Abstract

A phase-stepped double-grating lateral shearing interferometer to be used for wave-front analysis is presented. The resulting interference patterns are analyzed with a differential Zernike polynomial matrix-inversion method. Possible error sources are analyzed in the design stage, and it is shown that the inaccuracy can be kept within 2–5 mλ rms. The apparatus was tested and evaluated in practice. Comparison with a phase-stepped Twyman–Green interferometer demonstrates that the accuracy of the two methods is comparable. Lateral shearing interferometry scores better on reproducibility, owing to the stability and robustness of the method.

© 1996 Optical Society of America

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  1. M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of refraction,” Appl. Opt. 3, 853–857 (1964).
    [CrossRef]
  2. M. V. R. K. Murty, “The use of a single plane parallel plate as a lateral shearing interferometer with a visible gas laser source,” Appl. Opt. 3, 531–534 (1964).
    [CrossRef]
  3. M. V. R. K. Murty, E. C. Hagerott, “Rotational-shearing interferometry,” Appl. Opt. 5, 615–619 (1966).
    [CrossRef] [PubMed]
  4. S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
    [CrossRef]
  5. W. C. Sweatt, R. N. Shagam, “New configuration for the rotating shear-plate interferometer, a.k.a. SHEAR madness,” in Nonlinear Optical Properties of Organic Materials VD. J. Williams, ed., Proc. SPIE1775, 131–1371992).
  6. J.-P. Bétend-Bon, L. Wosinski, M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 55–69 (1992).
    [CrossRef]
  7. T. Yatagai, “Fringe scanning Ronchi test for aspherical surfaces,” Appl. Opt. 23, 3676–3679 (1984).
    [CrossRef] [PubMed]
  8. L. Ping, C. Xiaolan, L. Changjiang, “A modified shearing interferometer and interferogram analysis,” Optik 86, 61–63 (1990).
  9. J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12, 2057–2060 (1973).
    [CrossRef] [PubMed]
  10. P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
    [CrossRef]
  11. K. Patorski, “Grating shearing interferometer with variable shear and fringe orientation,” Appl. Opt. 25, 4192–4198 (1986)
    [CrossRef] [PubMed]
  12. J.-A. Lin, S.-W. Hsu, F.-T. Wu, “Double grating interferometer with large lateral shear,” Optik 84, 28–32 (1990).
  13. J. W. Hardy, A. J. MacGovern, “Shearing interferometry: a flexible technique for wavefront measurement,” in Interferometric Metrology, N. A. Massie, ed., Proc. SPIE816, 180–195 (1987)
  14. J. B. Saunders, “A simple, inexpensive wavefront shearing interferometer,” Appl. Opt. 6, 1581–1583 (1967).
    [CrossRef] [PubMed]
  15. W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. 59, 940–950 (1947).
    [CrossRef]
  16. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  17. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 349–392.
    [CrossRef]
  18. G. Harbers, P. J. Kunst, G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by the use of Zernike polynomials,” Appl. Opt. 35, 6162–6172 (1996).
    [CrossRef] [PubMed]
  19. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  20. J. D. Briers, “Interferometric testing of optical systems and components: a review,” Opt. Laser Technol. 4, 28–41 (1972).
    [CrossRef]
  21. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]

1996 (1)

1992 (1)

J.-P. Bétend-Bon, L. Wosinski, M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 55–69 (1992).
[CrossRef]

1990 (3)

L. Ping, C. Xiaolan, L. Changjiang, “A modified shearing interferometer and interferogram analysis,” Optik 86, 61–63 (1990).

S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
[CrossRef]

J.-A. Lin, S.-W. Hsu, F.-T. Wu, “Double grating interferometer with large lateral shear,” Optik 84, 28–32 (1990).

1987 (1)

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

1986 (1)

1984 (1)

1983 (1)

1974 (2)

1973 (1)

1972 (1)

J. D. Briers, “Interferometric testing of optical systems and components: a review,” Opt. Laser Technol. 4, 28–41 (1972).
[CrossRef]

1967 (1)

1966 (1)

1964 (2)

1947 (1)

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. 59, 940–950 (1947).
[CrossRef]

Bates, W. J.

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. 59, 940–950 (1947).
[CrossRef]

Bétend-Bon, J.-P.

J.-P. Bétend-Bon, L. Wosinski, M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 55–69 (1992).
[CrossRef]

Brangaccio, D. J.

Breidne, M.

J.-P. Bétend-Bon, L. Wosinski, M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 55–69 (1992).
[CrossRef]

Briers, J. D.

J. D. Briers, “Interferometric testing of optical systems and components: a review,” Opt. Laser Technol. 4, 28–41 (1972).
[CrossRef]

Bruning, J. H.

Burow, R.

Changjiang, L.

L. Ping, C. Xiaolan, L. Changjiang, “A modified shearing interferometer and interferogram analysis,” Optik 86, 61–63 (1990).

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 349–392.
[CrossRef]

Eiju, T.

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

Elssner, K.-E.

Gallagher, J. E.

Grzanna, J.

Hagerott, E. C.

Harbers, G.

Hardy, J. W.

J. W. Hardy, A. J. MacGovern, “Shearing interferometry: a flexible technique for wavefront measurement,” in Interferometric Metrology, N. A. Massie, ed., Proc. SPIE816, 180–195 (1987)

Hariharan, P.

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Herriott, D. R.

Hsu, S.-W.

J.-A. Lin, S.-W. Hsu, F.-T. Wu, “Double grating interferometer with large lateral shear,” Optik 84, 28–32 (1990).

Kasana, R. S.

S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
[CrossRef]

Kunst, P. J.

Leibbrandt, G. W. R.

Lin, J.-A.

J.-A. Lin, S.-W. Hsu, F.-T. Wu, “Double grating interferometer with large lateral shear,” Optik 84, 28–32 (1990).

MacGovern, A. J.

J. W. Hardy, A. J. MacGovern, “Shearing interferometry: a flexible technique for wavefront measurement,” in Interferometric Metrology, N. A. Massie, ed., Proc. SPIE816, 180–195 (1987)

Merkel, K.

Murty, M. V. R. K.

Oreb, B. F.

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

Patorski, K.

Ping, L.

L. Ping, C. Xiaolan, L. Changjiang, “A modified shearing interferometer and interferogram analysis,” Optik 86, 61–63 (1990).

Rosenfeld, D. P.

Saunders, J. B.

Schwider, J.

Shagam, R. N.

W. C. Sweatt, R. N. Shagam, “New configuration for the rotating shear-plate interferometer, a.k.a. SHEAR madness,” in Nonlinear Optical Properties of Organic Materials VD. J. Williams, ed., Proc. SPIE1775, 131–1371992).

Spolaczyk, R.

Srivastava, S. N.

S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
[CrossRef]

Steel, W. H.

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Sweatt, W. C.

W. C. Sweatt, R. N. Shagam, “New configuration for the rotating shear-plate interferometer, a.k.a. SHEAR madness,” in Nonlinear Optical Properties of Organic Materials VD. J. Williams, ed., Proc. SPIE1775, 131–1371992).

Tomar, M. S.

S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
[CrossRef]

White, A. D.

Wosinski, L.

J.-P. Bétend-Bon, L. Wosinski, M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 55–69 (1992).
[CrossRef]

Wu, F.-T.

J.-A. Lin, S.-W. Hsu, F.-T. Wu, “Double grating interferometer with large lateral shear,” Optik 84, 28–32 (1990).

Wyant, J. C.

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12, 2057–2060 (1973).
[CrossRef] [PubMed]

Xiaolan, C.

L. Ping, C. Xiaolan, L. Changjiang, “A modified shearing interferometer and interferogram analysis,” Optik 86, 61–63 (1990).

Yatagai, T.

Appl. Opt. (1)

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

Appl. Opt. (10)

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of refraction,” Appl. Opt. 3, 853–857 (1964).
[CrossRef]

M. V. R. K. Murty, “The use of a single plane parallel plate as a lateral shearing interferometer with a visible gas laser source,” Appl. Opt. 3, 531–534 (1964).
[CrossRef]

M. V. R. K. Murty, E. C. Hagerott, “Rotational-shearing interferometry,” Appl. Opt. 5, 615–619 (1966).
[CrossRef] [PubMed]

T. Yatagai, “Fringe scanning Ronchi test for aspherical surfaces,” Appl. Opt. 23, 3676–3679 (1984).
[CrossRef] [PubMed]

J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12, 2057–2060 (1973).
[CrossRef] [PubMed]

K. Patorski, “Grating shearing interferometer with variable shear and fringe orientation,” Appl. Opt. 25, 4192–4198 (1986)
[CrossRef] [PubMed]

J. B. Saunders, “A simple, inexpensive wavefront shearing interferometer,” Appl. Opt. 6, 1581–1583 (1967).
[CrossRef] [PubMed]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

G. Harbers, P. J. Kunst, G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by the use of Zernike polynomials,” Appl. Opt. 35, 6162–6172 (1996).
[CrossRef] [PubMed]

Opt. Commun. (1)

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Opt. Laser Technol. (2)

S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
[CrossRef]

J. D. Briers, “Interferometric testing of optical systems and components: a review,” Opt. Laser Technol. 4, 28–41 (1972).
[CrossRef]

Optik (2)

L. Ping, C. Xiaolan, L. Changjiang, “A modified shearing interferometer and interferogram analysis,” Optik 86, 61–63 (1990).

J.-A. Lin, S.-W. Hsu, F.-T. Wu, “Double grating interferometer with large lateral shear,” Optik 84, 28–32 (1990).

Proc. Phys. Soc. (1)

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. 59, 940–950 (1947).
[CrossRef]

Pure Appl. Opt. (1)

J.-P. Bétend-Bon, L. Wosinski, M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 55–69 (1992).
[CrossRef]

Other (3)

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 349–392.
[CrossRef]

J. W. Hardy, A. J. MacGovern, “Shearing interferometry: a flexible technique for wavefront measurement,” in Interferometric Metrology, N. A. Massie, ed., Proc. SPIE816, 180–195 (1987)

W. C. Sweatt, R. N. Shagam, “New configuration for the rotating shear-plate interferometer, a.k.a. SHEAR madness,” in Nonlinear Optical Properties of Organic Materials VD. J. Williams, ed., Proc. SPIE1775, 131–1371992).

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Figures (14)

Fig. 1
Fig. 1

Basic LS interferometer setup in which two gratings are used.

Fig. 2
Fig. 2

Cross-sectional view of the actual LS interferometer used in this investigation.

Fig. 3
Fig. 3

Interference pattern measured with the LS interferometer. The vertical fringes are the result of defocus. The straight fringes indicate a good wave-front quality.

Fig. 4
Fig. 4

Experimental setup designed to measure a test sample in both a T-G interferometer and a shearing interferometer. If the shutter is opened, the arrangement functions as a T-G interferometer, and if closed, the beam wave front can be analyzed with the LS interferometer.

Fig. 5
Fig. 5

Error propagation in the matrix-inversion method: expected effective errors in the calculated Zernike coefficients anm as a result of measurement errors of 1 mλ in the measured shear coefficients rjk .

Fig. 6
Fig. 6

Error propagation in the matrix-inversion method: relative errors in the calculated Zernike coefficients anm as a results of a shear ratio error Δs = 0.002.

Fig. 7
Fig. 7

(a) Perfect grating alignment, (b) alignment error of 1°.

Fig. 8
Fig. 8

Orders involved in the LS interferometer.

Fig. 9
Fig. 9

Ratio R(0, 1) of the zeroth order with respect to the first-order amplitude as a function of λ (nm) for a couple of rectangular gratings for λ0 = 700 nm.

Fig. 10
Fig. 10

Ratio R(0, 1) as a function of λ (nm) for a set consisting of a λ0 = 630 nm grating and a λ0 = 780 nm grating.

Fig. 11
Fig. 11

LS interference patterns in (a) the X-shear direction, (b) the Y-shear direction, recorded from a test sample in the setup of Fig. 4.

Fig. 12
Fig. 12

T-G interference pattern recorded from the same test sample as in Fig. 11.

Fig. 13
Fig. 13

Averages and distribution of five aberration terms in a wave front measured with the LS interferometer for varying shear ratios. Each measurement is performed 10 times. The error bars indicate the 3σ ranges. The dashed lines indicate the 3σ upper and lower boundaries as measured with a T-G interferometer.

Fig. 14
Fig. 14

Propagation of small alignment errors into wave-front determination errors; matrix dimension D = 12.

Tables (2)

Tables Icon

Table 1 Zernike Coefficients that are Primarily Affected by Misalignment, as Follows from the Model described in the Text a

Tables Icon

Table 2 Comparison Measurements of the LS Interferometer versus the T-G Interferometer a

Equations (14)

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

s = λ d R p .
s = q 2 r .
Δ s s = [ ( Δ r r ) 2 + ( Δ q q ) 2 ] 1 / 2 .
ϕ = 2 α λ p
a 1 , 1 = 2 R λ ϕ ,
υ = f ϕ M R ,
a 1 , + 1 = 2 R λ ϕ sin γ = 2 f s M λ ϕ 2 .
h = λ 0 ( k + ½ ) n 1 , k = 0 , 1 , 2 , ,
a 4 , 4 edge
a 4 , 4 edge
a 6 , 4 edge
a 4 , 4 edge
a 6 , 4 edge
a 8 , 4 edge

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