Abstract

Lateral shearing interferometry is a simple and accurate method of measuring the shape of an optical wave front. However, the complex mathematics required in reduction of the interferograms has limited the adoption of this interferometric technique. A data reduction technique using Fourier analysis methods has been developed which makes lateral shearing interferogram reduction more flexible. This technique is hypothetically applied to the measurement of wave fronts propagating through a turbulent medium, and demonstrates the simplicity and accuracy of this data reduction method.

© 1976 Optical Society of America

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  1. W. J. Bates, "A Wavefront Shearing Interferometer," Proc. Phys. Soc. 59, 940 (1947).
  2. L. C. Martin, Technical Optics II (Sir Isaac Pitman & Son, London, 1961), pp. 330–334.
  3. O. Bryngdahl, in Progress in Optics IV, edited by E. Wolf (North Holland, Amsterdam, 1965), pp. 39–47.
  4. J. B. Saunders, "A Wavefront Shearing Interferometer," J. Res. Nat. Bur. Stand., 65, 239 (1961).
  5. J. B. Saunders in Advanced Optical Techniques, edited by A. C. S. Van Heel (North Holland, Amsterdam, 1967), pp. 16–20.
  6. J. B. Saunders, "A Simple Interferometer Method for Work-shop Testing of Optics," Appl. Opt. 9, 1623 (1970).
  7. R. B. Herrick and J. R. Meyer-Arendt, "Interferometry through the Turbulent Atmosphere at an Optical Path Difference of 354 m," Appl. Opt. 5, 981 (1966).
  8. A. L. Buck, "Effects of the Atmosphere on Laser Beam Propagation," Appl. Opt. 6, 703 (1967).
  9. M. Carnevale, B. Crosignani, and P. Di Porto, "Influence of Laboratory Generated Turbulence on Phase Fluctuations of a Laser Beam," Appl. Opt. 7, 1121 (1968).
  10. M. Bertolotti, M. Carnevale, B. Crosignani, and P. Di Porto, "Influence of Thermal Turbulence in a Convective Ascending Stream on Phase Fluctuations of a Laser Beam," Appl. Opt. 8, 1111 (1969).
  11. M. Bertolotti, L. Muzii, and D. Sette, "Correlation Measurements on Partially Coherent Beams by Means of an Integration Technique," J. Opt. Soc. Am. 60, 1603 (1970).
  12. M. Bertolotti, M. Carnevale, L. Muzii, and D. Sette, "Reversing-Front Interferometer for Phase Correlations Measurements in the Turbulent Atmosphere," Appl. Opt. 9, 510 (1970).
  13. P. Burlamacchi, A. Consortini, and L. Ronchi, "Time-Resolved Measurements of the Phase Fluctuations of a Coherent Beam at the Emergence from a Turbulent Layer," Appl. Opt. 6, 1273 (1967).
  14. R. C. Ward and G. V. Berry, "Differential-Interferometer Seeing Meter," Appl. Opt. 6, 1136 (1967).
  15. K. E. Erickson, "Long Path Interferometry Through an Uncontrolled Atomosphere," J. Opt. Soc. Am. 52, 781 (1962).
  16. M. Bertolotti, M. Carnevale, L. Muzii, and D. Sette, "Interferometric Study of Phase Fluctuations of a Laser Beam Through the Turbulent Atomosphere," Appl. Opt. 7, 2246 (1968).
  17. M. Bertolotti, M. Carnevale, B. Daino, and D. Sette, "Optical Processing in the Phase Correlation Induced by a Turbulent Medium in a Laser Beam," Appl. Opt. 9, 962 (1970).
  18. S. F. Clifford, G. M. B. Bouricius, G. R. Ochs, M. H. Ackley, "Phase Variations in Atmospheric Optical Propagation," J. Opt. Soc. Am. 61, 1279 (1971).
  19. R. R. Shannon and W. S. Smith, "An Experiment for Measuring Effect of Atmospheric Turbulence on a Vertical Optical Path," Proc. SPIE 75, 44 (1976).
  20. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).
  21. J. W. Cooley, P. A. W. Lewis, and P. P. Welch, "The Fast Fourier Transform and its Applications," IEEE Trans. Educ. 12, 27 (1969).
  22. E. T. Whittaker, "On the Functions Which are Represented by the Expansions of the Interpolation Theory," Proc. R. Soc. Edinburgh Sect. A, 35, 181 (1915).
  23. C. E. Shannon, "Communication in the Presence of Noise," Proc. IRE 37, 10 (1949).
  24. D. A. Linden, "A Discussion of Sampling Theorems," Proc. IRE 47, 1219 (1959).
  25. R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959).

1976 (1)

R. R. Shannon and W. S. Smith, "An Experiment for Measuring Effect of Atmospheric Turbulence on a Vertical Optical Path," Proc. SPIE 75, 44 (1976).

1971 (1)

1970 (4)

1969 (2)

M. Bertolotti, M. Carnevale, B. Crosignani, and P. Di Porto, "Influence of Thermal Turbulence in a Convective Ascending Stream on Phase Fluctuations of a Laser Beam," Appl. Opt. 8, 1111 (1969).

J. W. Cooley, P. A. W. Lewis, and P. P. Welch, "The Fast Fourier Transform and its Applications," IEEE Trans. Educ. 12, 27 (1969).

1968 (2)

1967 (3)

1966 (1)

1962 (1)

1961 (1)

J. B. Saunders, "A Wavefront Shearing Interferometer," J. Res. Nat. Bur. Stand., 65, 239 (1961).

1959 (2)

D. A. Linden, "A Discussion of Sampling Theorems," Proc. IRE 47, 1219 (1959).

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959).

1949 (1)

C. E. Shannon, "Communication in the Presence of Noise," Proc. IRE 37, 10 (1949).

1947 (1)

W. J. Bates, "A Wavefront Shearing Interferometer," Proc. Phys. Soc. 59, 940 (1947).

1915 (1)

E. T. Whittaker, "On the Functions Which are Represented by the Expansions of the Interpolation Theory," Proc. R. Soc. Edinburgh Sect. A, 35, 181 (1915).

Ackley, M. H.

Bates, W. J.

W. J. Bates, "A Wavefront Shearing Interferometer," Proc. Phys. Soc. 59, 940 (1947).

Berry, G. V.

Bertolotti, M.

Blackman, R. B.

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959).

Bouricius, G. M. B.

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

Bryngdahl, O.

O. Bryngdahl, in Progress in Optics IV, edited by E. Wolf (North Holland, Amsterdam, 1965), pp. 39–47.

Buck, A. L.

Burlamacchi, P.

Carnevale, M.

Clifford, S. F.

Consortini, A.

Cooley, J. W.

J. W. Cooley, P. A. W. Lewis, and P. P. Welch, "The Fast Fourier Transform and its Applications," IEEE Trans. Educ. 12, 27 (1969).

Crosignani, B.

Daino, B.

Erickson, K. E.

Herrick, R. B.

Lewis, P. A. W.

J. W. Cooley, P. A. W. Lewis, and P. P. Welch, "The Fast Fourier Transform and its Applications," IEEE Trans. Educ. 12, 27 (1969).

Linden, D. A.

D. A. Linden, "A Discussion of Sampling Theorems," Proc. IRE 47, 1219 (1959).

Martin, L. C.

L. C. Martin, Technical Optics II (Sir Isaac Pitman & Son, London, 1961), pp. 330–334.

Meyer-Arendt, J. R.

Muzii, L.

Ochs, G. R.

Porto, P. Di

Ronchi, L.

Saunders, J. B.

J. B. Saunders, "A Simple Interferometer Method for Work-shop Testing of Optics," Appl. Opt. 9, 1623 (1970).

J. B. Saunders, "A Wavefront Shearing Interferometer," J. Res. Nat. Bur. Stand., 65, 239 (1961).

J. B. Saunders in Advanced Optical Techniques, edited by A. C. S. Van Heel (North Holland, Amsterdam, 1967), pp. 16–20.

Sette, D.

Shannon, C. E.

C. E. Shannon, "Communication in the Presence of Noise," Proc. IRE 37, 10 (1949).

Shannon, R. R.

R. R. Shannon and W. S. Smith, "An Experiment for Measuring Effect of Atmospheric Turbulence on a Vertical Optical Path," Proc. SPIE 75, 44 (1976).

Smith, W. S.

R. R. Shannon and W. S. Smith, "An Experiment for Measuring Effect of Atmospheric Turbulence on a Vertical Optical Path," Proc. SPIE 75, 44 (1976).

Tukey, J. W.

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959).

Ward, R. C.

Welch, P. P.

J. W. Cooley, P. A. W. Lewis, and P. P. Welch, "The Fast Fourier Transform and its Applications," IEEE Trans. Educ. 12, 27 (1969).

Whittaker, E. T.

E. T. Whittaker, "On the Functions Which are Represented by the Expansions of the Interpolation Theory," Proc. R. Soc. Edinburgh Sect. A, 35, 181 (1915).

Appl. Opt. (10)

R. B. Herrick and J. R. Meyer-Arendt, "Interferometry through the Turbulent Atmosphere at an Optical Path Difference of 354 m," Appl. Opt. 5, 981 (1966).

A. L. Buck, "Effects of the Atmosphere on Laser Beam Propagation," Appl. Opt. 6, 703 (1967).

M. Carnevale, B. Crosignani, and P. Di Porto, "Influence of Laboratory Generated Turbulence on Phase Fluctuations of a Laser Beam," Appl. Opt. 7, 1121 (1968).

M. Bertolotti, M. Carnevale, L. Muzii, and D. Sette, "Interferometric Study of Phase Fluctuations of a Laser Beam Through the Turbulent Atomosphere," Appl. Opt. 7, 2246 (1968).

M. Bertolotti, M. Carnevale, B. Crosignani, and P. Di Porto, "Influence of Thermal Turbulence in a Convective Ascending Stream on Phase Fluctuations of a Laser Beam," Appl. Opt. 8, 1111 (1969).

M. Bertolotti, M. Carnevale, B. Daino, and D. Sette, "Optical Processing in the Phase Correlation Induced by a Turbulent Medium in a Laser Beam," Appl. Opt. 9, 962 (1970).

J. B. Saunders, "A Simple Interferometer Method for Work-shop Testing of Optics," Appl. Opt. 9, 1623 (1970).

P. Burlamacchi, A. Consortini, and L. Ronchi, "Time-Resolved Measurements of the Phase Fluctuations of a Coherent Beam at the Emergence from a Turbulent Layer," Appl. Opt. 6, 1273 (1967).

M. Bertolotti, M. Carnevale, L. Muzii, and D. Sette, "Reversing-Front Interferometer for Phase Correlations Measurements in the Turbulent Atmosphere," Appl. Opt. 9, 510 (1970).

R. C. Ward and G. V. Berry, "Differential-Interferometer Seeing Meter," Appl. Opt. 6, 1136 (1967).

IEEE Trans. Educ. (1)

J. W. Cooley, P. A. W. Lewis, and P. P. Welch, "The Fast Fourier Transform and its Applications," IEEE Trans. Educ. 12, 27 (1969).

J. Opt. Soc. Am. (3)

J. Res. Nat. Bur. Stand. (1)

J. B. Saunders, "A Wavefront Shearing Interferometer," J. Res. Nat. Bur. Stand., 65, 239 (1961).

Proc. Phys. Soc. (1)

W. J. Bates, "A Wavefront Shearing Interferometer," Proc. Phys. Soc. 59, 940 (1947).

Proc. R. Soc. Edinburgh Sect. A (1)

E. T. Whittaker, "On the Functions Which are Represented by the Expansions of the Interpolation Theory," Proc. R. Soc. Edinburgh Sect. A, 35, 181 (1915).

Other (8)

C. E. Shannon, "Communication in the Presence of Noise," Proc. IRE 37, 10 (1949).

D. A. Linden, "A Discussion of Sampling Theorems," Proc. IRE 47, 1219 (1959).

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959).

J. B. Saunders in Advanced Optical Techniques, edited by A. C. S. Van Heel (North Holland, Amsterdam, 1967), pp. 16–20.

R. R. Shannon and W. S. Smith, "An Experiment for Measuring Effect of Atmospheric Turbulence on a Vertical Optical Path," Proc. SPIE 75, 44 (1976).

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

L. C. Martin, Technical Optics II (Sir Isaac Pitman & Son, London, 1961), pp. 330–334.

O. Bryngdahl, in Progress in Optics IV, edited by E. Wolf (North Holland, Amsterdam, 1965), pp. 39–47.

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