Abstract

When applied to a Fourier spectrometer, chirping raises questions about the resolution and contrast of the device. A theory is presented to show how the nonlinear phase affects the instrument profile and resolution; the penalty for chirping a high resolution interferometer is much smaller than had been believed. An algorithm is presented for recovering contrast; it is shown that the fast Fourier transform is still usable, allowing realization of full contrast. Systems bearing a residual nonlinear phase dispersion (accidental chirping) can take advantage of this theory.

© 1975 Optical Society of America

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References

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  1. J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
    [CrossRef]
  2. J. Connes, Rev. Opt. 40, 45, 116, 171, 231 (1961).
  3. A. A. Michelson, Light Waves and Their Uses (University of Chicago Press, 1902 and 1961).
  4. G. D. Bergland, IEEE Spectrum 6, 41 (1969).
    [CrossRef]
  5. J. Connes, Aspen International Conference on Fourier Spectroscopy, 1970 (Air Force Cambridge Research Laboratory Special Report No. 114, January5, 1971), Chap. 6.
  6. T. P. Sheahen, Appl. Opt. 13, 2907 (1974).
    [CrossRef] [PubMed]
  7. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  8. M. L. Forman, W. H. Steel, G. A. Vanasse, J. Opt. Soc. Am. 56, 59 (1966).
    [CrossRef]
  9. R. B. Sanderson, E. E. Bell, Appl. Opt. 12, 266 (1973).
    [CrossRef] [PubMed]
  10. W. H. Steel, Interferometry (Cambridge U. P., 1967).
  11. G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1967), Vol. 6, Chap. 7.
    [CrossRef]
  12. J. Chamberlain, Infrared Phys. 11, 25 (1971).
    [CrossRef]
  13. R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic Press, New York, 1972).
  14. Should the detector begin to saturate under high intensity, D must be written as D(ω,I), in which case we have an intractable integral equation for all but the simplest spectra. For a discussion of ways to treat saturation in a chirped interferometer, see T. P. Sheahen, J. Opt. Soc. Am. 64, 485 (1974).
    [CrossRef]
  15. The use of aliasing has its own difficulties. See D. A. Walmsley, T. A. Clark, R. E. Jennings, Appl. Opt. 11, 1148 (1972).
    [CrossRef] [PubMed]
  16. An equivalent approach is to add N extra zeros on the right of the digitized interferogram and perform a FFT to obtain the N nonzero real spectral points, ignoring the N imaginary points. This second method uses more computer core.
  17. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  18. L. Mertz, Infrared Phys. 7, 17 (1967).
    [CrossRef]
  19. I. Coleman, L. Mertz, “Experimental Study Program to Investigate Limits in Fourier Spectroscopy,” Block Engineering, Report AFCRL-68-0050 (January1968).
  20. Because the frequency dependence of A in Eq. (14) appears only parametrically through x0, it is tempting to invoke the Fourier shift theorem and the convolution theorem to manipulate Eq. (14) into a form suitable for a Fourier transform. However, the fixed upper limit of L prevents this because the compensating apodizer A[x − x0(ωj)] is not simply a sliding boxcar of constant length and shifting center; it is a boxcar that decreases in length as the phase delay [and x0(ωj)] increases.
  21. J. M. Dowling, Ref. 5, Chap. 4.
  22. G. F. Hohnstreiter, W. R. Howell, T. P. Sheahen, Ref. 5, Chap. 24.
  23. T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, I. Coleman, Ref. 5, Chap. 25.
  24. C. Flammer, Spheroidal Wave Functions (Stanford University Press, 1957).
  25. A. M. Despain, J. W. Bell, Ref. 5, Chap. 41.
  26. In retrospect, the start and end positions of the mirror scan might have been shifted to the left.

1974 (2)

1973 (1)

1972 (1)

1971 (1)

J. Chamberlain, Infrared Phys. 11, 25 (1971).
[CrossRef]

1969 (1)

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

1967 (1)

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

1966 (1)

1965 (1)

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

1961 (1)

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

Bell, E. E.

Bell, J. W.

A. M. Despain, J. W. Bell, Ref. 5, Chap. 41.

Bell, R. J.

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

Bergland, G. D.

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

Bracewell, R.

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

Chamberlain, J.

J. Chamberlain, Infrared Phys. 11, 25 (1971).
[CrossRef]

Clark, T. A.

Coleman, I.

T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, I. Coleman, Ref. 5, Chap. 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, Rev. Opt. 40, 45, 116, 171, 231 (1961).

J. Connes, Aspen International Conference on Fourier Spectroscopy, 1970 (Air Force Cambridge Research Laboratory Special Report No. 114, January5, 1971), Chap. 6.

Cooley, J. W.

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

Despain, A. M.

A. M. Despain, J. W. Bell, Ref. 5, Chap. 41.

Dowling, J. M.

J. M. Dowling, Ref. 5, Chap. 4.

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford University Press, 1957).

Forman, M. L.

Hohnstreiter, G. F.

T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, I. Coleman, Ref. 5, Chap. 25.

G. F. Hohnstreiter, W. R. Howell, T. P. Sheahen, Ref. 5, Chap. 24.

Howell, W. R.

G. F. Hohnstreiter, W. R. Howell, T. P. Sheahen, Ref. 5, Chap. 24.

T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, I. Coleman, Ref. 5, Chap. 25.

Jennings, R. E.

Mertz, L.

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

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

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

Michelson, A. A.

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

Sakai, H.

G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1967), Vol. 6, Chap. 7.
[CrossRef]

Sanderson, R. B.

Sheahen, T. P.

Steel, W. H.

Tukey, J. W.

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

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, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1967), Vol. 6, Chap. 7.
[CrossRef]

Walmsley, D. A.

Appl. Opt. (3)

IEEE Spectrum (1)

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

Infrared Phys. (2)

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

J. Chamberlain, Infrared Phys. 11, 25 (1971).
[CrossRef]

J. Opt. Soc. Am. (2)

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 (16)

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

J. Connes, Aspen International Conference on Fourier Spectroscopy, 1970 (Air Force Cambridge Research Laboratory Special Report No. 114, January5, 1971), Chap. 6.

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

G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1967), Vol. 6, Chap. 7.
[CrossRef]

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

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

Because the frequency dependence of A in Eq. (14) appears only parametrically through x0, it is tempting to invoke the Fourier shift theorem and the convolution theorem to manipulate Eq. (14) into a form suitable for a Fourier transform. However, the fixed upper limit of L prevents this because the compensating apodizer A[x − x0(ωj)] is not simply a sliding boxcar of constant length and shifting center; it is a boxcar that decreases in length as the phase delay [and x0(ωj)] increases.

J. M. Dowling, Ref. 5, Chap. 4.

G. F. Hohnstreiter, W. R. Howell, T. P. Sheahen, Ref. 5, Chap. 24.

T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, I. Coleman, Ref. 5, Chap. 25.

C. Flammer, Spheroidal Wave Functions (Stanford University Press, 1957).

A. M. Despain, J. W. Bell, Ref. 5, Chap. 41.

In retrospect, the start and end positions of the mirror scan might have been shifted to the left.

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

An equivalent approach is to add N extra zeros on the right of the digitized interferogram and perform a FFT to obtain the N nonzero real spectral points, ignoring the N imaginary points. This second method uses more computer core.

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

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Figures (5)

Fig. 1
Fig. 1

Simultaneous phase rotation and convolution. Dashed line: real part of profile. Dotted line: imaginary part of profile. Solid line: actual spectrum. Conceptually, as the complex sinc function slides past the spectrum, the spectrum rotates in the complex plane. The rate of rotation depends on the degree of chirping.

Fig. 2
Fig. 2

Response of a strongly chirped interferogram to a synthetic molecular vibration-rotation band with lines spaced by Δω = 4π/L. Top: original chirped interferogram, showing the amount of optical retardation for the frequency of the band center due to chirping. Center: spectrum recovered from interferogram. Bottom: reconstructed unchirped interferogram with no shift in zero position.

Fig. 3
Fig. 3

Interferogram and spectrum of strong CO2 radiation from low density gas containing weak HF radiation. Top: original chirped interferogram. Center: spectrum. Bottom: reconstructed unchirped interferogram. Resolving fringes of the P and R branches of the HF spectrum can be found as described in the text.

Fig. 4
Fig. 4

The relation between location of central fringes and resolving fringes of a band system. The resolving fringes on the right are never reached, so resolution of lines in the H2O spectrum is determined only by the left resolving fringe.

Fig. 5
Fig. 5

Interpolation applied to the H2O region of the spectrum in Fig. 4. Left: contrast obtained using normal boxcar apodization. Right: contrast obtained using compensating apodization.

Equations (20)

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I ( x ) = 2 0 Ω 0 B ( ω ) D ( ω ) { 1 + cos [ ω x + φ ( ω ) ] } d ω .
g ( ω j ) = 1 N x k = 0 L f ( x k ) A ( x k ) exp ( i ω j x k ) .
g ( ω j ) = 2 0 Ω 0 d ω B ( ω ) D ( ω ) { 1 N x k = 0 + L A ( x k ) cos [ ω x k + φ ( ω ) ] × exp ( i ω j x k ) } .
( exp [ + i φ ( ω ) ] 2 N { exp [ i ( ω j + ω ) L ] 1 exp [ i ( ω j + ω ) Δ x ] 1 } + exp [ i φ ( ω ) ] 2 N { exp [ i ( ω j ω ) L ] 1 exp [ i ( ω j ω ) Δ x ] 1 } ) .
g ( ω j ) = 0 Ω 0 d ω B ( ω ) exp [ i φ ( ω ) ] × { exp [ i ( ω j ω ) L ] 1 i ( ω j ω ) L } .
A ( x ) = 2 x / L , 0 < x < L / 2 , A ( x ) = 2 ( 1 x / L ) , L / 2 < x < L ,
g ( ω j ) = 1 2 0 Ω 0 d ω B ( ω ) D ( ω ) exp [ i φ ( ω ) ] × { exp [ i ( ω j ω ) L / 2 ] 1 i ( ω j ω ) L / 2 } 2 ,
{ [ exp ( i Δ ω L ) 1 ] / i Δ ω L } exp ( i ω L / 3 ) = { [ ( sin Δ ω L / 2 ) / ( Δ ω L / 2 ) ] exp ( i Δ ω L / 6 ) } exp ( i ω L / 3 ) .
f ( ξ ω j ) = [ exp i ( ξ ω j ) L 1 ] / i ( ξ ω j ) L
ω j = 0 Ω 0 g ( ω j ) { exp [ i ( ξ ω j ) L ] 1 i ( ξ ω j ) L } = 0 Ω 0 B ( ω ) D ( ω ) exp [ i φ ( ω ) ] { exp [ i ( ξ ω ) L ] 1 i ( ξ ω ) L } d ω
g ( ω ) = B ( ω ) D ( ω ) exp [ i φ ( ω ) ] .
ω j = 0 Ω 0 g ( ω j ) { exp [ i ( ξ ω j ) L ] 1 i ( ξ ω j ) L } = 0 Ω 0 B ( ω ) D ( ω ) exp [ i φ ( ω ) ] { exp [ i ( ξ ω ) L / 2 ] 1 i ( ξ ω ) L / 2 } 2 d ω ,
ω j = 0 Ω 0 { exp [ i ( ω j ω ) L ] 1 i ( ω j ω ) L } exp [ i φ ( ω j ) ] × { exp [ i ( ω ξ ) L ] 1 i ( ω ξ ) L } ,
ω j = 0 Ω 0 g ( ω j ) exp [ i φ ( ω j ) ] { exp [ i ( ω j ξ ) L ] 1 i ( ω j ξ ) L } = 0 Ω 0 B ( ω ) d ( ω ) exp ( i [ φ ( ξ ) φ ( ω ) ] ) × { exp [ i ( ω ξ ) L ] 1 i ( ω ξ ) L } d ω
B ( ω ) = exp { [ ( ω ω 0 ) / α ] 2 } ( 2 + cos γ ω ) .
A ( x ) = 1 / 2 ; L / 3 < x < + L / 3 , A ( x ) = 1 ; + L / 3 < x < + 2 / 3 L .
L / 3 + 2 L / 3 A ( x ) cos [ ω x + φ ( ω ) ] exp ( i ω x ) d x / L
1 Δ ω L { ( sin 2 3 Δ ω L ) + i [ cos 2 3 Δ ω L cos ( 1 3 Δ ω L ) ] } .
1 Δ ω L [ ( sin 2 3 Δ ω L + sin 1 3 Δ ω L ) + i ( cos 2 3 Δ ω L cos 1 3 Δ ω L ) ] .
g ( ω j ) = x k = 0 L f ( x k ) A [ x k x 0 ( ω j ) ] exp ( i ω j x k ) .

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