Abstract

The double grating shearing interferometry method for determination of the degree of light collimation is described. High accuracy is obtained by performing the observation of fringes in the area of the size twice as big as the one usually assumed in shearing interferometry experiments. The conditions under which such a detection mode is feasible are derived. They represent at the same time a very strong argument proving the highly diffractive (wave optics) character of the classical Ronchi test.

© 1976 Optical Society of America

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References

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  1. P. Langenbeck, Appl. Opt. 9, 2590 (1970).
    [CrossRef] [PubMed]
  2. S. Yokozeki, T. Suzuki, Appl. Opt. 10, 1575 (1971); A. W. Lohmann, D. E. Silva, Opt. Commun. 2, 413 (1971).
    [CrossRef] [PubMed]
  3. D. E. Silva, Appl. Opt. 10, 1980 (1971).
    [CrossRef]
  4. J. C. Fouéré, D. Malacara, Appl. Opt. 13, 1322 (1974).
    [CrossRef] [PubMed]
  5. S. Yokozeki, K. Patorski, K. Ohnishi, Opt. Commun. 14, 401 (1975).
    [CrossRef]
  6. M. V. R. K. Murty, Appl. Opt. 3, 531 (1964).
    [CrossRef]
  7. V. Ronchi, Appl. Opt. 3, 437 (1964).
    [CrossRef]
  8. V. Ronchi, Appl. Opt. 4, 1041 (1965).
    [CrossRef]
  9. J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. Lond. B76, 378 (1960).
    [CrossRef]
  10. D. Malacara, A. Cornejo, Bol. Inst. Tonantzintla 1, 193 (1974).
  11. J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. Lond. B70, 486 (1957).

1975 (1)

S. Yokozeki, K. Patorski, K. Ohnishi, Opt. Commun. 14, 401 (1975).
[CrossRef]

1974 (2)

J. C. Fouéré, D. Malacara, Appl. Opt. 13, 1322 (1974).
[CrossRef] [PubMed]

D. Malacara, A. Cornejo, Bol. Inst. Tonantzintla 1, 193 (1974).

1971 (2)

1970 (1)

1965 (1)

1964 (2)

1960 (1)

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. Lond. B76, 378 (1960).
[CrossRef]

1957 (1)

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

Cornejo, A.

D. Malacara, A. Cornejo, Bol. Inst. Tonantzintla 1, 193 (1974).

Cowley, J. M.

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. Lond. B76, 378 (1960).
[CrossRef]

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

Fouéré, J. C.

Langenbeck, P.

Malacara, D.

J. C. Fouéré, D. Malacara, Appl. Opt. 13, 1322 (1974).
[CrossRef] [PubMed]

D. Malacara, A. Cornejo, Bol. Inst. Tonantzintla 1, 193 (1974).

Moodie, A. F.

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. Lond. B76, 378 (1960).
[CrossRef]

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

Murty, M. V. R. K.

Ohnishi, K.

S. Yokozeki, K. Patorski, K. Ohnishi, Opt. Commun. 14, 401 (1975).
[CrossRef]

Patorski, K.

S. Yokozeki, K. Patorski, K. Ohnishi, Opt. Commun. 14, 401 (1975).
[CrossRef]

Ronchi, V.

Silva, D. E.

Suzuki, T.

Yokozeki, S.

Appl. Opt. (7)

Bol. Inst. Tonantzintla (1)

D. Malacara, A. Cornejo, Bol. Inst. Tonantzintla 1, 193 (1974).

Opt. Commun. (1)

S. Yokozeki, K. Patorski, K. Ohnishi, Opt. Commun. 14, 401 (1975).
[CrossRef]

Proc. Phys. Soc. Lond. (2)

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. Lond. B76, 378 (1960).
[CrossRef]

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

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of the Talbot shearing interferometer in the beam collimation test. S—point source; L—collimator objective; G1—beam splitter–grating; G2—detecting grating; p—diffuse plate; zp = md2/λ < zT; zT—walkoff distance.

Fig. 2
Fig. 2

Relative localization of the fringes in two-beam and three-beam interference regions as a function of the distance between the diffraction grating and observation plane. (A), (B), and (C) correspond to the interference fields at zp = md2/, z = (m + ½)d2/λ, and zp = (m + 1)d2/λ, respectively. m is any positive integer.

Fig. 3
Fig. 3

A double grating shearing interferometer optimally tuned for a beam collimation test. Symbols mean the same as in Fig. 1.

Fig. 4
Fig. 4

Two-beam interference fields obtained in the case of a collimated beam. (A) shows the moiré corresponding to the location of the detecting grating at a distance z = (m + ½)d2/λ from the first grid. In (B) the detecting grating was at a distance z = md2/λ. m is a positive integer.

Equations (16)

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U ( x , y , z ) = A 0 exp i 2 π λ [ x 2 + y 2 2 ( R + z ) ] + A - 1 exp i 2 π λ { [ x + ( λ / d ) R ] 2 + y 2 2 ( R + z ) - λ 2 2 d 2 R } + A 1 exp i 2 π λ { [ x - ( λ / d ) R ] 2 + y 2 2 ( R + z ) - λ 2 2 d 2 R } ,
R = ( f 2 / Δ f ) - f
I ( x , z ) = U ( x , z ) 2 = A 0 2 + 2 A 1 2 + 2 A 0 A - 1 cos 2 π λ [ 2 λ R d x + λ 2 R 2 2 ( R + z ) d 2 - λ 2 R 2 d 2 ] + 2 A 0 A 1 cos 2 π λ [ 2 λ R d x - λ 2 R 2 2 ( R + z ) d 2 + λ 2 R 2 d 2 ] + 2 A 1 2 cos 2 π x [ 2 λ R x ( R + z ) d ] .
I ( x , z ) = A 0 2 + 2 A 1 2 + 4 A 0 A 1 cos 2 π ( λ R 2 d 2 z R + z ) × cos 2 π d R ( R + z ) x + 2 A 1 2 cos 4 π d R R + z x .
I M ( x , z p ) = A + B cos ( 2 π d z p R + z p x + m π ) ,
F N = D 0 - ( 2 λ z / d ) ( R + z ) d / z .
Δ f = 2 λ ( f / D 0 ) 2 .
cos ( 2 π d z R + z x - π λ d 2 R z R + z ) = cos ( 2 π d z p R + z p x - m π ) ,
cos ( 2 π d z R + z x + π λ d 2 R z R + z ) = cos ( 2 π d z p R + z p x - m π ) .
w = D 0 for 0 < z z T W = D 0 - 2 λ d ( z - D 0 d 2 λ ) for z T z 2 z T ,
D 0 / [ d ( R + z ) / z ] for 0 < z z T [ D 0 - 2 λ ( z - D 0 d / 2 λ ) / d ] / [ d ( R + z ) / z ] ; for z T z 2 z T .
Δ f = λ ( f / D 0 ) 2 / 2.
θ = λ / 2 D 0 .
D 0 = 2 m d .
2 A 0 A - 1 cos ( 2 π d R R + z x - π λ d 2 R z R + z ) ,
2 A 0 A 1 cos ( 2 π d R R + z x + π λ d 2 R z R + z ) .

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