Abstract

In local reference holographic interferometry and radial shear interferometry, a small portion of the test beam is magnified and used as the reference for the rest of the wave front. The reference portion may be off-center or contain aberrations that complicate measurement of the wave front. An iterative method for correcting the measurement is described here. Information about the aberration in the reference is derived from the interferogram and is used to improve the estimate of the wave front shape. A computer simulation is used to verify operation of the algorithm and estimate the effect of certain input errors.

© 1986 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing (Wiley, New York, 1978).
  2. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).
  3. C. Roychoudhuri, B. J. Thompson, “Application of Local Reference Beam Holography to the Study of Laser Beam Parameters,” Opt. Eng. 13, 347 (1974).
    [CrossRef]
  4. J. C. Fouere, C. Roychoudhuri, “A Holographic Radial and Lateral Shear Interferometer,” Opt. Commun. 12, 29 (1974).
    [CrossRef]
  5. A. I. Kharitonov, V. A. Gorshikov, E. S. Simonova, “A Double-Beam Interferometer with Lateral and Radial Shearing of Wavefronts,” Sov. J. Opt. Technol. 42, 457 (1975).
  6. W. L. Howes, “Large-Aperture Interferometer with a Local Reference Beam,” Appl. Opt. 23, 1467 (1984).
    [CrossRef] [PubMed]
  7. M. V. R. K. Murty, “A Compact Radial Shearing Interferometer Based on the Law of Refraction,” Appl. Opt. 3, 853 (1964).
    [CrossRef]
  8. D. Malacara, “Mathematical Interpretation of Radial Shearing Interferometers,” Appl. Opt. 13, 1781 (1974).
    [CrossRef] [PubMed]
  9. We use the notation of M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 466.Malacara1 also discusses Zernike polynomials but uses a different index notation.

1984 (1)

1975 (1)

A. I. Kharitonov, V. A. Gorshikov, E. S. Simonova, “A Double-Beam Interferometer with Lateral and Radial Shearing of Wavefronts,” Sov. J. Opt. Technol. 42, 457 (1975).

1974 (3)

D. Malacara, “Mathematical Interpretation of Radial Shearing Interferometers,” Appl. Opt. 13, 1781 (1974).
[CrossRef] [PubMed]

C. Roychoudhuri, B. J. Thompson, “Application of Local Reference Beam Holography to the Study of Laser Beam Parameters,” Opt. Eng. 13, 347 (1974).
[CrossRef]

J. C. Fouere, C. Roychoudhuri, “A Holographic Radial and Lateral Shear Interferometer,” Opt. Commun. 12, 29 (1974).
[CrossRef]

1964 (1)

Born, M.

We use the notation of M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 466.Malacara1 also discusses Zernike polynomials but uses a different index notation.

Cathey, W. T.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

Fouere, J. C.

J. C. Fouere, C. Roychoudhuri, “A Holographic Radial and Lateral Shear Interferometer,” Opt. Commun. 12, 29 (1974).
[CrossRef]

Gorshikov, V. A.

A. I. Kharitonov, V. A. Gorshikov, E. S. Simonova, “A Double-Beam Interferometer with Lateral and Radial Shearing of Wavefronts,” Sov. J. Opt. Technol. 42, 457 (1975).

Howes, W. L.

Kharitonov, A. I.

A. I. Kharitonov, V. A. Gorshikov, E. S. Simonova, “A Double-Beam Interferometer with Lateral and Radial Shearing of Wavefronts,” Sov. J. Opt. Technol. 42, 457 (1975).

Malacara, D.

Murty, M. V. R. K.

Roychoudhuri, C.

J. C. Fouere, C. Roychoudhuri, “A Holographic Radial and Lateral Shear Interferometer,” Opt. Commun. 12, 29 (1974).
[CrossRef]

C. Roychoudhuri, B. J. Thompson, “Application of Local Reference Beam Holography to the Study of Laser Beam Parameters,” Opt. Eng. 13, 347 (1974).
[CrossRef]

Simonova, E. S.

A. I. Kharitonov, V. A. Gorshikov, E. S. Simonova, “A Double-Beam Interferometer with Lateral and Radial Shearing of Wavefronts,” Sov. J. Opt. Technol. 42, 457 (1975).

Thompson, B. J.

C. Roychoudhuri, B. J. Thompson, “Application of Local Reference Beam Holography to the Study of Laser Beam Parameters,” Opt. Eng. 13, 347 (1974).
[CrossRef]

Wolf, E.

We use the notation of M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 466.Malacara1 also discusses Zernike polynomials but uses a different index notation.

Appl. Opt. (3)

Opt. Commun. (1)

J. C. Fouere, C. Roychoudhuri, “A Holographic Radial and Lateral Shear Interferometer,” Opt. Commun. 12, 29 (1974).
[CrossRef]

Opt. Eng. (1)

C. Roychoudhuri, B. J. Thompson, “Application of Local Reference Beam Holography to the Study of Laser Beam Parameters,” Opt. Eng. 13, 347 (1974).
[CrossRef]

Sov. J. Opt. Technol. (1)

A. I. Kharitonov, V. A. Gorshikov, E. S. Simonova, “A Double-Beam Interferometer with Lateral and Radial Shearing of Wavefronts,” Sov. J. Opt. Technol. 42, 457 (1975).

Other (3)

D. Malacara, Optical Shop Testing (Wiley, New York, 1978).

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

We use the notation of M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 466.Malacara1 also discusses Zernike polynomials but uses a different index notation.

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Figures (5)

Fig. 1
Fig. 1

Simple interferometer with both radial and lateral shear. The wave front incident from the left has already been distorted by a test medium.

Fig. 2
Fig. 2

Coordinates in the interferogram plane. A, region used for the reference wave front.

Fig. 3
Fig. 3

Plot of 0.5 λ U 5 1 along the x axis. The phase in the reference region (for r0 = 0.25, M = 5) has a peak-to-valley distortion of 0.23 waves.

Fig. 4
Fig. 4

Error in measurement of a wave front for different iterations illustrating the covergence of the algorithm. Ordinates for j = 0 should be multiplied by 100. (All curves were set to zero at the origin.)

Fig. 5
Fig. 5

Root mean square error in wave front measurement after two iterations for various errors in the location of the reference.

Equations (9)

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p ( r ) = ϕ ( r 0 + r / M ) .
W 0 ( r ) = ϕ ( r ) ϕ ( r 0 + r / M ) .
W 0 ( r 0 + r / M ) .
W 1 ( r ) = W 0 ( r ) + W 0 ( r 0 + r / M ) .
W j ( r ) = W 0 ( r ) + W j 1 ( r 0 + r / M ) = W 0 ( r ) + n = 1 j W 0 ( r n ) ,
r n = ( M n 1 ) r 0 M n 1 ( M 1 ) + r M n .
W j ( r ) = ϕ ( r ) ϕ ( r j + 1 ) .
W 0 ( r ) = 0.5 [ U 5 1 ( r ) + U 5 1 ( r 0 + 0.2 r ) ] ,
U 5 1 ( r ) = ( 10 r 5 12 r 3 + 3 r ) cos ( θ ) .

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