Abstract

A structured source is used to illuminate the four-beam cross-grating interferometer to cause the interference fringes to localize at different planes. When a monochromatic spatially incoherent source is used, the amplitude transmittance of the first cross grating can be imaged to different locations in various planes. With white light extended source illumination, a cross-gratinglike pattern can be formed not only near but also far away from the optical axis simply by choosing four beams of equal path length to interfere. A multicross-grating interferometer is also analyzed with special emphasis on the three-cross-grating interferometer.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. J. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 227 (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 (1975).
    [CrossRef] [PubMed]
  4. Y. S. Cheng, “Fringe Formation in Incoherent Light with a Two-grating Interferometer,” Appl. Opt. 23, 3057 (1984).
    [CrossRef] [PubMed]
  5. G. J. Swanson, “Broad-source Fringes in Grating and Conventional Interferometers,” J. Opt. Soc. Am. A 1, 1147 (1984).
    [CrossRef]
  6. G. J. Swanson, E. N. Leith, “Lau Effect and Grating Imaging,” J. Opt. Soc. Am. 72, 552 (1982).
    [CrossRef]
  7. E. N. Leith, R. Hershey, “Transfer Functions and Spatial Filtering in Grating Interferometers,” Appl. Opt. 24, 237 (1985).
    [CrossRef] [PubMed]
  8. G. J. Swanson, E. N. Leith, “Analysis of the Lau Effect and Generalized Grating Imaging,” J. Opt. Soc. Am. A 2, 789 (1985).
    [CrossRef]
  9. Y. S. Cheng, “Fringe Formation with a Cross-Grating Interferometer,” Appl. Opt. 25, 4185 (1986).
    [CrossRef] [PubMed]
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 510.
  11. For example, A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, Englewood Cliffs, NJ, 1970).

1986

1985

1984

1982

1975

1959

F. J. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Alferness, R. C.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 510.

Chang, B. J.

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

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

Cheng, Y. S.

Durelli, A. J.

For example, A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, Englewood Cliffs, NJ, 1970).

Hershey, R.

Leith, E. N.

Parks, V. J.

For example, A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, Englewood Cliffs, NJ, 1970).

Swanson, G. J.

Weinberg, F. J.

F. J. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 510.

Wood, N. B.

F. J. Weinberg, N. B. Wood, “Interferometer Based on Four Diffraction Gratings,” J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

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 (6)

Fig. 1
Fig. 1

Young’s experiment with four pinholes.

Fig. 2
Fig. 2

Two-cross-grating interferometer illuminated with a plane wave.

Fig. 3
Fig. 3

First cross grating with aperture R00.

Fig. 4
Fig. 4

Division of the second cross grating into area elements Qij.

Fig. 5
Fig. 5

Division of the observation plane into area elements Pij.

Fig. 6
Fig. 6

Cascaded cross-grating interferometer.

Tables (11)

Tables Icon

Table I Spatial Frequencies of the Fringe Patterns and the Indices of the Four Interfering Beams at P00

Tables Icon

Table II Spatial Frequencies of the Fringe Patterns and the Indices of the Four Interfering Beams at P01

Tables Icon

Table III Spatial Frequencies of the Fringe Patterns and the Indices of the Four Interfering Beams at P02

Tables Icon

Table IV Spatial Frequencies of the Fringe Patterns and the Indices of the Four Interfering Beams at P03

Tables Icon

Table V Spatial Frequencies of the Fringe Patterns and the Indices of the Four Interfering Beams at P11

Tables Icon

Table VI Spatial Frequencies of the Fringe Patterns and the Indices of the Interfering Beams at P12

Tables Icon

Table VII Spatial Frequencies of the Fringe Patterns and the Indices of the Interfering Beams at P13

Tables Icon

Table VIII Spatial Frequencies of the Fringe Patterns and the Indices of the Interfering Beams at P22

Tables Icon

Table IX Spatial Frequencies of the Fringe Patterns and the Indices of the Interfering Beams at P23

Tables Icon

Table X Spatial Frequencies of the Fringe Patterns and the Indices of the Interfering Beams at P33

Tables Icon

Table XI Indices of the Four Interfering Beams In the Three-Cross-Grating Interferometer

Equations (38)

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

s p i p ¯ = [ L 1 2 + ( X / 2 + γ i ξ ) 2 + ( Y / 2 + ɛ i η ) 2 ] 1 / 2 + [ L 2 2 + ( x + γ i X / 2 ) 2 + ( y + ɛ i Y / 2 ) 2 ] 1 / 2 ,
s p i p ¯ L 1 + [ ( X / 2 + γ i ξ ) 2 + ( Y / 2 + ɛ i η ) 2 ] / 2 L 2 + L 2 + [ ( x + γ i X / 2 ) 2 + ( y + ɛ i Y / 2 ) 2 ] / 2 L 2 .
Φ i 2 π s p i p ¯ / λ Φ 0 + π ( γ i X ξ + ɛ i Y η ) / λ L 1 + π ( γ i X x + ɛ i Y y ) / λ L 2 ,
Φ 0 = 2 π λ [ L 1 + L 2 + ( X 2 + Y 2 ) / 8 L 1 + ( X 2 + Y 2 ) / 8 L 2 + ( ξ 2 + η 2 ) / 2 L 1 + ( x 2 + Y 2 ) / 2 L 2 ] ,
I ( x , y ; ξ , η ) = s ( ξ , η ) { 1 + cos [ 2 π X ( ξ / L 1 + x / L 2 ) / λ ] } × { 1 + cos [ 2 π Y ( η / L 1 + y / L 2 ) / λ ] } ,
I ( x , y ) = s ( ξ , η ) { 1 + cos [ 2 π λ X ( ξ / L 1 + x / L 2 ) ] } × { 1 + cos [ 2 π λ Y ( η / L 1 + y / L 2 ) ] } d ξ d η .
I ( x , y ) = s 1 ( ξ ) { 1 + cos [ 2 π X ( ξ / L 1 + x / L 2 ) / λ ] } d ξ × s 2 ( η ) { 1 + cos [ 2 π Y ( η / L 1 + y / L 2 ) / λ ] } d η .
I ( θ , φ , λ ; x , y , z ) = { ½ + ½ cos [ 2 π f 1 ( x - z sin θ cos φ ) ] } × { ½ + ½ cos [ 2 π f 1 ( y - z sin θ sin φ ) ] } .
I ( θ x , θ y ; x , y , z ) = { ½ + ½ cos [ 2 π f 1 ( x - z θ x ) ] } × { ½ + ½ cos [ 2 π f 1 ( y - z θ y ) ] } ,
I ( x , y , z ) = S ( θ x , θ y ) { ½ + ½ cos [ 2 π f 1 ( x - z θ x ) ] } × { ½ + ½ cos [ 2 π f 1 ( y - z θ y ) ] } d θ x d θ y ,
I ( x , y , z ) = ( Δ θ x Δ θ y / 4 ) [ 1 + sinc ( f 1 z Δ θ x ) cos ( 2 π f 1 x ) ] × [ 1 + sinc ( f 1 z Δ θ y ) cos ( 2 π f 1 y ) ] .
I ( x , y , z ) = ( Δ θ x Δ θ y / 16 ) ( 1 + sinc ( f m Δ θ x ) + { sinc ( f 1 z θ x ) + ½ C 0 sinc [ f 1 ( z - f m / f 1 ) Δ θ x + ½ C 0 sinc [ f 1 ( z + f m / f 1 ) Δ θ x ] } × cos ( 2 π f 1 x ) ) · ( 1 + sinc ( f n Δ θ y ) + { sinc ( f 1 z Δ θ y ) + ½ C 1 sinc [ f 1 ( z - f n / f 1 ) Δ θ y ] + ½ C 1 sinc [ f 1 ( z + f n / f 1 ) Δ θ y ] } × cos ( 2 π f 1 y ) ) .
I ( x , y , 0 ) = ( Δ θ x Δ θ y / 4 ) [ ½ + ½ cos ( 2 π f 1 x ) ] × [ ½ + ½ cos ( 2 π f 1 y ) ] ,
I ( x y , ± f m / f 1 ) = ( Δ θ x Δ θ y / 16 ) [ 1 + ½ C 0 cos ( 2 π f 1 x ) ] × [ 1 + ½ C 1 cos ( 2 π f 1 y ) ] .
g 1 ( x , y ) = m n 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 2 x ) exp ( i 2 π n f 2 y ) .
k = 2 π λ ( x ^ sin θ cos φ + y ^ sin θ sin φ + z ^ cos θ )
u = exp [ i 2 π ( f a x + f b y ) ]
u = m n m n a m n b m n exp [ i 2 π ( m f 1 + m f 2 ) x ] × exp [ i 2 π ( n f 1 + n f 2 ) y ] · exp [ - i π λ d 1 ( m 2 + n 2 ) f 1 2 ] × exp { - i π λ d 2 [ ( m f 1 + m f 2 ) 2 + ( n f 1 + n f 2 ) 2 ] } × exp { - i 2 π λ [ d 1 m f 1 + d 2 ( m f 1 + m f 2 ) ] f a } × exp { - i 2 π λ [ d 1 n f 1 + d 2 ( n f 1 + n f 2 ) ] f b } ,
d 1 m f 1 + d 2 ( m f 1 + m f 2 ) = M d 1 f 1 ,
d 1 n f 1 + d 2 ( n f 1 + n f 2 ) = N d 1 f 1 .
m = [ M - ( 1 + β 1 ) m ] / β 1 β 2 ,
n = [ N - ( 1 + β 1 ) n ] / β 1 β 2 .
u = m n a m n b [ M - ( 1 + β 1 ) m ] / β 1 β 2 , [ N - ( 1 + β 1 ) n ] / β 1 β 2 × exp [ i 2 π ( M - m ) f 1 x / β 1 ] · exp [ i 2 π ( N - n ) f 1 y / β 1 ] × exp { - i π λ d 1 f 1 2 [ m 2 + ( M - m ) 2 / β 1 + n 2 + ( N - n ) 2 / β 1 ] } .
u = m n a m n b [ M - ( 1 + β 1 ) m ] / β 1 β 2 , [ N - ( 1 + β 1 ) n ] / β 1 β 2 × exp ( - i 2 π m f 1 x / β 1 ) · exp ( - i 2 π n f 1 y / β 1 ) .
u = m n a m n b β 6 - β 5 m , β 7 - β 5 n · exp ( i 2 π m f 1 x / β 1 ) × exp ( i 2 π n f 1 y / β 1 ) ,
b β 6 - β 5 m , β 7 - β 5 n .
g i ( x , y ) = m i n i a m i n i ( i ) exp [ i 2 π ( m i f i x + n i f i y ) ] .
u = exp [ i 2 π ( f a x + f b y ) ] m 1 n 1 a m 1 m 1 ( 1 ) exp [ i 2 π ( m 1 f 1 x + n 1 f 1 y ) ] × exp { - i π λ d 1 } [ ( f a + m 1 f 1 ) 2 + ( f b + n 1 f 1 ) 2 ] }
u = m 1 n 1 m 2 n 2 m K N K a m 1 n 1 ( 1 ) a m 2 n 2 ( 2 ) a m K n K ( K ) + exp [ i 2 π ( m 1 f 1 + m 2 f 2 + + m K f K ) x ] × exp [ i 2 π ( n 1 f 1 + n 2 f 2 + + n K f K ) y ] × exp ( - i π λ { d 1 ( m 1 2 + n 1 2 ) f 1 2 + d 2 [ ( m 1 f 1 + m 2 f 2 ) 2 + ( n 1 f 1 + n 2 f 2 ) 2 ] + + d K [ ( m 1 f 1 + m 2 f 2 + + m K f K ) 2 + ( n 1 f 1 + n 2 f 2 + + n K f K ) 2 ] } ) × exp { - i 2 π λ [ d 1 m 1 f 1 + d 2 ( m 1 f 1 + m 2 f 2 ) + + d K ( m 1 f 1 + m 2 f 2 + + m K f K ) ] f a } × exp { - i 2 π λ [ d 1 n 1 f 1 + d 2 ( n 1 f 1 + n 2 f 2 ) + + d K ( n 1 f 1 + n 2 f 2 + + n K f K ) ] f b } .
u = m 1 n 1 m 2 n 2 m K N K a m 1 n 1 ( 1 ) a m 2 n 2 ( 2 ) a m K n K ( K ) × exp [ i 2 π ( m 1 + m 2 + + m K ) f 1 x ] × exp [ i 2 π ( n 1 + n 2 + + n K ) f 1 y ] × exp ( - i π λ d 1 f 1 2 { ( m 1 2 + n 1 2 ) + [ ( m 1 + m 2 ) 2 + ( n 1 + n 2 ) 2 ] + + [ ( m 1 + m 2 + + m K ) 2 + ( n 1 + n 2 + + n K ) 2 ] } ) × exp { - i 2 π λ d 1 f 1 [ m 1 + ( m 1 + m 2 ) + + ( m 1 + m 2 + + m K ) ] f a } × exp { - i 2 π λ d 1 f 1 [ n 1 + ( n 1 + n 2 ) + + ( n 1 + n 2 + + n K ) ] f b } .
m 1 + ( m 1 + m 2 ) + + ( m 1 + m 2 + + m K ) = M ,
n 1 + ( n 1 + n 2 ) + + ( n 1 + n 2 + + n K ) = N ,
m K = M - K m 1 - ( K - 1 ) m 2 - - 2 m K - 1 ,
n K = N - K n 1 - ( K - 1 ) n 2 - - 2 n K - 1 .
u = m 1 n 1 m 2 m 2 m K - 1 n K - 1 a m 1 n 1 ( 1 ) a m 2 n 2 ( 2 ) × a M - K m 1 - ( K - 1 ) m 2 - - 2 m K - 1 , N - K n 1 - ( K - 1 ) n 2 - - 2 n K - 1 × exp { - i 2 π [ ( K - 1 ) m 1 + ( K - 2 ) m 2 + + m K - 1 ] f 1 x } × exp { - i 2 π [ ( K - 1 ) n 1 + ( K - 2 ) n 2 + + n K - 1 ] f 1 y } .
I = | m 1 n 1 m 2 n 2 a m 1 n 1 ( 1 ) a m 2 n 2 ( 2 ) a M - 3 m 1 - 2 m 2 , N - 3 n 1 - 2 n 2 ( 3 ) × exp [ - i 2 π ( 2 m 1 + m 2 ) f 1 x ] · exp [ - i 2 π ( 2 n 1 + n 2 ) f 1 y ] × exp ( - i π λ f 1 2 { d 1 ( m 1 2 + n 1 2 ) + d 2 [ ( m 1 + m 2 ) 2 + ( n 1 + n 2 ) 2 ] + d 3 [ ( M - 2 m 1 - m 2 ) 2 + ( N - 2 n 1 - n 2 ) 2 ] } ) × exp { - i 2 π λ f 1 [ d 1 m 1 + d 2 ( m 1 + m 2 ) + d 3 ( M - 2 m 1 - m 2 ) ] f a } × exp { - i 2 π λ f 1 [ d 1 n 1 + d 2 ( n 1 + n 2 ) + d 3 ( N - 2 n 1 - n 2 ) ] f b } | 2
I = | m 1 n 1 a m 1 n ( 1 ) a - m 1 , - n 1 ( 2 ) a M - m 1 , N - n 1 ( 3 ) × exp [ - i 2 π ( m 1 f 1 x + n 1 f 1 y ) ] × exp ( - i π λ d 1 f 1 2 { ( m 1 2 + n 1 2 ) + [ ( M - m 1 ) 2 + ( N - n 1 ) 2 ] } | 2

Metrics