Abstract

We present a new, folded reversal interferometer. It is based on the reflective grating interferometer and can be applied for optical isolation and testing of coma aberration. The interferometer has several advantages in respect to other existing optical reversal configurations. A carrier can be easily added for phase retrieval in interferometric fringe patterns for mapping coma aberration. Furthermore, in an asymmetric optical configuration a lateral shear can also be added, transforming it in a reversal shearing interferometer. The principle of operation of the interferometer is described, and the application for measuring the coma aberration of a parabolic mirror used off axis is demonstrated.

© 1999 Optical Society of America

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References

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  1. O. Bryngdahl, “Applications of shearing interferometry,” in Progress in Optics, E. Wolf ed. (North-Holland, Amsterdam, 1965), Vol. 4, Chap. 2, pp. 34–83.
  2. D. Malacara, “Reversal shear interferometers,” in Optical Shop Testing, 2nd ed. D. Malacara, ed. (Wiley Interscience, New York, 1992), Sec. 5.4, pp. 198–203.
  3. P. Hariharan, D. Shen, “The separation of symmetrical and asymmetrical aberrations in the Twyman interferometer,” Proc. Phys. Soc. 77, 328–332 (1961).
    [CrossRef]
  4. D. Sen, P. N. Puntambekar, “An inverting Fizeau interferometer,” Opt. Acta 12, 137–141 (1965).
    [CrossRef]
  5. J. W. Gates, “Reverse-shearing interferometry,” Nature (London) 176, 359–360 (1955).
    [CrossRef]
  6. J. B. Saunders, “Inverting interferometer,” J. Opt. Soc. Am. 45, 133–135 (1955).
    [CrossRef]
  7. P. N. Puntambekar, D. Sen, “A simple inverting interferometer,” Opt. Acta. 18, 719–721 (1971).
    [CrossRef]
  8. V. Parthiban, C. Joenathan, R. S. Sirohi, “Inversion and folding shear interferometer with holoelements,” in Optics and the Information Age, H. H. Arsenault, ed., Proc. SPIE813, 211–213 (1987).
    [CrossRef]
  9. V. Parthiban, C. Joenathan, R. S. Sirohi, “Simple inverting interferometer with holoelements,” Appl. Opt. 27, 1913–1914 (1988).
    [CrossRef] [PubMed]
  10. S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 24, 491–494 (1995).
    [CrossRef]
  11. M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 24, 761–765 (1996).
  12. S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect,” Opt. Commun. 24, 432–436 (1996).
    [CrossRef]
  13. S. De Nicola, P. Ferraro, “Fourier transform calibration method for phase retrieval of carrier-coded fringe pattern,” Opt. Commun. 151, 217–221 (1998).
    [CrossRef]
  14. R. Kingslake, “The interferometer pattern due to the primary aberrations,” Trans. Opt. Soc. 27, 94–105 (1926).
    [CrossRef]

1998

S. De Nicola, P. Ferraro, “Fourier transform calibration method for phase retrieval of carrier-coded fringe pattern,” Opt. Commun. 151, 217–221 (1998).
[CrossRef]

1996

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 24, 761–765 (1996).

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect,” Opt. Commun. 24, 432–436 (1996).
[CrossRef]

1995

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 24, 491–494 (1995).
[CrossRef]

1988

1971

P. N. Puntambekar, D. Sen, “A simple inverting interferometer,” Opt. Acta. 18, 719–721 (1971).
[CrossRef]

1965

D. Sen, P. N. Puntambekar, “An inverting Fizeau interferometer,” Opt. Acta 12, 137–141 (1965).
[CrossRef]

1961

P. Hariharan, D. Shen, “The separation of symmetrical and asymmetrical aberrations in the Twyman interferometer,” Proc. Phys. Soc. 77, 328–332 (1961).
[CrossRef]

1955

J. W. Gates, “Reverse-shearing interferometry,” Nature (London) 176, 359–360 (1955).
[CrossRef]

J. B. Saunders, “Inverting interferometer,” J. Opt. Soc. Am. 45, 133–135 (1955).
[CrossRef]

1926

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

Bryngdahl, O.

O. Bryngdahl, “Applications of shearing interferometry,” in Progress in Optics, E. Wolf ed. (North-Holland, Amsterdam, 1965), Vol. 4, Chap. 2, pp. 34–83.

De Nicola, S.

S. De Nicola, P. Ferraro, “Fourier transform calibration method for phase retrieval of carrier-coded fringe pattern,” Opt. Commun. 151, 217–221 (1998).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect,” Opt. Commun. 24, 432–436 (1996).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 24, 491–494 (1995).
[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. 24, 761–765 (1996).

DeNicola, S.

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 24, 761–765 (1996).

Ferraro, P.

S. De Nicola, P. Ferraro, “Fourier transform calibration method for phase retrieval of carrier-coded fringe pattern,” Opt. Commun. 151, 217–221 (1998).
[CrossRef]

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 24, 761–765 (1996).

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect,” Opt. Commun. 24, 432–436 (1996).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 24, 491–494 (1995).
[CrossRef]

Finizio, A.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect,” Opt. Commun. 24, 432–436 (1996).
[CrossRef]

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 24, 761–765 (1996).

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 24, 491–494 (1995).
[CrossRef]

Gates, J. W.

J. W. Gates, “Reverse-shearing interferometry,” Nature (London) 176, 359–360 (1955).
[CrossRef]

Hariharan, P.

P. Hariharan, D. Shen, “The separation of symmetrical and asymmetrical aberrations in the Twyman interferometer,” Proc. Phys. Soc. 77, 328–332 (1961).
[CrossRef]

Joenathan, C.

V. Parthiban, C. Joenathan, R. S. Sirohi, “Simple inverting interferometer with holoelements,” Appl. Opt. 27, 1913–1914 (1988).
[CrossRef] [PubMed]

V. Parthiban, C. Joenathan, R. S. Sirohi, “Inversion and folding shear interferometer with holoelements,” in Optics and the Information Age, H. H. Arsenault, ed., Proc. SPIE813, 211–213 (1987).
[CrossRef]

Kingslake, R.

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

Malacara, D.

D. Malacara, “Reversal shear interferometers,” in Optical Shop Testing, 2nd ed. D. Malacara, ed. (Wiley Interscience, New York, 1992), Sec. 5.4, pp. 198–203.

Parthiban, V.

V. Parthiban, C. Joenathan, R. S. Sirohi, “Simple inverting interferometer with holoelements,” Appl. Opt. 27, 1913–1914 (1988).
[CrossRef] [PubMed]

V. Parthiban, C. Joenathan, R. S. Sirohi, “Inversion and folding shear interferometer with holoelements,” in Optics and the Information Age, H. H. Arsenault, ed., Proc. SPIE813, 211–213 (1987).
[CrossRef]

Pesce, G.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 24, 491–494 (1995).
[CrossRef]

Pierattini, G.

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 24, 761–765 (1996).

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect,” Opt. Commun. 24, 432–436 (1996).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 24, 491–494 (1995).
[CrossRef]

Puntambekar, P. N.

P. N. Puntambekar, D. Sen, “A simple inverting interferometer,” Opt. Acta. 18, 719–721 (1971).
[CrossRef]

D. Sen, P. N. Puntambekar, “An inverting Fizeau interferometer,” Opt. Acta 12, 137–141 (1965).
[CrossRef]

Saunders, J. B.

Sen, D.

P. N. Puntambekar, D. Sen, “A simple inverting interferometer,” Opt. Acta. 18, 719–721 (1971).
[CrossRef]

D. Sen, P. N. Puntambekar, “An inverting Fizeau interferometer,” Opt. Acta 12, 137–141 (1965).
[CrossRef]

Shen, D.

P. Hariharan, D. Shen, “The separation of symmetrical and asymmetrical aberrations in the Twyman interferometer,” Proc. Phys. Soc. 77, 328–332 (1961).
[CrossRef]

Sirohi, R. S.

V. Parthiban, C. Joenathan, R. S. Sirohi, “Simple inverting interferometer with holoelements,” Appl. Opt. 27, 1913–1914 (1988).
[CrossRef] [PubMed]

V. Parthiban, C. Joenathan, R. S. Sirohi, “Inversion and folding shear interferometer with holoelements,” in Optics and the Information Age, H. H. Arsenault, ed., Proc. SPIE813, 211–213 (1987).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

Nature (London)

J. W. Gates, “Reverse-shearing interferometry,” Nature (London) 176, 359–360 (1955).
[CrossRef]

Opt. Acta

D. Sen, P. N. Puntambekar, “An inverting Fizeau interferometer,” Opt. Acta 12, 137–141 (1965).
[CrossRef]

Opt. Acta.

P. N. Puntambekar, D. Sen, “A simple inverting interferometer,” Opt. Acta. 18, 719–721 (1971).
[CrossRef]

Opt. Commun.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, G. Pesce, “Reflective grating interferometer for measuring the refractive index of transparent materials,” Opt. Commun. 24, 491–494 (1995).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Reflective grating interferometer for measuring the focal length of a lens by digital moiré effect,” Opt. Commun. 24, 432–436 (1996).
[CrossRef]

S. De Nicola, P. Ferraro, “Fourier transform calibration method for phase retrieval of carrier-coded fringe pattern,” Opt. Commun. 151, 217–221 (1998).
[CrossRef]

Proc. Phys. Soc.

P. Hariharan, D. Shen, “The separation of symmetrical and asymmetrical aberrations in the Twyman interferometer,” Proc. Phys. Soc. 77, 328–332 (1961).
[CrossRef]

Pure Appl. Opt.

M. deAngelis, S. DeNicola, P. Ferraro, A. Finizio, G. Pierattini, “A reflective grating interferometer for measuring refractive index of liquids,” Pure Appl. Opt. 24, 761–765 (1996).

Trans. Opt. Soc.

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

Other

O. Bryngdahl, “Applications of shearing interferometry,” in Progress in Optics, E. Wolf ed. (North-Holland, Amsterdam, 1965), Vol. 4, Chap. 2, pp. 34–83.

D. Malacara, “Reversal shear interferometers,” in Optical Shop Testing, 2nd ed. D. Malacara, ed. (Wiley Interscience, New York, 1992), Sec. 5.4, pp. 198–203.

V. Parthiban, C. Joenathan, R. S. Sirohi, “Inversion and folding shear interferometer with holoelements,” in Optics and the Information Age, H. H. Arsenault, ed., Proc. SPIE813, 211–213 (1987).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

RGI; M, mirror; and G, diffraction grating. The wave-front W(x, y) is divided spatially by the RGI. The half-wave-front W 2(x, y) is folded and reversed onto the half-wave-front W 1(x, y). For clarity, the wave front W(x, y) has been represented with only coma aberration. OPD is the optical path difference produced by the two comatic half and reversed wave fronts.

Fig. 2
Fig. 2

Experimental configuration for testing coma of a parabolic mirror used at angle θ. G, grating; M, mirror; and PP, photoplate.

Fig. 3
Fig. 3

Fringe pattern recorded by the photoplate.

Fig. 4
Fig. 4

Wrapped phase map computed of the fringe pattern shown in Fig. 3.

Fig. 5
Fig. 5

Unwrapped phase map of the comatic interference pattern.

Equations (20)

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

Wx, y=Ax2+y22+Byx2+y2+Cx2+3y2+Dx2+y2+Ex+Fy,
W1x, y=Ax2+y22+Byx2+y2+Cx2+3y2+Dx2+y2+Ex+Fy,
W2x, y=Ax2+y22-Byx2+y2+Cx2+3y2+Dx2+y2+Ex-Fy.
OPDx, y=W1x, y-W2x, y-2Fy
OPDx, y=2Byx2+y2.
OPDx, y=W1x, y-W2x, S-y.
OPLSVH=SV+VH.
zH=cos2θz+sinθcosθy,
yH=sinθcosθz+sin2θy,
z=x2+y2/4f
VH=yH2+zH21/2
VH1-θ2/2z+θy,
OPLSVH=f+z+f-zθ22+θy.
OPLSAA=x2+y-η2+z-ξ21/2+z
zx, y=Byx2+y2+Cx2+3y2+Dx2+y2,
x=x,
y=y+θz,
Ix, y=I0x, y1+cos ΔΦx, y,
ΔΦx, y=2πλ OPDx, y,
OPDx, y=2B cosαyx2+cos2αy2.

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