Abstract

Current methods in shear interferometry provide shear only along one direction at a time. We propose a method in which interferograms with shear along the x, y, and xy directions can be obtained simultaneously from a single exposure recording via the concept of spatial multiplexing. The method utilizes holographic lenses, which have been recorded on a single plate with their optical centers translated along the x and y directions. The phase information is extracted through the Fourier transform method. In addition, this technique also provides a method to obtain interferograms with shear in multiples of the original shear along the x and y axis in one single frame capture due to the nonlinearity inherent in the phase holographic lens.

© 2013 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. (Elsevier, 1965), Vol. 4, pp. 34–83.
  2. D. Malacara, “Lateral shear interferometers,” in Optical Shop Testing, D. Malacara, ed. Wiley Series (Wiley, 2007), pp. 122–184.
  3. J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12, 2057–2060 (1973).
    [CrossRef]
  4. D. Malacara and S. Mallick, “Holographic lateral shear interferometer,” Appl. Opt. 15, 2695–2697 (1976).
    [CrossRef]
  5. C. Joenathan, R. K. Mohanty, and R. S. Sirohi, “Lateral shear interferometry with holo shear lens,” Opt. Commun. 52, 153–156 (1984).
    [CrossRef]
  6. R. P. Shukia, N. C. Das, V. Mantravadi Murty, D. Cook, and M. D. Aggrawal, “Design and fabrication of a variable frequency grating and its application as a lateral-shear interferometer having a variable shear,” Opt. Laser Technol. 39, 338–346 (2007).
    [CrossRef]
  7. C. Joenathan, G. Pedrini, and W. Osten, “Novel and simple lateral shear interferometer with holographic lens and spatial Fourier transform,” Opt. Eng. 51, 075601 (2012).
    [CrossRef]
  8. C. Joenathan, A. Bernal, G. Pedrini, and W. Osten, “Radial shear interferometer with holographic lenses coupled with a spatial Fourier transform method suitable for static and dynamic measurements,” Opt. Eng. 52, 035603 (2013).
    [CrossRef]
  9. C. Joenathan, A. Bernal, and R. S. Sirohi, “Lateral shear interferometer using multiplexed holographic lenses and spatial Fourier transform: varying spectrum position and phase fluctuations,” Opt. Eng. (to be published).
  10. V. Nercissian, I. Harder, K. Mantel, A. Berger, G. Leuchs, N. Lindlen, and J. Schwider, “Diffractive simultaneous bidirectional shearing interferometry using tailored spatially coherent light,” Appl. Opt. 50, 571–578 (2011).
    [CrossRef]
  11. H. I. Bjelkhagen, Selected Papers on Holographic Recoding Materials, SPIE milestone series, vol. 130 (SPIE, 1996).
  12. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005), pp. 343–351.

2013

C. Joenathan, A. Bernal, G. Pedrini, and W. Osten, “Radial shear interferometer with holographic lenses coupled with a spatial Fourier transform method suitable for static and dynamic measurements,” Opt. Eng. 52, 035603 (2013).
[CrossRef]

2012

C. Joenathan, G. Pedrini, and W. Osten, “Novel and simple lateral shear interferometer with holographic lens and spatial Fourier transform,” Opt. Eng. 51, 075601 (2012).
[CrossRef]

2011

2007

R. P. Shukia, N. C. Das, V. Mantravadi Murty, D. Cook, and M. D. Aggrawal, “Design and fabrication of a variable frequency grating and its application as a lateral-shear interferometer having a variable shear,” Opt. Laser Technol. 39, 338–346 (2007).
[CrossRef]

1984

C. Joenathan, R. K. Mohanty, and R. S. Sirohi, “Lateral shear interferometry with holo shear lens,” Opt. Commun. 52, 153–156 (1984).
[CrossRef]

1976

1973

Aggrawal, M. D.

R. P. Shukia, N. C. Das, V. Mantravadi Murty, D. Cook, and M. D. Aggrawal, “Design and fabrication of a variable frequency grating and its application as a lateral-shear interferometer having a variable shear,” Opt. Laser Technol. 39, 338–346 (2007).
[CrossRef]

Berger, A.

Bernal, A.

C. Joenathan, A. Bernal, G. Pedrini, and W. Osten, “Radial shear interferometer with holographic lenses coupled with a spatial Fourier transform method suitable for static and dynamic measurements,” Opt. Eng. 52, 035603 (2013).
[CrossRef]

C. Joenathan, A. Bernal, and R. S. Sirohi, “Lateral shear interferometer using multiplexed holographic lenses and spatial Fourier transform: varying spectrum position and phase fluctuations,” Opt. Eng. (to be published).

Bjelkhagen, H. I.

H. I. Bjelkhagen, Selected Papers on Holographic Recoding Materials, SPIE milestone series, vol. 130 (SPIE, 1996).

Bryngdahl, O.

O. Bryngdahl, “Applications of shearing interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1965), Vol. 4, pp. 34–83.

Cook, D.

R. P. Shukia, N. C. Das, V. Mantravadi Murty, D. Cook, and M. D. Aggrawal, “Design and fabrication of a variable frequency grating and its application as a lateral-shear interferometer having a variable shear,” Opt. Laser Technol. 39, 338–346 (2007).
[CrossRef]

Das, N. C.

R. P. Shukia, N. C. Das, V. Mantravadi Murty, D. Cook, and M. D. Aggrawal, “Design and fabrication of a variable frequency grating and its application as a lateral-shear interferometer having a variable shear,” Opt. Laser Technol. 39, 338–346 (2007).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005), pp. 343–351.

Harder, I.

Joenathan, C.

C. Joenathan, A. Bernal, G. Pedrini, and W. Osten, “Radial shear interferometer with holographic lenses coupled with a spatial Fourier transform method suitable for static and dynamic measurements,” Opt. Eng. 52, 035603 (2013).
[CrossRef]

C. Joenathan, G. Pedrini, and W. Osten, “Novel and simple lateral shear interferometer with holographic lens and spatial Fourier transform,” Opt. Eng. 51, 075601 (2012).
[CrossRef]

C. Joenathan, R. K. Mohanty, and R. S. Sirohi, “Lateral shear interferometry with holo shear lens,” Opt. Commun. 52, 153–156 (1984).
[CrossRef]

C. Joenathan, A. Bernal, and R. S. Sirohi, “Lateral shear interferometer using multiplexed holographic lenses and spatial Fourier transform: varying spectrum position and phase fluctuations,” Opt. Eng. (to be published).

Leuchs, G.

Lindlen, N.

Malacara, D.

D. Malacara and S. Mallick, “Holographic lateral shear interferometer,” Appl. Opt. 15, 2695–2697 (1976).
[CrossRef]

D. Malacara, “Lateral shear interferometers,” in Optical Shop Testing, D. Malacara, ed. Wiley Series (Wiley, 2007), pp. 122–184.

Mallick, S.

Mantel, K.

Mantravadi Murty, V.

R. P. Shukia, N. C. Das, V. Mantravadi Murty, D. Cook, and M. D. Aggrawal, “Design and fabrication of a variable frequency grating and its application as a lateral-shear interferometer having a variable shear,” Opt. Laser Technol. 39, 338–346 (2007).
[CrossRef]

Mohanty, R. K.

C. Joenathan, R. K. Mohanty, and R. S. Sirohi, “Lateral shear interferometry with holo shear lens,” Opt. Commun. 52, 153–156 (1984).
[CrossRef]

Nercissian, V.

Osten, W.

C. Joenathan, A. Bernal, G. Pedrini, and W. Osten, “Radial shear interferometer with holographic lenses coupled with a spatial Fourier transform method suitable for static and dynamic measurements,” Opt. Eng. 52, 035603 (2013).
[CrossRef]

C. Joenathan, G. Pedrini, and W. Osten, “Novel and simple lateral shear interferometer with holographic lens and spatial Fourier transform,” Opt. Eng. 51, 075601 (2012).
[CrossRef]

Pedrini, G.

C. Joenathan, A. Bernal, G. Pedrini, and W. Osten, “Radial shear interferometer with holographic lenses coupled with a spatial Fourier transform method suitable for static and dynamic measurements,” Opt. Eng. 52, 035603 (2013).
[CrossRef]

C. Joenathan, G. Pedrini, and W. Osten, “Novel and simple lateral shear interferometer with holographic lens and spatial Fourier transform,” Opt. Eng. 51, 075601 (2012).
[CrossRef]

Schwider, J.

Shukia, R. P.

R. P. Shukia, N. C. Das, V. Mantravadi Murty, D. Cook, and M. D. Aggrawal, “Design and fabrication of a variable frequency grating and its application as a lateral-shear interferometer having a variable shear,” Opt. Laser Technol. 39, 338–346 (2007).
[CrossRef]

Sirohi, R. S.

C. Joenathan, R. K. Mohanty, and R. S. Sirohi, “Lateral shear interferometry with holo shear lens,” Opt. Commun. 52, 153–156 (1984).
[CrossRef]

C. Joenathan, A. Bernal, and R. S. Sirohi, “Lateral shear interferometer using multiplexed holographic lenses and spatial Fourier transform: varying spectrum position and phase fluctuations,” Opt. Eng. (to be published).

Wyant, J. C.

Appl. Opt.

Opt. Commun.

C. Joenathan, R. K. Mohanty, and R. S. Sirohi, “Lateral shear interferometry with holo shear lens,” Opt. Commun. 52, 153–156 (1984).
[CrossRef]

Opt. Eng.

C. Joenathan, G. Pedrini, and W. Osten, “Novel and simple lateral shear interferometer with holographic lens and spatial Fourier transform,” Opt. Eng. 51, 075601 (2012).
[CrossRef]

C. Joenathan, A. Bernal, G. Pedrini, and W. Osten, “Radial shear interferometer with holographic lenses coupled with a spatial Fourier transform method suitable for static and dynamic measurements,” Opt. Eng. 52, 035603 (2013).
[CrossRef]

Opt. Laser Technol.

R. P. Shukia, N. C. Das, V. Mantravadi Murty, D. Cook, and M. D. Aggrawal, “Design and fabrication of a variable frequency grating and its application as a lateral-shear interferometer having a variable shear,” Opt. Laser Technol. 39, 338–346 (2007).
[CrossRef]

Other

O. Bryngdahl, “Applications of shearing interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1965), Vol. 4, pp. 34–83.

D. Malacara, “Lateral shear interferometers,” in Optical Shop Testing, D. Malacara, ed. Wiley Series (Wiley, 2007), pp. 122–184.

C. Joenathan, A. Bernal, and R. S. Sirohi, “Lateral shear interferometer using multiplexed holographic lenses and spatial Fourier transform: varying spectrum position and phase fluctuations,” Opt. Eng. (to be published).

H. I. Bjelkhagen, Selected Papers on Holographic Recoding Materials, SPIE milestone series, vol. 130 (SPIE, 1996).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005), pp. 343–351.

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Figures (8)

Fig. 1.
Fig. 1.

Schematic arrangement of the holographic element with three spatially recorded lenses forming shears along the x and y axes.

Fig. 2.
Fig. 2.

Vertical, horizontal, and diagonal carrier frequencies created in the x shear, y shear, and xy shear, respectively, are shown.

Fig. 3.
Fig. 3.

Fourier spectrum of a test lens. Filtering the frequency band along the x axis provides x slope changes and one along the y axis provides y slope changes. The frequency band located at 45° to the x and y axes provides xy-shear fringes. Note that on either side of all the frequency bands are extraneous bands that are caused by reflections in the recording process of the multiplexed holographic element.

Fig. 4.
Fig. 4.

Schematic of the focal spots formed at the focal plane of the multiplexed holographic element.

Fig. 5.
Fig. 5.

Fourier spectrum for the test lens showing the entire frequency spectrum obtained by Fourier transforming one of the images of all of the shears captured by the CCD camera.

Fig. 6.
Fig. 6.

Schematic of the experimental arrangement of the multiplexed x-y-lateral shear interferometer.

Fig. 7.
Fig. 7.

Phase maps for spherical aberrations created by the collimating lens: (a) x shear fringes, (b) y shear fringes, and (c) xy shear fringes.

Fig. 8.
Fig. 8.

Phase maps of the spherical aberration present in the test lens (a) first-order x-shear, (b) second-order x-shear, (c) first-order y-shear, (d) second-order y-shear, and (e) first-order xy-shear. Note that the second-order spectra have twice the shear when compared to the first-order spectra.

Equations (10)

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

Δx=Lα1,
Δy=Lα2,
ν1=Δx/λL,
ν2=Δy/λL,
ν3=(Δx2+Δy2)1/2/λL=Δξ/λL.
f(x,y)=I(x,y){1+Mexp(inϕ1+inϕ2+inϕ3)},
wx=mλLα1.
wy=mλLα2,
wξ=mλL(α12+α22)1/2,
wx=mλnLα1.

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