Abstract

Chirping introduced into a Michelson interferometer permits correction for nonlinearities, distortion, and intermodulation in the final spectrum. This is because the false harmonics of each frequency carry the characteristic phase of the original frequency, whereas true spectral components at higher frequencies have their own characteristic phase. In an unchirped interferogram, there is no such distinguishing phase. This paper explains how the distinguishing phase occurs, presents two algorithms for carrying out the distortion correction when the form of the distortion is known, and displays results of correcting selected distorted experimental interferograms. This capability is a new advantage of chirping in Fourier spectroscopy.

© 1974 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. P. Sheahen, Chirped Michelson Spectroscopy, (September1971). Copies obtainable from Bell Laboratories or from Defense Documentation Center, Alexandria, Va. 22314.
  2. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  3. H. J. Landau, Bell Syst. Tech. J. 39, 351 (1960).
    [Crossref]
  4. P. Franklin, Differential Equations for Electrical Engineers (Wiley, New York, 1933), p. 275.
  5. G. F. Hohnstreiter, W. R. Howell, and T. P. Sheahen, in Aspen International Conference on Fourier Spectroscopy, 1970, (1971), Ch. 24. Copies obtainable from National Technical Information Service, Springfield, Va.
  6. T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, and I. Coleman, in Ref. 5, Ch. 25.
  7. S. Goldman, Frequency Analysis, Modulation and Noise (McGraw–Hill, New York, 1948). Strictly speaking, the presence of harmonics falls within the category of intermodulation; but in this paper, such distortion is called detector nonlinearity and the term intermodulation is restricted to distortion that originates in the electronics, where neither phase nor amplitude can be calculated.
  8. J. F. McManamen (private communication).
  9. I. Coleman and L. Mertz, Experimental Study Program to Investigate Limits in Fourier Spectroscopy, (National Technical Information Service, Springfield, Va., 1968).
  10. R. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill, New York, 1965). The discussion on p. 110 states the problem quite succinctly.
  11. J. W. Cooley and J. W. Tukey, J. Math. Comput. 19, 297 (1965).
    [Crossref]

1965 (1)

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

1960 (1)

H. J. Landau, Bell Syst. Tech. J. 39, 351 (1960).
[Crossref]

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill, New York, 1965). The discussion on p. 110 states the problem quite succinctly.

Coleman, I.

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

I. Coleman and L. Mertz, Experimental Study Program to Investigate Limits in Fourier Spectroscopy, (National Technical Information Service, Springfield, Va., 1968).

Cooley, J. W.

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

Franklin, P.

P. Franklin, Differential Equations for Electrical Engineers (Wiley, New York, 1933), p. 275.

Goldman, S.

S. Goldman, Frequency Analysis, Modulation and Noise (McGraw–Hill, New York, 1948). Strictly speaking, the presence of harmonics falls within the category of intermodulation; but in this paper, such distortion is called detector nonlinearity and the term intermodulation is restricted to distortion that originates in the electronics, where neither phase nor amplitude can be calculated.

Hohnstreiter, G. F.

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

G. F. Hohnstreiter, W. R. Howell, and T. P. Sheahen, in Aspen International Conference on Fourier Spectroscopy, 1970, (1971), Ch. 24. Copies obtainable from National Technical Information Service, Springfield, Va.

Howell, W. R.

G. F. Hohnstreiter, W. R. Howell, and T. P. Sheahen, in Aspen International Conference on Fourier Spectroscopy, 1970, (1971), Ch. 24. Copies obtainable from National Technical Information Service, Springfield, Va.

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

Landau, H. J.

H. J. Landau, Bell Syst. Tech. J. 39, 351 (1960).
[Crossref]

McManamen, J. F.

J. F. McManamen (private communication).

Mertz, L.

I. Coleman and L. Mertz, Experimental Study Program to Investigate Limits in Fourier Spectroscopy, (National Technical Information Service, Springfield, Va., 1968).

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

Sheahen, T. P.

T. P. Sheahen, Chirped Michelson Spectroscopy, (September1971). Copies obtainable from Bell Laboratories or from Defense Documentation Center, Alexandria, Va. 22314.

G. F. Hohnstreiter, W. R. Howell, and T. P. Sheahen, in Aspen International Conference on Fourier Spectroscopy, 1970, (1971), Ch. 24. Copies obtainable from National Technical Information Service, Springfield, Va.

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

Tukey, J. W.

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

Bell Syst. Tech. J. (1)

H. J. Landau, Bell Syst. Tech. J. 39, 351 (1960).
[Crossref]

J. Math. Comput. (1)

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

Other (9)

T. P. Sheahen, Chirped Michelson Spectroscopy, (September1971). Copies obtainable from Bell Laboratories or from Defense Documentation Center, Alexandria, Va. 22314.

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

P. Franklin, Differential Equations for Electrical Engineers (Wiley, New York, 1933), p. 275.

G. F. Hohnstreiter, W. R. Howell, and T. P. Sheahen, in Aspen International Conference on Fourier Spectroscopy, 1970, (1971), Ch. 24. Copies obtainable from National Technical Information Service, Springfield, Va.

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

S. Goldman, Frequency Analysis, Modulation and Noise (McGraw–Hill, New York, 1948). Strictly speaking, the presence of harmonics falls within the category of intermodulation; but in this paper, such distortion is called detector nonlinearity and the term intermodulation is restricted to distortion that originates in the electronics, where neither phase nor amplitude can be calculated.

J. F. McManamen (private communication).

I. Coleman and L. Mertz, Experimental Study Program to Investigate Limits in Fourier Spectroscopy, (National Technical Information Service, Springfield, Va., 1968).

R. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill, New York, 1965). The discussion on p. 110 states the problem quite succinctly.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Chirped interferogram (top) and spectrum with no corrections applied for distortion (bottom). The false harmonics at 4600 cm−1 arise from the 2300 cm−1 CO2 band, due to nonlinearities in the detector.

Fig. 2
Fig. 2

Enlargement of the region of a spectrum containing false harmonics, similar to Fig. 1. The wiggles at 4600 cm−1 originate in distortion of a strong source component at 2300 cm−1.

Fig. 3
Fig. 3

Amplifier characteristic producing intermodulation and harmonics. Solid line: actual characteristic; dashed line: ideal linear characteristic. The term intermodulation in this paper refers only to distortion originating in such transfer characteristics.

Fig. 4
Fig. 4

Specially limited interferogram (above) and resulting spectrum, designed to isolate false harmonics (below). The trapezoidal line on the chirped interferogram is the limiting profile. By eliminating all but the brightest central fringes, true radiation near 4600 cm−1 was removed, leaving only false harmonics at frequencies above about 3000 cm−1. The residual phase clearly shows high-frequency components to be false.

Fig. 5
Fig. 5

Enlargement of the 4600-cm−1 region of the spectrum of Fig. 4, and its Fourier transform. This spectrum was selectively limited as denoted by the trapezoidal line, and was then Fourier transformed to give the interferogram shown at the bottom. This is the interferogram of the distortion, and is subtracted from the original chirped interferogram in Fig. 1. The interferogram would have been invisible were it not enlarged tenfold.

Fig. 6
Fig. 6

Final interferogram and phase-corrected spectrum resulting from Fig. 1 after the process shown in Figs. 4 and 5 was done. The regular but erroneous phase between 4400 and 4800 cm−1 has been obliterated; the remaining random phase fluctuations show that the S/N ratio is weak in that frequency range.

Fig. 7
Fig. 7

Successive interferograms and final spectrum to illustrate correction procedure. The process of Figs. 1 and 46 was used to obtain this spectrum. Top: original chirped interferogram; second: limited interferogram, retaining central fringes of low frequencies; third: interferogram attributable to distortion (scale expanded 10 times). Bottom: final spectrum corrected for nonlinearity. Because the S/N ratio near 4600 cm−1 is substantial, there is no random residual phase in this case.

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

V ( t ) = B cos ω 0 t
F ( t ) = I ( t ) 1 + b I ( t ) ,
V out ( t ) = V in ( t ) + A sin [ B V in ( t ) ] .
P ( ω ) FT [ g 2 ( t ) ] g 2 ( t ) e - i ω t d t
Q ( ω ) FT [ F 2 ( t ) ] F 2 ( t ) e - i ω t d t .
ϕ ( ω ) n = - 3 5 C n ω n ,
σ ( ω ) 2 φ ( ω / 2 ) - φ ( ω ) .
σ ( ω ) = n = - 3 5 C n ω n ( 1 2 n - 1 - 1 ) .
F ( t ) g ( t ) + α g 2 ( t ) ,
F 2 ( t ) g 2 ( t ) + 2 α g 3 ( t ) .
F ( ω ) = B ( ω ) e - i φ ( ω ) + α P ( ω ) .
A 2 ( ω / 2 ) = 2 P ( ω ) exp [ - i 2 φ ( ω / 2 ) ] .
F ( ω ) = B ( ω ) + ( α / 2 ) A 2 ( ω / 2 ) e - i σ ( ω ) .
φ ( ω ) = tan - 1 ( - sin [ σ ( ω ) ] cos [ σ ( ω ) ] + 2 B ( ω ) / α A 2 ( ω / 2 ) ) .
I ( t ) = A [ I 0 + g ( t ) ] ,
F ( t ) = A [ I 0 + g ( t ) ] 1 + b A [ I 0 + g ( t ) ] .
F ( t ) = A ( I 0 + g ( t ) 1 + b A [ I 0 + g ( t ) ] - ζ 0 ( A , I 0 , b , g ¯ ) ) ,
g ( t ) = F ( t ) / A + ζ 0 1 - b A ( F ( t ) / A + ζ 0 ) - I 0 .
g ( t ) = F ( t ) A ( 1 - b A ζ 0 ) ( 1 + b F ( t ) 1 - b A ζ 0 ) + ζ 0 1 - b A ζ 0 ( 1 + b F ( t ) 1 - b A ζ 0 ) - I 0 = F ( t ) A ( 1 - b A ζ 0 ) 2 ( 1 + b F ( t ) 1 - b A ζ 0 ) + ζ 0 1 - b A ζ 0 - I 0 .
B ( ω ) = 1 A ( 1 - b A ζ 0 ) 2 ( F ( ω ) + b ( 1 - b A ζ 0 ) Q ( ω ) e + i ϕ ( ω ) ) .
ζ 0 = 1 - [ ( 1 + b A I 0 ) 2 - ( b A ) 2 ] - 1 2 b A .
B ( ω ) = 1 A [ ( 1 + b A I 0 ) 2 - ( b A ) 2 ] × [ F ( ω ) + b Q ( ω ) e + i ϕ ( ω ) ] ,
Δ F pp = 2 A ( 1 + b A I 0 ) 2 - ( b A ) 2 .
B ( ω ) = 2 [ F ( ω ) Δ F pp + b ( 2 A Δ F pp ) 1 2 Q ( ω ) Δ F pp e + i φ ( ω ) ] .
F ( ω ) = B ( ω ) exp { i [ ϕ ( ω ) + θ ( ω ) ] } + α 1 2 A 2 ( ω / 2 ) exp { i [ 2 ϕ ( ω 2 ) + θ ( ω ) ] } .
Q ( ω ) = A 2 ( ω / 2 ) 2 exp { i [ 2 ϕ ( ω 2 ) + 2 θ ( ω 2 ) ] } .
ξ ( ω ) = 2 θ ( ω / 2 ) - θ ( ω ) .
g ( t ) = A 1 cos [ ( ω 0 - δ ) t + ϕ ( ω 0 - δ ) ] + A 2 cos [ ω 0 t + ϕ ( ω 0 ) ] + A 3 cos [ ( ω 0 + δ ) t + ϕ ( ω 0 + δ ) ] + B 1 cos [ ( 2 ω 0 - 2 δ ) t + ϕ ( 2 ω 0 - 2 δ ) ] + B 2 cos [ ( 2 ω 0 - δ ) t + ϕ ( 2 ω 0 - δ ) ] + B 3 cos [ ( 2 ω 0 ) t + ϕ ( 2 ω 0 ) ] + B 4 cos [ ( 2 ω 0 + δ ) t + ϕ ( 2 ω 0 + δ ) ] + B 5 cos [ ( 2 ω 0 + 2 δ ) t + ϕ ( 2 ω 0 + 2 δ ) ] .
ϕ ( ω 0 + δ ) = ϕ ( ω 0 ) + ( ϕ ω | ω 0 ) ( + δ ) , ϕ ( 2 ω 0 - 2 δ ) = ϕ ( 2 ω 0 ) + ( φ ω | 2 ω 0 ) ( - 2 δ ) ,
F ( t ) g ( t ) + α g 2 ( t ) .
g 2 ( t ) = A 1 2 cos 2 [ ( ω 0 - δ ) t + ϕ ( ω 0 ) - δ ϕ 0 ] + A 2 2 cos 2 [ ( ω 0 t ) + ϕ ( ω 0 ) ] + A 3 2 cos 2 [ ( ω 0 + δ ) t + ϕ ( ω 0 ) + δ ϕ 0 ] + 2 A 1 A 2 cos [ ( ω 0 - δ ) t + ϕ ( ω 0 ) - δ ϕ 0 ] cos [ ω 0 t + ϕ ( ω 0 ) ] + 2 A 1 A 3 cos [ ( ω 0 - δ ) t + ϕ ( ω 0 ) - δ ϕ 0 ] × cos [ ( ω 0 + δ ) t + ϕ ( ω 0 ) + δ ϕ 0 ] + 2 A 2 A 3 cos [ ( ω 0 t ) + ϕ ( ω 0 ) ] cos [ ( ω 0 + δ ) t + ϕ ( ω 0 ) + δ ϕ 0 ] + terms in A i B j + terms in B i B j } none of these results in frequencies near 2 ω 0 .
A 1 2 cos 2 ( etc . ) = 1 2 A 1 2 { 1 + cos [ ( 2 ω 0 - 2 δ ) t + 2 ϕ ( ω 0 ) - 2 δ ϕ 0 ] } , A 2 2 cos 2 ( etc . ) = 1 2 A 2 2 { 1 + cos [ ( 2 ω 0 t ) + 2 ϕ ( ω 0 ) ] } , A 3 2 cos 2 ( etc . ) = 1 2 A 3 2 { 1 + cos [ ( 2 ω 0 + 2 δ ) t + 2 ϕ ( ω 0 ) + 2 δ ϕ 0 ] } , 2 A 1 A 2 cos u cos v = A 1 A 2 { cos [ ( 2 ω 0 - δ ) t + 2 ϕ ( ω 0 ) - δ ϕ 0 ] + cos ( - δ t - δ ϕ 0 ) } , 2 A 1 A 3 cos u cos v = A 1 A 3 { cos [ ( 2 ω 0 t ) + 2 ϕ ( ω 0 ) ] + cos ( - 2 δ t - 2 δ ϕ 0 ) } , 2 A 2 A 3 cos u cos v = A 2 A 3 { cos [ ( 2 ω 0 + δ ) t + 2 ϕ ( ω 0 ) + δ ϕ 0 ] + cos ( - δ t - δ ϕ 0 ) } .
F ( t ) = A 1 cos [ ( ω 0 - δ ) t + ϕ ( ω 0 ) - δ ϕ 0 ] + A 2 cos [ ( ω 0 ) t + ϕ ( ω 0 ) ] + A 3 cos [ ( ω 0 + δ ) t + ϕ ( ω 0 ) + δ ϕ 0 ] + { B 1 cos [ ( 2 ω 0 - 2 δ ) t + ϕ ( 2 ω 0 ) - 2 δ ϕ 2 ] + α 1 2 A 1 2 cos [ ( 2 ω 0 - 2 δ ) t + 2 ϕ ( ω 0 ) - 2 δ ϕ 0 ] } + { B 2 cos [ ( 2 ω 0 - δ ) t + ϕ ( 2 ω 0 ) - δ ϕ 2 ] + α A 1 A 2 cos [ ( 2 ω 0 - δ ) t + 2 ϕ ( ω 0 ) - δ ϕ 0 ] } + { B 3 cos [ 2 ω 0 t + ϕ ( 2 ω 0 ) ] + α 1 2 A 2 2 cos [ ( 2 ω 0 ) t + 2 ϕ ( ω 0 ) ] + α A 1 A 3 cos [ ( 2 ω 0 ) t + 2 ϕ ( ω 0 ) ] } + { B 4 cos [ ( 2 ω 0 + δ ) t + ϕ ( 2 ω 0 ) + δ ϕ 2 ] + α A 2 A 3 cos [ ( 2 ω 0 + δ ) t + 2 ϕ ( ω 0 ) + δ ϕ 0 ] } + { B 5 cos [ ( 2 ω 0 + 2 δ ) t + ϕ ( 2 ω 0 ) + 2 δ ϕ 2 ] + α 1 2 A 3 2 cos [ ( 2 ω 0 + 2 δ ) t + 2 ϕ ( ω 0 ) + 2 δ ϕ 0 ] } .
cos ( y + z ) cos [ ( y + γ ) + ( z - γ ) ] = cos ( y + γ ) cos ( z - γ ) - sin ( y + γ ) sin ( z - γ ) ,
F ( t ) = A 1 ( etc . ) + A 2 ( etc . ) + A 3 ( etc . ) + { ( B 1 + α 1 2 A 1 2 cos [ 2 ϕ ( ω 0 ) - ϕ ( 2 ω 0 ) - 2 δ ( ϕ 0 - ϕ 2 ) ] ) cos [ ( 2 ω 0 - 2 δ ) t + ϕ ( 2 ω 0 ) - 2 δ ϕ 2 ] - α 1 2 A 1 2 sin [ 2 ϕ ( ω 0 ) - ϕ ( 2 ω 0 ) - 2 δ ( ϕ 0 - ϕ 2 ) ] × sin [ ( 2 ω 0 - 2 δ ) t + ϕ ( 2 ω 0 ) - 2 δ ϕ 2 ] } + { ( B 2 + α A 1 A 2 cos [ 2 ϕ ( ω 0 ) - ϕ ( 2 ω 0 ) - δ ( ϕ 0 - ϕ 2 ) ] ) cos [ ( 2 ω 0 - δ ) t + ϕ ( 2 ω 0 ) - δ ϕ 2 ] - α A 1 A 2 sin [ 2 ϕ ( ω 0 ) - ϕ ( 2 ω 0 ) - δ ( ϕ 0 - ϕ 2 ) ] × sin [ ( 2 ω 0 - δ ) t + ϕ ( 2 ω 0 ) - δ ϕ 2 ] } + { ( B 3 + α { P 2 } cos [ 2 ϕ ( ω 0 ) - ϕ ( 2 ω 0 ) ] ) × cos [ ( 2 ω 0 ) t + ϕ ( 2 ω 0 ) ] - α ( A 1 A 3 + 1 2 A 2 2 ) sin [ 2 ϕ ( ω 0 ) - ϕ ( 2 ω 0 ) ] × sin [ ( 2 ω 0 ) t + ϕ ( 2 ω 0 ) ] } + { ( B 4 + α A 2 A 3 cos [ 2 ϕ ( ω 0 ) - ϕ ( 2 ω 0 ) + δ ( ϕ 0 - ϕ 2 ) ] ) cos [ ( 2 ω 0 + δ ) t + ϕ ( 2 ω 0 ) + δ ϕ 2 ] - α A 2 A 3 sin [ 2 ϕ ( ω 0 ) - ϕ ( 2 ω 0 ) + δ ( ϕ 0 - ϕ 2 ) ] × sin [ ( 2 ω 0 + δ ) t + ϕ ( 2 ω 0 ) + δ ϕ 2 ] } + { ( B 5 + α ( 1 2 A 3 2 ) cos [ 2 ϕ ( ω 0 ) - ϕ ( 2 ω 0 ) + 2 δ ( ϕ 0 - ϕ 2 ) ] ) cos [ ( 2 ω 0 + 2 δ ) t + ϕ ( 2 ω 0 ) + 2 δ ϕ 2 ] - α ( 1 2 A 3 2 ) sin [ 2 ϕ ( ω 0 ) - ϕ ( 2 ω 0 ) + 2 δ ( ϕ 0 - ϕ 2 ) ] × sin [ ( 2 ω 0 + 2 δ ) t + ϕ ( 2 ω 0 ) + 2 δ ϕ 2 ] } .
σ ( ω ) 2 ϕ ( ω / 2 ) - ϕ ( ω ) 2 ϕ l - ϕ u .
F ( 2 ω 0 - δ ) = B 2 + α A 1 A 2 cos σ - 1 + i ( - α A 1 A 2 ) sin σ - 1
A ( 2 ω 0 - δ ) = B 2 2 + α 2 ( A 1 A 2 ) 2 + 2 α B 2 A 1 A 2 cos σ - 1 ϕ ( 2 ω 0 - δ ) = tan - 1 ( - sin σ - 1 cos σ - 1 + ( B 2 / α A 1 A 2 ) ) .