Abstract

Combined effects of amplitude and phase variations on the irradiance in the image of a periodic complex object, which has amplitude and phase distributions, have been investigated for an optical system with partially coherent object illumination. A general expression to obtain the illuminance in the image and the image contrast of such an object is derived by use of the concept of the effective source specifying the coherence condition. The irradiance in the image for the diffraction-limited aberration-free optical system illuminated with bounded effective sources that have uniform and nonuniform radiance, and also with annular illumination, have been calculated. Numerical calculations have been made to obtain the image, in which harmonics up to the thirteenth order are taken into account. One of the results is that abrupt amplitude and phase variations are extremely influential for the appearance of the image of a periodic, low-contrast, complex object for any mode of illumination. On the other hand, less abrupt changes of the amplitude and phase in a complex object produce less variation of the appearance and contrast of the image for any illumination mode except annular illumination. Images of pure amplitude and phase objects have also been obtained by use of the general treatment, and their characteristics are evaluated. It is suggested that a technique to manipulate the radiance distribution in the effective source is valuable in improving the fidelity of the image of a periodic complex object formed by an optical system under partially coherent illumination.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Kermisch, J. Opt. Soc. Am. 65, 887 (1975).
    [CrossRef]
  2. H. H. Hopkins, Proc. R. Soc. A 217, 408 (1953).
    [CrossRef]
  3. B. J. Thompson, “Image Formation with Partially Coherent Light,” Progress in Optics, edited by E. Wolf, Vol. VII (North-Holland, Amsterdam, 1969).
    [CrossRef]
  4. R. J. Becherer and G. B. Parrent, J. Opt. Soc. Am. 57, 1479 (1967).
    [CrossRef]
  5. R. E. Swing and J. R. Clay, J. Opt. Soc. Am. 57, 1180 (1967).
    [CrossRef]
  6. M. De and S. C. Som, J. Opt. Soc. Am. 53, 779 (1963).
    [CrossRef]
  7. R. Barakat, Opt. Acta 17, 337 (1970).
    [CrossRef]
  8. M. De and P. K. Mondal, Opt. Acta 17, 397 (1970).
    [CrossRef]
  9. R. E. Kinzly, J. Opt. Soc. Am. 55, 1002 (1965).
  10. Y. Ichioka, K. Yamamoto, and T. Suzuki, J. Opt. Soc. Am. 65, 892 (1975).
    [CrossRef]
  11. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 530.
  12. K. G. Birch, Opt. Acta 17, 43 (1970).
    [CrossRef]
  13. P. Jacquinot and B. Roizen-Dossier, “Apodisation,” in Ref. 3, Vol. III, p. 31.
  14. K. Yamamoto (private communication).

1975 (2)

1970 (3)

K. G. Birch, Opt. Acta 17, 43 (1970).
[CrossRef]

R. Barakat, Opt. Acta 17, 337 (1970).
[CrossRef]

M. De and P. K. Mondal, Opt. Acta 17, 397 (1970).
[CrossRef]

1967 (2)

1965 (1)

1963 (1)

1953 (1)

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

Barakat, R.

R. Barakat, Opt. Acta 17, 337 (1970).
[CrossRef]

Becherer, R. J.

Birch, K. G.

K. G. Birch, Opt. Acta 17, 43 (1970).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 530.

Clay, J. R.

De, M.

M. De and P. K. Mondal, Opt. Acta 17, 397 (1970).
[CrossRef]

M. De and S. C. Som, J. Opt. Soc. Am. 53, 779 (1963).
[CrossRef]

Hopkins, H. H.

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

Ichioka, Y.

Jacquinot, P.

P. Jacquinot and B. Roizen-Dossier, “Apodisation,” in Ref. 3, Vol. III, p. 31.

Kermisch, D.

Kinzly, R. E.

Mondal, P. K.

M. De and P. K. Mondal, Opt. Acta 17, 397 (1970).
[CrossRef]

Parrent, G. B.

Roizen-Dossier, B.

P. Jacquinot and B. Roizen-Dossier, “Apodisation,” in Ref. 3, Vol. III, p. 31.

Som, S. C.

Suzuki, T.

Swing, R. E.

Thompson, B. J.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 530.

Yamamoto, K.

J. Opt. Soc. Am. (6)

Opt. Acta (3)

R. Barakat, Opt. Acta 17, 337 (1970).
[CrossRef]

M. De and P. K. Mondal, Opt. Acta 17, 397 (1970).
[CrossRef]

K. G. Birch, Opt. Acta 17, 43 (1970).
[CrossRef]

Proc. R. Soc. A (1)

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

Other (4)

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

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 530.

P. Jacquinot and B. Roizen-Dossier, “Apodisation,” in Ref. 3, Vol. III, p. 31.

K. Yamamoto (private communication).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (17)

FIG. 1
FIG. 1

Arrangement of the image forming system.

FIG. 2
FIG. 2

Model of a sinusoidal complex object; (a) amplitude transmittance, C + A cos2πx0u, and (b) phase distribution, exp(iB cos2πjx0u).

FIG. 3
FIG. 3

Model of a trapezoidal gratinglike complex object; (a) amplitude transmittance, and (b) phase distribution. A, B, C, s1, s2, s3, and s4 are parameters describing the object characteristics. D is the period.

FIG. 4
FIG. 4

Six effective sources.

FIG. 5
FIG. 5

Images of square gratinglike complex objects that have different amplitude and phase variations illuminated with coherent source in Fig. 4(a). Object parameter A specifying the amplitude contrast is changed, from the left to the right, as 1.0, 0.7, 0.5, and 0 and the parameter B specifying the phase variation is changed, from the bottom to the top, as 0, 1 2 π, π, and 3 2 π, respectively.

FIG. 6
FIG. 6

Same as Fig. 5 but with the bounded effective source with uniform radiance with R = 0.5 in Fig. 4(b).

FIG. 7
FIG. 7

Same as Fig. 5 but with the bounded effective source with uniform radiance with R = 1.0 in Fig. 4(c).

FIG. 8
FIG. 8

Same as Fig. 5 but with the annular source in Fig. 4(d).

FIG. 9
FIG. 9

Same as Fig. 5 but with the bounded effective source with nonuniform radiance distribution specified by γ(x) = x2 in Fig. 4(e).

FIG. 10
FIG. 10

Same as Fig. 5 but with the bounded effective source with nonuniform radiance distribution specified by γ(x) = −x2 + 1 in Fig. 4(f).

FIG. 11
FIG. 11

Effects of phase variations on the image contrast for the fundamental and second harmonic components in the image of the square gratinglike complex object illuminated by six different effective sources in Fig. 4. Illuminating light sources to obtain contrast curves are; (a) the coherent source in Fig. 4(a); (b) the bounded effective source with uniform radiance with R = 0.5 in Fig. 4(b); (c) the bounded effective source with uniform radiance with R = 1.0 in Fig. 4(c); (d) the annular source in Fig. 4(d); (e) the bounded effective source with nonuniform radiance specified by γ(x) = x2; and (f) the bounded effective source with nonuniform radiance specified by γ(x) = −x2 + 1 in Fig. 4(f).

FIG. 12
FIG. 12

Generation of the notches in the images of the edge objects that have sharp boundaries of amplitude and phase under coherent illumination. The parameter A specifying amplitude contrast in Eq. (17) is 0.87 for every figure, and the parameter B is π, 1 2 π, and 0 for the top, middle, and bottom figures.

FIG. 13
FIG. 13

Variation of the irradiance in the images of the pure phase object [A = 1 and B = π in Eq. (17)] that has sharp boundary of phase as a change in the source size R in the annular source.

FIG. 14
FIG. 14

Variation of the irradiance at the location of sharp boundary (u′ = 0) in the image of the pure phase object used in Fig. 13 as a function of the source size in the annular source.

FIG. 15
FIG. 15

Images of the trapezoidal gratinglike complex object illuminated with the coherent source in Fig. 4(a). Parameters A and B describe the amplitude and phase contrast in the object model in Fig. 3, where C = 1.

FIG. 16
FIG. 16

Same as Fig. 13 but with the bounded effective source with uniform radiance with R = 1 in Fig. 4(c).

FIG. 17
FIG. 17

Images of the sinusoidal complex objects formed by an optical system illuminated with the bounded effective source with uniform radiance with R = 0.5 in Fig. 4(b). The complex transimttance of the object is ( 1 + 1 2 cos 2 π x 0 u ) exp ( i B cos 2 π x 0 u ).

Tables (1)

Tables Icon

TABLE I Object models with useful forms and Fourier coefficients of them. Rn and In mean the real and imaginary parts of Fourier coefficients An.

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

I ( u ) = γ ( x ) | o ( s ) f ( x - s ) e 2 π i s u d s | 2 d x ,
O ( u ) = n A n e 2 π i n x 0 u ,
o ( s ) = n A n δ ( s - n x 0 ) ,
I ( u ) = n m A n A m * T ( n , m ) e 2 π i ( n - m ) x 0 u ,
T ( n , m ) = γ ( x ) f ( x + n x 0 ) f * ( x + m x 0 ) d x T r ( n , m ) + i T i ( n , m ) ,
T r ( n , m ) = T r ( m , n ) ,             T i ( n , m ) = - T i ( m , n ) ,
A n = R n + i I n ,             A m * = R m - i I m ,
I ( u ) = k = 0 ( D k cos 2 π k x 0 u + D k sin 2 π k x 0 u ) = k = 0 E k cos ( 2 π k x 0 u - θ k ) ,
E k = 1 2 ( D k 2 + D k 2 ) ,
θ k = tan - 1 ( D k / D k ) ,
D k = n [ P n k T r ( n , n - k ) + Q n k T i ( n , n - k ) ] ,
D k = n [ Q n k T r ( n , n - k ) - P n k T i ( n , n - k ) ] ,
P n k = k ( R n R n - k + I n I n - k ) ,
Q n k = k ( R n I n - k - I n R n - k ) ,
O ( u ) = ( C + A cos 2 π x 0 u ) exp ( i j = 1 N B j cos 2 π j x 0 u ) ,
e i B j cos 2 π j x 0 u = n i n J n ( B j ) e 2 π i n j x 0 u ,
A n = j 1 j 2 j N i n - j 2 - 2 j 3 - ( N - 1 ) j N J j 2 ( B 2 ) J j 3 ( B 3 ) J j N ( B N ) × { C J n - α ( B 1 ) + 1 2 A i [ J n + 1 - α ( B 1 ) - J n - 1 - α ( B 1 ) ] } , α = 2 j 2 + 3 j 3 + N j N ,
A n = i n C J n ( B 1 ) + 1 2 A i n + 1 [ J n + 1 ( B 1 ) - J n - 1 ( B 1 ) ] ;
A n = m i n - m J m ( B 2 ) { C J n - 2 m ( B 1 ) + 1 2 A i [ J n - 2 m + 1 ( B 1 ) - J n - 2 m - 1 ( B 1 ) ] } .
R n = [ A s 1 sinc ( 2 n s 1 ) ] + ( ( A s 2 - C s 1 ) sinc [ 1 2 ( B - 2 n s 2 + 2 n s 1 ) ] cos [ 1 2 π ( B - 2 n s 2 - 2 n s 1 ) ] + ( C - A ) [ s 2 sin π ( B - 2 n s 2 ) + s 1 sin 2 π n s 1 ] π ( B - 2 n s 2 + 2 n s 1 ) - ( C - A ) ( s 2 - s 1 ) sinc [ 1 2 ( B - 2 n s 2 - 2 n s 1 ) ] sin [ 1 2 π ( B - 2 n s 2 - 2 n s 1 ) ] π ( B - 2 n s 2 + 2 n s 1 ) ) + [ C ( s 3 - s 2 ) sinc n ( s 2 - s 3 ) cos π ( n s 2 + n s 3 - B ) ] + ( ( C s 4 - A s 3 ) sinc [ 1 2 ( B - 2 n s 3 + 2 n s 4 ) ] cos [ 1 2 π ( B - 2 n s 4 - 2 n s 3 ) ] + ( A - C ) [ s 4 sin 2 π n s 4 + s 3 sin π ( B - 2 n s 3 ) ] π ( B - 2 n s 3 + 2 n s 4 ) - ( A - C ) ( s 3 - s 4 ) sinc [ 1 2 ( B - 2 n s 3 + 2 n s 4 ) ] sin [ 1 2 π ( B - 2 n s 4 - 2 n s 3 ) ] π ( B - 2 n s 3 + 2 n s 4 ) ) + [ A ( 1 - s 4 ) sinc n ( 1 - s 4 ) cos π n ( 1 + s 4 ) ] ,
I n = [ - A s 1 sinc ( n s 1 ) sin π n s 1 ] + ( ( A s 2 - C s 1 ) sinc [ 1 2 ( B - 2 n s 2 + 2 n s 1 ) ] sin [ 1 2 π ( B - 2 n s 2 - 2 n s 1 ) ] - ( C - A ) [ s 2 cos π ( B - 2 n s 2 ) - s 1 cos 2 π n s 1 ] π ( B - 2 n s 2 + 2 n s 1 ) + ( C - A ) ( s 2 - s 1 ) sinc [ 1 2 ( B - 2 n s 2 + 2 n s 1 ) ] cos [ 1 2 π ( B - 2 n s 2 - 2 n s 1 ) ] π ( B - 2 n s 2 + 2 n s 1 ) ) + C ( s 2 - s 3 ) sinc n ( s 2 - s 3 ) sin π ( n s 2 + n s 3 - B ) ] + ( ( C s 4 - A s 3 ) sinc [ 1 2 ( B - 2 n s 3 + 2 n s 4 ) ] sin [ 1 2 π ( B - 2 n s 4 - 2 n s 3 ) ] - ( A - C ) [ - s 4 cos 2 π n s 4 + s 3 cos π ( B - 2 n s 3 ) ] π ( B - 2 n s 3 + 2 n s 4 ) + ( A - C ) ( s 3 - s 4 ) sinc [ 1 2 ( B - 2 n s 3 + 2 n s 4 ) ] cos [ 1 2 π ( B - 2 n s 4 - 2 n s 4 ) ] π ( B - 2 n s 3 + 2 n s 4 ) ) + [ - A ( 1 - s 4 ) sinc n ( 1 - s 4 ) sin π n ( 1 + s 4 ) ] ,
C T k = E k / E 0 ,             θ k = tan - 1 ( D k / D k ) .
f ( x ) = 1 for x 1 = 0 otherwise .
( a ) γ ( x ) = δ ( x ) ( coherent source ) ; ( b ) γ ( x ) = 1 for x 0.5 ( R ) ( bounded effective source with uniform radiance ) ; = 0 otherwise , ( c ) γ ( x ) = 1 for x 1.0 ( R ) ( bounded effective source with uniform radiance ) ; = 0 otherwise , ( d ) γ ( x ) = 1 for 0.95 ( R ) x 1.0 ( R ) ( annular source ) ; = 0 otherwise , ( e ) γ ( x ) = x 2 for x 1.0 ( R ) ( bounded effective source with nonuniform radiance ) ; = 0 otherwise , ( f ) γ ( x ) = - x 2 + 1 for x 1.0 ( R ) ( bounded effective source with nonuniform radiance ) , = 0 otherwise
O e ( u ) = { 1 u > 0 , 1 2 ( 1 + A e i B ) u = 0 , A e i B u < 0.
o e ( x ) = 1 2 [ ( 1 + A e i B ) δ ( x ) - ( 1 - A e i B ) i / π x ] ,
γ ( x ) = 1 2 [ δ ( x - R ) + δ ( x + R ) ] .
I e ( u ) = ( 1 / 4 π 2 ) ( π 2 ( 1 + A 2 + 2 A cos B ) - 2 π ( A 2 - 1 ) × [ Si ( X 2 ) - Si ( X 1 ) ] + ( 1 + A 2 - 2 A cos B ) × { [ Si ( X 2 ) - Si ( X 1 ) ] 2 + [ Ci ( X 2 ) - Ci ( X 1 ) ] 2 } ) ,
I e ( u = 0 ) = 1 4 π 2 [ π 2 ( 1 + A 2 + 2 A cos B ) + ( 1 + A 2 - 2 A cos B ) ln 2 ( 1 + R 1 - R ) ] .
I e ( u ) = ( 1 / π 2 ) { [ Si ( X 2 ) - Si ( X 1 ) ] 2 + [ Ci ( X 2 ) - Ci ( X 1 ) ] 2 } ,
I e ( u = 0 ) = ( 1 / π 2 ) ln 2 [ ( 1 + R ) / ( 1 - R ) ] .
I R = 1 ( u = 0 ) = 1 π 2 0 1 ln 2 ( 1 + R 1 - R ) d R .
1 π 2 0 1 γ m ( R ) ln 2 ( 1 + R 1 - R ) d R = 1.