This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
ddzz)
e)
U(z).
(6.48)
Upon differentiation with respect to z and cross-substitution, these equations become d2U(z) _ d (lnic m dU(z) 2 2 o (S — n Z) U (z) + k d z dz dz2 )
(6.49)
viiTe general solution is a superposition of two waves: X(x) = cx,e`y`0x +cg2 e —i ch0x
Under the current consideration of one plane wave, only the first term is retained.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Stratified media
119
and Zk2
d2 v ( z ) — d[ln { z (; 1m —£e)] dv(z) -
dz
dz 2 dz
+ß(S2 —n 2 )v(z).
(6.50)
Since both U (z) and V (z) satisfy a second order differential equation, they can be expressed as linear combinations of two particular solutions, namely, U (z) = c 1 Ui (z) + c u2 U2 (z) and
V (z) = c 1 VI (z) + cV 2 V2 (z) . These particular solutions [Ui (z), Vi (z)] and [U2 (z), V2 (z)] are coupled by Eqs. (6.47) and (6.48), meaning Vl (z) dU2 (z) — dUl (z)
dz
dz
V2 (z) = 0
and Ul (z) dV2 (z) — dVl (z)
dz
dz
U2 (z) =0.
Together they imply C^Z
[Ul(z)v2(z)_U2(z)V1(z)] =0
(6.51)
or
Ul (z)V2 (z) — U2 (z) V, (z) = constant.
6.4.2 Characteristic matrix To simplify the mathematics, let us pick particular solutions that satisfy the following conditions: U1(0) = V2(0) = 1
and
U2(0)=V1(0)=0.
(6.52)
With these conditions, the solutions of Eqs. (6.49) and (6.50) with U(0) = Uo and V(0) = Vo can be written as U(z) = Ul (Z) Uo + U2 (Z)Vo
and V(z) =V1(z)Uo+V2(z)Vo•
We can express these equations in one matrix equation: Ul (z) U2(z)
Uo = NQ(0). vo [v (z) I — [vi(z) (z) V2
Q(z) = U(z)
(6.53)
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
120
Oblique Rays
The determinant of N is 1 (see Exercise 6.20). Inverting the above matrix equation
gives Q( 0 ) =N —' Q(z) _
1 Vi (z) Ui(z) —
)
1
(6.54)
IV (z)1 MQ(z)
The determinant of the matrix M is also 1. Given M, knowledge of the fields at a particular z-plane is sufficient for determining those at other planes. The matrix M is the characteristic matrix of the stratified medium. Let us derive the characteristic matrix of a homogeneous dielectric film, with a TE plane wave propagating in a direction making an angle 0 with the z-axis. In this situation, S = n sin 0 [see Eq. (6.46)], and Eqs. (6.49) and (6.50) become d'-U(z) 2 2 dz2 +kón cos 0U(z) = 0, d'-V(z) a d`2
+ kón cosa0 V (z) = 0.
The solutions are U(z) = cl cos(konzcos0) + c2 sin(konzcos0), V(z) = i
m
cos0[ —cl sin(konzcos0)+c2cos(konzcos0)].
The particular solutions that satisfy the conditions stated in Eq. (6.52) are
Ui(z) = cos(konzcos0), U2(z) = 1 ^m sin(konzcos0), icosøV E e Vi (z) = —i cos 0 £e sin(konz cos 0), V2(z) = cos(konzcos 0), and the characteristic matrix is
M (z)
-1
Ui (z) V(Z) Vi
-
)1
cos(konzcos6)
pose £e sin(konzcos6)
i cos 6m sin(konzcos 0)
cos(konzcos A)
(6.55)
j
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Stratified media
121
Since Maxwell's equations are invariant under the simultaneous substitutions .ii E H
and
Ee
-
/Lm,
the characteristic matrix for a TM wave is cos(konzcos0) MTM(z) _
—icos8 £e sin(konzcos0)
^os`e µm sin(konzcos0) cos(konzcos 0)
with Hy(x,z,t) = U(z)e` (5kox-0)r) and Ex (x z; t) = V
(z)ei(Skox—wt)
The sign change of the off-diagonal elements originate from replacing —icqu m by iWEe in Eq. (6.47). We can therefore define a general characteristic matrix as M(z) —
cos (konz cos0) [ixVsin(kOnzcosO)
, sin(konzcos8)1 cos(konz cos 0) JI '
(6.56)
with NJ = cos 8 £
e
= cos 8
in the TE polarization, and
/1m
(6.57)
yr = —cos 8 ^m = —rl cos 8 EQ
in the TM polarization.
Since the tangential components of the electric and magnetic vectors are continuous across dielectric interfaces (see Exercise 6.9), the characteristic matrix of a homogeneous dielectric film can be extended to describe a stratified medium comprising two materials, the first extending from z = 0 to z = zl and the second from z = zi to z = z2• If M1(z) and M2(z) are the characteristic matrices of the two materials, then and
Q(0) =Mi(zl)Q(zi)
Q(zi) =M2(z2 — zi)Q(z2),
and Q(0) =Mi(zi)M2(z2 — zi)Q(Z2).
This expression can be generalized to a stratified medium consisting of q layers with a total thickness of d, with di denoting the thickness of the ith layer: Q(0) = MI(di)M2( d2)...Me(dq)Q(d) nMi (di ) [
Q(d)
i=
(6.58)
= M(d)Q(d)
= `m 11 m 121 rU(d)^
LM21 m22
lV (d)
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
122
Oblique Rays
Wave transmission through the entire stratified medium can be characterized by the four matrix entries ml1, m12, m21, and m22.
6.4.3 Reflection and transmission Let us now examine the reflection and transmission of an electromagnetic wave upon incidence onto a stratified medium, as illustrated in Fig. 6.7. Consider first the TE polarization, in which the electric field vector is polarized in the y-direction. Following the approach taken in §6.3.2, the electric field in the incident medium is E =E + E refl = aIe i13inc + rle il^refl Y y y
The variables 1j and i3refl embed the spatial and temporal dependence of the incident and reflected waves. It follows that U(0) = a1 + rL. Using Eq. (1.26), we can write V(0) = 1 Vinc ( — a1 + r1),
where, as defined in Eq. (6.57), cos einc Winc =
Tlinc
The transmitted fields are U(d)=t1
and
V(d)=—yr.tL.
Equation (6.58) becomes [Vinc( a_L+r_L)l r — [m21 m22] ^—Want] . —
Solving this equation results in r1 _ Vine (m11 — 1IJtranml2) + (m21 — Wtranm22) (6.59) Nlinc(m11 — 19tranml2) — (m21 — Wtranm22) ' P1 = a1
_
t1
_
21Vinc
(6.60)
T1 a1 zinc (m11 — Wtranml2) — (m21 — Wtranm22)
In the situation where the stratified medium is absent (see Fig. 6.5),
M=[ 0 10 Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Stratified media
123
inc
refI
ninc
aL
Xr,
11
ereil ereil
,
a II
d
n P =n'ß,(1+ivP ) a P 0 r layerp 0 0
bt
0
n q =n'Q (1+
I
Z=O dp
k su sac
I dq J
ginc
ntran
etran
Yr X
tL
1
Z Z
til gtran
Figure 6.7: Reflection from and transmission through a stratified medium.
and —
Pi
Tltran COS e inc — 1linc COS 9tran 11tran COS ejnc + Tlinc COS etran
21jtran COS e inc 11tran COS Ainc + 1 1inc COS Atran
We recover Eq. (6.25). The matrix equation for a TM wave can be similarly derived: (a ll + rll)
cos 6inc(all — rll)
mll m12
1_"^tII
m21 m22
cos Ot tll
Solving this equation gives rij 1Vinc(ml1 — tranm12)+(m21 —ltranm22)
P11=— all
(6.61)
1 Vinc(MI1 — Nftranm12) — (m21 —1 Vtranm22)
tij —2tltran cos 6inc ( ill= — = tinc(mil — Wtranml2) — (m21 — Wtranm22)
(6.62)
all
where yr is defined in Eq. (6.57). Although Eqs. (6.59)—(6.62) have been derived under the assumption that the plane of incidence is the xz-plane, they are applicable for arbitrary planes of incidence, provided the electric field vector is decomposed into perpendicular and parallel components.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
124
Oblique Rays
6.5 Intensity distribution in photoresist`'nl Referring to Fig. 6.7, suppose that the photoresist, with a thickness dp, is the p th layer of a stratified medium consisting of q layers. To determine the resist image, let us define a substack that is a stratified medium comprising the layers (p + 1) to q of the original stratified medium, and an angle 6 p between the z-axis and the +z-propagating plane wave in the photoresist corresponding to an incident angle 6;ßc such that ninc sin eint = nimage Sin eine = np sin Op,
np = , (1 + iKp ) .
The amplitude of the +z-propagating TE wave in the photoresist a1 is related to the incident TE amplitude al by a P ,substack = a 1 I
stack 1 I
where the superscript "substack" refers to quantities related to the substack and the superscript "stack" denotes quantities associated with the entire stratified medium. The electric field within the photoresist layer arising from an incident TE plane wave of amplitude a 1 is stack
ikpfl•r _ ti1 ikP`•r E1— al substack [e + p1 substacke
I
1
stack
= ei(kPx +kP) a ti l ikp1—z [ e+(d1) +
1
substack
1
_ kP,
substack e — ikp(d,—z) p1 ]'
where kP e (kp, kP) and kp fl = ( kP, k, —kf) are the wave vectors of the +z traveling and —z-traveling plane waves in the photoresist. Since the parameters tack , tïbstack, , ,? stack , and kp are functions of B ob] and z only, the ratio of the photoresist electric field amplitude to the incident amplitude is stack
ELI
ti l ikz ( p — z) + p s bstacke— ikP(dp e+' a1 ,substack =[ 1
—z)1 = IF±(Pe, z) I.
(6.63)
1
Equations for the TM polarization can be similarly derived. According to the convention of Fig. 6.7, the x- and y-components of the reflected wave subtract from the incident wave, whereas the z-components are summed: E'Y a
ll
I=I =I
,stack
substack [ e
+ikp(dp—z) — p 5tack e — ikP(dP
—z)1 = IF (pe,z)I
(6.64)
IFI (p e z)I.
(6.65)
I
EZ ,stack
[e+d_z) +p llubstack e — ikP(dd —z)1 =
substack a ll Ts J v iii The
exposition in this section follows Yeung [42] and Flagello [43].
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Intensity distribution in photoresist
125
The effect of the stratified medium is a scaling of the amplitude of each imageforming ray of the spherical wavefront by the factors FL, Fr', and F, depending on the incident angle of the light ray and the z-position within the photoresist. The quantities PP11 and FIIj in Eq. (6.13) should be scaled by the appropriate factors. For example,
P X —^ F1PX j,
and
PYllz
Fj PYllz
The vector field at an image point P arising from a ray is a modification of Eq. (6.13):
Pxi Py F1 Fxy 0 0 01 ' x 'yMX E(P) = 0 0 Fi FxY 0 Px1y PYLY
0
0
Msxx Ms = Mss, Ms
0
0
FI
PxIIY PYIIY PXllz PYIIZ
^Ey J (6.66)
1 Ex = MstackEo,
M&Z Msy^
where Ms.(po;z) = FLPX +F Px l x Msyx(pe;z) = F1Py ix+F PYIIx Ms, (pe; z) = FLPXLy + FXYPxIIY , Msyy (00; z) = FLPy 1y + F PYIIY, (6.67)
Msxz (Po; z) = F^ Pp,
Msy, (Pe; z) = F, PYIIz.
The resist image intensity is
I(,9)=nP f. f iu,g) 7 (ƒ+f^,g+g)H * (f+f^^,g+g^) O(J ,g )Mstack(f +.f' , '
'
+S' ;z)Eo .6* (1" , ")M Ck(f + J", + ";z)E e -^ 2n[(f'-f")x+(g'- ')y] d fdgd f, d g d j" d g
=nP ...ƒju,g)H(f+.P,g+g)H*(f+f^^,g+g^) i ={x,y} j={x,y}
Msik(f + f^ g+g^;z)Ms;k (f + f" g+g";z)EiE^
k={x,y,z}
6 (.jß , S )O * (.Í n
g') e -
^2 ^ [( f_ f)X+(s^- )yl d fdgd J' d g d f"dg'.
(6.68)
Because of reflections from the photoresist interfaces (characterized by F L , FIx, and F^ ), the resist image exhibits a standing wave whose ratio decreases with resist
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
126
Oblique Rays
Figure 6.8: (a) Reflections from material interfaces create a standing wave in the resist image. (b) Post-exposure baking reduces the variation, resulting in a smoothed latent image. depth due to absorption (KP ). ïx An example resulting from the exposure conditions detailed in §D.4.1 is shown in Fig. 6.8(a). Post-exposure baking smooths any standing wave that may be present such that the latent image corresponding to the resist image of Fig. 6.8(a) may resemble that shown in Fig. 6.8(b).
6.6 Immersion imaging A reason for using light rays at large angles with respect to the optic axis is resolution, since both minimum half-pitch and minimum dimension decrease with increasing numerical aperture NA = n; n,age sin 8ob^ . In addition to using a large semi-aperture angle 9 obj, we can increase the numerical aperture by immersing the image space in a high-refractive-index medium [47].x Figure 6.9 shows a schematic representation of an immersion imaging system. This diagram is the same as Fig. 2.7, except that the image space refractive index is nage > 1 and that the focal length of L2 is increased by a factor of nimage• Following the analysis in §2.5 and noting that the angles (po and (pi in Fig. 6.9 are related by Snell's law [see Eq. (6.22)] sin (p0 = ni. ge sin pi, (
i "The
standing wave ratio can be reduced by decreasing reflection from both photoresist interfaces (using top antireflective coating and bottom antireflective coating) as well as increasing resist absorption [46]. The latter approach, however, degrades resist image quality. 'For example, water with a refractive index of 1.45 is a suitable medium at 193 nm.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Imaging with oblique rays
127
Figure 6.9: Representation of an immersion imaging system.
the lateral magnification is
yi
Mlateral = y
_ nimage fM sin (p i
f
sin (po
M'
the angular magnification Mang„lar is
sini5i _ y /nimagefM 1 an a ular — sin150 nimageM' y'/f '
g
and the longitudinal magnification is Mlongitudinal = Mlateral/Mangular = nimageM
2
Compared with an imaging system with the same numerical aperture in which the object and image media are the same [see Eq. (2.12)], light rays in an immersion system travel at shallower angles. The obliquity factor is modified from Eq. (4.11) accordingly: l^g^M2 pé = (6.69) E(0) = 4
1—pé
6.7 Imaging with oblique rays In the scenario shown in Fig. 6.10, where two light rays interfere to form an image, the intensity distribution depends not only on the angle between the rays [see Eq. (5.20)] but also on their polarization. With transverse electric polarization, the vibrational directions of both beams are the same. The total field is the scalar sum of all constituting fields. In the transverse magnetic polarization, however, the field
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
128
Oblique Rays
bi
a1
bll
all
Figure 6.10: The fields add in a scalar manner in the perpendicular polarization but
add as vector quantities in the parallel polarization. vectors do not align. The fields arising from all beams add in a vector manner to give the overall field distribution. Since
images resulting from the parallel polarization have lower contrast than those resulting from the perpendicular polarization. 7 Let us consider, as examples that are shown in Fig. 6.11, the aerial images of three equal line-space chromium-on-glass patterns [see Eq. (5.7)] of different periodicities. The illumination for each pattern is an optimized dipole described by Eq. (5.17): f(j,g) = S(f+ 1 / 2PX)S(S)•
°
With px = 1 [Fig. 6.11(a)], the TE image shows a higher contrast than the TM image. The unpolarized image is the average of the TE and TM images, a result consistent with Eq. (6.5). When the period decreases to px = 1/v [Fig. 6.11(b)], the two interfering orders [see Fig. 5.7(a)] are mutually perpendicular. The parallel field components do not interfere; the TM image has no modulation. But the TE image still has a healthy contrast. With further reduction of the period to px = 0.6, the TM image exhibits intensity reversal [Fig. 6.11(c)]. The nominally bright area becomes dark and the opaque region becomes bright. The unpolarized image contrast is merely 0.277 despite a TE image contrast of 0.906. Although the fraction of p-polarized light increases as a beam travels through a wafer stack (see §6.3.4 and Exercise 6.17), resist-image quality degradation is less severe than the aerial image contrast loss depicted in Fig. 6.11. As a consequence of Snell's law (see §6.3.1), light rays in photoresists, with a typical refractive index of 1.8, travel at shallower angles, thus alleviating TM image degradation. Figure 6.12 demonstrates the difference between aerial and resist images for the scenario of Fig. 6.11(b), the situation where the TM aerial image has no modulation. Contrast of the TM resist image is 0.747 [Fig. 6.12(b)]. Such degradation of transverse magnetic image quality is sometimes said to be caused by polarization-induced stray light.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Imaging with oblique rays
129
TE
TE TM
TM
0
T
cN
unpolarized
unpolarized^ 0 C
-0.50
-0.25
0.25
0.00
position (2/NA)
(a)
i3
=
0.50
-0.50
1
0.00
-0.25
0.25
0.50
position (A/NA)
(b)Px= 1 /\ TE TM
T
c
unpolarized
a, C
-0.50
0.00
-0.25
0.25
0.50
position (7/NA)
(C)
0.6
Figure 6.11: Transverse electric images show higher contrast than transverse magnetic images.
A TM resist image that has zero contrast or that exhibits intensity reversal is possible only if (see Exercise 6.24)
NA>
n^
(6.70)
Assuming a photoresist refractive index of 1.8, this condition means that NA> 1.273, a value that is not uncommon with immersion systems. In increasing the numerical aperture for resolution improvement, immersion imaging is more susceptible (than reducing the wavelength) to vector effects as light rays can travel at more oblique angles within the photoresist. In our analysis of image formation and adoption of the canonical coordinates [Eq. (4.30) in §4.5], we showed that optical imaging scales with XO/NA. But the 1 /NA dependence is first-order, accurate only if the fields add in a scalar manner. With oblique rays, vector addition of interfering light beams gives rise to a second-order phenomenon that increases in significance with vibration direction misalignment (namely, with NA) and that partly diminishes the first-order resolution improvement. The overall resolution improvement is sublinear with 1 /NA.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
130
Oblique Rays
9 TE
TM ..
TE
ó
TM _T
/
N
C N
\
/
/
\
0
-0.50
-0.25
0.00
0.25
0.50
-0.50
-0.25
0.00
0.25
0.50
position (A/NA)
position (AJNA)
(b) resist image
(a) aerial image
Figure 6.12: Resist image quality degradation is less severe than aerial image contrast loss. NA=0.1
R
NA=0.5
y, o
NA=0.9
N C 1
Since c is positive semi-definite, the intensity i(Po) attains its maximum value if the series in Eq. (7.16) has the single term Zj, implying Obal = cjZj. The Zernike polynomials are naturally balanced. Each polynomial combines terms in such a way that the normalized intensity at the geometrical image point is a Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
140
Aberrations
maximum. For example, spherical aberration is balanced by defocus in Z11, and tilt balances coma in Z7 and Z8. Besides the representation of Eq. (7.10), Zernike polynomials can be expressed according to the Fringe Zernike convention
Z^ (0, 0) = Rn (P)Yj ) • The coefficients of an aberration function expanded in terms of the Fringe Zernike set
c(0,0) = Y, cj, ZZ i
are related to those of Eq. (7.13) by c,=clan. Regardless of convention, an aberration function is in general an infinite sum of the Zernike set. In practice, the series is often truncated after J terms such that i
i
4)(p, 0) = 1 c1Z1 _ Y, c^Z^. i =1
(7.17)
1=1
The number of terms is usually 37 in photolithography applications. More terms are necessary in well-corrected (namely, low-aberration-level) systems where higherorder terms have similar magnitudes to the lower 37 terms.
7.4 Effects on imaging Although the exact impact of aberrations is specific to the object shape and illumination configuration, it is possible to describe the general effects of each Zernike term. In addition to the displacement theorem of Eq. (7.6), the concept of ray aberration is useful for our discussion. Consider an arbitrary imaging ray, say QQ'Pr P, in Fig. 7.1. It intersects the image plane at Pr = (x r , yr ), a point that is, in gen (xo, yo). The displacement of Pr from Po is the ray aberra--eral,difntomP= tion [51]: (Ax, Ay) = (xr
—
xo,Yr
—
Yo)
( a XcI aXcI In terms of the normalized variables (A9) = N o (AXI DY) _ NA RX( J: a(D ko R sin 8obi
51 'i •
ac:I: ac = afag
(7.18) )
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Effects on imaging
141
Piston
The first Zernike term Zl (p, 4)) = 1 represents the average phase of the wavefront; it does not affect the image.
Tilt The ray aberration corresponding to a wave aberration represented by = c2Z2 +c3Z3 = c225 cos4)+c32p sin0 is (A ,A9) _ ( 2 c2, 2 C3)The intensity distribution is shifted; but the image is otherwise unaffected. From Eq. (7.6), the amount of displacement is (&,A94) = ( 2 c2, 2 c3, 0 ). Defocus
With defocus aberration ( D(P)0)
= C4Z4 = C4V(2p 2 — 1),
the intensity distribution is unchanged except for a translation by (M,Af,OA) = ( 0,0,4Vc4).
(7.19)
Astigmatism
The effect of the aberration 'D(0 0) = C6Z6 = c« / 1
2 cos 24)
depends on the orientation of the object. A pattern varying only in the x- direction [such as that described by Eq. (5.5)] under coherent illumination with f(f,, g) = S(f , g) means that 0 = 0 or 0 _ Ir, so that (5, 4)) = c6 2 . This is a defocus. A pattern that varies in the 9-direction has a spectrum such that 4) = ir/2 or $ = 3rc/2, and c1(p, 0) = — c6\./p 2 . This is a defocus of the same magnitude in the opposite direction. Features that orient in the . - and 9-directions are focused onto different planes. The other astigmatism term
vp
0
(p, 0) = c5Z5 = c5V op t sin 20
affects patterns oriented at 45 deg and 135 deg with respect to the x-axis. In the presence of Z5 and Z6, different parts of an object experience different amounts of defocusing depending on their orientations. Illustrated in Fig. 7.2, a square opening having an elliptical image is one manifestation of astigmatism. Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Aberrations
142
Figure 7.2: The image of a square pattern [Eq. (5.26)] with d = 0.4, tf g = 1, and tb g = 0 under the influence of astigmatism (c6 = 0.1).
Coma The ray aberration corresponding to Z8 = c8 \(3p 3 — 2p) cos 4
is (Al, A9) = C8
/(9J2 + 3
g2 — 2 , 6.Jß).
For a pattern that varies only in the z-direction with J(f g) = 6(f, g), ( A9) =c8\(9fß -2,0).
While the —c82v term describes a constant translation (x-tilt), the c89^f^ factor represents an image shift that varies with the location of the light ray in the pupil, namely, the amount of displacement is a function of the object spectrum. In addition, the f^ dependence means that a ray at —Jo is shifted by the same amount as that at +fo. The image of a symmetrical object generally becomes asymmetrical. Such lateral asymmetry is orientation-dependent in the presence of Z7 and Z8. Nevertheless, the intensity distribution is symmetrical with respect to the z = 0 plane because of the absence of even power terms in p. Shamrock
Similar to coma, the Zernike polynomials Z 9 and Z 10 result in lateral image asymmetries. The 34 dependence means that patterns possessing a threefold rotational symmetry are the most prone.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Measurement
143
Figure 7.3: In the presence of spherical aberration, light rays from the pupil intersect
the optic axis at different points. Spherical aberration A wavefront with spherical aberration [Z11 — 6p 2 + 1) and higher-order polynomials in which YJ (cß) = 1] can be perceived as having a radius of curvature that varies as a function of the radial position in the pupil. An image is degraded because light rays from the pupil intersect the optic axis at different points, as depicted in Fig. 7.3. Such degradation is asymmetric with respect to the Z = 0 plane, although no lateral asymmetry is introduced. High-order polynomials The effects of high-order terms can be ascertained qualitatively similar to those considered above. Light is diverted over a larger area with increasing n values because the ray aberration increases; the diffraction pattern spreads farther. Objects with mo -fold rotational symmetry are the most susceptible to Zernike polynomials with m = mo. Terms in which m is odd result in lateral but not longitudinal asymmetry.
7.5 Measurement The ability to measure aberration is indispensable for the production of high-quality optical elements. Knowledge of a system's aberration is beneficial to its users as it allows them to ascertain its performance and to monitor possible system degradation. There are three categories of aberration measurement techniques. One determines the aberration from the interference pattern formed by the wavefront of interest and a reference wavefront. Another class of techniques deduces the wave aberration from a set of intensity distributions produced by the imaging system in
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
144
Aberrations
Light source
U
ice
Figure 7.4: Principle of the Michelson interferometer.
question. The third category measures ray aberrations from which wave aberrations are derived.
7.5.1 Interferometry Since we cannot observe phase difference by itself, we need to convert the optical path deviation into an observable quantity. Interferometry determines aberration by interfering the wavefront of interest with a reference wavefront. Phase information can be determined from the resulting fringes. There are many interferometry techniques."' As an example, we study the Michelson interferometer, whose principle is illustrated in Fig. 7.4. A quasi-monochromatic light source is collimated by lens Ll such that a plane wavefront impinges onto the glass plate P with a beamsplitting surface S. The fraction of the incident wave that transmits through S is reflected by mirror M1 and then by S, forming the reference wavefront W. The test wavefront is created by the portion of incident wave that is reflected from S. This light travels through the lens under test (L), being affected by aberrations that may be present, before reflecting from mirror M2 and transmitting through S, resulting in the test wavefront W'. The fringes produced by interference between W and W' can be observed through lens L2.
7.5.2 The extended Nijboer Zernike approach -
The aberration signature of a lens set is usually obtained using interferometry prior to system assembly and shipment. But this signature may not represent the aberra"'Malacara [52] provides a catalog of interferometry techniques and descriptions of their principles.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Measurement
145
tion of the optical system because a final adjustment is usually performed at installation. It is desirable to measure the aberration of the installed optical system and to track its evolution. When in-situ interferometry [53,54] is unavailable, we need to determine aberrations by less direct means. One approach is the extended Nijboer-Zernike method [55]. In this technique, the through-focus coherent transfer function is represented by a linear combination of basis functions expressed in Bessel series. The coefficients of the basis functions are the same as the Fringe Zernike coefficients representing the pupil function. These coefficients are estimated by matching the theoretical and measured intensity distribution through focus. For an aberration function described by
(D(P, 0) =1 c Rn (P) cosm4, the field resulting from a point object is U(P,2) = 2 UÓ (P, 2)+ 2 ^i
m+l cJ n
(P1Z)cosm4,
j
where
n
V (P,2) = e
+i2 (-2i2)u-1 q buvJin+u+2v(2TCP)' u=1
v=0
u(27EP)u
(-1)g • (m+u+2v) • (m+v+u— 1)!(v+u-1)!(p+v)!u! 1 v + v + u)! u-1 v.)!v-}-u— q — )!(q — )!(p buv ( m+v)!( !( 2=kz(1— 1—sine 8 obi),
n+m p= 2 n—m q= 2 and the relationship between j, n, and in are given by Eqs. (7.11) and (7.12). For low levels of aberration, the intensity distribution is approximately I( P,4,2) = 4 IVó (P,2)I 2 + 8 1 c,Re {im +1vó * (P, 2 )VV (P,2)}cosmi j
=x8(P,z) (7.20) j
where +1UÓ (P 2 X (P>z) = f,Re {i ) V,, (P,Z)1, *
n
_ 4 8
ifn=m=0, otherwise.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Aberrations
146
We need to estimate the coefficients c'j given a set of measured intensity distributions at various focus levels 2. Performing a Fourier analysis of the observed intensity distributions Imes (p, 2) with respect to the angular variable gives 21c
Wmea(P, 2 ) =
1
f Imea(p,^^z)cosm0Ci0.
(7.21)
0
Comparing Eq. (7.20) and Eq. (7.21) we obtain, form 0, the relationship Wmea(p, 2) — n
c'j (ß52).
(7.22)
Let us define an inner product
Po (Ymea,
X) = f Wmea(0, 2)X * (P, 2) PdO, z
0
with po being the range of p measurements and the summation covers all measured focus levels. The product of Eq. (7.22) with xn (p, 2) is Cj(in íX n') .
1 (mea,X )=
,
(7.23)
n
If we truncate the infinite series in Eq. (7.22) to N terms, and take the inner product of yrmea with the same N xn (p, 2) terms, Eq. (7.23) results in an N x N linear system
of equations. Its solution gives the least square approximation of the Fringe Zernike coefficients.
7.5.3 The Hartmann test The Hartmann test measures the relative shift of an array of imaged features [56, 57J. The ray aberration and subsequently the wave aberration can be deduced from the displacement data. Figure 7.5 illustrates the principle of the technique. The object contains an array of small openings, two of which are denoted by Ql and Q2 with their Gaussian images at Pl and P2. A screen S with a small aperture is inserted between the object and the first lens, such that the image of Ql (denoted by Pl) is formed by a small light bundle located at (a, b) in the pupil. According to Eq. (7.18), the displacement of Pl from Pl is proportional to the ray aberration at (a, b). Similarly, the shift of the image of Q2 (labeled P2) from its Gaussian image point P2 is proportional to the ray aberration at a different location of the pupil. Measuring the image displacements of all array openings allows us to sample the ray aberration over the pupil.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Measurement
147
plane Figure 7.5: Principle of the Hartmann technique.
To derive the wave aberration from the measured ray aberration, we assume that the aberration can be represented by the first N Zernike polynomials N
c1Z1(0 0).
_
,
i =1
The measured displacement and the Zernike coefficients are then related by N
(Axmea,Aymea) _
az;
N
aZ1 1
C^—, Y, C^ á g ^1 af =i
J
.
For an object array comprising M openings, the Above relation becomes the matrix equation Omea = Ac,
where A is a 2M x N matrix in which each entry is the gradient of a Zernike polynomial, c is the vector of Zernike coefficients, and Omea represents the measured dis -placemntd.Thisrxquaoncbeslvdigthaquremod.
7.5.4 Aberration monitor patterns For optical systems with low aberration levels the phase factor exp(+i2itc) can be approximated by the first two terms of its Taylor series expansion: e +i 2
1c4D(P,O) = 1 + i2t(D(P, 0) = 1 + i2i I c1Zj (0, 0) -
With an aberration monitor pattern [58] described by Ok(x,Y) = Zk(j,g)e 12 +g 2 4
7.4 A dielectric slab of refractive index n and thickness d is inserted between a spherical wavefront and its geometrical image point. Derive the equivalent aberration introduced by the slab. 7.5 What is the intensity perturbation at the center of an aberration monitor pattern if the pinhole transmittance is p = 0.5 and the Zernike coefficient is Ck = 0.01?
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Chapter 8
Numerical Computation The theoretical developments in the last chapters have provided us with equations that describe optical imaging in photolithography. With these equations, images of objects can be computed in lieu of carrying out exposures. The possibility to simulate allows us to harness the power of affordable computers to predict images of object patterns, and to optimize the object and exposure configuration given a desired image. In this chapter we discuss common numerical formulations for imaging simulation.
8.1 Imaging equations Summarizing results of previous chapters, the imaging equation for a system with lateral magnification M is [from Eqs. (6.14) and (6.68)]:
I(x,y,z)=K f ... f J(.f,k)
7
(f+.f',g+g)H * (.f+.f",g+g')
6(.f' ,g' )M(ƒ+f' , +g' )E0 . O* (.Í" ,g")M* (ƒ+.f" ,g+$" )EÓ
e - i 2 "9 d f dgd f'dg d f"dg'
=Kƒ ... f Jj.f,g)H7 (J+j+g )H * (.f +.P',g+g") , Muk(f +f',b'+8')Myk(f +.f„ b'+g")E^E^
i={x,y}
j{x,y} k={x,y,z}
6(f', g') Ô (J", g")e
-i2..p
d.Ídgd.f'dg d.f"dg', (8.1)
151
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
152
Numerical Computation
where
K=
nimage
for coupling image [see Eq. (6.14)],
nP
for resist image [see Eq. (6.68)],
is refractive index of image space, is real part of resist refractive index,
nimage
nP
H
i
(4.31)
is the effective source,
J(f,g) -
ni a
ge 2
e +^2^^(P,^)
if Pe < sin8obi,
(6.69)
otherwise,
0
(7.10)
0(010) = 1 cizi (P, 0), j=1
for coupling image [see Eq. (6.14)], ge [ ^ see E .6. 68], for res i s ta f ii mae l )
_ Mo (f , 9) M(f ' g) M s tack(,Í^g)
(
ro'^' M0yx MO =
Px.,+Pxllx
PYj+PYIIx
oxy
MOYY = Px.Ly+PxIIY PY1Y+PYIIY oz= M0 PXIIZ PYliz
ß2 Px
aß
1_ y2'
__
(6.13)
aß
Px1y 1-72'
Prix=
-
1-12 ,
a2
PyIy = 1—Y2 ,
(6.10)
Py =O,
Px1z = 0 ,
_ a2
Pxllx — 1-721
Y
PYllx = 1aß 2 2
aß7 PxIIY = 1— y2'
PY IY
Pxllz = al
PYllz = ß,
1—y2 '
(6.12)
—
—
MS. Msyx
Mstack =
Ms
Msxz MSY, F±Px +Fjy Pxllx FLPylx+F xyPYllx
(6.66)
= FIPxly+Fl P lly FL PY 1Y+F PYIIY
Fi Pxll z
Fj PYllz
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Transmission cross-coefficient integration
153
F1 (p o z) _ t stack 1 re+ikP(dp—z)+p1bstacke—ikp(dp—z)1 Z substack L
(6.63)
1
,stack
11 (Po,z)
^Ikubstack [e
+ikP(dp _z) — P llubstack e —ikf (dp—z)1
(6.64)
,stack F^ ( Po ,z ) = tis^ubstack [e+' Pi
z (dp —z) +
1,
Pbstacke—ikP(dP—z)]
(6.65)
(6.59)—(6.62)
pij,zij,
dp is the photoresist thickness, = f sin9 obj,
ß=gsin8obj, y= 1—(f2 +g2 )sin e 8abj= pe = psinO obj,
= f2 + g2 , (P= (j'- 1")x+(S — g^)ƒ+(kp r— kp
„ )(z —
zo).
For computation of the coupling image, kp is the z-component of the wave vector in the coupling medium, and zo is zero. If our interest is in the resist image, kP is the z-component of the wave vector in the photoresist and zo is geometrical focal plane relative to the top of the photoresist. In writing Eq. (8.1), we assumed that • the spectrum is independent of the source point except for a translation proportional to the location of the effective source point [Eq. (4.40)]; • the diffracted field arising from each source point is proportional to the object spectrum at the appropriate spatial frequencies (f, g); • polarization of the diffracted field is independent of the object and the effective source; and • the ratio of the perpendicular to the parallel components is preserved during diffraction and propagation through the imaging system.
8.2 Transmission cross-coefficient integration To calculate image intensity at any point, we need to evaluate the sixfold integral of Eq (8.1), a daunting computation task. We can reduce the effort by assuming that the object is periodic in both the z- and 9-directions with periods p x and py . The spectrum then comprises discrete diffraction orders (see Exercise 8.1):
ó(M)=Er 5 (f — m ,g — n )O mn. n m
Px
Py
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
(8.2)
Numerical Computation
154
An isolated pattern is approximated by a periodic object with a large period. Such an approximation is acceptable, except for coherent illumination, because the complex degree of coherence is vanishingly small if the separation between points is large. The image of an isolated pattern under coherent illumination can be computed using Eq. (4.39). With a discrete spectrum, Eq. (8.1) becomes an equation with a double integral: — i2ncp Om n O* m„ n„ e
Î(,9,z) = K
=
n!"n',m'
fff(.f,g)H(.f+m'/ß ,g+n/ßy)JJ*(. +fn"/p,g+n"/ß ) Mik (f', g )Mk (J", g")EiE dfdg (8.3) t={X1'}
k= {.r _ .Z}
Om'n, O n,^ nn e^i2' r TCCm of ;mrrn ri. z EiEY ,
K
j={.rp}
rt 1 .m^
k-{.r,y,z}
where +00
TCC W ;m„ n,,.Z =
LI
J(ƒ,g)H(J+m'/ß, + n'/ß)
H* (f + m"/ß, + n"/Py)Mik (5; z)MYk (Pë; z) d f dg. (8.4) Illumination in optical lithography is usually unpolarized. Since unpolarized light of intensity IQ is equivalent to two independent linearly polarized waves of amplitude IQ/2 vibrating in mutually perpendicular directions (see §6.1), I (x,Y, z) — 2 K
Om'n' Omnnn e —i2 '
TCCn„.mnnn. Z.
(8.5)
k={r.y.z}
To determine the intensity at a particular focus level z', we need to calculate the various transmission cross-coefficients TCC n ;m and sum up contributions from all relevant diffraction orders according to Eq. (8.3) or Eq. (8.5) [42, 61]. A possible computation procedure is: IIn°Z'
1. Calculate the object spectrum Ómn by Fourier transformation of Ô(,9) [see Eq. (4.33)]. 2. Compute the set of transmission cross-coefficients TCCm n,;m°n,,;Z by numerical quadrature. For each coefficient, the region of integration is defined by the effective source J(f,g) and the two displaced pupils H(f+m'/px ,g+n'/py ) and H(f +m"/px ,g+n"/p) ) (see Fig. 4.8). Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Source points integration 155
3. Sum contributions from different polarization couplings [indexes i, j, and k in Eq. (8.3) or indexes i and k in Eq. (8.5)] and diffraction orders [indexes (m', n') and (m", n") in the equations]. Computation complexity of this transmission cross-coefficient integration method (also called Hopkins' approach [19]) depends on the number of diffraction orders. The range of m' and m" is proportional to px and the range of n' and n" to p y , meaning that the computation time tcompute scales asymptotically with the square of the object area A m = px x py : 2
tcompute °C (C1tTCC+C2tsum)Am,
where tTCC is the computation time of one transmission cross-coefficient and tsum is proportional to the time needed to perform the multiple summations in Eq. (8.3) or Eq. (8.5). Since each transmission cross-coefficient calculation involves a 2D quadrature that requires evaluation of the computation intensive functions Mik(OO;Z)Mjk(Pë;z) and H(f,g)
while ts„m involves only a few additions and multiplications, tTCC » is im
and
tcompute « tTCCAm
For calculating images produced by a specific imaging system with objects having the same periods pX and py , the transmission cross-coefficients can be precomputed and stored, thereby shortening the computation time such that tcompute °` tsumA 1
8.3 Source points integration Instead of performing numerical quadrature to compute transmission cross-coefficients, the source points integration approach (also called Abbe's method) transforms the sixfold integral of Eq. (8.1) into a fourfold integral, which can subsequently be approximated by a summation. Such transformation is possible because source points comprising the light source are mutually incoherent. Intensities produced by all source points aggregate to give the final image. This addition property allows us to express Eq. (8.1) in the following form:
Î(î,z) — ^^j(.Í,g) ^ ^H(i+f ,g+g^)^(Í^ g ) J '
2
(
M.Eo)e-^2n[f'x+g'y+kA^(z -zo)] dpdg dfdg. (8.6)
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
156
Numerical Computation
The quantity delimited by the absolute value is the electric field, generalized from Eq. (4.36) for polarization effects, arising from a source point of unit strength located at (I, g). If we denote the square of this quantity by Ícon , we can rewrite Eq. (8.6) as +00
I(,9,z) = I. (f,g)Icon(f,g;z)dfdS,
where Icon(f,;z) = K ^^H(f +f^,g+g)O(f^^g )
(
M Eo e —i2rz [!'z+g 9+Oz ' (z—zo)] d .
)
f'dg 2
We can perceive the overall image as the sum of an infinite number of weighed coherent images Ícon, with each component image arising from an effective point source of intensity f(fs , gs ) located at (fs, gs). For a periodic object with a discrete spectrum described by Eq. (8.2), +00
LI
i i
i
i
H(f+f ,g+8)o(f ,g)(M'Eo)e
f'x+'y+
2
'(z—zo)]
df dg
• E e—i2 [mx/PX +ny/ny +kP (z —zo)]
y) O ^ mn ( _ I H(.f + m/ß, g+ n/ ß)
0)
m,n
and Icon =
KI l (f +m/px,b' +nl b )O y
mn (M
Eo)
m,n
e —i2n [in /Px+n9/Py+kP' (z—zo)
2 ]
. (8.7)
In numerical computation, we approximate the effective source function by a finite number of point sources:
j(f,9)=Jf(J,g)
(8.8) s
where a s is the effective strength of the discretized point source located at 6(f — fs, g — gs). The image becomes I(x,f,z) =1 aslcon(Ís,b's;z),
(8.9)
s
and a computation procedure for this source points integration method can be:
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
157
Coherent decomposition
1. Calculate the object spectrum Ómn from Ô(1,9) [see Eq. (4.33)]. 2. Approximate the effective source J(f, g) by the discretized source JA(f, g) [see Eq. (8.8)]. 3. For each discrete source point, compute the component image according to Eq. (8.7). 4. Sum component images to obtain total intensity [see Eq. (8.9)]. For an effective source of area A and an object of area A m , the computation time scales according to S
tcompute — As . Am
8.4 Coherent decomposition We can also reduce the sixfold integral of Eq. (8.1) into a fourfold integral by (inverse) Fourier transformation of the constituting functions into their spatial-domain counterparts. For scalar imaging we essentially recover Eq. (4.32) from Eq. (4.35) (see also Exercise 4.4):
I(x,Y) =
Nl
w (x — zá,Y — yo,x — xö,Y — Yo) o(xo,yo)O' ( 1o,9o)d1od9d1ád9ö,
where
W (.óY10;x ool ,Yo) = f((xo xo^,Ya yo)H(xo,Yo)H * (xo,Yu)• —
—
The function W can be decomposed into a series of its eigenvectors cpk [62-64]: W (xo}Yo;x11 40) _ ^k(Pkl-oJo1 ) (Pk(xoJo} k=1
where Xk is the eigenvalue corresponding to the eigenvector cpk. With this series representation, the image becomes
I(19)—k^kkffff (Pk(x—z Y—Y)Ô(xY) (P*(x—.x',9-9")O*(xß,3 ")d2'd9' dx"dy"
_ k=1
ff (Pk(.x-x,9-f)O(x,Y)d 'dy
+°°
1
,
2
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Numerical Computation
158
We can interpret each eigenvector as the transfer function of a coherent imaging system [compare with Eq. (4.1)]. The overall image is the weighed sum of the images produced by an infinite number of coherent systems. In situations where the magnitudes of a few eigenvalues are much larger than the rest, we can approximate the image as a finite sum of these dominant eigenvectors: K
1(1,9) =
2
(8.10)
kk (O ® Pk) (x, Y) I , (
k=1
where K is the number of significant eigenvectors, and the symbol ® denotes convolution. To calculate the image intensity at any point, we need to compute K 2D convolutions. Equation (8.10) is amenable to efficient computation of objects comprising polygons. For example, a rectangular pattern of foreground transmittance tf g and background transmittance tb g = 0 with vertices at (ui , 9i), (12, 91), (12, y2), and (11,92) can be described by O(x,Y) =tfg[Q( 1— zl,9 - 9l) — Q( 1-12,9 - 91)+
Q(1-12,9-92) — Q(x — x1) — y2)], where 1 Q(x ' Y) 0
ifx>0and9>0, otherwise,
is the quadrant function. An object consisting of N rectangles can be described by
N
o(x,Y) _ tfg ) [Q(x -1 1n) ,y — Yins ) — Q(.x — x2 'Y — Yins )+ n=1
Q(x—x2n^'Y—Y2n)) — Q(x — x1n ^,Y — Y2
where the superscript (n) indexes the rectangular shapes. Substituting this object description into Eq. (8.10) results in I(x,Y) = I Xk ( 6 ®^k)(xiY) 2 k=1 K IN
(n) (n) (n) ^ (n) 1vk(x—x1 ,9 91 ) — lk(X -12 'YY1 )+
_ ^?k k=1
-
I n=1 2 (n) (n) (n) (n) 1 1Vk(x —.z2 ,Y — Y2 ) — Mfk(x — x 1 ,y—y 2 ) J , (8.11)
where 11k(1,9) =
Q(x,y) ®(Pk(1,Y)
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Object spectrum
159
is the convolution of the quadrant function with the kt eigenvector. As the functions 1rk(1,9) are independent of the object, they can be precomputed (for each particular exposure configuration). Image calculation becomes a double summation over the significant eigenvectors [index k in Eq. (8.11)] and over the rectangular shapes [index n in Eq. (8.11)]. Each inner loop requires four function look-ups and three additions, and each outer loop needs two multiplications. The computation cost is tcompute = NK(4tlook-up + 3 tadd) + 2Ktmuit + (K —1)tadd [+Kty^], where tlook-ups tadd, tm„lt, and t, are respectively the time needed for function lookup, addition, multiplication, and computation of a convoluted function 111k (1,9). Assuming the functions yrk(1,9) are calculated in advance, the computation time scales linearly with the number of object vertices asymptotically. The procedure for image computation by the coherent decomposition approach is:
1. Determine the eigensystem of W, and the number of retained eigenvectors K. 2. Precompute the functions yrk (x, y) for all significant eigenvectors. 3. Compute the image according to Eq. (8.11).
8.5 Object spectrum The object spectrum is the Fourier transform of the field transmitted by the object under the illumination of a unit-amplitude wave. This field behaves in a complicated manner, varying continuously and oscillating in the vicinity of transitions between regions of different transmittance. To simplify image computation, we make the approximation, similar to Kirchhoff's in his derivation of the diffraction integral [Eq. (3.7)], that the transmitted field changes abruptly according to the transmittance of the object. This thin-object approximation (also called thin-mask approximation), of which Eqs. (5.5) and (5.26) are examples, suffices for patterns sizes large compared with the wavelength. When the object dimensions are on the order of or smaller than the wavelength, we should treat object transmission as a boundary value problem based on Maxwell's equations, and solve for the transmitted fields numerically. For example, for the space depicted in Fig. 8.1, we need to find the field across the line AB subject to an incident radiation. The computation should consider material properties, including the refractive indexes of the mask substrate nglass, the opaque layer n opa , and the object space medium nobject, as well as the thickness d opa of the light-blocking material.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Numerical Computation
160
Figure 8.1: The field along AB can be determined by solving Maxwell's equations. The thin-mask approximation simplifies object spectrum computation.
Figure 8.2: Time-domain finite-difference is one approach for computing the field along AB in Fig. 8.1.
Time-domain finite-difference is one approach for solving such a numerical problem [65-68].' Using the integral form of Maxwell's equations [Eq. (1.6)], we solve for the six electromagnetic field components
e = ( ex
,
ey , ez )
and
h = (hX hy , hZ ,
)
by decomposing the object into a grid and assigning one field component to each grid point, as illustrated in Fig. 8.2. For a cubic grid in which Ax = Ay = Az, the surface and line integrals in Eq. (1.6) are evaluated on squares. Referring to Fig. 8.3, the z-component of the electric field ez(iAx,joy,kAz) = ez(i,j,k) =e(,,)
can be expressed in terms of the x- and y-components of the magnetic field:
á
(£e t +óc)ez(i,j,k)
= [h,(i,j— 1/2,k) —h,(i, j+ 1/2,k)+ by (i+1/2,j,k)—by (i-1/2,j,k)]Ax. (8.12)
'Besides the time-domain finite-difference approach, the boundary value problem can be solved by finite-element [69] and frequency-domain methods [70-75].
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Object spectrum
161
hX (i,j+1 /2,k) Ay/2 b y (i -1 /2,j,k)—e, k)—h y (i +1 /2,j,k) Y hJJ-1/2,k)
ZLX
Ax/2
Figure 8.3: The z-component of the electric field is expressed in terms of the x- and y-components of the magnetic field.
This equation is derived under the assumption that ez (i, j, k) is constant over the square surface and that the magnetic field components are constant along each line segment comprising the square. To obtain a time discretization of Eq. (8.12), we assume that the electric field components are constant within the time period [nAt, (n + 1)At), and that the magnetic field components are fixed within the duration [(n — 1/2)At, (n + 1/2)At). This formulation results in the time-domain finite-difference equation: eZ
+1(> >) =
aez (, ,) + ß [hx +1/2 (, —1 j2, ) — hz + 1/2
where
_
a
(,1/2,) +hy
+i/2 ( 1 / 2
,,) — by
+i /2 (
-1 / 2
,,)11 (8.13)
2Ee — G At _ At 2 and 2£e + 6cAt ß dx 2Ee + 6,Ot
Following the same procedure, the equations for the other five components are ey +1 ( 1/2,1/2)= aey(,1 /2,1/2) +ß[hz +i /2 (-1/2,1/2,1/2)— h 1/2 (1/2 1/2,1/2) +h 112 ( 1/2,1) —hX+z /2(,1/2,)], (8.14)
e
x +l (1 /2„ 1 /2) = aex (1 /2„ 1 /2) + [hy+i
/2 (1 / 2
„) —
by +1/2 ( 1/2 1) +hz +i /2 (1/2,1/2,1/2) —hz +1/2 (1/2,-1/2,1/2)1, (8.15) hz +1/2(1/2 1/2,1/2) = hz -1 / 2 (1/2 1/2,1/2)—
At1 [e,'(1/2„ 1/2) —eX(1/2,1, l/2)+ey(1,1/2,1/2) —ey(,1 /2,1/2)], (8.16) Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Numerical Computation
162
n-1/2
n+ l2
by
(1/2 )=hy l „ At 1
hx+l/2 (,
(1/2,,)—
[e?(„)—ez(1„)+ey(1/2„1/2)—ey(1/2„-1/2),, (8.17)
1 / 2 ,) = hX—' 12 (^ 1/2,)— At
I ez(,1,)—ez(„),. (8.18)
The field transmitted by the object can be computed by iteration of the above equations. To confine the computation volume, absorbing boundary conditions are needed on the border surfaces of the simulation domain. These conditions result in a different set of iteration equations compared with Eqs. (8.13)—(8.18). Please refer to the literature [76] [77] for details. Electromagnetic simulation using the time-domain finite-difference approach thus comprises two primary steps: 1. Discretize simulation volume; assign (material) parameters to each grid point. 2. Iterate field components according to Eqs. (8.13)—(8.18) or boundary conditions until convergence.
8.6 Remarks The formulation of Eq. (8.1) assumes that the wafer stack is a stratified medium. In situations where there is significant wafer topography effects, the analysis of §6.4 is inadequate. There is generally no close-form solution. The resist image must be computed rigorously by consideration of light coupling into the photoresist in the presence of wafer dielectrics [78, 79]. The time-domain finite-difference technique described in §8.5 is one method for such calculation. Besides images in photoresists, we are often interested in modeling photoresist dissolution. The effects of resist processing can be simulated by two classes of techniques. Rigorous approaches [80-84] attempt to describe the photoresist dissolution process with first-principle formulation involving reaction and kinetics of chemical species within the resist. Phenomenological methods [85-88], on the other hand, seek to model photoresist behaviors heuristically by calibrating actual resist performance with model parameters that may bear little relationship with resist chemistry and physics. Interested readers may wish to consult these references for details.
Exercises 8.1 Show that the spectrum of a periodic object with periods px and py in the i- and 9-directions is given by Eq. (8.2).
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Exercises
163
8.2 Derive the spectrum of a periodic rectangular feature with periods pX and py in the x- and 9-directions. The coordinates of the lower left and upper right corners of one of the patterns are (io,9o) and (xl,yl). Each rectangle has a transmittance of tfg with the background transmittance being tbg . 8.3 Is it possible to simulate immersion imaging with a simulator that assumes the refractive index of the coupling medium to be one? If so, how? 8.4 Using Stoke's theorem (see Exercise 1.5), develop a formula to compute the area of a polygon.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Chapter 9
Variabilities In replicating an integrated circuit layout during fabrication, the same object shapes are often delineated numerous times. Because of unavoidable variabilities of a manufacturing process, the delineated shapes are generally different from the nominal shapes and from one another. This variability should be kept under the specified tolerance according to which integrated circuits are designed. Too much deviation causes circuit failure. An understanding of the various causes of variation is helpful in devising means to reduce, stabilize, and compensate for the undesirable variation. Although all processing steps (such as deposition, lithography, etching, and chemical-mechanical polishing) contribute to patterning nonuniformity, we focus on variabilities arising from optical imaging, since, with both layout shapes and image tolerance shrinking rapidly compared with Xo/NA, control of image variabilities is of increasing concern. Lithography becomes more difficult with decreasing kl and k1h_P;tch.
9.1 Categorization We can classify the causes of lithography variability into two categories. One affects object shapes located in identical environments, and the other impacts the same object shapes situated within distinct configurations of neighboring shapes. Let us call the effects of the former fluctuations and those of the latter inherent variations. The same cause can result in both fluctuation and inherent variation. On the other hand, a fluctuation and an inherent variation may have similar manifestations. To distinguish fluctuation from inherent variation we must first expound the meaning of environment. Let us define afeature as an object shape within a particular configuration of neighboring shapes. The shapes A and B in Fig. 9.1, although identical, are distinct features because their environments of shapes are different. The periodic space described by Eq. (5.5) is a feature that is fully specified by its foreground and background transmittance tf g and tb g , size d, and period ßx. A peri-
165
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Variabilities
166
Al
A2
A
gi
B B2 B2
Á
A
B^
l A2
Bj
Figure 9.1: A feature is an object shape within a particular configuration of neighboring shapes. A and A' are the same feature if we identify edge Al with Ai and A2 with A. odic line feature is similarly determined by these four parameters:
OX(i) = tbg if Ix—npx l < cl/2, n E Z, g otherwise.
ti
(9.1)
The orientation of a feature does not constitute its environment. For example, a y varying periodic line described by -
^
tb g if IY — nPy I < d/2, n E Z, y ^ y ^ — tfg otherwise,
is the same feature as that described by Eq. (9.1), provided they have the same tb g , tfg , and cl, and px = py . With this understanding of "feature," we can define fluctuation as the variability of the same feature, and inherent variation as the difference, excluding fluctuation, between distinct features .of the same object shape. Referring to Fig. 9.1, the difference between the delineated shapes of A and A' (and that between those of B and B') is fluctuation. So are differentiations between their delineated shapes across an exposure field, from wafer to wafer, and from lot to lot. The distinction between the averaged image of all replicas of A and A' (and those of B and B') is inherent variation. By this demarcation, we can perceive fluctuations as arising from engineering imperfection while inherent variations as consequences of the laws of physics. The variation in the images of periodic spaces with the same nominal dimension d but different periodicities follows from Eq. (4.35) because of differences in their spectra. But no physical law dictates that images of a feature at two points in the field of an optical instrument should differ. The difference one may observe can be caused by aberration fluctuations of the imaging system.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Proximity effect
167
A
pthreshold
w
O
-1.0
-0.5
0.0
0.5
1.0
x (7JNA)
Figure 9.2: Method for determining the size and placement of a space image.
To quantify image variability, it is useful to convert each image into a number or a set of numbers that represents a relevant quality of the image. Metrics for this purpose include contrast, depth of focus, exposure latitude, normalized image logslope [89], exposure-defocus window [38], and total window [90]. These metrics can be determined from measured or simulated data. As an example, Fig. 9.2 illustrates a method that derives image size and placement from a computed image. The image of a feature, a periodic space in this case, is first computed using the techniques described in Chapter 8. The size iv of the image is the distance between the intersections of the simulated image and a threshold intensity 'threshold.' The placement error 0 can be defined as the distance between the center of the object shape and the midpoint of the two intersections. We use this method in our investigation of variabilities in subsequent sections of this chapter.
9.2 Proximity effect The spectrum of an object spreads as its size decreases. For dimensions on the order of or smaller than the wavelength, the pupil cuts off spatial frequencies that carry a sizable fraction of the light energy transmitted through the object. The image becomes distorted. One manifestation of optical distortion is proximity effect, an inherent variation where the image size w of a shape changes depending on its environment. A typical scenario is shown in Fig. 9.3, which plots the image sizes of nominally d = 0.6 spaces [Eq. (5.5)] and lines [Eq. (9.1)] as a function of period. As the period changes 'We can relate !threshold to the exposure dose Dprint that is supplied in an exposure. Denoting the dose to clear a positive resist or to harden a negative resist with a completely transmitting mask by Do, the dose D pri ❑t is Dprint
Do 'threshold
This approximation becomes more accurate with increasing photoresist contrast [85].
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
168
Variabilities
❑--❑
space
Q o Z
0-0
.
line
N
N_ (0
O N ^ N
.1
O
1.0
2.0
3.0
4.0
period (RINA)
Figure 9.3: Proximity effect is the change in image size depending on the environ-
ment of an object shape. The features are nominally d = 0.6 spaces and lines with different periods.
the image sizes of periodic lines vary by as much as 20%. ü Figure 9.3 suggests that periodic spaces exhibit less proximity effect than periodic lines. We can use Eq. (4.32) to investigate whether this is generally true. Let us first contrast the imaging of two periodic spaces of the same size dbut different periods: pi = 2d and P2 = 4d. Imagine that the object is divided into atomic regions of width d, as illustrated in Fig. 9.4(a). The integral in Eq. (4.32) can be converted into a sum involving interaction of these atomic regions. For tfg = 1 and tb g = 0, the intensity at space Ao is roughly +00
I2d 0e ^ 1 J(0, 0)H(2nd, 0)uY* (2nó, 0) } n=--
I, l (2 [i — j]d, o) H(2id, 0)R* (2 jd, 0) (9.2) iii
and +00
,space f(0, 0)H(4nd, 0)H* (4nd, 0)+ n=--
J(4[i— j]d,0)H(4id,0)PI*(4jó,0). (9.3) i0i
Except for coherent illumination, the mutual intensity J(x,9) generally decreases with increasing 1.xJ and y^. For circular illumination with a partial coherence factor 6 [Eq. (5.13)], the first dark ring of f(î,9) occurs at [see Eq. (4.29)] r
_
z2 + y2
_ 0.61 6
"Proximity effect generally means any type of image shape distortion including pattern shortening and corner rounding. But it has the specific meaning of image size variation with period in this context.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Proximity effect
169
2d 0
0
— — 1ine — —
0
0
0
O 0 0 0 0 0
Ap
A2
A4
A -1 A l
pl =2c1
pl = 2d
4d O
A3 A5
' line \ —
O
0
Ao
A4
00 000
OOO 00
A - ,A - ,A -1 A l A 2 A3
A5 A6
pl =4J
(a) space Figure 9.4:
(b) line
Spaces generally exhibit less proximity effect than lines.
If we assume 6= 0.8 and d=0.4, f(î,9)0
for r > 0.8 = 2d.
(9.4)
With the approximation of Eq. (9.4), the intensities expressed by Eqs. (9.2) and (9.3) become I2ppace.
+00 N
j(0, 0)H(2nd, 0)H* (2nd, 0),
and
n=--
Isppace
+00
N
f(0,0)hH(4nd0)H*(4nó,0). n=--
Since the point spread function H(z, y) generally decreases with increasing r, with the first zero at P = 0.61 for a circular aperture [Eq. (3.24)], only the first term is significant in both sums. The intensities of the two spaces are approximately equal. The situation is different for periodic lines. Dividing the object into atomic units in a similar manner [Fig. 9.4(b)], the image of a line of size d and period pl = 2d is roughly
72ó
e
—
If(o, 0)R( — [n2 + 1], 0)R ( — { n2 + l], ) +
n=--
ll,
([i— j]2d,0)H(— [i2+1]d,0)H*(—[j2+1]d,0).
i0i
The sum over i and j can again be neglected due to Eq. (9.4). But the double sum is non-negligible for a line of period p2 = 4d. For this feature, the atomic units such as A 1 and A2 are separated only by d. They interact and contribute to an image that can be much different from the image of a line of period pi = 2d. Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Variabilities
170
In addition to mask tone, proximity effect also depends on illumination configuration, mask technology, and wafer stack through their presence in Eq. (8.1). Although the exact behavior depends on the detailed interaction between the object spectrum and optical system [Eq. (5.1)], proximity effect generally increases with decreasing kl and k1h_p;«h factors, and with an increasing degree of illumination coherence. For example, the first zero of the complex degree of coherence under partially coherent illumination [Eq. (5.13)] is, according to Eq. (4.29), r=0.61 ko aNA The range of non-negligible optical interaction increases with decreasing partial coherence factor.
9.3 Object variabilities (photomask errors) 9.3.1 Dimensional error Photomask dimensional error is the deviation of a mask-feature size from the designed size. From a mask maker's viewpoint, dimensional error comprises both fluctuations and inherent variations. For the purpose of our discussion, all mask dimensional-error components are considered fluctuations. When an object pattern is large, its image is scaled exactly, except for a constant bias bo, by the magnification of the projection optics M. The image size wo varies linearly with the nominal mask dimension do with a slope equal to the magnification M: wo = Mdo + bo. Deviation of mask dimension from the nominal (Ad) results in a scaled linewidth error (Ow): Aw=w —wo = M(d — do) + (bo — bo) = MAd.
A unit change in the mask dimension corresponds to a unit change in the image size scaled by the magnification of the exposure system, a phenomenon illustrated in that portion of Fig. 9.5(a) where mask dimensions are large. The effect of mask dimensional errors is diminished in a reduction system. Image sensitivity to mask variation generally increases with decreasing feature size. When the object size is small, the effective mask dimensional error MAd is magnified [mask dimensions smaller than 0.5Xo/(NA x M) in Fig. 9.5(a)]. The severity of this amplification is described by the mask -error factor (MEF) [also
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Object variabilities (photomask errors)
171
U,
U,
0—o space
Z o
ó ^
N
w
0—❑
Ü
N N
square
Q
a)
N
^
E ^? 0
N
co
(C
E
1.0
0.5
1.5
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
d (GINA)
mask dimension [X/(NA x M)j
(b)
(a)
Figure 9.5: (a) Image size varies linearly with mask dimension for large features, but sensitivity to mask variation increases with decreasing features size. The features are periodic spaces with ßx = 2d [Eq. (5.7)]; and the magnification factor is assumed to be 0.25. (b) The mask-error factor typically increases with decreasing feature size, although the exact behavior depends on all aspects of the imaging process, including illumination configuration and mask technology. [From A. Wong, et al., "The mask error factor in optical lithography," IEEE Transactions on Semiconductor Manufacturing, vol. 13, no. 1, pp. 76-87 (Feb. 2000), copyright (2004) IEEE.]
called mask-error enhancement factor (MEEF)] [91, 92], such that the relation between mask dimensional error and image size fluctuation becomes Aw = MEF x MAd. As an example of mask-error factor dependence on object size cl, Fig. 9.5(b) plots MEF for the same spaces as those in Fig. 9.5(a) as well as for a square [Eq. (5.26)]. The MEF is I for large features, but increases from unity as d decreases below 0.75 for the square and 0.5 for the space. Because of its amplification for small object sizes, mask dimensional error is a significant fluctuation contributor in low-k1 optical lithography. i^i "'In addition to its dependence on object size, the mask error factor also depends on feature type and its environment, photomask technology (chromium-on-glass, attenuated phase-shifting mask, and alternating phase-shifting mask), and illumination configuration (on-axis and off-axis). In general, the factor increases with decreasing kl and k1half-P,«h. One exception is the imaging of a narrow, sparse line by alternating phase-shifting masks in which the MEF approaches zero with reduction of the chromium size [931.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
172
Variabilities
0
d=0.10 ?/NA
CR ó
---------------d=0.051JNA
>'(0 to o
N •_
t Ó
N O
-2
-1
0
1
2
x (AJNA)
Figure 9.6: The image intensity is quartered when the size of a small isolated space is halved [23].
Mask-error amplification arises from image integrity degradation. Consider a 1D isolated space of width d: _ 1
^x(x) 0
if 1 xß < J/2,
otherwise.
Its spectrum is ô(f) =dsinc(fd). In the limit of small d, lim Óx (f) = d, d-*O
and the image is [see Eq. (4.37)] I () d
ff
TCCX(t ; ") e -i21r [íf'-f")z1 d f' d J"
—
(9.5)
= d^L(J, H). The image shape is determined by L(J, H), a quantity sometimes called the line spread function. Since L(J, H) is a function of the effective source intensity dis J and the coherent transfer function H, the image shape depends only on-tribuon exposure system parameters. Modification of the object has no effect on the image shape, except for scaling of the intensity by the square of its size. Halving the size quarters the intensity. This phenomenon is illustrated in Fig. 9.6 for two spaces of sizes d= 0.1 and 0.05. The quadratic dependence of image intensity on size described by Eq. (9.5) is the primary physical cause of the mask error factor. We can qualitatively ascertain
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Object variabilities (photomask errors)
173
O
N Y c0 Q
Eq. (9.5) W) Eq. (4.38)
—
Ó .^ 6 0 O 0 d r
a) .^ 0 D)
O
0.0
0.2
0.4
0.6
0.8
1.0
d (XJNA)
Figure 9.7: Severity of mask error amplification can be ascertained by comparing the peak image intensity predicted by Eq. (4.37) and Eq. (9.5). [From A. Wong, et al., "The mask error factor in optical lithography," IEEE Transactions on Semiconductor Manufacturing, vol. 13, no. 1, pp. 76-87 (Feb. 2000), copyright (2004) IEEE.]
the severity of error amplification by comparing the square root of the actual image peak intensity with that of Eq. (9.5). A large difference means that mask errors are not amplified much, whereas closeness in the numbers indicate large effective magnification. Plotted in Fig. 9.7, the difference between the straight line, which represents Eq. (9.5), and the curve, which represents the actual image peak intensity, is big for large spaces. As the size decreases from d = 0.5, the curve and the line begins to converge. Below d = 0.2, changing the object no longer changes the image shape. Image control is lost. A similar consideration applies to squares with minor changes of the imaging equation (see Exercise 5.10):
I^x,9) _ [
yJ (r)
r
J
Z d4 •
In the limit of small cd, the image shape is solely determined by the point spread function while the intensity varies with the size. Rather than a quadratic dependence, the intensity is proportional to the fourth power of the linear dimension of the square contact hole. Because of this fourth power dependence, the MEF of contact holes starts to increase from unity at a larger dimension and has a higher value than that of spaces at the same dimension. The MEF can be as high as 4 for 0.5X0/NA contact holes, offsetting the advantage of 4X reduction systems. Although loss in coupling image integrity is the primary cause of the MEF, all factors causing image quality degradation, whether coupling, resist, or latent, worsens the MEF. Focus error, aberrations, excessive diffusion of chemical species in photoresists, and low photoresist contrast all increase the MEF.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
174
Variabilities
incident radiation
Figure 9.8: An alternating phase-shifting mask fabricated using a subtractive process.
9.3.2 Phase and transmission errors Phase and transmission errors are results of nonuniformities of phase-shifting mask dielectrics. They can arise from mask topography, imperfection of defect repair, and thickness nonuniformity of photomask films. As an example of phase and transmission errors, consider an alternating phaseshifting mask fabricated on a chromium-on-glass substrate using a subtractive process. Illustrated in Fig. 9.8, the 180 deg regions are etched areas of the fused silica substrate. The thickness of material removed detch is determined by requiring an optical path difference of
2n+1 ^
aEZ 2 (9.6)
between light passing through the 0deg and 180 deg phase regions. For a path difference of 2/2, a first-order estimation of the required etch depth detch is
xo
nglassdetch — nobjectdetch =
i
2nobject
detch =
(9.7)
2nobject(nglass — nobject)
where nglass is the refractive index of the fused silica substrate at the exposure wavelength and nobject is the refractive index of the object space. For wavelengths of interest in optical lithography (436, 365, 248, 193, and 157 nm),
nglass
=
1.5,
nobject = 1 ,
and detch = kO
The etch depth is approximately equal to the exposure wavelength. The phase regions are asymmetric because the 180 deg areas are etched while the 0 deg are not [94-96]. This imbalance causes transmission and phase errors. Figure 9.9 shows the simulated field within an alternating mask using the time-domain finite-difference technique described in §8.5. The incident light is reflected from Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Object variabilities (photomask errors)
175
Figure 9.9: The Odeg and 180 deg phase regions are asymmetric because of mask edge scattering [23]. the chromium areas, resulting in the standing waves. Transmitted fields through the two openings are not identical. The darker color in the unetched 0 deg region on the left compared with the etched 180 deg opening on the right suggests higher transmittance of the 0deg region. Detail analysis also reveals that the phase difference between the regions is not exactly 180 deg, even when the etch depth is that given by Eq. (9.7). The effects of phase and transmission errors on images are demonstrated in Fig. 9.10. For an alternating phase-shifting mask with only transmission but no phase error [Fig. 9.10(a)], the region with lower transmission gives a lower peak intensity for all focus levels. With only phase error [Fig. 9.10(b)], there is no intensity imbalance at nominal focus. The peak intensity of one opening is greater than the other with focus error in one direction, and the difference is reversed when the focus varies in the other direction. The effect of pure phase error is a focusdependent peak intensity difference. With a combination of transmission and phase errors [Fig. 9.10(c)], the peak intensities at best focus are not balanced because of transmission error. The imbalance is not maintained through focus due to phase error. With intensity imbalance, spaces that have the same nominal dimension print with different sizes in positive photoresists. Lines are shifted; but their sizes remain mostly unchanged. The effects are reversed in negative photoresists: resist islands differ in size while clear areas are displaced.
9.3.3 Edge roughness We mean by edge roughness the minute and random undulation of the contours of a photomask feature. By analyzing its spectrum, we can classify roughness into two categories: low-frequency unevenness that is resolved and replicated onto the wafer, and high-frequency coarseness where spatial frequencies are beyond resolution by the exposure configuration in use (on the order of 2NA/X0; see §5.6). Although these high-frequency components are not reproduced directly, they may collectively cause an effective mask dimensional error.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
176
Variabilities
0
k
ó
positive
focus
O
4 O
C N O
C N
r
d
Q)
-0.8 -0.4
0.0
0.4
-0.8 -0.4 0.0
0.8
x (a/NA)
0.4
0.8
x ()/NA)
40.00.40.8
x (AJNA)
Lo n ó
nominal focus
_T mus,
_T
C
'(/1 C
C N
C
W O d
0
m ó
d
C N ó
ó -0.8 -0.4
T
0.0
0.4
0.8 -0.4
0.8
x ()JNA)
x ()/NA)
0.0
0.4
0.8
0.4
0.8
x (AMA) N r ó
_T í1 O7
negative focus
^N O
C N
^N O
. O d N
C N
C N
ó
O
-0.8 -0.4
0.0
0.4
0.8
x ()/NA)
(a) transmission
-0.8 -0.4
0.0
0.4
0.8
-0.8 -0.4
x (A/NA)
(b) phase
0.0
x (2JNA)
(c) both
Figure 9.10: Effects of phase and transmission errors [23].
9.4 Polarization effects Not only does photomask topography cause phase and transmission errors, as demonstrated in §9.3.2, it also changes the degree of polarization of light. Let us begin by studying transmission of linearly-polarized light through an opening, as illustrated in Fig. 9.11(a). The incident light can be polarized with the electric field vibrating parallel or perpendicular to the edge of the opaque layer, which is assumed to be chromium. Figure 9.11(b) plots the fraction of energy transmitted through the opening normalized to its width for both polarizations, where TE (transverse electric) denotes electric field vibration parallel to the chromium edge, and TM (transverse magnetic) represents perpendicular vibration. The TM polarization shows higher transmission for all opening widths; the fractional difference increases with decreasing opening size [97]. Higher transmittance of the TM polarization may be traced to the electromagnetic boundary conditions at the air-chromium interface. Because the absorbing chromium layer is metallic-like, the electric field component parallel to the chromium edge is close to zero, whereas the perpendicular component is not (see
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Illumination
177
0 ^Q
TE O
TM
a,=248 nm
TE O —TM nglass ...:..:.::
C
cs
L
y
0
0
ir_
U
.
c chromium
N cs
0
250
500
750
1000
opening width (nm) (a)
(b)
Figure 9.11: (a) Structure of an opening. (b) Fraction of energy transmitted through the opening normalized to its width for the TE and TM polarizations [97]. Exercise 6.9). Unpolarized light becomes partially TM polarized upon transmission through photomasks. The degree of polarization increases with decreasing opening width. Since images of TM-polarized light have lower contrast than those of the TE polarization (see §6.7), changes in the degree of polarization degrades images by emphasizing the TM component [98]. Despite adversely affecting image quality, photomask-induced partial polarization does not cause inherent variation or fluctuation for unpolarized illumination. Under polarized illumination, however, the degree of polarization of transmitted light is a function of feature orientation. Variability would manifest as orientationdependent image shapes, convolving with the effects of astigmatism (see §7.4).
9.5 Illumination When the effective source intensity distribution is asymmetric with respect to the optic axis, the exposure system suffers from nontelecentricity. Since an asymmetric effective source implies a preponderance of light incident from some off-axis direction, we can understand the fluctuation arising from nontelecentricity by studying imaging with an off-axis point source, as illustrated by the schematics of Fig. 9.12. Imagine an image formed from a bundle of rays centered at the undiffracted direction indicated by the bold arrow in the figure. The lateral position of this center ray shifts with the longitudinal (focus) planes. Nontelecentricity causes image placement error that increases with defocusing.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
178
Variabilities
out of focus^ \i
best focus
Figure 9.12: The center image-forming ray shifts with focus for an off-axis source point [23].
A remedy for such focus-dependent image shift is to maintain source symmetry, such that the image shift corresponding to one source point is compensated by that of the mirror source, resulting in a final image that has no net displacement. This is the rationale behind dipole illumination (see §5.3). One can observe the effective source intensity distribution on the image plane using the method illustrated in Fig. 9.13 [99]. Rays from the light source impinge upon a clear reticle with an opaque spot P on the back. With Köhler illumination, the intensity on the object plane would have been uniform in the absence of the opaque spot. Let us denote this intensity by I. With the opaque spot, the radiance on the object plane diminishes. For example, point B receives light from all parts of the source except for the blocked bundle AP. The intensity distribution on the object plane is Iobject(x,y) = Jo
— ff
J(.f, g) d.fdg,
where S2 is the solid angle that P subtends at (x, y), and f and g define the propagation direction of the light beam AP. For a small opaque spot, Iobject(x,y)
= IO — C J(f, g)
with c being a constant. The relationship between (x, y) and (f, g) is (see Exercise 9.1): nob'ectsin 15
nglass r
_(9.8)
t 2 + r2
where r = x2 +y2 , sine = \/ f2 + g2/f, and t is the thickness of the reticle. If we define Jmax as the maximum intensity of the effective source and Ia n as the corresponding minimum intensity on the object plane, then Irvin = Jo — cJ , and f(f,g) Jmax
Io I(x,y) — lo — 'min —
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Pupil
179
object plane
Figure 9.13: An opaque spot on the back of a clear reticle creates a reverse-tone image of the effective source on the image plane.
The intensity distribution of the effective source is replicated in the object, which is in turn reproduced onto the image plane.
9.6 Pupil Aberration contributes to fluctuation in two ways: image variability caused by aberration itself, and fluctuation resulting from across-field aberration changes. In the presence of coma, for example, the images of a pair of nominally identical spaces described by if d/2 < Ill < 3d/2 tfg (9.9) Óx(z) _ tb g otherwise, are different. This asymmetry changes as the amount of coma varies across the image field.'° Vignetting is another defect that can cause fluctuation. Usually the result of cursory lens design, some image-forming light rays are not captured by the lens when vignetting occurs. The peripheral of the field is especially prone to this defect.
9.7 Focus Focus variation arises from focal plane deviation (variation of Z4 across the image field), astigmatism fluctuation (variation of Z5 and Z6 across the exposure field), wafer topography, mask flatness variation, focus setting and auto-leveling errors, wafer and chuck nonflatness, lens heating, barometric pressure and other environ1 °Brunner [100], Progler and Wheeler [101], and Flagello, et al. [102] provide more discussions on the lithography effects of aberrations.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Variabilities
180
mental variations. A defocused image is blurred, resulting in image quality degradation. Although decreasing X and increasing NA improve resolution of imaging systems, these measures adversely impact depth of focus—the maximum amount of tolerable focus variation. We can estimate how depth of focus scales with 4 and NA using Rayleigh's criterion [103]. Consider the situation shown in Fig. 9.14, where two light rays, one from the center of the pupil and the other from the edge of the aperture, interfere to form a sharp image on the plane zo. This image degrades as the observation plane moves away from zo. At a plane a distance z from zo, the relative phase change between the two rays is 21t
nimage (z — z cos 0obj)
rad.
Rayleigh used the heuristic criterion that the image is sufficiently blurred when the phase change is 90 deg, namely, the optical path difference is a quarter wavelength. According to this criterion, the Rayleigh unit of depth of focus is a distance R. U. such that (see Exercise 9.2) R. U. = -0 1
2 I
1 + 1— sin e 8obi /
4 nimage sin B obt \ 1 1
4 NA • sin S abi
(9.10) 1 — sin2 9abj .
In improving the resolution by reducing X and increasing NA the focus tolerance diminishes. The decrease is linear with wavelength, but depends on the manner in which the numerical aperture changes. Increasing the image space refractive index decreases depth of focus linearly with 1/ni mage , namely, linearly with 1/NA. On the other hand, increasing the semi-aperture angle reduces focus tolerance faster than 1 / sin 2 9 obj, namely, quadratically with 1/NA. When the semi-aperture angle is small, the Rayleigh unit is approximately R. U. =
2 sint 0obj
(9.11)
As sin0 obi approaches one, the Rayleigh unit approaches (X/4) rather than (?/2). The depth of focus is further halved because of the (1 + 1— sin e A obi) factor. Obtaining a larger numerical aperture by increasing nimge impacts depth-of-focus less adversely than increasing sine o b^. °Although defocus is one form of aberrations, we consider it separately because of the varied causes and its historical importance in optical lithography.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Focus
181
pupil ^,' 1 0 1 — 0 2 1= ^/4
eob; nimage
I 01 1 z o z-zo nimage
Figure 9.14: Rayleigh's depth of focus criterion.
Figure 9.15: The focus monitor comprises lines bordered by transmitting regions
that are 90deg out of phase. Similar to the definition of kl for minimum image size [see Eq. (5.27)], we use the k2 factor to denote the depth of focus of a lithography process [104]: depth of focus = k2 R. U. k2
2
for low-NA imaging. 2 sin 9 obj
Focus sensitivity of a process and hence its difficulty increase with decreasing k2. We can measure focus crudely by exposing isolated lines of decreasing dimensions onto a positive photoresist through a series of focus levels [105]. The focus value that prints the narrowest line is the best focus. For finer determination of focus, we can use the focus monitor [106]. The measurement structure consists of a line bordered by two transmitting regions that are 90 deg out of phase, as shown in Fig. 9.15. The 90 deg phase makes the lines Ll and L2 shift in opposite directions with focus changes (see §9.3.2), and the displacement, being insensitive to dose fluctuation, is linear with respect to defocus at small deviations from the diffraction focus. Together with the lines L3 and L4 this structure can be used to characterize focus and astigmatism.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
182
Variabilities
á^ f°o E m
E
x (7/NA)
- f (NA/A)
(a) transmissivity
x (a/NA)
(b) spectrum
(c) approximation
Figure 9.16: (a) A transmission wedge for monitoring dose variation. (b) Its spectrum. (c) Its approximation.
9.8 Dose Dose change can cause both inherent variation and fluctuation. On the one hand, dose nonuniformity arising from wafer topography, wafer stack, and exposure dose variation, including actual dose fluctuation across exposure field or across the slit of step-and-scan exposure systems, and effective dose variation caused by nonuniform hot plate temperature during post-exposure bake, results in fluctuation because of finite exposure latitude. On the other hand, a uniform dose change such as a deliberate dose modification or a dose setting error affects proximity effect because dose sensitivity (exposure latitude) differs among features. The ideal structure for dose monitoring should be sensitive to dose variation but not to other process detractors. One candidate is a transmission wedge described by Ôx(x) — m(1—mpzI)
0
if ^zI < m,m>0, (9.12) otherwise,
with the spectrum (see Exercise 9.4) m2 [1—cos(27tf/m)]. Ó,f (f) = 22f2
(9.13)
The transmission function and spectrum of an example structure with m = 0.2 is shown in Fig. 9.16(a) and (b). Because of the inverse f^ dependence the spectrum is sizable only around f = 0. Concentration of energy in low-spatial-frequency components implies low sensitivity to aberrations including defocusing. But the gradual transmissivity change ensures responsiveness to dose variations.
The transmission function of Eq. (9.12) is difficult to realize in practice. We can approximate the gradation by a structure comprising an array of spaces at a fixed period but decreasing duty ratio from the middle, as illustrated in Fig. 9.16(c) [1071. This pattern is sensitive to actual and effective dose variations including maskdimensional error.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Flare
183
We can determine dose and focus concurrently using the two structures shown in Fig. 9.17 [108]. One of the patterns, shown in Fig. 9.17(a), consists of two gratings situated in a clear background. The grating period p is beyond resolution by the optical metrology system, namely, p
metrology o < 2NAmetrology
The image varies only in the horizontal direction [along AB in Fig. 9.17(a)], as plotted in Fig. 9.17(c); and we can associate with this structure a width w s . An analogous quantity wl, corresponding to two gratings situated in an opaque background [Fig. 9.17(b)], can be similarly determined [Fig. 9.17(d)]. With increasing defocus, the grating features of both structures shorten, causing ws and w1 to increase. Their focus dependence is approximately quadratic around optimum focus, as shown in Fig. 9.17(e). With dose increase, however, the grating features of Fig. 9.17(b) lengthen linearly (on positive photoresists) while those of Fig. 9.17(a) shorten, causing the behavior illustrated in Fig. 9.17(f). Based on the similarity in focus response and difference in dose behavior, we can develop a parametric model such as the following [109]: wti = ao ; + ai 1 E + (a21 + a3 ; E) (z — zo), i E {l, s}, where the coefficients ao 1 ,...,3, and the best focus position zo are parameters determined by experimental calibration. With these parameters, the focus [or defocus (z — zo)] and dose E can be solved for each pair of measurements w s and wl:
E_
—b+ b 2 — 4ac
2a
and
(z — zo) 2 =
wi
— (ao i +a1 E) ;
a2, +a3 E ;
where
a = al,a3, — a3 1 a1, b = a3 1 WS — a3 wt +ao 1 a3 +al 1 a2, — a2 1 al s — a3,ao., c = a2 1 ws —a2swt +ao l a2, —a2 1 aos . S
5
9.9 Flare Also called stray light and scattered light, flare is light existing in the image beyond that captured in the physics described by Eq. (8.1). Figure 9.18 illustrates the primary sources of stray light. Consider image formation of an object point A with the geometrical image point B. Because of diffraction and possible focus error, the intensity distribution around B is blurred. A fraction of this halo is reflected from the wafer toward the reticle and then back to the wafer from the reticle, adding to the original image intensity. Each additional round-trip reflection increases the amount
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
184
Variabilities
Figure 9.17: The similarity in focus response (e) and difference in dose behavior (f) between the structures shown in (a) and (b) enables simultaneous determination of focus and dose.
of stray light, with the halo increasing in size but decreasing in brightness. Depicted by the dotted rays in Fig. 9.18, we can reduce this cause of flare by decreasing the reflectivity of the wafer or the reticle or both. Light scattering from optical elements also contribute to flare. Scattering is caused by surface roughness of optical elements. An ideal optical element has smooth surfaces; it performs its designed function without scattering. For example, a thin lens with smooth surfaces focuses a parallel light beam toward a point on the focal plane within the approximations of geometrical optics (see §2.4). There may be undesired specular reflection if the lens anti-reflection coatings are imperfect;
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Flare
185
wafer and reticle
-
reflection
B A ^ ^ y
, -
------------surface
D
- aberration JC
scattering
Figure 9.18: Primary sources of flare.
but no light is diffused into other directions. However, a physical optical element has (microscopically) rough surfaces that scatter light. The amount of scattering depends on surface roughness and wavelength according to e(4na/X)2 — 1
where a is the standard deviation of the assumed Gaussian-rough surface [110]. Any light (including image-forming beams and rays bouncing between the wafer and the reticle) that impinges onto a surface is scattered; and the amount of scattered light increases with the number of optical elements. Stray light also arises from high-order aberrations. Recall the discussion in §7.4 that the ray aberration generally increases with the Zernike term number, causing the diffraction pattern to spread. If a wavefront contains significant components of high-order aberrations that are not included in the truncated Zernike-series representation of Eq. (7.17), the diverted light contributes to flare. This source is depicted by the light ray ACD in Fig. 9.18. We can surmise that stray light generally increases with the average transmission of the object, and the amount of flare at a particular field point depends on the magnitudes of these sources and their characteristic light spreading distances [111]. Scattered light has the longest range (on the order of the field size), while ray aberration has the shortest (on the order of micrometers). The amount of flare is a function of object pattern density over a large area. By raising the overall image intensity, stray light reduces contrast. But flare variation across the field causes more fluctuation than contrast loss. As an example, assume a feature prints at a nominal dose of Dprint = 1OD0/3, where Do is the dose to clear a large area of positive photoresist. The corresponding normalized intensity level (threshold intensity) is 3/10 = 0.3. If the flare decreases by 3%, the feature would be printing at a normalized intensity level of 0.30 + 0.03 = 0.33 of the original image. This 3% flare level change translates to a 10% dose variation; Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Variabilities
186
the effect of flare variation is amplified. Let us call this effective magnification the flare-amplification factor (FAF): FAF = D^ón
t
Given by the ratio of dose-to-print to dose-to-clear, the flare-amplification factor causes large amounts of fluctuation especially for features with little exposure latitude. By direct addition of light to images, flare causes features to print at a different intensity level, resulting in gross fluctuation. A simple way to measure flare is to expose an opaque object, so large that its nominal interior intensity is zero in the absence of flare, in a sequence of dose steps and observe the dose Ddisappear at which its image formed in positive photoresist completely disappears [Ill]. The ratio of dose to clear D o to Ddisappear is the amount of flare: flare = D0 x 100%. (9.14) Ddisappear
A more detailed characterization of flare can be accomplished by using a large opaque pattern with small openings placed at regular intervals [111], as shown in Fig. 9.19(a). We again expose this test pattern in a sequence of dose steps onto a positive photoresist. As dose increases the edge of the developed photoresist recedes from the designed edge. From the dependence of the resist edge position, determined relative to the small openings, on the exposure dose, we can deduce, using an equation similar to Eq. (9.14), the amount of flare as a function of distance into an opaque region. Figure 9.19(b) shows the result of such a measurement. Three distinct regions are observed. At large distances, the amount of stray light steadies around 3.2%; the background flare (or long-range flare) is 3.2%. At intermediate distances between 1µm and a few tens of micrometers, flare decreases gradually with distance. Sometimes called midrange flare, this behavior is attributable to high-order aberrations [ 112] and wafer-reticle reflections. At distances smaller than 1µm the change in spatial characteristics indicates flare arising from different phenomena. We cannot draw definite conclusions, however, because accuracy of this receding edge method worsens with increasing proximity between the photoresist and geometrical edges due to diffraction and lateral photoresist development effects.
9.10 Remarks We can simulate, with a program such as one developed using the equations of Chapter 8, the effects of most causes of variabilities. Their impact is naturally object-dependent. Even for a specific technology for which only one critical dimension is of primary concern (namely, a single kl factor), image sensitivity can vary greatly among different pattern configurations. Nevertheless, it is possible to
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
187
Remarks
0
CO
0 large opaque pattern
\
mid-range
0 00
CO
O Q0
long-range
0000
V small openings
short-range
Co 4
0
N -
0.1
1.0
10.0
100.0
distance from designed edge (gm) (a) test pattern
(b) measurement results
Figure 9.19: (a) Test pattern for the receding edge method. (b) An example of mea-
surement results [111 ]. estimate a representative sensitivity for each detractor and subsequently a corresponding control requirement. An exercise performed for a kl = 0.35 technology gives the following estimations: Detractor Proximity effect Mask dimensional error Phase error Transmission error Mask edge roughness Degree of polarization Non-telecentricity Aberrations Focus Dose Flare
Control Requirements OPC ±0.008/M (?o/NA) ±2deg 0.005 (Do)
— < 0.05 (for unpolarized illumination) ±0.01 (p) 0.015 (2 rms) ±0.5 (R. U.) [see Eq. (9.10)] ±2% (Dprint) ±0.5% (Do)
These requirements are determined separately for each variability contributor. During actual processing all detractors affect imaging and their net effects are quantified by metrics such as across-chip linewidth variation (ACLU). The manner in which variabilities from various causes combine is a question that often arises in tasks such as estimating variability, developing control requirements, and deciding how the different components of each cause mix together. (An example of the latter task is the determination of how focal plane deviation, wafer topography, mask flatness variation, focus setting error, and wafer nonflatness combine into an overall focus error.) In one extreme we can assume that variabilities caused by all
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Variabilities
188
contributors add linearly. But this conservative and unrealistic assumption results in unnecessarily stringent control requirements. The other extreme assumes that the effects are uncorrelated and random, such that variance of the total across-chip linewidth variation is the sum of the individual detractor variances: t
ótotal =
6i
je {detractors}
This latter assumption is usually adopted in practice. We should beware, however, that effects of independent causes may unwittingly reinforce or cancel one another such that the uncorrelation assumption is violated. For example, the spatial linewidth variation signatures resulting from mask dimensional error, aberrations, and flare fluctuation can have much semblance [113]. In general, individual variabilities and their possible correlation is process dependent. They can be determined only by thorough empirical characterization [113-115]. A well-characterized process would also allow us to perform optical and process correction (OPC), i the technique of predistorting mask patterns to reduce image variability [116-125]. Being a photomask approach, OPC can only remedy inherent variations and systematic and stable fluctuations. We should minimize random variability components for efficacious correction. Even with exact engineering control, however, noise is one random fluctuation that cannot be eliminated. In the context of optical lithography, shot noise refers to the statistical variation in the number of photons used to define a feature. The impact of shot noise depends on the photoresist, the photoacid generation mechanism, and the post-exposure bake and dissolution processes [126-128]. To understand qualitatively the effects of shot noise, consider 193 -nm exposure of a contact hole of area A = 45 nm x 45 nm with a dose of Dp int = 1 mJ cm 2 . The number of photons absorbed Nphotons is approximately h
Nphotons
Dprint X Ithreshold X fabsorbed
xA
Ephoton
where fabsorbed is the fraction of exposure energy deposited into the photoresist, Ithreshold is the threshold intensity of the contact exposure process, and Ephoton is the energy carried by each photon. Assuming the resist absorbs 10% of the exposure energy and Ithreshold = 0.3, 10-3•0.3•0.1•2025 x 10 Nphotons =
-
14
x 10 +$
6.6256 X 10-34. 3193x10- 9 590.
There are only 590 = 24 photons per side, with each corresponding to a substantial 4.1% of the critical dimension. Stochastic variation in the number of photons `''OPC was originally shorthand for optical proximity correction. But the correction has since been applied to compensate for nonoptical processing effects.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Exercises
189
absorbed can cause resist line-edge roughness, resulting in higher variability. Such line-edge roughness can be reduced by increasing the number of absorbed photons, thereupon lowering the photoresist speed.
Exercises 9.1 Derive Eq. (9.8). 9.2 Derive Eq. (9.10). 9.3 In the low-NA limit, what is the Zernike series representation of one Rayleigh unit? 9.4 Derive Eq. (9.13).
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Appendix A
Birefringence The theories and equations explicated in the text were based on two distinct sets of fundamental equations: Maxwell's equations [Eqs. (1.1)—(1.4)], which governs wave propagation; and constitutive relations [Eqs. (1.7)—(1.9)], which describe the interaction of electromagnetic waves with materials. We have assumed in our developments that materials are optically isotropic such that the electric displacement D and electric field E are related by the dielectric constant £e , and the magnetic field H and magnetic induction B are related by the permeability µ,n : D = £ e (w)E
and
Bim(CO)H.
(A.1)
The permittivity and permeability are frequency dependent; however, they can be considered constants since we are concerned with monochromatic waves. The constitutive relations described by Eq. (A. 1) indicate that the electric field is aligned with the displacement, and the magnetic field with the induction. In an anisotropic material, however, these pairs of vectors are not parallel. For electrically anisotropic but magnetically isotropic materials, we can retain the relationship between the magnetic field and induction in Eq. (A. 1), but the simplest connection between the electric field and displacement becomes a linear matrix transformation: D=
Dx
[EJ
£ems £e Xy E e l eyx £ eyy LeyZ Ey
= I£
£ezy £e z ,
= £e E,
EZ
where the nine quantities £ejj constitute the dielectric tensor £e .' Let us investigate consequences of misalignment between the electric field and displacement on wave propagation. For a monochromatic plane wave of angular frequency co, the field phasors are proportional to exp(ice[ --r.s] —t). 'Born and Wolf [129] contains detailed discussions on optical behavior in anisotropic materials.
191
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
192
Birefringence
B H
S
E
E1
'
D Figure A.1: The electric field E and electric displacement D are not parallel in an electrically anisotropic medium.
We can express the partial derivatives of the field vectors as
á
E = —iwwE,
and
all
= — iwH,
V x E = tw—ss x E,
VxH=iwnsxH.
In the absence of currents, Eqs. (1.1) and (1.2) become H=— n 9xE
and
JLmCO
D= n9xH. co .
(A.2)
Substituting the first expression of Eq. (A.2) into the second and using the vector identity Ax(BxC)- (A.C)B—(A•B)C gives n2
n2
n2
m C0
mC0
UmC0
D =— 2 sx(s"xE)= 2 [E—(9 -E)s]= 2 E 1 .
( A.3)
The quantity E1 denotes the vector component of E perpendicular to in the plane formed by E and s, as illustrated in Fig. A. 1. Since, from Eq. (A.2), the vectors H and B are perpendicular to E, D, and s, the latter three vectors are coplanar, with D being orthogonal to "s. The magnetic field H and the electric displacement D are transverse to the normal of the wavefront 9, as in an isotropic medium; but the electric field E is not. To understand the manner in which wave propagates in anisotropic materials, we revisit the expressions for the electric and magnetic energy densities [Eq. (1.14)]
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Birefringence
193
and the Poynting vector [Eq. (1.28)]: 1 2
1 we = 1 E • D = 1 Ei£eijEj,
2i
(A.4)
={x,y,z} j={x,y,z}
w m = ZH•B= 2 IH^ 2 ,
and
S=ExH. Performing mathematical manipulation similar to that leading to Eq. (1.12), we derive that —V.(ExH)=E.-D+H aB at at =
aEj u„
óM1 2
Ei£e `^ 2 + 2 at i ={x,y,z} j={x,y,z}
The term on the left side is the rate of total energy density increase. In the absence of absorption or amplification, it should be the sum of the rate of electric and magnetic energy density changes. For the first term to represent the rate of electric energy density change, it must equal the time derivative of Eq. (A.4): aEj1
aE j aEi
[^
1 Ei£e at = 2 L £e"( Ei —+E —) at at ° i ={x,y,z} j={x,y,z} i ={x,y,z} j={x,y,z}^' 1
meaning 1 aEi
aE j
1
at
i ={x,y,z} j={x,y,z}
£e`' (E —
El
at
=
^+
(EejjEejj)Ei
L
0.
i ={x,y,z} j={x,y,z}
For nonzero electric field components, the above expression is zero if, and only if, £e jj = £ ejj .
The dielectric tensor is symmetric. It has six instead of nine independent components. We can now write the electric energy density as
2
£ E Z £e E? £ E?
We = e 2 x +
Y
+
e
2
+ £eyz EyEZ + £exz ExE2 + Eeg ExE9.
(A.5)
Since the energy density must be greater than or equal to zero, Eq. (A.5) is a semipositive-definitive quadratic form. Spatial points with the same energy density form
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Birefringence
194
an ellipsoid surface. There exists a coordinate system (x',y',z') such that the energy density is We = Eex,Ex +£ey Ey, + Eez E^, where C ex„ E ey„ and Eeg, are the principal dielectric constants (or principal permittivities). In this system of principal dielectric axes, the material equations are simplified: D'
=
E ez, ExI ,
Dyi = Eey, Ey ,
and
DZ' = Cez, Ezi .
(A.6)
Because the principal dielectric constants are distinct for an anisotropic material, the electric field E and displacement D are not parallel unless E coincides with one of the principal axes." Substituting Eq (A.6) into Eq. (A.3) gives, after some straightforward algebra, skEk
n 2 ( g
E )sk2
, — n2 —^um£ekCO
k E {x, y', z' }.
(A.7)
Adding this set of three equations together and dividing the sum by (s • E) results in k={ ',z'}
Subtracting both sides by
n2s2 k um6ekCO
n2_
sk = 1, we obtain k={x' y',z'}
2 2
2 2
(A.8)
f ____OSk v = 2 psk 2 2 2 k={x',Y ,z'} n —,Um£ek CO k={x',y',z'} vk — Vp
where V P = co/n and vx, _
1
vy1 _
1
um£e^
and
vz,
1 ,umEeZ,
are the three principal propagation velocities. Equation (A.8) is a quadratic equation in vp. There are two values of v p for every direction s". With each of the two values of v p , we can solve Eq. (A.7) for the ratios between the electric field components, and subsequently find the electric displacement components using Eq. (A.6). Since the ratios between the field components are real, the electric field and displacement are linearly polarized. We can conclude that an anisotropic medium permits two monochromatic plane waves with two different linear polarizations propagating with two different velocities in any given direction. The electric field also aligns with the displacement if E e,, _ £ey, = se . But this is the isotropic case.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
195
Birefringence
A ramification of having two propagation velocities is a phenomenon known as double refraction or birefringence. When a plane wave incidents from an isotropic medium onto an anisotropic one, there are generally two refracted rays on the plane of incidence, since, according to Snell's law [Eq. (6.22)],
e
Si n int
_ Cinc
sin 8tran Ctran ctr and hence 6tran can assume two values.' In photolithography lenses made of birefringent materials (such as calcium fluoride for 157-nm lithography), each light ray entering an imaging system with N lens elements gives rise to 2N beams in the exit pupil. Such birefringence can be described by a 2 x 2 matrix that characterizes coupling between the different waves [130, 131 ].
"'In addition to intrinsically anisotropic materials, birefringence can be caused by anisotropy induced in nominally isotropic media resulting from mechanical stress, or by an orderly arrangement of particles of isotropic materials. The former phenomenon is called stress birefringence, while the latter is called form birefringence.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Appendix B
Stationarity and Ergodicity l In computing time averages of optical fields, we regard the ensembles to be stationary and ergodic. Stationarity means that all ensemble averages are independent of the time origin; and ergodicity implies that each ensemble average is equal to the time average involving a typical member of the ensemble. If we denote the field at a point Pi arising from a point source i by Ul (Pi, t), < U(P1,t) >= f Uip(Ui)dUi = 0,
since Ul (Pi, t) can be viewed as a random process on a time scale much longer than the coherence time of the light. Now consider another field generated by a different point source j. Let us denote the field at P2 arising from j by U1 (P2, t). The time average of U, (Pl , t) U1 (P2, t) is +00
< Ui(P1,t)Uu(P2,t) >= ffuuJp(u,uJ)duduj.
Since U (Pl , t) and U^ (Pa, t) are statistically independent, P(U, U.i) = P(Ui)P(U1),
and < Ui(Pi,t)Uj(P2 i t) > = fUp(U)dUiJUjp(Uj)dU1 =< U(P,,t) >< U;(P2 i t) > =0. 'This appendix is contributed by Anthony Yen. 197
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Appendix C
Some Zernike Polynomials j
n
m
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
0 1 1 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8
0 1 1 0 2 2 1 1 3 3 0 2 2 4 4 1 1 3 3 5 5 0 2 2 4 4 6 6 1 1 3 3 5 5 7 7 0
an
R,`, (0)YJ (0)
vÏ \ V6 \
v
V 10 10 10
viö 12 12 12 12 VÏ2 12 14 14 14 14 14 16 16 16 16 16 16 VÎ 6 16
(+1) (+p)cosrp (+p)sin4 (+2 2 _1) (+p 2 )sin24 (+p 2 )cos24 (+3p 3 -25)sin4 (+3p3 — 2p)cos4 (+p3)sin34 (+53)cos34 (+6p 4 -65 2 +1) (+40 4 - 3p 2 )cos2¢ (+40 4 — 3p 2 )sin24 (+p 4 )cos44 (+p 4 )sin44 (+10p 5 —12pá +3p)cos4 (+100 5 —120 3 +3p)sin4 (+50 5 -4p 3 )cos3c (+50 5 —4p 3 )sin30 (+p 5 )cos54 (+p 5 )sin54 (+2006 — 3004 + 120 2 — 1) (+150 —200 4 +6p 2 )sin20 (+150 6 — 200 4 + 6p 2 )cos 20 (+60 6 - 5p 4 )sin44 (+60 6 — 5p 4 )cos 44 (+p6)sin64 (+p )cos64 (+350 7 — 600 5 + 305 3 — 4p)sin 4 (+350 7 —600 5 +300 3 —4p)cos4 (+210 7 — 300 5 + 10p 3 )sin34 (+210 7 — 300 5 + 100 3 )cos 34 (+70 7 - 615 5 )sin50 (+70 7 - 60 5 )cos54 (+p 7 )sin7m (+p 7 )cos74 (+7008-14006+9004-2002+1)
199
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Some Zernike Polynomials
200
JÎ8
38 39 40 41 42 43 44 45 46 47 48 49 50
8 8 8 8 8 8 8 8 9 9 9 9 9
2 2 4 4 6 6 8 8 1 1 3 3 5
51
9
5
52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93
9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13
7 7 9 9 0 2 2 4 4 6 6 8 8 10 10 1 1 3 3 5 5 7 7 9 9 11 11 0 2 2 4 4 6 6 8 8
10 10 12 12
1 1
18 iT8 18 VÎ 8 18 18 18 20 20 20 20 20 20 20 20 20 20 11 22 22 22 22 22 22 22 22 22 22 24 24 24 24 24 24 24 24 24 24 Vi 24 VT3 26 v26 26 26 26 26 26 26 26 26 26 26 28 28
(+560 8 —10515 6 +600 4 —10p 2 )cos24 (+5615 8 —1050 6 +6004 —10p 2 )sin 2cß (+2815 8 - 420 6 +1515 4 )cos40 (+280 8 — 420 6 + 15p 4 )sin44 (+80 8 - 715 6 )cos6c (+80 8 —7p 6 )sin60 (+15 8 )cos80 (+p 8 )sin84 (+1260 9 -2800 7 +2100 5 - 600 3 +5p)cos4 (+1260 9 - 2800 7 +2100 5 - 600 3 +5p)sin0 (+840 9 —1680 7 + 1050 5 — 200 3 )cos 34 (+840 9 —1680 7 + 1050 5 — 200 3 ) sin 30 (+360 9 - 560 7 +2115 5 )cos54 (+360 9 - 560 7 +2115 5 )sin5 0 (+90 9 — 8p 7 )cos74 (+90 9 -80 7 )sin70 (+15 9 )cos90 (+15 9 )sin90 (+2520 10 — 6300 8 + 5600 6 — 2100 4 + 300 2 —1) (+2100 10 —5040 8 +4200 6 —1400 4 + 15p 2 )sin24 (+2100 10 — 50415 8 +4200 6 —1400 4 + 15p 2 )cos 20 (+1200 10 - 2520 8 +1680 6 - 35p 4 )sin44 (+ 1200 10 — 25215 8 + 1680 6 — 350 4 )cos 40 (+450 10 - 720 8 +28p 6 )sin60 (+45 0 10 — 720 8 + 28p 6 )cos 60 (+10p 10 - 90 8 )sin80 (+10p 10 - 90 8 )cos80 (+p 10 )sinl04 (+15 10 )cos 100 (+4620 11 12600 9 + 126015 7 — 5600 5 + 1051 3 — 615) sin 0 (+4620 11 -12600 9 + 12600 7 — 5600 5 + 1050 3 — 6p)cos 4 (+3300 11 — 8401 9 + 75615 7 — 2800 5 + 350 3 ) sin 34 (+3300 11 -84015 9 +75615 7 -28015 5 +35p 3 )cos30 (+1650 11 — 3600 9 + 2520 7 — 560 5 ) sin 50 (+1650 11 3600 9 +2520 7 - 56p 5 )cos51 (+55p 11 — 900 9 + 360 7 ) sin 70 (+550 11 — 900 9 + 36 5 7 ) cos 70 (+110 11 100 9 )sin90 (+110 11 1015 9 )cos94 +p l l )sin 114 (+P 11 )cos 110 (+9240 12 — 27720 10 + 31500 8 —168015 6 +42015 4 _4215 2 +1) -
-
-
-
(
(+7920 12 — 23100 10 + 25200 8 —12600 6 +2805 — 210 2 )cos 20 (+7920 12 — 23100 10 + 25200 8 —12600 6 + 2800 4 — 21 15 2 )sin 21 (+49515 12 -13200 10 +12600 8 -5040 6 +7015 4 )cos4 0 (+4950 12 —132015 10 + 126015 8 — 5040 6 + 700 4 ) sin 40 (+2200 12 — 4950 10 + 3600$ — 840 6 )cos 60 (+2200 12 - 4950 1 0 +360p 8 - 8415 6 )s1n64 (+660 12 -110p 10 + 45p 8 )cos80 (+660 12 -110p 10 + 450 8 )sin80 (+120 12 —11p 10 )cos 100 (+ 120 12 —11p 10 )sin 100 (+15 12 )cos 120 (+0 12 )sin 120 + 12600 5 —1680 3 + 7p)cos 0 (+17160 13 — 55440 11 + 69300 9 —420015+ (+17160 13 — 554415 11 + 69300 9 — 42000 7 + 12600 5 —16ßp 3 + 715) sin 0
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Some Zernike Polynomials
94 13 3 95 13 3 96 13 5 97 13 5 98 13 7 99 13 7 100 13 9 101 13 9 102 13 11 103 13 11 104 13 13 105 13 13 106 14 0 107 14 2 108 14 2 109 14 4 110 14 4 111 14 6 112 14 6 113 14 8 114 14 8 115 14 10 116 14 10 117 14 12 118 14 12 119 14 14 120 14 14 121 15 1 122 15 1 123 15 3 124 15 3 125 15 5 126 15 5 127 15 7 128 15 7 129 15 9 130 15 9 131 15 11 132 15 11 133 15 13 134 15 13 135 15 15 136 15 15
201
28 (+12870 1 3 -39600"+46200 9 -25200 7 +630p á -560 3 )cos30 28 (+12870 13 —3960" +46200 9 —25200 7 +6300 5 — 563 3 )sin34 28 (+7150 13 —19800" + 19800 9 — 8400 7 + 1260 5 )cos 50 28 (+71áp 1 ' — 19800" + 19800 9 — 8400 7 + 126p 5 )sin50 28 (+2860 1 ^ — 6600" +4950 9 —1200 7 )cos 70 28 (+2860 13 -6600 11 +4950 9 -120p 7 )sin70 28 (+780" 1 3 -1325 11 +55p 9 )cos94 28 (+780 13 —1320" +55p 9 )sin94 28 (+130" 13 —120")cos 110 28 (+130 1 3 — 120 11 )sin 114 28 (+p' 3 )cos 134 28 (+p 13 )sin 130 vTi5 (+34320 14 -120120 12 +166320 10 -115500 8 +42000 6 -7560 4 +560 2 -1) 30 (+30030 14 — 1029615 12 + 138600 10 — 92400 8 + 31500 6 —50415 —5040 4 +28 0 2 ) sin 20 30 (+30030 34 — 102960 12 + 1386015 10 — 92400 8 + 31500 6 — 5ß4p 4 + 28p z )cos 24 30 (+20020 14 — 64350 12 +792015 10 — 46200 8 + 12600 6 — 12615 4 )sin40 30 (+20020 14 -64350 12 +79205 10 -46200 8 +12600 6 -126p 4 )cos441 30 (+ 100115 14 -28600' 2 +29700 10 -13200 8 +2101 )sin60 30 (+10010 14 -28600' 2 +297015 10 -13200 8 +21015 6 )cos641 30 (+3640 14 — 8580 12 + 66015 10 —1650 8 )sin 80 30 (+36415 14 — 8580 12 + 6600 10 —16515 8 )cos 8i 30 (+910 14 -1560 12 +660 10 )sin100 30 (+910 14 —1560' 2 +66p 10 )cos 100 30 (+140 14 -1315' 2 )sin120 30 (+140 14 -1315' 2 )c0s120 30 (+p 14 )sin140 30 (+15 14 )cos140 32 (+64350 15 _2402415 13 +3603615h 1 — 277201 9 + 1155007 —2520i5 +25215 — 8p)sin 0 32 (+64350 15 — 240240 13 +360361511 —2772015 0 + 1155015v —252015 +25215 — 815)cos o 32 (+500515 15 — 180180 13 +25740151 —184800 9 +693015 v — 12601 5 + 8415 3 )sin 341 32 (+50050 15 — 180180 13 + 257400 11 — 184800 9 + 69300 7 - 12600 5 + 840 3 )cos 30 32 (+30030 15 —100100 13 + 128700 11 — 79200 9 +231015 7 — 25215 5 )sin50 32 (+30ß3p 15 — 100100 13 + 128700 11 —792015 +23100 7 — 252p 5 )cos50 32 (+13650 15 -40040' 3 +429ßp"-19800 9 +33013 7 )sin74 32 (+136515 15 -40040 13 +42900 11 -19800 9 +330p 7 )cos7, 32 (+45515 15 -10920 13 +8580"-22015 9 )sin94 32 (+45515 15 — 109215 13 + 8580 11 — 2200 9 )cos 90 32 (+10515' 5 -1820 13 +7815")sin110 32 (+ 1050' 5 —1825 13 +780")cos 110 32 (+150 1 5 —1415 1 3 )sin 130 32 (+ 1515 15 — 14p 13 )cos 134 32 (+15 15 )sin 150 32 (+15 15 )cos 154
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
202
Some Zernike Polynomials
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Some Zernike Polynomials
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
203
I
Appendix D
Simulator Accuracy Tests It is often desirable to assess the accuracy of an imaging simulation program. A test suite for fidelity evaluation should be comprehensive, and the cases should admit analytic solutions. In this chapter we construct a set of tests, based on the physics encapsulated in Eq. (8.1), which can be used to gauge the accuracy of simulation programs.
D.1 Blank mask For a mask that is completely transmitting,
Ô(1,9)=1
6 (f,) = S(Í)S(8)
and
Because of the obliquity factor (see §4.2), the coupling image intensity, although constant, is not necessarily unity: +00
I^1,9,z) —
2 1
ff
—
J(f,)
nimM age M2"2e
_ P e
dfd8•
For radially symmetric illumination described by i 1(1,g)
if Ginner P C bouter,
100'ute
0
otherwise,
the intensity is
_
bouter Sln eobj
2^ S
— 7C Sin t bob ( outer uter
=
6inner 2 )
J
1 — n i2tnageM2I5 1— p2
PedPe
Ginner Slrieobj
1-6 ener Slri z eobi
nimageM
1 + a dx,
sin t Bobj (óouter — 6 inner)
i — bouter Sin t eobj
x
205
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
206
Simulator Accuracy Tests
where x = 1-
pé and a = 1 / ( n
ge M 2 )
- 1. The indefinite integral is analytic:
1 + a dx = z+ a ln(a/2+x+z), with z = x(a+x). The intensity is
2
I = 2 ai geM sin eobj (bouter -
Z+ a ln(a/2+x+z) ó inner) 2
1-6 ener s in t 9 obj 1-6o„ters1n2Oobj
Table D. 1 tabulates the intensities of various imaging systems. Table D.1: Coupling image intensities of a 100% transmitting mask imaged by various systems.
nimage
1
NAóinner
NAóouter
sin9 o bi 0.0
sinO o bj 0.0
1
1
1
0.2 0.4 0.6 0.8 0.4 0.6 0.8 0.6 0.8 0.8
1 1 1 1 1 1 1 1 1 1
1.0095713537 1.0409095183 1.1046046472 1.2363625785 1.0513555731 1.1164838089 1.2514819934 1.1555607504 1.3015135985 1.4057656329
1.0097995793 1.0418649964 1.1069527619 1.2412943339 1.0525534688 1.1190969098 1.2567273175 1.1590229744 1.3077707797 1.4140192121
0.0 0.2 0.4 0.6 0.8 0.4 0.6 0.8
1 0.9870433935 0.9415567288 0.8274118142
1 1.0087783694 1.0375828099 1.0963999155 1.2190366429 1.0471842901 1.1073526087 1.2330538611
1 1.0092923179 1.0397401291 1.1017257759 1.2303000474 1.0498893995 1.1132799582 1.2450338960
0.2
0.4 0.6 1.5
0.0
0.2
M=1
0.9263945072 0.8074578668
-
M
= 0.25
M
= 0.2
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
Images of M = 1 systems
207
D.2 Images of M = 1 systems For aberration-free unity-magnification systems in which the object and imagespace media are the same, the scalar-coupling image is given by a simplified form of Eq. (8.3): —i2 Î(1,.9,z) = I I Cm' W WOmn n, e
TCCm' n'; mx nrf,
n",m" n',m'
where / mr nr m` 0 l --- 1^circ —,-,1 dfdg, Py ( P.„ Icy +_
TCCmn,m"n" =ff f(J)circ
px
(D.1)
and circ(fo go 6) = 1 0
if ^(f —fo)2+(g—go)2 < 6, otherwise
is the circle function. For effective sources of uniform intensity, each transmission cross-coefficient is proportional to the overlapping area of the effective source and the two displaced pupils. The integral of Eq. (D. 1) can be computed analytically [132,133]. With circular sources, for example, we can determine the overlapping area of three circles by geometry or by using Stoke's theorem [ 134] (see Exercise 8.4). In the following subsections we present aerial images of representative objects produced by various optical systems.
D.2.1 Chromium-on-glass mask under on-axis illumination Object and imaging configuration: g if
Óx{z) = tf
tbg
Imaging parameters 500 nm 0.5 J(f,g) circ(0,0,0.5) ko NA
(z—npx I < d/2 n E Z, otherwise. Object parameters px 1.5 d 0.5 tfg 1 tb g 0
The image and select intensity values are shown in Fig. D.1.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
(5.5)
Simulator Accuracy Tests
208
ó I
position (ko /NA)
L (
0
ó
-0.50
0.00
-0.25
0.25
PL
0.50
position (A/NA)
intensity 0.702211 0.684056 0.631798 0.551715 0.453194 0.347301 0.245127 0.156188 0.087137 0.040992 0.017003
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Figure D.1: Coupling intensity distribution of a line -space on a chromium-on-glass mask under partially coherent illumination.
D.2.2 Dipole illumination of attenuated phase-shifting mask Object and imaging configuration:
X0
Object [Eq. (5.5)]
Imaging parameters 500 nm
NA
0.5
J(f,g)
circ(+1/2p ,,0,0.2)
Ax d tfg
-
0.8 0.4 1
tb g -(TC
2 )/( it i 2)
The image and select intensity values are shown in Fig. D.2. position (ao /NA) intensity
I
a) ó
-0.4
-0.2
0.0
0.2
0.4
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0.605236 0.582200 0.516601 0.418425 0.302618 0.186811 0.088635 0.023035 0.000000
position (A/NA)
Figure D.2: Coupling intensity distribution of a line-space on an attenuated phaseshifting mask under dipole illumination.
Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms
209
Images of M = 1 systems
D.2.3 Equal line-space on alternating phase-shifting mask Object and imaging configuration: 1
if
-2np,1 < (p,- 2),
Ô(î) = -1 if (j3 +Z) < 1- 2nß I < (2px
0
(5.22)
otherwise.
Imaging parameters 500 nm Xo NA 0.5
J(f g)
-i),
Object parameters 0.5 A, 0.25 d
(f,8); circ (0, 0, 0.3 )
z
Onm
This is the scenario investigated in Exercise 5.8. The images and select intensity values under the two illumination configurations are shown in Fig. D.3.
r
T
52 5
-0.25
-0.50
0.25
0.00
0.50
position (Xo/NA)
a=0
a = 0.3
0.00
0.810569
0.189713
0.05 0.10 0.15 0.20
0.733167 0.530525 0.280045 0.077403
0.189713 0.189713 0.189713 0.189713
0.25
0.000000
0.189713
position (2./NA)
Figure D.3: Coupling intensity distributions of an equal line-space on an alternating phase-shifting mask under coherent and partially coherent illumination.
D.2.4 Periodic contacts on chromium-on-glass mask Object and imaging configuration: tfg if
1z-mpx ^
