Abstract

Both on- and off-axis four-beam interference patterns are analyzed using ray tracing. The cross gratinglike interference pattern is accompanied by an extra term which consists of two orthogonal two-beam interference patterns. When partially coherent light is used, the extra term generally degrades the contrast of the cross gratinglike pattern unless some special kinds of source are utilized. With gratings of high spatial frequencies, the amplitude of the extra term can become large compared with the desired term. Consequently, the localized cross gratinglike pattern is changed to be periodic in different directions.

© 1991 Optical Society of America

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References

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  1. F. J. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 277 (1959).
    [CrossRef]
  2. B. J. Chang, “Grating-Based Interferometers,” Ph.D. Dissertation, U. Michigan (1974); University Microfilm 74-23-170.
  3. B. J. Chang, R. C. Alferness, E. N. Leith, “Space-Invariant Achromatic Grating Interferometers: Theory,” Appl. Opt. 14, 1592–1600 (1975).
    [CrossRef] [PubMed]
  4. G. J. Swanson, E. N. Leith, “Lau Effect and Grating Imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
    [CrossRef]
  5. E. N. Leith, R. Hershey, “Transfer Functions and Spatial Filtering in Grating Interferometers,” Appl. Opt. 24, 237–239 (1985).
    [CrossRef] [PubMed]
  6. G. J. Swanson, E. N. Leith, “Analysis of the Lau Effect and Generalized Grating Imaging,” J. Opt. Soc. Am. A 2, 789–793 (1985).
    [CrossRef]
  7. Y.-S. Cheng, “Fringe Formation in Incoherent Light with a Two-Grating Interferometer,” Appl. Opt. 23, 3057–3059 (1984).
    [CrossRef] [PubMed]
  8. G. J. Swanson, “Broad-Source Fringes in Grating and Conventional Interferometers,” J. Opt. Soc. Am. A 1, 1147–1153 (1984).
    [CrossRef]
  9. Y.-S. Cheng, “Fringe Formation with a Cross-Grating Interferometer,” Appl. Opt. 25, 4185–4191 (1986).
    [CrossRef] [PubMed]
  10. Y.-S. Cheng, “Analysis of the Interference Pattern in a Cross-Grating Interferometer,” Appl. Opt. 27, 3025–3034 (1988).
    [CrossRef] [PubMed]
  11. Y.-S. Cheng, “Interference Patterns in Cross-Grating Interferometers: Further Analysis,” Appl. Opt. 28, 556–564 (1989).
    [CrossRef] [PubMed]
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 128.
  13. H. W. Lippincott, H. Stark, “Optical-Digital Detection of Dents and Scratches on Specular Metal Surfaces,” Appl. Opt. 21, 2875–2881 (1982).
    [CrossRef] [PubMed]

1989

1988

1986

1985

1984

1982

1975

1959

F. J. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 277 (1959).
[CrossRef]

Alferness, R. C.

Chang, B. J.

B. J. Chang, R. C. Alferness, E. N. Leith, “Space-Invariant Achromatic Grating Interferometers: Theory,” Appl. Opt. 14, 1592–1600 (1975).
[CrossRef] [PubMed]

B. J. Chang, “Grating-Based Interferometers,” Ph.D. Dissertation, U. Michigan (1974); University Microfilm 74-23-170.

Cheng, Y.-S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 128.

Hershey, R.

Leith, E. N.

Lippincott, H. W.

Stark, H.

Swanson, G. J.

Weinberg, F. J.

F. J. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 277 (1959).
[CrossRef]

Wood, N. B.

F. J. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 277 (1959).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Sci. Instrum.

F. J. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 277 (1959).
[CrossRef]

Other

B. J. Chang, “Grating-Based Interferometers,” Ph.D. Dissertation, U. Michigan (1974); University Microfilm 74-23-170.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 128.

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Figures (1)

Fig. 1
Fig. 1

Higher-order analysis for the phase delays of four different, paths a 1 b 1 p ¯ , a 2 b 2 p ¯ , a 3 b 3 p ¯, and a 4 b 4 p ¯.

Equations (54)

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u = exp [ i 2 π ( f a x + f b y ) ]
u i = exp i [ 2 π ( f a + α i f 1 ) x + 2 π ( f b + β i f 2 ) y ] ,
θ i = sin 1 [ ( sin θ cos φ + α i λ f 1 ) 2 + ( sin θ sin φ + β i λ f 2 ) 2 ] 1 / 2 ,
φ i = tan 1 [ ( sin θ sin φ + β i λ f 2 ) / ( sin θ cos φ + α i λ f 1 ) ] .
θ j = sin 1 [ ( sin θ cos φ + α i λ f 1 + α j λ f 3 ) 2 + ( sin θ sin φ + β i λ f 2 + β j λ f 4 ) 2 ] 1 / 2 ,
φ j = tan 1 [ ( sin θ sin φ + β i λ f 2 + β j λ f 4 ) / ( sin θ cos φ + α i λ f 1 + α j λ f 3 ) ] ,
b i ( x d 2 tan θ j cos φ j , y d 2 tan θ j sin φ j , d 2 ) ,
a i [ x d 2 tan θ j cos φ j d 1 tan θ i cos φ i , y d 2 tan θ j sin φ j d 1 tan θ i sin φ i , ( d 1 + d 2 ) ] .
Φ i = 2 π f a ( x d 2 tan θ j cos φ j d 1 tan θ i cos φ i ) + 2 π f b ( y d 2 tan θ j sin φ j d 1 tan θ i sin φ i ) + 2 π α i f 1 ( x d 2 tan θ j cos φ j d 1 tan θ i cos φ i ) + 2 π β i f 2 ( y d 2 tan θ j sin φ j d 1 tan θ i sin φ i ) + 2 π d 1 / ( λ cos θ i ) + 2 π α j f 3 ( x d 2 tan θ j cos φ j ) + 2 π β j f 4 ( y d 2 tan θ j sin φ j ) + 2 π d 2 / ( λ cos θ j ) .
Φ i = 2 π ( f a x + f b y ) + 2 π ( α i f 1 + α j f 3 ) x + 2 π ( β i f 2 + β j f 4 ) y + 2 π d 1 cos θ i / λ + 2 π d 2 cos θ j / λ ,
I = 4 ( 1 + cos ψ 1 ) ( 1 + cos ψ 2 ) 4 ( cos ψ 1 + cos ψ 2 ) ( 1 cos ψ 3 ) ,
ψ 1 = 1 2 ( Φ 1 Φ 2 Φ 3 + Φ 4 ) ,
ψ 2 = 1 2 ( Φ 1 + Φ 2 Φ 3 Φ 4 ) ,
ψ 3 = 1 2 ( Φ 1 Φ 2 + Φ 3 Φ 4 ) .
ψ 1 = 4 π ( f 3 f 1 ) x 4 π f 1 d 1 sin θ cos φ { F 1 + 1 2 [ F 1 3 + λ 2 ( f 1 2 cos 2 φ + 3 f 2 2 sin 2 φ ) F 1 5 ] sin 2 θ + Ө ( sin 4 θ ) } + 4 π ( f 3 f 1 ) d 2 sin θ × cos φ ( F 2 + 1 2 { F 2 3 + λ 2 [ ( f 3 f 1 ) 2 cos 2 φ + 3 ( f 4 f 2 ) 2 sin 2 φ ] F 2 5 } sin 2 θ + Ө ( sin 4 θ ) ) ,
ψ 2 = 4 π ( f 4 f 2 ) y 4 π f 2 d 1 sin θ sin φ { F 1 + 1 2 [ F 1 3 + λ 2 ( f 1 2 cos 2 φ + 3 f 2 2 sin 2 φ ) F 1 5 ] sin 2 θ + Ө ( sin 4 θ ) } + 4 π ( f 4 f 2 ) d 2 sin θ × sin φ ( F 2 + 1 2 { F 1 3 + λ 2 [ 3 ( f 3 f 1 ) 2 cos 2 φ + ( f 4 f 2 ) 2 sin 2 φ ] F 2 5 } sin 2 θ + Ө ( sin 4 θ ) ,
ψ 3 = 4 π λ f 1 f 2 d 1 sin 2 θ cos φ sin φ { F 1 3 + [ 3 F 1 5 / 2 + 5 λ 2 ( f 1 2 cos 2 φ + f 2 2 sin 2 φ ) F 1 7 / 2 ] sin 2 θ + Ө ( sin 4 θ ) } 4 π λ ( f 3 f 1 ) ( f 4 f 2 ) d 2 sin 2 θ × cos φ sin φ ( F 2 3 + { 3 F 2 5 / 2 + 5 λ 2 [ ( f 3 f 1 ) 2 cos 2 φ + ( f 4 f 2 ) 2 sin 2 φ ] F 2 7 / 2 } sin 2 φ + Ө ( sin 4 θ ) ) ,
F 1 = [ 1 λ 2 ( f 1 2 + f 2 2 ) ] 1 / 2 , F 2 = { 1 λ 2 [ ( f 3 f 1 ) 2 + ( f 4 f 2 ) 2 ] } 1 / 2 .
ψ 1 = 4 π ( f 3 f 1 ) x 4 π f 1 d 1 sin θ [ F 1 + 1 2 ( F 1 3 + λ 2 f 1 2 F 1 5 ) sin 2 θ + Ө ( sin 4 θ ) ] ± 4 π ( f 3 f 1 ) d 2 sin θ { F 2 + 1 2 [ F 2 3 + λ 2 ( f 3 f 1 ) 2 F 2 5 ] × sin 2 θ + Ө ( sin 4 ) } ,
ψ 2 = 4 π ( f 4 f 2 ) y ,
ψ 3 = 0 .
I = ( 1 + cos ψ 1 ) ( 1 + cos ψ 2 ) .
d 2 = d 1 f 1 f 3 f 1 F 1 F 2 .
ψ 1 = 4 π ( f 3 f 1 ) x ,
ψ 2 = 4 π ( f 4 f 2 ) y 4 π f 2 d 1 sin θ [ F 1 + 1 2 ( F 1 3 + λ 2 f 2 2 F 1 5 ) sin 2 θ + Ө ( sin 4 θ ) ] ± 4 π ( f 4 f 2 ) d 2 sin θ { F 2 + 1 2 [ F 2 3 + λ 2 ( f 4 f 2 ) 2 F 2 5 ] × sin 2 θ + Ө ( sin 4 θ ) } ,
ψ 3 = 0 .
d 2 = d 1 f 2 f 4 f 2 F 1 F 2 .
ψ 1 = 4 π f 1 x + 4 π f 1 z sin θ cos φ { F 1 + 1 2 [ F 1 3 + λ 2 ( f 1 2 cos 2 φ + 3 f 2 2 sin 2 φ ) F 1 5 ] sin 2 θ + Ө ( sin 4 θ ) } ,
ψ 2 = 4 π f 2 y + 4 π f 2 z sin θ sin φ { F 1 + 1 2 [ F 1 3 + λ 2 ( 3 f 1 2 cos 2 φ + f 2 2 sin 2 φ ) F 1 5 ] sin 2 θ + Ө ( sin 4 θ ) } ,
ψ 3 = 4 π λ f 1 f 2 ( 2 d 1 + z ) sin 2 θ cos φ sin φ { F 1 3 + [ 3 F 1 5 / 2 + 5 λ 2 ( f 1 2 cos 2 φ + f 2 2 sin 2 φ ) F 1 7 / 2 ] sin 2 θ + Ө ( sin 4 θ ) } ,
ψ 1 = 4 π f 1 x ± 4 π f 1 z sin θ [ F 1 + 1 2 ( F 1 3 + λ 2 f 1 2 F 1 5 ) sin 2 θ + Ө ( sin 4 θ ) ] ,
ψ 2 = 4 π f 2 y ,
ψ 3 = 0 ,
4 π f 1 z M sin θ { [ 1 ( λ ¯ + 1 2 Δ λ ) 2 ( f 1 2 + f 2 2 ) ] 1 / 2 [ 1 ( λ ¯ 1 2 Δ λ ) 2 ( f 1 2 + f 2 2 ) ] 1 / 2 } = 2 π ,
z M = ( 1 / 2 f 1 sin θ ) { [ 1 ( λ ¯ + 1 2 Δ λ ) 2 ( f 1 2 + f 2 2 ) ] 1 / 2 [ 1 ( λ ¯ 1 2 Δ λ ) 2 ( f 1 2 + f 2 2 ) ] 1 / 2 } 1 .
z M = [ 1 λ ¯ 2 ( f 1 2 + f 2 2 ) ] 3 / 2 2 λ ¯ Δ λ f 1 ( f 1 2 + f 2 2 ) sin θ ,
z M = 1 2 λ ¯ Δ λ f 1 ( f 1 2 + f 2 2 ) sin θ .
Δ z = [ 1 λ 2 ( f 1 2 + f 2 2 ) ] 1 / 2 2 f 1 Δ θ ,
ψ 1 = 4 π f 1 x ,
ψ 2 = 4 π f 2 y ± 4 π f 2 z sin θ [ F 1 + 1 2 ( F 1 3 + λ 2 f 2 2 F 1 5 ) sin 2 θ + Ө ( sin 4 θ ) ] ,
ψ 3 = 0 ,
g 1 ( x , y ) = m m A m n exp ( i 2 π m f 1 x ) exp ( i 2 π n f 1 y ) ,
g 2 ( x , y ) = m n B m n exp ( i 2 π m f 1 x ) exp ( i 2 π n f 1 y ) .
θ i = sin 1 [ ( sin θ cos φ + α i λ f 1 ) 2 + ( sin θ sin φ + β i λ f 1 ) 2 ] 1 / 2 ,
φ i = tan 1 [ ( sin θ sin φ + β i λ f 1 ) / ( sin θ cos φ + α i λ f 1 ) ] ,
θ j = sin 1 { [ sin θ cos φ + ( α i + α j ) λ f 1 ] 2 + [ sin θ sin φ + ( β i + β j ) λ f 1 ] 2 } ,
φ j = tan 1 { [ sin θ sin φ + ( β i + β j ) λ f 1 ] / [ sin θ cos φ + ( α i + α j ) λ f 1 ] } ,
Φ i = 2 π ( f a x + f b y ) + 2 π ( α i + α j ) f 1 x + 2 π ( β i + β j ) y + ( 2 π / λ ) d 1 cos θ i + ( 2 π / λ ) d 2 cos θ j .
ψ 1 = 2 π f 1 x + ( π / λ ) z { ( 1 F 3 1 ) + λ f 1 ( F 3 + F 4 ) sin θ cos φ + λ f 1 ( F 3 F 4 ) sin θ sin φ + 1 2 ( F 3 1 ) sin 2 θ + 1 2 λ 2 f 1 2 [ F 3 3 ( cos φ + sin φ ) 2 + F 4 3 ( cos 2 φ sin 2 φ ) ] sin 2 θ + Ө ( sin 3 θ ) } ,
ψ 2 = 2 π f 1 y + ( π / λ ) z { ( 1 F 3 1 ) + λ f 1 ( F 3 F 4 ) sin θ cos φ + λ f 1 ( F 3 + F 4 ) sin θ sin φ + 1 2 ( F 3 1 ) sin 2 θ + 1 2 λ 2 f 1 2 [ F 3 3 ( cos φ + sin φ ) 2 + F 4 3 ( cos 2 φ sin 2 φ ) ] sin 2 θ + Ө ( sin 3 θ ) } ,
ψ 3 = ( π / λ ) ( 2 d 1 + z ) { ( 1 + F 3 1 2 F 4 1 ) λ f 1 ( F 3 F 4 ) sin θ ( cos φ + sin φ ) 1 2 ( 1 + F 3 2 F 4 ) sin 2 θ 1 2 λ 2 f 1 2 [ F 3 3 ( cos φ + sin φ ) 2 F 4 3 ] sin 2 θ + Ө ( sin 3 θ ) } ,
ψ 1 = 2 π f 1 x + π z [ 1 ( 1 2 λ 2 f 1 2 ) 1 / 2 ] / λ ,
ψ 2 = 2 π f 1 y + π z [ 1 ( 1 2 λ 2 f 1 2 ) ] 1 / 2 / λ ,
ψ 3 = π ( 2 d 1 + z ) [ 1 + ( 1 2 λ 2 f 1 2 ) 1 / 2 2 ( 1 λ 2 f 1 2 ) 1 / 2 ] / λ .

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