Abstract

The Fresnel diffraction pattern of a grating periodically gives an exact reproduction of the grating. By superposing a second grating on such an image, we form a Moiré pattern that depends upon the degree of parallelism of the incident beam. We present here the possibility of using this technique for studying the focusing errors in a collimating lens or a mirror.

© 1974 Optical Society of America

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References

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  1. H. Talbot, Philos. Mag. 9, 401 (1836).
  2. Rayleigh, Philos. Mag. 11, 196 (1881).
  3. J. M. Burch, Progr. Opt. 2, 75 (1963).
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    [CrossRef]
  5. G. L. Rogers, Brit. J. Appl. Phys. 14, 657 (1963).
    [CrossRef]
  6. S. Yokozeki, T. Suzuki, Appl. Opt. 10, 1575 (1971).
    [CrossRef] [PubMed]
  7. D. Malacara (to be published).
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    [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1971 (1)

1970 (2)

1965 (1)

1964 (2)

1963 (2)

J. M. Burch, Progr. Opt. 2, 75 (1963).

G. L. Rogers, Brit. J. Appl. Phys. 14, 657 (1963).
[CrossRef]

1962 (1)

1957 (1)

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. (London) 70, 486, 497, 505 (1957).
[CrossRef]

1881 (1)

Rayleigh, Philos. Mag. 11, 196 (1881).

1836 (1)

H. Talbot, Philos. Mag. 9, 401 (1836).

Burch, J. M.

J. M. Burch, Progr. Opt. 2, 75 (1963).

Cornejo, A.

Cowley, J. M.

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. (London) 70, 486, 497, 505 (1957).
[CrossRef]

Langebeck, P.

Malacara, D.

Meier, R. W.

Moodie, A. F.

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. (London) 70, 486, 497, 505 (1957).
[CrossRef]

Murty, M. V. R. K.

Rayleigh,

Rayleigh, Philos. Mag. 11, 196 (1881).

Rogers, G. L.

G. L. Rogers, Brit. J. Appl. Phys. 14, 657 (1963).
[CrossRef]

Suzuki, T.

Talbot, H.

H. Talbot, Philos. Mag. 9, 401 (1836).

Vargady, L. O.

Yokozeki, S.

Appl. Opt. (5)

Brit. J. Appl. Phys. (1)

G. L. Rogers, Brit. J. Appl. Phys. 14, 657 (1963).
[CrossRef]

J. Opt. Soc. Am. (2)

Philos. Mag. (2)

H. Talbot, Philos. Mag. 9, 401 (1836).

Rayleigh, Philos. Mag. 11, 196 (1881).

Proc. Phys. Soc. (London) (1)

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. (London) 70, 486, 497, 505 (1957).
[CrossRef]

Progr. Opt. (1)

J. M. Burch, Progr. Opt. 2, 75 (1963).

Other (1)

D. Malacara (to be published).

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Figures (8)

Fig. 1
Fig. 1

Optical arrangement.

Fig. 2
Fig. 2

The grating acting as a shearing interferometer.

Fig. 3
Fig. 3

Geometrical interpretation of defocusing.

Fig. 4
Fig. 4

Calculation of the optical path at a distance l from the grating.

Fig. 5
Fig. 5

Experimental assembling.

Fig. 6
Fig. 6

moiré fringes observed for a linear grating with a period d = 0.25 mm, a wavelength λ = 0.6328 μ, and a collimator objective of 360-mm focal length. For this grating DR = 396 mm: (a) defocus term (Δf)/f is 0.5%; (b) 1.5%; (c) the period of the grating is d = 0.125 mm, DR = 1584 mm, and the defocus term (Δf)/f = 0.5%.

Fig. 7
Fig. 7

moiré fringes observed with a circular grating: (a) the defocus term (Δf)/f = 1%; (b) the defocus term (Δf)/f = 0, but the mirror is slightly tilted. Note: In order to get a higher contract, the pictures were taken from the side. Thus they appear elliptical.

Fig. 8
Fig. 8

Collimation testing using Moiré technique and projection by autocollimation.

Equations (23)

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p = d 1 d 2 / ( d 2 - d 1 ) .
W 0 ( y ) = y 2 / ( 2 R ) + a y ,
W 1 ( y ) = [ ( y - s ) 2 / ( 2 R ) ] + b y ,
S = l λ / d .
OPD ( y ) = W 0 ( y ) - W 1 ( y ) = ( S / R ) y + ( a - b ) y .
( a - b ) y ( λ / d ) y .
OPD ( y ) = ( λ / d ) [ ( l / R ) + 1 ] y .
OPD ( y = d ) = ( λ / d ) [ ( l / R ) + 1 ] d = λ ,
( 1 / d ) - ( 1 / d ) = 1 / ( R d ) .
( Δ d ) / d ( Δ d ) / d = l / R .
R = f 2 / ( Δ f )
l = 2 m ( d 2 / λ ) ,
( Δ d ) / d = 2 m · [ d 2 / ( λ f ) ] · [ ( Δ f ) / f ] .
( d - d ) / d = ( Δ d ) / d l / R
R = f 2 / ( Δ f ) ,
( Δ d ) / d ( l / f ) · ( Δ f ) / f ,
OPD ( l ) = R + l - ( R 2 + l 2 + 2 R l cos θ ) 1 / 2 = ( R + l ) { 1 - [ 1 - 2 R l ( 1 - cos θ ) ( R + l ) 2 ] 1 / 2 } ,
OPD ( l ) [ λ 2 / ( 2 d 2 ) ] · [ ( R l ) / ( R + l ) ] .
OPD ( l = D R * ) = λ 2 / ( 2 d 2 ) · [ ( R · D R * ) / ( R + D R * ) ] = λ .
( R D R * ) / ( R + D R * ) = 2 ( d 2 / λ ) = D R ,
( D R * - D R ) / D R * = ( Δ D R ) / D R * = D R / R .
p = d · d / ( Δ d ) = ( 1 / ( 2 m ) · ( λ f ) / d · [ f / ( Δ f ) ] .
S = ( λ l ) / d = 2 m d .

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