We introduce an approximation to the Hopkins integral for partially coherent optical systems. The average coherence approximation yields results close to the Hopkins integral for a wide range of coherent transfer functions and illumination functions and is far less computationally demanding than the full Hopkins integral

Rosen, Joseph

Salik, Boaz

Yariv, Amnon

English

Caltech Authors - Main

2086 J. Opt. Soc. Am. A/Vol. 13, No. 10 /October 1996 Salik et al.Average coherence approximation for partially
coherent optical systems
Boaz Salik, Joseph Rosen, and Amnon Yariv
California Institute of Technology, M.S. 128-95, Pasadena, California 91125
Received January 2, 1996; accepted April 19, 1996; revised manuscript received May 21, 1996
We introduce an approximation to the Hopkins integral for partially coherent optical systems. The average
coherence approximation yields results close to the Hopkins integral for a wide range of coherent transfer func-
tions and illumination functions and is far less computationally demanding than the full Hopkins integral.
© 1996 Optical Society of America.Monochromatic illumination systems always possess com-
plete spatial coherence.1 Since all physical illumination
sources have a nonzero linewidth, their radiation is more
generally described as partially coherent. Thus partially
coherent imaging is important in a variety of fields, in-
cluding astronomy, photolithography, and medicine.
Given a coherent impulse response K, propagation
through a system (Fig. 1) is described by the Hopkins
equation2
uE0~x8, y8!u2 5 E E E E E~x, y !E* ~ x̃, ỹ !J~x, y; x̃, ỹ !
3 K~x, y; x8, y8!
3 K* ~ x̃, ỹ; x8, y8!dxdydx̃dỹ, (1)
where E is the field, J is the mutual intensity function,
(x, y) are the system’s input coordinates, and (x8, y8) are
the system’s output coordinates. This equation, though
general, is tedious to compute, both analytically and nu-
merically; still, it is used almost universally in analysis
and simulation of partially coherent systems.3–5 There
have been several attempts to simplify this expression
and derive other formulas that are less computationally
demanding,6–9 but they have all addressed special cases
of the impulse response or mutual intensity functions.
Here we introduce an approximation to the Hopkins
equation that is valid for a wide range of coherence func-
tions and impulse responses and considerably reduces the
computation time required for determining the output in-
tensity. This approximation is particularly useful when,
given J and K, the output intensity is desired for varying
input fields, e.g., in iterative routines that optimize the
input field to yield a desired output intensity.3
For simplicity we begin with one transverse dimension,
then generalize to two-dimensional problems. The one-
dimensional Hopkins integral is, analogously to Eq. (1),
uE0~x8!u2 5 E E E~x !E* ~ x̃ !J~x, x̃ !K~x, x8!
3 K* ~ x̃, x8!dxdx̃. (2)0740-3232/96/1002086-05$10.00Our strategy is to decompose the contribution to
E0(x8) by each point E(x) into a coherent component and
an incoherent component, then sum the contributions of
all points in the x plane to the field intensity at x8. The
coherent radiation component interferes coherently with
the coherent radiation from all other points x, whereas
the incoherent component interferes incoherently with all
other radiation. To determine the coherent and incoher-
ent components of radiation from each point, we must
somehow average its coherence J(x, x̂) with all other
points x̂ (Fig. 2). This average should be weighted by the
contribution of the points to the intensity at x8, and this
contribution is given by uK(x, x8)u2. Thus we can define
a function f(x, x8) that gives the fraction of the intensity
at x that interferes coherently with all other coherent
field contributions:
f~x, x8! 5
E uK~x8, x̂ !u2m~x, x̂ !dx̂
E uK~x8, x̃ !u2dx̃ ,
m~x, x̂ ! 5
J~x, x̂ !
@J~x, x !J~ x̂, x̂ !#1/2
, (3)
where x̂ and x̃ are dummy variables at the input plane
and the functions K and J are normalized to ensure that
0 < f < 1 (this condition is imposed because f is by defi-
nition the fraction of incident power that is spatially co-
herent).
We are now able to consider the coherent and incoher-
ent contributions to the field at x8 independently, using
the following relations:
uE0~x8!u2 5 uE~x ! * K~x, x8!u2, E~x ! coherent,
uE0~x8!u2 5 uE~x !u2 * uK~x, x8!u2, E~x ! incoherent,
(4)
where f(x) * g(x, x8) [ * f(x)g(x, x8)dx is a general
superposition integral.© 1996 Optical Society of America
Salik et al. Vol. 13, No. 10 /October 1996 /J. Opt. Soc. Am. A 2087In our case, K(x) is assumed to have two components,
one coherent and one incoherent, i.e.,
uK~x, x8!u2 5 uKc~x, x8!u2 1 uKi~x, x8!u2
5 u@ f~x, x8!#1/2K~x, x8!u2
1 u@1 2 f~x, x8!#1/2K~x, x8!u2, (5)
so the output intensity is given by
uE0~x8!u2 5 uE~x ! * Kc~x, x8!u2
1 uE~x !u2 * uKi~x, x8!u2
5 E~x ! * u@f~x, x8!#1/2K~x, x8!u2
1 uE~x !u2 * u@1 2 f~x, x8!#1/2K~x, x8!u2
(6)
and we have effectively reduced the double integral of the
one-dimensional Hopkins equation to a sum of two single
integrals. When we use the form of f(x) defined in Eq.
(3), Eq. (6) implies that
uE0~x8!u2 5 U E E~x !@ f~x, x8!#1/2K~x, x8!dxU2
1 E uE~x !u2u@1 2 f~x, x8!#uuK~x, x8!u2dx.
(7)
The two-dimensional generalization is straightforward;
Eq. (3) becomes
Fig. 1. Schematic diagram of the general optical system de-
scribed by the Hopkins equation.
Fig. 2. Schematic representation of the system configuration in
the average coherence approximation ( x can be an
n-dimensional vector, in general).f~x, y, x8, y8!
5
E E uK~x8, y8, x̂, ŷ !u2m~x, y, x̂, ŷ !dx̂dŷ
E E uK~x8, y8, x̃, ỹ !u2dx̃dỹ ,
(8)
and Eq. (7) is now
uE0~x8, y8!u2 5 U E E E~x, y !@f~x, y, x8, y8!#1/2
3 K~x, y, x8, y8!dxdyU2
1 E E uE~x, y !u2u@1
2 f~x, y, x8, y8!#uuK~x, y, x8, y8!u2dxdy.
(9)
Note that, although computing f requires an integra-
tion, this need be done only once, given the system pa-
rameters K and J, after which computing the output in-
tensity requires only one integration dimension per
transverse dimension. Furthermore, if our system is
space invariant, i.e., if
K~x, y, x8, y8! 5 K~x 2 x8, y 2 y8!,
J~x, y, x̂, ŷ ! 5 J~x 2 x̂, y 2 ŷ !, (10)
then the integrals of Eqs. (3) and (7)–(9) become convolu-
tions, which are efficiently calculated by use of fast Fou-
rier transforms [because of the space invariance, Eqs. (3)
and (7) become one-dimensional convolutions and Eqs. (8)
and (9) become two-dimensional convolutions]. Further
simplification is possible if we assume specific forms for
K and J. For example, if we assume one-dimensional
imaging by a rectangular aperture of size D, then
K~x, x8! 5 sincS x 2 x8d D 5 dp~x 2 x8! sinFp~x 2 x8!d G ,
(11)
where d 5 lf/D is the system’s resolution radius, so that
uK~x, x8!u2 5 sinc2S x 2 x8d D , (12)
and, if our illumination source is rectangular,8
J~x, x̃ ! 5 sincS x 2 x̃a D 5 ap~x 2 x̃ ! sinFp~x 2 x̃ !a G ,
(13)
where a is the coherence radius. Thus we have
f~x, x8! 5
1
E sinc2S x̃ 2 xd Ddx̃
E sinc2S x̃ 2 x8d D
3 sincS x̃ 2 xa Ddx̃. (14)
Using the substitutions x̂ 5 x̃ 2 x8 and x̄ 5 x 2 x8
simplifies Eq. (14) to
2088 J. Opt. Soc. Am. A/Vol. 13, No. 10 /October 1996 Salik et al.f~x, x8! 5
1
E sinc2S x̂d Ddx̂
E sinc2S x̂d D sincS x̂ 2 x̄a Ddx̂,
(15)
Fig. 3. One-dimensional slit in (a), after imaging through a fi-
nite aperture (b), yields the patterns in (c) when either the Hop-
kins integral (solid curve) or the ACA (dashed curve) is used.
Here J(x, x8) is a sinc function half as wide as the image.which, when we Fourier transform and assume that
d . 2a, becomes
f~x, x8! } sinc2S x 2 x8d D . (16)
Relation (16) can now be used a priori in Eq. (7), elimi-
nating the integration that is due to f(x, x8).
To evaluate the error of the average coherence approxi-
mation we recall that J is unity for coherent imaging sys-
tems and is a delta function for incoherent systems.
Therefore (in one transverse dimension) Eq. (7) is identi-
cal to a Hopkins integral, where J(x, x̂) is replaced by
J8~x, x̃, x8! 5 @ f~x, x8!f~ x̃, x8!#1/2 1 $@1 2 f~x, x8!#
3 @1 2 f~ x̃, x8!#%1/2d~x 2 x̃ !. (17)
Therefore the error in field intensity as given by Eq. (7) is
e~x8! 5 E E~x !E* ~ x̃ !K~x, x !8K* ~ x̃ x8!J~x, x̃ !dxdx̃
2 E E~x !E* ~ x̃ !K~x, x8!K* ~ x̃, x8!
3 J8~x, x̃, x8!dxdx̃
5 E E~x !E* ~ x̃ !K~x, x8!K* ~ x̃, x8!@J~x, x̃ !
2 J8~x, x̃, x8!#dxdx̃. (18)
This expression is easier to compute than a direct sub-
traction of Eq. (7) from Eq. (2), and is used in the ex-
amples below.
The error will in general depend on the spatial coordi-
nates, input field, system kernel K, and coherence func-
tion J; specifically, the error becomes smaller as
uK(x8, x)u becomes sharper [its energy is concentrated
near some (x8, x)]. If our system is an imaging system,
for example, then as its numerical aperture becomes
larger uK(x, x8)u2 → d (x 2 x8) and therefore (assuming
that J is normalized)
Fig. 4. Pattern in (a) after imaging through a finite aperture be-
comes (b) under the average coherence approximation and (c) by
use of the full Hopkins equation. The aperture size is half of the
image bandwidth, and coherence diameter is half of the image
size; average error/pixel, 0.95%.
Salik et al. Vol. 13, No. 10 /October 1996 /J. Opt. Soc. Am. A 2089f~x, x8! 5 E d~ x̂ 2 x8!J~x, x̂ !dx̂ 5 J~x, x8!⇒
e~x8! 5 E E E~x !E* ~ x̃ !K~x, x8!
3 K* ~ x̃, x8!J~x, x̃ !dxdx̃
2 H U E E~x !@J~x, x8!#1/2K~x, x8!dxU2
1 E uE~x !u2u1 2 J~x, x8!uuK~x, x8!u2dxJ
5 uE~x8!u2@1 2 J~x8, x8! 2 1 1 J~x8, x8!#
5 0. (19)
Furthermore, our error goes to zero as we approach full
coherence and full incoherence. In the fully coherent
case, J(x, y, x8, y8) 5 1; thus f(x, y, x8, y8) 5 1. From
Eq. (6) it is clear then that
uE0~x8!u2 5 uE~x8!*K~x8!u2, (20)
which is the exact expression for coherent propagation.
For the incoherent case,
J~x, x8! 5 d~x 2 x8!⇒
f~x, x8! 5
uK~x, x8!u2
J~x, x !E uK~x, x8!u2dx 5 0, (21)
and, again using Eq. (6), we arrive at
uE0~x8!u2 5 uE~x8!u2* uK~x8!u2, (22)
which is the exact expression for incoherent illumination.
A notable case in which the error does not approach zero
occurs when the impulse response K becomes very wide;
then our averaging of the coherence actually loses infor-
mation, and the results in general may vary from the
Hopkins calculation.
Empirically, the average coherence approximation
(ACA) yields excellent agreement with the exact Hopkins
integral for a wide range of input fields and coherence
Fig. 5. ACA error versus normalized coherence diameter for a
one-dimensional finite aperture undergoing Fresnel diffraction.functions. Figure 3 compares the ACA and the Hopkins
integral results for a finite, asymmetric one-dimensional
aperture after imaging by a finite aperture ( J is a sinc
function with a coherence diameter half of the image
size). We can see that there is agreement to within 5% of
maximum intensity throughout the output plane, and the
average error/pixel is 0.15%. Figure 4 compares the out-
put intensities for a two-dimensional input pattern im-
aged through a finite aperture. Finally, Fig. 5 plots the
ACA’s average error/pixel (compared to the Hopkins inte-
gral) over the output plane for various widths of the co-
herence function J, assuming Rayleigh diffraction with
z 5 16 pixels. As expected, the error minima lie at full
coherence and full incoherence (they are nonzero because
of quantization error).
We can estimate the relative computation times of the
Hopkins integral and the average coherence approxima-
tion by using order-of-magnitude considerations. We
assume a grid of npoints (i.e., a j 3 k two-dimensional
grid would have n 5 jk). We can compute the Hopkins
integral, assuming a shift-invariant system, by using a
triple correlation for the inner integral,
* E( x̃)J(x, x̃)K* ( x̃, x8)dx̃, which requires 6n2 log n
1 2n2 1 2n2 log n complex operations, assuming that
fast Fourier transforms (FFT’s) with O(n log n) are used.
The outer integral imposes n iterations of the inner
integral plus two complex multiplications, which implies
a computational order of n(8n2 log n 1 2n2 1 2n2)
5 4n3(2 log n 1 1). The average coherence approxima-
tion requires 2 FFT’s, a complex multiplication, and an
inverse FFT to compute f(x, x8), plus two integrals, each
involving 2 FFT’s, a complex multiplication, and an in-
verse FFT. Thus the total computational order is
2~2n2 log n ! 1 n2 1 2n2 log n 1 2@2~2n2 log n ! 1 n2
1 2n2 log n# 5 3n2~5 log n 1 1 !.
We can see that the ACA scales approximately with
n2 log n, whereas the Hopkins integral scales as
n3 log n, which for a two-dimensional image is a differ-
ence of 2 orders of magnitude. Empirically, the ACA has
yielded results 10–50 times faster than the Hopkins inte-
gral, depending on the size of the input pattern.
In summary, we have introduced an approximation to
the Hopkins integral for propagation of partially coherent
fields, that is computationally simpler, both analytically
and numerically, and yields good agreement with the
Hopkins integral for a wide range of input patterns and
system transfer functions. We have presented several
analytical conditions for the accuracy of this approxima-
tion and numerically analyzed its performance for a range
of system parameters. We have shown that the average
coherence approximation can be computed significantly
faster than the Hopkins integral.
ACKNOWLEDGMENT
This paper is based on research supported in part by an
Intel Foundation graduate research fellowship and by
grants from the Advanced Research Projects Agency and
the U. S. Air Force Office of Scientific Research.
2090 J. Opt. Soc. Am. A/Vol. 13, No. 10 /October 1996 Salik et al.REFERENCES
1. J. W. Goodman, Introduction to Fourier Optics (McGraw-
Hill, San Francisco, Calif., 1968), p. 106.
2. H. H. Hopkins, ‘‘The concept of partial coherence in optics,’’
Proc. R. Soc. London Ser. A 208, 263–277 (1951).
3. Y. Liu, A. K. Pfau, and A. Zakhor, ‘‘Systematic design of
phase-shifting masks with extended depth of focus and/or
shifted focus plane,’’ IEEE Trans. Semicond. Manuf. 6,
1–21 (1993).
4. D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag,
‘‘Derivation and simulation of higher numerical aperture
scalar aerial images,’’ Jpn. J. Appl. Phys. 31, 4110–4119
(1992).5. Y. C. Pati and T. Kailath, ‘‘Phase-shifting masks for mi-
crolithography: automated design and mask require-
ments,’’ J. Opt. Soc. Am. A 11, 2438–2452 (1994).
6. J. Perina, Coherence of Light, 2nd ed. (Reidel, Dordrecht,
The Netherlands, 1985), Chap. 4.
7. L. Mandel, ‘‘Concept of cross-spectral purity in coherence
theory,’’ J. Opt. Soc. Am. 51, 1342–1350 (1961).
8. M. Born and E. Wolf, Principles of Optics (Pergamon, Ox-
ford, 1965), p. 526.
9. W. H. Carter and E. Wolf, ‘‘Coherence and radiometry with
quasihomogeneous planar sources,’’ J. Opt. Soc. Am. 67,
785–796 (1977).


Average coherence approximation for partially coherent optical systems

2086 J. Opt. Soc. Am. A/Vol. 13, No. 10 /October 1996 Salik et al.Average coherence approximation for partially
coherent optical systems
Boaz Salik, Joseph Rosen, and Amnon Yariv
California Institute of Technology, M.S. 128-95, Pasadena, California 91125
Received January 2, 1996; accepted April 19, 1996; revised manuscript received May 21, 1996
We introduce an approximation to the Hopkins integral for partially coherent optical systems. The average
coherence approximation yields results close to the Hopkins integral for a wide range of coherent transfer func-
tions and illumination functions and is far less computationally demanding than the full Hopkins integral.
© 1996 Optical Society of America.Monochromatic illumination systems always possess com-
plete spatial coherence.1 Since all physical illumination
sources have a nonzero linewidth, their radiation is more
generally described as partially coherent. Thus partially
coherent imaging is important in a variety of fields, in-
cluding astronomy, photolithography, and medicine.
Given a coherent impulse response K, propagation
through a system (Fig. 1) is described by the Hopkins
equation2
uE0~x8, y8!u2 5 E E E E E~x, y !E*~ x˜, y˜ !J~x, y; x˜, y˜ !
3 K~x, y; x8, y8!
3 K*~ x˜, y˜; x8, y8!dxdydx˜dy˜, (1)
where E is the field, J is the mutual intensity function,
(x, y) are the system’s input coordinates, and (x8, y8) are
the system’s output coordinates. This equation, though
general, is tedious to compute, both analytically and nu-
merically; still, it is used almost universally in analysis
and simulation of partially coherent systems.3–5 There
have been several attempts to simplify this expression
and derive other formulas that are less computationally
demanding,6–9 but they have all addressed special cases
of the impulse response or mutual intensity functions.
Here we introduce an approximation to the Hopkins
equation that is valid for a wide range of coherence func-
tions and impulse responses and considerably reduces the
computation time required for determining the output in-
tensity. This approximation is particularly useful when,
given J and K, the output intensity is desired for varying
input fields, e.g., in iterative routines that optimize the
input field to yield a desired output intensity.3
For simplicity we begin with one transverse dimension,
then generalize to two-dimensional problems. The one-
dimensional Hopkins integral is, analogously to Eq. (1),
uE0~x8!u2 5 E E E~x !E*~ x˜ !J~x, x˜ !K~x, x8!
3 K*~ x˜, x8!dxdx˜. (2)0740-3232/96/1002086-05$10.00Our strategy is to decompose the contribution to
E0(x8) by each point E(x) into a coherent component and
an incoherent component, then sum the contributions of
all points in the x plane to the field intensity at x8. The
coherent radiation component interferes coherently with
the coherent radiation from all other points x, whereas
the incoherent component interferes incoherently with all
other radiation. To determine the coherent and incoher-
ent components of radiation from each point, we must
somehow average its coherence J(x, xˆ) with all other
points xˆ (Fig. 2). This average should be weighted by the
contribution of the points to the intensity at x8, and this
contribution is given by uK(x, x8)u2. Thus we can define
a function f(x, x8) that gives the fraction of the intensity
at x that interferes coherently with all other coherent
field contributions:
f~x, x8! 5
E uK~x8, xˆ !u2m~x, xˆ !dxˆ
E uK~x8, x˜ !u2dx˜ ,
m~x, xˆ ! 5
J~x, xˆ !
@J~x, x !J~ xˆ, xˆ !#1/2
, (3)
where xˆ and x˜ are dummy variables at the input plane
and the functions K and J are normalized to ensure that
0 < f < 1 (this condition is imposed because f is by defi-
nition the fraction of incident power that is spatially co-
herent).
We are now able to consider the coherent and incoher-
ent contributions to the field at x8 independently, using
the following relations:
uE0~x8!u2 5 uE~x ! * K~x, x8!u2, E~x ! coherent,
uE0~x8!u2 5 uE~x !u2 * uK~x, x8!u2, E~x ! incoherent,
(4)
where f(x) * g(x, x8) [ * f(x)g(x, x8)dx is a general
superposition integral.© 1996 Optical Society of America
Salik et al. Vol. 13, No. 10 /October 1996 /J. Opt. Soc. Am. A 2087In our case, K(x) is assumed to have two components,
one coherent and one incoherent, i.e.,
uK~x, x8!u2 5 uKc~x, x8!u2 1 uKi~x, x8!u2
5 u@ f~x, x8!#1/2K~x, x8!u2
1 u@1 2 f~x, x8!#1/2K~x, x8!u2, (5)
so the output intensity is given by
uE0~x8!u2 5 uE~x ! * Kc~x, x8!u2
1 uE~x !u2 * uKi~x, x8!u2
5 E~x ! * u@f~x, x8!#1/2K~x, x8!u2
1 uE~x !u2 * u@1 2 f~x, x8!#1/2K~x, x8!u2
(6)
and we have effectively reduced the double integral of the
one-dimensional Hopkins equation to a sum of two single
integrals. When we use the form of f(x) defined in Eq.
(3), Eq. (6) implies that
uE0~x8!u2 5 U E E~x !@ f~x, x8!#1/2K~x, x8!dxU2
1 E uE~x !u2u@1 2 f~x, x8!#uuK~x, x8!u2dx.
(7)
The two-dimensional generalization is straightforward;
Eq. (3) becomes
Fig. 1. Schematic diagram of the general optical system de-
scribed by the Hopkins equation.
Fig. 2. Schematic representation of the system configuration in
the average coherence approximation ( x can be an
n-dimensional vector, in general).f~x, y, x8, y8!
5
E E uK~x8, y8, xˆ, yˆ !u2m~x, y, xˆ, yˆ !dxˆdyˆ
E E uK~x8, y8, x˜, y˜ !u2dx˜dy˜ ,
(8)
and Eq. (7) is now
uE0~x8, y8!u2 5 U E E E~x, y !@f~x, y, x8, y8!#1/2
3 K~x, y, x8, y8!dxdyU2
1 E E uE~x, y !u2u@1
2 f~x, y, x8, y8!#uuK~x, y, x8, y8!u2dxdy.
(9)
Note that, although computing f requires an integra-
tion, this need be done only once, given the system pa-
rameters K and J, after which computing the output in-
tensity requires only one integration dimension per
transverse dimension. Furthermore, if our system is
space invariant, i.e., if
K~x, y, x8, y8! 5 K~x 2 x8, y 2 y8!,
J~x, y, xˆ, yˆ ! 5 J~x 2 xˆ, y 2 yˆ !, (10)
then the integrals of Eqs. (3) and (7)–(9) become convolu-
tions, which are efficiently calculated by use of fast Fou-
rier transforms [because of the space invariance, Eqs. (3)
and (7) become one-dimensional convolutions and Eqs. (8)
and (9) become two-dimensional convolutions]. Further
simplification is possible if we assume specific forms for
K and J. For example, if we assume one-dimensional
imaging by a rectangular aperture of size D, then
K~x, x8! 5 sincS x 2 x8d D 5 dp~x 2 x8! sinFp~x 2 x8!d G ,
(11)
where d 5 lf/D is the system’s resolution radius, so that
uK~x, x8!u2 5 sinc2S x 2 x8d D , (12)
and, if our illumination source is rectangular,8
J~x, x˜ ! 5 sincS x 2 x˜a D 5 ap~x 2 x˜ ! sinFp~x 2 x˜ !a G ,
(13)
where a is the coherence radius. Thus we have
f~x, x8! 5
1
E sinc2S x˜ 2 xd Ddx˜
E sinc2S x˜ 2 x8d D
3 sincS x˜ 2 xa Ddx˜. (14)
Using the substitutions xˆ 5 x˜ 2 x8 and x¯ 5 x 2 x8
simplifies Eq. (14) to
2088 J. Opt. Soc. Am. A/Vol. 13, No. 10 /October 1996 Salik et al.f~x, x8! 5
1
E sinc2S xˆd Ddxˆ
E sinc2S xˆd D sincS xˆ 2 x¯a Ddxˆ,
(15)
Fig. 3. One-dimensional slit in (a), after imaging through a fi-
nite aperture (b), yields the patterns in (c) when either the Hop-
kins integral (solid curve) or the ACA (dashed curve) is used.
Here J(x, x8) is a sinc function half as wide as the image.which, when we Fourier transform and assume that
d . 2a, becomes
f~x, x8! } sinc2S x 2 x8d D . (16)
Relation (16) can now be used a priori in Eq. (7), elimi-
nating the integration that is due to f(x, x8).
To evaluate the error of the average coherence approxi-
mation we recall that J is unity for coherent imaging sys-
tems and is a delta function for incoherent systems.
Therefore (in one transverse dimension) Eq. (7) is identi-
cal to a Hopkins integral, where J(x, xˆ) is replaced by
J8~x, x˜, x8! 5 @ f~x, x8!f~ x˜, x8!#1/2 1 $@1 2 f~x, x8!#
3 @1 2 f~ x˜, x8!#%1/2d~x 2 x˜ !. (17)
Therefore the error in field intensity as given by Eq. (7) is
e~x8! 5 E E~x !E*~ x˜ !K~x, x !8K*~ x˜ x8!J~x, x˜ !dxdx˜
2 E E~x !E*~ x˜ !K~x, x8!K*~ x˜, x8!
3 J8~x, x˜, x8!dxdx˜
5 E E~x !E*~ x˜ !K~x, x8!K*~ x˜, x8!@J~x, x˜ !
2 J8~x, x˜, x8!#dxdx˜. (18)
This expression is easier to compute than a direct sub-
traction of Eq. (7) from Eq. (2), and is used in the ex-
amples below.
The error will in general depend on the spatial coordi-
nates, input field, system kernel K, and coherence func-
tion J; specifically, the error becomes smaller as
uK(x8, x)u becomes sharper [its energy is concentrated
near some (x8, x)]. If our system is an imaging system,
for example, then as its numerical aperture becomes
larger uK(x, x8)u2 ! d (x 2 x8) and therefore (assuming
that J is normalized)
Fig. 4. Pattern in (a) after imaging through a finite aperture be-
comes (b) under the average coherence approximation and (c) by
use of the full Hopkins equation. The aperture size is half of the
image bandwidth, and coherence diameter is half of the image
size; average error/pixel, 0.95%.
Salik et al. Vol. 13, No. 10 /October 1996 /J. Opt. Soc. Am. A 2089f~x, x8! 5 E d~ xˆ 2 x8!J~x, xˆ !dxˆ 5 J~x, x8!⇒
e~x8! 5 E E E~x !E*~ x˜ !K~x, x8!
3 K*~ x˜, x8!J~x, x˜ !dxdx˜
2 H U E E~x !@J~x, x8!#1/2K~x, x8!dxU2
1 E uE~x !u2u1 2 J~x, x8!uuK~x, x8!u2dxJ
5 uE~x8!u2@1 2 J~x8, x8! 2 1 1 J~x8, x8!#
5 0. (19)
Furthermore, our error goes to zero as we approach full
coherence and full incoherence. In the fully coherent
case, J(x, y, x8, y8) 5 1; thus f(x, y, x8, y8) 5 1. From
Eq. (6) it is clear then that
uE0~x8!u2 5 uE~x8!*K~x8!u2, (20)
which is the exact expression for coherent propagation.
For the incoherent case,
J~x, x8! 5 d~x 2 x8!⇒
f~x, x8! 5
uK~x, x8!u2
J~x, x !E uK~x, x8!u2dx 5 0, (21)
and, again using Eq. (6), we arrive at
uE0~x8!u2 5 uE~x8!u2*uK~x8!u2, (22)
which is the exact expression for incoherent illumination.
A notable case in which the error does not approach zero
occurs when the impulse response K becomes very wide;
then our averaging of the coherence actually loses infor-
mation, and the results in general may vary from the
Hopkins calculation.
Empirically, the average coherence approximation
(ACA) yields excellent agreement with the exact Hopkins
integral for a wide range of input fields and coherence
Fig. 5. ACA error versus normalized coherence diameter for a
one-dimensional finite aperture undergoing Fresnel diffraction.functions. Figure 3 compares the ACA and the Hopkins
integral results for a finite, asymmetric one-dimensional
aperture after imaging by a finite aperture ( J is a sinc
function with a coherence diameter half of the image
size). We can see that there is agreement to within 5% of
maximum intensity throughout the output plane, and the
average error/pixel is 0.15%. Figure 4 compares the out-
put intensities for a two-dimensional input pattern im-
aged through a finite aperture. Finally, Fig. 5 plots the
ACA’s average error/pixel (compared to the Hopkins inte-
gral) over the output plane for various widths of the co-
herence function J, assuming Rayleigh diffraction with
z 5 16 pixels. As expected, the error minima lie at full
coherence and full incoherence (they are nonzero because
of quantization error).
We can estimate the relative computation times of the
Hopkins integral and the average coherence approxima-
tion by using order-of-magnitude considerations. We
assume a grid of npoints (i.e., a j 3 k two-dimensional
grid would have n 5 jk). We can compute the Hopkins
integral, assuming a shift-invariant system, by using a
triple correlation for the inner integral,
* E( x˜)J(x, x˜)K* ( x˜, x8)dx˜, which requires 6n2 log n
1 2n2 1 2n2 log n complex operations, assuming that
fast Fourier transforms (FFT’s) with O(n log n) are used.
The outer integral imposes n iterations of the inner
integral plus two complex multiplications, which implies
a computational order of n(8n2 log n 1 2n2 1 2n2)
5 4n3(2 log n 1 1). The average coherence approxima-
tion requires 2 FFT’s, a complex multiplication, and an
inverse FFT to compute f(x, x8), plus two integrals, each
involving 2 FFT’s, a complex multiplication, and an in-
verse FFT. Thus the total computational order is
2~2n2 log n ! 1 n2 1 2n2 log n 1 2@2~2n2 log n ! 1 n2
1 2n2 log n# 5 3n2~5 log n 1 1 !.
We can see that the ACA scales approximately with
n2 log n, whereas the Hopkins integral scales as
n3 log n, which for a two-dimensional image is a differ-
ence of 2 orders of magnitude. Empirically, the ACA has
yielded results 10–50 times faster than the Hopkins inte-
gral, depending on the size of the input pattern.
In summary, we have introduced an approximation to
the Hopkins integral for propagation of partially coherent
fields, that is computationally simpler, both analytically
and numerically, and yields good agreement with the
Hopkins integral for a wide range of input patterns and
system transfer functions. We have presented several
analytical conditions for the accuracy of this approxima-
tion and numerically analyzed its performance for a range
of system parameters. We have shown that the average
coherence approximation can be computed significantly
faster than the Hopkins integral.
ACKNOWLEDGMENT
This paper is based on research supported in part by an
Intel Foundation graduate research fellowship and by
grants from the Advanced Research Projects Agency and
the U. S. Air Force Office of Scientific Research.
2090 J. Opt. Soc. Am. A/Vol. 13, No. 10 /October 1996 Salik et al.REFERENCES
1. J. W. Goodman, Introduction to Fourier Optics (McGraw-
Hill, San Francisco, Calif., 1968), p. 106.
2. H. H. Hopkins, ‘‘The concept of partial coherence in optics,’’
Proc. R. Soc. London Ser. A 208, 263–277 (1951).
3. Y. Liu, A. K. Pfau, and A. Zakhor, ‘‘Systematic design of
phase-shifting masks with extended depth of focus and/or
shifted focus plane,’’ IEEE Trans. Semicond. Manuf. 6,
1–21 (1993).
4. D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag,
‘‘Derivation and simulation of higher numerical aperture
scalar aerial images,’’ Jpn. J. Appl. Phys. 31, 4110–4119
(1992).5. Y. C. Pati and T. Kailath, ‘‘Phase-shifting masks for mi-
crolithography: automated design and mask require-
ments,’’ J. Opt. Soc. Am. A 11, 2438–2452 (1994).
6. J. Perina, Coherence of Light, 2nd ed. (Reidel, Dordrecht,
The Netherlands, 1985), Chap. 4.
7. L. Mandel, ‘‘Concept of cross-spectral purity in coherence
theory,’’ J. Opt. Soc. Am. 51, 1342–1350 (1961).
8. M. Born and E. Wolf, Principles of Optics (Pergamon, Ox-
ford, 1965), p. 526.
9. W. H. Carter and E. Wolf, ‘‘Coherence and radiometry with
quasihomogeneous planar sources,’’ J. Opt. Soc. Am. 67,
785–796 (1977).


This paper is based on research supported in part by an Intel Foundation graduate research fellowship and by grants from the Advanced Research Projects Agency and the U. S. Air Force Office of Scientific Researc

Boaz Salik

Joseph Rosen

Amnon Yariv

CiteSeerX

Average Coherence Approximation for Partially Coherent Optical Systems

Coherence and radiometry with quasihomogeneous planar sources,’’

Coherence of Light,

Concept of cross-spectral purity in coherence theory,’’

Derivation and simulation of higher numerical aperture scalar aerial images,’’

Introduction to Fourier Optics (McGrawHill,

Phase-shifting masks for microlithography: automated design and mask requirements,’’

Principles of Optics (Pergamon,

Systematic design of phase-shifting masks with extended depth of focus and/or shifted focus plane,’’

The concept of partial coherence in optics,’’

http://authors.library.caltech.edu/9557/1/SALjosaa96.pdf

Average coherence approximation for partially coherent optical systems

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main

Caltech Authors - Main

CiteSeerX