Abstract

In a Michelson interferometer used as a Fourier spectrometer, whenever the detector system is nonlinear, then the self-convolution of the spectrum occurs enroute to the correction for distortion. In that event, it is important to treat phase data carefully, especially when the interferogram is asymmetric, as in the commonplace cases of imperfect compensation or mild chirping due to electronics and filters. We investigate the distinction between the inverse transform of the interferogram and the original spectrum with phase; in general, asymmetric interferograms show a phase discrepancy that becomes important whenever the self-convolution of the spectrum is involved. Examples of correct and erroneous calculations illustrate this point.

© 1975 Optical Society of America

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References

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  1. J. Connes, Rev. d’Optique 40, 45 (1961); Rev. d’Optique 40, 116 (1961); Rev. d’Optique 40, 171 (1961); Rev. d’Optique 40, 231 (1961).
  2. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  3. A. A. Michelson, Light Waves and Their Uses (University of Chicago Press, Chicago, 1902, 1961).
  4. E. V. Loewenstein, Appl. Opt. 5, 846 (1966).
    [Crossref]
  5. J. W. Cooley and J. W. Tukey, J. Math. Comput. 19, 297 (1965).
    [Crossref]
  6. G. D. Bergland, IEEE Spectrum 6, 41 (1969).
    [Crossref]
  7. E. V. Loewenstein, in Aspen International Conference on Fourier Spectroscopy, 1970, (1971), Ch. 1. Copies obtainable from National Technical Information Service, Springfield, Va.
  8. T. P. Sheahen, J. Opt. Soc. Am. 64, 485 (1974).
    [Crossref]
  9. R. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill, New York, 1965).
  10. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sec. 10.2.
  11. J. Connes, Spectroscopic Studies Using Fourier Transforms, translated by C. A. Flanagan, U. S. Naval Ordnance Test Station, China Lake, Calif. (January1963). Copies obtainable from National Technical Information Service, Springfield, Va.
  12. R. B. Sanderson, in Molecular Spectroscopy: Modern Research, edited by K. N. Rao and C. W. Mathews (Academic, New York, 1972), p. 301.
  13. W. H. Steel, Interferometry (Cambridge U. P., Cambridge, 1967).
  14. M. L. Forman, W. H. Steel, and G. A. Vanasse, J. Opt. Soc. Am. 56, 59 (1966).
    [Crossref]
  15. H. Sakai, G. A. Vanasse, and M. L. Forman, J. Opt. Soc. Am. 58, 84 (1968).
    [Crossref]
  16. G. A. Vanasse and H. Sakai, in Progress in Optics, Vol. 6, edited by E. Wolf (North–Holland, Amsterdam, 1967), Ch. 7.
    [Crossref]
  17. R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).
  18. H. Sakai and G. A. Vanasse, J. Opt. Soc. Am. 56, 131 (1966); W. H. Steel and M. L. Forman, J. Opt. Soc. Am. 56, 982 (1966).
    [Crossref]
  19. Following the usual convention, complex quantities are denoted by a circumflex (ˆ).
  20. J. Chamberlain, J. E. Gibbs, and H. A. Gebbie, Infrared Phys. 9, 185 (1969).
    [Crossref]
  21. E. E. Bell, Infrared Phys. 6, 57 (1966).
    [Crossref]
  22. H. Ziegler, Infrared Phys. 15, 19 (1975).
    [Crossref]
  23. Equation (11) is produced by a FFT applied to g2(t), if and only if g(t) is zero outside the range of the sampled interferogram; this is equivalent to requiring that the interferogram be long enough to eliminate concern over resolution.
  24. Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (Dover, New York, 1965).
  25. T. P. Sheahen, Appl. Opt. 13, 2907 (1974); Appl. Opt. 14, 1004 (1975).
    [Crossref] [PubMed]

1975 (1)

H. Ziegler, Infrared Phys. 15, 19 (1975).
[Crossref]

1974 (2)

1969 (2)

G. D. Bergland, IEEE Spectrum 6, 41 (1969).
[Crossref]

J. Chamberlain, J. E. Gibbs, and H. A. Gebbie, Infrared Phys. 9, 185 (1969).
[Crossref]

1968 (1)

1966 (4)

1965 (1)

J. W. Cooley and J. W. Tukey, J. Math. Comput. 19, 297 (1965).
[Crossref]

1961 (1)

J. Connes, Rev. d’Optique 40, 45 (1961); Rev. d’Optique 40, 116 (1961); Rev. d’Optique 40, 171 (1961); Rev. d’Optique 40, 231 (1961).

Bell, E. E.

E. E. Bell, Infrared Phys. 6, 57 (1966).
[Crossref]

Bell, R. J.

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).

Bergland, G. D.

G. D. Bergland, IEEE Spectrum 6, 41 (1969).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sec. 10.2.

Bracewell, R.

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

Chamberlain, J.

J. Chamberlain, J. E. Gibbs, and H. A. Gebbie, Infrared Phys. 9, 185 (1969).
[Crossref]

Connes, J.

J. Connes, Rev. d’Optique 40, 45 (1961); Rev. d’Optique 40, 116 (1961); Rev. d’Optique 40, 171 (1961); Rev. d’Optique 40, 231 (1961).

J. Connes, Spectroscopic Studies Using Fourier Transforms, translated by C. A. Flanagan, U. S. Naval Ordnance Test Station, China Lake, Calif. (January1963). Copies obtainable from National Technical Information Service, Springfield, Va.

Cooley, J. W.

J. W. Cooley and J. W. Tukey, J. Math. Comput. 19, 297 (1965).
[Crossref]

Forman, M. L.

Gebbie, H. A.

J. Chamberlain, J. E. Gibbs, and H. A. Gebbie, Infrared Phys. 9, 185 (1969).
[Crossref]

Gibbs, J. E.

J. Chamberlain, J. E. Gibbs, and H. A. Gebbie, Infrared Phys. 9, 185 (1969).
[Crossref]

Loewenstein, E. V.

E. V. Loewenstein, Appl. Opt. 5, 846 (1966).
[Crossref]

E. V. Loewenstein, in Aspen International Conference on Fourier Spectroscopy, 1970, (1971), Ch. 1. Copies obtainable from National Technical Information Service, Springfield, Va.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Michelson, A. A.

A. A. Michelson, Light Waves and Their Uses (University of Chicago Press, Chicago, 1902, 1961).

Sakai, H.

Sanderson, R. B.

R. B. Sanderson, in Molecular Spectroscopy: Modern Research, edited by K. N. Rao and C. W. Mathews (Academic, New York, 1972), p. 301.

Sheahen, T. P.

Steel, W. H.

M. L. Forman, W. H. Steel, and G. A. Vanasse, J. Opt. Soc. Am. 56, 59 (1966).
[Crossref]

W. H. Steel, Interferometry (Cambridge U. P., Cambridge, 1967).

Tukey, J. W.

J. W. Cooley and J. W. Tukey, J. Math. Comput. 19, 297 (1965).
[Crossref]

Vanasse, G. A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sec. 10.2.

Ziegler, H.

H. Ziegler, Infrared Phys. 15, 19 (1975).
[Crossref]

Appl. Opt. (2)

IEEE Spectrum (1)

G. D. Bergland, IEEE Spectrum 6, 41 (1969).
[Crossref]

Infrared Phys. (3)

J. Chamberlain, J. E. Gibbs, and H. A. Gebbie, Infrared Phys. 9, 185 (1969).
[Crossref]

E. E. Bell, Infrared Phys. 6, 57 (1966).
[Crossref]

H. Ziegler, Infrared Phys. 15, 19 (1975).
[Crossref]

J. Math. Comput. (1)

J. W. Cooley and J. W. Tukey, J. Math. Comput. 19, 297 (1965).
[Crossref]

J. Opt. Soc. Am. (4)

Rev. d’Optique (1)

J. Connes, Rev. d’Optique 40, 45 (1961); Rev. d’Optique 40, 116 (1961); Rev. d’Optique 40, 171 (1961); Rev. d’Optique 40, 231 (1961).

Other (13)

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

A. A. Michelson, Light Waves and Their Uses (University of Chicago Press, Chicago, 1902, 1961).

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

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sec. 10.2.

J. Connes, Spectroscopic Studies Using Fourier Transforms, translated by C. A. Flanagan, U. S. Naval Ordnance Test Station, China Lake, Calif. (January1963). Copies obtainable from National Technical Information Service, Springfield, Va.

R. B. Sanderson, in Molecular Spectroscopy: Modern Research, edited by K. N. Rao and C. W. Mathews (Academic, New York, 1972), p. 301.

W. H. Steel, Interferometry (Cambridge U. P., Cambridge, 1967).

G. A. Vanasse and H. Sakai, in Progress in Optics, Vol. 6, edited by E. Wolf (North–Holland, Amsterdam, 1967), Ch. 7.
[Crossref]

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).

Following the usual convention, complex quantities are denoted by a circumflex (ˆ).

E. V. Loewenstein, in Aspen International Conference on Fourier Spectroscopy, 1970, (1971), Ch. 1. Copies obtainable from National Technical Information Service, Springfield, Va.

Equation (11) is produced by a FFT applied to g2(t), if and only if g(t) is zero outside the range of the sampled interferogram; this is equivalent to requiring that the interferogram be long enough to eliminate concern over resolution.

Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (Dover, New York, 1965).

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Equations (25)

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g ( x ) = 2 0 B ( σ ) cos [ σ x + φ ( σ ) ] d σ ,
B ( σ ) exp { i φ ( σ ) } = - g ( x ) exp { - i σ x } d x
g ( x ) = - B ( σ ) exp { + i [ σ x + φ ( σ ) ] } d σ .
B ( - σ ) = + B ( + σ ) ,
φ ( - σ ) = - φ ( + σ ) ,
Ĝ ( - σ ) = Ĝ * ( σ ) .
g ( x ) = 0 B ( σ ) exp { - i φ ( σ ) } exp { - i σ x } d σ + 0 B ( σ ) exp { i φ ( σ ) } exp { i σ x } d σ ,
g ( x ) = - 0 B ( - σ ) exp { - i φ ( - σ ) } exp { + i σ x } d σ + 0 B ( σ ) exp { i φ ( σ ) } exp { i σ x } d σ .
g ( x ) = - Ĝ ( σ ) exp { i σ x } d σ .
G ( σ ) = { B ( σ ) exp { i φ ( σ ) } , σ > 0 B ( - σ ) exp { - i φ ( - σ ) } , σ < 0.
M ( x ) = g ( x ) + g 2 ( x ) + K g 3 ( x ) + .
P ˆ ( σ ) - g 2 ( x ) exp { - i σ x } d x .
P ˆ ( σ ) = - d η Ĝ ( η ) Ĝ ( σ - η ) .
P ˆ ( σ ) = 0 d η B ( η ) B ( σ + η ) exp { - i φ ( η ) } exp { + i φ ( σ + η ) } + 0 σ d η B ( η ) B ( σ - η ) exp { + i φ ( η ) } exp { + i φ ( σ - η ) } + σ d η B ( η ) B ( η - σ ) exp { + i φ ( η ) } exp { - i φ ( η - σ ) } .
A ( σ ) = - d η B ( η ) B ( η - σ ) exp { + i φ ( η ) } exp { - i φ ( η - σ ) } ,
Ŝ ( σ ) = - d η B ( η ) B ( σ - η ) exp { + i φ ( η ) } exp { + i φ ( σ - η ) } .
P ˆ ( σ ) = exp ( - σ / α ) 2 α { ( 1 + σ α ) + ( σ / α ) 3 3 exp ( + i 2 φ 0 ) } .
B ( σ ) exp { i φ ( σ ) } = 2 α π exp ( - [ σ - σ 0 ] 2 / α 2 ) exp { i φ 0 } .
A ( σ ) = 4 ( 2 π α 2 ) 1 / 2 exp ( - σ 2 / 2 α 2 )
Ŝ ( σ ) = 4 ( 2 π α 2 ) 1 / 2 exp [ - ( 2 σ 0 - σ ) 2 2 α 2 ] exp ( + i 2 φ 0 ) .
P ˆ ( σ ) = 4 ( 2 π α 2 ) 1 / 2 [ exp ( - σ 2 / 2 α 2 ) erfc { ( σ - 2 σ 0 ) / α 2 } + exp ( + i 2 φ 0 ) exp [ - ( σ - 2 σ 0 ) 2 / 2 α 2 ] erf { σ / α 2 } ] .
P ˆ 0 ( σ ) = 4 exp ( - σ 2 / 2 α 2 ) ( 2 π α 2 ) 1 / 2 [ erfc ( σ α / 2 ) + exp ( + i 2 φ 0 ) erf ( σ α 2 ) ] .
P ˆ ( 2 σ 0 ) = 4 ( 2 π α 2 ) 1 / 2 [ exp ( - 2 σ 0 2 / α 2 ) + exp ( i 2 φ 0 ) erf { 2 σ 0 / α } ] .
P ˆ ( 2 σ 0 ) = 4 ( 2 π α 2 ) 1 / 2 [ 0.018 + 0.995 exp ( + i 2 φ 0 ) ] ,
P ˆ ( 1.5 σ 0 ) = 4 ( 2 π α 2 ) 1 / 2 [ 0.05 + 0.65 exp ( + i 2 φ 0 ) ] .