Abstract

The fringe characteristics in four-beam cross-grating interferometers, illuminated by a source at finite distance, are analyzed. From the measurements of the fringe visibilities and the fringe shifts on discrete planes, the spatial irradiance distribution of the illuminating source which is separable in two orthogonal directions can be synthesized. When the line structures of the two cross gratings are not parallel, the fringe pattern has lower contrast which suggests that four line gratings should be used to replace the second cross grating in order to have high contrast fringes. The possibility of forming cross-gratinglike patterns using either a combination of one cross grating and two line gratings or a combination of two line-grating interferometers is discussed. Finally, moire interferometry using a cross-grating interferometer is suggested.

© 1989 Optical Society of America

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References

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  1. Y. S. Cheng, “Fringe Formation with a Cross-Grating Interferometer,” Appl. Opt. 25, 4185 (1986).
    [CrossRef] [PubMed]
  2. Y. S. Cheng, “Analysis of the Interference Pattern in a Cross-Grating Interferometer,” Appl. Opt. 27, 3025 (1988).
    [CrossRef] [PubMed]
  3. E. N. Leith, “Simultaneous Position and Velocity Measurement by Interferometric Imaging,” Opt. Lett. 6, 261 (1981).
    [CrossRef] [PubMed]
  4. E. N. Leith, B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217 (1977).
    [CrossRef]
  5. A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, Englewood Cliffs, NJ, 1970).
  6. M. L. Basehore, D. Post, “Moire Method for In-Plane and Out-of-Plane Displacement Measurements,” Exp. Mech. 21, 321 (1981).
    [CrossRef]
  7. Y. S. Cheng, “Two-Dimensional Grating Interferometric Imaging by Computed Tomography,” Opt. Lett. 12, 230 (1987).
    [CrossRef] [PubMed]

1988

1987

1986

1981

M. L. Basehore, D. Post, “Moire Method for In-Plane and Out-of-Plane Displacement Measurements,” Exp. Mech. 21, 321 (1981).
[CrossRef]

E. N. Leith, “Simultaneous Position and Velocity Measurement by Interferometric Imaging,” Opt. Lett. 6, 261 (1981).
[CrossRef] [PubMed]

1977

E. N. Leith, B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217 (1977).
[CrossRef]

Basehore, M. L.

M. L. Basehore, D. Post, “Moire Method for In-Plane and Out-of-Plane Displacement Measurements,” Exp. Mech. 21, 321 (1981).
[CrossRef]

Chang, B. J.

E. N. Leith, B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217 (1977).
[CrossRef]

Cheng, Y. S.

Durelli, A. J.

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

Leith, E. N.

E. N. Leith, “Simultaneous Position and Velocity Measurement by Interferometric Imaging,” Opt. Lett. 6, 261 (1981).
[CrossRef] [PubMed]

E. N. Leith, B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217 (1977).
[CrossRef]

Parks, V. J.

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

Post, D.

M. L. Basehore, D. Post, “Moire Method for In-Plane and Out-of-Plane Displacement Measurements,” Exp. Mech. 21, 321 (1981).
[CrossRef]

Appl. Opt.

Exp. Mech.

M. L. Basehore, D. Post, “Moire Method for In-Plane and Out-of-Plane Displacement Measurements,” Exp. Mech. 21, 321 (1981).
[CrossRef]

Opt. Commun.

E. N. Leith, B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217 (1977).
[CrossRef]

Opt. Lett.

Other

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

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Figures (10)

Fig. 1
Fig. 1

Cross-grating interferometer illuminated by a point source at (x0,y0).

Fig. 2
Fig. 2

Three coordinate systems.

Fig. 3
Fig. 3

Four line gratings to replace cross grating G2.

Fig. 4
Fig. 4

Another four line gratings to replace cross grating G2.

Fig. 5
Fig. 5

Top view of the interferometer with one cross grating and four line gratings.

Fig. 6
Fig. 6

Side view of the interferometer with one cross grating and four line gratings.

Fig. 7
Fig. 7

Top view of the interferometer which consists of two orthogonal line-grating interferometers.

Fig. 8
Fig. 8

Side view of the interferometer which consists of two orthogonal line-grating interferometers.

Fig. 9
Fig. 9

Top view of the interferometric setup for in-plane displacement measurement.

Fig. 10
Fig. 10

Side view of the interferometric setup for in-plane displacement measurement.

Equations (71)

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g 1 ( x , y ) = m n a m n exp [ i 2 π ( m f 1 x + n f 2 y ) ]
g 2 ( x , y ) = m n b m n exp [ i 2 π ( m f 3 x + n f 4 y ) ] .
u = exp { i π λ d 0 [ ( x - x 0 ) 2 + ( y - y 0 ) 2 ] } ,
I ( x 0 , y 0 , λ ; x , y , z ) = { ½ + ½ cos [ 4 π ( f 3 - f 1 ) ( d x + z x 0 ) / ( d + z ) ] } × { ½ + ½ cos [ 4 π ( f 4 - f 2 ) ( d y + z y 0 ) / ( d + z ) ] } ,
I ( x , y , z ) = s ( x o , y 0 ) { ½ + ½ cos [ 4 π ( f 3 - f 1 ) ( d x + z x 0 ) / ( d + z ) ] } × { ½ + ½ cos [ 4 π ( f 4 - f 2 ) ( d y + z y 0 ) / ( d + z ) ] } d x 0 d y 0 .
s ( x 0 , y 0 ) = [ m = 0 A m cos ( 2 π m x 0 / X + φ m ) n = 0 B n × cos ( 2 π n y 0 / Y + ψ n ) ] rect ( x 0 / X ) rect ( y 0 / Y ) .
α 1 = 4 π ( f 3 - f 1 ) d x / ( d + z ) ,
β 1 = 2 ( f 3 - f 1 ) z / ( d + z ) ,
α 2 = 4 π ( f 4 - f 2 ) d y / ( d + z ) ,
β 2 = 2 ( f 4 - f 2 ) z / ( d + z ) .
I ( x , y , z ) = m = 0 n = 0 A m B n ( X Y / 4 ) { sinc ( m ) cos φ m + ½ sinc ( β 1 X + m ) × cos ( α 1 + φ m ) + ½ sinc ( β 1 X - m ) cos ( α 1 - φ m ) } × { sinc ( n ) cos ψ n + ½ sinc ( β 2 Y + n ) cos ( α 2 + ψ n ) + ½ sinc ( β 2 Y - n ) cos ( α 2 - ψ n ) } .
I ( x , y , 0 ) = A 0 B 0 ( X Y / 4 ) ( 1 + cos [ 4 π ( f 3 - f 1 ) x ] } × { 1 + cos [ 4 π ( f 4 - f 2 ) y ] } .
I ( x , y , - m 1 d / [ 2 ( f 3 - f 1 ) X + m 1 ] } = ( X Y / 4 ) ( A 0 + ½ A m 1 cos { 2 π [ 2 ( f 3 - f 1 ) + m 1 / X ] x + φ m 1 } ) ( B 0 + ½ B n 1 cos { 2 π ( 2 ( f 4 - f 2 ) + n 1 / Y ] y + ψ n 1 } ) .
I { x , y , - m d / [ 2 ( f 3 - f 1 ) X + m ] } = ( X Y / 4 ) ( A 0 + ½ A m cos { 2 π [ 2 ( f 3 - f 1 ) + m / X ] x + φ m } ) ( B 0 + ½ B m cos { 2 π [ 2 ( f 4 - f 2 ) + m / Y ] y + ψ m } ) .
s ^ ( x 0 , y 0 ) = [ m = 0 m c A m cos ( 2 π m x 0 / X + φ m ) n = 0 n c B n cos ( 2 π n y 0 / Y + ψ n ) ] × rect ( x 0 / X ) rect ( y 0 / Y ) .
I ( x 0 , y 0 , λ ; x , y , z ) = cos 2 ( ½ α 1 + π β 1 x 0 ) + cos 2 ( ½ α 2 + π β 2 y 0 ) + 2 cos ( ½ α 1 + π β 1 x 0 ) × cos ( ½ α 2 + π β 2 y 0 ) cos ( π λ γ ) ,
I ( x , y , z ) = s ( x 0 , y 0 ) [ cos 2 ( ½ α 1 + π β 1 x 0 ) + cos 2 ( ½ α 2 + β 2 y 0 ) + 2 cos ( ½ α 1 + π β 1 x 0 ) cos ( ½ α 2 + π β 2 y 0 ) × s ( λ ) cos ( π λ γ ) d λ ] d x 0 d y 0 ,
I ( x , y , z ) = s ( x 0 , y 0 ) { cos 2 ( ½ α 1 + π β 1 x 0 ) + cos 2 ( ½ α 2 + π β 2 y 0 ) + 2 cos ( ½ α 1 + π β 1 x 0 ) cos ( ½ α 2 + π β 2 y 0 ) cos ( π λ ¯ γ ) × sinc ( 1 / 2 Δ λ γ ) } d x 0 d y 0 .
I ( x , y , z ) = s ( x 0 , y 0 ) [ 1 + cos [ ½ ( α 1 + α 2 ) + π β 1 ( x + y 0 ) ] } × [ 1 + cos [ ½ ( α 1 - α 2 ) + π β 1 ( x 0 + y 0 ) ] } d x 0 d y 0 ,
I ( x , y , z ) = [ s ( x 0 , y 0 ) d y 0 ] [ ½ + ½ cos ( α 1 + 2 π β 1 x 0 ) ] } d x 0 + { s ( x 0 , y 0 ) d x 0 } [ ½ + ½ cos ( α 2 + 2 π β 2 y 0 ) ] d y 0 .
I ( x , y , z ) = B 0 Y m = 0 A m ( X / 2 ) [ sinc ( m ) cos φ m + ½ sinc ( β 1 X + m ) cos ( α 1 + φ m ) + ½ sinc ( β 1 X - m ) cos ( α 1 - φ m ) ] + A 0 X n = 0 B n ( Y / 2 ) [ sinc ( n ) × cos ψ n + ½ sinc ( β 2 Y + n ) cos ( α 2 + ψ n ) + ½ sinc ( β 2 Y - n ) cos ( α 2 - ψ n ) ] .
g 1 ( x , y ) = m A m exp ( i 2 π m f 1 x ) · n B n exp ( i 2 π n f 1 y 1 )
g 2 ( x , y ) = m a m exp ( i 2 π m f 2 x ) · n b n exp ( i 2 π n f 2 y 2 ) .
x 1 = x cos σ 1 + y sin σ 1 , y 1 = - x sin σ 1 + y cos σ 1 , }
x 2 = x cos σ 2 + y sin σ 2 , y 2 = - x sin σ 2 + y cos σ 2 . }
I ( θ , φ , λ ; x , y , z = 0 ) = 4 cos 2 δ 1 + 4 cos 2 δ 2 + 8 cos δ 1 cos δ 2 cos δ 3 ,
δ 1 = 2 π ( f 2 - f 1 ) x ,
δ 2 = 2 π [ ( f 2 cos σ 2 - f 1 cos σ 1 ) y - ( f 2 sin σ 2 - f 1 sin σ 1 ) x ] + 2 π f 1 [ d 1 f 2 / ( f 2 - f 1 ) ] [ sin θ cos φ ( sin σ 2 - sin σ 1 ) + sin θ sin φ ( cos σ 1 - cos σ 2 ) ] ,
δ 3 = 2 π λ d 1 [ f 1 2 f 2 / ( f 2 - f 1 ) ] [ 1 - cos ( σ 1 - σ 2 ) ] .
I ( θ , φ , λ ; x , y , 0 ) = 4 { 1 + cos [ 2 π ( f 2 - f 1 ) ( x + y ) ] } × { 1 + cos [ 2 π ( f 2 - f 1 ) ( x - y ) ] } ,
I ( x , y , z = 0 ) = 4 { 1 + cos [ 2 π ( f 2 - f 1 ) ( x + y 1 ) ] } × { 1 + cos [ 2 π ( f 2 - f 1 ) ( x - y 1 ) ] } ,
I ( θ , φ , λ ; x , y , z ) = 4 cos 2 δ 4 + 4 cos 2 δ 5 + 8 cos δ 4 cos δ 5 cos δ 6 ,
δ 4 = 2 π ( f 2 - f 1 ) ( x - z sin θ cos φ ) + ½ ( Φ 1 + Φ 2 ) ,
δ 5 = 2 π ( f 2 - f 1 ) [ y 1 - z sin θ sin ( φ - σ 1 ) ] + ½ ( Φ 3 + Φ 4 ) ,
δ 6 = ½ ( Φ 1 - Φ 2 - Φ 3 + Φ 4 ) .
I ( θ , φ , λ ; x , y , z ) = 4 cos 2 δ 7 + 4 cos 2 δ 8 + 8 cos δ 7 cos δ 8 cos δ 9 ,
δ 7 = 2 π ( f 2 - f 1 ) ( 1 - sin σ 1 ) ( x - z sin θ cos φ ) + 2 π ( f 2 - f 1 ) cos σ 1 ( y - z sin θ sin φ ) + ½ ( Φ 1 + Φ 2 ) ,
δ 8 = 2 π ( f 2 - f 1 ) ( 1 + sin σ 1 ) ( x - z sin θ cos φ ) - 2 π ( f 2 - f 1 ) cos σ 1 ( y - z sin θ sin φ ) + ½ ( Φ 3 + Φ 4 ) ,
δ 9 = 4 π λ d 1 f 1 f 2 sin σ 1 + 4 π λ z ( f 2 - f 1 ) 2 sin σ 1 - ½ ( Φ 1 - Φ 2 - Φ 3 + Φ 4 ) .
I ( x , y , z = 0 ) = 4 cos 2 { 2 π ( f 2 - f 1 ) [ ( 1 - sin σ 1 ) x + ( cos σ 1 ) y ] + ½ ( Φ 1 + Φ 2 ) } + 4 { cos 2 2 π ( f 2 - f 1 ) [ ( 1 + sin σ 1 ) x - ( cos σ 1 ) y ] + ½ ( Φ 3 + Φ 4 ) } .
g 1 ( x , y ) = m n a m n exp [ i 2 π ( m f 1 x + n f 1 y ) ] .
g 2 x ( x , y ) = ½ + ½ cos ( 4 π f 1 x ) ,
g 2 y ( x , y ) = ½ + ½ cos ( 4 π f 1 y ) .
u 1 2 = exp [ i 2 π f 1 ( x - z sin θ cos φ ) ,
u 3 4 = exp { i 2 π f 1 [ y - ( z - 2 2 ) sin θ sin φ ] } ,
I ( θ , φ , λ ; x , y , z ) = 4 ( 1 + cos { 2 π f 1 [ 2 x - z sin θ cos φ - ( z - 2 2 ) sin θ sin φ ] } ) × ( 1 + cos { 2 π f 1 [ 2 y + z sin θ cos φ - ( z - 2 2 ) sin θ sin φ ] } ) ,
I ( x , y , 0 ) = 4 [ 1 + cos ( 2 2 π f 1 x ) ] [ 1 + cos ( 2 2 π f 1 y ) ] .
u 1 2 3 = exp [ i 2 π ( f a x + f b y ) ] · { ( 1 / 4 ) exp ( i 2 π f 1 x ) ( 1 / 4 ) exp ( - i 2 π f 1 x ) ½ ,
u 1 2 = exp i { 2 π ( f a x + f b y ) ± 2 π f 1 x - π λ 1 [ ( f a ± f 1 ) 2 + f b 2 ] } ,
u 3 4 = exp i [ 2 π ( f a x + f b y ) ± 2 π f 1 y - π λ 1 ( f a 2 + f b 2 ) ] .
u 1 2 = exp i [ 2 π ( f 2 - f 1 ) ( x - z λ f a ) - π λ [ 1 f 1 2 + 2 ( f 2 - f 1 ) 2 ] } ,
u 3 4 = exp i [ 2 π ( f 2 - f 1 ) ( y - z λ f b ) ± 2 π λ f b ( 1 f 1 - 2 f 2 ) - π λ 2 f 1 2 ] ,
I ( θ , φ , λ ; x , y , z ) = 2 cos [ 2 π ( f 2 - f 1 ) ( x - z sin θ cos φ ) ] × exp { - i π λ [ 1 f 1 2 + 2 ( f 2 - f 1 ) 2 ] } + 2 cos ( 2 π ( f 2 - f 1 ) { y - [ z - ( 2 f 2 - 1 f 1 ) / × ( f 2 - f 1 ) ] sin θ sin φ } ) exp ( - i π λ 2 f 1 2 ) 2 .
2 = 1 f 1 2 / [ ( 2 f 1 - f 2 ) f 2 ] ( for 1 0 and f 2 2 f 1 ) .
I ( θ , φ , λ ; x , y , z ) = 4 ( 1 + cos { 2 π f 1 [ 2 x - z sin θ cos φ - ( z - 2 2 ) sin θ sin φ ] } ) × ( 1 + cos { 2 π f 1 [ 2 y + z sin θ cos φ + ( z - 2 2 ) sin θ sin φ ) } ) .
I ( θ , φ , λ ; x , y , z ) = 4 ( 1 + cos [ 2 π ( f 2 - f 1 ) { 2 x - z sin θ cos φ - [ z - 1 f 1 / ( 2 f 1 - f 2 ) ] sin θ sin φ } ] ) × ( 1 + cos [ 2 π ( f 2 - f 1 ) { 2 y + z sin θ cos φ - [ z - 1 f 1 / ( 2 f 1 - f 2 ) ] sin θ sin φ } ] ) .
I ( x , y ) = 4 { 1 + cos [ 2 2 π ( f 2 - f 1 ) x ] } { 1 + cos [ 2 2 π ( f 2 - f 1 ) y ] } .
g 1 ( x , y ) = m n a m n exp [ i 2 π f 1 ( m x + n y ) ]
g 2 ( x , y ) = m n b m n exp [ 2 π f 2 ( m x + n y ) ] .
s ( x 0 , y 0 ) = [ A 0 + A 1 cos ( 2 π m 1 x / X ) ] [ A 0 + A 1 cos ( 2 π m 1 y / X ) ] × rect ( x 0 / X ) rect ( y 0 / X )
I ( x , y , z ) = { A 0 + [ A 0 sinc ( β 1 X ) + ½ A 1 sinc ( β 1 X + m 1 ) + ½ A 1 sinc ( β 1 X - m 1 ) ] cos α 1 } × { A 0 + [ A 0 sinc ( β 1 X ) + ½ A 1 sinc ( β 1 X + m 1 ) + ½ A 1 sinc ( β 1 X - m 1 ) ] cos α 2 } ,
α 1 = 4 π ( f 2 - f 1 ) d x / ( d + z ) ,
β 1 = 2 ( f 2 - f 1 ) z / ( d + z ) ,
α 2 = 4 π ( f 2 - f 1 ) d y / ( d + z ) .
I ( x , y , 0 ) = A 0 2 { 1 + cos [ 4 π ( f 2 - f 1 ) x ] } { 1 + cos [ 4 π ( f 2 - f 1 ) y ] } .
I { x , y , ± m 1 d / [ 2 ( f 2 - f 1 ) X m 1 ] } = ( A 0 + ½ A 1 cos [ 2 π [ 2 ( f 2 - f 1 ) m 1 / X ] x } ) · ( A 0 + ½ A 1 cos { 2 π [ 2 ( f 2 - f 1 ) m 1 / X ] y } ) .
g 0 ( x , y ) = { ½ + ½ cos [ 2 π ( f 2 - f 1 + Δ f ) x ] } × { ½ + ½ cos [ 2 π ( f 2 - f 1 + Δ f ) y ] }
I ( x , y ) = { A 0 + ½ A 1 cos [ 2 π ( 2 Δ f + m 1 / X ) x ] } × { A 0 + ½ A 1 cos [ 2 π ( 2 Δ f + m 1 / X ) y ] } .
g 0 ( x , y ) = ( ½ + ½ cos { 2 π f 3 [ x - v x ( x , y ) ] } ) × ( ½ + ½ cos { 2 π f 3 [ y - v y ( x , y ) ] } ) ,
I 1 ( x , y ) = ½ + ½ cos { 4 π ( f 3 - f 2 + f 1 ) [ x - f 3 v x / ( f 3 - f 2 + f 1 ) ] } .
I 2 ( x , y ) = ½ + ½ cos { 4 π ( f 3 - f 2 + f 1 ) × [ y - f 3 v y / ( f 3 - f 2 + f 1 ) ] } .

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