mode fiber. We discuss the two different fiber types in more detail later; for now, note that the typical multimode fiber has a core diameter of approximately 50 애m to 62.5 애m, whereas a single-mode fiber has a core diameter of approximately 10 애m. Numerical aperture is a frequently quoted specification of optical fiber; for multimode fiber NA is typically between 0.2 and 0.3, while for singlemode fibers it is around 0.1. In addition to being a measure of the fiber’s ability to collect light, NA is also a good measure of the light-guiding properties of the fiber. Larger NA corresponds to more modes and greater fiber dispersion. Larger NA also means that the fiber can undergo tighter bends (smaller bend radius) before bending loss becomes a problem. Furthermore, in single-mode fiber, higher NA means a higher contrast in refractive index between the core and cladding; this is often due to higher dopant concentrations in the cladding material. This doping increases the attenuation of the fiber; since a large portion of the light in single-mode fibers travels in the cladding, there will often be higher attenuation in single-mode fiber with larger NA. The ray model for light propagation may lead us to believe that there are an infinity of possible paths for light rays to follow within the fiber, provided they are launched within the NA. However, this is not actually the case; light rays are drawn perpendicular to the optical wavefront, and are valid only for plane waves in which the diameter of the light beam is much larger than the wavelength. To fully understand light propagation within a fiber, we must return to the wave description of light. We assume a harmonically time-varying wave propagating in the z direction of Fig. 5 with phase constant 웁; the electric field can be expressed as E = E0 (x, y) cos(ωt − βz)

(4)

(6)

where v is the propagation velocity, or phase velocity, defined by v = c/n

(7)

c is the speed of light, and n is the refractive index. Note that the phase velocity describes the speed of the wavefront as it propagates along the fiber; this is slightly different than the group velocity, which describes the propagation of information such as a modulated wave packet along the fiber and is given by vg = dω/dβ

(8)

If there is no dispersion in the medium then group velocity and phase velocity are the same; otherwise, phase velocity is slightly greater. Since light is an electromagnetic wave, it is governed by Maxwell’s equations (1); starting from these fundamental expressions, it is possible to derive the wave equation that describes propagation of the electric and magnetic fields in the z direction defined in Fig. 5. The wave equation for the longitudinal electric field component in the z direction is given by t2 Ez (x, y) + βt2 Ez (x, y) = 0

(9)

where we have introduced the transverse Laplacian notation: t2 =

d2 d2 + dx2 dy2

(10)

and the transverse phase constant βt2 = (2πn/λ)2 − β 2

(11)

When light is confined in space, boundary conditions imposed on the light restrict the phase constant to a limited number of values. Each possible phase constant represents a mode; thus, when light is confined within a fiber it can propagate only in a limited number of ways. Under ideal conditions, all

x φ r

a

z y

Figure 5. Typical optical fiber geometry. Reprinted from Ref. 3, courtesy of Academic Press.

OPTICAL COMMUNICATION

of the propagating modes within a fiber are orthogonal, and there is no power transfer between modes or interference between one mode and another. In practice this often does not hold true, as we see in the discussion of link budgets. In almost all practical fibers, the refractive index difference between the core and cladding is small and the modes are called weakly guided; some modes enter the cladding, either due to the launch conditions or as a result of bends in the fiber. Still others called leaky modes satisfy the marginal case between being bound and being a cladding mode; these modes propagate for some distance before being lost from the core and cladding; in a short link, however, they may reach the optical receiver and create excess dispersion because their group velocities are much slower than the bound modes. Some link designers incorporate several tight bends in the fiber, intended to pass all bound modes and eliminate leaky ones; this practice is called mode stripping. To find how many modes can propagate in a fiber, their phase constants, and their spatial profile, we must solve the wave equation for a specific fiber geometry. This is often done by transforming the wave equation into cylindrical coordinates to accommodate the geometry of a glass fiber. The solution depends on the refractive index profile of the fiber. The simplest case is a step index fiber (shown in Fig. 6), for which there is an abrupt transition of the refractive index at the core/cladding boundary. For step index fibers, a complete set of analytical solutions to the wave equation can be given; the solutions can be grouped into different types of modes, known as transverse electric (TE), transverse magnetic (TM), and hybrid modes. In practice the refractive index difference between core and cladding is very small, only approximately 0.005, so most of these modes are degenerate and it is sufficient to use a single notation for all types, calling them lateral polarization (LPlm) modes (where the subscripts l and m refer to the number of radial and azimuthal zeros for a particular mode). To determine if a given LP mode will propagate, it is very useful to define the normalized frequency, or V number, of the fiber,




V = ka n21 − n22

(12)

n1

241

n1(r) n2

r=0 ~50 µ m ~120 µ m

n2

r=0 ~50 µ m ~120 µ m

(a)

(b) ~10 µ m

~10 µ m

(c)

(d) ~5 µ m

~10 µ m

(e)

(f)

~6 µ m

~6 µ m

(g)

(h) ~6 µ m

~6 µ m

(i)

(j)

Figure 6. Refractive index profiles of (a) step-index multimode fibers, (b) graded index multimode fibers, (c) match cladding singlemode fibers, (d,e) depressed cladding single-mode fibers, (f–h) dispersion shifted fibers, and (i,j) dispersion flattened fibers. Reprinted from Ref. 3, courtesy of Academic Press.

and the normalized propagation constant for a particular mode, b, give by b=

(β 2 /k2 ) − n22 n21 − n22

1

(13) 0.8

where a is the core radius. Therefore, a mode cannot propagate in the fiber if its wavelength is longer than the cut-off wavelength. Cut-off values for the V number have been tabulated (11). It is also possible to determine the spatial intensity distributions of these modes; to a very close approximation,

01 0.6

11

b(V)

Rather than use the formal definition for b given above, for LP modes, Gloge et al. (11) has derived a series of analytical formulas to determine b for different modes to a very good approximation. Using these expressions, we can plot b(V) as shown in Fig. 7. It can be seen that for guided modes, b varies between 0 and 1; the wavelength for which b ⫽ 0 is called the cut-off wavelength, √ b(V ) = 0 → λco = (2π/V )a n1 2 (14)

21 31 02 12

0.4

0.2

0

0

2

4

6

8

10

Figure 7. Cutoff frequencies for the lowest order LP modes; b(V ) is the normalized propagation constant as a function of V number. Reprinted from Ref. 3, courtesy of Academic Press.

242

OPTICAL COMMUNICATION

approximation (13,14). The phase constants for different modes can be shown to obey the following relationship:

E0 e–1E0

βm = n k

r

MFD




1 − 2(m/n)q/q+2

for m = 1, 2, . . ., N

(20)

Optical Coupling

Figure 8. Definition of mode field diameter in an optical fiber. Reprinted from Ref. 3, courtesy of Academic Press.

Some typical properties of optical fibers are shown in Table 1. For an incoherent light source such as a light emitting diode (LED), the total power accepted by the fiber, P, is given by P = πBA(NA)2

the fundamental mode can be described as a Gaussian function, E(r) = E0 exp[−(r/w)2 ]

(15)

where E0 is the amplitude and 2w is called the mode field diameter (MFD). The MFD is another important parameter; it represents the optical spot size that carries most of the optical power (mathematically, the diameter of the spot for which the power has decreased by e⫺2 or the amplitude has decreased by e⫺1). As illustrated in Fig. 8, the MFD is not necessarily the same dimension as the fiber core; it is larger than the core of a single-mode fiber, thus much of the optical power in a single-mode fiber is carried by the cladding. The MFD is much smaller than the core diameter of a typical multimode fiber. For wavelengths between 0.8 and 2 times the cutoff wavelength, the optimum MFD (12) is given by w = 0.65 + 1.69V −3/2 + 2.87V −6 a

n2 (r) = n21 (1 − 2) = n22

for r ≥ a

(22)

where Pin is the power launched into the fiber and Plm is the power accepted by the LPlm mode. For optical link budget ␩ analysis, it is more convenient to express coupling loss in decibels, α = 10 log η

(23)

For coherent light sources, such as lasers, which can be approximated by a Gaussian beam, the coupling efficiency for the LP01 mode in single-mode fiber may be calculated analytically (12,15) 4D exp B




−

AC B



(24)

where

(17)

(18)

This expression is only valid for large V numbers; for a step index fiber q is infinite and N = V 2 /2

η = Plm /Pin

(16)

where q is called the profile exponent. The optimum profile for minimum dispersion is given for q slightly less than 2. For multimode fibers, the total number of modes, N, which can propagate is given by q N = 12 V 2 q+2

where B is the LED’s radiance in units of watts per area and steradian and A is the cross-sectional area of the fiber. Given the total power accepted by the fiber, the coupling efficiency may be defined as

η=

The radial distribution for higher order modes can be obtained using Bessel functions (13). The preceding discussion is valid only for step index fibers; in practice, the refractive index of the core material is usually graded as shown in Fig. 6. This causes different modes to propagate with similar velocities. The various graded index profiles are described by the expression

n2 (r) = n21 (1 − 2(r/a)q ) for 0 ≤ r ≤ a

(21)

(19)

The different ray paths possible in multimode fiber can be calculated from geometric optics (1). A more accurate approach is to solve the wave equation using the so-called WKB

A=

(2π n s1 /λ)2 2

(25)

B = G2 + (D + 1)2

(26)

C = (D + 1)F 2 + 2DFG sin θ + D(G2 + D + 1) sin2 θ

(27)

D=


s 2 2

(28)

s1

F=

2 2π n s21 /λ

(29)

G=

2Z 2π n s21 /λ

(30)

Table 1. Typical Properties of Single-mode Optical Fiber Compliant with CCITT Recommendation G.652a Parameters Cladding diameter Mode field diameter Cutoff wavelength ␭co 1550 nm bend loss Dispersion

Dispersion slope

Specifications 125 애m 9–10 애m 1100–1280 nm ⱕ1 dB for 100 turns of 7.5 cm diameter ⱕ3.5 ps/nm · km between 1285 and 1330 nm ⱕ6 ps/nm · km between 1270 and 1340 nm ⱕ20 ps/nm · km at 1550 nm ⱕ0.095 ps/nm2 · km

Reprinted from Ref. 3, courtesy of Academic Press.

OPTICAL COMMUNICATION

Optical fiber cores

(a) ∆

(b) ∆z

(c)

θ (d) Figure 9. Different coupling cases for optical fiber cores. (a) Mismatch in core size and mode field diameter (spot size); (b) transverse offset; (c) lack of physical contact or lateral offset; (d) angular misalignment. Reprinted from Ref. 3, courtesy of Academic Press.

where ⌬Z is the separation distance between source and fiber, ⌬ is the lateral displacement, ␪ is the angular displacement, and s1 and s2 are the mode field radii or spot sizes of the source field and fiber field, respectively. This expression takes into account four different coupling cases, shown in Fig. 9; if only one condition is present at a time, the expression simplifies as follows: Spot size mismatch (s1 not equal to s2):




η=

2s1 s2 s21 + s22

2

(31)

243

NA, a so-called equilibrium mode launch. Typically, the optical fiber is threaded into a ceramic ferrule, which in turn is part of a connector assembly. The ferrule is then inserted into bores that house the optical transmitter and receiver. Two kinds of bores are available, solid and split sleeve. A solid sleeve is a cylinder of rigid ceramic material, as shown in Fig. 10, whereas a split sleeve is made of plastic or other flexible material and is designed to enlarge slightly to accommodate the ferrule. Split sleeves are most commonly used in fiber-tofiber couplers, whereas solid bores are often found in optical transceivers. When inserting a ferrule into an optical transceiver, the position of the optical axis relative to the mechanical axis of the bore is known as the beam centrality (BC). The corresponding misalignment between the centerlines of the ferrule and the fiber, due to imperfect ferrule manufacturing, is known as the ferrule–core eccentricity (FCE). It is desirable to have both BC and FCE equal to zero for optimal alignment. Even when this is the case, the laser beam may still possess some small tilt, known as the pointing angle, that adversely affects coupling. Furthermore, the ferrule endface may not be polished exactly flat, or may be deliberately polished at a 5⬚ to 10⬚ angle to minimize reflections. Optical sources and detectors are often packaged together to form transceivers capable of bidirectional communication over a pair of optical fibers. Such transceivers accept a duplex optical connector, which compounds the alignment problems further. The axial misalignment of the entire optical subassembly and bore from its desired position in the transceiver is called subassembly misalignment (SAM). To partially compensate for alignment errors, most optical connectors are designed with some amount of lateral movement or float, which enables the ferrule to be moved slightly within the connector assembly when it is plugged into a transceiver. Other characteristics of the transceiver that affect optical coupling are the connector body dimensions and transceiver receptacle dimensions. Poor control of the mating tolerances can lead to two basic problems. First, the same connector plugged into the same transceiver may exhibit excessive variation in the launched power; this is known as plug repeatability. Second,

Transverse offset, ⌬ η = exp[−(/s2 )2 ]

(32)

Longitudinal offset ⌬Z η=

1 1 + (Z/2Z0 )2

(33) Ceramic solid bore

Angular misalignment, ␪ η = exp[−(θ/θ0 )2 ]

(a)

(34)

All of these expressions must be modified if a lens system is used to couple light into a fiber; they must also be corrected for reflection losses at the fiber endface (16). These expressions are important in the design of optical connectors and splices to minimize loss in the optical link budget. Both mechanical and optical characteristics must be considered in the design of optical interfaces between fibers and active devices, or between a pair of fibers. It is desirable to control the launch conditions from a source into a fiber such that at least 70% of the light is captured in 70% of the fiber’s

Metal split-sleeve bore (b) Figure 10. Solid and split-sleeve bores. (a) Ceramic solid bore. (b) Metal split-sleeve bore. Reprinted from Ref. 3, courtesy of Academic Press.

OPTICAL COMMUNICATION

Table 2. Typical Values Affecting Optical Metrology Influence on CPR Element BD BC PA SBS FD FCE EFA FBS SAM FF SD CD

Typical Value 2.5010–2.5025 mm 3.3 ⫾ 2.0 애m ⬍3⬚ 5–20 애m 2.4992–2.5000 mm ⬍1.6 애m (0.7 애m) 0.1⬚–0.2⬚ 9.5 ⫾ 0.5 애m (SM only) spec. ⫾ 0.34 mm spec. ⫾ 0.34 mm spec. ⫾ 0.125 mm spec. ⫾ 0.125 mm

Solid Bore X XX XX XX X X X X

Split Sleeve XX XX XX

received optical power levels. Some power is lost by connections, splices, and bulk attenuation in the fiber. There may also be optical power penalties due to dispersion, modal noise, or other effects in the fiber and electronics. The optical power levels define the signal-to-noise ratio at the receiver, Q; this is related to the bit error rate (BER) by the Gaussian integral

1 BER = √ 2π




∞




exp −

Q

Q2 2



dQ ∼ =




Q2 1 √ exp − 2 Q 2π



(35) X X X X X X X

Reprinted from Ref. 4, courtesy of Academic Press.

when multiple connectors are mated with the same transceiver the difference between the highest and lowest coupled power levels is known as the cross-plug range; it is desirable to keep this as low as possible. When designing the interface between an optical fiber and a light source, an important parameter is the coupled power range (CPR) or the allowable difference between minimum and maximum optical power levels. For example, the maximum power level may be determined by laser safety standards, while the minimum power is determined by the need to achieve a desired bit error rate when measured with a given receiver sensitivity. Establishing the CPR for a given application requires an understanding of the optical link budget, which is discussed in the following section. There are about a dozen key characteristics of the optical interface metrology that affect coupled power ratio; these are summarized in Table 2 along with typical values. For solid bore transceivers, only eight parameters have a significant effect on alignment; the first four parameters in Table 2 are related to the transceiver design and the second four are related to the connector design. The relative effect of each parameter on CPR is indicated by the number of X’s shown; a blank column indicates no effect, one X indicates some effect, and two X’s indicate a significant effect. When all the components of radial, axial, and angular misalignment are present together, they can interact in such a way that some parameters become even more critical in determining CPR. Analytic solutions for misalignments in the same plane are given by Nemoto and Makimoto (15,17); out-of-plane alignments are more difficult to analyze. Transceiver manufacturers typically employ some combination of active alignment (which involves alignment of a source and detector while both are operational) and passive alignment (which depends on the mechanical tolerances of the design) in different products.

From Eq. (35), we see that a plot of BER and the received optical power yields a straight line on a semilog scale, as illustrated in Fig. 11. Nominally, the slope is approximately 1.8 dB/decade; deviations from a straight line may indicate the presence of nonlinear or non-Gaussian noise sources. Some effects, such as fiber attenuation, are linear noise sources; they can be overcome by increasing the received optical power, as seen from Fig. 11, subject to constraints on maximum optical power (laser safety) and the limits of receiver sensitivity. There are other types of noise sources, such as mode partition noise or relative intensity noise (RIN), which are independent of signal strength. When such noise is present, no amount of increase in transmitted signal strength will affect the BER; a noise floor is produced, as shown by curve B in Fig. 11. This type of noise can be a serious limitation on link performance. If we plot BER against receiver sensitivity for increasing optical power, we obtain a curve similar to Fig. 12 which shows that for very high power levels, the receiver will go into saturation. The characteristic bathtub-shaped curve illustrates a window of operation with both upper and lower limits on the received power. There may also be an upper limit on optical power because of eye safety considerations. We can see from Fig. 11 that receiver sensitivity is specified at a given BER, which is often too low to measure directly in a reasonable amount of time (for example, a 200 Mbit/s link operating at a BER of 10⫺15 will only take one error every 57 days on average, and several hundred errors are recommended for a reasonable BER measurement). For practical

10–5

10–6 Bit error rate

244

B

10–7 10–8

A

10–9 10–10

LINK BUDGET An optical transmitter is capable of launching a limited amount of optical power into the fiber; there is a limit to how weak a signal can be detected by the receiver in the presence of noise. Thus, a fundamental consideration is the optical link power budget, or the difference between the transmitted and

10–11 10–12

Incident optical average power (dBm)

Figure 11. Bit error rate as a function of received optical power. Curve A shows typical performance, whereas curve B shows a BER floor. Reprinted from Ref. 3, courtesy of Academic Press.

OPTICAL COMMUNICATION

10–4

Bit error rate (errors/bits)

10–5 10–6

Saturation Minimum sensitivity

10–7 10–8 10–9 10–10 10–11 10–12

–35 –30 –15 Receiver sensitivity (dBm)

–10

Figure 12. Bit error rate as a function of received optical power illustrating range of operation from minimum sensitivity to saturation. Reprinted from Ref. 3, courtesy of Academic Press.

reasons, the BER is typically measured at much higher error rates, where the data can be collected more quickly (such as 10⫺4 to 10⫺8) and then extrapolated to find the sensitivity at low BER. This assumes the absence of nonlinear noise floors, as cautioned previously. The relationship between optical input power, in watts, and the BER, is the complimentary Gaussian error function BER = 1/2 erfc(Pout − Psignal/rms noise)

(36)

where the error function is an open integral that cannot be solved directly. Several approximations have been developed for this integral, which can be developed into transformation functions that yield a linear least-squares fit to the data (1). The same curve-fitting equations can also be used to characterize the eye window performance of optical receivers. Clock position/phase and BER data are collected for each edge of the eye window; these data sets are then curve fitted with the above expressions to determine the clock position at the desired BER. The difference in the two resulting clock positions on either side of the window gives the clear eye opening (2–4). In describing Figs. 11 and 12, we also have made some assumptions about the receiver circuit. Most data links are asynchronous and do not transmit a clock pulse along with the data; instead, a clock is extracted from the incoming data and used to retime the received datastream. We have made the assumption that the BER is measured with the clock at the center of the received data bit; ideally, this occurs when we compare the signal with a preset threshold to determine if a logical ‘‘1’’ or ‘‘0’’ was sent. When the clock is recovered from a receiver circuit such as a phase lock loop, there is always some uncertainty about the clock position; even if it is centered on the data bit, the relative clock position may drift over time. The region of the bit interval in the time domain where the BER is acceptable is called the eyewidth; if the clock timing is swept over the data bit using a delay generator, the BER will degrade near the edges of the eye window. Eyewidth measurements are an important parameter in link

245

design, which is discussed further in the section on jitter and link budget modeling. To design a proper optical data link, the contribution of different types of noise sources should be assessed when developing a link budget. There are two basic approaches to link budget modeling. One method is to design the link to operate at the desired BER when all the individual link components assume their worst-case performance. This conservative approach is desirable when very high performance is required, or when it is difficult or inconvenient to replace failing components near the end of their useful lifetimes. The resulting design has a high safety margin; in some cases, it may be overdesigned for the required level of performance. Because it is very unlikely that all the elements of the link will assume their worst-case performance at the same time, an alternative is to model the link budget statistically. For this method, distributions of transmitter power output, receiver sensitivity, and other parameters are either measured or estimated. They are then combined statistically using an approach such as the Monte Carlo method, in which many possible link combinations are simulated to generate an overall distribution of the available link optical power. A typical approach is the threesigma design, in which the combined variations of all link components are not allowed to extend more than three standard deviations from the average performance target in either direction. The statistical approach results in greater design flexibility and generally increased distance compared with a worst-case model at the same BER. Installation Loss It is convenient to break down the link budget into two areas: installation loss and available power. Installation or DC loss refers to optical losses associated with the fiber cable plant, such as connector loss, splice loss, and bandwidth considerations. Available optical power is the difference between the transmitter output and receiver input powers, minus additional losses from optical noise sources on the link. With this approach, the installation loss budget may be treated statistically and the available power budget as worst case. First, we consider the installation loss budget. Some of the important factors to be considered include the following: fiber cable loss (transmission loss), fiber attenuation as a function of wavelength, connector loss, and splice loss. Transmission Loss. Transmission loss is perhaps the most important property of an optical fiber; it affects the link budget and maximum unrepeated distance. The number and separation between optical repeaters and regenerators is largely determined by this loss. The mechanisms responsible for this loss include material absorption as well as both linear and nonlinear scattering of light from impurities in the fiber (1– 5). Figure 13 illustrates some of the contributions to transmission loss in silica-based fibers as a function of wavelength in the near-infrared region, where optical loss is minimal and most optical communication systems operate. Typical loss for single-mode optical fiber is approximately 2 dB/km to 3 dB/ km near a 800 nm wavelength, 0.5 dB/km near a 1300 nm wavelength, and 0.2 dB/km near a 1550 nm wavelength. Multimode fiber loss is slightly higher, and bending loss will only increase the link attenuation further.

OPTICAL COMMUNICATION

1.0

1.1

1.2

Wavelength (µ m) 1.4 1.5 1.6 1.7 1.8 1.3

3.0 2.0

2.5

0.5

Attenuation (dB)

246

Total loss (experimental)

2.0 1.5 1.0 0.5

0.3

1.2

Loss (dB/km)

Total loss (estimated) 0.2

1.3

1.4 1.5 Wavelength (µ m)

1.6

1.7

Figure 14. Attenuation versus wavelength of a singlemode fiber for different impurity levels. Attenuation peak near 1.4 애m increases with impurity concentration in fiber. Reprinted from Ref. 3, courtesy of Academic Press.

Rayleigh scattering loss 0.1 Infrared absorption loss Infrared absorption loss

0.05 Loss due to imperfections of waveguide 0.03

0.02 1.2

1.1

1.0 0.9 0.8 Photon energy (eV)

0.7

0.6

Figure 13. Transmission loss in silica-based optical fibers. Reprinted from Ref. 3, courtesy of Academic Press.

Attenuation versus Wavelength. Since fiber loss varies with wavelength, changes in the source wavelength or use of sources with a spectrum of wavelengths will produce additional loss. Transmission loss is minimized near the 1550 nm wavelength band, which unfortunately does not correspond with the dispersion minimum at around 1310 nm. An accurate model for fiber loss as a function of wavelength has been developed by Walker (18); this model accounts for the effects of linear scattering, macrobending, and material absorption from ultraviolet and infrared band edges, hydroxide (OH) absorption, and absorption from common impurities such as phosphorus. Using this model, it is possible to calculate the fiber loss as a function of wavelength for different impurity levels; an example of such a plot is shown in Fig. 14. Using this method, the fiber properties can be specified along with the acceptable wavelength limits of the source to limit the fiber loss over the entire operating wavelength range; design tradeoffs are possible between center wavelength and fiber composition to achieve the desired result. Typical loss due to wavelength-dependent attenuation for laser sources on single-mode fiber can be held below 0.1 dB/km. Connector and Splice Loss. There are also installation losses associated with fiber optic connectors and splices; both of these are inherently statistical in nature and can be charac-

terized by a Gaussian distribution. There are many different kinds of standardized optical connectors, some of which are shown in Fig. 15. Many different models which have been published for estimating connection loss from fiber misalignment (19,20); most of these treat loss from misalignment of fiber cores, offset of fibers on either side of the connector, and angular misalignment of fibers. The loss due to these effects is then combined into an overall estimate of the connector performance. There is no general model available to treat all types of connectors, but typical connector loss values average about 0.5 dB worst case for multimode, slightly higher for singlemode (see Table 3). Optical splices are required for longer links, because fiber is usually available in spools of 1 to 5 km, or to repair broken fibers. There are two basic types, mechanical splices (which involve placing the two fiber ends in a receptacle that holds them close together, usually with epoxy) and the more commonly used fusion splices (in which the fibers are aligned, then heated sufficiently to fuse the two ends together). Typical splice loss values are given in Table 3.

IBM duplex single-mode connector

FC connector MIC (FDDI) connector

ST connector Bionic nonphysical-contact connector

Fiber channel standard SC-duplex connector

Figure 15. Common fiber optic connectors. Reprinted from Ref. 3, courtesy of Academic Press.

OPTICAL COMMUNICATION

247

Table 3. Typical Fiberoptic Cable Plant Optical Losses Component

Description

Connector a

Physical contact

Connector a

Nonphysical contact (multimode only)

Splice

Mechanical

Splice

Fusion

Cable

Multimode jumper Multimode jumper Single-mode jumper Trunk Trunk Trunk

Size (애m)

Mean loss

Variance (dB2)

62.5–62.5 50.0–50.0 9.0–9.0 b 62.5–50.0 50.0–62.5 62.5–62.5 50.0–50.0 62.5–50.0 50.0–62.5 62.5–62.5 50.0–50.0 9.0–9.0 b 62.5–62.5 50.0–50.0 9.0–9.0 b 62.5 50.0 9.0 62.5 50.0 9.0

0.40 dB 0.40 dB 0.35 dB 2.10 dB 0.00 dB 0.70 dB 0.70 dB 2.40 dB 0.30 dB 0.15 dB 0.15 dB 0.15 dB 0.40 dB 0.40 dB 0.40 dB 1.75 dB/km 3.00 dB/km at 850 nm 0.8 dB/km 1.00 dB/km 0.90 dB/km 0.50 dB/km

0.02 0.02 0.06 0.12 0.01 0.04 0.04 0.12 0.01 0.01 0.01 0.01 0.01 0.01 0.01 NA NA NA NA NA NA

a The connector loss value is typical when attaching identical connectors. The loss can vary significantly at attaching different connector types. b Single-mode connectors and splices must meet a minimum return loss specification of 28 dB. Reprinted from Ref. 3, courtesy of Academic Press.

Optical Power Penalties Next, we consider the assembly loss budget, which is the difference between the transmitter output and receiver input powers, allowing for optical power penalties from noise sources in the link. Contributing factors include the following: Dispersion (modal and chromatic) Mode partition noise Mode hopping Extinction ratio Multipath interference Relative intensity noise (RIN) Timing jitter Radiation-induced darkening Modal noise Nonlinear effects (stimulated Raman and Brillouin scattering, frequency chirping) Dispersion. The most important of these effects, and the most important fiber characteristic after transmission loss, is dispersion, or intersymbol interference. This refers to the broadening of optical pulses as they propagate along the fiber. As pulses broaden, they tend to interfere with adjacent pulses; this limits the data rate. In multimode fibers, there are two dominant kinds of dispersion: modal and chromatic. Modal dispersion refers to the fact that different modes travel at different velocities and cause pulse broadening. The fiber’s modal bandwidth in units of megahertz per kilometer, is specified according to the expression BWmodal = BW1 /Lγ

(37)

where BWmodal is the modal bandwidth for a length L of fiber, BW1 is the manufacturer-specified modal bandwidth of a 1 km section of fiber, and 웂 is a constant known as the modal bandwidth concatenation length-scaling factor. The term 웂 usually assumes a value between 0.5 and 1, depending on details of the fiber manufacturing and design as well as the operating wavelength; it is conservative to take 웂 ⫽ 1.0. Modal bandwidth can be increased by mode mixing, which promotes the interchange of energy between modes to average out the effects of modal dispersion. Fiber splices tend to increase the modal bandwidth, although it is conservative to discard this effect when designing a link. There have been many attempts to fabricate fibers with enhanced modal bandwidth, most of which have not yet produced commercially viable products (1,21). The other major contribution is chromatic dispersion, BWchrom, which occurs because different wavelengths of light propagate at different velocities in the fiber. Put another way, the refractive index of the fiber is wavelength dependent; the fiber index depends on the fiber composition and may be calculated from the Sellmeier equation (1,3). For multimode fiber, chromatic bandwidth is given by an empirical model of the form Lγ c BWchrom = √ λw (a0 + a1 |λc − λeff |)

(38)

where L is the fiber length in km; ␭c is the center wavelength of the source in nm; ␭w is the source FWHM spectral width in nanometers; 웂c is the chromatic bandwidth length scaling coefficient, a constant; ␭eff is the effective wavelength, which combines the effects of the fiber zero dispersion wavelength

OPTICAL COMMUNICATION

104 L−0.69 BWchrom = √ λw (1.1 + 0.0189|λc − 1370|)

(39)

For this expression, the center wavelength was 1335 nm and ␭eff was chosen midway between ␭c and the water absorption peak at 1390 nm; although ␭eff was estimated in this case, the expression still provides a good fit to the data. For 50/125 애m fiber, the expression becomes 104 L−0.65 BWchrom = √ λw (1.01 + 0.0177|λc − 1330|)

BW2t

=

1 BW2chrom

+

1 BW2modal



Bit Rate(Mb/s) BWt (MHz)

10 9 8 7 6 5 4 0

1

2 3 4 Distance (km)

5

6

(a) 10

(41)

Once the total bandwidth is known, the dispersion penalty can be calculated for a given data rate. One expression for the dispersion penalty in decibels is Pd = 1.22

500 MHz-km 600 MHz-km 800 MHz-km 1000 MHz-km

11

(40)

For this case, ␭c was 1313 nm and the chromatic bandwidth peaked at ␭eff ⫽ 1330 nm. Recall that this is only one possible model for fiber bandwidth (23). The total bandwidth capacity of multimode fiber BWt is obtained by combining the modal and chromatic dispersion contributions, according to 1

12 Optical power penalty (dB)

and spectral loss signature; and the constants a1 and a0 are determined by a regression fit of measured data. From Ref. 22, the chromatic bandwidth for 62.5/125 애m fiber is empirically given by

Optical power penalty (dB)

248

500 MHz-km 600 MHz-km 800 MHz-km 1000 MHz-km

9 8 7 6 5 4 0

1

2 3 4 Distance (km)

2

5

6

(b)

(42)

For typical telecommunication grade fiber, the dispersion penalty for a 20 km link is about 0.5 dB. The graph of Fig. 16 shows the dispersion penalty as a function of the fiber bandwidth and length. Dispersion is usually minimized at wavelengths near 1310 nm; special types of fiber have been developed that manipulate the index profile across the core to achieve minimal dispersion near 1550 nm, which is also the wavelength region of minimal transmission loss. Unfortunately, this dispersionshifted fiber suffers from some practical drawbacks, including susceptibility to certain kinds of nonlinear noise and increased interference between adjacent channels in a wavelength-multiplexing environment. There is a new type of fiber that minimizes dispersion while reducing the unwanted cross-talk effects, called dispersion optimized fiber. By using a very sophisticated fiber profile, it is possible to minimize dispersion over the entire wavelength range from 1300 to 1550 nm, at the expense of very high loss (around 2 dB/km); this is known as dispersion-flattened fiber. Yet another approach is called dispersion-compensating fiber; this fiber is designed with negative dispersion characteristics, so that when used in series with conventional fiber it will undisperse the signal. Dispersion-compensating fiber has a much narrower core than standard single-mode fiber, which makes it susceptible to nonlinear effects; it is also birefringent and suffers from polarization mode dispersion, in which different states of polarized light propagate with very different group velocities. Note that standard single-mode fiber does not preserve the polarization state of the incident light; there is yet

Figure 16. Dispersion penalty versus distance for different fiber bandwidths: (a) 50 애m fiber, (b) 62.5 애m fiber. Reprinted from Ref. 3, courtesy of Academic Press.

another type of specialty fiber, with asymmetric core profiles, capable of preserving the polarization of incident light over long distances. By definition, single-mode fiber does not suffer modal dispersion. Chromatic dispersion is an important effect, though, even given the relatively narrow spectral width of most laser diodes. The dispersion of single-mode fiber corresponds to the first derivative of group velocity ␶g with respect to wavelength and is given by D=

S dτg = 0 dλ 4


λc −

λ40 λ3c

 (43)

where D is the dispersion [in ps/km ⭈ nm⫺1] and ␭c is the laser center wavelength. The fiber is characterized by its zero dispersion wavelength, ␭0, and zero dispersion slope, S0. Usually, both center wavelength and zero dispersion wavelength are specified over a range of values; it is necessary to consider both upper and lower bounds in order to determine the worstcase dispersion penalty. This can be seen from Fig. 17, which plots D against wavelength for some typical values of ␭0 and ␭c; the largest absolute value of D occurs at the extremes of this region. Once the dispersion is determined, the intersymbol interference penalty as a function of link length, L, can be deter-

OPTICAL COMMUNICATION

The power penalty caused by mode partition noise was first calculated by Ogawa (25) as

5

Attenuation (dB)

4

249

Zero dispersion wavelength = 1300 nm Zero dispersion wavelength = 1320 nm

2 Pmp = 5 log(1 − Q2 σmp )

3

(45)

2

where

1 0

2 σmp =

–1

1 2 k (πB)4 [A41 λ4 + 42A21 A22 λ6 + 48A42λ8 ] 2 A1 = DL

–2 –3

(46) (47)

and

–4

A2 =

–5 –6 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 Wavelength (nm)

Figure 17. Single-mode fiber dispersion as a function of wavelength. Reprinted from Ref. 3, courtesy of Academic Press.

mined to a good approximation from a model proposed by Agrawal (24): Pd = 5 log(1 + 2π (BD λ)2 L2 )

A1 2(λc − λ0 )

(48)

The mode partition coefficient, k, is a number between 0 and 1 that describes how much of the optical power is randomly shared between modes; it summarizes the statistical nature of mode partition noise. According to Ogawa, k depends on the number of interacting modes and rms spectral width of the source, the exact dependence being complex. However, subsequent work (26) has shown that Ogawa’s model tends to underestimate the power penalty from mode partition noise, because it does not consider the variation of longitudinal mode power between successive baud periods

;yyy ;y;; ;y;y ;yy; ;; y;yyy; ; y ;; yy ; y ; y ;; yy ; y y;yy y;y;y; ;; y;yy y;y;y; yy ;; ;y;y y;yy ;y;; ;y;y y;yy ;y;; ;y;y ;y;yyy ; y ;; yy ; y ;; ; y y;yy ;y;; ;y;y ;yyy ;y;y ;y;; y;y; yy ;; yy y;y; ;;

(44)

where B is the bit rate and ⌬␭ is the root mean square (rms) spectral width of the source. By maintaining a close match between the operating and zero dispersion wavelengths, this penalty can be kept to a tolerable 0.5 dB to 1.0 dB in most cases.

Mode Partition Noise. Group velocity dispersion contributes to another optical penalty that remains the subject of continuing research: mode partition noise and mode hopping. This penalty is related to the properties of a Fabry–Perot type laser diode cavity; although the total optical power output from the laser may remain constant, the optical power distribution among the laser’s longitudinal modes will fluctuate. This is illustrated by the model depicted in Fig. 18; when a laser diode is directly modulated with injection current, the total output power stays constant from pulse to pulse; however, the power distribution among several longitudinal modes will vary between pulses. We must be careful to distinguish this behavior of the instantaneous laser spectrum, which varies with time, from the time-averaged spectrum, which is normally observed experimentally. The light propagates through a fiber with wavelength-dependent dispersion or attenuation, which deforms the pulse shape. Each mode is delayed by a different amount because of group velocity dispersion in the fiber; this leads to additional signal degradation at the receiver, in addition to the intersymbol interference caused by chromatic dispersion alone, which was discussed earlier. This is known as mode partition noise; it is capable of generating BER floors, such that additional optical power into the receiver will not improve the link BER. This is because mode partition noise is a function of the laser spectral fluctuations and wavelength-dependent dispersion of the fiber, so the signal-to-noise ratio due to this effect is independent of the signal power.

λ3

O

λ2

T

λ1

O

T

λ2 λ1 λ3

O

T

(a)

O

T

O

T

(b)

Figure 18. Model for mode partition noise; an optical source emits different wavelengths, illustrated by different shaded blocks: (a) wavelength-dependent loss, (b) chromatic dispersion. Reprinted from Ref. 3, courtesy of Academic Press.

250

OPTICAL COMMUNICATION

and because it assumes a linear model of chromatic dispersion rather than the nonlinear model given in the above equation. A more detailed model has been proposed by Campbell (27), which is general enough to include effects of the laser diode spectrum, pulse shaping, transmitter extinction ratio, and statistics of the datastream. While Ogawa’s model assumed an equiprobable distribution of zeros and ones in the datastream, Campbell showed that mode partition noise is data dependent as well. Recent work based on this model (28) has rederived the signal variance: 2 2 2 σmp = Eav (σ02 + σ+1 + σ−1 )

(49)

where the mode partition noise contributed by adjacent baud periods is defined by 2 2 σ+1 + σ−1 =

1 2 k (πB)4 2 [1.25A41λ4 + 40.95A21A22 λ6 + 50.25A42λ8 ]

(50)

and the time-average extinction ratio Eav ⫽ 10 log(P1 /P0), where P1, P0 represent the optical power by a ‘‘1’’ and ‘‘0,’’ respectively. If the operating wavelength is far from the zero dispersion wavelength, the noise variance simplifies to σm2 p = 2.25

k2 2 Eav (1 − e−β L )2 2

(51)

which is valid provided that β = (πBDλ)2  1

(52)

Many diode lasers exhibit mode hopping or mode splitting in which the spectrum appears to split optical power between two or three modes for brief periods of time. The exact mechanism is not fully understood, but stable Gaussian spectra are generally only observed for continuous operation and temperature-stabilized lasers. During these mode hops the above theory does not apply, because the spectrum is non-Gaussian, and the model will overpredict the power penalty; hence, it is not possible to model mode hops as mode partitioning with k ⫽ 1. There is no currently published model describing a treatment of mode hopping noise, although recent papers (29) suggest approximate calculations based on the statistical properties of the laser cavity. In a practical link, some amount of mode hopping is probably unavoidable as a contributor to burst noise; empirical testing of link hardware remains the only reliable way to reduce this effect. Extinction Ratio. The receiver extinction ratio also contributes directly to the link penalties. The receiver BER is a function of the modulated ac signal power; if the laser transmitter has a small extinction ratio, the dc component of total optical power is significant. Gain or loss can be introduced in the link budget if the extinction ratio at which the receiver sensitivity is measured differs from the worst-case transmitter extinction ratio. If the extinction ratio Et at the transmitter is defined as the ratio of optical power when a one is transmitted versus when a zero is transmitted, Et =

Power(1) , Power(0)

(53)

then we can define a modulation index at the transmitter Mt according to Mt =

Et − 1 Et + 1

(54)

Similarly, we can measure the linear extinction ratio at the optical receiver input and define a modulation index Mr. The extinction ratio penalty is given by Per = −10 log


M  t

Mr

(55)

where the subscripts t and r refer to specifications for the transmitter and receiver, respectively. Multipath Interference. Another important property of the optical link is the amount of reflected light from the fiber endfaces that returns up the link back into the transmitter. Whenever there is a connection or splice in the link, some fraction of the light is reflected back; each connection is thus a potential noise generator, because the reflected fields can interfere with one another to create noise in the detected optical signal. The phenomenon is analogous to the noise caused by multiple atmospheric reflections of radio waves and is known as multipath interference noise. To limit this noise, connectors and splices are specified with a minimum return loss. If there are a total of N reflection points in a link and the geometric mean of the connector reflections is 움, then based on the model of Ref. 30 the power penalty due to multipath interference (adjusted for BER and bandwidth) is closely approximated by Pmpi = 10 log(1 − 0.7Nα)

(56)

Multipath noise can usually be reduced well below 0.5 dB with available connectors, whose return loss is often better than 25 dB. Relative Intensity Noise. Stray light reflected back into a Fabry–Perot type laser diode gives rise to intensity fluctuations in the laser output. This is a complicated phenomenon, strongly dependent on the type of laser; it is called either reflection-induced intensity noise or relative intensity noise (RIN). This effect is important, because it can also generate BER floors. The power penalty due to RIN is the subject of ongoing research; since the reflected light is measured at a specified signal level, RIN is data dependent, although it is independent of link length. Since many laser diodes are packaged in windowed containers, it is difficult to correlate the RIN measurements on an unpackaged laser with those of a commercial product. There have been several detailed attempts to characterize RIN (31,32); typically, the RIN noise is assumed Gaussian in amplitude and uniform in frequency over the receiver bandwidth of interest. The RIN value is specified for a given laser by measuring changes in the optical power when a controlled amount of light is fed back into the laser; it is signal dependent and is also influenced by temperature, bias voltage, laser structure, and other factors that typically influence laser output power (32).

OPTICAL COMMUNICATION

If we assume that the effect of RIN is to produce an equivalent noise current at the receiver, then the additional receiver noise ␴r may be modeled as σr = γ 2 S2g B

(57)

where S is the signal level during a bit period, B is the bit rate, and g is a noise exponent that defines the amount of signal-dependent noise. If g ⫽ 0 noise power is independent of the signal, whereas if g ⫽ 1 noise power is proportional to the square of the signal strength. The coefficient 웂 is given by γ 2 = S2(1−g) i 10 (RINi /10)

(58)

where RINi is the measured RIN value at the average signal level Si, including worst-case backreflection conditions and operating temperatures. The Gaussian BER probability due to the additional RIN noise current is given by 
S − S  
1 S1 − S0 1 0 1 0 P Perror = + Pe (59) 2 e 2σ1 2σ0 where ␴1, ␴0 represent the total noise current during transmission of a digital ‘‘1’’ and ‘‘0,’’ respectively and Pe1, Pe0 are the probabilities of error during transmission of a ‘‘1’’ or ‘‘0,’’ respectively. The power penalty due to RIN may then be calculated by determining the additional signal power required to achieve the same BER with RIN noise present as without RIN present. One approximation for the RIN power penalty is given by 
1 2  2 2g RIN10 (60) PRIN = −5 log 1 − Q (BW)(1 + Mr ) (10 ) Mr where the RIN value is specified (in decibels per Hertz), BW is the receiver bandwidth, Mr is the receiver modulation index, and the exponent g is a constant varying between 0 and 1 that relates the magnitude of RIN noise to the optical power level. Jitter. Another important area in link design deals with the effects of timing jitter on the optical signal. In a typical optical link, a clock is extracted from the incoming data signal, which is used to retime and reshape the received digital pulse; the received pulse is then compared with a threshold to determine if a digital 1 or 0 was transmitted. So far, we have discussed BER testing with the implicit assumption that the measurement was made in the center of the received data bit; to achieve this, a clock transition at the center of the bit is required. When the clock is generated from a receiver timing recovery circuit, it will have some variation in time and the exact location of the clock edge will be uncertain. Even if the clock is positioned at the center of the bit, its position may drift over time. There will be a region of the bit interval, or eye, in the time domain where the BER is acceptable; this region is defined as the eyewidth (1–3). Eyewidth measurements are an important parameter for evaluation of fiber optic links; they are intimately related to the BER, as well as the acceptable clock drift, pulse width distortion, and optical power. At low optical power levels, the receiver signal-to-noise ratio is reduced; increased noise

251

causes amplitude variations in the received signal. These amplitude variations are translated into time domain variations in the receiver decision circuitry, which narrows the eyewidth. At the other extreme, an optical receiver may become saturated at high optical power, reducing the eyewidth and making the system more sensitive to timing jitter. This behavior results in the typical bathtub-shaped curve shown in Fig. 12; for this measurement, the clock is delayed from one end of the bit cell to the other, with the BER calculated at each position. Near the ends of the cell, a large number of errors occur; toward the center of the cell, the BER decreases to its true value. The eye opening may be defined as the portion of the eye for which the BER remains constant; pulse width distortion occurs near the edges of the eye, which denotes the limits of the valid clock timing. Uncertainty in the data pulse arrival times causes errors to occur by closing the eye window and causing the eye pattern to be sampled away from the center. This is one of the fundamental problems of optical and digital signal processing, and a large body of work has been published in this area (e.g., 33,34). In general, multiple jitter sources will be present in a link, which tend to be uncorrelated. However, jitter on digital signals, especially resulting from a cascade of repeaters, may be coherent. The Commission for International Communications by Telephone and Telegraph (CCITT) (now known as the International Telecommunications Union, or ITU) has adopted a standard definition of jitter (34) as short-term variations of the significant instants (rising or falling edges) of a digital signal from their ideal position in time. Longer-term variations are described as wander; in terms of frequency, the distinction between jitter and wander is somewhat unclear. The predominant sources of jitter include the following: 1. Phase noise in receiver clock recovery circuits, particularly crystal-controlled oscillator circuits may be aggravated by filters or other components that do not have a linear phase response. Noise in digital logic resulting from restricted rise and fall times may also contribute to jitter. 2. Imperfect timing recovery in digital regenerative repeaters is usually dependent on the data pattern. 3. Different data patterns may contribute to jitter when the clock recovery circuit of a repeater attempts to recover the receive clock from inbound data. Data pattern sensitivity can produce as much as a 0.5 dB penalty in receiver sensitivity. Higher data rates are more susceptible (⬎1 Gbit/s); data patterns with long run lengths of 1s or 0s, or with abrupt phase transitions between consecutive blocks of 1s and 0s, tend to produce worstcase jitter. 4. At low optical power levels, the receiver signal-to-noise ratio, Q, is reduced; increased noise causes amplitude variations in the signal, which may be translated into time domain variations by the receiver circuitry. 5. Low frequency jitter, also called wander results from instabilities in clock sources and modulation of transmitters. 6. Very low frequency jitter can be caused by variations in the propagation delay of fibers, connectors, and so on, typically resulting from small temperature variations.

252

OPTICAL COMMUNICATION

This can make it especially difficult to perform longterm jitter measurements. In general, jitter from each of these sources will be uncorrelated; jitter related to modulation components of the digital signal may be coherent, and cumulative jitter from a series of repeaters or regenerators may also contain some well-correlated components. There are several parameters of interest in characterizing jitter performance. Jitter may be classified as either random or deterministic, depending on whether it is associated with pattern-dependent effects; these are distinct from the duty cycle distortion which often accompanies imperfect signal timing. Each component of the optical link (data source, serializer, transmitter, encoder, fiber, receiver, retiming/clock recovery/deserialization, decision circuit) will contribute some fraction of the total system jitter. If we consider the link to be a black box (but not necessarily a linear system) then we can measure the level of output jitter in the absence of input jitter; this is known as the intrinsic jitter of the link. The relative importance of jitter from different sources may be evaluated by measuring the spectral density of the jitter. Another approach is the maximum tolerable input jitter (MTIJ) for the link. Finally, since jitter is essentially a stochastic process, we may attempt to characterize the jitter transfer function (JTF) of the link, or estimate the probability density function of the jitter. When multiple traces occur at the edges of the eye, this can indicate the presence of data-dependent jitter or duty-cycle distortion; a histogram of the edge location will show several distinct peaks. This type of jitter can indicate a design flaw in the transmitter or receiver. By contrast, random jitter typically has a more Gaussian profile and is present to some degree in all data links. All of these approaches have their advantages and drawbacks. One of the first attempts to model the optical power penalty due to jitter (35) considered the general case of a receiver whose input was a raised cosine signal of the form St =

1 [1 + cos(πBt)] 2

(61)

where B is the bit rate. A decision circuit samples this signal at some interval t ⫽ n/B. In the presence of random timing jitter, the sampling point fluctuates and the jitter-induced noise depends on the probability density function (PDF) of the random timing fluctuations. Determination of the actual PDF is quite difficult; if we can approximate Bt Ⰶ 1, then it has been derived (36) that for a uniform PDF, the jitter induced noise ␴j is given by σj2 =

4 (πBt/4)4 5

(62)

Whereas for the less conservative case of a Gaussian PDF in the same limit, σj2

= 2(πBt/4)

4

(63)

The optical power penalty in decibels due to jitter noise is then given by Pj = −5 log(1 − 4Q2 σi2 )

(64)

where Q is the Gaussian error function. Based on this expression, the penalty for a Gaussian system is much larger than for a uniform PDF; when Bt ⫽ 0.35, a BER floor appears at 1e-9. However, it was subsequently shown that the approximation on Bt is very restrictive, and the actual PDF is far from Gaussian; indeed, it is given by


Perror =




1/B −1/B

(PDFj )Q

A cos(πBt) 2σj

 dt

(65)

where the probability density function of the jitter is included under the integral. Numerical integration of this PDF shows that the approximate results given above tend to underestimate the effects of Gaussian jitter and overestimate the effects of uniform jitter by over 2 dB. Using similar approximations, jitter power penalties have been derived for both PIN and avalanche photodiodes (37). An alternative modeling approach has been to derive a worst-case distribution for the PDF, which will provide an upper bound on the performance of the optical link and can be more easily evaluated for analytical purposes. The problem of jitter accumulation in a chain of repeaters becomes increasingly complex; however, we can state some general rules of thumb. It has been shown that jitter can be generally divided into two components, one due to repetitive patterns and one due to random data (38). In receivers with phase-lock loop timing recovery circuits, repetitive data patterns tend to cause jitter accumulation, especially for long run lengths. This effect is commonly modeled as a second-order receiver transfer function; the jitter accumulates according to the relationship Jitter ∝ N +


N 2 ξ

L

(66)

where N is the number of identical repeaters and ␰ is the loop-damping factor, specific to a given receiver circuit. For large ␰, jitter accumulates almost linearly with the number of repeaters, whereas for small ␰ the accumulation is much more rapid. Jitter will also accumulate when the link is transferring random data; jitter due to random data is of two types, systematic and random. The classic model for systematic jitter accumulation in cascaded repeaters was published by Byrne (39). The Byrne model assumes cascaded identical timing recovery circuits; the general expression for the jitter power spectrum is given by

JsN

2  N N      = Js  H + k( f ) + Hk ( f ) + · · · + Hk ( f )  k=1 k=2

(67)

where Js is the systematic jitter generated by each repeater and H( f) is the jitter transfer function of each timing recovery circuit. The corresponding model for random jitter accumulation is

 JsN

= Jr

N  k=1

|Hk ( f )| + 2

N  k=2

 |Hk ( f )| + · · · + |Hk ( f )| 2

2

(68)

OPTICAL COMMUNICATION

where Jr is the random jitter generated by each repeater. The systematic and random jitter can be combined as rms quantities, Jt2 = (JsN )2 + (JrN )2

(69)

so that Jt, the total jitter due to random jitter, may be obtained. This model has been generalized to networks consisting of different components (40), and to non-identical repeaters (41). Despite these considerations, for well-designed practical networks the basic results of the Byrne model remain valid for N nominally identical repeaters transmitting random data; that is, systematic jitter accumulates in proportion to N1/2 and random jitter accumulates in proportion to N1/4. Modal Noise. An additional effect of lossy connectors and splices is modal noise. Because high-capacity optical links tend to use highly coherent laser transmitters, random coupling between fiber modes causes fluctuations in the optical power coupled through splices and connectors; this phenomena is known as modal noise (42). As one might expect, modal noise is worst when using laser sources in conjunction with multimode fiber; recent industry standards have allowed the use of short-wave lasers (750 nm to 850 nm) on 50 애m fiber which may experience this problem. Modal noise is usually considered to be nonexistent in single-mode systems. However, modal noise in single-mode fibers can arise when higher order modes are generated at imperfect connections or splices. If the lossy mode is not completely attenuated before it reaches the next connection, interference with the dominant mode may occur. The effects of modal noise have been modeled previously (42), assuming that the only significant interaction occurs between the LP01 and LP11 modes for a sufficiently coherent laser. For N sections of fiber, each of length L in a single-mode link, the worst case sigma for modal noise can be given by σm =

√ 2 Nη(1 − η)e−aL

(70)

where a is the attenuation coefficient of the LP11 mode, and ␩ is the splice transmission efficiency, given by η = 10−(η 0 /10)

253

data rate. Optical fiber with a pure silica core is least susceptible to radiation damage; however, almost all commercial fiber is intentionally doped to control the refractive index and dispersion properties of the core and cladding. Trace impurities are also introduced, which become important only under irradiation; among the most important are germanium dopants in the core of graded index (GRIN) fibers, in addition to flourine, chlorine, phosphorus, boron, and hydroxide content, and the alkali metals. In general, radiation sensitivity is worst at lower temperatures, and is also made worse by hydrogen diffusion from materials in the fiber cladding. Because of the many factors involved, there does not exist a comprehensive theory to model radiation damage in optical fibers. The basic physics of the interaction has been described elsewhere (43,44); there are two dominant mechanisms, radiation-induced darkening and scintillation. First, high-energy radiation can interact with dopants, impurities, or defects in the glass structure to produce color centers that absorb strongly at the operating wavelength. Carriers can also be freed by radiolytic or photochemical processes; some of these become trapped at defect sites, which modifies the band structure of the fiber and causes strong absorption at infrared wavelengths. This radiation-induced darkening increases the fiber attenuation; in some cases, it is partially reversible when the radiation is removed, although high levels or prolonged exposure will permanently damage the fiber. A second effect is caused if the radiation interacts with impurities to produce stray light or scintillation. This light is generally broadband but will tend to degrade the BER at the receiver; scintillation is a weaker effect than radiation-induced darkening. These effects will degrade the BER of a link. However, they can be prevented by shielding the fiber or partially overcome by a third mechanism, photobleaching. The presence of intense light at the proper wavelength can partially reverse the effects of darkening in a fiber. It is also possible to treat silica core fibers by briefly exposing them to controlled levels of radiation at controlled temperatures, which increases the fiber loss, but makes the fiber less susceptible to future irradiation. These so-called radiation-hardened fibers are often used in environments where radiation is anticipated to play an important role. Recently, several models have been advanced for the performance of fiber under moderate radiation levels (44); the effect on BER is a power law model of the form

(71)

where ␩0 is the mean splice loss (typically, splice transmission efficiency will exceed 90%). The corresponding optical power penalty due to modal noise is given by

BER = BER0 + A(dose)b

(73)

(72)

where BER0 is the link BER prior to irradiation, the dose is given in rads, and the constants A and b are empirically fitted. The loss due to normal background radiation exposure over a typical link lifetime can be held below about 0.5 dB.

Radiation-Induced Loss. Another important environmental factor as mentioned earlier is exposure of the fiber to ionizing radiation damage. There is a large body of literature concerning the effects of ionizing radiation on fiber links (43,44). Many factors can affect the radiation susceptibility of optical fiber, including the type of fiber, type of radiation (gamma radiation is usually assumed to be representative), total dose, dose rate (important only for higher exposure levels), prior irradiation history of the fiber, temperature, wavelength, and

Stimulated Brillouin and Raman Scattering. At high optical power levels, nonlinear effects in the fiber may limit the link performance. The dominant effects are stimulated Raman and Brillouin scattering. When incident optical power exceeds a threshold value, a significant amount of light scatters from small imperfections in the fiber core. The scattered light is frequency shifted, since the scattering process involves the generation of phonons (45). This is known as stimulated Brillouin scattering; under these conditions, the output light intensity becomes nonlinear, as well. When the scattered light experiences frequency shifts outside the acoustic phonon

P = −5 log(1 − Q2 σm2 ) where Q corresponds to the desired BER.

254

OPTICAL COMMUNICATION

range, due instead to modulation by impurities or molecular vibrations in the fiber core, the effect is known as Raman scattering. Stimulated Brillouin scattering will not occur below the optical power threshold defined by Pe = 21

A W Gb L e

(74)

where Le is the effective interaction length, A is the crosssectional area of the guided mode, and Gb is the Brillouin gain coefficient. Similarly, Raman scattering will not occur unless the optical power exceeds the value given by Pr = 16

A W Gr L

(75)

where Gr is the Raman gains coefficient. Brillouin scattering has been observed in single-mode fibers at wavelengths greater than cutoff with optical power as low as 5 mW; a good rule of thumb is that the Raman scattering threshold is approximately three times larger than the Brillouin threshold. As a general guide, the optical power threshold for Brillouin scattering in singlemode fiber is approximately 10 mW and for Raman scattering is approximately 3.5 W. These effects rarely occur in multimode fiber, where the thresholds for Brillouin and Raman scattering are approximately 450 mW and 150 W, respectively. Frequency Chirp. The final nonlinear effect that we consider is frequency chirping of the optical signal. Chirping refers to a change in frequency with time, and takes its name from the sound of an acoustic signal whose frequency increases or decreases linearly with time. There are three ways in which chirping can affect a fiber optic link. First, the laser transmitter can be chirped as a result of physical processes within the laser (46); the effect has its origin in carrier-induced refractive index changes, making it an inevitable consequence of high power direct modulation of semiconductor lasers. For lasers with low levels of relaxation oscillation damping, a model has been proposed for the chirped power penalty:




P = 10 log 1 +

 πB2 λ2 LDa 4c

(76)

where c is the speed of light, B is the fiber bandwidth, ␭ is the wavelength of light, L is the length of the fiber, D is the dispersion, and a is the linewidth enhancement factor (a typical value is ⫺4.5); this model is only a first approximation, because it neglects the dependence of chirp on extinction ratio and nonlinear effects such as spectral hole burning. Secondly, a sufficiently intense light pulse is chirped by the nonlinear process of self-phase modulation in an optical fiber (47). This carrier-induced phase modulation arises from the interaction of the light and the intensity dependent portion of the fiber’s refractive index (known as the Kerr effect); it is thus dependent on the material and structure of the fiber, polarization of the light, and the shape of the incident optical pulse. If different wavelengths are present in the same fiber, the effect caused by one signal can induce cross-phase modulation in the others. Based on a model by Stolen and Lin (47), the max-

imum optical power level before the spectral width increases by 2 nm is given by P=

n2 A W 377n2 κLe

(77)

where Le = (1/a0 )[1 − exp(−a0 L)]

(78)

where n is the fiber’s refractive index, ␬ is the propagation constant (a typical value is 7 ⫻ 104 at a wavelength of 1.3 애m), a0 is the fiber attenuation coefficient, A is the fiber core cross-sectional area, n2 is the nonlinear coefficient of the fiber’s refractive index (a typical value is 6.1 ⫻ 10⫺19), and Le is the effective interaction length for the nonlinear interaction, which is related to the actual length of the fiber, L, by Eq. (78). This expression should be multiplied by 5/6 if the fiber is not polarization preserving. Typically, this effect is not significant for optical power levels less than 950 mW in a singlemode fiber at 1.3 애m wavelength. Finally, there is a power penalty arising from the propagation of a chirped optical pulse in a dispersive fiber, because the new frequency components propagate at different group velocities. This may be treated as simply a much worse case of the conventional dispersion penalty, provided that one of the first two effects exists to chirp the optical signal. OPTICAL LINK STANDARDS In the past 10 years there have been several international standards adopted for optical communications. This section presents a brief overview of several major industry standards, including the following: ESCON/SBCON (Enterprise System Connection / Serial Byte Connectivity), FDDI (Fiber Distributed Data Interface), Fibre Channel Standard, ATM (Asynchronous Transfer Mode)/SONET (Synchronous Optical Network), and Gigabit Ethernet. ESCON The Enterprise System Connection (ESCON) architecture was introduced on the IBM System/390 family of mainframe computers in 1990 (ESCON is a registered trademark of IBM Corporation) as an alternative high-speed input–output channel attachment (48,49). The ESCON interface specifications were adopted in 1996 by the ANSI X3T1 committee as the Serial Byte Connection (SBCON) standard (50). The ESCON/SBCON channel is a bidirectional, point-topoint 1300 nm fiber optic data link with a maximum data rate of 17 Mbyte/s (200 Mbit/s). ESCON supports a maximum unrepeated distance of 3 km using 62.5 애m multimode fiber and LED transmitters with an 8 dB link budget, or a maximum unrepeated distance of 20 km using singlemode fiber and laser transmitters with a 14 dB link budget. The laser channels are also known as the ESCON Extended Distance Feature (XDF). Physical connection is provided by an ESCON duplex connector (Fig. 15). Recently, the singlemode ESCON links have adopted the SC duplex connector as standardized by Fibre Channel (Fig. 15). With the use of repeaters or switches, an ESCON link can be extended up to 3 to 5 times these distances; however, performance of the attached devices

OPTICAL COMMUNICATION

typically falls off quickly at longer distances due to the longer round-trip latency of the link, making this approach suitable only for applications that can tolerate a lower effective throughput, such as remote backup of data for disaster recovery. ESCON devices and CPUs may communicate directly through a channel-to-channel attachment, but more commonly attach to a central nonblocking dynamic cross-point switch. The resulting network topology is similar to a starwired ring, which provides both efficient bandwidth utilization and reduced cabling requirements. The switching function is provided by an ESCON Director, a nonblocking circuit switch. Although ESCON uses 8B/10B encoded data, it is not a packet switching network; instead, the data frame header includes a request for connection that is established by the Director for the duration of the data transfer. An ESCON data frame includes a header, payload of up to 1028 bytes of data, and a trailer. The header consists of a two character start-of-frame delimiter, two byte destination address, two byte source address, and one byte of link control information. The trailer is a two byte cyclic redundancy check (CRC) for errors and a three character end-of-frame delimiter. ESCON uses a DC-balanced 8B/10B coding scheme developed by IBM. FDDI The fiber distributed data interface (FDDI) was among the first open networking standards to specify optical fiber. It was an outgrowth of the ANSI X3T9.5 committee proposal in 1982 for a high-speed token passing ring as a back-end interface for storage devices. Although interest in this application waned, FDDI found new applications as the backbone for local area networks (LANs). The FDDI standard was approved in 1992 as ISO standards IS 9314/1-2 and DIS 9314-3; it follows that the architectural concepts of IEEE standard 802 (although it is controlled by ANSI, not IEEE, and therefore has a different numbering sequence) and is among the family of standards including token ring and Ethernet which are compatible with a common IEEE 802.2 interface. FDDI is a family of four specifications, namely the physical layer (PHY), physical media dependent (PMD), media access control (MAC), and station management (SMT). These four specifications correspond to sublayers of the Data Link and Physical Layer of the OSI reference model; as before, we concentrate on the physical layer implementation. The FDDI network is a 100 Mbit/s token passing ring, with dual counter-rotating rings for fault tolerance. The dual rings are independent fiber optic cables; the primary ring is used for data transmission, and the secondary ring is a backup in case a node or link on the primary ring fails. Bypass switches are also supported to reroute traffic around a damaged area of the network and prevent the ring from fragmenting in case of multiple node failures. The actual data rate is 125 Mbit/s, but this is reduced to an effective data rate of 100 Mbit/s by using a 4B/5B coding scheme. This high-speed allows FDDI to be used as a backbone to encapsulate lower speed 4, 10, and 16 Mbit/s LAN protocols; existing Ethernet, token ring, or other LANs can be linked to an FDDI network via a bridge or router. Although FDDI data flows in a logical ring, a more typical physical layout is a star configuration with all nodes connected to a central hub or concentrator rather than to the backbone itself. There are two types of FDDI nodes, either dual attach (connected to both rings) or single attach; a net-

255

work supports up to 500 dual attached nodes, 1000 single attached nodes, or an equivalent mix of the two types. FDDI specifies 1300 nm LED transmitters operating over 62.5 애m multimode fiber as the reference media, although the standard also provides for the attachment of 50, 100, 140, and 85 애m fiber. Using 62.5 애m fiber, a maximum distance of 2 km between nodes is supported with an 11 dB link budget; because each node acts like a repeater with its own phase-lock loop to prevent jitter accumulation, the entire FDDI ring can be as large as 100 km. However, an FDDI link can fail due to either excessive attenuation or dispersion; for example, insertion of a bypass switch increases the link length and may cause dispersion errors even if the loss budget is within specifications. For most other applications, this does not occur because the dispersion penalty is included in the link budget calculations or the receiver sensitivity measurements. The physical interface is provided by a special Media Interface Connector (MIC), illustrated in Fig. 15. The connector has a set of three color-coded keys which are interchangeable depending on the type of network connection (1); this is intended to prevent installation errors and assist in cable management. An FDDI data frame has variable length, and contains up to 4500 8-bit bytes, or octets, including a preamble, start of frame, frame control, destination address, data payload, CRC error check, and frame status/end of frame. Each node has a MAC sublayer that reviews all the data frames looking for its own destination address. When it finds a packet destined for its node, that frame is copied into local memory, a copy bit is turned on in the packet, and it is then sent on to the next node on the ring. When the packet returns to the station that originally sent it, the originator assumes that the packet was received if the copy bit is on; the originator will then delete the packet from the ring. As in the IEEE 802.5 token ring protocol, a special type of packet called a token circulates in one direction around the ring and a node can only transmit data when it holds the token. Each node observes a token retention time limit, and also keeps track of the elapsed time since it last received the token; nodes may be given the token in equal turns, or they can be given priority by receiving it more often or holding it longer after they receive it. This allows devices having different data requirements to be served appropriately. Because of the flexibility built into the FDDI standard, many changes to the base standard have been proposed to allow interoperability with other standards, reduce costs, or extend FDDI into the MAN or WAN. These include a singlemode PMD layer for channel extensions up to 20 km to 50 km. An alternative PMD provides for FDDI transmission over copper wire, either shielded or unshielded twisted pairs; this is known as copper distributed data interface, or CDDI. A new PMD is also being developed to adapt FDDI data packets for transfer over a SONET link by stuffing approximately 30 Mbit/s into each frame to make up for the data rate mismatch (we will discuss SONET as an ATM physical layer in a later section). An enhancement called FDDI-II uses time division multiplexing to divide the bandwidth between voice and data; it would accommodate isochronous, circuit-switched traffic as well as existing packet traffic. Recently, an option known as low cost FDDI has been adopted. This specification uses the more common SC duplex connector instead of the expensive

256

OPTICAL COMMUNICATION

MIC connectors, and a lower cost transceiver with a 9 pin footprint similar to the singlemode ESCON parts. Fibre Channel Standard Development of the ANSI Fibre Channel Standard (FC) began in 1988 under the X3T9.3 Working Group, as an outgrowth of the Intelligent Physical Protocol Enhanced Physical Project. The motivation for this work was to develop a scaleable standard for the attachment of both networking and input–output devices using the same drivers, ports, and adapters over a single channel at the highest speeds currently achievable. The standard applies to both copper and fiber optic media, and uses the English spelling ‘‘fibre’’ to denote both types of physical layers. In an effort to simplify equipment design, FC provides the means for a large number of existing upper level protocols (ULPs), such as IP, SCI, and HIPPI, to operate over a variety of physical media. Different ULPs are mapped to FC constructs, encapsulated in FC frames, and transported across a network; this process remains transparent to the attached devices. The standard consists of five hierarchical layers (51), namely a physical layer, an encode– decode layer which has adopted the dc-balanced 8B/10B code, a framing protocol layer, a common services layer (at this time, no functions have been formally defined for this layer) and a protocol mapping layer to encapsulate ULPs into FC. The second layer defines the FC data frame; frame size depends upon the implementation, and is variable up to 2148 bytes long. Each frame consists of a 4 byte start of frame delimiter, a 24 byte header, a 2112 byte payload containing from 0 to 64 bytes of optional headers and 0 to 2048 bytes of data, a 4 byte CRC and a 4 byte end of frame delimiter. In October

1994, the FC physical and signaling interface standard FCPH was approved as ANSI standard X3.230-1994. Logically, FC is a bidirectional point-to-point serial data link. Physically, there are many different media options (see Table 4) and three basic network topologies. The simplest, default topology is a point-to-point direct link between two devices, such as a CPU and a device controller. The second, Fibre Channel Arbitrated Loop (FC-AL), connects between 2 and 126 devices in a loop configuration. Hubs or switches are not required, and there is no dedicated loop controller; all nodes on the loop share the bandwidth and arbitrate for temporary control of the loop at any given time. Each node has equal opportunity to gain control of the loop and establish a communications path, but once the node relinquishes control, a fairness algorithm insures that the same node cannot win control of the loop again until all other nodes have had a turn. As networks become larger, they may grow into the third topology, an interconnected switchable network or fabric in which all network management functions are taken over by a switching point, rather than each node. An analogy for a switched fabric is the telephone network; users specify an address (phone number) for a device with which they want to communicate, and the network provides them with an interconnection path. In theory there is no limit to the number of nodes in a fabric; practically, there are only about 16 million unique addresses. Fibre Channel also defines three classes of connection service, which offer options such as guaranteed delivery of messages in the order they were sent and acknowledgment of received messages. As shown in Table 4, FC provides for both singlemode and multimode fiber optic data links using long-wave (1300 nm)

Table 4. Fibre Channel Media Performance Media Type Single-mode fiber

50 애m multimode fiber

62.5 애m multimode fiber

105 ⍀ type-1 shielded twisted-pair electrical 75 ⍀ mini coax

75 ⍀ video coax

150 ⍀ twinax or STP

Data Rate (Mb/s) 100 50 25 100 50 25 12.5 100 50 25 12.5 25 12.5 100 50 25 12.5 100 50 25 12.5 100 50 25

Reprinted from Ref. 3, courtesy of Academic Press.

Max. Distance 10 10 10 500 1 2 10 300 600 1 2 50 100 10 20 30 40 25 50 75 100 30 60 100

km km km m km km km m m km km m m m m m m m m m m m m m

Signaling Rate (Mbaud)

Transmitter

1062.5 1062.5 1062.5 1062.5 531.25 265.625 132.8125 1062.5 531.25 265.625 132.8125 265.125 132.8125 1062.5 531.25 265.625 132.8125 1062.5 531.25 265.625 132.8125 1062.5 531.25 265.625

LW laser LW laser LW laser SW laser SW laser SW laser LW LED SW laser SW laser LW LED LW LED ECL ECL ECL ECL ECL ECL ECL ECL ECL ECL ECL ECL ECL

OPTICAL COMMUNICATION

lasers and LEDs as well as short-wave (780 to 850 nm) lasers. The physical connection is provided by an SC duplex connector defined in the standard (see Fig. 15), which is keyed to prevent misplugging of a multimode cable into a singlemode receptacle. This connector design has since been adopted by other standards, including ATM, low cost FDDI, and singlemode ESCON. The requirement for international class 1 laser safety is addressed using open fiber control (OFC) on some types of multimode links with short-wave lasers. This technique automatically senses when a full duplex link is interrupted, and turns off the laser transmitters on both ends to preserve laser safety. The lasers then transmit low duty cycle optical pulses until the link is reestablished; a handshake sequence then automatically reactivates the transmitters.

257

5, SONET also contains provisions to carry sub-OC-1 data rates, called virtual tributaries, which support telecom data rates including DS-1 (1.544 Mbit/s), DS-2 (6.312 Mbit/s), and DS1C (3.152 Mbit/s). The basic SONET data frame is an array of nine rows with 90 bytes per row, known as a synchronous-transport-signal level 1 (STS-1) frame. In an OC-1 system, an STS-1 frame is transmitted once every 125 애s (810 bytes per 125 애s yields 51.84 Mbit/s). The first three columns provide overhead functions such as identification, framing, error checking, and a pointer to identify the start of the 87 byte data payload. The payload floats in the STS-1 frame, and may be split across two consecutive frames. Higher speeds can be obtained either by concatenation of N frames into an STS-Nc frame (the ‘‘c’’ stands for ‘‘concatenated’’) or by byte interleaved multiplexing of N frames into an STS-N frame. ATM technology incorporates elements of both circuit and packet switching. All data are broken down into a 53 byte cell, which may be viewed as a short-fixed-length packet. Five bytes make up the header, providing a 48 byte payload. The header information contains routing information (cell addresses) in the form of virtual path and channel identifiers, a field to identify the payload type, an error check on the header information, and other flow control information. Cells are generated asynchronously; as the data source provides enough information to fill a cell, it is placed in the next available cell slot. There is no fixed relationship between the cells and a master clock, as in conventional time division multiplexing schemes; the flow of cells is driven by the bandwidth needs of the source. ATM provides bandwidth on demand; for example, in a client-server application the data may come in bursts; several data sources could share a common link by multiplexing during the idle intervals. Thus, the ATM adaptation layer allows for both constant and variable bit rate services. The combination of transmission options is sometimes described as a pleisosynchronous network, meaning that it combines some features of multiplexing operations without requiring a fully synchronous implementation. Note that the

ATM/SONET Developed by the ATM Forum, this protocol has promised to provide a common transport medium for voice, data, video, and other types of multimedia. ATM is a high level protocol, which can run over many different physical layers including copper; part of ATM’s promise to merge voice and data traffic on a single network comes from plans to run ATM over the Synchronous Optical Network (SONET) transmission hierarchy developed for the telecommunications industry. SONET is really a family of standards defined by ANSI T1.105-1988 and T1.106-1988, as well as by several CCITT recommendations (52–55). Several different data rates are defined as multiples of 51.84 Mbit/s, known as OC-1. The numerical part of the OC-level designation indicates a multiple of this fundamental data rate, thus, 155 Mbit/s is called OC-3 (see Table 5). The standard provides for seven incremental data rates, OC-3, OC-9, OC-12, OC-18, OC-24, OC-36, and OC-48 (2.48832 Gbit/s). Both singlemode links with laser sources and multimode links with LED sources are defined for OC-1 through OC-12; only singlemode laser links are defined for OC-18 and beyond. In addition to the specifications in Table

Table 5. SONET Physical Layer Optical Specifications Parameter Data rate Bit rate Tolerance Transmitter type ␭Wmin ␭Wmax ⌬␭max PTmax PTmin rcmin Optical path system ORLmax DS Rmax Max sndr. to rcvr. reflectance Receiver Prmax Prmin P0

Units

OC-1

OC-3

Mb/s ppm

51.84

155.52

OC-9

OC-12

OC-24

OC-36

OC-48

933.12

1244.16

1866.24

2488.32

nm nm nm dBm dBm dB

MLM/LED 1260 1360 80 ⫺14 ⫺23 8.2

MLM/LED 1260 1360 40/80 ⫺8 ⫺15 8.2

455.56 100 MLM/LED 1260 1360 19/45 ⫺8 ⫺15 8.2

dB ps/nm dB

na na na

na na na

na 31/na na

na 13/na na

20 13 ⫺25

20 13 ⫺25

24 13 ⫺27

24 12 ⫺27

dBm dBm dBm

⫺15 ⫺31 1

⫺8 ⫺23 1

⫺8 ⫺23 1

⫺8 ⫺23 1

⫺8 ⫺23 1

⫺5 ⫺20 1

⫺3 ⫺18 1

⫺3 ⫺18 1

Reprinted from Ref. 3, courtesy of Academic Press.

622.08

OC-18

MLM/LED 1260 1360 14.5/35 ⫺8 ⫺15 8.2

MLM 1260 1360 9.5 ⫺8 ⫺15 8.2

MLM 1260 1360 7 ⫺5 ⫺12 8.2

MLM 1260 1360 4.8 ⫺3 ⫺10 8.2

MLM 1265 1360 5 ⫺3 ⫺10 8.2

258

OPTICAL COMMUNICATION

fixed cell length allows the use of synchronous multiplexing and switching techniques, while the generation of cells on demand allows flexible use of the link bandwidth for different types of data, characteristic of packet switching. Higher level protocols may be required in an ATM network to insure that multiplexed cells arrive in the correct order or to check the data payload for errors (given the typical high reliability and low BER of modern fiber optic technology, it was considered unnecessary overhead to replicate data error checks at each node of an ATM network). If an intermediate node in an ATM network detects an error in the cell header, cells may be discarded without notification to either end user. Although cell loss priority may be defined in the ATM header, for some applications the adoption of unacknowledged transmission may be a concern. ATM data rates were intended to match SONET rates of 51, 155, and 622 Mbit/s; an FDDI compliant data rate of 100 Mbit/s was added, to facilitate emulation of different types of LAN traffic over ATM. To provide a low-cost copper option and compatibility with 16 Mbit/s token ring LANs to the desktop, a 25 Mbit/s speed has also been approved. For premises wiring applications, ATM specifies the SC Duplex connector, color coded beige for multimode links and blue for singlemode links. At 155 Mbit/s, multimode ATM links support a maximum distance of 3 km while single-mode links support up to 20 km. Gigabit Ethernet Ethernet is a local area network (LAN) communication standard originally developed for copper interconnections on a common data bus; it is an IEEE standard 802.3 (56). The basic principle used in Ethernet is carrier sense multiple access with collision detection (CSMA/CD). Ethernet LANs are typically configured as a bus, often wired radially through a central hub. A device attached to the LAN that intends to transmit data must first sense whether another device is transmitting. If another device is already sending, then it must wait until the LAN is available; thus, the intention is that only one device will be using the LAN to send data at a given time. When one device is sending, all other attached devices receive the data and check to see if it is addressed to them; if it is not, then the data is discarded. If two devices attempt to send data at the same time (for example, both devices may begin transmission at the same time after determining that the LAN is available; there is a gap between when one device starts to send and before another potential sender can detect that the LAN is in use), then a collision occurs. Using CSMA/CD as the media access control protocol, when a collision is detected attached devices will detect the collision and must wait for different lengths of time before attempting retransmission. Since it is not always certain that data will reach its destination without errors or that the sending device will know about lost data, each station on the LAN must operate an end-to-end protocol for error recovery and data integrity. Data frames begin with an 8 byte preamble used for determining start-of-frame and synchronization, and a header consisting of a 6 byte destination address, 6 byte source address, and 2 byte length field. User data may vary from 46 to 1500 bytes, with data shorter than the minimum length padded to fit the frame; the user data is followed by a

2 byte CRC error check. Thus, an Ethernet frame may range from 70 bytes to 1524 bytes. The original Ethernet standard, known also as 10Base-T (10 Mbit/s over unshielded twisted pair copper wires) was primarily a copper standard, although a specification using 850 nm LEDs was also available. Subsequent standardization efforts increased this data rate to 100 Mbit/s over the same copper media (100Base-T), while once again offering an alternative fiber specification (100Base-FX). Recently, the standard has continued to evolve with the development of gigabit Ethernet (1000Base-FX), which will operate over fiber as the primary medium; this has the potential to be the first networking standard for which the implementation cost on fiber is lower than on copper media. Currently under development as IEEE 802.3z, the gigabit Ethernet standard is scheduled for final approval in late 1998. Gigabit Ethernet will include some changes to the MAC layer in addition to a completely new physical layer operating at 1.25 Gbit/s. Switches rather than hubs are expected to predominate, since at higher data rates throughput per end user and total network cost are both optimized by using switched rather than shared media. The minimum frame size has increased to 512 bytes; frames shorter than this are padded with idle characters (carrier extension). The maximum frame size remains unchanged, although devices may now transmit multiple frames in bursts rather than single frames for improved efficiency. The physical layer will use standard 8B/10B data encoding. The physical layer specifications have not been finalized as of this writing. The standard does not specify a physical connector type for fiber; at this writing there are several proposals, including the SC duplex and various small-form factor connectors approximately the size of a standard RJ-45 jack, such as the MT-RJ, SC/DC, and SG connectors. Transceivers may be packaged as gigabit interface converters, or GBICs, which allows different optical or copper transceivers to be plugged onto the same host card. There is presently a concern with proposals to operate long-wave (1300 nm) laser sources over both singlemode and multimode fiber. When a transmitter is optimized for a singlemode launch condition, it will underfill the multimode fiber; this causes some modes to be excited and propagate at different speeds than others, and the resulting differential mode delay significantly degrades link performance. One proposed solution involves the use of special optical cables with offset ferrules to simulate an equilibrium mode launch condition into multimode fiber. The issue has not been resolved as of this writing. ADVANCED TOPICS In the following sections, several advanced topics under development in optical communications are presented, including the following: parallel optical interconnections, plastic optical fiber, wavelength division multiplexing (WDM), solitons, and optical amplifiers. Parallel Optical Interconnect In the quest for 2.5 Gbit/s data rates and beyond, one approach is to develop very-high-speed serial transceivers. Another approach, which may be more cost effective, involves striping several lower speed fibers in parallel; this space-division multiplexed approach is the motivation behind parallel

optical interconnects (POI). Recent advances in vertical cavity surface emitting laser (VCSEL) arrays, optical fiber ribbon connectors, and receiver arrays have contributed to the commercialization of this technology. The first commercially available POI transceiver was the Motorola Optobus, announced in 1995; it provides a 10 fiber ribbon between transmitter and receiver arrays, with an aggregate data rate of 2.4 Gbit/s over 300 m. Since then, there have been many vendors who have demonstrated prototype POIs (3). Although there is not yet a common standard for these devices, most are based on short wavelength VCSELs and are limited to a few tens or hundreds of meters. There is no standardization on the ribbon connectors; although many products are based on the MTP/ MPO 12 fiber ferrule, different manufacturers have developed at least a half dozen proprietary implementations which are not compatible. Applications for POI include highly parallel supercomputers or clustered multiprocessors for large servers, large telecom or datacom switches, channel extensions for existing parallel data buses such as the Small Computer Interface (SCI), and options for emerging parallel bus standards such as HIPPI 6400. Research into VCSELs capable of operating at longer wavelengths (1300 nm) promise to extend the distance of future parallel optical interconnects; development of two-dimensional arrays and so-called smart pixel technology are also very promising areas of future research. Plastic Fiber The first research in optical fiber transmission was conducted around 1955 using crude plastic fibers. There has been recent research into the use of large-core plastic fiber as a low-cost optical data link. Because of the much higher volumes of glass fiber manufactured today, the cost of plastic optical fiber remains higher than glass in the short term. Plastic fibers are currently limited by their high loss (120 dB/km to 150 dB/ km) to fairly short distances (around 50 to 100 m). The bandwidth at 100 m is approximately 125 MHz. Despite this, there is ongoing interest in this technology because plastic fiber links can use very-low-cost visible light sources; a 680 nm (red) LED transceiver could be as inexpensive as $5 to $10, compared with over $100 for a 1300 nm transceiver. Plastic fiber links have already found some applications in laser printers and microprocessor control systems for automobiles. Fusion splices in plastic fiber exhibit very high loss, approximately 5 dB. Several plastic fiber connectors have been proposed, including variations on the F07, SMA 905 and 906, EIAJ Digital Audio, ST, and FC connectors. Step index plastic fiber has a core diameter of 980 애m and a 20 애m cladding, over 100 times larger than singlemode fiber; its NA is approximately 0.3. The most common material used today is called for poly(methyl methylacrylate) (PMMA). While the attenuation of plastic fiber remains higher than glass, transmission with visible sources at 570 nm and 650 nm is feasible. Another type of specialty fiber is hard polymer clad fiber (HPCF), which is a glass fiber with a hard plastic cladding. It attempts to combine the advantage of glass and plastic fiber while overcoming some of their drawbacks. Typical HPCF fiber has a core diameter of 200 애m, cladding diameter of 225 애m, attenuation of 0.8 dB/100 m, NA of 0.3, and bandwidth of 10 Mhz/km. If we use visible wavelengths of approximately

Fiber attenuation (dB/km)

OPTICAL COMMUNICATION

259

25 20 15 10 5 0 500

600

700 800 Wavelength (nm)

900

Figure 19. Attenuation of HPCF optical fiber plotted against wavelength.

650 nm, attenuation is relativly high; all HPCF is step index only. The attenuation of HPCF fiber is shown in Fig. 19. Wavelength Multiplexing Optical wavelength division multiplexing (WDM) is a way to take advantage of the high bandwidth of fiber optic cables without requiring extremely high modulation rates at the transceiver. Because different wavelengths do not interfere with each other, multiple datastreams may be combined in a single optical fiber. Coarse WDM schemes utilize only a few wavelengths spaced far apart; some systems use only two signals at 1300 nm and 1550 nm. More complex dense WDM systems may employ tens of wavelengths spaced apart 1 nm or less, typically in the region of minimal dispersion close to 1550 nm. Such systems require that the wavelength of the optical transmitter be held very stable, often incorporating temperature and voltage control of the source. The optical signals at different wavelengths may be combined using passive optical components such as splitters, couplers, prisms, or diffraction gratings. For systems with a small number of channels, phase array multiplexers are preferred; they consist of an array of dispersive waveguides with variable pathlengths that can be used to focus light from an input port to a number of different output ports depending on the wavelength of the light. It is also possible to employ sources that are wavelength tunable; examples include external cavity lasers with a frequency selective component such as a filter or diffraction grating. Two- or three-section distributed feedback lasers may also be tuned over a narrow wavelength range by varying an applied electric current (57). When the optical signals are demultiplexed, they must be separated at the optical receiver; this can be done using filters or by means of tunable receivers that select the desired wavelength in response to an applied electric current. The simplest form of a tunable receiver is a single cavity Fabry–Perot interferometer with a movable mirror that creates a tunable resonant cavity. Cascaded Mach– Zender interferometers with variable phase delay in one branch may also be used. Switchable gratings such as the monolithic grating spectrometer or acousto-optic tunable filter may also be employed. There is ongoing research in the area of all-optical networks incorporating these components. One possible network architecture is the broadcast-and-select network (3), which consists of nodes interconnected via a star coupler. Optical fiber carries signals from each node to the star, where they are combined and distributed to all other nodes equally; examples include the Lambda (3) and Rainbow (3) networks. A wavelength-routed network consists of either

260

OPTICAL COMMUNICATION

static or reconfigurable wavelength selective routers interconnected by fiber links. Currently there are no standards for WDM networking, although the International Telecommunications Union is developing a draft standard G.692 entitled ‘‘Optical interfaces for multichannel systems using optical amplifiers.’’

Gain

16 14 12 10

Solitons Next to attenuation, dispersion is the most important factor in determining the ultimate length of an optical link. Solitons have received strong interest in the optical communications area because they represent a solution to the wave equation (a hyperbolic secant pulse shape) which propagates without dispersion on standard optical fiber. This is achieved by balancing two effects: self-phase modulation and chromatic dispersion (also known as group velocity dispersion) in the fiber. Self-phase modulation is caused when optical pulses have sufficiently high intensity to modify the refractive index of the fiber via the nonlinear Kerr effect. This causes a chirp in the optical signal such that longer wavelengths move toward the beginning of the optical pulse and shorter wavelengths to the end. In standard optical fiber at wavelengths greater than 1310 nm (the so-called anomalous dispersion regime), chromatic dispersion causes shorter wavelengths to travel faster than longer ones, such that shorter wavelengths tend to move toward the beginning of the pulse (the opposite direction from self-phase modulation). Thus, if these two effects exactly balance each other, the pulse shape is retained during propagation; of course, pulses still suffer attenuation and must be periodically amplified. The effect only occurs at wavelengths greater than 1310 nm, for which the group velocity dispersion in standard fiber is negative; otherwise, the pulses would suffer increased dispersion. This raises the possibility of optical communication over very long distances, and at very high data rates when combined with time division multiplexing techniques. Related areas of research include dark soliton; a small gap within an unbroken, high power optical beam or a very long pulse will behave exactly like a regular soliton. It is also possible for the soliton effect to be spatial rather than temporal; high-power optical beams can be used to modify the refractive index of dispersive media in such a way that they create their own waveguides. Such spatial solitons remain localized in space as if they were confined in a waveguide by balancing the effects of diffraction (spatial dispersion) and self-phase modulation. Optical Amplifiers An optical amplifier is a device that amplifies the optical signal directly without ever converting it into an electrical signal. This makes the signal amplification independent of the type of data encoding used, since retiming of the signal is not required as in a digital regenerator. There are many possible types of optical amplifiers. Almost any semiconductor laser can be converted into an amplifier by increasing the length of the gain region; such devices are called semiconductor optical amplifiers, and are most useful at wavelengths near 1300 nm. In the late 1980s, researchers developed fiber-based amplifiers, which amplify optical signals by passing them through a specially doped length of fiber (approximately 10 m) which couples light from a separate optical pump source into the

1530 1540 1550 1560 Wavelength (nm)

Figure 20. Gain curve of a typical erbium doped fiber amplifier.

desired optical signal. The fibers are typically doped with rare earth elements such as erbium, neodynium, or praseodymium, and operate on the same principle as a laser. Rare earth ions are pumped into elevated energy states by using a relatively high-powered optical source (10 mW to 200 mW) at a different wavelength than the desired signal (980 nm or 1400 nm optical pumps are most common). A photon from the desired signal will then cause stimulated emission at the same wavelength as the desired signal (1300 nm or 1550 nm). In this way, a band of wavelengths approximately 24 nm wide can be optically amplified. Reflections back into the amplifier from the attached fiber must be controlled using optical isolators to insure stable operation of the amplifier. Erbium-doped amplifiers operating near 1550 nm are most commonly in use today; they offer low insertion loss, low noise, and minimal interchannel crosstalk and polarization sensitivity. The gain characteristic of a typical erbium-doped optical amplifier is shown in Fig. 20 (note the logarithmic scale). The gain is not uniform over the passband; it is approximately 3 dB higher at 1560 nm than at 1540 nm. If several amplifiers are cascaded in series, this effect is cumulative and results in significant nonuniform amplification at different wavelengths. A further complication is that the gain profile changes with signal power levels, so that the amplifier response will be different depending on the number of channels being amplified at once in a WDM system. Most optical amplifiers operate in gain saturation or beyond the point where further increase in the input power does not result in a corresponding increase in output power (in this respect, optical amplifiers behave very differently from electrical amplifiers, which are subject to nonlinear distortion when they saturate). This is possible because erbium amplifiers respond to changes in average power (fluctuations over the course of a few milliseconds) rather than to instantaneous changes in power levels as for electrical amplifiers. Current research is directed toward flattening the gain curve; this can be done in many ways, for example by introducing other dopants including aluminum or ytterbium, controlling the pump power through a feedback loop, gain clamping (addition of an extra WDM channel locally at the amplifier to absorb excess power), or incorporating preemphasis filters and blazed diffraction gratings into the system. These amplifiers also suffer from amplified spontaneous emission noise and significant broadening of the amplified signal spectra; secondary effects such as stimulated Raman scattering must also be controlled at the system level. CONCLUSION Since the technology for optical communications over fiber first emerged in the 1970s, the need for higher bandwidth,

OPTICAL COMMUNICATION

faster speed, and longer distances has driven a host of new applications. This area is experiencing rapid growth, however, estimated by some to be as much as 25% compounded annually through the turn of the century. As of this writing, there is enough fiber installed in the world today to stretch between the earth and the moon 28 times. The past decade has seen a reduction in the size and cost of optical transceivers comparable to the transition from vaccum tubes to solidstate semiconductor chips in the electronics industry. As fiber optic technology continues to evolve, several trends have emerged; the use of smaller form factors and nonhermetic transceiver packaging, transceivers that operate at lower voltages to dissipate less power, advances in optoelectronic packaging including plastic housings, sleeves, fiber, and ferrules, and growing interest in WDM, optical amplifiers, and parallel optics as ways to increase the available bandwidth of a communication link. Predicting the future of optical communications is a proposition fraught with uncertainty. As the datacom industry has continued to evolve, many people have tried to speculate what the future will bring. In 1965, Gordon Moore (founder and current chairman of Intel Corporation) projected that computing power as measured by the logic density of silicon integrated circuits would grow exponentially, roughly doubling every 12 months to 18 months (3). This has held true for the past 30 years, and has come to be known as Moore’s Law. Recent presentations by Moore (3) forecast that this trend will continue through the next decade or so, as feature sizes approach 0.18 애m and as the economic problems associated with fabricating such chips begin to be felt. Observing that the usefulness of this computer power depends on the number of networked users, in 1980 Bob Metcalfe argued that the value of a network can be measured by the square of the number of users (3). Metcalfe’s Law is more fully described by George Gilder, also the namesake for the Gilder Paradigm (3), which predicts in part that future communication system designs will be influenced by a key scarcity of bandwidth. In this same spirit, similar predictions can be made concerning the growth of bandwidth requirements in data communications following the recent introducton of fiber optics (before the widespread use of fiber, data rates appear to have increased only incrementally over a relatively long period of time). During a recent technical conference (1996 Optical Society of America Annual Meeting), for example, the author introduced a graph showing the extrapolated growth of input–output bandwidth on mainframes and large servers as a measure of leading-edge application requirements. This bandwidth has been growing exponentially since about 1988, when optical fiber first became available as an option on the IBM System/390. This trend, labeled by one of the meeting attendees as DeCusatis’ Law, is projected to continue for at least the next several generations of CMOS-based large mainframe computer systems, and perhaps beyond.

BIBLIOGRAPHY 1. S. E. Miller and A. G. Chynoweth (eds.), Optical Fiber Telecommunications, New York: Academic Press, 1979. 2. J. Gowar, Optical Communication Systems, Englewood Cliffs, NJ: Prentice-Hall, 1984.

261

3. C. DeCusatis et al. (eds.), Handbook of Fiber Optic Data Communication, New York: Academic Press, 1998. 4. R. Lasky, U. Osterberg, and D. Stigliani (eds.), Optoelectronics for Data Communication, New York: Academic Press, 1995. 5. J. Senior, Optical Fibre Communications: Principles and Practice, 2nd ed., Englewood Cliffs, NJ: Prentice-Hall, 1982. 6. P. Green, Fiber Optic Networks, Englewood Cliffs, NJ: PrenticeHall, 1993. 7. J. Laude, Wavelength Division Multiplexing, Englewood Cliffs, NJ: Prentice-Hall, 1993. 8. United States laser safety standards are regulated by the Dept. of Health and Human Services (DHHS), Occupational Safety and Health Administration (OSHA), Food and Drug Administration (FDA), Code of Radiological Health (CDRH), 21 Code of Federal Regulations (CFR) subchapter J; the relevant standards are ANSI Z136.1, Standard for the Safe Use of Lasers (1993 revision) and ANSI Z136.2, Standard for the Safe Use of Optical Fiber Communication Systems Utilizing Laser Diodes and LED Sources (1996–97 revision); elsewhere in the world, the relevant standard is International Electrotechnical Commission (IEC/CEI) 825 (1993 revision). 9. Electronics Industry Association/Telecommunications Industry Association (EIA/TIA) commercial building telecommunications cabling standard (EIA/TIA-568-A). Detail specification for 62.5 애m core diameter/125 애m cladding diameter class 1a multimode graded index optical waveguide fibers (EIA/TIA-492AAAA), detail specification for class IV-a dispersion unshifted single-mode optical waveguide fibers used in communication s systems (EIA/ TIA-492BAAA) Electronics Industry Association, New York. 10. J. P. Powers, An Introduction to Fiber Optic Systems, Homewood, IL: Aksen, 1993. 11. D. Marcuse, D. Gloge, and E. A. J. Marcatiti, Guiding properties of fibers, in Optical Fiber Telecommunications, S. E. Miller and A. G. Chynoweth (eds.), New York: Academic Press, 1979. 12. D. Marcuse, Loss analysis of single mode fiber splices, Bell System Tech. J., 56: 703–718, 1977. 13. T. Okoshi, Optical Fibers, New York: Academic Press, 1982. 14. P. K. Cheo, Fiber Optics and Optoelectronics, 2nd ed., Englewood Cliffs, NJ: Prentice-Hall, 1990. 15. S. Nemoto and T. Makimoto, Analysis of splice loss in singlemode fibers using a Gaussian field approximation, Opt. Quantum Electron., 11: 447–457, 1979. 16. J. R. Webb and U. L. Osterberg, Fiber, cable, and coupling, in R. Lasky, U. Osterberg, and D. Stigliani (eds.), Optoelectronics for Data Communication, New York: Academic Press, 1995. 17. E. G. Neumann, Singlemode Fibers: Fundamentals, Berlin: Springer-Verlag, 1988. 18. S. S. Walker, Rapid modeling and estimation of total spectral loss in optical fibers, IEEE J. Lightwave Technol., 4: 1125–1132, 1996. 19. D. Gloge, Propagation effects in optical fibers, IEEE Trans. Microwave Theory Tech., MTT-23: 106–120, 1975. 20. P. M. Shanker, Effect of modal noise on single-mode fiber optic network, Opt. Commun., 64: 347–350, 1988. 21. C. DeCusatis and M. Benedict, Method for fabrication of high bandwidth optical fiber, IBM Tech. Disc. Bulletin, May 1992. 22. J. J. Refi, LED bandwidth of multimode fiber as a function of source bandwidth and LED spectral characteristics, IEEE J. Lightwave Technol., LT-14: 265–272, 1986. 23. S. E. Miller and I. P. Kaminow (eds.), Optical Fiber Telecommunications II, New York: Academic Press, 1988. 24. G. P. Agrawal et al., Dispersion penalty for 1.3 micron lightwave systems with multimode semiconductor lasers, IEEE J. Lightwave Technol., LT-6: 620–625, 1988.

262

OPTICAL FILTERS

25. K. Ogawa, Analysis of mode partition noise in laser transmission systems, IEEE J. Quantum Electron., QE-18: 849–9855, 1982. 26. K. Ogawa, Semiconductor laser noise; mode partition noise, in R. K. Willardson and A. C. Beer (eds.), Semiconductors and Semimetals, Vol. 22C, New York: Academic Press, 1985. 27. J. C. Campbell, Calculation of the dispersion penalty of the route design of single-mode systems, IEEE J. Lightwave Technol., LT6: 564–573, 1988. 28. M. Ohtsu et al., Mode stability analysis of nearly single-mode semiconductor laser, IEEE J. Quantum Electron., GE-24: 716– 723, 1988. 29. M. Ohtsu and Y. Teramachi, Analysis of mode partition and mode hopping in semiconductor lasers, IEEE J. Quantum Electron., 25: 31–38, 1989. 30. D. Duff et al., Measurements and simulation of multipath interference for 1.7 Gbit/s lightwave systems utilizing single and multifrequency lasers, Proc. Optical Fiber Conf., San Diego, CA, 1989, p. 128. 31. J. Radcliff, Fiber Optic Link Performance in the Presence of Internal Noise Sources, IBM Tech. Rep., Glendale Labs, Endicott, NY, 1989. 32. L. L. Xiao, C. B. Su, and R. B. Lauer, Increase in laser RIN due to asymmetric nonlinear gain, fiber dispersion, and modulation, IEEE Photon. Technol. Lett., PTL-4: 774–777, 1992. 33. CCITT, Geneva, Switzerland, Recommendations G.824, G.823, O.171, and G.703 on Timing Jitter in Digital Systems, 1984. 34. A. A. Manalino, Time domain eyewidth measurements of an optical data link operating at 200 Mbit/s, IEEE Trans. Instrum. Meas., IM-36, 551–553, 1987. 35. F. M. Gardner, Phaselock Techniques, 2nd ed., New York: Wiley, 1979. 36. G. P. Agrawal and T. P. Shen, Power penalty due to decision time jitter in optical communication systems, Electron. Lett., 22: 450–451, 1986.

48. D. Stigliani, Enterprise systems connection fiber optic link, in C. DeCusatis et al. (eds.), Handbook of Optoelectronics for Fiber Optic Data Communications, New York: Academic Press, 1998. 49. ESCON I/O Interface Physical Layer Document, 3rd ed. (IBM document number SA23-0394), Mechanicsburg, PA: IBM Corporation, 1995. 50. ANSI Single Byte Command Code Sets CONnection architecture (SBCON), draft ANSI standard X3T11/95-469 (rev. 2.2), 1996, available from Global Engineering Documents, Santa Ana, CA. 51. ANSI X3.230-1994 rev. 4.3, Fibre Channel—Physical and Signaling Interface (FC-PH), ANSI X3.272-199x, rev. 4.5, Fibre Channel—Arbitrated Loop (FC-AL), June 1995, ANSI X3.269-199x, rev. 012, Fibre Channel Protocol for SCSI (FCP), May 30, 1995. 52. ANSI T1.105-1988, Digital Hierarchy Optical Rates and Format Specification, 1988, available from Global Engineering Documents, Santa Ana, CA. 53. CCITT Recommendation G.707, Synchronous Digital Hierarchy Bit Rates, 1988, available from International Telecommunications Union, Geneva, Switzerland. 54. CCITT Recommendation G.708, Network Node Interfaces for the Synchronous Digital Hierarchy, 1988, available from International Telecommunications Union, Geneva, Switzerland. 55. CCITT Recommendation G.709, Synchronous Multiplexing Structure, 1988, available from International Telecommunications Union, Geneva, Switzerland. 56. IEEE 802.3z, Draft Supplement to Carrier Sense Multiple Access with Collision Detection (CSMA/CD) Access Method and Physical Layer Specifications: Media Access Control (MAC) Parameters, Physical Layer, Repeater and Management Parameters for 1000 Mb/s Operation, June 1997, available from Institute of Electrical and Electronics Engineers, Piscataway, NJ. 57. H. Dutton, Optical Communication Technology: An Introduction and Tutorial, New York: Academic Press, 1998.

CASIMER DECUSATIS

37. T. M. Shen, Power penalty due to decision time jitter in receiver avalanche photodiodes, Electron. Lett., 22: 1043–1045, 1986. 38. P. R. Trischitta and E. L. Varma, Jitter in Digital Transmission Systems, Boston: Artech House, 1989. 39. C. J. Byrne, B. J. Karafin, and D. B. Robinson Jr., Systematic jitter in a chain of digital regenerators, Bell Syst. Tech. J., 43: 2679–2714, 1963. 40. R. J. S. Bates and L. A. Sauer, Jitter accumulation in token passing ring LANs, IBM J. Res. Develop., 29: 580–587, 1985. 41. C. Chazmas, Accumulation of jitter, a stochastic model, AT&T Tech. J., 64: 120–134, 1985. 42. P. M. Shanker, Effect of modal noise on single-mode fiber optic networks, Opt. Commun., 64: 347–350, 1988. 43. J. B. Haber et al., Assessment of radiation induced loss for AT& T fiber optic transmission systems in the terrestrial environment, IEEE J. Lightwave Technol., 6: 150–154, 1988. 44. M. Kyoto et al., Gamma ray radiation hardened properties of pure silica core single-mode fiber and its data link system in radioactive environments, IEEE J. Lightwave Technol., 10: 289– 294, 1992. 45. D. Cotter, Observation of stimulated Brillouin scattering in low loss silica fibers at 1.3 microns, Electron. Lett., 18: 105–106, 1982. 46. H. J. A. diSilva and J. J. O’Reily, System performance implications of laser chirp for long haul high bit rate direct detection optical fiber systems, Proc. IEEE Globecom., 1989, pp. 19.5.1– 19.5.5. 47. R. H. Stolen and C. Lin, Self-phase modulation in silicon optical fiber, Phys. Rev. A, 17: 1448, 1978.

IBM Corporation

OPTICAL COMMUNICATION EQUIPMENT. See DEMULTIPLEXING EQUIPMENT.

OPTICAL COMPONENTS. See PACKAGING OF OPTICAL COMPONENTS AND SYSTEMS.

OPTICAL CONSTANTS. See OPTICAL PROPERTIES. OPTICAL CORRELATION. See MATCHED FILTERS. OPTICAL DATA LINK PACKAGING. See PACKAGING OF OPTICAL COMPONENTS AND SYSTEMS.

OPTICAL FIBERS, ATTENUATOR. See ATTENUATION MEASUREMENT.

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Wiley Encyclopedia of Electrical and Electronics Engineering Optical Materials Standard Article Anand K. Kulkarni1 1Michigan Technological University, Houghton, MI Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W4401 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (242K)

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Abstract The sections in this article are Ferroelectrics Semiconductors Glass Polymers Optical Properties Optical Coatings About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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296

OPTICAL MATERIALS

OPTICAL MATERIALS Optical materials have emerged as an extremely important class of materials in the modern world. From simple mirrors in everyday use to complex optical communication systems using lasers, optical fibers, and photodetectors, optical materials have gained a unique position in the advancement of technology. A number of optical properties such as refractive index, absorption coefficient, nonlinear dielectric susceptibility, J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

OPTICAL MATERIALS

and their variations with respect to applied electric and magnetic fields are exploited in a variety of applications. These applications include ferroelectric oxides as electro-optic modulators, semiconductors as light sources and detectors, polymers as holographic optical storage devices and image processing devices, and glass as an optical fiber. However, future optical materials will largely be in the form of optical coatings, i.e., thin and thick films of optical materials designed and fabricated to suit particular applications. FERROELECTRICS Ferroelectrics have nonlinear optical properties that are exploited for a number of applications. These nonlinear properties are attributed to (1) crystalline structure lacking a center of inversion symmetry, (2) large nonlinear susceptibilities, and (3) large spontaneous birefringence. Electro-optic modulators have been successfully fabricated in several materials, such as barium titanate (BaTiO3), lithium titanate (LiTiO3), and lead zirconium titanate (PLZT ceramics) (1). The PLZT compositions contain 9% or more lanthanum and are called slim-loop materials because they have negligible remanent polarization. These electro-optic modulators have applications in industrial and military eye protection devices, stereoscopic three-dimensional television, and one-dimensional optical memories. At present, titanium-diffused bulk lithium niobate (LiNbO3) channel waveguide structures are the most widely used optical modulators because of their relative ease of fabrication (2). Figure 1 shows optical-grade LiNbO3 boules and wafers of 7.5 cm to 10 cm (3 in. to 4 in.) diameter grown by the Czochralski technique. The waveguides are characterized by low propagation losses (usually in the range of 0.1 to 0.2 dB/cm) and exhibit very low coupling losses (0.5 dB/facet). In practical modulators the optical properties should not degrade when subjected to a light beam and a radio-frequency field. Certain materials, such as strontium barium niobates (Sr1⫺xBaxNb2O6), have extremely large electro-optic coefficients, but their properties degrade in the presence of modulation fields (3). Both potassium hydrogen phosphate (KH2PO4, KDP) and potassium deuterium phosphate

297

(KD2PO4, DKDP) are widely used as modulator materials, but both of them are hygroscopic and require a water-free environment for their operation (3). Table 1 gives a number of ferroelectric materials and their optical properties. SEMICONDUCTORS Semiconductors are the backbone of the modern electronics industry. The electronic properties of semiconductors are of utmost importance in the design and fabrication of integrated circuits used in computers, video games, and electronic instruments. On the other hand, the optical properties of semiconductors, such as their emission and absorption characteristics, are used extensively in light sources and detectors for optical communication systems. Semiconductors are also used as solar cell materials to convert the sun’s light into electricity and as optoelectronic integrated circuit materials. Figure 2 shows a number of semiconductors with their energy bandgaps and their corresponding wavelengths of emission. As seen in this figure, the compound semiconductor gallium arsenide (GaAs) emits at a wavelength of 0.85 애m, one of the useful wavelengths for light sources used in optical communication. The transmission windows for silica fibers (the wavelengths at which the losses and dispersion are a minimum in silica fibers) are at 0.85, 1.30, and 1.55 애m. Since there are no elemental or compound semiconductors that emit radiation at 1.30 and 1.55 애m, it has become necessary to employ ternary and quaternary III–V alloys to produce materials of appropriate bandgaps. However, the choice of materials is limited by the fact that high-quality epitaxial materials can be grown only on those substrates that have a lattice match to the grown material. Since gallium arsenide (GaAs) and indium phosphide (InP) substrates are available readily, the ternary and quaternary alloys such as aluminum gallium arsenide (AlGaAs) and indium gallium arsenide phosphide (InGaAsP) are grown on GaAs and InP respectively. In spite of the good lattice match between GaAs and AlAs, the thermal expansion coefficients of GaAs and AlGaAs differ substantially, resulting in dislocations at the interface during cooling. However, very efficient luminescent devices (lightemitting diodes and lasers) radiating in the range 0.8 애m to 0.9 애m are fabricated of materials in the AlGaAs–GaAs system and are used as light sources in optical communication systems, compact disc players, and pointing devices. The details on the design and fabrication of these devices can be found in the excellent Refs. 4 to 8. Light Detection Devices

Figure 1. Lithium niobate boules and wafers (2).

In a receiver set of an optical communication system, the light signal is converted to an electrical signal. The detection devices used for this purpose are usually photoconductors or photodiodes. Silicon photodiodes are commonly used to detect the light from GaAs–AlGaAs laser sources at 0.85 애m. Here the absorption coefficient of the material, 움, determines the thickness of the semiconductor to be used. The absorption coefficient of silicon is of the order of 103 cm⫺1 requiring at least 50 애m thick material. For optical detection at 1.30 애m and 1.55 애m wavelengths heterojunction photodiodes are more commonly used. In this case, a large-bandgap semiconductor such as InP is used as a substrate material, and smallerbandgap alloys such as gallium indium arsenide (GaInAs)

298

OPTICAL MATERIALS

Table 1. Ferroelectric Materials and Their Properties Static Dielectric Constant

Material BaTiO3 (barium titanate)

3600

LiNbO3 (lithium niobate)

⑀3 ⫽ 32 ⑀1 ⫽ 78

ADP (ammonium dihydrogen phosphate)

56

1640 (unclamped)

2.49

820 (clamped) 30.8 (clamped)

2.29

34.0 (unclamped)

8.5

1.53

r63 ⫽ ⫺10.5 r41 ⫽ 8.6

1.51

DKDP (potassium dideuterium phosphate)

⑀3 ⫽ 50

r63 ⫽ 26.4 r41 ⫽ 8.8

1.51

51

LiTaO3 (lithium tantalate) PLZT (lead zirconium titanate)

r33 ⫽ 1340

2.31

⑀3 ⫽ 45 ⑀1 ⫽ 51 —

r33 ⫽ 30.3

2.18

102 to 612

—

Red Yellow Green Blue Violet

Visible

Infrared

Ultraviolet

GaAs GaP Si CdSe CdS SiC

1

2 1

ZnS

3 0.5

4 0.35

36

750

and GaInAsP are grown epitaxially on InP. Detailed information on heterojunction photodiodes can be found in Refs. 9 and 10. Photoconductors are used in detecting visible (0.4 애m to 0.7 애m) and infrared (0.7 애m to 1 mm) radiations. In these devices, the conductivity of the material is altered by the photon irradiation and is detected as a photocurrent. Cadmium sulfide (CdS), with an energy bandgap of 2.42 eV, has a very high sensitivity at about 0.5 애m in the visible region of the spectrum. To improve the gain of this photoconductor, effective trap states for holes are introduced using dopants such as iodine or chlorine. The detection sensitivity of a CdS photoconductor can be extended to higher wavelengths of the visible spectrum by adding copper to create trap states in the bandgap of the material to allow absorption of low-energy photons. Such detectors in the visible range find applications in illumination-level measuring instruments, in exposure meters for cameras, and as control devices to switch light on and

75 3 2

r33 ⫽

再 再

⑀3 ⫽ 21 ⑀1 ⫽ 42

Sr0.75Ba0.25Nb2O6 (strontium barium niobate)

0

r42 ⫽

Refractive Index

KDP (KH2PO4) (potassium dihydrogen phosphate)

Ba2NaNb5O15 (barium sodium niobate)

InSb Ge

Electro-Optic Constants (pm/V)

Eg (eV)

λ ( µ m)

Figure 2. Bandgaps of some common semiconductors relative to the optical spectrum (10).

off inside a home by measuring the light intensity. Narrowbandgap semiconductors such as Ge (Eg ⫽ 0.67 eV), InSb (Eg ⫽ 0.18 eV), PbSe (Eg ⫽ 0.27 eV), and PbS (Eg ⫽ 0.41 eV) can be used as photoconductors in the near infrared range (0.7 애m to 2 애m). The Ge devices are typically operated at liquid nitrogen temperature (77 K) to reduce thermal noise. For the detection of mid-infrared radiation (2 애m to 10 애m), ternary alloys of II–VI compounds such as HgCdTe can be fabricated with appropriate bandgaps. However, the crystal growth technology of these alloys with well-controlled stoichiometry (appropriate ratios of elements in an alloy) is yet to be realized (11). To obtain photoconductors to detect very long wavelengths (10 애m to 100 애m), germanium doped with gallium, silicon doped with phosphorus, and GaAs doped with selenium or tellurium are commonly used. Again these materials have to be operated at low temperatures (77 K) to ensure that the free carriers are not excited from the impurity levels by thermal energies. Lastly, infrared radiation in the range 0.1 mm to 1 mm is detected with free electron absorption. Such devices are cooled to liquid helium temperature (4 K). Solar cells, or photovoltaic devices that convert light energy into electrical energy, use a wider range of semiconducting materials than any other optical device. The most important optical properties of solar cell materials are (1) absorption coefficient, (2) energy bandgap, and (3) quantum efficiency, defined as the number of electron–hole pairs generated per incident photon. Two important parameters of a solar cell are Vo, the opencircuit voltage, and Is, the short-circuit current. The maximum efficiency of a solar cell is obtained by maximizing the output power, which is typically 80% of the product VoIs. Hence, considerable research effort has gone into the design

OPTICAL MATERIALS

50 AM 1.5 300 K

Si InP GaAs CdTe AlSb

40

Efficiency (%)

Ga

Like electronic monolithic integrated circuits, optoelectronic monolithic integrated circuits with lasers, modulators, photodetectors, and electronic components all on a single chip have been fabricated, and limited success has been achieved in the actual implementation of these chips. One of the major hurdles in high-speed computing is the delay in the electrical interconnects. If photons instead of electrons are used to carry the message from one device to another device on the chip or from one chip to another chip in a microprocessor system, the speed of the system can be significantly enhanced. Photons are preferred over electrons because they are not subject to propagation delays caused by the finite resistance, capacitance, and inductance of the interconnects. The optical materials being developed for optical waveguides to replace electrical interconnects are GaAs–AlGaAs, InP–GaAsP, photorefractive materials, and certain organic materials having nonlinear effects. Details on optoelectronic integrated circuits can be found in Refs. 7, 14, and 15.

Cu2O GaP

C=1

20

CdS

10

0

0

1

heterojunction solar cells can withstand temperatures up to 100 ⬚C. Comprehensive reviews on photovoltaic materials are found in Refs. 4, 10, 12, and 13. Optoelectronic Integrated Circuits

30 C=1000

299

2 Eg (eV)

3

Figure 3. Ideal solar cell efficiency at 300 K for 1 sun and for 1000 sun concentration (10).

of solar cells and choice of materials to achieve large values of Vo and Is simultaneously. Again, pn junctions are the obvious choice for devices, and semiconductors are the best choice for materials. The dependence of the solar cell efficiency on the energy bandgap of a semiconductor is shown in Fig. 3. A maximum efficiency of about 28% occurs around 1.4 eV to 1.5 eV, corresponding to the bandgaps of Cu2S, InP, GaAs, and CdTe. Because of the low material costs and availability of advanced technology, silicon solar cells are most commonly used today. One important material consideration in the fabrication of solar cells is to obtain highly pure starting materials devoid of metallic impurities such as chromium, copper, iron, and sodium, which degrade the minority carrier lifetime, thus reducing Is. On the other hand, the stringent material requirements (e.g., the crystal quality) needed in the fabrication of integrated circuits can be relaxed in the manufacture of solar cells. Large-grain polycrystalline and amorphous silicon are extensively used to reduce the material costs in manufacturing solar cells. Mass production of polycrystalline silicon ribbons and sheets has been accomplished by dendritic web growth or edge-defined film-fed growth or chemical vapor deposition. A 10% efficient solar cell can provide roughly 100 W/m2 in a sunny location. In order to obtain higher power it is possible to use solar concentrator systems where 100 or even 1000 times the sun’s actual energy flux can be incident on the cell, using mirrors and lenses. Although silicon cells are not very suitable (their efficiency decreases with increasing temperature due to concentrated sunlight), GaAs and GaAs–AlGaAs

Multiple-Quantum-Well Structures With the availability of computer-controlled molecular beam shutters in molecular beam epitaxy (MBE) systems since the early 1980s, it has been possible to grow extremely thin (1 nm to 5 nm) layer-by-layer structures in a variety of III–V and II–VI semiconducting alloys such as GaAs–AlGaAs, GaInAs–AlInAs, GaSb–GaAlSb, GaInAs–InP, CdTe– CdMnTe, GaAsSb–GaAlSb, and ZnSe–ZnMnSe (16). Quantum well lasers constructed of alternate layers of GaAs and AlGaAs of thicknesses of a few nanometers have better spontaneous quantum efficiency than conventional double heterojunction lasers made of GaAs and AlGaAs. This results in a reduction of the threshold current density and its temperature dependence. Recently, a few more semiconducting optical material systems have gained significant importance. They are: (1) porous silicon (17), (2) gallium nitride (GaN) and related compounds (18), (3) Si–Ge alloys (19), (4) diamond (20), (5) silicon carbide (SiC) (21), and wide-bandgap II–VI semiconductors (22). GLASS Glass is one of the most important optical materials, used in many applications, including optical fibers to transmit information across continents, lenses to improve the vision of millions of people, and a variety of display devices. As one of the most widely used manufactured materials, glass is used in everyday life (mirrors, telescopes, electric light bulbs, buildings, bottles and tumblers for drinking, and cookware). A few other interesting applications of glass are found in nose cones of missiles, crowns for teeth, beads in human organs to help them tolerate radiation doses, and even in nuclear waste disposal. Glass is essentially an inorganic liquid with the rigidity of a solid. Glass has a number of unique properties, e.g., it is transparent and durable, it can be shaped to different forms

300

OPTICAL MATERIALS

easily, and it is inexpensive. The use of glass in fiber optics and liquid crystal displays is a 16 billion dollar industry in the US (23). Tempered glass produced using a chemical process rather than by quick cooling can withstand pressures up to 150 MPa (22,000 lb/in.2) (23). This special glass is used in eyeglasses, oven cookware, basketball backboards, and car windows. Lead oxide (PbO) and barium oxide (Ba2O3) are added to the basic sand–soda–lime (SiO2 –Na2CO3 –CaO) mixture to obtain sparkling glass, boric oxide (B2O3) is added to obtain heat-resistant glass, and chromium (Cr) and copper (Cu) are added to make green sunglasses. Certain other oxides such as aluminum oxide (Al2O3) add durability to glass, magnesium oxide (MgO) reduces the melting temperature, boric oxide (B2O3) reduces the viscosity, and lead oxide (PbO) increases the refractive index (23,24). The refractive index of a glass can be varied from 1.517 for borosilicate glass to 1.980 for phosphate glass by adding different oxides. Corning manufactures 750 different glasses and glass related products. Notable among these are Macor, machinable glass used to make glass nuts and bolts and window frames for the Space Shuttle, and Dicor, used in dental crowns that are plaqueresistant and highly translucent. Optical Fibers Long-distance transmission of optical signals requires lowloss optical fibers to maximize the distance between repeaters. Apart from the wavelength of the optical signal (the loss is only 0.2 dB/km at 1.55 애m, but 0.5 dB/km at 1.30 애m), the purity of the silica (SiO2) used in the construction of the fiber determines the losses of the signal. Glasses containing less than 1 ppm of transition metal ions have attenuations as low as 0.2 dB/km. Figure 4 shows the attenuation in optical fibers as a function of wavelength. As indicated already, low transmission losses occur at 1.30 애m and 1.55 애m. Another important aspect of the choice of wavelength of transmission is the dispersion of the fiber. Dispersion is the variation of the refractive index of the optical fiber with wavelength. When material absorption and dispersion are reduced to a minimum at ␭ ⫽ 1.55 애m by proper variation of the refractive index of the fiber, Rayleigh scattering, which is due to the granularity of silica, dominates the absorption in the medium (25). One way to reduce Rayleigh scattering is to use crystalline materials. Crystalline zirconium fluoride (ZrF), arsenic

100

1 50

0.1

0

Dispersion (ps/km⋅ nm)

Attenuation (dB/km)

10

Table 2. Rare Earth Ions Used in Laser Glasses (28) Ion

Host Glass (Oxides)

␭ (애m)

Nd3⫹

K–Ba–Si La–Ba–Th–B Na–Ca–Si Li–Ca–Al–Si

1.060 1.370 0.920 1.060

Yb3⫹

Li–Mg–Al–Si K–Ba–Si

1.015 1.060

Ho3⫹

Li–Mg–Al–Si

2.100

Er3⫹

Yb–Na–K–Ba–Si Li–Mg–Al–Si Yb–Al–Zn–P

1.543 1.550 1.536

Tm3⫹

Li–Mg–Al–Si Yb–Li–Mg–Al–Si

1.850 2.105

triselenide (AsSe3), and potassium iodide (KI) have very low attenuations (앑10⫺2 dB/km) in the 2 애m to 5 애m range (26). Heavy-metal fluoride glasses (ZrF2, PbF2, BaF2) with low phonon energies and chalcogenide glasses based on sulfides, selenides, and tellurides and their mixtures are being investigated as candidates for optical transmission (27). Optical Amplifiers Instead of using electronic amplifiers at repeater stages to boost the signal strength, modern optical communication systems use amplifiers made of silica optical fiber doped with erbium (Er3⫹). These amplifiers can be laser-diode-pumped at 0.80 애m, 0.98 애m, and 1.48 애m wavelengths. However, this amplifier suffers from a short fluorescence lifetime due to nonradiative decay. Pr3⫹-doped amplifiers use fluoride glasses as host materials with fluorescence lifetime of 100 애s. Laser Glasses The first lasers were made of crystals, e.g., ruby lasers (Al2O3 containing Cr3⫹ ions) and YAG lasers (Y3Al5O12 containing Nd3⫹ ions) emitting at 0.69 애m and 1.06 애m respectively. Table 2 shows the rare earth ions used in various types of glasses and their corresponding emission wavelengths (28). Although Nd3⫹ is the most favored ion in a variety of host glasses, the durability and ease of fabrication favor alkali metal and alkaline earth silicates (28). The main difference between glass lasers and crystal lasers is that large, isotropic, and homogeneous volumes with good optical quality glass lasers can be easily fabricated. Furthermore, the disordered structure of glass allows many more sites for ions to occupy than in ordered structures. However, the spectra of glass lasers contain broader fluorescent lines (앑30 nm) than the spectra of lasers made of crystalline materials (YAG has 1 nm broadening) (28). POLYMERS

0.9

1.1 1.3 1.5 Wavelength ( µ m)

1.7

Figure 4. Optical attenuation and dispersion behavior of standard silica fiber (25).

Several nonlinear optical phenomena have been observed in organic materials (29–31). These phenomena include: (1) third-harmonic generation, (2) two-photon absorption, (3) electric-field-induced second-harmonic generation, (4) inten-

OPTICAL MATERIALS Table 3. Figure of Merit for Inorganic and Organic Materials (31)

Material Gallium arsenide (GaAs) Barium titanate Lithium niobate (LiNbO3) Sr0.75Ba0.25Nb2O6 Organic crystal Organic polymer Photorefractive polymer

Electro-Optic Coefficient r (pm/V) 1.430 1640 34 216 67 30 3.1

Dielectric Refractive Constant Index n ⑀

Figure of Merit Q ⫽ n3r/ ⑀ (pm/V)

3.40

12

4.7

2.49 2.29

3600 32

7.0 12.8

2.3 2.0 1.6 1.7

750 3.2 4.0 7.0

3.5 168 31 2.2

301

A comparison of the figures of merit of polymers and organics with inorganic materials is given in Table 3 (31). As seen in this table, in spite of low values of electro-optic coefficients, the figures of merit of the polymers can be significantly higher than those of ferroelectric oxides. One major problem that exists in the development of polymers is their temperature stability and mechanical ruggedness. To date, polymers are used in high-speed modulators, directional couplers, second- and third-harmonic generators, programmable optical interconnects, multiple-image processors and optical data storage systems. A recent issue of the MRS Bulletin has articles on polymer electroluminescent devices and conjugated polymer surfaces and interfaces for light-emitting devices (32–34). OPTICAL PROPERTIES

sity-dependent index of refraction, and (5) stimulated Raman scattering. The major difference between inorganic and organic nonlinear optical materials is the origin of the nonlinearity. In inorganic materials the optical nonlinearity is mainly due to the ionic polarizability resulting from the displacement of the ions. Hence large electro-optic coefficients in materials such as BaTiO3 and KNbO3 are also associated with large dielectric constants. Since the optical nonlinearity and dielectric constant are inextricably linked, the figure of merit (see Table 3) of inorganic materials does not change significantly from one material to the other. On the contrary, in organic materials the optical nonlinearity is a molecular property arising from the asymmetry of the electronic charge distributions in the ground and excited states of the individual molecules. Thus, a large figure of merit (an increase of a factor of ten over inorganics) can be achieved in organic materials. Another major advantage of organics over inorganics is the ease of fabrication and processing of polymers and their compatibility with integrated circuit processing techniques. The polymers can be easily fabricated in the form of thin films of high optical quality and can be modified by means of chemical doping. Three main classes of polymers have emerged as the most promising materials in optoelectronics. They are: (1) guest– host systems, (2) side-chain systems, and (3) cross-linked systems (30). In guest–host systems, a nonlinear chromophore is dissolved in a host polymer without any chemical bond between the dye and the polymer. The dye concentration in the polymer determines its optical quality and electro-optic efficiency. Since the dyes are disordered in the host system, a center of symmetry exists and no second-order response is possible. However, the dye molecules can be aligned in an intense static electric field at a temperature above the glass transition temperature of the polymer, and a second-order effect can be generated. Unfortunately, the electro-optic coefficient has so far been less than 10 pm/V. The electro-optic coefficient can be enhanced by creating a chemical attachment or a bond between the dye molecule and the polymer molecule. This can be accomplished either axially (main-chain) or on a side of the polymer molecule (side-chain). The latter configuration has better orientation stability, leading to stable optical properties.

The principal optical properties of materials are related to the phenomena of absorption, reflection, refraction, transmission, dispersion, polarization, emission, and birefringence. These properties are the (1) refractive index n, (2) complex dielectric function ⑀, (3) extinction coefficient k, (4) absorption coefficient 움, (5) reflection coefficient R, (6) transmission coefficient T, (7) electro-optic coefficient rij, (8) dispersion coefficient d␭, and (9) Abbe number vd, which is a measure of the chromatic aberration in an optical material. These optical phenomena and their relationships to the optical properties of materials are described in detail below. Absorption When light interacts with matter, the matter may partially absorb the light, partially transmit the light, and partially reflect the light. The proportions of light absorbed A, transmitted T, and reflected R, are essentially determined by the refractive index and extinction coefficient of the material. Our current understanding of this interaction of light with matter is based on Maxwell’s equations that describe light as electromagnetic waves. It is the variation of the magnitudes and directions of electric and magnetic field vectors in a medium with respect to space and time that allows us to determine the optical properties of the medium. In a one-dimensional case (assuming that light travels in one direction), the dependence of the electric field phasor on space and time is given as:
αx  E = E0 exp[ jω(nx/c − t)] exp − 2

(1)

In this equation, E is the electric field, E0 the amplitude, 웆 the frequency, n the real part of the refractive index, c the velocity of light in vacuum, t the time, 움 the absorption coefficient of the medium, and x the propagation direction. If 움 ⫽ 0 (no absorption), the wave propagates without any attenuation in the medium. If 움 ⬆ 0 (absorbing medium), the intensity of light, I, decays exponentially in the distance x: I = I0 exp(− αx)

(2)

where I0 is the intensity of light on the surface of the medium. The intensity I is related to the electric field E by I ⫽ E*E, where E* is the complex conjugate of E. The absorption coef-

302

OPTICAL MATERIALS

of free space, m*n the effective mass of a conduction electron, and ␶ the relaxation time. Separating the complex dielectric function into its real and imaginary parts yields

107

106 Absorption coefficient (cm–1)

1 = n2 − k2 = ∞ − 10

GaAs

5

Ge

 ω2 +

1 τ2

−1 (5)

and

104

2 = 2nk =

Si 103

Nq2 0 m∗n ωτ

 ω2 +

1 τ2

−1 (6)

The electron gas plasma frequency is defined as the value of 웆 for which ⑀1 ⫽ 0. Solving Eq. (5) for 웆 ⫽ 웆p, the plasma frequency, we obtain

102 300 K

ωp2 =

77 K

101

100 0.6 0.8

Nq2 0 m∗n

1

2 3 hν (eV)

4

5

λp =

Figure 5. Measured absorption coefficients near and above the fundamental absorption edge for pure Ge, Si, and GaAs (10).

ficient 움 is related directly to the extinction coefficient as 움 ⫽ 4 앟 k/ ␭. Short-Wave Cutoff—Bandgap Effects. The cause of the shortwavelength cutoff in semiconductors, shown in Fig. 5, is the bandgap Eg of the semiconductor. The cutoff wavelength is related to the bandgap by a simple equation (3)

where ␭c is the cutoff wavelength in micrometers and Eg the energy bandgap in electron volts. As seen in Fig. 5, ␭c shifts to the left (higher values) as Eg of the material decreases. In degenerately doped semiconductors the lowest energy levels of the conduction band are completely filled. This gives an effective bandgap larger than that of the intrinsic material. The shift in ␭c, known as Burstein–Moss shift, is directly related to the free electron concentration, since the free electrons block the lowest states of the conduction band. Long-Wave Cutoff—Electron Gas Effects. Longer-wavelength photons of incident radiation are usually absorbed by lattice ions in resonance with incident radiation. However, in metals and conducting oxides the photons interact with the electron gas. Drude developed the theory of this interaction. According to this theory, the complex dielectric function of a medium, ⑀*, is expressed in terms of the refractive index n and extinction coefficient k as shown below:  ∗ = (n − jk)2 = 1 − j2 = ∞ −

Nq2 0 m∗n

 ω2 −

jω τ

(7)

where ⑀s ⫽ ⑀앝⑀0. The electron gas plasma frequency is converted to wavelength in micrometers using

6 9 8 9 10

λc = 1.24/Eg

Nq2 1 − 2 s m∗n τ

−1 (4)

Here ⑀1 and ⑀2 are the real and imaginary parts of the dielectric function, ⑀앝 the static dielectric function, N the free electron concentration, q the electron charge, ⑀0 the permittivity

1.24q ~ω p

(8)

Here ប is h/2앟, where h is Planck’s constant. From Eqs. (7) and (8) it is seen that the plasma wavelength cutoff ␭p depends on the material properties (e.g., ⑀s, m*n , and ␶) and the doping properties (e.g., N). The values of m*n and ␶ do depend slightly on the doping concentration. Reflection Two important laws govern reflection of light at an interface: (1) the angle of incidence is equal to the angle of reflection, (2) the incident ray, reflected ray, and normal to the interface all lie in the same plane. Based on the classical theory, the reflectivity R of a material having an interface with air (vacuum) under normal incidence conditions is expressed as R=

E r |2 |E (n − 1)2 + k2 = 2 E i| |E (n + 1)2 + k2

(9)

where 兩Er兩 and 兩Ei兩 are the magnitudes of the reflected and incident electric field vectors respectively. For nonconducting materials (k ⫽ 0), R given in Eq. (9) reduces to R=

(n − 1)2 (n + 1)2

(10)

For conducting materials the reflection is approximately given as (35)  R≈1−

2ωs πσ

1/2 (11)

where ␴ is the conductivity of the material. Hence metals with conductivities of the order of 108 ⍀⫺1 ⭈ m⫺1 reflect over 90% of the incident light at visible frequencies (웆 앑 1015 rad/s). Total Internal Reflection. A very interesting phenomenon called total internal reflection occurs whenever light is inci-

OPTICAL MATERIALS

dent at an angle greater than the critical angle, ␪c, from a medium of higher refractive index, n1, into a medium of lower refractive index, n2. The critical angle, ␪c, is given as: θc = sin−1 (n2 /n1 )

(12)

This concept is used in guiding light through optical fibers. A cladding material of higher refractive index than that of optical fiber core enables light to be guided along the fiber by total internal reflection mechanism. The details on optical fibers can be found in a section on glass. Refraction When electromagnetic waves (light) pass from one medium of refractive index, n1, to a second medium of refractive index, n2, the direction and speed of the wavefronts change. This phenomena is known as refraction. According to Snell’s law: sin θi / sin θr = n2 /n1 = v1 /v2

(13)

where ␪i is the angle of incidence, ␪r the angle of refraction, v1 the speed of light in medium 1 (c/n1) and v2 the speed of light in medium 2 (c/n2). Since no refraction occurs in vacuum, its refractive index is set equal to one. The index of refraction of air is 1.0003 and hence it is used as a reference standard to determine the refractive indices of optical materials. Transmission The ability to transmit information via optical fibers rather than a metallic cable has revolutionized the modern communication technology. The optical fibers made of silica (SiO2) have extremely low optical attenuation of less than 0.5 dB/ km. The transmission capacity of these fibers is increased by increasing the bit rate (⬎109 bits/s). The reflection and transmission coefficients of a material are dependent on the polarization of the light and are given in the next section.

negative quantity. Circular polarization occurs only if the ratio E02 /E01 is equal to one and the phase difference ␾b ⫺ ␾a ⫽ ⫾앟/2 radians. The wave represented by Eq. (14) is elliptically polarized if it is neither linearly nor circularly polarized. Many light sources in real life (e.g., sunlight or light from an electric lamp) are unpolarized (randomly polarized). A glare (reflection of sunlight from the surface of water or a sheet of metal) is partially polarized. In the automobile industry, design and implementation of glareless headlight systems has been a major task. Orthogonal polarizations have been considered in the communication industry to double the capacity of a fixed frequency band allocated to a radio or a television station. Reflection and Transmission Coefficients. The reflection and transmission coefficients are defined for a parallel polarized wave (the direction of electric field vector is parallel to the plane of incidence) and a perpendicularly polarized wave (the direction of electric field vector is perpendicular to the plane of incidence). For a parallel polarized wave they are (36) R1 =

(n2 /n1 ) cos θi − cos θt (n2 /n1 ) cos θi + cos θt

(15a)

T1 =

[2(n2/n1 ) cos θi ] (n2 /n1 ) cos θi + cos θt

(15b)

Here it is assumed that the two media have identical permeabilities (애1 ⫽ 애2). The angles ␪i and ␪t are the angles of incidence and transmission respectively. ␪t is obtained from Snell’s law of refraction [Eq. (13)]. Brewster’s Angle. For a certain angle of incidence ␪i ⫽ ␪b, called Brewster’s angle in the case of a parallel polarized wave, the reflection coefficient R1 given by Eq. (15a) is zero. For this particular angle of incidence the wave is totally transmitted. This angle is determined by the refractive indices of the two media and is expressed as θb = tan−1 (n2 /n1 )

Polarization Polarization is another important property of light that can be exploited for practical applications. Polarization is defined as the locus of the tip of the electric field vector as a function of time. Depending on the light source and optics involved, light can be linearly polarized (locus is a straight line), circularly polarized (locus is a circle), or elliptically polarized (locus is an ellipse). According as the motion of the tip of the electric field vector is clockwise or counterclockwise, circular and elliptical polarization are described as left-handed or righthanded. Mathematically, for a wave traveling in the z direction the x and y components of the electric field are given as Ex = E01 cos(ω t − kz + φa )

(14a)

Ey = E02 cos(ω t − kz + φb )

(14b)

Here E01 and E02 are the amplitudes of Ex and Ey respectively, ␾a and ␾b are the phases associated with the two components, and k is the magnitude of the wave vector, equal to 2앟/ ␭. The requirement for linear polarization is ␾b ⫺ ␾a ⫽ 0 or 앟 radians. In this case the ratio of Ey to Ex is a fixed positive or

303

(16)

A gas laser fitted with a Brewster’s-angle window produces linearly polarized light. Details can be found in Ref. 36. For many optical materials and devices, the polarization of light is important because the polarization state of light is manipulated and controlled by the material properties. Electronic Polarization. The velocity of a high-frequency electromagnetic wave (visible light) is reduced in a material because of the electronic polarization (electrons respond to the rapidly varying electric and magnetic fields, whereas ions cannot respond). The high-frequency dielectric constant of the material is given by ⑀s ⫽ n2. It is possible to relate the index of refraction of a crystal to the physical properties of the crystal as shown below (37): N β M n2 − 1 = 0 ρ n2 + 2 30

(17)

Here N0 is the Avagadro’s number, M the molecular weight, ␳ the density, and 웁 the polarizability of a molecule. In summary, most materials are polarizable in different ways according to the frequency of the applied electric and

304

OPTICAL MATERIALS

Cu, Ag, and Mn to emit the primary additive colors red, green, and blue. Three electron beams are swept together to excite appropriate phosphor dots to produce a broad range of colors by addition of red, green, and blue.

0 2

Radio

Microwave

Visible

f

Electroluminescence. Light-emitting diodes (LED) and lasers consist of pn diodes where injection of minority carriers by an electric current results in light emission when the minority carriers recombine with the majority carriers. A detailed description of light-emitting semiconductors has been given in the section on semiconductors.

f

Birefringence

1

Figure 6. Typical variation of ⑀1 and ⑀2 with frequency (26).

magnetic fields. An important medium, water, has ⑀s ⫽ 80 at radio frequencies, but its refractive index is not equal to (80)1/2 as expected, but 1.3. At optical frequencies ⑀s 앒 2, indicating contribution only from the electronic polarization. At low frequencies, the orientational polarization of a water molecule is dominant because the ions respond to the slowly varying fields. Typical variation of the real and imaginary parts of the complex dielectric function given in Eqs. (5) and (6) as a function of frequency is shown in Fig. 6. As seen in this figure, the peaks at different frequencies in ⑀2 correspond to absorption by the medium due to either orientational or electronic polarizations. A discontinuity observed in ⑀1 at the same frequencies is also a result of the polarization of the medium. These frequencies correspond to natural resonance frequencies of the system, corresponding to rotational, vibrational, or electronic energy transitions in the system. Materials such as glass that transmit visible light usually absorb strongly in the ultraviolet and infrared regions because of the time difference between the induced and applied fields. Emission The general phenomenon of light emission (energy given out) by an electronic transition from a higher energy state to a lower energy state is called luminescence. Depending upon the excitation process, the emission can be photoluminescence (free carriers are excited by photon absorption) or cathodoluminescence (free carriers are excited by high-energy electron bombardment) or electroluminescence (free carriers are excited by passing a current). If the crystal emits light almost simultaneously (within 10⫺8 s), the emission is called fluorescence. At the other extreme, if it takes more than a microsecond to emit light, the emission is called phosphorescence and the materials are called phosphors. In direct bandgap semiconductors such as GaAs, the light emission is fluorescent and the emission wavelength is determined by the bandgap of the material [see Eq. (3)]. In indirect bandgap materials such as zinc sulfide (ZnS), the wavelength of light (i.e., its color) depends on the impurity levels present. Cathodoluminescence. Display systems such as cathode ray tubes (CRT) in television sets or flat panel display systems in a laptop computer make use of a high-energy electron beam to excite electrons in a phosphor-coated screen. As an example, in color television, the screen is coated with ZnS, a phosphor. The ZnS is doped with different impurities such as

Tetragonal and hexagonal crystals do not have the symmetry of the cubic crystals and hence possess anisotropic properties (the refractive index changes with direction inside the crystal) except for the c axis, which is called the optic axis. Ordinary light entering these anisotropic crystals in a direction other than that of the optic axis is split into two linearly polarized beams, which travel through the crystal in different directions. This phenomenon, known as birefringence or double refraction, is observed in many transparent crystals, such as calcite and quartz. As shown in Fig. 7, one ray, called the ordinary ray or O ray, traverses the crystal obeying Snell’s law, whereas the other ray, called the extraordinary ray or E ray, violates Snell’s law. Irrespective of the polarization of the incident light, the E ray has its electric field variations in a plane containing the optic axis of the crystal and the incident ray. The O ray is always polarized in a direction perpendicular to the E ray. The velocities of the two rays are different, depending upon the refractive indices in two different directions. Thus, a phase difference of 90⬚ or 180⬚ between the rays can be created by choosing an appropriate thickness of the crystal. This is the basis of electro-optic modulation, described in the next subsection. Isotropic materials are described by only one relative permittivity, i.e., ⑀r. The refractive index n is equal to ⑀1/2 r . For anisotropic media, the relative permittivity is a dielectric tensor with nine components. The diagonal terms, ⑀r1, ⑀r2, and ⑀r3, are known as the principal dielectric permittivities. In anisotropic materials, if ⑀r1 ⫽ ⑀r2 ⬆ ⑀r3, the material is called uniaxial, whereas if ⑀r1 ⬆ ⑀r2 ⬆ ⑀r3, it is called biaxial. In uniaxial materials, if ⑀r3 ⬍ ⑀r1, the material is said to be negative uniaxial (e.g., LiNbO3), and if ⑀r3 ⬎ ⑀r1, the material is said to be positive uniaxial (e.g., quartz, LiTaO3, and TiO2). A very interesting application of birefringence is in liquid crystals, used extensively as display devices. There are three types of liquid crystal structures: nematic, cholesteric, and smectic. The molecules of these materials are easily polariz-

Optic axis (c axis)

Incident light E ray O ray Figure 7. O ray and E ray inside a crystal (37).

OPTICAL MATERIALS

able under the influence of an electric field. By aligning the molecules in an appropriate manner it is possible to either allow light of a particular polarization to pass through (creating a bright spot) or block it (creating a dark spot). Many material- and device-related challenges remain in the development of liquid crystal display devices. Materials that possess low threshold voltages, high nonlinear responses, high stability, high resistivity, and low absorption are being developed.

305

Table 4. Quadratic Electro-Optic Coefficient g11 ⴚ g12 for Various Polar Materials (1) g11 ⫺ g12 (m4⭈C⫺2)

Material BaTiO3 LiNbO3 LiTaO3 PLZT ceramics Ba2NaNb5O15 SrTiO3 K(TaNb)O3

0.14 0.12 0.14 0.010 to 0.018 0.12 0.14 0.16 to 0.20

Electro-Optic Effects Two important applications of optical materials—high-speed modulators and second-harmonic generators—make use of the electro-optic effects observed in many crystals. There are two electro-optic effects: (1) the Pockels effect, (2) the Kerr effect. Pockels Effect. For a given material, the application of an electric field E alters the dielectric permittivity tensor (nine components) in a linear fashion. A set of 27 components rij that satisfy the equation shown below completely determine these electro-optic coefficients: (1/r )i = ri j E j

(18)

Only certain components rij are large in magnitude and are exploited in practical applications. Typical rij values are from 1 ⫻ 10⫺12 m/V to 35 ⫻ 10⫺12 m/V; values for many ferroelectrics are given in Table 1. For example, in LiNbO3, one of the ferroelectric oxides, r33 ⫽ 3.08 ⫻ 10⫺11 m/V (26). From Eq. (18), using ⑀r ⫽ n2, one can obtain an expression for the change in refractive index, ⌬n, as n = − 12 n3 r33 E

(19)

Using n ⫽ 2.29 and E ⫽ 106 V/m, we have ⌬n ⫽ 1.86 ⫻ 10⫺4. This small change in the refractive index causes a change in the polarization of the material, leading to intensity modulation. Kerr Effect. The quadratic electro-optic effect is also known as the Kerr effect. Here the change in refractive index is given as ni j = Si jkl Ek El

(20)

The second-order coefficients Sijkl are significantly enhanced when the energy of the optical signal is close to the bandgap energy of the material. In simple terms,

the difference between g11 and g12. The quadratic electro-optic coefficients for various ferroelectrics are given in Table 4 (1). Instead of the electric field dependence as shown in Eq. (20), the Kerr effect is easier to express in terms the intensity of light as (37) n = n0 + n2 I

(22)

where I is the intensity of the light, n0 the refractive index without the light, and n2 the refractive index due to the nonlinear effect. For waveguide modulation based on the Kerr effect an appropriate figure of merit is n2I/ ␭움, where ␭ is the wavelength and 움 the absorption coefficient. The Kerr effect is mostly observed in noncentrosymmetric crystals such as AgGaS2, Ti3AsSe3, and polymers that contain asymmetric molecules. Other Nonlinear Effects. The development of the laser has led to the study of nonlinear optics. Nonlinear light scattering, (e.g., Rayleigh, Brilliouin, and Raman scattering), though known before the advent of lasers, became more pronounced with the high intensity of the optical field of the laser. Thirdorder nonlinear effects are observed in centrosymmetric materials with high density of polarizable electrons. Hence, metals and organic molecules with extended electron wavelengths show strong third-order nonlinear effects (38). These materials find applications in liquid crystal displays, high-speed optical gates, real time holography, and optical transistors. Acousto-Optic Effects An acoustic wave propagating in a dielectric medium can introduce strain in the medium, causing variations in its refractive index. With this principle, acoustical holograms have been fabricated in display systems made of nematic liquid crystal panels. These devices are used in medical diagnostics. An acoustically tunable laser and acousto-optical amplifying tunable filter have been demonstrated using LiNbO3 (Ref. 2, p. 328). This concept can be used in the modulation of an optical beam as well. A list of materials used for acousto-optic devices is found in Ref. 26, pp. 383–386. OPTICAL COATINGS

n =

− 12 g n 3 P2

(21)

where P is the magnitude of polarization, and g, a fourthrank tensor, is proportional to the Sijkl of Eq. (20). For most materials the largest quadratic electro-optic effect is due to

Optical coatings or thin films of thickness 10 nm to 1000 nm deposited by evaporation or sputtering techniques in highvacuum systems have wide applications. These applications include transparent conducting coatings, antireflective coatings, reflective coatings (cold light and heat mirrors), and a

306

OPTICAL MATERIALS Table 5. Physical Properties of Transparent Conducting Oxides (39)

Material

Electron Mobility (cm2 /V · s)

Electron Carrier Concentration (cm⫺3)

Resistivity (⍀ · cm)

Bandgap (eV)

CdO Cd2SnO4 In2O3

2 to 120 8 to 73 15 to 70

5 ⫻ 1016 to 1 ⫻ 1021 1 ⫻ 1017 to 1 ⫻ 1021 1 ⫻ 1019 to 2 ⫻ 1021

5 ⫻ 10⫺4 5 ⫻ 10⫺4 1 ⫻ 10⫺2 to 2 ⫻ 10⫺4

SnO2 ZnO

10 to 50 앒15

1 ⫻ 1018 to 1 ⫻ 1021 앑5 ⫻ 1020

1 ⫻ 10⫺1 to 4 ⫻ 10⫺4 앒8 ⫻ 10⫺4

2.3 to 2.7 2.06 to 2.85 3.6 to 3.85 (direct) 2.6 to (indirect) 3.97 to 4.63 3.3

Properties of Optical Coatings For light incident from air or vacuum (n ⫽ 1) onto a partially absorbing film of thickness t, refractive index nf and extinction coefficient kf in contact with a thick, isotropic, weakly absorbing substrate with refractive ns and extinction coefficient ks, the coefficients T and R are given as (39)

R=

16ns (n2f + k2f ) + b2 e−α + b3 cos η + b4 sin η

(23)

a1 eα + a2 e−α + a3 cos η + a4 sin η b1 eα + b2 e−α + b3 cos η + b4 sin η

(24)

b1

eα

α = 4πkf t/λ,

η = 4πnf t/λ

2.0 n

ITO In2O3

1.0

k 0.0 0.1

0.1 Wavelength ( µm)

n 2.0

(25)

The coefficients a1, a2, a3, a4, b1, b2, b3, b4 are essentially dependent on nf , kf , ns, and ks, and the algebraic expressions can be found in Ref. 40. The optical constants are strongly dependent on the wavelength, making the algebraic manipulation of Eqs. (23) and (24) quite difficult. However, some generalizations can be made. The index of refraction of most substrates is in the range of 1.4 to 1.6, and extinction coefficient is quite small (39). This results in negligible absorption by the substrate. Oxide films have higher refractive indices (앒2.0) and low extinction coefficients. These types of conditions lead to reflection being the dominant source of light loss. Transparent Conducting Coatings Transparent conducting oxides, or TCOs, are a special kind of materials that are transparent to visible light and conducting as well. However, their conductivities lie in between the conductivities of metals and dielectrics. Such materials include cadmium oxide (CdO), cadmium stannate (Cd2SnO4), indium

Optical constants n, k

T=

oxide (In2O3), tin oxide (SnO2), and zinc oxide (ZnO). These materials are also doped at cation sites with Sb, Cd, In, P, Te, Sn, Ti, or W and/or at anion sites with C1 or F to enhance their conductivities (39). Table 5 shows the physical properties of transparent conducting oxides (41). Indium oxide or indium oxide doped with tin (ITO) has been the material of choice for a variety of applications requiring transparent elec-

Optical constants n, k

variety of filters. Deposition process parameters such as the deposition rate, substrate temperature, and pressure during deposition significantly affect the microstructure (grain size, grain orientation, and grain morphology) of the thin films and hence the optical properties of the coatings. It is basically the electronic structure of the atoms or molecules that determines the optical properties of thin films. However, optical properties of thin films differ from those of bulk materials because of structural imperfections that dominate in thin films. These imperfections are voids, microcracks, pinholes, and dislocations. The transmittance T and reflectance R of these thin films are primarily determined by the refractive index nf and the extinction coefficient kf . Based on electromagnetic theory, expressions for T and R in terms of nf and kf are obtained in the following subsections.

ITO In2O3

1.0

k 0.0

0.2

0.5 1.0 2.0 2.5 Wavelength ( µm)

Figure 8. (a) Estimates of the refractive index and extinction coefficient for In2O3 and ITO; (b) measured refractive index and extinction coefficient for In2O3 and ITO (49)

OPTICAL MATERIALS

100

(a) 100 T

80

R

A

Transmission (%)

T, R, A (%)

80 60 40 20 0

Experimental results 60 40 20 Estimated results

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Wavelength ( µ m)

0 300

(b) 100

400

600 500 Wavelength ( µ m)

700

800

Figure 10. Measured and estimated transmission for an annealed ITO sample (44)

90 80

307

T

R

60

A

50 40 30 20 10 0

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Wavelength ( µ m)

Figure 9. (a) Estimates of the transmission, reflection, and absorption of an ITO thin film; (b) published results on transmission, reflection, and absorption of spray-deposited ITO film (50)

trodes. The electrical and optical properties of ITO thin films have been investigated in detail to optimize simultaneously the conductivity and the transparency (40–42). Based on an ab initio Hartree–Fock calculation of the energy band diagram of In2O3 and ITO, an empirical relationship of the effective electron mass as a function of the ideal free carrier concentration is established (42). The importance of the varying electron effective mass in the prediction of complex dielectric function, the refractive index, extinction coefficient, and the optical transmittance based on Drude’s theory is shown by comparing the estimated results with measured and published results (42). Figures 8(a) and (b) show the estimated and measured refractive index and extinction coefficient for In2O3 and ITO. Figure 9(a) shows estimates of the transmission, reflection, and absorption of an ITO film with the following properties: thickness 0.75 애m, carrier concentration 4.9 ⫻ 1020 cm⫺3, electron effective mass 0.413m0, mobility 5.15 cm2 /V ⭈ s, resistivity 앒2.5 ⫻ 10⫺4 ⍀ ⭈ cm, and average transmittance 0.80 to 0.90 over wavelengths from 0.4 애m to 1.0 애m (40). These estimated results can be compared with the measured data shown in Fig. 9(b). The transmission and reflection are calculated using Eqs. (23) and (24). The absorption A is determined by the equation A=1−T −R

(26)

Figure 10 shows the measured and estimated transmission for an annealed ITO sample (40). The calculated figures of merit of ITO thin films based on estimates of transmission and conductivity are shown in Fig. 11 (43). The optimum carrier concentrations predicted from all figures of merit range from 앒6 ⫻ 1020 cm⫺3 to 앒1.6 ⫻ 1021 cm⫺3. This range is well within the practical doping limits of ITO thin films (44). Recently, ITO thin films deposited on glass and polymer substrates by RF sputter deposition are characterized to correlate the sheet resistivity and transmission of these films with their microstructure (45).The refractive index and extinction coefficients are related to the microstructure of a thin film via the packing density, defined as the ratio of the volume of the solid part of the film (i.e., columns) to the total volume of the film (46). Typical values of packing density are 0.7 to 1.0 for optical films. The transmission and reflection are then related to the microstructure via n and k, using Eqs. (23) and (24). The refractive index n decreases with decreasing film density. The density is a function of the porosity of the material depending upon the content of solid material, e.g., grains and voids. Deposition process parameters such as substrate tem-

1.0

0.8 Normalized units

T, R, A (%)

70

0.6

0.4

Transmission, T Conductivity, σ Tσ T(i) σ σ /(1 – T) – σ /In T

0.2

0.0 1018

1019 1020 1021 –3 Carrier concentration (cm )

1022

Figure 11. Calculation of the figures of merit of ITO thin films based on estimates of transmission and conductivity

308

OPTICAL MATERIALS

Table 6. Optical Properties of Thin Films (50) Material NaF LiF CaF2 Na3AlF6 AlF3 MgF2 ThF4 LaF3 CeF3 SiO2 Al2O3 MgO Y2O3 La2O3 CeO2 ZrO2 SiO ZnO TiO2 ZnS CdS ZnSe PbTe

Refractive Index 1.29 1.30 1.23 1.32 1.23 1.32 1.50 1.55 1.63 1.45 1.54 1.7 1.89 1.98 2.2 1.97 2.0 2.1 1.9 2.3 2.5 2.57 5.6

to 1.30 to 1.46 to 1.35 to 1.39

to 1.46

Transmittance Range (애m) ⱖ0.2 0.11 to 7 0.15 to 12 0.2 to 14 ⱖ0.2 0.11 to 4 0.2 to 15 0.25 to 2 0.3 to 5 0.2 to 9 0.2 to 7 0.2 to 8 0.3 to 12 ⱖ0.3 0.4 to 12 0.34 to 12 0.7 to 9 ⱖ0.4 0.4 to 3 0.4 to 14 0.55 to 7 0.55 to 15 3.5 to 20

perature and deposition rate affect the grain growth during e-beam or sputter deposition and need to be controlled carefully to maximize n and minimize k. The purity of the materials used and the cleanliness of the vacuum system affect the optical properties of thin films significantly (47). Table 6 shows the optical properties of several thin film materials. Details on transparent conducting coatings can be found in Refs. 39 to 45, 47, 49, and 50. An excellent reference for optical properties of thin films is Ref. 48. BIBLIOGRAPHY 1. J. C. Burfoot and G. W. Taylor, Polar Dielectrics and their Applications, Berkeley: University of California Press, 1979. 2. J. Saulnier, Lithium niobate for optoelectronic applications. In M. Quillec (ed.), Materials for Optoelectronics, Boston: Kluwer Academic, 1996. 3. M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectric Materials, Oxford: Clarendon, 1977. 4. C. R. M. Grovenor, Microelectronic Materials, Adam Hilger, 1989. 5. J. Singh, Optoelectronics, New York: McGraw-Hill, 1996. 6. B. G. Streetman, Solid State Electronic Devices, 4th ed., Englewood Cliffs, NJ: Prentice-Hall, 1995. 7. M. A. Pollack, Advances in materials for optoelectronic and photonic integrated circuits, Materials Sci. & Eng. B, B6: 233–245, 1990. 8. S. M. Sze (ed.), High Speed Semiconductor Devices, New York: Wiley, 1990. 9. P. Bhattacharya, Semiconductor Optoelectronic Devices, Englewood Cliffs, NJ: Prentice-Hall, 1994. 10. S. M. Sze, Physics of Semiconductor Devices, 2nd ed., New York: Wiley, 1981. 11. R. M. Brody and V. J. Mazurczyk, In R. K. Willardson and A. C. Beer (eds.), Semiconductors and Semimetals, 18, New York: Academic, 1981.

12. H. Fritzsche, Noncrystalline semiconductors, Phys. Today, 37 (10): 34–43, 1984. 13. J. L. Stone, Photovoltaics: Unlimited electrical energy from the sun, Phys. Today, 46 (9): 22–31, 1993. 14. S. R. Forrest, Optoelectronic integrated circuits, Proc. IEEE, 75: 1488–1497, 1987. 15. T. E. Bell, Electronics and the stars, IEEE Spectrum, 32 (8): 16– 24, 1995. 16. C. Weisbuch, in R. Dingle (ed.), Semiconductors and Semimetals, 24, New York: Academic, 1987. 17. R. J. Collins, P. M. Fauchet, and M. A. Tischler, Porous silicon: From luminescence to LEDs, Phys. Today, 50 (1): 24–33, 1997. 18. MRS Bull., 22 (2): 17–57, 1997. 19. T. P. Pearsall, Silicon/germanium optoelectronic materials. In M. Quillec (ed.), Materials for Optoelectronics, Boston: Kluwer Academic, 1996. 20. R. F. Davis, Diamond Films and Coatings, Park Ridge, NJ: Noyes Publications, 1993. 21. M. A. Capano and R. J. Trew, Silicon carbide electronic materials and devices, MRS Bull., 22 (3): 19–24, 1997. 22. R. L. Gunshor, The wide band gap II–VI semiconductors. In M. Quillec (ed.), Materials for Optoelectronics, Boston: Kluwer Academic, 1996. 23. W. S. Ellis, Glass: Capturing the dance of light, National Geographic, 184 (6): 37–69, 1993. 24. S. Musikant, Optical Materials, New York: Marcel Dekker, 1985. 25. P. C. Becker and M. R. X de Barros, Optical fibers for telecommunications: Transmission and amplification. In M. Quillec (ed.), Materials for Optoelectronics, Boston: Kluwer Academic, 1996. 26. L. Solymar and D. Walsh, Lectures on the Electrical Properties of Materials, 4th ed., New York: Oxford University Press, 1988. 27. J. S. Sanghera and I. D. Aggarwal, J. Non-crystalline Solids, 213, 214: 126–136, 1997. 28. J. Zarzyzki, Glass and Vitreous State, New York: Cambridge University Press, 1991. 29. S. P. Karna and A. T. Yeates, Nonlinear optical materials: Theory and modeling. In S. P. Karna and A. T. Yeates (eds.), Nonlinear Optical Materials, Washington, D.C.: American Chemical Society, 1996. 30. R. Levenson and J. Zyss, Polymer based optoelectronics from molecular optics to device technology. In M. Quillec (ed.), Materials for Optoelectronics, Boston: Kluwer Academic, 1996. 31. D. D. Nolte, Photorefractive Effects and Materials, Boston: Kluwer Academic, 1995. 32. Y. Yang, Polymer electroluminescent devices, MRS Bull., 22 (6): 31–38, 1997. 33. T. Tsutsui, Progress in electroluminescent devices using molecular thin films, MRS Bull., 22 (6): 39–45, 1997. 34. W. R. Slaneck and J. L. Bre’das, Conjugated polymer surfaces and interfaces for light-emitting devices, MRS Bull., 22 (6): 46– 51, 1997. 35. R. A. Colclaser and S. Diehl-Nagle, Materials and Devices, New York: McGraw-Hill, 1985. 36. L. C. Shen and J. A. Kong, Applied Electromagnetism, 3rd ed., Boston: PWS, 1995. 37. L. V. Aza’roff and J. J. Brophy, Electronic Processes in Materials, New York: McGraw-Hill, 1963. 38. D. Bloor, Nonlinear optical materials, an overview. In D. Bloor, R. J. Brook, M. C. Flemings, and S. S. Mahajan (eds.), Encyclopedia of Advanced Materials, New York: Pergamon, 1994. 39. J. L. Vossen, Transparent conducting films. In G. Hass, M. H. Francombe, and R. W. Hoffman (eds.), Physics of Thin Films, New York: Academic, 1977.

OPTICAL NEURAL NETS 40. S. A. Knickerbocker and A. K. Kulkarni, Estimation and verification of the optical properties of indium tin oxide based on the energy band diagram, J. Vac. Sci. Technol., A14 (3): 757–761, 1996. 41. K. L. Chopra and S. R. Das, Thin Film Solar Cells, New York: Plenum, 1983. 42. A. K. Kulkarni and S. A. Knickerbocker, Estimation and verification of the electrical properties of indium tin oxide based on the energy band diagram, J. Vac. Sci. Technol., A14 (3): 1709– 1713, 1996. 43. S. A. Knickerbocker and A. K. Kulkarni, Calculation of the figure of merit for indium tin oxide films based on basic theory, J. Vac. Sci. Technol., A13 (3): 1048–1052, 1995. 44. S. A. Knickerbocker, Estimation and verification of the electrical and optical properties of indium tin oxide based on the energy band diagram, Ph.D. Dissertation, Michigan Technological University, 1995. 45. A. K. Kulkarni et al., Electrical, optical and structural characteristics of indium tin oxide thin films deposited on glass and polymer substrates, Thin Solid Films, 308–309: 1–7 (1997). 46. H. A. Macleod, Relationship of microstructure to optical properties of thin films, presented at Topical Conference on Basic Optical Properties of Materials, 1985. 47. T. S. Lim, Electrical, optical and structural properties of indium tin oxide thin films deposited on glass, PET and polycarbonate substrates by RF sputtering, M.S. Thesis, Michigan Technological University, 1997. 48. M. Ohring, The Materials Science of Thin Films, New York: Academic, 1992. 49. K. L. Chopra, S. Major, and D. K. Pandya, Transparent conductors: A status review, Thin Solid Films, 102: 1–46, 1983. 50. T. Nagatomo, Y. Maruta, and O. Omoto, Electrical and optical properties of vacuum-evaporated indium–tin–oxide films with high electron mobility, Thin Solid Films, 192: 17–25, 1990.

ANAND K. KULKARNI Michigan Technological University

309

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Wiley Encyclopedia of Electrical and Electronics Engineering Optical Properties Standard Article Olaf Stenzel1 and Alexander Stendal2 1Technische Universität Chemnitz, 09107 Chemnitz, Germany 2Technische Universität Chemnitz, 09107 Chemnitz, Germany Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W4406 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (184K)

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Abstract The sections in this article are Light Propagation in Classical Physics Selected Problems of Optical Characterization About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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OPTICAL PROPERTIES

Incoming light

327

Signal Sample

(Input)

(System)

(Output)

Figure 1. Optical signal as the result of interaction of an electromagnetic wave with the sample.

OPTICAL PROPERTIES Materials science may be characterized by a tremendous diversity of techniques, including mechanical characterization, electrical methods, chemical analysis, electron microscopy, and others. This article presents an introduction to the optical characterization of solid state materials.

In the framework of classical electrodynamics, any kind of light (which is used in optics) may be regarded as a superposition of electromagnetic waves. The idea of optical material characterization is quite simple: If we have an object to be investigated (a sample), we have to bring it into interaction with electromagnetic waves (light). As the result of the interaction with the sample, certain properties of the light will be modified. The specific modification of the properties of electromagnetic waves resulting from the interaction with the sample should give us information about the nature of the sample of interest. For sufficiently low light intensities, the interaction process does not result in sample damage. Therefore, the majority of optical characterization techniques belongs to the nondestructive analytical tools in materials science. This is one of the advantages of optical methods. Although the main idea of optical characterization is quite simple, it may be an involved task to turn it into practice. In fact, one has to solve two problems. The first one is of an entirely experimental nature: The modifications in the light properties (which represent our signal) must be detected. For standard tasks, this part of the problem may be solved with the help of commercially available equipment. The second part is more closely related with mathematics: From the signal (which may be simply a curve in a diagram) one has to conclude on concrete quantities characteristic for the sample. Despite the researcher’s intuition, this part may include severe computational efforts. Thus, the solution of the full problem requires the researcher to be skilled in experiment and theory as well. Let us now have a look at Fig. 1. Imagine the very simplest case—a monochromatic light wave impinging on a sample which is to be investigated. The parameters characterizing the incoming light (wavelength, intensity, polarization of the light, propagation direction) are supposed to be known. Further imagine that as the result of the interaction with the sample, we are able to detect an electromagnetic wave with modified properties. With properties of the electromagnetic wave may have changed as the result of the interaction with the sample? In principle, all of them may have changed. It is possible that the interaction with the sample leads to changes in the wavelength of the light. Typical examples are provided by Raman scattering, or by several nonlinear optical processes. The polarization of the light may change as well. Ellipsometric techniques detect polarization changes and use them to judge the sample properties. Clearly, the light intensity may change (in most cases the light will be attenuated). This gives rise to numerous photometric methods analysing the sample properties based on the measurement of intensity changes. And finally, we know that the refraction of light may lead to changes in the propagation direction. Any refractome-

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

328

OPTICAL PROPERTIES

law. In practice, a certain fraction of the light intensity is diffusely scattered. That leads us to the definition of the optical scatter S as the ratio of the intensity of the light participating in scattering processes and the incoming intensity. Analogously, we define the absorptance A as the ratio of the absorbed intensity and the incoming one. In the presence of absorption and scatter, the energy conservation law may be written as

S

T

ϕ

T +R+A+S=1

R Figure 2. The optical signal may be generated by transmitting (T), specularly reflecting (R), or scattering (S) the incident light. ␸ is the angle of incidence.

ter makes use of this effect to determine the refractive index of a sample. The diversity of parameters characterizing electromagnetic radiation (in practice they are more than those mentioned here) may give rise to quite diverse optical characterization techniques. Clearly, in the framework of this paper only a few of them may be discussed. Any practical situation is more complicated than that shown in Fig. 1. After interaction with the sample, light may leave the sample in several directions (Fig. 2). From the phenomenological point of view, the light may either be

(1)

So that these four quantities are not independent from each other, and accurate knowledge of three of them allows the fourth to be immediately calculated. Nevertheless, all four quantities T, R, S, and A may, in principle, be measured independently from each other. The algebraic sum of absorption and scatter is often called optical loss, and the measurement of the optical loss represents one of the most challenging experimental tasks of optical material characterization (2). Let us make a final remark in this context. The values T, R, S, and A are characteristic for a sample in specific experimental conditions. This means that both sample material and its geometry (including the experiment geometry) are responsible for the signal. If one is interested in the pure material properties, the geometrical influences on the signal have to be eliminated, experimentally or by calculations. Important Spectral Regions

• Transmitted through the sample (in a well-defined direction), or • Specularly reflected from the sample, or • Diffusely scattered at the sample surfaces or in its volume, or • Absorbed at the sample surfaces or in its volume

Table 1 provides an overview on important spectral regions and the origin of characteristic material absorption structures (3,4).

Let us for simplicity focus on the intensities of the signals. It is a common practice to define the transmittance T of the sample as the ratio of the intensity of the transmitted light and that of the incoming light (1). Accordingly, we define the specular reflectance R as the ratio of the specularly reflected intensity and the incoming one. If we deal with a sample that does neither diffusely scatter nor absorb the irradiation, then the thus defined transmittance and reflectance must sum up to the value one—simply as a result of the energy conservation

In terms of classical physics, the propagation of electrodynamic waves may be described starting from Maxwell’s equations (5,6). In order to simplify the description here, we restrict ourselves to optically homogeneous and isotropic media. Furthermore, we will regard the medium as nonmagnetic. In this case, Maxwell’s equations lead to

LIGHT PROPAGATION IN CLASSICAL PHYSICS The Wave Equation in the Homogeneous and Isotropic Case

E ≡ graddivE E − E E = −µ0 rotrotE

∂2 D ∂t2

(2)

Table 1. Important Spectral Regions

Spectral Region Far Infrared (FIR) Middle Infrared (MIR) Near Infrared (NIR) Visible (VIS) Ultraviolet (UV) X-ray (X)

Wavenumber v v ⫽ 1/␭ (cm⫺1)

Angular Frequency 웆 웆 ⫽ 2앟vc (s⫺1)

Photon Energy W ⫽ hcv (eV)

10 ⫺ 200

1.9 ⫻ 1012 ⫺ 3.8 ⫻ 1013

1 ⫻ 10⫺3 ⫺ 2 ⫻ 10⫺2

Free carriers; orientation

200 ⫺ 4000

3.8 ⫻ 1013 ⫺ 7.5 ⫻ 1014

2 ⫻ 10⫺2 ⫺ 5 ⫻ 10⫺1

Free carriers; vibrations

2.5 ⫻ 103 ⫺ 8 ⫻ 102

4000 ⫺ 12500

7.5 ⫻ 1014 ⫺ 2.4 ⫻ 1015

5 ⫻ 10⫺1 ⫺ 1.6

8 ⫻ 102 ⫺ 4 ⫻ 102 4 ⫻ 102 ⫺ 10

12500 ⫺ 25000 25000 ⫺ 106

2.4 ⫻ 1015 ⫺ 4.7 ⫻ 1015 4.7 ⫻ 1015 ⫺ 1.9 ⫻ 1017

1.6 ⫺ 3.1 3.1 ⫺ 124

Free carriers; vibrational overtones Excitation of valence Electrons

Vacuum Wavelength ␭ (nm) 106 ⫺ 5 ⫻ 104 5 ⫻ 104 ⫺ 2.5 ⫻ 103

10 ⫺ 0.005

106 ⫺ 2 ⫻ 109 (unusual)

1.9 ⫻ 1017 ⫺ 3.8 ⫻ 1020

124 ⫺ 25000

Origin of Absorption (Examples)

Excitation of core electrons

OPTICAL PROPERTIES

P

P≈P(1)>>P(2), P(3)

Dispersion of Optical Constants

P(1)

Let us now postulate the final form of the relation Eq. (7), valid for linear optics. The polarization at the moment t in the point characterized by the vector r may be written as

P(2)

 P (t, r ) = 0

P(3)

0 Linear optics

Nonlinear optics

Ε

with E the vector of electric field strength, and D that of the induction. Per definition, the induction is connected with the electric field via (3)

where P is the so-called polarization of the medium. It is defined as the dipole moment per volume. From the viewpoint of optical material characterization, it plays the key role here because it contains the specific optical properties of the medium where the electromagnetic wave will propagate. Combining Eq. (2) and Eq. (3), we obtain E − E E + µ0  0 graddivE

∂2 ∂2 E = −µ0 2 P 2 ∂t ∂t

(4)

Equation (4) completely describes the propagation of electromagnetic waves in a homogeneous, isotropic, and nonmagnetic medium. It is now essential to define a relationship between E and P to solve Eq. (4). Considering only induced (by the field) dipole moments, an arbitrary dependence P(E) may be formally written as Taylor’s sum (7,8): P = P (1) + P (2) + P (3) + · · ·

(5)

P(1) ∝ E, P(2) ∝ E 2 , P(3) ∝ E 3 , . . .

(6)

with

and so on. The principal E-dependence of these terms is illustrated in Fig. 3 (9). From Fig. 3 it is essential to remark that for weak field strength values the polarization must be dominated by the first (linear) term in Eq. (5). In this field strength region, Eq. (4) may be solved regarding the simple dependence P ≈ P (1) ∝ E



∞

E (t − τ , r − R ) dR R κ (τ , R )E

dτ 0

(8)

V

Equation (8) may be understood in the following way: The polarization (at the time t in the point r), which represents the response of a system on the stimulating field, may depend on

Figure 3. The polarization of a medium as a function of the electric field strength contains linear as well as nonlinear contributions. For weak electric fields, the linear contribution is dominant, thus determining the field of linear optics.

D = 0 E + P

329

(7)

Then Eq. (4) describes the processes of the so-called linear optics. The more general case [Eqs. (5, 6)] is called nonlinear optics. In the forthcoming, we will focus on the linear case.

• The spatial field strength distribution in the volume V surrounding the point of interest • At all previous moments (causality principle) • The specifics of the medium, which are hidden in the response function ␬ The combination of Eq. (4) and Eq. (8) may be satisfied by a transversal propagating wave according to kr kr) E (t, r ) = E 0 e−i(ωt−kr

(9)

This may be checked substituting Eq. (8) and Eq. (9) into Eq. (4). One will then find that Eq. (9) indeed is a solution of Eq. (4) and Eq. (8), if the absolute value of the wavevector k is equal to √ 2π √ ω −k2 + µ0 0 ω2 = 0 ⇒ k = ±  = ±  c λ

(10)

where the value ⑀ is called the dielectric function of the medium and must satisfy 



∞

 = (ω, k ) = 1 +

kR kR) R κ (τ , R )ei(ωτ −kR dR

dτ 0

(11)

V

From our treatment it turns out that the introduced dielectric function • Has to be regarded as a complex function, and consequently has a real and an imaginary part • May depend on the frequency of the electric field. This effect is called time dispersion or frequency response • May depend on the wavevector k (and therefore on the wavelength of the light). The latter effect is called spatial dispersion or wavelength response While time dispersion occurs, when the light frequency comes close to a characteristic eigen-frequency of the medium investigated, spatial dispersion will be found when the light wavelength equals a characteristic spatial dimension of the medium. In materials science practice, spatial dispersion is often negligible, and we will concentrate on the time dispersion often simply called dispersion in the forthcoming section. Keeping in mind these remarks, we finally obtain from Eq. (9) √ ω √  c r −i (ωt−Re  ωc r )

E (t, r ) = E 0 e−Im

e

(12)

330

OPTICAL PROPERTIES

Ee

which represents an attenuated wave if the imaginary part of the square root of the dielectric function is positive. This leads us to the definition of the absorption coefficient 움: ω √ α(ω) = 2 Im (ω) c

√ n(ω) = Re (ω)

√ (ω)

(15)

Its real part describes the phase velocity of the wave (and consequently the refraction), and its imaginary part (often: extinction index) the attenuation. The frequency dependence of the optical constants is closely related to the frequency dependence of the corresponding sample spectra. Generally, in optical spectra theory, there exist two different tasks. The first one is the calculation of spectra, when the optical constants and the system geometry are known. This kind of calculation is called a forward search, and in many situations this may be accomplished in terms of explicit formulae. The second one is what we often have in practice: The spectra are known (they have been measured), and the optical constants have to be calculated. This so-called reverse search is a much more complicated procedure, because usually no explicit formulae allow the calculation of optical constants from the spectra. The typical way is to perform forward calculations with different assumed sets of optical constants, until one of the thus generated spectra fits the experimental one. In this sense the reverse search strategy bases on forward search calculations. It is therefore essential to start our further discussion with examples of spectra calculations. We will then describe a few basic experimental techniques before coming to the question of reverse search procedures. Interfaces Up to now we have dealt with the description of a light wave within a homogeneous and isotropic medium. As already suggested from Fig. 2, the surfaces and interfaces of the sample will however play an important role for transmission and reflection processes. The influence of an interface may be mathematically taken into account introducing electrical field transmission and reflection coefficients according to (see Fig. 4) t=

Et (z = 0) ; Ee (z = 0)

r=

ϕ ϕ x

0 Medium 2 N2

ψ Et z

Figure 4. Incident (e), transmitted (t), and reflected (r) beam at a smooth interface. The z-axis is perpendicular to the interface.

(14)

As a result of the dispersion of ⑀, both the refractive index and the absorption coefficient are frequency dependent. The refractive index and the absorption coefficient are a pair of so-called optical constants (characteristic for a given medium), that completely describe the process of wave propagation within a homogeneous and isotropic medium. Alternatively, often the complex refractive index N is defined N(ω) = n(ω) + iK(ω) =

Medium 1 N1

(13)

It is given in reciprocal centimeters, where its inverse value represents the penetration depth of light into matter. On the contrary, the real part of 兹⑀ affects the phase velocity of the light. We define the dimensionless refractive index n via

Er

Er (z = 0) Ee (z = 0)

(16)

In accordance with Fig. 4, ␸ is the incidence angle, and ␺ that of refraction. In the following, medium 1 will always mark the medium from where the light is incident. The transmission and reflection coefficients are found to be polarization-dependent, where the subscript p denotes parallel polarization (the electric field vector is linearly polarized in the plane of incident beam and surface normal, which is the x–z plane in our description), while subscript s marks perpendicular polarization (electric field vector perpendicular to incident beam and surface normal). In these terms, the transmission and reflection coefficients are given by Fresnel’s formulae (1):

p–Polarization: N cos ϕ − N1 cos ψ ; rp = 2 N2 cos ϕ + N1 cos ψ s–Polarization: N cos ϕ − N2 cos ψ rs = 1 ; N1 cos ϕ + N2 cos ψ

tp =

2N1 cos ϕ N2 cos ϕ + N1 cos ψ

ts =

2N1 cos ϕ N1 cos ϕ + N2 cos ψ

(17)

N sin ϕ = 2 sin ψ N1 The combination of Eq. (12) and Eq. (17) allows the calculation of light propagation through homogeneous samples with smooth surfaces. As intensities are proportional to the square of the absolute values of the complex field amplitudes, the transmittance, the reflectance, and the absorption become accessible to calculations, if only the geometry of the sample (and illumination geometry), and all optical constants are known. Thus, for example, one may calculate the transmission and reflection spectra of a thick absorbing wafer (medium 2) with parallel smooth surfaces, embedded in a nonabsorbing environment (medium 1), taking into account all multiple internal reflections. In fact this is again the situation as shown in Fig. 2, and it allows an exact calculation of transmittance and reflectance. As a result of summarizing the intensities of all internal waves, one obtains the expressions (9) ω

T=

q

|t12 |2 |t21 |2 e−2 c dIm

N 2 −sin2 ϕ

q

ω 2 −4 c dIm

1 − |r12 |2 |r21 | e

2

N 2 −sin2 ϕ 2

ω

R = |r12 | + 2

;

q 2 2 N −sin ϕ 2 q 2 2

|t12 |2 |r21 |2 |t21 |2 e−4 c dIm ω

1 − |r12 |2 |r21 |2 e−4 c dIm

(18)

N −sin ϕ 2

Here, the r and t values are the Fresnel coefficients from Eq. (17), and d is the thickness of the wafer. The subscripts explain the direction of the wave (thus t12 means that the wave

OPTICAL PROPERTIES

(ω) = const. +

5 1 J π j=1 j




ω0 j

1 1 + − ω − iγ j ω0 j + ω + iγ j



(19)

where the constant parameters J, 웆0, and 웂 have been chosen to model five mutually overlapping absorption lines in the

2600

2800

3000

3200

3400

2600

2800

3000

3200

3400

Refractive index

1.736

1.734

1.732

1.730

Absorption coefficient in cm–1

250 200

1.0

2600

2800

3000

3400

0.6 T R A

0.4

0.2

0.0

2600

2800 3000 3200 Wavenumber in cm–1

3400

Figure 6. Transmittance, reflectance, and absorption (normal incidence) calculated for a 0.1 mm thick wafer according to Eq. (18). Optical constants as shown in Fig. 5. This is an example for a forward search procedure (spectra calculation). In such a sample, an absorption feature may clearly be identified from the transmission spectrum only.

MIR. Figure 5 shows the optical constants calculated via Eq. (13) and Eq. (14) assuming Eq. (19) for the dielectric function. The normal incidence spectra according to Eq. (18) are shown in Fig. 6 and demonstrate that the assumed absorption structure may well be identified from the spectra. The inclusion of scatter losses requires a refinement of our expressions. The scatter losses may occur at the surfaces and in the bulk. In many practically relevant cases, the surface scatter losses are dominant, because the optical constant discontinuities are usually larger at the surface than in the bulk inhomogeneities. However, for absorption losses we will assume that the surface absorption losses are negligible compared with the bulk losses. In experimental situations, where surface and bulk loss contribution are of the same order of magnitude, they may be separated from each other by investigating samples with different thickness values. The simplest way to consider surface scatter losses in the equations for specular transmittance and reflectance is to multiply the Fresnel-coefficients with a roughness-dependent attenuation factor. For normal incidence, the corresponding formulae are: σ 2

1

t12 → t12 e− 2 [2π (n 1 −n 2 ) λ ] ; 1

σ 2

r12 → r12 e− 2 [4π n 1 λ ] ;

150

r21 → r21 e

100 50 0

3200

0.8 T,R,A

transmits the interface coming from medium 1 and entering medium 2). The reflectance and transmittance are thus completely determined by the incidence angle, polarization, frequency, sample thickness, and optical constants. Because (up to now) in our assumptions scatter mechanisms have been excluded (homogeneous media, smooth surfaces), the bulk absorption loss may simply be calculated from T and R using Eq. (1) and neglecting S. Equation (18) represent the simplest example of a forward search procedure, where the sample’s properties (material, geometry) are known, as well as all parameters of the (optical) input (polarization, frequency, amplitude, propagation direction). From these data, the optical output (viz., transmittance and reflectance) are calculated. For illustration, Figs. 5 and 6 demonstrate a set of optical constants as may be calculated from a multioscillator model, and the corresponding spectra obtained from Eq. (18) assuming normal incidence and a sample thickness of 0.1 mm. The dielectric function of the multioscillator model has been described by

331

2600

2800

3000

3200

3400

Wavenumber in cm–1 Figure 5. The frequency dependence of the refractive index and the absorption coefficient are strongly correlated to each other. This figure shows the refractive index (upper graph) and the absorption coefficient (lower) for the special case of a multioscillator model (see text). This model is frequently used to describe optical constants in the presence of single absorption lines.

(20)

− 12 [4π n 2 σλ ] 2

Here, ␴ is the root mean square (rms) interface roughness. Clearly, with increasing roughness, the specular reflectance and transmittance are attenuated (and consequently, the scatter loss increases). Please note that surface roughness may decrease the specular reflectance down to zero, if the wavelength is sufficiently small. This is in contrast to the effects of bulk absorption, because the latter cannot depress the finite reflection signal from the first surface. Measurements of Optical Loss Today, transmission spectrophotometers belong to the standard equipment in many university and industrial labs. Typi-

332

OPTICAL PROPERTIES

refractive indices), and the incidence angle satisfies the condition:

Reference LS

MC

CH

D

A

sin ϕ ≥

Sample 1

1

2

2

3

n2 n1

(21)

3 1

2

3

4

5

6

7

8

9

10

SC Figure 7. Principal scheme of a double beam dispersive spectrophotometer; (LS) light source, (MC) monochromator, (CH) chopper, (SC) sample compartment, (D) detector, (A) amplifier.

cal spectrophotometers are either designed for the UV/VIS region (UV/VIS spectrometer, which often work with spectrally dispersive monochromators as shown schematically in Fig. 7) or for the MIR (so-called IR spectrometers). IR spectrometers are produced today to an increasing extent as Fourier transform spectrometers. This has led to the abbreviation FTIR (Fourier transform infrared). These spectrometers do not contain monochromators. Instead, a built-in interferometer (often of Michelson type) records the interferogram of the polychromatic light coming from the sample or a reference. From the intensity distribution in the interferogram, the spectrum of the incoming light may be calculated by means of a fast Fourier transformation (FFT) procedure. Thus, the output represents the same type of spectra (for example, transmittance) as supplied by a dispersive spectrophotometer. An important advantage of the FTIR technique is the much faster spectra registration. The NIR region is usually accessible in so-called UV/VIS/ NIR-dispersive spectrometers, or as an optional upgrading of FTIR spectrometers. The latter type of spectrometer also allows upgrading for FIR measurements. In its standard version, a transmission spectrometer performs measurements of the transmittance with an absolute measurement error of approximately 0.002 to 0.01, depending on the quality of the spectrometer and the wavelength range. Usually, a suitable specular reflectance attachment is optionally available, so that T and R may be measured. From that, the optical loss may be calculated from Eq. (1) for the relevant light incidence geometry. Particularly, one may design measurement situations where the transmittance completely vanishes. Examples are provided by IRAS (infrared reflection absorption spectroscopy) and ATR (attenuated total reflection). The first one has been developed to investigate the optical loss of thin adsorbate layers on metal substrates. As the metal substrate does not transmit light, the optical loss may be directly determined from the reflectance via Eq. (1). It may be shown from a detailed analysis of Fresnel’s formula that the best sensitivity for adsorbate absorption loss detection demands the application of grazing incidence light with linear parallel polarization. Usually, the incidence angle is 85⬚ or larger. The method is based on the specific behavior of the IR optical constants of metals, therefore its application is restricted to the IR, as indicated by its name. Although the ATR method works in larger spectral regions, it assumes that the sample has a lower refractive index than the medium from where the light is incident. As a special conclusion from Fresnel’s coefficients, the reflectance becomes 1 when both media 1 and 2 are nonabsorbing (purely real

This situation is called total reflection of light. If, however, the refractive index of the sample has a nonvanishing imaginary part (lossy sample medium), then the reflectance decreases, and one speaks about attenuated total reflection. So one can calculate the sample material extinction index simply from the reflectance attenuation. In multiple reflection arrangements, ATR is a sensitive tool to determine small absorption losses (1). The physical reason for the total reflection attenuation is the following: Even in ideal total reflection conditions the second medium is penetrated by a surface wave, which travels along the interface and damps quickly into the depth of the sample. In the geometry of Fig. 4, its electric field is proportional to the wave function: ω

e−i (ωt− c n 1 x sin ϕ ) e− c ω

q

n 2 sin2 ϕ−n 2 z 1

2

(22)

If the sample medium is lossy, then this surface wave needs to be permanently fed by the incident light, which leads to the detectable reflection attenuation. Clearly, in any real situation, from the knowledge of two data (T and R) only the full optical loss may be determined. A discrimination between absorption and scatter losses is then impossible without additional model assumptions on the nature of the sample and their realization in refined mathematical spectra fitting procedures. An indication of surface scatter at the first sample surface may be drawn from the specular reflectance: If the first surface is rough, the specular reflectance gradually decreases down to zero with increasing frequency, in agreement with Eq. (20). Another principal problem occurs in connection with the measurement of small loss values. As T and R are measured with a finite accuracy, the measurement of small losses (typically below 0.01) becomes impossible by this method. This is simply a consequence of the high under-ground signal, provided by the transmittance and the reflectance spectra. In such cases, one should directly measure the optical loss, and not conclude on it from T and R measurements (6,10). Occasionally, corresponding attachments may also be combined with the previously mentioned spectrophotometers. Backscattering losses (back into medium 1) and forward scatter can be measured in so-called integrating sphere attachments, where the diffusely scattered light is collected and brought to the detector. Thus, a scatter signal integrated over all scattering angles is measured, and this signal is commonly called total integrated scatter (TIS). These spheres are commercially provided for the NIR/VIS/UV spectral regions (coated with BaSO4 or Spectralon) or for the MIR (coated with Infragold). From the viewpoint of their size, these spheres reach from minispheres (a few centimeters in diameter) up to devices with more than one meter in diameter. As already mentioned, scatter losses may principally originate from the volume and the surfaces of the sample. In the case that the scatter loss originates only from one surface with a surface roughness ␴, the latter may be determined from a measure-

OPTICAL PROPERTIES

ment of the diffuse backscatter loss and the specular reflectance. For small scatter losses, we obtain from Eq. (20):

λ σ= 4πn1


S

(back into medium 1)

R

(23)

The accurate measurement of absorption losses is based on the idea that the energy absorbed in the sample must either leave the sample (with a certain time delay) or enhance its temperature. In other words: The absorbed energy portion will participate in relaxation processes, and this is our chance to detect it. In order to detect very small absorption losses, absorption measurements are often accomplished with high incident light intensities, reliably supplied from laser sources. The nature of the sample and its environmental conditions (e.g., temperature) will determine which of the relaxation channels works most rapidly. If radiative relaxation is fast enough, the fluorescence intensity allows us to conclude on the previously absorbed energy, and thus to determine the absorption. This is what is done by the fluorescence method. If nonradiative relaxation is faster, then the absorbed energy will finally lead to sample heating. As the temperature increase may be conveniently measured, the absorption of the sample may be determined. Thus we have calorimetric methods of absorption measurements. Other absorption measurement techniques make use of the sample heating without direct temperature measurements. Thus, the optoacoustical measurements detect the sound wave generated in a medium as a result of the absorption of pulsed light due to thermal expansion. Further methods detect the deformation of the sample surface, caused by thermal expansion due to light absorption. This deformation may be optically detected by the angular deflection of a weak probe beam. The corresponding method is called photothermal deflection spectroscopy (PDS). Alternatively, the thermal expansion of the embedding medium surrounding the sample surface may be detected through its refractive index change. If the probe beam is of grazing incidence, the refractive index gradient in the vicinity of the heated surface leads to an angular deflection of the probe beam, which may be detected. SELECTED PROBLEMS OF OPTICAL CHARACTERIZATION Reverse Search Procedures In many cases, one is not only interested in a measurement of the optical loss, but in a more complete investigation of the optical properties of a sample and, in particular, of the optical properties of the sample material. The latter are hidden in its optical constants, so that the task may be to determine the optical constants of a medium from measured experimental spectra. In the simplest case this may be measurements of the transmittance and/or the reflectance. In other cases ellipsometric data are to be discussed. The general philosophy, however, does not depend on whether photometric or ellipsometric data or a combination of them are available. As exemplified in the previous sections, it is possible to provide explicit expressions for the spectra of any sample if geometry and optical constants are known. A more complicated problem is the so-called reverse search procedure, where the spectra and more or less complete data on the geometry are given, and the optical constants of the medium are to be

333

determined. Additionally, complementation of geometrical information (e.g., sample thickness) may be required. It is often impossible to obtain explicit expressions for the optical constants as a function of the measured data [see, e.g., Eq. (18)]. This makes the numerical side of the reverse search more complicated than the forward search, because it may be necessary to apply involved iteration procedures to find the result. As a further complication, unambiguity and numerical stability of the result may not be guaranteed. From the formal point of view, the reverse search procedures may be classified into single wavelength methods and multiwavelength methods (11). The latter include the Kramers–Kronig methods as well as curve-fitting techniques. Often, the reverse search bases on the numerical minimization of an appropriately defined error function F, for example:

F=

M  Z 

wi (ω j )[i,theor. (N(ω j ), ω j , . . .) − i,measurement(ω j )]2g

i=1 j=1

(24) Here, M is the number of available experimental spectra ⌽i,measurement, each containing Z values recorded at the angular frequency values 웆j. g is an integer, often g ⫽ 1. The ⌽i,theor. values represent theoretical expressions for the spectra, they are dependent on the concrete physical model chosen to describe the system. As examples, the expressions in Eq. (18) may be used, if a wafer with plane boundaries is investigated. The weighting functions wi are usually inversely proportional to the corresponding measurement error at a power 2g. The numerical minimization of Eq. (24) represents a purely mathematical problem, and the corresponding skills will not be discussed here. In the ideal case, a set of optical constants may be found which generates theoretical spectra equal to those measured, so that F becomes zero. In practice this is impossible, and it makes no sense to minimize the error function Eq. (24) below a threshold value determined by the measurement accuracy ⌬⌽. Thus, we may regard that the minimization was successful when the condition:

F
). This will be an ideal realization of the phenomenon of localization uncomplicated by many-body effects present in the case of electron localization. Another interesting effect is that zero-point fluctuations, which are present even in vacuum, are absent for frequencies within a photonic gap. Over the past decade, there has been rapid development in this field in the fabrication of photonic band gap materials. Unlike the case of electron waves, which usually have wavelengths on the atomic scale, the wavelengths of electromagnetic waves of interest are several orders of magnitude larger, varying between hundreds of nanometers for visible light to meters and centimeters for radio and microwaves. While the periodic lattice for electron waves is constrained by the crystal structure, the periodic dielectric

structures for photonic band gap materials are artificial structures that can be designed and fabricated to provide a desired electromagnetic response. Therefore, there is much interest in theoretical calculations for these systems, and advances in the field have been characterized by a close collaboration between theorists and experimentalists. The absence of the photon-photon interaction makes photonic crystals an ideal testing ground for theoretical simulation, bypassing the complications with electron-electron interactions inherent in the analogous electronic case. One-dimensional photonic crystals have been well known for several decades as the distributed Bragg reflector (DBR), and they are the basis of many devices, such as dielectric mirrors, Fabry-Perot filters, and distributed feedback lasers (13–15). A 1-D photonic crystal (PC) is shown in Fig. 1(a). A typical example of a 1-D PC with periodicity a, is a superlattice of alumina layers (with dielectric constant, ε = 9.61) and air layers (?=1). We consider alumina thicknesses of 0.4375 mm and air layers with thickness of 1.3125 mm, designed for millimeter region of the electromagnetic spectrum. Simulations for waves incident on the structure at three different incident angles (0◦ , 30◦ , and 60◦ ) are shown in Fig. 1(b) and 1(c). For normal incidence (solid lines in Fig. 1), there is a drop of the transmission from about 36 to 77 GHz. The transmission at the center of the gap is almost five orders (almost −50 dB) of magnitude less than the incident wave. Calculations are based on the real space transfer matrix method described later. The gap results from the multiple interference of waves, from the different layers of the structure, and there is destructive interference of transmitted waves within the band gap region. As is well-known for geometrical optics the wavelength of the gap appears when the wavelength λ is 2neff a where neff is the effective refractive index of the multilayer. We expect the gap to disappear for other incident angles. Indeed, by increasing the incident angle, the gap increases for the polarization with the electric field out of the plane of incidence [see Fig. 1(b)], but the gap tends to disappear for the wave polarized in the plane of incidence [see Fig. 1(c)]. A physical realization of a 2-D photonic crystal is obtained using infinitely long cylinders arranged in the 2-D triangular lattice (1–4) with the cross section of this structure shown in Fig. 2(a). The cylinders may be composed of air residing in a dielectric background or of dielectric material residing in an air background. The transmission for EM waves with incident k vector in the x,z plane is shown in Fig. 2. We use air cylinders with radius 0.805 mm surrounded by a dielectric with ε = 12.25 (the dielectric constant of GaAs); the distance between the center of the cylinders is 1.75 mm, and the total thickness of the system along the z direction is 9.1 mm. For waves with E field parallel to the cylinders [Fig. 2(b)], there is a small gap at around 48 GHz and a much wider gap at around 80 GHz. As the incident angle increases and the k vector is perpendicular to the axis of the cylinders, the second gap moves to smaller frequencies. For the polarization with the E field in the x,z plane [Fig. 2(c)], there is a gap at around 70 GHz for all the angles and for k vectors perpendicular to the axis of the cylinders. For both polarizations and for k vectors in the x,z plane, there is a gap from 70 to 80 GHz.

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright © 2007 John Wiley & Sons, Inc.

2

Photonic Band Gap Materials

Figure 1. The transmission for EM waves propagating in a onedimensional photonic crystal [see panel (a)] for incident angle of 0◦ , 30◦ , and 60◦ (solid, dotted, and dashed lines, respectively). The incident k vector is always in the x,z plane. Panels (b) and (c) correspond to E fields parallel to the y axis and in the x,z plane, respectively.

This structure of the triangular lattice with air columns in a dielectric background is the only known 2-D PC that has a gap for both polarizations and has been used extensively for applications. As in the 1-D case, we expect that the gap will disappear as the k vector moves out of the x,z plane because the system is homogeneous along the y axis. An important property of Maxwell’s equations for dielectric systems in the absence of absorption or non-linearities is that the lattice spacing a can be scaled to any length scale by any factor γ, (a = γa), and the frequencies will scale as f  = f /γ [3]. Although we have described the behavior of photonic crystals at millimeter wave frequencies, exactly the same electromagnetic response will occur when the structure is created at optical length scales. This scaling behavior allows photonic crystal structures to be designed, synthesized and measured at much more convenient microwave/millimeter wave frequencies to understand the behavior of the same photonic crystals at optical length scales. It is clear that we need a 3-D structure with periodicity along three directions in order to have a complete photonic band gap where transmission of waves is forbidden for all polarizations and all the incident directions in a certain frequency range (the band gap). Intense research in the beginning of the 1990s described periodic structures a photonic band gaps (PBG) (<xref target="W4410-bib-0001

Figure 2. The transmission for EM waves propagating in a twodimensional photonic crystal. The cross section of the structure is shown in panel (a). The incident angle is 0◦ , 30◦ , and 60◦ (solid, dotted, and dashed lines, respectively). The incident k vector is always in the x,z plane. Panels (b) and (c) correspond to E fields parallel to the y axis and in the x,z plane, respectively.

W4410-bib-0002 W4410-bib-0003 W4410-bib-0004 W4410bib-0016 W4410-bib-0017" style="unformatted"/>). In fact, the first 3-D photonic crystal built by Yablonovitch and Gmitter (18) did not have a complete PBG. This structure consisted of air spheres embedded in an Al2 O3 material forming a face-centered-cubic (fcc) lattice. We can visualize this crystal by placing the spheres at the edges and at the center of the faces of a cube. It was constructed by drilling hemispherical cavities on dielectric plates that were stacked together. The whole structure can be constructed by periodically displacing the cube into the space. In contrast to the transmission measurements, which showed a complete PBG for this structure because of a degeneracy of modes at a particular direction (19–21). At this point, Ho, Chan, and Soukoulis (21,22) theoretically proved that the diamond structure consisting of air or dielectric spheres posesses a complete PBG. A diamond structure is similar to the fcc structure, but instead of placing one sphere in each fcc lattice point, we place one more sphere in each lattice point displaced parallel to the body diagonal of the cube by one quarter of the length of the diagonal. The first photonic crystal with a complete PBG was built by Yablonovitch et al. (23). They devised an ingenious way of constructing a diamond lattice. They noted

Photonic Band Gap Materials

3

Figure 3. Layer-by-layer structure constructed by orderly stacking of dielectric rods. The periodicity is four layers in the stacking direction. Layers in the second neighbor layer are shifted by a/2 in the plane.

that the diamond lattice is a very open structure characterized by an open channel along the [110] directions. This structure was named the three cylinder structure, was can be constructed by drilling holes on the surface of a materials’ slab. The holes form a triangular array. Three drilling operations are conducted through each hole, 35.26◦ off normal incidence and spread out 120◦ on the azimuth. The structure had a complete PBG centered at 14 GHz and the width of the forbidden gap was 19% of its center frequency. Subsequently, the Iowa State group designed and fabricated the layer-by-layer structure shown in Fig. 3 (24). The structure is assembled by stacking layers consisting of parallel rods with a center-to-center separation of a. The rods are rotated by 90◦ in each successive layer. Starting at any reference layer, the rods of every second neighboring layer are parallel to the reference layer but shifted by a distance 0.5a perpendicular to the rod axes. This results in a stacking sequence that repeats every four layers. This lattice has face-centered-tetragonal (fct) lattice symmetry with a basis of two rods. This structure has a robust photonic band gap when both the filling ratio and the dielectric contrast meet certain requirements. The photonic band gap is not sensitive to the cross-sectional shape of the rods. Several different structures have been constructed with midgap frequencies at 13, 100, and 450 GHz using etching techniques and Al2 O3 or Si as materials (25–30). The structure has been fabricated with a measured PBG at around 2 THz using laser-induced direct-write deposition from the gas phase (31). Figures 4 and 5 show the transmission for a layer-bylayer structure with rods with a circular cross section; the radius of each rod is 20 µm, the in-plane separation of the rods is 160 µm, and the dielectric constant of the rods is 9.61. The crystal contains 12 layers of rods (3 unit cells). For propagation along the stacking direction (k parallel to the z axis), there is gap between 0.9 and 1.25 THz, and the transmitted intensity at the center of the gap is more than six orders of magnitude smaller than the incident intensity (−60 dB). By increasing the incident angle, the gap becomes smaller, and the transmission at the center of the gap increases, but there is a complete PBG for all the angles and polarizations between 0.9 and 1.05 THz. By creating small distortions (or defects) in these photonic crystals, we can also create defect states inside the PBG, which give rise to sharp peaks of the transmission inside the PBG. This is analogous to donors or acceptors in

Figure 4. The transmission for EM waves propagating in a threedimensional photonic crystal similar to the one shown in Fig. 3. The incident angle is 0◦ , 30◦ , and 60◦ (solid, dotted, and dashed lines, respectively). The incident k vector is always perpendicular to the y axis. Panels (a) and (b) correspond to E fields parallel to the y axis and in the x,z plane, respectively. The rods of the first layer are parallel to the y axis.

Figure 5. The same as in Fig. 4 except that the k vector is always perpendicular to the x axis.

semiconductors that introduce impurity states in the electronic energy gap. Such systems can be used as filters. Let us first study a 1-D photonic crystal with a defect (Fig. 6). The structure is the same as the one studied in Fig. 1, with one defect. In the third unit cell, the air slab has a thickness 0.8 times its original thickness, and the dielectric slab has a thickness 1.2 times its original thickness. As a result of that difference, the transmission for normal incidence has a peak inside the gap at 71 GHz (compare the solid lines in Figs. 1 and 6). However, when the incident angle increases, the transmission peak moves to higher frequencies and it appears at different frequencies for each polarization (com-

4

Photonic Band Gap Materials

Figure 7. The transmission for EM waves propagating in a twodimensional photonic crystal (similar to the one in Fig. 2) with a defect. The incident angle is 0◦ and 30◦ (solid and dotted, respectively). The incident k vector is always in the x,z plane. The E field is parallel to the y axis.

THEORETICAL METHODS Plane Wave Method Figure 6. The transmission for EM waves propagating in a onedimensional photonic crystal (similar to the one in Fig. 1) with a defect for incident angle of 0◦ , 30◦ , and 60◦ (solid, dotted, and dashed lines, respectively). The incident k vector is always in the x,z plane. Panels (a) and (b) correspond to E fields parallel to the y axis and in the x,z plane, respectively.

pare the different lines in Fig. 6). Even for the 30◦ angle, the transmission peak appears at 76 GHz (for E field perpendicular to the plane of incidence), 5 GHz higher than its value at normal incidence. The strong dependence of the defect mode on the incident angle is again related to the fact that the 1-D photonic crystal is actually homogeneous along the x and y directions. It has been shown theoretically and experimentally that 3-D photonic crystals can solve this problem (29–34). For simplicity, we will illustrate this with a 2-D photonic crystal similar to the one in Fig. 2. Recall that 2-D photonic crystals suffer from the same disadvantage as the 1-D photonic crystals discussed in the previous paragraph. The reason for that problem is the homogeneity of the structure along the y axis [see Fig. 2(a)]. For that reason, we will show results only for k vectors in the x,z plane [see Fig. 2(a)] and for the E field parallel to the axis of the cylinders (y axis). The defect is introduced by decreasing the radius of one cylinder in the center of the structure. The distorted radius is 0.7 times its original value. For normal incidence (solid line in Fig. 7), there are three transmission peaks inside the gap, at 71, 76, and 78 GHz. For incident angle of 30◦ , the peaks remain at almost the same frequencies. The small changes in the frequency are actually an artifact of the calculations. We will return to this point when we discuss the computational method (transfer matrix method). We can tune the position of the transmission peak inside the gap, by changing the radius of the distorted cylinder. The flexibility in tuning defect modes makes photonic crystals a very attractive medium for the design of novel types of filters, couplers, laser microcavities, etc. (1–4).

To study the behavior of electromagnetic waves in photonic band gap crystals, we must solve Maxwell’s equations for a media characterized by a spatially varying dielectric function ε(r):

These may be decoupled to generate an equation only in the magnetic field,

and

At this point, we note that the vector nature of the wave equation is of crucial importance. Early attempts (35) at adopting the scalar wave approximation led to qualitatively wrong results. The simplest case happens when ε(r) is a real and periodic function of r, and we assume that it is frequency-independent in the range of interest. We also assume the magnetic permeability µ is 1. In this case, the solution of the problem scales with the period of ε(r). For example, reducing the size of the structure by a factor of two will not change the spectrum of electromagnetic modes other than scaling all frequencies up by a factor of two. Because of the periodicity of the problem, we can make use of Bloch’s theorem to expand the electric and magnetic fields in terms of Bloch waves:

where K = k + G. k is a vector in the Brillouin zone, and G is a reciprocal lattice vector. The solution for the magnetic

Photonic Band Gap Materials

5

field has the form of an eigenvalue problem:

The corresponding equation for the E field does not have the form of a simple eigenvalue problem because the dielectric function enters into the frequency-dependent righthand side:

Hence, we obtain photonic band structure by solving Eq. (9) for the magnetic fields. Here εK,K = ε(G − G ) is the Fourier transform of the dielectric function. Dielectric functions with sharp spatial discontinuities require an infinite number of plane waves in the Fourier expansion. To avoid this problem, we smear out the interfaces of the dielectric objects in the unit cell. For example, for modeling a cylinder of radius a and dielectric ε, we employ the smeared dielectric function

where the width w of the interface is chosen as a small fraction of the radius a (≈ 0.01–0.05 a). In practice, we incorporate the smearing and define the dielectric function ε(r) over a grid in real space. The Fourier transform of the dielectric function in our finite plane wave basis set is computed to obtain ε(G − G ). The dielectric matrix in Fourier space is then inverted to obtain ε−1 (G − G ). This procedure yields much better convergence than the alternative method of determining ε−1 (G − G ). The transverse components of the magnetic field are hK,1 , that is,

where the unit vectors e1 and e2 form an orthogonal triad (e1 , e2 , K). The solution in Eq. (9) for the magnetic field reduces to the eigenvalue problem:

The matrix M is defined by

In practice, the photonic band structure given by the frequencies ω(K, λ) is computed over several high symmetry points in the Brillouin zone or on a grid in the Brillouin zone if the density of states is needed. Plane wave convergence is closely checked. The first structure (19–21) considered by researchers was the fcc structure composed of low dielectric spheres in a high dielectric (ε) background. This simple structure with close packed spheres has the band structure shown in Fig. 8. There is no fundamental gap between the second and third bands—the bands are degenerate at the W point of the zone. There is a region of low densities of states between bands 2 and 3—the pseudogap, which may have interesting consequences. Another very interesting feature is a sizable complete gap between the 8 and 9 bands (8–9

Figure 8. Photonic band structure for the fcc structure composed of air spheres in a high dielectric background (dielectric constant e = 9.61). The geometry is for close packed spheres (i.e., 74% filling ratio). The bands are shown along the 110 axis of the Brillouin zone.

Figure 9. Size of the three-dimensional photonic band gap measured by the gap/midgap ratio for a diamond structure with spheres on the diamond sites and its conjugate structure. Band gaps are plotted as a function of the filling ratio. The dielectric contrast of 12.96 is used.

gap), which exists over the entire zone (i.e., for all directions of propagation of the EM wave). The size of this gap is about 8% for a refractive index contrast of 3.1. The direct fcc structure (high dielectric spheres in a low contrast background), however, does not possess the dip in the photonic DOS. The diamond structure has been the subject of much investigation (<xref target="W4410-bib-0001 W4410-bib0002 W4410-bib-0003 W4410-bib-0004 W4410-bib-0021 W4410-bib-0022" style="unformatted"/>) because it has a full three-dimensional photonic band gap between the fundamental bands (2–3 gap between the second and third bands). This gap exists for (1) high dielectric spheres on the sites of the diamond lattice (Fig. 9), (2) the diamond structure with low dielectric spheres on the diamond sites [Fig. 9; conjugate of (i)], and (3) the diamond structure connected by dielectric rods (Fig. 10). The best performing gap (29%) is reached for the diamond structure with 89% air spheres (i.e., a multiply connected sparse structure). A similar large gap (30%) is also found for the diamond structure connected with dielectric rods with about 30% dielectric filling fraction. These gap magnitudes are for a refractive index contrast of 3.6, appropriate for GaAs. A novel layer-by-layer structure was designed and fabricated at Iowa State University (24–30) (Fig. 3) with a full three-dimensional fundamental PBG (Fig. 11). The agreement between the calculated bands and the experimental measurements are excellent (Fig. 11) for both EM-wave

6

Photonic Band Gap Materials

Figure 10. Size of the three-dimensional photonic band gap when the diamond structure is connected by dielectric rods (e = 12.96). Rectangular rods connect the sites in 111 planes, whereas cylindrical rods connect the rods along the 111 axis. Figure 13. Photonic densities of states for the layer-by-layer structure using the experimentally fabricated geometry with a filling ratio of 0.26 and silicon (e = 11.67) as the dielectric material.

Transfer Matrix Method

Figure 11. Calculated (lines) and measured photonic band structure for the layer-by-layer structure composed of stacked alumina cylinders (e = 9.61) with a full bandgap between 12 and 14 GHz. The circles represent measurements for the E-field polarization parallel to the rod axes, whereas the squares represent measurements performed with an E field perpendicular to the rod axes.

Figure 12. Size of the gap in the layer-by-layer structure as a function of filling ratio for different c/a ratios. c is the repeat distance, and a is the rod separation. A dielectric contrast of 12.96 has been used to facilitate comparison.

polarizations. This particularly robust structure has been fabricated at length scales providing gaps ranging from 13 to 500 GHz (24–30). As is typical for the diamond structure, the magnitude of the gap is maximized at about 25% (Fig. 12) for the contrast of n = 3.6. the densities of photon states for the experimentally fabricated structure with silicon micromachining (Fig. 13) provides a picture of both the fundamental gap and other frequency regions that display depleted or enhanced DOS. The photonic band gap depends on (1) the local connectivity of the dielectric structure, (2) the contrast between the two media, and (3) the filling ratio. A minimum dielectric contrast (ε > 4) is usually needed to observe the band gaps. The photonic band structure method is a systematic way to search for the existence of band gaps in dielectric structures (3, 36).

Although the method described in the last section focuses on a particular wavevector, there are complementary methods that focus on a single frequency. In the transfer matrix method (TMM), first introduced by Pendry and MacKinnon (37), Eqs. (3) and (<xref target="W4410-mdis-0004" style="unformatted">4) are discretized, and the z components of the fields can be eliminated, so we derive the following equations:

Photonic Band Gap Materials

7

The derivatives in the Maxwell’s equations are approximated with finite differences and the electromagnetic fields components are located on a Yee cell (43). In the Yee cell, the E-field components at time nt are located on the sides of a cube. The magnetic field H components at times (n + 1/2)t are located at the face-centered points of the Yee cell. This results in both spatial and temporal offsets of the two fields when the Maxwell curl equations are solved on each face of the cube. The system is described by a spatial grid. The time step is chosen such that an EM wave will propagate less than a grid spacing during the time step. ε(i, j, k) and µ(i, j, k) are the dielectric constant and the magnetic permeability at the subcell (i, j, k). a, b, and c are the dimensions of each subcell along the x, y, and z directions. Equations (15)–(18) are connecting the fields at the k + 1 plane with the fields at the k plane. Using TMM, the band structure of an infinite periodic system can be calculated, but the main advantage of this method is for the calculation of transmission and reflection properties of EM waves of various frequencies incident on a finite thickness slab of PBG material. Such calculations are extremely useful in the interpretation of experimental measurements of transmission and reflection data. The TMM method can also be applied to calculate PBG structures containing absorptive and metallic materials. The TMM has previously been applied to defects in 2-D PBG structures (38), photonic crystals with complex and frequency dependent dielectric constants (39), metallic PBG materials (40,41), and angular filters (42). In all these examples, the agreement between theoretical calculations and experimental measurements was very good. At this point, we return to the discussion of Fig. 7. There, we used a rectangular conventional unit cell consisting of 15 × 26 subcells. In order to create the triangular lattice, we placed cylinders at the corners and at the center of the rectangular unit cell. The system is finite along the z direction [see Fig. 2(a) having a thickness.of three unit cells. Along the x direction, we use a supercell consisting of three conventional unit cells, and we assume periodic boundary conditions at the edges of the supercell. So, there are infinitely many defects along the x direction with separation of 3a. This is the reason for the small change in the frequency as we change the angle. Calculations with larger supercells (consisting of five conventional unit cells) show negligible angular dependence. Finite Difference Time Domain Method Even though the preceding transfer matrix method is employed for steady state solutions, the finite difference time domain (FDTD) method is used for general time-dependent solutions including transient behavior. In this method, the Maxwell curl equations are numerically solved:

8

Photonic Band Gap Materials

En x (i, j, k) is the x component of the electric field at the n time step in the (i, j, k) subcell. Finite size systems can be easily modeled. This widely used technique (44,45) can be used to find either the steady state or transient response of arbitrary systems containing dielectric or metallic components, as well as materials with nonlinear dielectric properties. In the perfect metallic code, the E field vanishes inside the metal. The FDTD can be used with a Gaussian pulse source. The fields are numerically integrated to obtain the fields at long times (>1000 time steps). The Fourier transform of the scattered and incident fields generates the frequency-dependent response of the system. Alternatively, the system may be subject to a source field with a single frequency ω, and one can determine the steady state response of the system at that frequency. Such steady state calculations may then be repeated at desired frequencies. At the edges of the FDTD cell, outer radiation boundary conditions are frequently employed. Here the incident wave at the boundary is absorbed. Methods to transform the near fields to radiating far fields are then employed. This is particularly necessary for antenna problems where far-field radiation patterns are desired. The FDTD method is a very powerful design tool in simulating the electromagnetic response of systems, covering a broad range of frequencies. We have used this method to calculate the radiation patterns of dipole antennas placed on PBG crystals (Fig. 14). The dipole antenna is driven by either a voltage pulse or a steady-state sinusoidal excitation, and the radiated far fields are determined. The symmetry of our PBG crystal has been used to develop a computational cell that is one fourth the size of the actual system (46). The calculations are in very good agreement with measurements (46). It is also possible to calculate the currents flowing in the antenna and to calculate the gain of the system. We have driven a finite length dipole oscillator on the surface of the PBG crystal at a frequency of 13 GHz near the center of the band gap. The dipole is at the intersection of the first and second layers, and the radiation patterns are calculated and measured at different heights z above the surface (Fig. 14). The agreement in both the E and H planes with measurements is very good.

Scattering matrix method- Fourier space transfer matrix method While early calculations were performed with this real space transfer matrix method, it has been found to be more convenient and accurate to use the transfer matrix method in a plane wave basis where Maxwell’s equations are solved in Fourier space [48]. The structure is divided into slices (along the z axis). In each slice the dielectric function ε(r) is a periodic function of the planar coordinates (x,y). Hence the dielectric function and its inverse are expressed as a Fourier expansion with coefficients ε(G) or ε−1 (G), where G are the reciprocal lattice vectors of the 2-dimensional lattice. The electric and magnetic fields also have Fourier

Figure 14. Measured (a) and FDTD calculations (b) of the antenna radiation in the E and H planes for a dipole antenna driven at 13 GHz at the center of the PBG. The three curves are for different heights z of the antenna above the surface, expressed as a ratio of z/d where d is the diameter of the dielectric rod. The antenna is at the intersection of the first and second layers, perpendicular to the first layer.

coefficients E(G) and H(G) defined through Ek (r) =




EG (z)ei(k+G)x||

G

where k is a Bloch wave-vector. From the six Maxwell equations for each Cartesian component of the E and H fields, the z-component of the E and H field is eliminated leading to four equations for the zderivatives of the x,y components of the E and H fields. We adopt a recently developed formulation [47,48] where the transfer matrix is computed in Fourier space. In each layer a transfer matrix M1 relates the z derivative of the electric field to the fourier components of the H field and a similar equation for the z-derivative of the H field with a transfer matrix M2 . This results (28) in a single eigenvalue equation for the E field in each layer involving the transfer matrix M. The transfer matrix M is diagonalized to obtain the eigenmodes within each layer. Both polarizations are included. The transfer matrix M in each layer is calculated that relates the z-dependence of the E, H fields in each layer. ∂ ∂ E = M1 H; H = M2 H ∂z ∂z 2 ∂ E = M1 M2 E = ME ∂z2 We have employed a compact notation where E and H in (28) represents the matrix of Fourier coefficients E(G), H(G) and M, M1 and M2 are corresponding square matrices. The propagation wave vectors (kz ) from this eigenmode problem determine whether the mode is propagating or decaying. The boundary conditions are that the parallel components of E and H are continuous at each interface and this leads to the individual scattering matrices si of each layer. A standard recursion algorithm [48,50] combines the scattering matrices of each layer into the scattering matrix S for the entire structure. Using the total S-matrix, we

Photonic Band Gap Materials

Figure 15. Simulated transmission spectrum for a 8-layer layerby-layer photonic crystal slab along the 001 direction. The polarization of the incident wave is such that the electric field is parallel to the rods in the first layer. The dielectric constant is | ∗ epsilion ∗ | = 12, the filling ratio is f=0.3, and the lattice spacing is 1| ∗ mu ∗ |.

simulate the reflection and transmission of the structure when fields are incident from the left. The advantage of this Fourier space approach is that any number of layers of differing width can be easily simulated since a real-space grid is not necessary. Since the solutions of Maxwell’s equations are independent for each frequency, the computational algorithm has been parallelized on massively parallel systems, where the transmission/reflection for each frequency is solved on a separate processor. This method relies on the convergence of the plane wave expansion of the fields, which depends on the type of structure being simulated. For structures where each layer has 2-dimensional periodicity both polarizations are solved for leading to a matrix size of N = 2NG where NG is the number of plane waves. For dielectric systems the number of plane waves (NG ) can be less than ∼150 for reasonable convergence. For metallic systems, considerable larger number of plane waves (>500) are necessary [49]. For the layer-by-layer structure, each layer has 1dimensional periodicity. This leads to the transfer matrix solution decomposing into separate solutions for the TE and TM modes, and a matrix of size NG is to be solved. We illustrate this method by the calculated transmission for a layer-by-layer photonic crystal (Fig. 15) with a fourier expansion of 13 × 13 plane waves. By enforcing periodic boundary conditions along the crystal axes the photonic band structure can also be computed with this method [47].

EXPERIMENTAL TECHNIQUES FOR FABRICATION OF PHOTONIC BAND GAPS There have been intensive efforts to build and test photonic band gap structures, dating back to the original efforts of Yablonovitch shortly after his first proposal for PBG crystals (51). Fabrication can be either easy or extremely difficult, depending on the desired wavelength of the band gap and the level of dimensionality. Because the wavelength of the band gap scales directly with the lattice constant of the photonic crystal (3, 52), lower-frequency structures that require larger dimensions will be easier to fabricate. At microwave frequencies, where the wavelength is on the order of 1 cm, the photonic crystals are decidedly macro-

9

scopic, and simple machining techniques or rapid prototyping methods can be employed in building the crystals. At the other extreme, optical wavelength PBGs require crystal lattice constants less than 1 µm. Building PBGs in the optical regime requires methods that push current stateof-the-art micro- or nanofabrication techniques. In a similar manner, the dimensionality of the PBG has a big impact on the ease or difficulty of fabrication. Because onedimensional PBGs require periodic variation of the dielectric constant in only one direction, they are relatively easy to build at all length scales. One-dimensional PBG mirrors (more commonly known as distributed Bragg reflectors (DBR)) have been used in optical and near-infrared photonic devices for many years. Two common examples of devices using 1-D PBGs are distributed feedback lasers and vertical-cavity surface-emitting lasers. Two-dimensional PBGs require somewhat more fabrication, but relatively mainstream fabrication techniques can be employed to achieve such structures. There are several examples of 2-D PBGs operating at mid- and near-IR wavelengths. Clearly, the most challenging PBG structures are fully 3-D structures with band gaps in the IR or optical regions of the spectrum. The fabrication of 3-D PBGs is complicated by a need for large dielectric contrast between the materials that make up the PBG crystal and the relatively low filling fractions that are required. The large dielectric contrast means that the materials must be dissimilar, and often the low-dielectric material is air with the other material being a semiconductor or a high-dielectric ceramic. The low filling fraction means that the PBG crystal with air as one dielectric will be relatively empty and the high dielectric material must be formed into a thin network or skeleton. When these difficulties are combined with a need for micron or submicron dimensions to reach into the optical region, the fabrication becomes very difficult, indeed. This area of PBG research has been one of the most active, and perhaps most frustrating, in recent years. The various methods to synthesize photonic crystals can be divided into two main groups: (1) methods that use (or extend) conventional semiconductor microfabrication techniques and (2) non-semiconductor techniques. The following list of processing methods is meant only to give some flavor for current avenues toward achieving PBGs. We focus on 3-D photonic crystals and 1-D and 2-D PBGs are not be included. Semiconductor-Based Methods Advanced semiconductor processing. A novel method to fabricate 3-D photonic crystals using state-of-the-art semiconductor processing techniques was pioneered by the Sandia group of S. Lin, J. Fleming and co-workers (53), and has been most successful in fabricating the layer-by-layer structure, first at a pitch of 4.2 µ (53) -corresponding to a gap wavelength of 10 µ, and then (54) at a pitch of 0.6 µ (with a gap wavelength of 1.5 µ). The 1.5 µ wavelength is critical for fiber-optic telecommunications applications. In this technique, the vertical topology of the 3D lattice structure is built by the repetitive deposition and etching of multiple dielectric films and a systematic multi-layer stacking process was developed. Within each layer, SiO2

10

Photonic Band Gap Materials

structure with electron beam lithography and spin on glass planarization (56). A rod spacing of 0.66 µ was achieved with a rod width of 0.22 µ for a five-layer structure. Rod spacings of 0.5 µ and 0.55 µ were also achieved. A reflectance peak spanning the near-IR range from 1.2-1.5 µ was observed for this family of structures, indicative of the band gap. Similar methods have also been employed (57) to fabricate metallic photonic crystals by deposition of tungsten into the patterned SiO2 layer and repeating the procedure. The SiO2 was also etched off to produce a 5-layer metallic layer-by-layer photonic crystal with a pitch of 4.2 µ. The optical properties of tungsten photonic crystals fabricated with this technique have been further measured by Seager et al. (58).

Figure 16. Transmission electron micrograph of the 3-D layerby-layer photonic crystal with a bar spacing of a = 4.2| ∗ mu ∗ |, fabricated by advanced silicon processing methods (Ref 53).

was first deposited, patterned in the required 1-d pattern, and etched to the desired depth. The resulting trenches were then filled with polycrystalline silicon. Following this, the surface of the wafer was planarized using chemical mechanical polishing, and the entire process was then repeated. The planarization step is critical for producing a flat surface for the deposition of the next layer and a repetition of the processing scheme. After the multi-layer process was completed, the wafer was immersed in a HF/water solution for the removal of the SiO2 . Samples with 5-layers of the layer-by-layer structure with a complete band gap were fabricated. Figure 16 shows a scanning electron micrograph (SEM) top view of a completed four-layer structure. The strongest attenuation occurs at 10-11 µ, with an attenuation strength of 12 dB per unit cell. The dependences of the band edges with angle of incidence confirm that the structure has a complete photonic gap. A variation of this planarization method (54) using filet processing was able to double the periodicity possible with a deposited SiO2 pattern, and was used to create a photonic crystal with a minimum feature size of 0.18 µ, and a band gap around 1.5 µ. Closely following these developments the group of Noda et al (55) fabricated a III-V photonic band gap crystal with a midgap wavelength of 1.4 µ. In this method, III-V semiconductor stripes (of GaAs or InP) were fabricated and then stacked with the wafer-fusion method. Precise alignment was achieved by laser-beam assisted aligning and a pitch of 0.7 µ was achieved. An attenutation of 40 dB was achieved with a 4-layer structure. Wafer fusion was best achieved at temperatures around 500 C. Because the crystal is constructed with III-V semiconductors, these are very suitable for optoelectronic devices. A sharp waveguide bend was also fabricated (55) by removing two rod segments. These pioneering achievements opened the field for further advanced semiconductor processing of the layer-bylayer structure at optical and near infrared length scales. Subramania and Lin (56), fabricated a 5-layer by layer

Vertical Reactive-Ion Etching. Vertical reactive-ion etching to form 2-D PBGs (59–62) is perhaps most straightforward technique because it derives directly from current microfabrication methods. The 2-D PBG is formed in a GaAs/(Al, Ga)As dielectric waveguide grown on a GaAs substrate using epitaxial growth techniques. The 2-D PBG consists of a square or triangular array of holes that are etched through the dielectric waveguide. Either electron beam lithography (59–61) or holographic patterning was used to define the 2-D pattern on the surface of the wafer. Then dry etching techniques are used to etch the holes down 1 µm or more. Lattice constants for the works cited here were in the range of 190–480 nm. The measured optical properties of the waveguide-PBG system in 59–61 showed clear transmission stop bands or strong reflection bands in the IR spectrum. O’Brien et al. (60) used the PBG as one mirror of a QW laser. Superlattice Disordering and Selective Oxidation. Use of superlattice disordering and selective oxidation to form 2D PBGs (63) achieves structures that are similar to those described in the previous section. The starting material is an epitaxially grown GaAs/AlAs multilayer structure. Silicon nitride is deposited on the top surface, and holes are etched into the silicon to expose the top surface of the wafer. In the reported work, the holes were arrayed in a triangular lattice pattern, with a hole diameter of 2 µm and a lattice constant of 8 µm. After the holes were etched, zinc was diffused into the semiconductor crystal through the openings. The diffusing zinc disorders the multilayer structure, converting it to a homogeneous (Al, Ga)As alloy. After the diffusion, the structure is exposed to an oxidizing ambient, and the disordered regions are selectively converted to aluminum oxide. Thus, the resulting structure is a GaAs/AlAs multilayer with a regular array of oxide posts inserted. The authors used this structure to build a photo-pumped semiconductor laser. Deep Anodic Etching of Silicon Wafers. Deep anodic etching of silicon wafers to form 2-D PBGs (64) is similar in concept to vertical reactive-ion etching, differing in starting materials and scale. The material is silicon. A 2-D array of small starter etch pits is formed on the surface using standard patterning and etching techniques. Typical dimensions and lattice constants are on the order of 2–10

Photonic Band Gap Materials

11

µm. Then, a photo-induced anodic etching procedure is employed. Macropores can be formed in the 110 direction in a ndoped Si wafer if it is anodized in an HF solution and illuminated from the backside. The illumination creates electronhole pairs and the electrons migrate to the growing etch pits. With careful control of the anodic etching, the pattern of holes can be extended through the entire thickness of the wafer—more than 400 µm—leading to a much thicker 2-D structure than can be achieved using the GaAs techniques described previously. The transmission spectrum of the fabricated structures showed distinct stop bands in the mid-IR region. This technique has been repeated for submicron lattice constants (65). It should be noted that without the use of a patterned silicon wafer, a random array of etch pits is produced in a porous silicon structure (66) which also has very interesting 1-D photonic crystal properties. True three-dimensional structures have not yet been demonstrated using semiconductor techniques, although there are several proposed routes toward optical PBGs.

solvent, the samples were pyrolyzed at 1100 ◦ C under N2 atmosphere. The resist decomposes into CO2 , CH4 , CO, and H2 O, whereas polyvinylsilazane is transformed into SiCN ceramic. A lattice of ceramic rods corresponding to the holes in the resist structure remained. Calculation using the transfer matrix method showed that the dielectric constant of the ceramic should be around 3.5 in order to fit the measured band gap centered at 2.5 THz. This is an indication that the ceramic was quite porous. Further work is needed for the fabrication of photonic crystals with more compact ceramic material, which hopefully will give higher dielectric constants. More recently, two- and threedimensional nanostructures of TiO2 were fabricated (69) using x-ray lithography and liquid-phase deposition. Using deep X-ray lithography a PMMA template was formed with an array of nano-order holes having a high aspect ratio. This template was filled with dense TiO2 by liquidphase deposition. A novel 3-D photonic crystal structure of slated pores that could be achieved with X-ray lithography, was theoretically predicted (70) to have a band gap of 28%.

Three-Cylinder PBG Using Directional Ion Beam Etching. The basic approach (67) uses a scaled-down version of Yablonovitch’s three-cylinder PBG proposed and demonstrated at microwave frequencies (23). The starting material is again GaAs. A masking layer of SiN and AlO is formed on the surface of a GaAs wafer, and then a triangular array of holes is etched through the masking layer to expose the underlying GaAs. Then a series of three angled ion beam etching steps is performed. Each etch step forms a deep array of etch pits angled at 35◦ off normal. After the first etch step, the GaAs substrate is rotated by 120◦ , a second is etched, the substrate is rotated again, and the third set of holes is drilled. The result is a set of intersecting air cylinders that form a diamond-like lattice structure in the GaAs crystal. Using electron-beam lithography, submicron dimensions are possible. Structurally, the photonic crystal clearly exhibits two or three unit cells. However, optical measurement has not shown the expected photonic band gap. The most likely cause of the lack of band gap is non-uniformity in the etching profile. Further research continues on this avenue of PBG fabrication.

Holographic Methods. Holography is a complementary approach to layer-by-layer methods in that the entire 3-D photonic crystal can be fabricated at the same time and has been an extremely active area of ongoing research (71–76). In holography a multiple beam of lasers is incident on a thick photoresist layer, providing a three-dimensional interference pattern. The photoresist is then developed. Negative photoresists exhibit a certain exposure dose threshold, above which the resist is not soluble in the developing process. Thus the spatial intensity of the dose gets transferred into the distribution of matter, resulting in a porous polymeric structure, the shape of which is tailored by the laser interference pattern. For positive photoresists, the underexposed regions remain after development. The photoresists have low refractive index contrast making them unsuitable for full photonic band gaps. Thus infiltration of higher index material (such as silicon) is necessary to improve the refractive index contrast, followed by removal of the resist altogether. The mathematical process of the interference pattern has been investigated in detail (73, 75). Generally four laser beams are necessary for formation of threedimensional photonic crystals with complete band gaps. Structures close to the three-hole structure having rhombohedral symmetry have been predicted from 4-beam holography. Such structures can have complete band gaps of 5% or larger for suitable filling ratios (38%) and structural parameters (75). Five holographic beams propagating from the same half-space has been predicted (75) to yield band gaps of 25% when the exposed photoresist is replicated with silicon. Genetic algorithms have also been utilized (76) to predict holographic structures with band gaps of 28%, similar to a rod-connected diamond structure.

Nonsemiconductor Methods Laser Rapid Prototyping. (31) Laser chemical vapor deposition has been used to fabricate a layer-by-layer structure similar to the one described in Fig. 3. The photonic crystal consisted of aluminum oxide rods, and the measured photonic band gap was centered at 2 THz. The index contrast may be lower than expected due to the porosity of the material. Further experiments on this promising direction are needed. Deep X-Ray Lithography. (68) PMMA resist layers with thickness of 500 µm were irradiated in order to form a three-cylinder structure. Because the dielectric constant of the PMMA is not enough for the formation of a photonic band gap, a molding step must be applied. The holes in the resist structure were filled with solution of polyvinylsilazane in tetrahydrofuran. After the evaporation of the

Self- Assembled Photonic Crystals. Theoretical studies (77, 78) have identified the “inverse” face-centered cubic (fcc) structure as one of the best suited for photonic band gaps. This consists of a periodic array of close-packed low dielectric spheres with refractive index n1 in a high dielectric background with refractive index n2 , generating

12

Photonic Band Gap Materials

then back-filled with silicon to generate an inverse opal diamond-like lattice in silicon (94). Micromanipulation of bars of InP were performed to build up the layer-by-layer photonic crystal with a pitch of 1.4 µ (95). A band gap at wavelengths between 3-4 µ was observed in a 4-layer sample that measured 15 µ × 15 µ in size.

Figure 17. Densities of photon states for the fcc structure of air spheres in a dielectric medium of refractive index 3.6. The filling ratio of air is 74%.

the refractive index contrast n=n2 /n1 . The spheres may be air (n1 =1) enclosed by the interconnected higher dielectric background. Calculations have found the inverse fcc structure to have both a low-frequency pseudogap between bands 2 and 3 where the density of photon states reaches zero, and a high frequency three-dimensional band gap between higher bands 8 and 9 (Fig. 17; Fig. 8) . The gap is ∼5% in magnitude for a refractive index contrast of 3.5. Generally a higher refractive index contrast (>2.4) is necessary to observe the full band gap. The inverse structure has much more desirable photonic gaps than the direct structure of close-packed dielectric spheres. Other stacking sequences such as hexagonal close-packed and double-hcp also yield high band gaps (79). . We have also achieved this inverse opal structure by a somewhat different ceramic technique where the ordering and filling process was performed simultaneously (88, 89). The starting point was a slurry of nanocrystalline titania which was mixed with a suspension of polystyrene spheres (88, 89). On drying of this slurry on a slide, ordering of the spheres was observed in a region of the slide evidenced by a band of color. The spheres were ordered and the interstitial regions were filled by the titania nanoparticles. The spheres were then removed by calcination above 200 C, leaving a macroporous solid where close-packed spherical air cavities are enclosed by a high dielectric matrix. It was necessary to sinter the titania near 550 C to densify it and improve the refractive index contrast (88, 89). The inverse opal structure was preceded by ordered structures of monodisperse spheres which did not have appreciable photonic band gaps but did exhibit stop bands (90–93) from the band gaps along the stacking direction. Robotic Micromanipulation. Robotic micromanipulation is a very specialized technique for fabricating precise three-dimensional photonic crystals of small sizes. Micromanipulation of two different size spheres has been employed to buld up a diamond lattice (94). This was

Soft Lithographic Methods. Soft lithography using microtransfer transfer molds has evolved into a very popular technique for generating economical large area photonic crystals, down to infrared and optical length scales. The basic principle is based on transferring a whole layer of polymer pattern using an elastomer mold onto a substrate or multilayer polymer template (96–98). The starting point is to create a master stamp which is typically a silicon wafer on which a patterned photo resist is created by standard photolithographic methods. Typically photo resist is spun on to a silicon wafer and exposed to a patterned mask and then developed and baked. For the layer-by-layer structure this pattern consists of a onedimensional layer of rods. The next step is to create an elastomeric mold from the master stamp. After the PDMS is cured, it is removed gently from the master stamp resulting in a relief structure on the elastomeric mold. The third step is to fill the troughs of the PDMS with epoxy. Care is taken to not overfill the epoxy; otherwise excess epoxy can spill over into the underlying layers. One way to fill with epoxy is to put a tiny drop of epoxy on the PDMS mold and drag the drop with a wire across the surface. The epoxy can be oven cured. In the final step the epoxy filled PDMS is placed in contact with a glass or silicon substrate. After the epoxy has hardened the PDMS is peeled off leaving a set of parallel epoxy rods on the substrate and one layer of the polymer template is thus created. The second and subsequent layers are fabricated in the same manner, except the epoxy filled elastomeric mold is applied to a one-layer or multi-layer structure on the substrate. By repeating this synthesis a multi-layer structure can be fabricated. Alignment of third and subsequent layers is a major concern in this method. It has proved to be very fruitful to align using the technique of Moire fringes and identify well-aligned regions in the sample. Layer-bylayer photonic crystals with pitch of 2.5 and 1 µ have been fabricated with this method, which can be extended into the optical length scale. Dielectric and metallic structures have been fabricated with this technique. 3-D PHOTONIC CRYSTALS WITH DIELECTRICS In this section, we study some of the recent achievements in the field, and we point out some of the difficulties that may rise in the future especially for photonic crystals operating at the optical frequencies. We start with the defect cases. We study 3-D layer-by-layer photonic crystals (24–30). The structure is made of layers of cylindrical alumina rods with a stacking sequence that repeats itself every four layers with repeat distance, c = 1.272 cm. Within each layer, the rods are arranged with their axes parallel and separated by a distance a = 1.123 cm. The orientations of the axes are rotated by 90◦ between adjacent layers. To obtain

Photonic Band Gap Materials

Figure 18. The transmission of EM waves propagating through a 3-D layer-by-layer PBG consisting of 16 layers of rods. The dotted lines correspond to the periodic case, whereas the solid lines correspond to the defect case in which every other rod from the eighth layer has been removed. Panels (a) and (b) correspond to the polarization with the electric field parallel and perpendicular to the first layer of rods.

the periodicity of four layers in the direction of stacking, the rods of the second neighbor layers are shifted by a distance of a/2 in the direction perpendicular to the rods axes (24–30). In order to simulate this structure with the real space TMM, we divide the unit cell into 7 × 7 × 8 subcells assuming that the z axis is along the stacking direction. Figure 18 shows the transmission of EM waves incident on a layer-by-layer photonic crystal with four unit cell thickness (16 layers of rods). The k vector of the incident wave is along the stacking direction (z axis). For the periodic case (dotted lines in Fig. 18), there is a gap between 11 and 15.7 GHz for both polarizations. We introduce a defect in this structure by removing every other rod in the eighth layer. A defect peak appears at 12.58 GHz. The width of the peak (0.016 GHz) is almost the same for both polarizations, and the transmission at the top of the peak (−3.4 and −29.7 dB for each polarization) is higher for the polarization where the incident electric field is parallel to the axis of the removed rods. In general, the transmission for the parallel polarized waves is more affected by the defect than the perpendicular polarized waves. The Q factor and the defect frequency are in very good agreement with measurements in the same configuration (30). However, the measured transmission at the top of the peak is about 10 dB smaller than that calculated, most probably because of some small absorption of the alumina rods (30). By increasing the thickness to eight unit cells (32 layers of rods), the width of the peak becomes 10−6 GHz, which corresponds to Q greater than 106 , whereas the defect frequency and the transmission at the top of the peak remain almost the same (12.61 GHz and −3.8 dB, respectively).

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Figure 19. The transmission of EM waves propagating in a system similar to the one described in Fig. 18. The dielectric constant of the rods is | ∗ epsilion ∗ | = 9.61 + ix. E field is parallel to the axis of the removed cylinders. Curves are for different values of the absorption x.

In the optical wavelength region, the dielectric constant of most materials has an appreciable imaginary part. In order to study the effect of the absorption on the peak transmission resulting from defects, we calculated the transmission of a layer-by-layer structure with a defect (Fig. 19). The structure is similar to the one described in Fig. 18, in which we removed every second rod from the eighth layer (the system contains 16 layers of rods). We assume that the dielectric constant is given by = 9.61 + ix. Increasing the imaginary part of the dielectric constant, the Q, as well as the transmission at the peak, decreases. In particular, Q = 800, 262, 163, and 90 for x = 0, 0.05, 0.1, and 0.2, respectively, whereas the transmission at the peak is −3.4, −14.0, −18.7, and −24.0 dB (Fig. 16). The introduction of the absorption makes the peak wider and the transmission on the peak smaller. Even in periodic structures, the effect of the absorption could significantly change the transmission. Figure 20 show the transmission of a periodic layer-by-layer structure similar to the one described in Fig. 18 with three unit cells thickness. The real part of the dielectric constant is 9.61. By increasing the imaginary part, the transmission decreases at all the frequencies. Especially at the upper edge of the gap, the transmission has dropped by almost 10 dB compared to the case with zero imaginary part. In the non-absorbing cases, it is commonly accepted that the photonic crystal must be as thick as possible because the transmission inside the gap decreases as the thickness of the crystal increases. However, in photonic crystals constructed of materials with significant absorption, the transmission is thickness-dependent at all frequencies. So, it is possible that we will not be able to measure the transmission at the upper edge of the gap, which is more affected by the absorption, if it is less than the noise level of our measurements.

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Photonic Band Gap Materials

Figure 20. The transmission of EM waves propagating in a layerby-layer system similar to the one in Fig. 4 with three unit cells thicknesses. The dielectric constant of the rods is | ∗ epsilion ∗ | = 9.61 + ix, where x is 0 and 0.2 (dotted and solid lines, respectively).

Figure 21. Measured transmission of electromagnetic waves incident a) along the x-axis (in the plane of the rods) and b) along the stacking direction (z-axis) of the layer-by-layer structure made with alumina rods. A rod spacing of 1.1 cm was used.

We show for comparison the measured transmission (Fig. 21) of EM waves incident from the top (z-axis) and from the side (x-axis) of the layer-by-layer photonic crystal composed of dielectric rods, with a spacing of 1.1 cm (25). The wider band gap along the stacking direction as predicted from the theoretical calculations is clearly evident. Very sharp defect states with high Q can be created with defects within the 3-D photonic crystal. Various point defects in the InP layer-by-layer photonic crystal have been synthesized by S. Noda et al (99) at infrared frequencies, by either removing potions of rods or adding material. Cavity modes were observed at these point defect sites. It should

be notedfor comparison that the highest Q defect with a Q exceeding 105 has been achieved by S. Noda and collaborators (<xref target="W4410-bib-0099 W4410-bib-0100" style="unformatted"/>) in a two-dimensional heterophotonic crystal. This 2-D photonic crystal has a defect region of slightly different lattice constant (100, 101), sandwiched between the regular lattice constant material. The field of 3-D photonic crystals has been very rich with examples of alternative photonic crystal structures with complete photonic band gaps. The simple-cubic lattice has a fundamental photonic gap between the lowest bands 2 and 3 with a magnitude of ∼6% for a refractive index contrast of 3.6. When spheres are introduced on the lattice sites and connect to each other with narrow dielectric rods, a higher band gap between bands 5 and 6 opens up and reaches a maximum value of 12% (102) for a refractive index contrast of 3.6. The simple cubic lattice of rods was also fabricated at infrared frequencies, showing the expected band gap (103). An alternative 3-D photonic crystal was designed by Johnson and Joannopoulos (104). This 3-D periodic dielectric structure with a large complete photonic band gap (PBG) consists of a structure with a sequence of planar layers, identical except for a horizontal offset, and repeating every three layers to form an fcc lattice. The layers can be thought of as an alternating stack of the two basic twodimensional (2D) PBG slab geometries: rods in air and air cylinders in dielectric. These high-symmetry planar crosssections may simplify the integration of optical devices and components by allowing modification of only a single layer, using simple defects of the same form as in the corresponding 2D systems. Gaps of over 21% are obtained for Si/air substrates. Reasonable gaps, over 8%, were achieved even for the moderate index ratio of 2.45 (Si/SiO2 ). A different 3D photonic crystal layered photonic crystal structure was also designed (105). This has a connectivity that is different from diamond and possesses square symmetry within each layer. This structure has a complete photonic band gap of 18% of the midgap frequency with a dielectric contrast of 12:1. A waveguide in this crystal was created by removing a row of rods from a single layer. A planar diamond structure with triangular lattice meshes, supported by vertical rods was also developed by us (106) with complete band gaps exceeding 29% for refractive index contrasts of 3.6 (106). This structure is amenable to fabrication with layer-by-layer methods.

Selected Applications of Photonic Crystals. Photonic crystals have been known to create cavities with defect modes within the photonic band gap. A planar Fabry-Perot type of cavity was created in the layer-by-layer crystal by separating the unit cells by a displacement d. The displacement d was adjusted to produce the defect mode within the band gap (107). A dipole antenna was placed within the cavity and driven at frequencies within the band gap. At the frequency of the defect mode an exceptionally directional pattern can emerge from the dipole antenna. Using an asymmetric cavity with different unit cells on the two sides we obtained (107) a source radiating in a forward di-

Photonic Band Gap Materials

rection with a full width of 14◦ in the E-plane and 12◦ in the H-plane (Fig. 19). It would be extremely difficult to obtain such an exceptionally directive source with conventional antenna array and we estimate >300 antennas in a phased array would be required to achieve such directionality. The simulated radiation pattern from the FDTD method agreed very well with the measurements performed on microwave-scale photonic crystals (107, 108). Very directional antennas have also been created by locating the antenna sources inside the photonic crystal and measuring the radiation emerging from the surface (109). This work demonstrates that altering the densities of photonic states by a photonic crystal can drastically alter the emissive properties of sources, one of the original motivations for photonic crystals. Modifying the spontaneous emission of atomic sources by altering the photonic densities of states in 2-D and 3-D photonic crystals is an area of much activity. Waveguides can easily be created in the layer-by-layer photonic crystal by removing an entire rod (X-waveguide), portions of a rod along the y-axis (Y-guide), or rod segments along the z-axis (Z-guide) (110). The X-waveguide has a wide band in the photonic band gap where the modes are transported. Waveguide bends are a novel application where EM waves can be turned through sharp bends without loss/scattering inherent in traditional waveguide geometries. Since the waveguide mode is confined within the bandgap, there is no loss to radiation modes in the bend region. We have carefully optimized the geometry of the bend to achieve near 100% bending efficiency and show (Fig. 23) the bending of a beam through 90◦ between two X-waveguides, simulated with the FDTD method. These simulations agree very well with measurements on microwave-scale photonic crystals(111, 112). Although such waveguide bends have been extensively investigated in 2-D photonic crystals for optical circuits, there can be significant loss in the z-direction at the bend region in the 2-D photonic crystals. Another novel application is the add-drop filter using 3-D photonic crystals. In telecommunications applications using wavelength division multiplexing to carry dense streams of data, the input stream consists of various frequency channels. It is critical to select or drop one frequency from this stream to an output guide. Conversely it is also necessary to add a particular frequency channel to an in input stream. We have achieved such an add-drop filter with our 3-D photonic layer-by-layer photonic crystal. The configuration consists of an input and output waveguides which are separated by L and uncoupled ( since L is several unit cells). There is a defect cavity located in a layer one unit cell above the waveguides that can support localized defect modes. When the input frequency matches the frequency of the defect that mode is excited in the defect cavity and transported to the output waveguide (113). Other waveguide modes are unaffected. Such 3-D add-drop filters can be an alternative to the extensively studied adddrop filters investigated with 2-D photonic crystals. It is interesting to note that the concept of impedance which is critical to matching waveguides and has played a very important role in microwave engineering, can be de-

15

Figure 22. Radiation pattern in the (a) E-plane and the (b) Hplane for a dipole radiator placed inside an asymmetric cavity formed inside a photonic crystal. 2 unit cells are separated from 3 unit cells with this planar cavity.

Figure 23. Bending of an electromagnetic wave around a 90 degree bend formed by two X-waveguides in the layer-by-layer photonic crystal, using FDTD simulation. The frequency is chosen to lie within the band gap of the photonic crystal.

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Photonic Band Gap Materials

Figure 24. Measured reflection and transmission of the metallic layer-by-layer structure as a function of the number of layers, compared with simulations with the scattering matrix method.

fined analogously for photonic crystals as a ratio of the energy density to the power flow at each frequency (114). This impedance concept can account for reflection and transmission of waves from photonic crystals. Metallic Photonic Crystals. Metallic photonic crystals have been an active subfield. As mentioned earlier the layer-by-layer photonic crystal has been fabricated with tungsten (58, 59) at rod separations from a=2−5 µ using advanced silicon processing methods (58, 59) or more recently with soft lithographic methods (97). The basic characteristic of the metallic layer-by-layer photonic crystal is that the reflectance is high (near 100%) and the transmission negligible at long wavelengths λ>a where the details of the structure are not resolved. At a wavelength λ∼a the reflectance dips and the transmission increases (Fig. 24) for shorter wavelengths. There is an absorption peak located near the wavelength λ∼a. Since the emission is the absorption modulated by the black-body emissivity, the thermal emissivity can be significantly altered and consists of an emission peak located near λ∼a. In contrast to connected metallic structures it is instructive to compare the case of isolated metallic scatterers. In this case the system is highly transmitting with little reflection for low frequencies upto a cutoff frequency defined by the lattice spacing. Above the cut-off frequency the transmission decreases with increase of reflection and absorption in the structure. An example of this behavior is the EM wave propagation in isolated metallic scatterers embedded in air [cermet topology (115, 116). Figure 25 shows the transmission and absorption of EM waves propagating in simple cubic (s.c.) lattice consisting of metallic spheres with filling ratio f = 0.03. The system is infinite along the x and y directions, whereas its thickness along the z axis is L = 4a, and the incident waves with k along the z axis. The results for both polarizations are the same because of the lattice symmetry. For the present as well for all the following cases, each unit cell is divided into 10 × 10 × 10 cells. Calculations with more subcells

show that the convergence is better than 5% for the periodic cases, and better than 15% for the defect cases. There are two drops in the transmission (Fig. 25); the first around νa/c = 0.45 and the second (and sharpest) one around 0.85. The wavevector k, parallel to the z axis corresponds to the − X direction in the k space. In this case, we expect the first gap to appear at the edge of the zone (in the X point) for νa/c about 0.5, which is slightly higher than the frequency where the first drop in the transmission appears in this direction (Fig. 25). Because of the small filling ratio, there is no full band gap because the gaps in different directions do not overlap. We find similar results for fcc, bcc, and diamond structures with isolated metallic spheres or cubes. For the cases where the metal forms isolated scatterers, the results are similar to those of the dielectric PBG materials. The present results for the isolated metallic scatterers are in agreement with the results of a recent work (117) in which monolayers consisting of metallic spheres with radius between 10 and 100 nm were studied. The frequency-dependent Drude dielectric function (118) was used in these simulations Perspectives and Future Directions. We briefly describe some applications of PBG materials on waveguides, lightemitting diodes, nonlinear effects, and quantum electrodynamics. Recent studies of two-dimensional photonic band gap waveguides have shown encouraging results for the use of photonic crystals in order to improve waveguide efficiency (119, 120). In one of the studies (119), a two-dimensional square lattice consisting of dielectric cylinders was used. A line of cylinders was removed in order to create the waveguide geometry. Numerical simulations using the FDTD method revealed complete transmission at certain frequencies and very high transmission (>95%) over wide frequency ranges. High transmission is observed even for 90◦ bends with zero radius of curvature, with maximum transmission of 98% as opposed to 30% for analogous conventional dielectric waveguides. More studies, especially on 3-D structures, are needed. Also, measurements on similar systems are highly desirable. In another interesting theoretical study (121), a thin slab of two-dimensional photonic crystal was shown to alter drastically the radiation pattern of spontaneous emission. By eliminating all guided modes at the transition frequencies, spontaneous emission can be coupled entirely to free space modes, resulting in a greatly enhanced extraction efficiency. Such structures might provide a solution to the long-standing problem of poor light extraction from high refractive-index semiconductors in light-emitting diodes (121). Extension of these studies into 3-D photonic crystals will be very useful. Two-dimensional photonic crystals have been combined with an array of holes in a metal sheet (49) to enhance the emission of infrared wavelengths from this structure. There has been research on nonlinear photonic band gap materials, focused on 1-D photonic crystals (122–125). Under certain circumstances, there may be nonlinear wave propagation within the photonic band gap. For a large-scale photonic band gap material, the propagation of high intensity, nonlinear solitary waves may provide a practical way

Photonic Band Gap Materials

17

Figure 25. Transmission and absorption vs. the dimensionless frequency | ∗ nu ∗ |a/c for EM waves propagating in a 3-D s.c. lattice consisting of metallic spheres with f = 0.03, L = 4a, and | ∗ theta ∗ | = 0◦ . Solid and dotted lines correspond to a = 1.27 and 12.7 | ∗ nu ∗ |m, respectively.

of coupling large amounts of optical energy into, and out of, the otherwise impenetrable photonic band gap. A stationary solitary wave may be regarded as a self-localized state. In a perfectly periodic material, the high light-intensity itself creates a localized dielectric defect through the nonlinear Kerr coefficient. Unlike the localized state induced by static disorder, the localized dielectric defect is free to move with the light intensity field. The result is a solitary wave that can move through the bulk photonic band gap material with any velocity ranging from zero up to the average speed of light in the medium. Using a variational method, John and Akozbek (126) found a variety of different solitary wave solutions in two dimensions. Their work suggests that photonic band gap materials in higher dimensions may have a variety of interesting bistable switching properties that go beyond the simple characteristic of one-dimensional dielectrics. Because an exact solution is no longer possible in two and three dimensions, numerical methods are required to solve this problem. The FDTD method described earlier with implementation of the formalism described in 127 is very promising for the solution of this problem. There are also very interesting implications of photonic crystals on quantum electrodynamics. For a single excited atom with transition frequency ωo to the ground state, which lies within the band gap, there is no true spontaneous emission of light. A photon that is emitted by the atom finds itself within the classically forbidden energy gap of the photonic crystal. The result is a coupled eigenstate of the electronic degrees of freedom of the atom and the electromagnetic modes of the photonic crystal. This photon-atom bound state (128–130) is the optical analog of an electron-impurity level bound state in the gap of a semiconductor. When a collection of atoms is placed into the photonic crystal, a narrow photonic impurity band is formed within the larger photonic band gap. This may lead to new effects in nonlinear optics and laser physics (128). Photonic crystals have created such a major revolution in manipulating electromagnetic waves that several new

fields of active research have emerged from them. One of the most active subfields are meta-materials or left-handed materials. Another subfield is the area of sub-wavelength hole arrays. Left-handed materials can be realized by having ? and ? negative in a certain frequency range. Higher frequency bands on photonic crystals where the group velocity is opposite to the propagation k-vector are an example of left-handed behavior. Sub-wavelength arrays of holes in metallic layers are another sub-field where metallic components with strong diffractive effects come into play. While many worldwide research groups have engineered very impressive 2-D photonic crystals, a goal of much present research is to fabricate three-dimensional photonic crystals at optical wavelengths and generate new applications to light-emitting devices and lasers. ACKNOWLEDGMENTS It is a pleasure to thank our colleagues C. T. Chan, E. Ozbay, W. Leung, K. Constant, J.H. Lee, and J. S. McCalmont for their insights and collaboration. We thank D. Crouch for providing FDTD simulations. This work was partially supported by the Director for Energy Research, Office of Basic Energy Sciences and Advanced Energy Projects. We also acknowledge support from the Department of Commerce through the Center for Advanced Technology (CATD). One of us (R.B) also acknowledges National Science Foundation funding through grants ECS-0601377 and DMR-0346508. The Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. W7405-Eng-82. BIBLIOGRAPHY 1. Development and applications of materials exhibiting photonic band gaps, J. Opt. Soc. Am. B 10: 1993(a special fea-

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Photonic Band Gap Materials ture issue edited by C. M. Bowden, J. P. Dowling, and H. O. Everitt). For a review seeC. M. Soukoulis (ed.), Photonic Bandgaps and Localization, New York: Plenum, 1993. J. D. Joannopoulos R. D. Meade J. N. Minn Photonic Crystals, Princeton, NJ: Princeton University Press, 1995. For a review seeC. M. Soukoulis, (ed.), Photonic Band Gap Materials, papers in Proc. NATO Advanced Study Institute, Kluwer, 1996. E. Yablonovitch Phys. Rev. Lett., 58: 2059, 1987. J. Martorell N. M. Lawandy Phys. Rev. Lett., 66: 887, 1991. J. Martorell N. M. Lawandy Phys. Rev. Lett., 65: 1877, 1990. B. Y. Tong et al. J. Opt. Soc. Am. B, 10: 356, 1993. S. John Phys. Rev. Lett., 58: 2486, 1987; S. John, Comments Cond. Mat. Phys., 14: 193, 1988. A. Z. Genack N. Garcia J. Opt. Soc. Am. B, 10: 408, 1993. G. Kurizki A. Z. Genack Phys. Rev. Lett., 66: 1850, 1991. S. John Physics Today, 44: 32, 1991. L. M. Brekhovskikh Waves in Layered Media, New York: Academic Press, 1960. P. Yeh Optical Waves in Layered Media, New York: John Wiley & Sons, 1988. A. Thelen Design of Optical Interference Coatings, New York: McGraw-Hill, 1989. C. M. Bowden J. P. Dowling H. O. Everitt, (eds.) Development and applications of materials exhibiting photonic band gaps, J. Opt. Soc. Am. B, 10: 1993. E. Yablonovitch, Photonic crystals,. J. Mod. Opt., 41: 209, 1994. E. Yablonovitch T. J. Gmitter Phys. Rev. Lett., 63: 1950, 1989.

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M. M. SIGALAS K. M. HO C. M. SOUKOULIS R. BISWAS G. TUTTLE

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Photonic Band Gap Materials Ames Laboratory-U.S. DOE and Department of Physics and Astronomy, Ames, IA, 50011 Ames Laboratory-U.S. DOE, Department of Physics and Astronomy, Microelectronics Research Center, Ames, IA, 50011 Microelectronics Research Center, Department of Electrical and Computer Engineering, Ames, IA, 50011