Abstract

Chirping is the deliberate dispersion of the frequencies in a signal to remove a strong central peak. In a Fourier spectrometer, chirping improves dynamic range. For typical applications, the improvement is equivalent to about 16 dB in SNR. A very large nonlinear phase correction is required, but this is shown to be surprisingly simple to achieve in practice.

© 1974 Optical Society of America

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References

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  1. J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, Bell Syst. Tech. J. 39, 745 (1960).
  2. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  3. T. P. Sheahen, J. Opt. Soc. Am. 64, 485 (1974).
    [CrossRef]
  4. T. P. Sheahen, Appl. Spectrosc. 28, 283 (1974).
    [CrossRef]
  5. W. H. Steel, Interferometry (Cambridge U. P., Cambridge, 1967).
  6. E. V. Loewenstein, Aspen International Conference on Fourier Spectroscopy, 1970; Air Force Cambridge Research Laboratory Special Report 114 (5January1971), Ch. 1.
  7. T. P. Sheahen, “Chirped Fourier Spectroscopy. 2: Theory of Resolution and Contrast” (submitted to Appl. Opt.).
  8. A. A. Michelson, Light Waves and Their Uses (Univ. of Chicago Press, Chicago, 1902, 1961).
  9. M. L. Forman, W. H. Steel, G. A. Vanasse, J. Opt. Soc. Am. 56, 59 (1966).
    [CrossRef]
  10. J. Connes, Rev. Opt. 40, 45, 116, 171, 231 (1961).
  11. P. Bouchareine, P. Connes, J. Phys. Radium 24, 134 (1963).
  12. Chirped interferometry must not be confused with amplitude spectroscopy, which measures the index of refraction of an unknown medium by observing the phase of the spectrum produced by an interferometer containing the unknown in one arm. See E. E. Bell, Ref. 6, Ch. 5.
  13. G. A. Vanasse, H. Sakai, Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1967), vol. 6, Ch. 7.
    [CrossRef]
  14. T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, I. Coleman, Ref. 6, Ch. 25.
  15. R. Curbelo, C. Foskett, Ref. 6, Ch. 21.
  16. J. Connes, P. Connes, J. P. Maillard, J. Phys. (Paris) 28C2, 120 (1967); Atlas des Spectres Planetaires Infrarouges (Editions du CNRS, Paris, 1969).
  17. P. Connes, Ref. 6, Ch. 8.
  18. J. E. Hoffman, Ref. 6, Ch. 15.
  19. R. B. Blackman, J. W. Tukey, Measurement of Power Spectra (Dover, New York, 1959).
  20. W. R. Howell, Digilab, Inc. (Cambridge, Mass.) private communication.
  21. G. F. Hohnstreiter, W. R. Howell, T. P. Sheahen, Ref. 6, Ch. 24.
  22. J. W. Cooley, J. W. Tukey, Math. Comput, 19, 297 (1965).
    [CrossRef]
  23. L. Mertz, Infrared Phys. 7, 17 (1967).
    [CrossRef]
  24. R. Bracewell, Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  25. I. Coleman, L. Mertz, “Experimental Study Program to Investigate Limits in Fourier Spectroscopy,” Block Engineering, Report AFCRL-68-0050 (January1968).
  26. A convenience of FORTRAN IV evades the problem of shifting each point in the reconstructed interferogram by a few points. The data array is slightly overdimensioned (viz., 2060 locations for a 211 = 2048 point interferogram); and typically the first ten points are repeated at the end. This is legitimate since the unchirped interferogram is the FFT of 1024 complex eigenvalues and is periodic over 2048 points. Then, if the central fringe is found at location 7, the FFT subroutine is called with the argument X(7). In this way, the change of an address in the computer in effect performs a rotation in frequency space, showing a rather unexpected relationship between computers and mathematical operations.
  27. M. F. A’Hearn, F. J. Ahern, D. M. Zipoy, Appl. Opt. 13, 1147 (1974).
    [CrossRef] [PubMed]
  28. A. Yariv, Quantum Electronics (Wiley, New York, 1967).
  29. A. S. Filler, Ref. 6, Ch. 42.

1974 (3)

1967 (2)

L. Mertz, Infrared Phys. 7, 17 (1967).
[CrossRef]

J. Connes, P. Connes, J. P. Maillard, J. Phys. (Paris) 28C2, 120 (1967); Atlas des Spectres Planetaires Infrarouges (Editions du CNRS, Paris, 1969).

1966 (1)

1965 (1)

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

1963 (1)

P. Bouchareine, P. Connes, J. Phys. Radium 24, 134 (1963).

1961 (1)

J. Connes, Rev. Opt. 40, 45, 116, 171, 231 (1961).

1960 (1)

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, Bell Syst. Tech. J. 39, 745 (1960).

A’Hearn, M. F.

Ahern, F. J.

Albersheim, W. J.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, Bell Syst. Tech. J. 39, 745 (1960).

Bell, E. E.

Chirped interferometry must not be confused with amplitude spectroscopy, which measures the index of refraction of an unknown medium by observing the phase of the spectrum produced by an interferometer containing the unknown in one arm. See E. E. Bell, Ref. 6, Ch. 5.

Blackman, R. B.

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

Bouchareine, P.

P. Bouchareine, P. Connes, J. Phys. Radium 24, 134 (1963).

Bracewell, R.

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

Coleman, I.

T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, I. Coleman, Ref. 6, Ch. 25.

I. Coleman, L. Mertz, “Experimental Study Program to Investigate Limits in Fourier Spectroscopy,” Block Engineering, Report AFCRL-68-0050 (January1968).

Connes, J.

J. Connes, P. Connes, J. P. Maillard, J. Phys. (Paris) 28C2, 120 (1967); Atlas des Spectres Planetaires Infrarouges (Editions du CNRS, Paris, 1969).

J. Connes, Rev. Opt. 40, 45, 116, 171, 231 (1961).

Connes, P.

J. Connes, P. Connes, J. P. Maillard, J. Phys. (Paris) 28C2, 120 (1967); Atlas des Spectres Planetaires Infrarouges (Editions du CNRS, Paris, 1969).

P. Bouchareine, P. Connes, J. Phys. Radium 24, 134 (1963).

P. Connes, Ref. 6, Ch. 8.

Cooley, J. W.

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

Curbelo, R.

R. Curbelo, C. Foskett, Ref. 6, Ch. 21.

Darlington, S.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, Bell Syst. Tech. J. 39, 745 (1960).

Filler, A. S.

A. S. Filler, Ref. 6, Ch. 42.

Forman, M. L.

Foskett, C.

R. Curbelo, C. Foskett, Ref. 6, Ch. 21.

Hoffman, J. E.

J. E. Hoffman, Ref. 6, Ch. 15.

Hohnstreiter, G. F.

T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, I. Coleman, Ref. 6, Ch. 25.

G. F. Hohnstreiter, W. R. Howell, T. P. Sheahen, Ref. 6, Ch. 24.

Howell, W. R.

G. F. Hohnstreiter, W. R. Howell, T. P. Sheahen, Ref. 6, Ch. 24.

T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, I. Coleman, Ref. 6, Ch. 25.

W. R. Howell, Digilab, Inc. (Cambridge, Mass.) private communication.

Klauder, J. R.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, Bell Syst. Tech. J. 39, 745 (1960).

Loewenstein, E. V.

E. V. Loewenstein, Aspen International Conference on Fourier Spectroscopy, 1970; Air Force Cambridge Research Laboratory Special Report 114 (5January1971), Ch. 1.

Maillard, J. P.

J. Connes, P. Connes, J. P. Maillard, J. Phys. (Paris) 28C2, 120 (1967); Atlas des Spectres Planetaires Infrarouges (Editions du CNRS, Paris, 1969).

Mertz, L.

L. Mertz, Infrared Phys. 7, 17 (1967).
[CrossRef]

I. Coleman, L. Mertz, “Experimental Study Program to Investigate Limits in Fourier Spectroscopy,” Block Engineering, Report AFCRL-68-0050 (January1968).

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

Michelson, A. A.

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

Price, A. C.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, Bell Syst. Tech. J. 39, 745 (1960).

Sakai, H.

G. A. Vanasse, H. Sakai, Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1967), vol. 6, Ch. 7.
[CrossRef]

Sheahen, T. P.

T. P. Sheahen, Appl. Spectrosc. 28, 283 (1974).
[CrossRef]

T. P. Sheahen, J. Opt. Soc. Am. 64, 485 (1974).
[CrossRef]

T. P. Sheahen, “Chirped Fourier Spectroscopy. 2: Theory of Resolution and Contrast” (submitted to Appl. Opt.).

T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, I. Coleman, Ref. 6, Ch. 25.

G. F. Hohnstreiter, W. R. Howell, T. P. Sheahen, Ref. 6, Ch. 24.

Steel, W. H.

M. L. Forman, W. H. Steel, 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, J. W. Tukey, Math. Comput, 19, 297 (1965).
[CrossRef]

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

Vanasse, G. A.

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

G. A. Vanasse, H. Sakai, Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1967), vol. 6, Ch. 7.
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1967).

Zipoy, D. M.

Appl. Opt. (1)

Appl. Spectrosc. (1)

Bell Syst. Tech. J. (1)

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, Bell Syst. Tech. J. 39, 745 (1960).

Infrared Phys. (1)

L. Mertz, Infrared Phys. 7, 17 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. (Paris) (1)

J. Connes, P. Connes, J. P. Maillard, J. Phys. (Paris) 28C2, 120 (1967); Atlas des Spectres Planetaires Infrarouges (Editions du CNRS, Paris, 1969).

J. Phys. Radium (1)

P. Bouchareine, P. Connes, J. Phys. Radium 24, 134 (1963).

Math. Comput (1)

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

Rev. Opt. (1)

J. Connes, Rev. Opt. 40, 45, 116, 171, 231 (1961).

Other (19)

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

I. Coleman, L. Mertz, “Experimental Study Program to Investigate Limits in Fourier Spectroscopy,” Block Engineering, Report AFCRL-68-0050 (January1968).

A convenience of FORTRAN IV evades the problem of shifting each point in the reconstructed interferogram by a few points. The data array is slightly overdimensioned (viz., 2060 locations for a 211 = 2048 point interferogram); and typically the first ten points are repeated at the end. This is legitimate since the unchirped interferogram is the FFT of 1024 complex eigenvalues and is periodic over 2048 points. Then, if the central fringe is found at location 7, the FFT subroutine is called with the argument X(7). In this way, the change of an address in the computer in effect performs a rotation in frequency space, showing a rather unexpected relationship between computers and mathematical operations.

Chirped interferometry must not be confused with amplitude spectroscopy, which measures the index of refraction of an unknown medium by observing the phase of the spectrum produced by an interferometer containing the unknown in one arm. See E. E. Bell, Ref. 6, Ch. 5.

G. A. Vanasse, H. Sakai, Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1967), vol. 6, Ch. 7.
[CrossRef]

T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, I. Coleman, Ref. 6, Ch. 25.

R. Curbelo, C. Foskett, Ref. 6, Ch. 21.

P. Connes, Ref. 6, Ch. 8.

J. E. Hoffman, Ref. 6, Ch. 15.

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

W. R. Howell, Digilab, Inc. (Cambridge, Mass.) private communication.

G. F. Hohnstreiter, W. R. Howell, T. P. Sheahen, Ref. 6, Ch. 24.

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

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

E. V. Loewenstein, Aspen International Conference on Fourier Spectroscopy, 1970; Air Force Cambridge Research Laboratory Special Report 114 (5January1971), Ch. 1.

T. P. Sheahen, “Chirped Fourier Spectroscopy. 2: Theory of Resolution and Contrast” (submitted to Appl. Opt.).

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

A. Yariv, Quantum Electronics (Wiley, New York, 1967).

A. S. Filler, Ref. 6, Ch. 42.

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Figures (4)

Fig. 1
Fig. 1

Optical retardation introduced by typical chirping plates. Each frequency’s point of stationary phase is defined as the location of its central fringe within the interferogram.

Fig. 2
Fig. 2

Actual experimental data from the strongly chirped channel of the interferometer. Top: original chirped interferogram. Center: phase corrected spectrum. Bottom: reconstructed unchirped interferogram. The origin of the bottom interferogram has been shifted outward from the left-hand for ease of visualization. Also, the vertical scale of the bottom interferogram has been greatly compressed. Its central peak is much higher than that of the top interferogram. The requirement that the total power be the same in both cases suggests the extent of shrinkage in the unchirped plot

Fig. 3
Fig. 3

Defective phase correction; a discontinuity of +2π was misinterpreted as −2π.

Fig. 4
Fig. 4

Almost correct phase correction; slight error due to weighting phases of spectral points in proportion to their amplitudes in a least-squares determination of the linear phase correction term.

Equations (4)

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I ( x ) = 0 B ( ω ) { 1 + cos [ ω x + φ ( ω ) ] } d ω ,
φ ( ω ) = ω [ [ n 2 ( z , ω ) 1 ] d z [ n 1 ( y , ω ) 1 ] d y ] ,
φ f ( ω ) = n = 3 5 C n ω n
D c ( ω ) | D ( ω ) | exp [ i φ ( ω ) ] .

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