EDITORIAL ADVISORY BOARD G.S. Agarwal
Stillwater, USA
T. Asakura
Sapporo, Japan
M.V. Berry
Bristol, England
C. Brosseau
Brest, France
A.T. Friberg
Stockholm, Sweden
F. Gori
Rome, Italy
D.F.V. James
Toronto, Canada
P. Knight
London, England
G. Leuchs
Erlangen, Germany
P. Milonni
Los Alamos, NM, USA
J.B. Pendry
London, England
J. Peˇrina
Olomouc, Czech Republic
J. Pu
Quanzhou, PR China
W. Schleich
Ulm, Germany
PROGRESS IN OPTICS VOLUME 53
EDITED BY
E. Wolf University of Rochester, N.Y., U.S.A.
Contributors U. L. Andersen, B. Crosignani, E. DelRe, M. R. Dennis, P. Di Porto, R. Filip, G. S. He, U. Leonhardt, M. Martínez-Corral, K. O’Holleran, M. J. Padgett, T. G. Philbin, G. Saavedra
Amsterdam • Boston • Heidelberg • London • New York • Oxford • Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo
Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2009 Copyright © 2009 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; e-mail:
[email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/ permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Catalog Card number: 61-19297 ISBN: 978-0-444-53360-9 ISSN: 0079-6638 For information on all Elsevier publications visit our web site at elsevierdirect.com Typeset by: diacriTech, India Printed and bound in Hungary 09 10 11 12 10 9 8 7 6 5 4 3 2 1
PREFACE
This volume contains articles which present reviews of current research in six areas of classical and quantum optics. Some of them deal with theory, others with experiments. The first article by M. Martinez-Corral and G. Saavedra gives an account of modern techniques that have been developed to provide high quality microscope images. The techniques include single photon confocal microscopy, the so-called confocal theta microscopy, standing wave microscopy, and two-photon excitation scanning microscopy. The second article by U. Leonhardt and T. G. Philbin reviews researches concerning the use of so-called meta-materials for application in devices that perform geometrical transformations. They have potential applications to production of perfect lenses and cloaking devices. The article by E. Del Re, B. Crosignani, and P. DiPorto which follows, reviews developments in a field which originated about a decade ago, namely photorefractive solitons. They are forerunners in the field of optical soliton physics, providing the foundation of a rich field in the physics of nonlinear waves. The article traces and brings together various elements that form our present day understanding of the underlying physics. The next article, by Guang S. He, gives an overview of developments in the field of stimulated light scattering with intense coherent light. It provides a review of the subject and discusses the principles of the materials and experimental features and applications of various scattering processes. The fifth article by M. Dennis, M. Padgett, and K. O’Holleran presents a comprehensive review of a class of interesting phenomena in the field of “singular optics” which originated in the 1970’s: namely of “optical vortices and polarization singularities”. The article presents a review of the development since the subject originated and includes discussions of topics such as the orbital angular momentum of light in the context of singular optics. The concluding article on quantum feedback control of light by U. Andersen and R. Filip is concerned with processes of gaining information about the dynamics of physical systems and about using the information to change the system in real time. Such processes are of v
vi
Preface
importance in various branches of traditional engineering but, more recently they have proved to be of interest in the quantum optics domain. Some examples of such developments are presented. As previous volumes in the series, the present one shows again that optics continues to be a dynamic field in which interesting new developments are taking place. Emil Wolf Department of Physics and Astronomy and The Institute of Optics, University of Rochester, Rochester, New York 14627 February 2009
CHAPTER
1 The Resolution Challenge in 3D Optical Microscopy Manuel Martínez-Corral* and Genaro Saavedra*
Contents
1 Introduction 2 Basic Theory for Microscope Imaging 2.1 Three-Dimensional Imaging as the Result of Axial Scanning 2.2 The Virtual 3D PSF 2.3 The Optical-Sectioning Capability 2.4 Metrics for the Optical-Sectioning Capability 3 The High-Numerical-Aperture Approach 3.1 Calculation of the PSF 3.2 Calculation of the OTF 3.3 Metrics for Resolution Improvement 3.4 Sampling Expansion 3.5 Spherical Aberration 4 Optical-Sectioning Microscopy 4.1 Confocal Scanning Microscopy 4.2 Structured Illumination Microscopy 4.3 Axially-Oriented Structured Illumination Microscopy 4.4 Two-Photon Excitation Scanning Microscopy 5 Conclusions Acknowledgments References
1 3 8 9 10 11 18 19 22 24 28 31 33 34 43 53 62 63 64 65
1. INTRODUCTION The spatial resolution of optical microscopes is mainly restricted by diffraction, that is, by their capability to produce a tight diffraction spot when imaging a light point source. Since Abbe (1873) formulated his wave theory for microscopic imaging, the improvement of resolution of optical * Department of Optics, University of Valencia, E46100 Burjassot, Spain Progress in Optics, Volume 53, ISSN: 0079-6638, DOI: 10.1016/S0079-6638(08)00201-1. Copyright © 2009 Elsevier B.V. All rights reserved.
1
2
The Resolution Challenge in 3D Optical Microscopy
microscopes has been the aim of many research efforts. The need for smaller spot sizes has induced fruitful researches not only in the case of optical frequencies (Sales, 1998) but also in other spectral ranges like X-ray microscopy (Miao et al., 2002a) or electron microscopy (Miao et al., 2002b). As pointed out by Gustafsson, Agard, and Sedat (1999), the light microscope remains an irreplaceable research tool for modern biology. Unlike any other microscope, it allows the study of samples in vivo or in their native hydrated environments, a highly specific labeling of multiple components, and the observation of detailed internal structures of 3D samples. In the search for maximum spatial resolution, emphasis has generally been placed on the use of modern microscope objectives with the highest available numerical aperture (NA) (Blanca and Hell, 2002). However, since lens design is by now a mature technology, the light-collecting angle of commercially available objective lenses is close to its practical maximum, with little if any room for further improvement. Another problem of conventional optical microscopy is the fact that even with the best optical elements it is not possible to obtain sharp images of 3D biological samples, since any image focused at a certain depth in the sample contains blurred information from the rest of the sample. This fact gives rise to 3D images with deteriorated contrast. In the last few years, it has been realized that the classical resolution limits, even though imposed by physical laws, can in fact be exceeded (Gustafsson, 1999). Three particularly important assumptions of Abbe theory are the following: (i) observation takes place in the conventional geometry in which light is collected by a single objective lens; (ii) the excitation light is uniform throughout the sample; (iii) fluorescence takes place through normal, linear absorption, and emission of a single photon. The negation of any these assumptions lead to new basic concepts for resolution extension. The negation of assumption (ii) permitted the design of a new class of optical microscope, which not only allows the improvement of lateral spatial resolution but also the achievement of important optical sectioning. We refer to the single-photon confocal scanning microscope (CSM) in which the monochromatic light emanating from a point source is focused onto a small region of the 3D fluorescent sample by a high-NA objective (Brakenhoff, Blom, and Barends, 1979; McCutchen, 1967; Minsky, 1988; Sheppard and Choudhury, 1977). The fluorescent light emitted by the sample is collected by the same objective, and it passes through a pinhole centered at the conjugated point, and is finally detected by a large-area sensor. An interesting feature of CSMs is that they deliver an electronic image that is available for digital signal processing (Pawley, 1995). The main advantage of CSMs is their sectioning power due to the light rejection from out-of-focus parts of the sample, but since the diffraction pattern of a focused bright spot
Basic Theory for Microscope Imaging
3
is naturally prolated for any NA, CSM suffers from anisotropic imaging, which in turn limits the effective lateral resolution. As a classical result, the related point-spread function (PSF) is the product of the two PSFs of the illuminating and collecting lenses, thereby providing not only an increase of lateral resolution, but mainly the ability to transmit axial frequencies. Another realization of non-uniform illumination is through patterned illumination techniques. In the so-called structured illumination microscopy, a periodic pattern transverse to the optical axis is projected onto the specimen and a stack of 2D images is recorded after scanning the object axially. By proper decoding procedures, it is possible to improve the lateral resolution, as well as to obtain an important capability of optical sectioning. The advantages of structured illumination over CSM are in imaging speed and light efficiency (Heintzmann and Ficz, 2006). The negation of assumptions (i) and (ii) has led to the invention of a new scanning imaging technique, which produces 3D images with quasiisotropic resolution. We refer to 4Pi scanning microscopy, in which the sample is illuminated with a periodic pattern parallel to the optical axis and the fluorescence light is collected by two opposed high-NA objectives (Hell and Stelzer, 1992a). A serious drawback of confocal architectures is photobleaching, which appears since the entire sample is bleached when any single plane is imaged. Another disadvantage of this technique when used in biomedical imaging is its poor depth penetration. To solve these problems, negation of assumption (iii) was invoked in proposing two-photon excitation (TPE) scanning microscopy (Denk, Strickler, and Webb, 1990). This non-linear imaging technique relies on the simultaneous absorption of two photons, whereby a single fluorescence photon is emitted (Göppert-Mayer, 1931). The overall fluorescent light is collected, and finally, the image is synthesized from the 3D sampling of the object. Two-photon excitation generally uses near-infrared light, which suffers less from absorption and scattering by biological tissues, allowing deeper penetration of the excitation beam. Besides, since photobleaching depends here on the time-averaged square of the intensity distribution, it is restricted to the neighborhood of the imaged plane. Two-photon, or even multiphoton, excitation fluorescence microscopy is even more attractive both because the imaging process can be considered as self-pinholed, due to the square dependence of the up-converted signal, and because contrast is dramatically enhanced by the subsequent side-lobe lowering in the illuminating lens PSF.
2. BASIC THEORY FOR MICROSCOPE IMAGING For an intuitive understanding of the optical principles of microscope imaging, it is convenient to start by analyzing a configuration (Figure 1)
4
The Resolution Challenge in 3D Optical Microscopy
that schematizes a conventional wide-field optical microscope by means of a telecentric arrangement (Born and Wolf, 1999a). Take into account that modern microscope objectives are usually designed for infinite conjugate ratio (Juskaitis, 2003). A second lens of higher focal length, the tube lens, is used to provide a real image in the neighborhood of its back focal plane (BFP). The aperture stop of such an arrangement is just the aperture stop of the objective, which is usually placed at its BFP. In Figure 1, we have represented the telecentric system in the simplest form. It consists of two thin converging lenses, ideally of infinite diameter, that are coupled in an afocal manner. To allow the system to be telecentric at both the object and the image side, the aperture stop of amplitude transmittance p(xp ) is placed just at the common focal plane. We calculate the image of 3D samples that are labeled with fluorescent materials. When the 3D fluorescent sample is uniformly illuminated by a monochromatic light beam whose wavelength is within the excitation band of the fluorescent dyes, the 3D intensity distribution is proportional to the 3D function O(x, z) that describes the spatial distribution of fluorescence generation in the sample. The fluorescence radiated by the sample can be considered, in quite good approximation, to be quasi-monochromatic and spatially incoherent. To calculate the image of such 3D intensity
(x0 ,y
0 , z0 )
y p (x p,
L1
y p)
yp
x
L2
xp F1
y9 F F91; 2
x9
f1
(x 9 , y 0 9 ,z 0 9 ) 0
f1
F92
f2 f2
FIGURE 1 Schematic of a telecentric, wide-field optical microscope. The light emanating from the object is collected by the objective (L1 ) and focused by the tube lens (L2 ).
Basic Theory for Microscope Imaging
5
distribution, we start by calculating the intensity distribution in the image of an arbitrary point of the sample, for example that at (x0 , z0 ); see Figure 1. We perform at this stage the analysis in a paraxial context. From the point source emanates a monochromatic, spherical (in fact paraboloidal in the paraxial approximation) wavefront. At the front focal plane (FFP) of L1 , the amplitude distribution of the impinging spherical wave is given by (Goodman, 1996)
U(x, z = 0) = −
1 −ikz0 k e exp −i |x − x0 |2 , z0 2z0
(1)
with wave number k = 2π/λ and writing x = (x, y). Any axial distances involved here and in the forthcoming reasoning are, of course, oriented. This is, for example, the case for the focal lengths in Figure 1; their direction and sign are determined by the orientation of the arrow. In the case of z0 , a negative value corresponds to a point placed to the left of F1 . Note that we are considering only wavefronts that propagate in the positive z-direction. Therefore for positive values of z0 , Eq. (1) represents a virtual distribution of amplitudes (Lukosz, 1967; Ojeda-Castañeda and Gómez-Sarabia, 1989). Next, we make use of the well-known fact that the amplitude distribution at the BFP of a thin lens is related to the amplitude distribution at the FFP through a 2D Fourier transformation, namely (Goodman, 1996)
1 ˜ xp U Up (xp ) = , 0 p(xp ), iλf1 λf1
(2)
˜ is the Fourier transform of U. In this equation, we have multiplied where U the impinging wavefront by the amplitude transmittance of the aperture stop, p(xp ), and have omitted the irrelevant constant phase factor exp(i2kf1 ). The amplitude distribution at the BFP of L2 is obtained from Up (xp ) again through a 2D Fourier transformation, that is,
x x 1 1 ˜ x , 0 ⊗2 p˜ Up = U , U (x , z = 0) = iλf2 λf2 M M λf2
(3)
where ⊗2 represents the 2D convolution product, and M = −f2 /f1 stands for the lateral magnification of the telecentric system. From Eq. (3) it is clear that telecentric arrangements provide, stricto senso, 2D images, since the output is obtained as the 2D convolution between the uniformly scaled input and a 2D function. That function can then be named the PSF of the imaging system. Note however that such ability is not exclusive to telecentric arrangements, since any single lens can produce 2D images.
6
The Resolution Challenge in 3D Optical Microscopy
The specific feature of telecentric arrangements becomes apparent when one analyzes their response to any arbitrary light point source within 3D object space. If we introduce Eq. (1) into Eq. (3), we straightforwardly find that
M −e−ikz0 p˜ z0
U (x , z = 0) =
k x 2 ⊗2 exp −i |x − x0 | , λf2 2z0
(4)
where x0 = Mx0 and z0 = M2 z0 are, respectively, the transverse and axial coordinates of the conjugate of the object point source through the telecentric system (see Figure 1). It is remarkable that both the lateral and the axial magnification are independent of the coordinates of the point object. The amplitude distribution at any arbitrary plane within the image space, for example at a distance z from the BFP of L2 , is obtained after 2D convolution between the function U (x , z = 0) and the amplitude PSF of free-space propagation, namely
k 2 eikz exp i |x | . iλz 2z
(5)
After straightforward mathematical manipulation, we obtain
U (x , z ) = Mp˜
x k eik(z −z0 ) 2 |x ⊗2 , exp i − x | 0 λf2 z − z0 2(z − z0 )
(6)
which can be rewritten as
U (x , z ) = M h (x − x0 , z − z0 ),
(7)
k 2 x eikz ⊗2 exp i |x | h (x , z ) = p˜ λf2 z 2z
(8)
where
is just the 3D amplitude distribution in the image space generated by a monochromatic point source placed just at F1 . Let us now extend our analysis to the whole 3D fluorescent sample, whose spatial distribution of intensity is given by O(x, z). Note that in general one should take into account that the radiation emitted by distant object points might be scattered or reflected by nearer object points.
7
Basic Theory for Microscope Imaging
Such radiation blockage will, of course, affect the intensity distribution in the image and, unless the object is an aerial image, will always be present. However, we will assume that the first-order Born approximation holds (Born and Wolf, 1999b) so that the multiple scattering and depletion of the incident beam are negligible. In such case, the image intensity can be summed over the elemental slices that constitute the 3D object, because the first Born approximation assumes that the principle of superposition is valid. Then,
∞
I (x , z ) = M
2
2 O(x0 , z0 )h (x − M x0 , z − M2 z0 ) d2 x0 dz0 ,
(9)
−∞
or, in terms of (x0 , z0 ),
1 I (x , z ) = 2 M
∞ 2 x0 z0 , 2 h (x − x0 , z − z0 ) d2 x0 dz0 O M M
−∞
2 1 x z , 2 ⊗3 h (x , z ) . = 2O M M M
(10)
This equation demonstrates that a paraxial telecentric arrangement, when imaging 3D spatially incoherent objects in absence of radiation blockage, has the property of 3D linearity and stationarity. The proof is the fact that the 3D image is obtained as the 3D convolution between a uniformly scaled copy of the object and a 3D function. That function is naturally recognized as the 3D intensity PSF of the telecentric system. The intensity PSF is then obtained as the squared modulus of the function in Eq. (8), which in the integral form is
ikz
∞
h (x , z ) = iλe
p(xp ) exp −i
−∞
k z 2f22
|xp |
2
2π exp −i xp x d2 xp . λf2
(11) Although we are analyzing an incoherent imaging process, it can easily be shown that telecentric arrangements have the property of 3D linearity and stationarity in the case of coherent imaging as well. Thus, we can call the function h (x , z ) the amplitude PSF. Based on the above, one can state that telecentric systems have the ability to produce, stricto senso, 3D images. This capability is exclusive to telecentric arrangements, since focal arrangements produce in the image space
8
The Resolution Challenge in 3D Optical Microscopy
3D intensity distributions in which both the magnification and the impulse response depend on the axial coordinate.
2.1. Three-Dimensional Imaging as the Result of Axial Scanning Note that the matrix sensors, such as charge-coupled devices (CCD) or complementary metal-oxide semiconductors (CMOS), usually employed for recording the images provided by optical microscopes are 2D. This implies that the intensity distribution calculated in Eq. (10) does not correspond to any real experimental situation, since no 3D matrix sensor is available for recording the 3D image. Instead, in actual microscopy, a 2D matrix sensor is placed just at the BFP of the tube lens, and then, a stack of 2D images is recorded while stepping the object through the in-focus plane. According to Eq. (9), for the primary axial position of the object, the intensity distribution at the sensor plane, z = 0, is
I0 (x )
∞
2 = I x ,z = 0 = M O(x0 , z0 )h x − Mx0 , −M2 z0 d2 x0 dz0 . (12) −∞
2
By stepping the sample an axial distance −zS , the object plane of former axial coordinate zS is now in focus, and the intensity at the detector plane is
∞
I (x , zS ) = M
2 O(x0 , z0 + zS )h x − Mx0 , −M2 z0 d2 x0 dz0 . (13) −∞
2
This can be rewritten after performing the mapping
z = M 2 zS ,
z0 = M2 (z0 + zS ),
and
x0 = Mx0 ,
(14)
with the result
1 I (x , z ) = 2 M
=
∞ x0 z0 , |h (x − x0 , z − z0 )|2 d2 x0 dz0 O M M2
−∞
1 x z , ⊗3 |h (x , z )|2 . O M M2 M2
(15)
We have now shown that the same 3D convolution as in Eq. (10) can be obtained in a realistic imaging experiment. In this hybrid experiment, the 3D image is constructed by computer from a stack of 2D images
Basic Theory for Microscope Imaging
9
recorded when axially scanning the sample. At this point, we define as “wide-field” those microscopes that use a 2D matrix sensor for the acquisition, sometimes after axial scanning, of the image.
2.2. The Virtual 3D PSF If one reads Eq. (10) from left to right, one can understand the imaging process as the result of two successive phenomena, that is, the uniform scaling inherent in typical geometrical-optics imaging, and the diffractive effects due to the limited size of the apertures. The diffractive effects are introduced into the equation by the convolution with the intensity PSF. However, this form of “reading” the 3D imaging is not comfortable for microscopists, since the properties of the image seem to be associated with the characteristics of the tube lens. For this reason, it could be convenient to calculate the 3D image by switching the order of the successive phenomena. Thus, one can first calculate the virtual intensity distribution in the object space, which already incorporates the diffraction effects, and then the uniformly scaled copy of the virtual object. The virtual intensity distribution is calculated as
Iv (x, z) = M2 I (M x, M2 z).
(16)
After scaling Eq. (10) accordingly, we obtain the following expression for the virtual object:
Iv (x, z) = O(x, z) ⊗3 |h(x, z)|2 ,
(17)
where
eikz h(x, z) = z
∞
p(xp ) exp −i
−∞
kz 2f12
|xp |
2
2π exp −i xp x d2 xp λf1
(18)
will be called the virtual amplitude PSF. We have found, then, that the image formation capability of a microscope is fully characterized by the intensity PSF, |h |2 , or equivalently by its virtual conjugate in the object space, |h|2 . The latter is more intuitive for microscopists, because it relates the imaging properties of a microscope with the characteristics of the objective. Consequently, the forthcoming analysis will be performed in terms of the virtual intensity PSF, that is, in terms of the intensity distribution obtained when a microscope objective is illuminated by a monochromatic plane wave; see Figure 2. In what follows, we will use Eqs. (17) and (18) to describe the imaging features of optical microscopes, but simplifying notation by omitting the subscripts from the focal length and the intensity.
10
The Resolution Challenge in 3D Optical Microscopy
yp y
xp
x F
f f
F⬘
FIGURE 2 The imaging properties of an optical microscope are mainly determined by the focusing characteristics of the objective.
2.3. The Optical-Sectioning Capability From the aforementioned statements, one could believe that telecentric arrangements are especially qualified for 3D imaging, since they have 3D linearity and stationarity. However, as explicitly shown in Eq. (13), at any transverse section of the 3D image, the sharp image of the in-focus section of the 3D object is accompanied by the blurred images of out-of-focus sections of the 3D object. This fact significantly affects the contrast in the 3D image. At this point, we introduce a new concept for the evaluation of the ability of a system to form 3D images: the optical-sectioning capability is the ability of an imaging system to provide sharp, high-contrast 2D images of the different transverse sections of a 3D object (Wilson, 1990). To evaluate the optical-sectioning capability of telecentric systems, we start by analyzing the imaging features in the spectral domain. Since we are dealing with 3D signals, we can calculate the spectrum of the image by 3D-Fourier-transforming Eq. (17), namely
˜ I(Q) =
I(R) exp(−i2πQ · R) d3 R,
(19)
image
where R = (x, y, z), while Q = (u, v, w) represents the triplicate of spatial frequencies. For later use, we define u = (u, v). Next, we focus our attention on the axial-frequency content of the image,
˜ 0, w) = I(0,
I(x, z) exp(−i2πwz) d2 x dz.
(20)
image
Taking into account that total power,
∞ I(x, z) d2 x = P, −∞
(21)
11
Basic Theory for Microscope Imaging
must be conserved when varying z, we find that
˜ 0, w) = P δ(w), I(0,
(22)
where δ is de Dirac delta function. Equation (22) shows that, independently of the nature of the object and of the pupil function, I˜ is singular at Q = (0, 0, 0). In other words, the axial frequencies of 3D objects are not transmitted by wide-field optical microscopes, and therefore, they are absent from the 3D image. Now, we can refine the definition of the optical-sectioning capability: it is the degree of efficiency in transferring the different axial-frequency components of the 3D object. Consequently, we can state that conventional microscopes do not have optical-sectioning capability. It could be, however, surprising to a microscopist to hear about this lack of optical-sectioning capability, as different sections of a probe can be viewed with reasonable image quality simply by changing the axial position of the slide. The response is that, while it is possible to see the 2D sections of 3D objects, these sections have very poor contrast; see Figure 3a. The lack of optical sectioning becomes apparent when one tries to obtain images of structures oriented in the axial direction. Such structures are totally absent from the image; see Figure 3b.
2.4. Metrics for the Optical-Sectioning Capability To work with a general formulation that does not depend on the particular parameters of the microscope, it is usual to define a set of normalized, dimensionless coordinates. To this end, we take into account that pupil functions have compact support, and we identify by rm the outermost radius of the pupil. In such case, we can rewrite Eq. (18) as
1 h(x, z) =
2 i2πz w0 iλrm e
p(xp ) exp −i πz|xp |2 exp −i 2πxp x d2 xp ,
−1
(23) where the normalized coordinate at the pupil plane is defined as xp = xp /rm . The transverse and axial normalized coordinates within the focal volume are defined as
x=
rm x λf
and
z=
2 rm z. λf 2
(24)
2 . There is also a phase factor that depends on the parameter w0 = f 2 /rm
12
The Resolution Challenge in 3D Optical Microscopy
(a)
y L1
) yp p (xp, yp
x
L2
xp y⬘
F1
f1 f1
F⬘1⬅
F2
x⬘
f2 f2
F⬘2
(b) y L1
yp
x
y ) p (xp, p
L2
xp
y⬘
F1
f1 f1
F F⬘1⬅ 2
x⬘
f2 f2
F⬘2
FIGURE 3 Conventional microscopes provide images in which (a) transverse sections are reproduced, albeit with poor contrast, but (b) axial structure is lost.
In the very typical case in which the pupil function has rotational symmetry, the amplitude PSF can be rewritten as
1 h(r, z) =
2 i2πz w0 i2πλrm e
p(rp ) exp −iπz r2p J0 2π r rp rp drp ,
(25)
0
where rp = rp /rm and r = (rm /λf )r, r being the radial coordinate at the focal volume. As an example, Figure 4 plots the square modulus of Eq. (25) for the case of a circular clear aperture. We can recognize the typical Airy-disk profile in the section z = 0. Along the optical axis one
13
Basic Theory for Microscope Imaging
Normalized lateral coordinate: x
2
1
0
⫺1
⫺2 ⫺3
⫺2
0 1 ⫺1 Normalized axial coordinate: z
2
3
FIGURE 4 Meridian section of the intensity PSF of a wide-field microscope in the typical case of a clear circular aperture.
can recognize the expected sinc2 variation corresponding to the circular aperture.
2.4.1. The Integrated Intensity Function The optical-sectioning capability of an imaging system can be evaluated from its intensity PSF through the calculation of a function known as the integrated intensity (Sheppard and Wilson, 1978), which is defined as
∞ |h(x, z)|2 d2 x.
Iint (z) =
(26)
−∞
This function evaluates, section by section, the total power in the 3D image of a point. The integrated intensity function tells us how out-of-focus parts of 3D objects contribute to the 2D image of the in-focus section. Taking into account the power-conservation law, it is clear that the integrated intensity is constant in wide-field microscopy and therefore that all sections of the 3D object contribute with the same weight to the in-focus 2D image. To have a heuristic understanding of the integrated intensity, let us perform a conceptual experiment. As drawn in Figure 5, in the experiment, a laminar fluorescent layer is uniformly illuminated and axially scanned in the object space. When the layer is at arbitrary distance z0 from the objective, the fluorescence-generation function is
O(x, z) = δ(z − z0 ),
(27)
14
The Resolution Challenge in 3D Optical Microscopy
t escen Fluor r laye
y L1
yp p (x p, yp
x
)
L2
xp
y 9Sensor
F1
z0
f1 f1
F F91; 2
x9
f2 f2
F9 2
FIGURE 5 Conceptual experiment to define the integrated intensity function. The thin fluorescent layer is axially scanned and a stack of 2D images is captured with the sensor.
where a normalization factor has been omitted. The 2D image (the virtual object indeed) at the detector plane is given by the function
I(x, z) = δ(z − z0 ) ⊗3 |h(x, z)|2 ,
(28)
evaluated at z = 0, namely
I(x, 0) =
|h(x0 , z0 )|2 d2 x0 ,
(29)
which is just the formula of the integrated intensity. Consequently, we can understand the integrated intensity function as the response of the imaging system to an axially scanned fluorescent layer. Heuristically, the opticalsectioning capability is the ability of the imaging system to determine the axial position of the layer. In wide-field microscopes, this function is constant, which implies that these microscopes cannot discriminate the axial position of the layer.
2.4.2. The 3D OTF The function of an optical microscope is to provide magnified images of objects in which the details are too fine to be seen by the naked eye or to be resolved by a matrix image sensor. The fine details of the object correspond to the high spatial frequencies. The efficiency with which such periodic components are transmitted to the image depends on the optical system. The function that accounts for this efficiency is the so-called optical
Basic Theory for Microscope Imaging
15
transfer function (OTF), which will be denoted here as H(u, w). It can be calculated as the 3D Fourier transform of the intensity PSF. Naturally, since the intensity PSF is the square modulus of the amplitude PSF, the OTF can be obtained as the self-correlation
˜ w) ∗3 h(u, ˜ w). H(u, w) = h(u,
(30)
Although we are dealing with spatially incoherent imaging processes, ˜ w) as the coherent transfer function (CTF) we can identify the function h(u, of the system, which is calculated as the Fourier transform of the amplitude PSF, namely
∞
˜ w) = h(u,
h(x, z) exp{−i2π(x u + z w)}d2 x dz.
(31)
−∞
The normalized frequencies are related to the actual ones through
u=
λf u rm
and
w=
λf 2 w. 2 rm
(32)
Note that the normalization factor ρc = rm /λf is just the radial cutoff frequency of the system. By substituting Eq. (23) into Eq. (31), we obtain
˜ w) = h(u,
∞
p xp d2 xp
−∞
∞ × −∞
∞
exp −i2πx u + xp d2 x
−∞
1 2 dz, exp −i2πz w − w0 + xp 2
(33)
where some irrelevant factors have been omitted. It can immediately be found that
˜h(u, w) = p(u) δ w − w0 + 1 |u|2 . 2
(34)
The 3D CTF of a telecentric imaging system is confined onto a parabolic shell, as shown in Figure 6. Although the shell is axially symmetric, the value of the CTF at any point of the shell is given by the pupil function p(u), which in general may not be axially symmetric.
16
The Resolution Challenge in 3D Optical Microscopy
v
w0
u
w 0.5
FIGURE 6 The 3D CTF is confined onto a shell of a paraboloid of revolution about the w axis. The shell is axially shifted by w 0 .
To obtain the OTF, we perform the self-correlation of the parabolic shell. To simplify calculations, we consider the case of an axially symmetric pupil function in which case the OTF is axially symmetric as well. Therefore, it is not necessary to perform the correlation along all the transverse Cartesian frequencies, but only along the positive values in one Cartesian direction, namely
H(ρ, w) = H u+ , ρ, 0, w =
˜h α − ρ , β, γ − w 2 2 ρ w ∗ ˜ × h α + , β, γ + dα dβ dγ. 2 2
(35)
Taking into account Eq. (34) we obtain
ρ ρ p α − , β p∗ α + , β δ(w + αρ)dα dβ 2 2 = P(ρ, α) δ(w + αρ)dα,
H(ρ, w) =
(36)
where
ρ ρ ∗ P(ρ, α) = p α − , β p α + , β dβ 2 2
(37)
is just the projection onto the α axis of the product of two radially symmetric pupil functions mutually displaced by a distance ρ along the mentioned axis (Frieden, 1966). By solving the integral in Eq. (36), we obtain
1 w H(ρ, w) = P ρ, − . ρ ρ
(38)
Basic Theory for Microscope Imaging
17
In the case of the circular aperture, the function P(ρ, α) is simply given, for each value of ρ, by the projection onto α of the common area of two circles (see Figure 7), namely
⎫ ⎧ ⎨ ρ 2⎬ , P(ρ, α) = 2 Re 1 − |α| + ⎩ 2 ⎭
(39)
and therefore (Sheppard and Gu, 1992)
⎫ ⎧ 2 ⎨ |w| ρ 2 ⎬ . + H(ρ, w) = Re 1− ⎩ ρ ρ 2 ⎭
(40)
In Figure 8, we have represented the above equation on a meridian section. Note that the OTF is symmetric about the axial-frequency axis. As one could have easily predicted from Eq. (22), the OTF exhibits a singularity at the origin. The OTF is a compact-support function confined to the region defined by the parabolic curves
w = ±ρ(1 − ρ/2).

 5 12(␣ 1 /2)
(41)
2
1 ␣
FIGURE 7 The function P(ρ, α) is obtained after projecting onto the α axis the product of two circle functions mutually displaced by a distance ρ.
OTF
w Missing cone
0.5
1.0
2.0
FIGURE 8 Meridian section of the 3D OTF of a circular aperture.
u
18
The Resolution Challenge in 3D Optical Microscopy
It is emblematic that there exists, in the neighborhood of the origin, a cone in which the OTF is zero. This cone is known as the missing cone (Streibl, 1984). Neither the axial frequencies nor the oblique frequencies included in the missing cone are transferred to the 3D image. In other words, no depth information is present in the image of 3D samples. This explains, in the OTF context, the lack of optical sectioning of conventional wide-field microscopes when imaging 3D objects.
2.4.3. Relation Between the Axial OTF and the Integrated Intensity We have shown that the optical-sectioning capability of an imaging system can be investigated through two different functions: the integrated intensity and the axial component of the 3D OTF. In fact, both functions constitute different representations of the same information, since they are a Fourier-transform pair. To show this, we take into account that the OTF is the Fourier transform of the 3D intensity PSF,
H(u, w) =
|h(x, z)|2 exp{−i2π(u · x + w z)}d2 x dz,
(42)
and particularize to the axial frequencies,
|h(x, z)|2 exp{−i2πw z} d2 x dz
H(0, w) = =
Iint (z) exp{−i2πw z}dz.
(43)
In conventional microscopy, the integrated intensity is constant, and therefore, the axial OTF is a delta function centered in the origin. Only zero-order axial frequencies are transferred to the image.
3. THE HIGH-NUMERICAL-APERTURE APPROACH Up to now we have considered imaging systems for which the paraxial approximation holds, and therefore spherical wavefronts have been approximated by parabolic wavefronts. One should take into account, however, that in actual microscopy the aim of obtaining images with high spatial resolution makes it necessary to use microscope objectives with the maximum available NA, which cannot be approximated as a thin lens. On the contrary, an accurate analysis should take account of the principal surfaces of the microscope objective, as shown in Figure 9. The back principal surface is, as in the paraxial case, a plane surface (S2 in Figure 10).
19
The High-Numerical-Aperture Approach
Princ
ipal s
y
urfac
es ) yp p (xp, yp
x
L2
xp
F1
y9 D CC
F 19 ≡ F2
f1
x9
f1 f2 f2
F92
FIGURE 9 An actual optical microscope is accurately schematized through a telecentric arrangement with the objective represented by its principal surfaces. S2
S1
p (rp) P1
ss fq
P R
F
f
f
FIGURE 10 When a high-NA objective is illuminated by a plane wave, the amplitude transmittance of the aperture stop is projected onto the spherical principal surface.
The front principal surface, S1 , is a sphere of radius f centered at the front focal point. As already stated, in most high-NA microscope objectives, the aperture stop is inserted just at the BFP. From this sketch, one can understand that a microscope objective, ideally free of aberrations, transforms an impinging monochromatic plane wave into a truncated spherical wavefront centered at the focal point, F. The amplitude transmittance of the aperture stop is mapped onto S1 .
3.1. Calculation of the PSF As stated by McCutchen (1963), the objective, such as a cookie cutter, chops out a chunk of the spherical wavefront, which can be regarded as a Huygenian source. The amplitude at any point in the vicinity of the focus is calculated by integrating contributions from this source, taking into account their relative phases. To calculate the amplitude distribution in
20
The Resolution Challenge in 3D Optical Microscopy
the neighborhood of the focus, we proceed by making use of the first equation of Rayleigh–Sommerfeld (Born and Wolf, 1999c), which reconstructs the amplitude distribution in the vicinity of the focus as the superposition of the secondary spherical wavelets that originated at the spherical wavefront, namely
h(R) = −
i λ
U(R1 )
eiks 2 d S. s
S1
(44)
The positions of a typical point, P1 , of the wavefront, and of observation point, P, respect to F are given by vectors R1 and R, respectively. Vectors qˆ and sˆ are, respectively, the unit vectors in such directions. The amplitude of any secondary wavelets is given by
U(R1 ) = p(R1 )
e−ikf , f
(45)
where p/f is the amplitude of the Huygenian source. The factor exp(−ikf ) is required to move the zero of phase from the Huygenian source, where it would otherwise be, to the geometric focus. In the vicinity of the focus, we can approximate
ˆ ≈s−f qR
and
d2 S ≈ f 2 d2 ,
(46)
where d2 is the solid angle that d2 S subtends at F. Then, Eq. (44) can be rewritten as
h(R) = −
i λ
ˆ ˆ eikqR p(q) d2 .
(47)
The above equation constitutes the so-called Debye scalar integral representation of strongly focused fields (Debye, 1909) and expresses the field as a coherent superposition of monochromatic plane wavefronts. The directions of propagation of the wavefronts fall inside the geometrical cone defined by the focus and by the projection of the pupil function onto the spherical principal surface. Since in most objective lenses the amplitude transmittance of the aperture stop has axial symmetry, it is convenient to express the positions in the reference sphere in terms of a set of spherical coordinates centered at the focus:
qˆ = (−sin θ cos ϕ, −sin θ sin ϕ, cos θ) ,
(48)
21
The High-Numerical-Aperture Approach
and
d2 = sin θ dθ dϕ.
(49)
Besides, we express the position of point P in terms of a set of cylindrical coordinates centered again at the focus:
R = (r cos ψ, r sin ψ, z).
(50)
Therefore, the amplitude distribution in the focal volume can be written as
i h(r, ψ, z) = − λ
2πα p(θ, ϕ) exp{−ikr sin θ cos(ϕ − ψ)} 0 0
ikz cos θ
×e
sin θ dθ dϕ.
(51)
Assuming axial symmetry for the pupil amplitude transmittance, the focal amplitude has axial symmetry as well,
2π h(r, z) = −i λ
α
p(θ) J0 (kr sin θ) eikz cos θ sin θ dθ.
(52)
0
Again in this case, one commonly uses normalized coordinates, defined by
r=
r sin α λ
and
z=
2z sin2 α/2 λ
(53)
so that we can write the amplitude distribution in the focal volume as
α 2π sin θ z exp iπ 2 h(r, z) = −i p(θ) J0 2πr λ sin α sin α/2 × exp −i2πz
sin2 θ/2 sin2 α/2
0
sin θ dθ.
(54)
We must recall at this point that energy consideration should be taken into account in the projection of the incident plane wavefront onto the emanated spherical wavefront; see Figure 10. Most microscope objectives are designed to fulfill the aplanatic condition, also known as sine condition,
22
The Resolution Challenge in 3D Optical Microscopy
to produce images with transverse invariance (Gu, √ 2000; Sheppard and Gu, 1993). In this case, an apodizing factor g(θ) = cos θ should be included in the integrand of Eq. (54). This equation is exact for scalar waves and is a good approximation for light if the NA is small enough that different parts of the arriving wavefront do not have their polarizations significantly twisted relative to one another on the way to focus. In many cases, it is useful to express Eq. (54) in terms of the normalized radial coordinate at the pupil plane. In case of aplanatic systems, the angular and the radial coordinates are related through rp = sin θ/ sin α. In that case
2π h(r, z) = −i λ
1
p rp J0 2πr rp
0
⎧ ⎪ ⎨
× exp −i2πz ⎪ ⎩
⎫ ⎬ 1 − r2p sin2 α ⎪ sin2 α/2
sin2 α rp drp . ⎪ ⎭ 1 − r2p sin2 α
(55)
3.2. Calculation of the OTF The calculation of the 3D OTF can easily be performed by taking into account that its coherent counterpart, the 3D CTF, is related with the amplitude PSF through a 3D Fourier transform, namely
h(x, z) =
˜ w) exp{i2π(u · x + wz)}d2 u dw. h(u,
(56)
The normalized frequencies are related with the actual ones through the scaling
u=u
λ sin α
and w = w
λ 2 sin2 α/2
.
(57)
The amplitude PSF can be rewritten as
∞ h(x, z) =
˜ ei2πQ·R d2 |Q|2 d|Q|, h(Q)
(58)
0
where R = (x, z) and Q = (u, w). From this equation, one can derive the Debye integral of Eq. (47), as expressed in normalized coordinates,
The High-Numerical-Aperture Approach
23
u
sin ␣
cos ␣ 1 w
FIGURE 11 The 3D CTF of a high-NA microscope is confined onto a spherical shell of unit radius.
provided that
˜ ˆ δ |Q|2 − 1 . h(Q) = p(q)
(59)
Then, the 3D CTF is confined onto the surface of a sphere, which, when expressed in normalized frequencies, has unit radius; see Figure 11. This is the result reported by McCutchen (1963) in his mythical paper in which he stated that the amplitude distribution in the focal volume is obtained as the 3D Fourier transform of the Huygenian source. Therefore, the Huygenian source is the CTF of the system. In the usual case of an axially symmetric pupil function, the 3D CTF is (Sheppard et al., 1994).
˜h(Q) = h(ρ, ˜ w) = p(ρ) δ w − 1 − ρ2 . 1 − ρ2
(60)
Naturally, the 3D OTF is obtained, as in the paraxial case, through the 3D self-correlation of the 3D CTF, as illustrated in Figure 12a. Similarly to the paraxial result, the high-NA OTF has a doughnut structure in which the most remarkable feature is the existence of a cone of nontransmitted spatial frequencies; see Figure 12b. Note that the mathematics that describe the high-NA focalization and imaging are different from the those describing the paraxial phenomena. However, the two situations are conceptually equivalent. By this, we mean that in both cases, the 3D intensity PSF exhibits a sharp central peak surrounded by low sidelobes. Also, in both cases, the 3D OTF has a doughnut shape in which the missing cone is responsible for the lack of optical-sectioning capability of conventional microscopes. As in the paraxial case, the high-NA integrated intensity can
24
(a)
The Resolution Challenge in 3D Optical Microscopy
(b) OTF
12cos a w
Missing cone
2sin α u
FIGURE 12 (a) The 3D OTF in the high-NA case is obtained through the self-correlation of a spherical shell of unit radius. (b) Also, in the high-NA case, the 3D OTF has a doughnut structure with a missing cone. For the calculation we set α = 3π/8.
be calculated through the 1D Fourier transform of the axial component of the 3D OTF, and therefore it is constant. We can conclude that, although paraxial equations do not provide an accurate description of the imaging properties of optical microscopes, they are relatively simple and easy to handle, and provide good conceptual ideas about the functioning of optical microscopes.
3.3. Metrics for Resolution Improvement Resolution is the key feature of optical microscopes. The ability of optical microscopes to provide sharp images of the finest details of samples has been usually evaluated through the OTF or, alternatively, through the intensity PSF by applying the two-point resolution Rayleigh criterion. In past years, an important research effort has been addressed at developing the so-called PSF-engineering techniques (Ando, 1992; Barakat, 1962; de Juana et al., 2003; Dorn et al., 2003; Mills and Thompson, 1986; OjedaCastañeda, Andrés, and Díaz, 1986; Sherif and Török, 2004; Toraldo di Francia, 1952). Such techniques aim to modify the shape of the intensity PSF so that it is narrowed in the transverse and/or in the axial direction. Here, we concentrate on the engineering of the axial PSF. We start by particularizing Eq. (54) to points in the optical axis,
α 2π z exp iπ 2 h(0, z) = −i p(θ) λ sin α/2 × exp −i2πz
sin2 θ/2 sin2 α/2
0
sin θ dθ.
(61)
25
The High-Numerical-Aperture Approach
Following McCutchen’s approach, Eq. (61) can be converted into a 1D Fourier transform provided that one performs the non-linear mapping
ζ=
sin2 θ/2 2
sin α/2
−
1 2
q(ζ) = p(θ).
and
(62)
After the mapping, we find that
0.5 h(0, z) =
q(ζ) exp(−i2πzζ) dζ,
(63)
−0.5
where some irrelevant pre-multiplying factors have been omitted. Note that, apart from a scaling and shifting, the non-linear mapping is the same as suggested by McCutchen,1 and therefore the function q(ζ) is nothing but the projection of p(θ) on the optical axis, as illustrated in Figure 13. It is remarkable that independent of the value of the NA, the projection of a clear circular aperture onto the optical axis has a rectangular form, and therefore q(ζ) = rect(ζ). Thus, for any NA, the axial intensity PSF is
sin2 πz = sinc2 (z). (πz)2
|h(0, z)|2 =
S2
(64)
S1
p (rp)
p ()
f
q ()
F
f
FIGURE 13 Independently of the numerical aperture of the imaging system, a clear circular aperture projects as a rectangle on the optical axis.
1 In his paper, McCutchen (1963) suggested the non-linear mapping = cos θ , which allows the projection
of the Huygenian source onto the optical axis.
26
The Resolution Challenge in 3D Optical Microscopy
The actual form of the axial PSF, expressed in actual spatial coordinates, is obtained after undoing the coordinate mapping of Eq. (53). Note from this mapping that the higher the NA the narrower the axial PSF. Next, we consider the case in which one inserts a diffractive element, usually called a pupil filter, in the aperture stop with the aim of engineering the axial PSF. To estimate the ability of such filter to improve the axial resolution, one needs an analytical tool for the easy evaluation of the width of the central lobe of the PSF without needing to compute all the focal intensity. Since the parabolic term in the power-series expansion for the axial PSF dominates within the central peak, an interesting method for evaluating the resolution is one originally suggested in a paraxial context by Sheppard and Hegedus (1988). They defined the gain in axial resolution as the ratio between the parabolic intensity fall-off provided by the filter and that provided by a circular aperture. In the case of pupil filters whose amplitude transmittance is real, if we expand the normalized axial PSF in power series up to second order, we find that
|h(0, z)|2
z ≈ 1 − 4π IN (z) = 2 1/σ |h(0, 0)|
2 ,
(65)
where
m2 − m0
σ=
m1 m0
2 (66)
is the standard deviation of the function q(ζ), and
0.5 q(ζ) ζ n dζ
mn =
(67)
−0.5
represents its nth statistical moment. Now, one can define the gain in axial resolution as
GA =
σF , σC
(68)
where the subscripts C and F correspond, respectively, to the circular aperture and the pupil filter. Obtaining the gain in transverse resolution is not as simple as obtaining its axial counterpart. This is because, as pointed out by McCutchen (1963),
The High-Numerical-Aperture Approach
S2
27
S1
p (rp)
qF () F p ()
f
f
FIGURE 14 The lateral PSF is obtained through the 1D Fourier transform of the projection of the pupil onto one axis perpendicular to the optical axis.
the shape of the function obtained as the projection of the circular aperture onto an axis perpendicular to the optical axis (see Figure 14) depends on the NA of the objective. The fall-off in intensity of the squared modulus of the 1D Fourier transform of such projection is the reference figure for the calculation of the gain in transverse resolution. Therefore, the gain strongly depends on the value of α. Proceeding in the same way as in the axial case, we find, after straightforward calculations, that
GT =
2(1 − cos α)/3 (3 + α) − GA , (3 + α) − (1 − cos α)/3
(69)
where we have considered the simplest case of q(ζ) being an even function. Imaging systems for which the paraxial approximation holds have GT ≈ 1. The gains in resolution have constituted an interesting tool for the design of many pupil profiles for the improvement of the lateral and/or the axial resolution of imaging systems. Among these, we single out the so-called shaded-ring (SR) pupil filters (Martínez-Corral et al., 2003). As we show below, SR filters have the ability to narrow the central lobe of the axial PSF. This narrowing is produced at the expense of only an insignificant deterioration of the lateral resolution and a small enlargement of the axial sidelobes. SR filters simply consist of a purely absorbing ring with constant transmittance, centered on the objective-lens aperture (see Figure 15). Depending on the width of the ring, different degrees of axial-peak compression can be achieved. The value of the axial gain in resolution is
GA =
1 − ημ3 , (1 − ημ)
(70)
28
The Resolution Challenge in 3D Optical Microscopy
q ()
/2
⫺0.5
0.5
FIGURE 15 1D projection and actual 2D transmittance of a SR pupil filter.
with the parameters η and μ identified in Figure 15. Note that all the pairs (η, μ) fulfilling Eq. (70) for a given value of GA, correspond to filters with the same axial gain but different sidelobe energies, as illustrated in Figure 16. In this figure, we have plotted, first, several curves for different values of GA. Every point of a curve corresponds to a different (η, μ) pair. The leftmost point of a curve corresponds to the dark-ring filter (Blanca and Hell, 2002; Martínez-Corral et al., 1995). For an intensity simulation, we selected GA = 1.20. The minimum of the curve (marked with a circle in the figure) corresponds to the SR filter with μ = 0.67 and η = 0.66. In terms of the sidelobe energy, the selected SR filter is 25% better than the opaque-ring one.
3.4. Sampling Expansion Point-spread function engineering constitutes an interesting tool for improving the performance of optical microscopes. Apart from the gains in resolution, which provide an easy method for estimating the superresolving abilities of pupil filters, other analytical tools are useful for the fast computation of the 3D PSF. Sampling expansions of the PSF have been used in the past for the computation of 2D or even 3D diffraction patterns (Barakat, 1980; Jacquinot and Roizen-Dossier, 1964; Li and Wolf, 1984; Piestun, Spektor, and Shamir, 1996), but always within the frame of the paraxial approximation. To be able to use this tool for calculating 3D PSF of optical microscopes, it is necessary to extend the Landgrave and Berriel-Valdos (1997) approach to a non-paraxial context.2 To this end, we start by rewriting Eq. (54) as
2π h(r, z) = −i λ
1 p(cos θ) J0 cos α
× exp i2πz
sin θ 2πr sin α
cos θ 2 sin2 α/2
d(cos θ).
2 Preliminary results in this direction were reported by Arsenault and Boivin (1967).
(71)
The High-Numerical-Aperture Approach
(a)
2.4
29
GA ⫽ 1.15 GA ⫽ 1.20 GA ⫽ 1.25
Sidelobe-to-peak ratio
2.0
GA ⫽ 1.30
1.6
1.2
0.8
0.4 0.2
(b)
0.4 0.6 0.8 Transmittance parameter:
1.0
1.0
Axial intensity PSF
0.8
0.6
0.4
0.2
0.0 ⫺4
0 2 ⫺2 Normalized axial coordinate: z
4
FIGURE 16 (a) Sidelobe-to-peak energy ratio for families of SR filters with the same axial gain. (b) Axial PSFs corresponding to the SR filter with GA = 1.20 (solid curve) and to a circular aperture (dashed curve).
Let us now suppose that the kernel of the above transformation,
K(θ; r, z) = J0
sin θ 2πr sin α
exp i2πz
cos θ 2 sin2 α/2
,
(72)
30
The Resolution Challenge in 3D Optical Microscopy
can be expanded in a Fourier series as
K(θ; r, z) =
∞
fm (r, z) K(θ; 0, m).
(73)
∞ 2π fm (r, z) h(0, m). λ m=−∞
(74)
m=−∞
In that case,
h(r, z) = −i
Following the Landgrave and Berriel-Valdos approach, now the problem consists in finding the coefficients of the kernel expansion. To solve this problem, we notice the following orthogonal property,
1
K(θ; 0, m) K ∗ (θ; 0, m ) d(cos θ) = (1 − cos α) δm,m ,
(75)
cos α
which permits us to find that
1 fm (r, z) =
cos α
K(θ; r, z) K ∗ (θ; 0, m) d(cos θ) 1 − cos α
=
hC (r, z − m) , 1 − cos α
(76)
where hC (r, z) is the 3D PSF corresponding to the circular aperture. Finally, we obtain
h(r, z) = −i
∞ 2π hC (r, z − m) h(0, m). λ(1 − cos θ) m=−∞
(77)
This important formula, which represents the non-paraxial form of the axial sampling theorem (Arsenault and Boivin, 1967), indicates that the 3D amplitude PSF of an optical microscope with a pupil filter inserted into the objective aperture results from the coherent superposition of an infinite number of axially shifted PSFs that correspond to the circular aperture. The shifts are equal to integer numbers. The weighting-factor set of this superposition is obtained by sampling the axial PSF of the pupil filter in the axial nulls of a circular-aperture PSF.
The High-Numerical-Aperture Approach
31
3.5. Spherical Aberration In many microscopy realizations, the specimen is embedded in a medium that does not match the refractive index of the immersion liquid. This fact creates important phase distortion in the, otherwise spherical, wavefronts emitted by the fluorophores. Consequently, the PSF is also distorted. In our formalism, the phase distortions are incorporated by modifying the Huygenian source by the term exp{−i2πW(θ)} so that
α h(r, z) =
p(θ) exp[−i2πW(θ)] J0 0
× exp −i2πz
sin2 θ/2 sin2 α/2
sin θ 2πr sin α
sin θ dθ,
(78)
where we have omitted some constant and a phase factor external to the integral. To evaluate the phase distortions, we can view, as explained in Section 2.2, this diffraction process from the opposite direction, that is, as the case in which a high-NA objective is illuminated by a monochromatic plane wave and the corresponding emerging beam is focused deeply through a planar interface between two media of different refraction index. In that case, we can assume that each plane-wave component of the field emerging from the objective obeys Snell’s law, n1 sin θ = n2 sin θ , when refracted at the interface. The resulting field is reconstructed as the superposition of refracted plane waves. In Figure 17, a plane-wave component is represented through a light-ray normal to the wavefront. The phase delay suffered by the rays is proportional to the optical path difference (Török et al., 1995),
W(θ) =
1 d
n2 cos θ − n1 cos θ . [n1 l1 (θ) − n2 l2 (θ)] = λ λ
(79)
Following the classical approach of Sheppard and Cogswell (1991), we expand this expression into power series of sin(θ/2), up to fourth order. We obtain
! 2 n 2n1 d W(θ) = (n1 − n2 ) 1 + sin2 (θ/2) + 2(n1 + n2 ) 13 sin4 (θ/2) . λ n2 n2 (80)
32
The Resolution Challenge in 3D Optical Microscopy
n1 n2 A