Abstract

Investigations have been carried out on the combined effects of amplitude and phase variations in a periodic complex object on the image formed in a partially coherent optical system in the presence of defocusing. To examine these effects, images of one-dimensional periodic complex objects and the two-dimensional Siemens Star are simulated with various different complex transmittances. A number of significant phenomena are clarified through observation of the simulated images and evaluation of the image contrast for the fundamental and second harmonic spatial frequency components. One of the results is that object phase variations are extremely important to the image appearance in the defocused system and to the change of contrast reversal frequency with defocusing under incoherent and partially coherent illumination, but not in the coherent case.

© 1976 Optical Society of America

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References

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  1. Y. Ichioka, K. Yamamoto, and T. Suzuki, J. Opt. Soc. Am. 65, 892 (1975).
    [CrossRef]
  2. Y. Ichioka and T. Suzuki, preceeding paper, J. Opt. Soc. Am. 66, 921–932 (1976).
    [CrossRef]
  3. H. H. Hopkins, Proc. R. Soc. A 23, 91 (1955a).
    [CrossRef]
  4. W. H. Steel, Opt. Acta 3, 65 (1956).
    [CrossRef]
  5. W. H. Steel, J. Opt. Soc. Am. 47, 405 (1957).
    [CrossRef]
  6. Bo Moller, Opt. Acta 15, 223 (1968).
    [CrossRef]
  7. R. Barakat, Opt. Acta 16, 205 (1969).
    [CrossRef]
  8. R. Barakat, Opt. Acta 17, 337 (1970).
    [CrossRef]
  9. H. H. Hopkins, Proc. R. Soc. A 217, 408 (1953).
    [CrossRef]
  10. H. Okuyama, Y. Ichioka, and T. Suzuki, Oyoboturi (Jpn.) 42, 1091 (1973).

1976 (1)

1975 (1)

1973 (1)

H. Okuyama, Y. Ichioka, and T. Suzuki, Oyoboturi (Jpn.) 42, 1091 (1973).

1970 (1)

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

1969 (1)

R. Barakat, Opt. Acta 16, 205 (1969).
[CrossRef]

1968 (1)

Bo Moller, Opt. Acta 15, 223 (1968).
[CrossRef]

1957 (1)

1956 (1)

W. H. Steel, Opt. Acta 3, 65 (1956).
[CrossRef]

1955 (1)

H. H. Hopkins, Proc. R. Soc. A 23, 91 (1955a).
[CrossRef]

1953 (1)

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

Barakat, R.

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

R. Barakat, Opt. Acta 16, 205 (1969).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, Proc. R. Soc. A 23, 91 (1955a).
[CrossRef]

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

Ichioka, Y.

Moller, Bo

Bo Moller, Opt. Acta 15, 223 (1968).
[CrossRef]

Okuyama, H.

H. Okuyama, Y. Ichioka, and T. Suzuki, Oyoboturi (Jpn.) 42, 1091 (1973).

Steel, W. H.

Suzuki, T.

Yamamoto, K.

J. Opt. Soc. Am. (3)

Opt. Acta (4)

W. H. Steel, Opt. Acta 3, 65 (1956).
[CrossRef]

Bo Moller, Opt. Acta 15, 223 (1968).
[CrossRef]

R. Barakat, Opt. Acta 16, 205 (1969).
[CrossRef]

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

Oyoboturi (Jpn.) (1)

H. Okuyama, Y. Ichioka, and T. Suzuki, Oyoboturi (Jpn.) 42, 1091 (1973).

Proc. R. Soc. A (2)

H. H. Hopkins, Proc. R. Soc. A 23, 91 (1955a).
[CrossRef]

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

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Figures (10)

FIG. 1
FIG. 1

Models of complex objects. (a) Square gratinglike complex object, and (b) trapezoidal gratinglike one.

FIG. 2
FIG. 2

Effects of phase variations in the square gratinglike complex objects on the image contrast for the fundamental and second harmonic spatial frequency components in defocused optical systems under six different illumination modes. Curves in the left column are obtained when the amount of defocusing is D F = 1 2 λ, and those in the right column when DF = λ. Light sources used to obtain images are (a) coherent source; (b) bounded source with uniform radiance, where source size is R = 0.5; (c) bounded source with uniform radiance, where source size is R = 1.0; (d) annular source [γ(x) = 1.0 for 0.95 ≦ |x| ≦ 1.0, and 0 otherwise]; (e) bounded source with nonuniform radiance [γ(x) = x2 for |x| ≦ 1.0, and 0 otherwise]; (f) bounded source with nonuniform radiance [γ(x) = −x2 + 1 for |x| ≦ 1.0, and 0 otherwise].

FIG. 3
FIG. 3

Variations of appearances in defocused images of square gratinglike complex objects that have various phase variations. Light sources used to form images in (a)–(f) correspond to those in (a)–(f) in Fig. 2, respectively. The amount of defocusing is set to be D F = 1 2 λ. Images in the left column are for pure phase objects [A = C = 1 in Fig. 1(a)], and those in the right column are for semitransparent objects [A = 0.25 and C = 1 in Fig. 1(a)].

FIG. 4
FIG. 4

Same as Fig. 3 but the amount of defocusing is D F = 1 4 λ.

FIG. 5
FIG. 5

Same as Fig. 3 but of trapezoidal gratinglike complex objects, where D F = 1 2 λ ..

FIG. 6
FIG. 6

Images of a pure phase object focused in three different image planes under coherent illumination. (a) Image in the best focal planes, (b) and (c) defocused images in different image planes.

FIG. 7
FIG. 7

Image of the edge object (pure phase object) that has phase step of π. (a) Complex amplitude in the image, and (b) image irradiance.

FIG. 8
FIG. 8

Simulated images of the Siemens stars in the defocused optical system ( D F = 1 2 λ). Illumination conditions are (a) coherent illumination, (b) partially coherent illumination with the bounded source with uniform radiance (R = 0.5), (c) partially coherent illumination with the bounded source with uniform radiance (R = 1.0), (d) annular illumination. Object complex transmittance is 1 2 for dark parts and 1.0 for bright ones (pure amplitude object).

FIG. 9
FIG. 9

Same as Fig. 8 but object complex transmittance is 1 2 exp ( 1 2 i π ) for dark parts and 1.0 for bright ones (complex object).

FIG. 10
FIG. 10

Same as Fig. 8 but object complex transmittance is 1 2 exp ( i π ) for dark parts and 1.0 for bright ones (complex object).

Equations (2)

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f ( x ) = exp [ i ( 2 π / λ ) D F x 2 ] ,
O ( u ) = { 1 , u > 0 , 0 , u = 0 , e i π ( = - 1 ) , u < 0 ,