Abstract

Three methods of digital simulation of partially coherent imagery are presented and compared. The first method is a direct discretization of imaging equations. In the second, the computations are performed in the Fourier domain. The third method is based on a modal expansion of the imaging as an incoherent sum of a number of coherent modes; this allows full utilization of FFT algorithms. It is shown that when the imaging is of narrow point spread function, the modal expansion method is very efficient, especially for relatively high coherence. Examples of 1-D and 2-D images are shown.

© 1982 Optical Society of America

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References

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  1. R. E. Swing, J. Opt. Soc. Am. 62, 199 (1972).
    [CrossRef]
  2. R. E. Kinzly, J. Opt. Soc. Am. 62, 386 (1972).
    [CrossRef]
  3. G. O. Reynolds, A. E. Smith, Appl. Opt. 12, 1259 (1973).
    [CrossRef] [PubMed]
  4. D. Nyyssonen, Appl. Opt. 16, 2223 (1977).
    [CrossRef] [PubMed]
  5. R. E. Burge, J. C. Dainty, Optik 46, 229 (1976).
  6. W. G. Oldham, S. Subramanian, A. R. Neureuther, Solid-State Electron. 24, 975 (1981).
    [CrossRef]
  7. D. Kermisch, J. Opt. Soc. Am. 65, 887 (1975).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 10.
  9. M. J. Beran, G. B. Parrent, The Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).
  10. B. J. Thompson, “Image Formation with Partially Coherent Light,” in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1969), Vol. 7.
    [CrossRef]
  11. G. O. Reynolds, J. B. DeVelis, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 2 (1979).
  12. Y. Ichioka, K. Yamamoto, T. Suzuki, J. Opt. Soc. Am. 65, 8921975).
    [CrossRef]
  13. Y. Ichioka, K. Yamamoto, T. Suzuki, J. Opt. Soc. Am. 66, 932 (1976).
    [CrossRef]
  14. R. E. Kinzly, J. Opt. Soc. Am. 56, 526 (1966).
    [CrossRef]
  15. E. C. Kintner, Appl. Opt. 17, 2747 (1978).
    [CrossRef] [PubMed]
  16. E. H. Hopkins, Proc. R. Soc. London Ser. A 217, 408 (1953).
    [CrossRef]
  17. S. Subramanian, Appl. Opt. 20, 1854 (1981).
    [CrossRef] [PubMed]
  18. See also M. M. O’Toole, “Simulation of Optically Formed Image Profiles in Positive Resist,” Ph.D. Dissertation, U. California, Berkeley (1979).
  19. E. Wolf, Opt. Commun. 38, 3 (1981).
    [CrossRef]
  20. E. Wolf, J. Opt. Soc. Am. 68, 1597 (1978).
    [CrossRef]
  21. B. E. A. Saleh, Opt. Acta 26, 777 (1979).
    [CrossRef]
  22. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  24. H. L. Althaus, R. J. Leake, IEEE Trans. Inf. Theory IT-15, 1973 (1969).
  25. D. E. Knuth, “The Art of Comuter Programming,” in Fundamental Algorithms, Vol. 1 (Addison-Wesley, Reading, Mass., 1969).

1981

W. G. Oldham, S. Subramanian, A. R. Neureuther, Solid-State Electron. 24, 975 (1981).
[CrossRef]

E. Wolf, Opt. Commun. 38, 3 (1981).
[CrossRef]

S. Subramanian, Appl. Opt. 20, 1854 (1981).
[CrossRef] [PubMed]

1979

B. E. A. Saleh, Opt. Acta 26, 777 (1979).
[CrossRef]

G. O. Reynolds, J. B. DeVelis, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 2 (1979).

1978

1977

1976

1975

1973

1972

1969

H. L. Althaus, R. J. Leake, IEEE Trans. Inf. Theory IT-15, 1973 (1969).

1966

1953

E. H. Hopkins, Proc. R. Soc. London Ser. A 217, 408 (1953).
[CrossRef]

Althaus, H. L.

H. L. Althaus, R. J. Leake, IEEE Trans. Inf. Theory IT-15, 1973 (1969).

Beran, M. J.

M. J. Beran, G. B. Parrent, The Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 10.

Burge, R. E.

R. E. Burge, J. C. Dainty, Optik 46, 229 (1976).

Dainty, J. C.

R. E. Burge, J. C. Dainty, Optik 46, 229 (1976).

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 2 (1979).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hopkins, E. H.

E. H. Hopkins, Proc. R. Soc. London Ser. A 217, 408 (1953).
[CrossRef]

Ichioka, Y.

Kermisch, D.

Kintner, E. C.

Kinzly, R. E.

Knuth, D. E.

D. E. Knuth, “The Art of Comuter Programming,” in Fundamental Algorithms, Vol. 1 (Addison-Wesley, Reading, Mass., 1969).

Leake, R. J.

H. L. Althaus, R. J. Leake, IEEE Trans. Inf. Theory IT-15, 1973 (1969).

Neureuther, A. R.

W. G. Oldham, S. Subramanian, A. R. Neureuther, Solid-State Electron. 24, 975 (1981).
[CrossRef]

Nyyssonen, D.

O’Toole, M. M.

See also M. M. O’Toole, “Simulation of Optically Formed Image Profiles in Positive Resist,” Ph.D. Dissertation, U. California, Berkeley (1979).

Oldham, W. G.

W. G. Oldham, S. Subramanian, A. R. Neureuther, Solid-State Electron. 24, 975 (1981).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Parrent, G. B.

M. J. Beran, G. B. Parrent, The Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 2 (1979).

G. O. Reynolds, A. E. Smith, Appl. Opt. 12, 1259 (1973).
[CrossRef] [PubMed]

Saleh, B. E. A.

B. E. A. Saleh, Opt. Acta 26, 777 (1979).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Smith, A. E.

Subramanian, S.

W. G. Oldham, S. Subramanian, A. R. Neureuther, Solid-State Electron. 24, 975 (1981).
[CrossRef]

S. Subramanian, Appl. Opt. 20, 1854 (1981).
[CrossRef] [PubMed]

Suzuki, T.

Swing, R. E.

Thompson, B. J.

B. J. Thompson, “Image Formation with Partially Coherent Light,” in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1969), Vol. 7.
[CrossRef]

Wolf, E.

E. Wolf, Opt. Commun. 38, 3 (1981).
[CrossRef]

E. Wolf, J. Opt. Soc. Am. 68, 1597 (1978).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 10.

Yamamoto, K.

Appl. Opt.

IEEE Trans. Inf. Theory

H. L. Althaus, R. J. Leake, IEEE Trans. Inf. Theory IT-15, 1973 (1969).

J. Opt. Soc. Am.

Opt. Acta

B. E. A. Saleh, Opt. Acta 26, 777 (1979).
[CrossRef]

Opt. Commun.

E. Wolf, Opt. Commun. 38, 3 (1981).
[CrossRef]

Optik

R. E. Burge, J. C. Dainty, Optik 46, 229 (1976).

Proc. R. Soc. London Ser. A

E. H. Hopkins, Proc. R. Soc. London Ser. A 217, 408 (1953).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

G. O. Reynolds, J. B. DeVelis, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 2 (1979).

Solid-State Electron.

W. G. Oldham, S. Subramanian, A. R. Neureuther, Solid-State Electron. 24, 975 (1981).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 10.

M. J. Beran, G. B. Parrent, The Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

B. J. Thompson, “Image Formation with Partially Coherent Light,” in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1969), Vol. 7.
[CrossRef]

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

D. E. Knuth, “The Art of Comuter Programming,” in Fundamental Algorithms, Vol. 1 (Addison-Wesley, Reading, Mass., 1969).

See also M. M. O’Toole, “Simulation of Optically Formed Image Profiles in Positive Resist,” Ph.D. Dissertation, U. California, Berkeley (1979).

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Figures (9)

Fig. 1
Fig. 1

Discrete expansion of a partially coherent system as an incoherent sum of coherent modes.

Fig. 2
Fig. 2

Kernel qlk as described by Eq. (17). Hatched area shows the region for which qlk is nonzero.

Fig. 3
Fig. 3

Transmission cross-coefficient Qlk shown over one period as described by Eq. (28). Hatched area represents the nonzero regions of Qlk.

Fig. 4
Fig. 4

Estimation of the computing time C as a function of coherence: (a) using the direct method; (b) using the Fourier method; (c) using the modal expansion method. Dashed lines represent the general case when no assumption regarding the extent of hn is made. Solid lines: (a), (c) hn is of small extent Nh; (b) Hk is of small bandwidth NH.

Fig. 5
Fig. 5

Partially coherent optical system.

Fig. 6
Fig. 6

Results of example 1: (a) coherent case; (b) partially coherent case with NΓ = 7; (c) partially coherent case with NΓ = 9; (d) incoherent case.

Fig. 7
Fig. 7

Two-dimensional object distribution used in examples 2 and 3.

Fig. 8
Fig. 8

Images of object of Fig. 7 using a focused imaging system: (a) coherent case; (b) partially coherent case with NΓ = 9; (c) incoherent case.

Fig. 9
Fig. 9

Images of object of Fig. 7 using a defocused imaging system: (a) coherent case; (b) partially coherent case with NΓ = 3; (c) partially coherent case with NΓ = 5; (d) incoherent case.

Equations (59)

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Γ ( r 1 , r 2 ) = I 0 ( r 1 ) I 0 ( r 2 ) γ ( r 1 - r 2 ) ,
I i ( r ) = - I 0 ( r 1 ) I 0 ( r 2 ) γ ( r 1 - r 2 ) × h * ( r - r 1 ) h ( r - r 2 ) d r 1 d r 2 .
g ( r ) = - f ( r 1 ) f ( r 2 ) q ( r - r 1 , r - r 2 ) d r 1 d r 2 ,
q ( r 1 , r 2 ) = h * ( r 1 ) h ( r 2 ) γ ( r 1 - r 2 ) .
g ( r ) = f ( r ) h ( r ) 2 ,
g ( r ) = f 2 ( r ) h ( r ) 2 .
G ( ω ) = - F ( ω 1 ) F ( ω - ω 1 ) Q ( ω 1 , ω - ω 1 ) d ω 1 / 2 π ,
Q ( ω 1 , ω 2 ) = - Γ ( ω ) H * ( ω - ω 1 ) H ( ω + ω 2 ) d ω / ( 2 π ) 2 .
γ ( r ) = m Γ m exp ( j ω 0 m · r ) ,             r A γ , Γ m = 1 D 2 A γ γ ( r ) exp ( - j ω 0 m · r ) d r ,
g ( r ) = m Γ m f ( r ) h ( m ) ( r ) 2 ,
h ( m ) ( r ) = h ( r ) exp ( j ω 0 m · r ) .
g ( r ) = - Γ ( ω ) f ( r ) [ h ( r ) exp ( j ω · r ) ] 2 d ω .
γ ( r ) = 1 , Γ m = 1 ,             m = ( 0 , 0 ) = 0 ,             otherwise
γ ( r ) = δ ( r ) ,             Γ m = 1 ,             m .
γ ( r ) = sinc ( x / 2 D C ) sinc ( y / 2 D C ) ,
Γ m = 1 , - D / 2 D C m x , m y D / 2 D C , = 0 , elsewhere .
H m ( ω ) = H ( ω - m ω 0 ) ;
g n = l = 0 N - 1 k = 0 N - 1 f l f k q n - l , n - k ,             n = 0 , , N - 1 ,
q l , k = h l * h k γ l - k ,
C = N 3 .
h n = 0 ,             n > ( N h - 1 ) / 2 ,
γ n = 0 ,             n > ( N γ - 1 ) / 2 ,
g n = l , k = n - ( N h - 1 ) / 2 l - k ( N γ - 1 ) / 2 n + ( N h - 1 ) / 2 f l f k h n - l * h n - k γ l - k , n = 0 , 1 , , N - 1.
N h 2 = 2 N h - 1
C N N h , = N γ = 1 ( incoherent case ) = N ( N h N γ - N γ 2 - 1 4 ) , N γ N h 2 , = N N h 2 , N γ N h 2 .
g n = | l = n - ( N h 2 - 1 ) / 2 n + ( N h 2 - 1 ) f l h n - l | 2 ,
C = N N h
X k = n = 0 N - 1 x n W N k n ,             x n = 1 N k = 0 N - 1 X k W N - k n ,
G k = l = 0 N 2 - 1 F l F k - l Q l , k - l ,
Q l , k = m = 0 N 2 - 1 Γ m H m - l * H m + k ,
C = N 2 2 + 2 N 2 log 2 N 2 .
C 4 N 2 + 4 N log 2 2 N .
H k = 0 ,             k > ( N H - 1 ) / 2.
Γ k = 0 ,             k > ( N Γ - 1 ) / 2 ,
C = N H 2 + ( N Γ - 1 ) ( 2 N H - 1 ) + 2 N 2 log 2 N 2 ,             N Γ < N 2 - N H + 1 , C = N H 2 + ( N 2 - N H ) ( 2 N H - 1 ) + 2 N 2 log 2 N 2 ,             N Γ N 2 - N H + 1.
C = N H 2 + 2 N 2 log 2 N 2 .
g n = 1 N 2 m = 0 N 2 - 1 Γ m g n ( m )             n = 0 , 1 , , N 2 - 1 ,
g n ( m ) = | k = 0 N 2 - 1 f k h n - k ( m ) | 2 ,             h n ( m ) = h n exp ( - j 2 π n m / N 2 ) .
C = N Γ N 2 + ( N Γ + 1 ) N 2 log 2 N 2 ,             N Γ N 2 .
g n = 1 N h 2 m = 0 N h 2 - 1 Γ m g n ( m ) ,
g n ( m ) = | k = n - ( N h - 1 ) / 2 n + ( N h - 1 ) / 2 f k h n - k ( m ) | 2 ,             h n ( m ) = h n exp ( - j 2 π n m / N h 2 ) .
C = N Γ N 2 + ( N Γ + 1 ) N 2 log 2 N 2 ,             N 2 = N + N h ,             N Γ N h 2 ,
C = N Γ N h N , N Γ N h 2 , C = N h 2 N h N , N Γ > N h 2 .
C = N N S ( 1 + 2 log 2 N S ) / ( N S - N h + 1 ) .
C = 2 N ( 1 + 2 log 2 N h 2 ) 4 N log 2 N h .
C = 4 N Γ N log 2 N h 2 ,             N Γ = N h 2 .
γ ( x ) = sinc ( ω Γ x / π ) ,             Γ ( ω ) = rect ( ω / 2 ω Γ ) ,
H ( ω ) = rect ( ω / 2 ω H ) exp [ j π 2 w λ ( ω / ω H ) 2 ]
f ( x , y ) = 1 , 0 x X / 2 , f ( x , y ) = 3 , X / 2 < x X .
f n = 1 0 n 127 , f n = 3 , 127 < n 255.
h n = sinc ( 3 n / 16 ) .
N h = 31.
Γ k = 1 / N Γ ,             k ( N Γ - 1 ) / 2 ,
h n = sinc ( 3 n x / 16 ) sinc ( 3 n y / 16 ) ,             n x , n y < 16 , = 0 ,             otherwise , Γ k = 1 / N Γ 2 ,             k x , k y ( N Γ - 1 ) / 2 ,
H k = exp [ j π ( k x 2 + k y 2 ) / ( N H - 1 ) 2 ] ,             × k x ,             k y ( N H - 1 ) / 2 , = 0 ,             elsewhere .
n = 0 N - 1 x n W N k n = 0 for k = 0 , 1 , , N - 1.
n = 0 N - 1 x n W N k n = α k for k = 0 , 1 , , N - 1 ,
α k = - N N - 1 x n W N k n
[ W N 0 W N 0 W N 0 W N 0 W N W N N - 1 , W N 0 W N N - 1 W N ( N - 1 ) ( N - 1 ) ]

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