Abstract

The theoretical interpretation of the shearing interferometer based on the moiré method using the fourier image of the grating is described. To obtain a pattern with good contrast, the observing plane must coincide with the normal fourier image plane of the grating or with the reversed fourier image plane. The information obtained by this method is the first partial derivative and under certain conditions the second partial derivative of the distortion from the reference wavefront, which is planar or spherical. Applications to measurement of the phase gradient and lens aberration are shown.

© 1971 Optical Society of America

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References

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  1. R. Kraushaar, J. Opt. Soc. Amer. 40, 480 (1950).
    [CrossRef]
  2. F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36, 227 (1959).
    [CrossRef]
  3. V. Ronchi, Atti Fond. G. Ronchi 13, 368 (1958).
  4. G. Toraldo di Francia, Natl. Bur. Stand. Circ. 526 (U.S. GPO, Washington, D. C., 1954), p. 161.
  5. J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. B70, 486 (1957).
  6. S. Yokozeki, T. Suzuki, Japan J. Appl. Phys. 9, 1011 (1970).
    [CrossRef]
  7. J. P. Duncan, P. G. Sabin, Exp. Mech. 5, 22 (1965).
    [CrossRef]
  8. S. Yokozeki, T. Suzuki, Appl. Opt. 9, 2804 (1970).
    [PubMed]
  9. J. Zimmerman, Appl. Opt. 2, 759 (1963).
    [CrossRef]
  10. L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
    [CrossRef]
  11. M. V. R. K. Murty, Appl. Opt. 3, 531 (1964).
    [CrossRef]

1970

S. Yokozeki, T. Suzuki, Japan J. Appl. Phys. 9, 1011 (1970).
[CrossRef]

S. Yokozeki, T. Suzuki, Appl. Opt. 9, 2804 (1970).
[PubMed]

1965

J. P. Duncan, P. G. Sabin, Exp. Mech. 5, 22 (1965).
[CrossRef]

1964

1963

1960

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

1959

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

1958

V. Ronchi, Atti Fond. G. Ronchi 13, 368 (1958).

1957

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. B70, 486 (1957).

1950

R. Kraushaar, J. Opt. Soc. Amer. 40, 480 (1950).
[CrossRef]

Cowley, J. M.

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. B70, 486 (1957).

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

Duncan, J. P.

J. P. Duncan, P. G. Sabin, Exp. Mech. 5, 22 (1965).
[CrossRef]

Kraushaar, R.

R. Kraushaar, J. Opt. Soc. Amer. 40, 480 (1950).
[CrossRef]

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

Moodie, A. F.

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. B70, 486 (1957).

Murty, M. V. R. K.

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

Ronchi, V.

V. Ronchi, Atti Fond. G. Ronchi 13, 368 (1958).

Sabin, P. G.

J. P. Duncan, P. G. Sabin, Exp. Mech. 5, 22 (1965).
[CrossRef]

Suzuki, T.

S. Yokozeki, T. Suzuki, Appl. Opt. 9, 2804 (1970).
[PubMed]

S. Yokozeki, T. Suzuki, Japan J. Appl. Phys. 9, 1011 (1970).
[CrossRef]

Toraldo di Francia, G.

G. Toraldo di Francia, Natl. Bur. Stand. Circ. 526 (U.S. GPO, Washington, D. C., 1954), p. 161.

Weinberg, F. J.

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Wood, N. B.

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Yokozeki, S.

S. Yokozeki, T. Suzuki, Japan J. Appl. Phys. 9, 1011 (1970).
[CrossRef]

S. Yokozeki, T. Suzuki, Appl. Opt. 9, 2804 (1970).
[PubMed]

Zimmerman, J.

Appl. Opt.

Atti Fond. G. Ronchi

V. Ronchi, Atti Fond. G. Ronchi 13, 368 (1958).

Exp. Mech.

J. P. Duncan, P. G. Sabin, Exp. Mech. 5, 22 (1965).
[CrossRef]

IRE Trans. Information Theory

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

J. Opt. Soc. Amer.

R. Kraushaar, J. Opt. Soc. Amer. 40, 480 (1950).
[CrossRef]

J. Sci. Instrum.

F. J. Weinberg, N. B. Wood, J. Sci. Instrum. 36, 227 (1959).
[CrossRef]

Japan J. Appl. Phys.

S. Yokozeki, T. Suzuki, Japan J. Appl. Phys. 9, 1011 (1970).
[CrossRef]

Proc. Phys. Soc.

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. B70, 486 (1957).

Other

G. Toraldo di Francia, Natl. Bur. Stand. Circ. 526 (U.S. GPO, Washington, D. C., 1954), p. 161.

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Figures (11)

Fig. 1
Fig. 1

Orthogonal coordinate system for the analysis.

Fig. 2
Fig. 2

Moiré pattern for a slide glass by the shearing interferometer using the grating when α = 0.

Fig. 3
Fig. 3

Interferogram for the same slide glass by the Mach-Zehnder interferometer.

Fig. 4
Fig. 4

Moiré pattern obtained by superposition of the two copies of Fig. 3.

Fig. 5
Fig. 5

Moiré patterns for the slide glass by the shearing interferometer when the grating G was shifted every 10 mm along the z axis.

Fig. 6
Fig. 6

Moiré patterns for the slide glass when α ≠ 0. (A) zPz0 = 100 mm, (B) zPz0 = 200 mm, (C) zPz0 = 300 mm.

Fig. 7
Fig. 7

Moiré patterns for the slide glass when zPz0 = constant. (A) α = 0°, (B) α = 1°, (C) α = 2°, (D) α = 3°, (E) α = 4°, (F) α = 5°

Fig. 8
Fig. 8

Orthogonal coordinate system for the analysis.

Fig. 9
Fig. 9

Moiré patterns for the camera lens whose focal length and f/number are 80 mm and 2.8, respectively.

Fig. 10
Fig. 10

Moiré patterns for the camera lens whose focal length and f/number are 50 mm and 1.8, respectively, when the grating G was shifted at intervals of about 10 mm along the z axis.

Fig. 11
Fig. 11

Moiré pattern for the planoconvex lens (f = 148 mm) by the shearing interferometer when α = 0.

Equations (25)

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U ( x , y , z P ) = A 0 exp [ ( 2 π i ) / λ ] [ g ( x , y , z P ) ] + A exp [ ( 2 π i ) / λ ] { ( λ / d ) x + g ( x - a , y , z P ) - [ λ 2 / ( 2 d 2 ) ] z P } + A exp [ ( 2 π i ) / λ ] { - ( λ / d ) x + g ( x + a , y , z P ) - [ λ 2 / ( 2 d 2 ) ] z P } ,
I ( x , y , z P ) = A 0 2 + 2 A 2 + 4 A 0 A cos [ ( 2 π ) / d ] × { x - ( z P - z 0 ) [ g ( x , y , z P ) / ( x ) ] } cos [ ( π λ z P / d 2 ) z P ] + 2 A 2 cos [ 2 π / ( d / 2 ) ] { x - ( z P - z 0 ) [ g ( x , y , z P ) / ( x ) ] } .
z P = ( d 2 / λ ) l ,
I ( x , y , z P ) = A 0 2 + 2 A 2 ± 4 A 0 A cos ( 2 π / d ) { x - ( z P - z 0 ) [ g ( x , y , z P ) / ( x ) ] } + 2 A 2 cos [ 2 π / ( d / 2 ) ] { x - ( z P - z 0 ) [ g ( x , y , z P ) / ( x ) ] } .
I m ( x , y ) = 1 2 [ 1 + cos ( 2 π / d ) ( x - y tan α ) ] ,
F ( x , y ) = I ( x , y , z P ) I m = ( A 0 2 + 2 A 2 ) / 2 ± 2 A 0 A cos ( 2 π / d ) { x - ( z P - z 0 ) [ g ( x , y , z P ) / x ] } + A 2 cos [ 2 π / ( d / 2 ) ] { x - ( z P - z 0 ) [ g ( x , y , z P ) / x ] } + [ ( A 0 2 + 2 A 2 ) / 2 ] cos ( 2 π / d ) ( x - y tan α ) ± A 0 A cos 2 π { ( 1 / d + 1 / d ) x - ( y / d ) tan α - [ ( z P - z 0 ) / d ] [ g ( x , y , z P ) / x ] } ± A 0 A cos 2 π × { ( 1 / d - 1 / d ) x + ( y / d ) tan α - [ ( z P - z 0 ) / d ] × [ g ( x , y , z P ) / x ] } + ( A 2 / 2 ) cos 2 π { ( 2 / d + 1 / d ) x - ( y / d ) tan α - [ 2 ( z P - z 0 ) / d ] [ g ( x , y , z P ) / x ] } + ( A 2 / 2 ) cos 2 π { ( 2 / d - 1 / d ) x + ( y / d ) tan α - [ 2 ( z P - z 0 ) / d ] [ g ( x , y , z P ) / x ] } .
F ( x , y ) = ( A 0 2 + 2 A 2 ) / 2 ± A 0 A cos { ( 2 π / d ) ( z P - z 0 ) × [ g ( x , y , z P ) / x ] } .
( z P - z 0 ) [ g ( x , y , z P ) / x ] = m d ,
F ( x , y ) = ( A 0 2 + 2 A 2 ) / 2 ± A 0 A cos 2 π { [ ( d - d ) / d d ] x + ( y / d ) tan α - [ ( z P - z 0 ) / d ] [ g ( x , y , z P ) / x ] } .
F ( x , y ) = ( A 0 2 + 2 A 2 ) / 2 ± A 0 A cos 2 π { [ ( d - d ) / d d ] x + ( y / d ) tan α - [ ( z P - z 0 ) / d ] [ g ( x , y , z P ) / x ] } .
Δ l x ( x , y , z P ) = ( z P - z 0 ) [ g ( x , y , z P ) / x ] .
Δ l y ( x , y , z P ) = ( z P - z 0 ) [ g ( x , y , z P ) / y ] .
Δ ϕ = ( 2 π / λ ) ( B C ¯ - A D ¯ ) ( π λ / d 2 ) ( f + z g ) .
U ( x , y , z P ) = A 0 exp ( 2 π i / λ ) { ( x 2 + y 2 ) / [ 2 ( z P - f ) ] - w ( x , y , z P ) } + A exp ( 2 π i / λ ) { [ ( x - a ) 2 + y 2 ] / 2 ( z P - f ) - w ( x - a z P / f , y , z P ) + ( λ 2 / 2 d 2 ) ( f + z g ) } + A exp ( 2 π i / λ ) × { [ ( x + a ) 2 + y 2 ] / 2 ( z P - f ) - w ( x + a z P / f , y , z P ) + ( λ 2 / 2 d 2 ) ( f + z g ) } .
I ( x , y , z P ) = A 0 2 + 2 A 2 + 4 A 0 A cos ( 2 π / d ) [ f / z P - f ) ] × { x - [ ( z P - f ) / f ] z P [ w ( x , y , z P ) / x ] } × cos { ( π λ / d 2 ) [ f 2 / ( z P - f ) + f + z g ] } + 2 A 2 cos [ 2 π / ( d / 2 ) ] [ f / ( z P - f ) ] { x - [ ( z P - f ) / f ] × z P [ w ( x , y , z P ) / x ] } .
( π λ / d 2 ) [ f 2 / ( z P - f ) + f + z g ] = π l ,
- z g = f z P / ( z P - f ) - ( d 2 / λ ) l ,
I ( x , y , z P ) = A 0 2 + 2 A 2 ± 4 A 0 A cos ( 2 π / d ) f / ( z P - f ) × { x - [ ( z P - f ) / f ] z P [ w ( x , y , z P ) / x ] } + 2 A 2 cos [ 2 π / ( d / 2 ) ] × [ f / ( z P - f ) ] { x - [ ( z P - f ) / f ] z P [ w ( x , y , z P ) / x ] } .
F ( x , y ) = I ( x , y , z P ) I m .
F ( x , y ) = ( A 0 2 + 2 A 2 ) / 2 ± A 0 A cos 2 π × [ ( f d ( z P - f ) - 1 d ) x + tan α d y - z P d w ( x , y , z P ) x ] .
y = cos α - f / ( z P - f ) sin α x + z P sin α w ( x , y , z P ) x + d sin α m , ( α 0 ) .
F ( x , y ) = ( A 0 2 + 2 A 2 ) / 2 ± A 0 A cos 2 π d [ 2 f - z P z P - f x - z P w ( x , y , z P ) x ] .
Δ l x = z P [ w ( x , y , z P ) / x ] .
I ( x , y , z P ) = A 0 2 + 2 A 2 ± 2 A 0 A cos 2 π d [ f z P - f x - z P w ( x , y , z P ) x + λ 2 d z P 2 × 2 w ( x , y , z P ) x 2 ] ± 2 A 0 A cos 2 π d [ f z P - f x - z P w ( x , y , z P ) x - 2 d z P 2 2 w ( x , y , z P ) x 2 ] + 2 A 2 cos 2 π d [ 2 f z P - f x - 2 z P w ( x , y , z P ) x ] .
F ( x , y ) = ( A 0 2 + 2 A 2 ) / 2 ± A 0 A cos 2 π d [ 2 f - z P z P - f x - z P w ( x , y , z P ) x ] × cos [ π λ d 2 z P 2 2 w ( x , y , z P ) x 2 ] .

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