Abstract

The performance of a standard Michelson interferometer is degraded by disturbances that cause the interferogram signal to be sampled at nonconstant time intervals. A formula that shows how the power spectrum of the random disturbances interacts with the signal to contaminate different regions of the measured spectrum is derived for the spectral noise. The sampling noise does not look conventionally noiselike because it is correlated over large regions of the measured spectrum, and adjustment of the unbalanced background interferogram to match the size of the balanced background interferogram minimizes the sampling-noise amplitude.

© 1999 Optical Society of America

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  1. D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).
  2. D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).
  3. W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).
  4. J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979), p. 98.
  5. P. H. Wirsching, T. L. Paez, H. Ortiz, Random Vibrations Theory and Practice (Wiley-Interscience, John Wiley & Sons, Inc., New York, 1995), p. 354.
  6. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 50–57, 201.
  7. T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, “New sampling method in Fourier spectroscopy,” Jpn. J. Appl. Phys. 14, 688–689 (1972).
  8. A. S. Zachor, “Drive nonlinearities: their effects in Fourier spectroscopy,” Appl. Opt. 16, 1412–1424 (1977).
    [CrossRef] [PubMed]
  9. A. S. Zachor, S. M. Aaronson, “Delay compensation: its effect in reducing sampling errors in Fourier spectroscopy,” Appl. Opt. 18, 68–75 (1979).
    [CrossRef] [PubMed]
  10. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991), p. 102.
  11. E. E. Bell, R. B. Sanderson, “Spectral errors resulting from random sampling-position errors in Fourier transform spectroscopy,” Appl. Opt. 11, 688–689 (1972).
    [CrossRef] [PubMed]
  12. L. Forman, W. H. Steel, G. A. Vanasse, “Correction of asymmetric interferograms obtained in Fourier spectroscopy,” J. Opt. Soc. Am. 50, 59–63 (1966).
    [CrossRef]
  13. D. Cohen, “Performance degradation of a Michelson interferometer when its misalignment angle is a rapidly varying, random time series,” Appl. Opt. 36, 4034–4042 (1997).
    [CrossRef] [PubMed]
  14. D. R. Hearn, “Fourier Transform Interferometry,” , (Lincoln Laboratory, MIT, Cambridge, Mass., 1995), p. 21.
  15. L. Rade, B. Westergren, eds., Beta β Mathematics Handbook, 2nd ed. (CRC Press, Boca Raton, Fla., 1990), p. 175.
  16. P. Haschberger, “Impact of the sinusoidal drive on the instrument line shape function of a Michelson interferometer with rotating retroreflector,” Appl. Spectrosc. 48, 307–315 (1994).
    [CrossRef]

1997 (1)

1994 (1)

1979 (1)

1977 (1)

1972 (2)

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, “New sampling method in Fourier spectroscopy,” Jpn. J. Appl. Phys. 14, 688–689 (1972).

E. E. Bell, R. B. Sanderson, “Spectral errors resulting from random sampling-position errors in Fourier transform spectroscopy,” Appl. Opt. 11, 688–689 (1972).
[CrossRef] [PubMed]

1966 (1)

L. Forman, W. H. Steel, G. A. Vanasse, “Correction of asymmetric interferograms obtained in Fourier spectroscopy,” J. Opt. Soc. Am. 50, 59–63 (1966).
[CrossRef]

Aaronson, S. M.

Bell, E. E.

Bicknell, W. E.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Bold, D. R.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Burnside, J. W.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Cafferty, M. S.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Candell, L. M.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Chamberlain, J.

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979), p. 98.

Cohen, D.

Cohen, D. L.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Colao, A. A.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Feinstein, H. W.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Filip, A. E.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Forman, L.

L. Forman, W. H. Steel, G. A. Vanasse, “Correction of asymmetric interferograms obtained in Fourier spectroscopy,” J. Opt. Soc. Am. 50, 59–63 (1966).
[CrossRef]

Forman, S. E.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 50–57, 201.

Haschberger, P.

Hearn, D. R.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

D. R. Hearn, “Fourier Transform Interferometry,” , (Lincoln Laboratory, MIT, Cambridge, Mass., 1995), p. 21.

Jimenez, H. J.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Kerekes, J. P.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Malyak, P. H.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Masutani, K.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, “New sampling method in Fourier spectroscopy,” Jpn. J. Appl. Phys. 14, 688–689 (1972).

Miller, R. W.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Mooney, D. L.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Morii, M.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, “New sampling method in Fourier spectroscopy,” Jpn. J. Appl. Phys. 14, 688–689 (1972).

Nishiyama, T.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, “New sampling method in Fourier spectroscopy,” Jpn. J. Appl. Phys. 14, 688–689 (1972).

Ohno, M.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, “New sampling method in Fourier spectroscopy,” Jpn. J. Appl. Phys. 14, 688–689 (1972).

Ortiz, H.

P. H. Wirsching, T. L. Paez, H. Ortiz, Random Vibrations Theory and Practice (Wiley-Interscience, John Wiley & Sons, Inc., New York, 1995), p. 354.

Paez, T. L.

P. H. Wirsching, T. L. Paez, H. Ortiz, Random Vibrations Theory and Practice (Wiley-Interscience, John Wiley & Sons, Inc., New York, 1995), p. 354.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991), p. 102.

Persky, M. J.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Pillsbury, A. D.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Plaut, A. B.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Ryan-Howard, D. P.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Sanderson, R. B.

Scouler, W. J.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Steel, W. H.

L. Forman, W. H. Steel, G. A. Vanasse, “Correction of asymmetric interferograms obtained in Fourier spectroscopy,” J. Opt. Soc. Am. 50, 59–63 (1966).
[CrossRef]

Ura, N.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, “New sampling method in Fourier spectroscopy,” Jpn. J. Appl. Phys. 14, 688–689 (1972).

Vanasse, G. A.

L. Forman, W. H. Steel, G. A. Vanasse, “Correction of asymmetric interferograms obtained in Fourier spectroscopy,” J. Opt. Soc. Am. 50, 59–63 (1966).
[CrossRef]

Weidler, D. E.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Wirsching, P. H.

P. H. Wirsching, T. L. Paez, H. Ortiz, Random Vibrations Theory and Practice (Wiley-Interscience, John Wiley & Sons, Inc., New York, 1995), p. 354.

Yamauchi, T.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, “New sampling method in Fourier spectroscopy,” Jpn. J. Appl. Phys. 14, 688–689 (1972).

Zachor, A. S.

Appl. Opt. (4)

Appl. Spectrosc. (1)

J. Opt. Soc. Am. (1)

L. Forman, W. H. Steel, G. A. Vanasse, “Correction of asymmetric interferograms obtained in Fourier spectroscopy,” J. Opt. Soc. Am. 50, 59–63 (1966).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, “New sampling method in Fourier spectroscopy,” Jpn. J. Appl. Phys. 14, 688–689 (1972).

Other (9)

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991), p. 102.

D. R. Hearn, “Fourier Transform Interferometry,” , (Lincoln Laboratory, MIT, Cambridge, Mass., 1995), p. 21.

L. Rade, B. Westergren, eds., Beta β Mathematics Handbook, 2nd ed. (CRC Press, Boca Raton, Fla., 1990), p. 175.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979), p. 98.

P. H. Wirsching, T. L. Paez, H. Ortiz, Random Vibrations Theory and Practice (Wiley-Interscience, John Wiley & Sons, Inc., New York, 1995), p. 354.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 50–57, 201.

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Figures (4)

Fig. 1
Fig. 1

Signal diagram shows the flow of information from the input scene radiance entering the FTIR system to the digital signal leaving the FTIR system. Although the Michelson interferometer is drawn with flat return mirrors, the equations derived in this paper still hold true when corner cubes are used to return the signal to the beam splitter.

Fig. 2
Fig. 2

Top curve: simulated spectrum shows two noise-free Lorentz emission lines. Middle three curves: simulated spectra show the same two emission lines taken from interferograms contaminated by sampling noise where the sampling errors have standard deviations of 4.4E-5, 2.2E-5, and 1.1E-5 cm. Bottom curve: simulated spectrum shows the two emission lines taken from an interferogram contaminated by additive white noise. The magnitude of the white noise is adjusted to give a rms spectral error that is the same size as the rms spectral error in the curve with sampling errors having a standard deviation of 4.4E-5.

Fig. 3
Fig. 3

Dotted curves are eight simulated FTIR spectral measurements of the same two emission lines shown in Fig. 2. The power spectrum of the sampling noise contaminating these eight curves is nonzero only for 30 cm-1 ≤ |σ| ≤ 40 cm-1, so its primary effect is to produce ghosts at wave-number offsets of approximately 35 cm-1 from the two emission lines. The solid curve is the standard deviation of the sampling error as a function of the wave number.

Fig. 4
Fig. 4

Boxes show the calculated variances from 1200 ghost curves generated from the same sampling-noise power spectrum used in Fig. 3. Standard statistical theory suggests that approximately two thirds of the boxes should lie between the two solid curves.

Equations (112)

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

zt=0 Btotσcos2πσutdσ,
Btotσ=1/2AdησRστaσΩscτfσBscσ+ΩfBfσ-ΩaBaσ,
x=ut,
f=uσ,
Zx=zx/u=-dσZσexp2πiσx,
Zσ=Zsσ+Zbσ,
Zsσ=1/4AdΩscη|σ|R|σ|τa|σ|τf|σ|Bsc|σ|,
Zbσ=1/4Adη|σ|R|σ|τa|σ|ΩfBf|σ|-ΩaBa|σ|.
St=ht*zt=ht*Zut,
ft*gt=- ftgt-tdt=- ft-tgtdt.
Hf=-dthtexp-2πift
ht=0  for t<0
Gx=Sx/u=-dxu hx-xuZx,
Gσ=-exp-2πiσxGxdx=HuσZσ.
Gmσ=- Πx, LGxexp-2πiσxdx,
Πx, L=1for |x|L0for |x|>L,
Gmσ=2L sinc2πσL*Gσ=2L sinc2πσL*HuσZσ,
sincz=sinz/z,
2L sinc2πσL=-dxΠx, Lexp±2πiσx.
δz=-exp±2πiyzdy,
ab δz-z0fz=fz0when z0 lies betweena and b0when z0 does not liebetween a and b,
xj=jΔx  for j=0, ±1, ±2,.
GmNxj=Gmxj+rxj=Gmxj+nsxj,
nsxrxGmIx,
GmIx=Πx, LdGdx=2πiΠx, L-Gσσ exp2πiσxdσ.
GmNAx=sincπxΔx*j=- GmNxjδx-xj=sincπxΔx*GmNxj=- δx-xj,
GmNx=Gmx+nsx.
GmNσ=- GmNxexp-2πiσxdx=Gmσ+nsσ,
nsσ=- Πx, Lnsxexp-2πiσxdx.
j=- δx-xj=j=- δx-jΔx
-dx exp-2πiσxj=- δx-xj=-dx exp-2πiσxj=- δx-jΔx=1Δxk=- δσ-kΔx.
GmNAσ=Πσ, 12ΔxGmNσ* k=- δσ-kΔx=GmAσ+nsAσ,
GmAσ=Πσ, 12Δxk=-Gmσ-kΔx,
nsAσ=Πσ, 12Δxk=-nsσ-kΔx.
Zσ=Gσ=0  for |σ|>σb, Zσ=Gσ=0  for |σ|<σa.
GmAσGmσGσ  for σa|σ|σb<12Δx=σNy.
Erx=0.
Erxrx=Rrx-x=Erxrx=Rrx-x,
Rrx=Rr-x.
Srσ=- Rrxexp-2πiσxdx,
Rrx=-Srσexp2πiσxdσ,
Sr-σ=Srσ,
ImSrσ=0.
Srσ0.
Sr is negligible for |σ|>σr,
σb+σr<1/2Δx=σNy.
GmσHuσZmσ,
Zmσ=2L sinc2πσL*Zσ.
GmσHuσZmbσ+HuσZmsσ,
Zmsσ=2L sinc2πσL*Zsσ,
Zmbσ=2L sinc2πσL*Zbσ.
GmNAσ=HuσZmbσ+HuσZmsσ+nsAσ.
EnsAσ=Πσ, 12Δxk=- Ensσ-kΔx=Πσ, 12Δxk=-- Πx, LEnsx×exp-2πiσxdx=0,
EGmNAσ=HuσZmbσ+HuσZmsσ.
EGmNAσ|space=HuσZmbσ.
GmNAσ-HuσZmbσ=HuσZmsσ+nsAσ.
PθσGmNAσ-HuσZmbσ=HuσZmsσ+PθσnsAσ,
ns-σ=nsσ*.
nsA-σ=Πσ, 12Δxk=-ns-σ-kΔx=Πσ, 12Δxk=-nsσ-kΔx*=nsAσ*.
|nσ|2+|nσ|2=|nsAσ|2,
E|nσ|2+E|nσ|2=E|nsAσ|2.
E|nσ|2=E|nσ|2=E|PθσnsAσ|2.
E|PθσnsAσ|2=1/2E|nsAσ|2.
NSR2=|Huσ|Zmsσ-11/2E|nsAσ|21/2.
NSR1=|Huσ|Zmsσ-1E|nsAσ|21/2.
ts=2L/ufor a double-sided interferogramL/ufor a single-sided interferogram,
NSR=Luts1/2E|nsAσ|21/2|Huσ|Zmsσ.
E|nsAσ|2E|nsσ|2=4π2-dσSrσ×σGσ*2L sinc2πσL|σ=σ-σ×σGσ**2L sinc2πσL|σ=σ-σ4π2Srσ*σ2|Huσ|2Zσ2.
NSR=2πLuts1/2|Huσ|Zsσ-1×Srσ*σ2|Huσ|2Zσ21/2.
NEdNBscσ=NSR,
NEdN=2πLuts1/2(Srσ*σ|Huσ|η|σ|R|σ|τa|σ|2Ωscτf|σ|Bsc|σ|+ΩfBf|σ|-ΩaBa|σ|2)1/2Ωscη|σ|R|σ|τa|σ|τf|σ||Huσ|.
Πσ, σ0, σM=0for |σ|<σ01for σ0|σ|σ0+σM0for |σ|>σ0+σM,
Erxrx=Rr0=-Srσdσ.
Srσr22σM Πσ, σ0, σM,
r2=Erxrx=Er2.
NEdN=π2Lr2utsσM1/2×(Π(σ,σ0,σM)*σ|Huσ|η|σ|R|σ|τa|σ|2Ωscτf|σ|Bsc|σ|+ΩfBf|σ|-ΩaBa|σ|2)1/2Ωscη|σ|R|σ|τa|σ|τf|σ||Huσ|.
NEdN2πLr2uts1/21Ωscτf×12σrσ-σrσ+σrdσσ2ΩscτfBsc|σ|+ΩfBf|σ|-ΩaBa|σ|21/22π|σ|Lr2uts1/212σrσ-σrσ+σrdσBsc|σ|21/2,
NEdNπ2Lr2uts1/21Ωscτf1σwσ+σ0σ+σ0+σwdσσ2×ΩscτfBsc|σ|+ΩfBf|σ|-ΩaBa|σ|2+1σwσ-σ0-σwσ-σ0dσσ2ΩscτfBsc|σ|+ΩfBf|σ|-ΩaBa|σ|21/2.
NEdNghostπ|σc|2Lr2uts1/2×1σwσc-σw/2σc+σw/2 Bsc|σ|2dσ1/2,
ΩfBfσΩaBaσ,
EnsAσ1nsAσ2*4π2-dσSrσσ1-σ×Gσ1-σσ2-σGσ2-σ*.
Bf=Ba=0,
η=τa=τf=1,
14 ΩscAdR|σ|Bsc|σ|=S0|σ|-σ12+α12+S0|σ|-σ22+α22
S0=1.0 signal units/cm,σ1=1000 cm-1,  σ2=1100 cm-1,α1=2 cm-1,  α2=3 cm-1.
Srσ=Σr2600 cm-1Πσ, 0, 300 cm-1,
Srσ=1×10-9 cm220 cm-1Πσ, 30 cm-1, 10 cm-1.
NEdNghostπ 1000 cm-11×10-9 cm21/2 ×110 cm-1995 cm-11005 cm-1 Bscσ2dσ1/20.0138
2L sinc2πσL*HuσZσHuσ2L sinc2πσL*Zσ,
ILσ=2L sinc2πσL*HuσZσ,
IRσ=Huσ2L sinc2πσL*Zσ,
ILσ=-dx exp-2πiσxΠx, L×-dxu hx-xuZx,
IRσ=-dx exp-2πiσx×-dxu hx-xuΠx, LZx,
ILσ-LLdx exp-2πiσx×-L-2ΔxLdxu hx-xuZx,
IRσ-LL+2Δxdx exp-2πiσx×-LLdxu hx-xuZx.
EnsAσ1nsAσ2*=j=-k=- Ensσ1+jΔx×nsσ2+kΔx*Ensσ1nsσ2*,
Ensσ1nsσ2*=4π2- Πx, Ldx-Πx, Ldx×- σGσdσ- σGσ*dσ×exp2π xσ-σ1-xσ-σ2Rrx-x=4π2-dσSrσKmσ1-σKmσ2-σ*,
Kmσ=-dσσGσ2L sinc2πLσ-σ=σGσ*2L sinc2πσL.
KmσσGσ.
EnsAσ1nsAσ2*=4π2-dσSrσKmσ1-σ×Kmσ2-σ*+4π2-dσSrσKmσ1-σ×k=±1,±2,Kmσ2-σ+kΔx*+4π2-dσSrσKmσ2-σ*×j=±1,±2,Kmσ1-σ+jΔx+4π2-dσSrσ×j=±1,±2,Kmσ1-σ+jΔx×k=±1,±2,Kmσ2-σ+kΔx*.
k=±1,±2,Kmσ1,2-σ+kΔx=-dσσGσ×k=±1,±2,sin2πLσ1,2-σ+kΔx-σπσ1,2-σ+kΔx-σ.
k=±1,±2,Kmσ1,2-σ+kΔx=1π-dσσGσsin2πLσ1,2-σ-σ×k=±1,±2,-1Nkσ1,2-σ-σ+kΔx=2Δx2π-dσσσ+σ-σ1,2Gσsin2πL×σ1,2-σ-σgNΔxσ+σ-σ1,2,
k=±1,±2,-1Nks+kΔx=k=1-1Nk1s+kΔx+1s-kΔx =-2sΔx2k=1-1Nkk2-s2Δx2,
gNs=k=1-1Nkk2-s2.
|σ-σ-σ1,2|2σb+σrσb+12Δx<1Δx,
|gNΔxσ+σ-σ1,2| =k=1-1Nk1-σ+σ-σ1,22Δx2k21-σ+σ-σ1,22Δx2k2×11-σ+σ-σ1,22Δx2|1-σ+σ-σ1,22Δx2|-1k=11k2=π261-σ+σ-σ1,22Δx2|-1,
1-σ+σ-σ1,22Δx2/1-σ+σ-σ1,22Δx2/k21  for k=1, 2, .
k=±1,±2,Kmσ1,2-σ+kΔxπΔx3|1-2σb+σr2Δx2|-1-dσ|σGσ|=π3|1-2σb+σr2/4σNy2|-1σb-σaσNy |σG|,
|σG|=12σb-σa-σb-σa|σGσ|dσ+σaσb|σGσ|dσ.
dgNds=2s k=1-1Nkk2-s22,
SN=k=1-1Nkk2-s22=0.
k=±1,±2,Kmσ1,2-σ+kΔx

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