Abstract

A third-order aberration analysis for a reflective grating interferometer (RGI) is developed for a noncollimated configuration. In such a configuration the RGI is still a folded and reversing interferometer that is sensitive only to coma aberration, as it is in the collimated configuration. However, for an unaberrated input beam converging on a plane grating, the reflective grating of the interferometer introduces self-aberrations. Consequently, a nonnull fringe pattern is obtained. Nevertheless, a RGI in the noncollimated configuration has the potential to be applied for isolating and measuring coma, and a possible configuration for this application is proposed. As an example of the application, the coma of a large mirror could be isolated and measured by use of a converged configuration to avoid the main limitation in using the RGI in a configuration with a nearly collimated beam.

© 2000 Optical Society of America

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References

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  1. D. Post, “Moiré interferometry,” in Handbook of Experimental Mechanics, A. S. Kobayashi, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 7.
  2. K. Patorski, Handbook of The Moirè Fringe Technique (Elsevier, Amsterdam, 1993), Chap. 5, pp. 196–207.
  3. S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 118, 491–494 (1995).
    [CrossRef]
  4. M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 5, 761–765 (1996).
    [CrossRef]
  5. S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lenses by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
    [CrossRef]
  6. M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
    [CrossRef]
  7. S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer: a folded reversal wave-front interferometer,” Appl. Opt. 38, 4845–4849 (1999).
    [CrossRef]
  8. T. Namioka, “Theory of the concave grating. I,” J. Opt. Soc. Am. 49, 446–465 (1959).
    [CrossRef]
  9. W. T. Welford, “Aberration theory of gratings and grating mountings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, pp. 241–280.
    [CrossRef]
  10. W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1991), Chap. 11, pp. 214–217.
  11. M. V. R. K. Murty, “Use of convergent and divergent illumination with plane gratings,” J. Opt. Soc. Am. 52, 768–773 (1962).
    [CrossRef]
  12. D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, New York, 1992), pp. 456–461.
  13. V. Parthiban, C. Joenathan, R. S. Sirohi, “Simple inverting interferometer with holoelements,” Appl. Opt. 27, 1913–1914 (1988).
    [CrossRef] [PubMed]
  14. R. Kingslake, “The interferometer pattern due to the primary aberrations,” Trans. Opt. Soc. 27, 94–105 (1926).
    [CrossRef]
  15. T. Kreis, “Computer-aided evaluation of holographic interferograms,” in Holographic Interferometry, P. K. Rastogi, ed. (Springer-Verlag, Berlin, 1994), Chap. 6, pp. 185–192.

1999 (2)

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer: a folded reversal wave-front interferometer,” Appl. Opt. 38, 4845–4849 (1999).
[CrossRef]

1996 (2)

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 5, 761–765 (1996).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lenses by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

1995 (1)

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 118, 491–494 (1995).
[CrossRef]

1988 (1)

1962 (1)

1959 (1)

1926 (1)

R. Kingslake, “The interferometer pattern due to the primary aberrations,” Trans. Opt. Soc. 27, 94–105 (1926).
[CrossRef]

de Angelis, M.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

De Nicola, S.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

deAngelis, M.

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 5, 761–765 (1996).
[CrossRef]

DeNicola, S.

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer: a folded reversal wave-front interferometer,” Appl. Opt. 38, 4845–4849 (1999).
[CrossRef]

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 5, 761–765 (1996).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lenses by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 118, 491–494 (1995).
[CrossRef]

DeVore, S. L.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, New York, 1992), pp. 456–461.

Ferraro, P.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer: a folded reversal wave-front interferometer,” Appl. Opt. 38, 4845–4849 (1999).
[CrossRef]

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 5, 761–765 (1996).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lenses by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 118, 491–494 (1995).
[CrossRef]

Finizio, A.

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer: a folded reversal wave-front interferometer,” Appl. Opt. 38, 4845–4849 (1999).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 5, 761–765 (1996).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lenses by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 118, 491–494 (1995).
[CrossRef]

Joenathan, C.

Kingslake, R.

R. Kingslake, “The interferometer pattern due to the primary aberrations,” Trans. Opt. Soc. 27, 94–105 (1926).
[CrossRef]

Kreis, T.

T. Kreis, “Computer-aided evaluation of holographic interferograms,” in Holographic Interferometry, P. K. Rastogi, ed. (Springer-Verlag, Berlin, 1994), Chap. 6, pp. 185–192.

Malacara, D.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, New York, 1992), pp. 456–461.

Murty, M. V. R. K.

Namioka, T.

Parthiban, V.

Patorski, K.

K. Patorski, Handbook of The Moirè Fringe Technique (Elsevier, Amsterdam, 1993), Chap. 5, pp. 196–207.

Pesce, G.

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 118, 491–494 (1995).
[CrossRef]

Pierattini, G.

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer: a folded reversal wave-front interferometer,” Appl. Opt. 38, 4845–4849 (1999).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 5, 761–765 (1996).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lenses by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 118, 491–494 (1995).
[CrossRef]

Post, D.

D. Post, “Moiré interferometry,” in Handbook of Experimental Mechanics, A. S. Kobayashi, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 7.

Sirohi, R. S.

Welford, W. T.

W. T. Welford, “Aberration theory of gratings and grating mountings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, pp. 241–280.
[CrossRef]

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1991), Chap. 11, pp. 214–217.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Opt. Commun. (3)

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 118, 491–494 (1995).
[CrossRef]

S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lenses by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

Pure Appl. Opt. (1)

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 5, 761–765 (1996).
[CrossRef]

Trans. Opt. Soc. (1)

R. Kingslake, “The interferometer pattern due to the primary aberrations,” Trans. Opt. Soc. 27, 94–105 (1926).
[CrossRef]

Other (6)

T. Kreis, “Computer-aided evaluation of holographic interferograms,” in Holographic Interferometry, P. K. Rastogi, ed. (Springer-Verlag, Berlin, 1994), Chap. 6, pp. 185–192.

D. Post, “Moiré interferometry,” in Handbook of Experimental Mechanics, A. S. Kobayashi, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 7.

K. Patorski, Handbook of The Moirè Fringe Technique (Elsevier, Amsterdam, 1993), Chap. 5, pp. 196–207.

W. T. Welford, “Aberration theory of gratings and grating mountings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, pp. 241–280.
[CrossRef]

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1991), Chap. 11, pp. 214–217.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, New York, 1992), pp. 456–461.

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Figures (8)

Fig. 1
Fig. 1

A diverging beam incident upon a plane diffraction grating is diffracted in an aberrated wave front W ab(x, y); for clarity, only the coma aberration and its caustic are shown.

Fig. 2
Fig. 2

RGI obtained in Fig. 1 by insertion of a mirror along the z axis at a right angle to the plane grating. The mirror acts as a reversing and folding element, producing the optical path difference (OPD).

Fig. 3
Fig. 3

Schematic drawing for calculating aberrated wave front W ab(x, y) as in Ref. 11.

Fig. 4
Fig. 4

Reflective grating interferometer: M, mirror; G, diffraction grating; A, diverging wave front divided spatially by the RGI into two half-wave fronts at a distance R from the origin (O) of the reference frame. The fringe pattern is projected onto diffuser screen S and imaged on the CCD array.

Fig. 5
Fig. 5

Fringe pattern produced by a RGI for a diverging input beam.

Fig. 6
Fig. 6

Wrapped phase map of the fringe pattern shown in Fig. 5.

Fig. 7
Fig. 7

Unwrapped phase map of the fringe pattern of Fig. 5.

Fig. 8
Fig. 8

Proposed optical setup for testing a large mirror: M, mirror; G, diffraction grating; B.E., beam expander; P.M., parabolic mirror; S, diffusing screen.

Equations (17)

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sinα+sinβ=λf,
W1x, y=Wabx, y,
W2x, y=Wabx, -y.
OPDx, y=W1x, y-W2x, -y.
Wabx, y=Ax2+y22+Byx2+y2+Cx2+3y2+Dx2+y2,
OPDx, y=2Byx2+y2.
Wx, y=AP+BP+mλfy,
AP=x2+y-η2+ζ21/2,
BP=x2+y-η2+ζ21/2.
AP=R2+x2+y2-2yR sinα1/2,
BP=R2+x2+y21/2.
Wabx, y=R+R+x221R+1R+y22cos2αR+1R+x2y2sinαR2+y32sinαcos2αR2.
OPDx, y=2Byx2+y2 cos2α,
B=sinα2R2.
W˜x, y=Ãx2+y22+B˜yx2+y2+C˜x2+3y2+D˜x2+y2,
OPDx, y=2B˜ cosαyx2+y2 cos2α.
OPDx, y=2B+B˜ cosαyx2+y2 cos2α.

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