Abstract

Engineers designing optical alignment servo systems for Michelson interferometers and Fourier-transform infrared spectrometers need to predict the amount of noise expected from the small and randomly varying amounts of misalignment that occur as the servo attempts to maintain alignment while taking data. A formula is derived for the noise-equivalent change in radiance due to this effect and the formula’s accuracy is demonstrated by comparison of its predictions to the errors found in simulated interferometer measurements contaminated by misalignment noise.

© 2003 Optical Society of America

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Errata

Douglas L. Cohen, "Noise-equivalent change in radiance for misalignment noise in a double-sided interferogram: erratum," Appl. Opt. 43, 1221-1222 (2004)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-43-6-1221

References

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  1. J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979), Chap. 8, p. 237.
  2. C. S. Williams, “Mirror misalignment in Fourier spectroscopy using a Michelson interferometer with circular aperture,” Appl. Opt. 5, 1084–1085 (1966).
    [CrossRef] [PubMed]
  3. L. W. Kunz, D. Goorvitch, “Combined effects of a converging beam of light and mirror misalignment in Michelson interferometry,” Appl. Opt. 13, 1077–1079 (1974).
    [CrossRef] [PubMed]
  4. 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]
  5. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991), Chap. 5, p. 102.
  6. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (Institute of Electrical and Electronics Engineers, New York, 1987), Chap. 4, p. 60.
  7. H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the high-resolution interferometer sounder,” Appl. Opt. 27, 3210–3218 (1988).
    [CrossRef] [PubMed]
  8. M. Evans, N. Hastings, B. Peacock, Statistical Distributions, 2nd ed. (Wiley, New York, 1993), Chaps. 2 and 29, pp. 12, 13, and 114.

1997 (1)

1988 (1)

1974 (1)

1966 (1)

Buijs, H.

Chamberlain, J.

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979), Chap. 8, p. 237.

Cohen, D.

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (Institute of Electrical and Electronics Engineers, New York, 1987), Chap. 4, p. 60.

Evans, M.

M. Evans, N. Hastings, B. Peacock, Statistical Distributions, 2nd ed. (Wiley, New York, 1993), Chaps. 2 and 29, pp. 12, 13, and 114.

Goorvitch, D.

Hastings, N.

M. Evans, N. Hastings, B. Peacock, Statistical Distributions, 2nd ed. (Wiley, New York, 1993), Chaps. 2 and 29, pp. 12, 13, and 114.

Howell, H. B.

Kunz, L. W.

LaPorte, D. D.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991), Chap. 5, p. 102.

Peacock, B.

M. Evans, N. Hastings, B. Peacock, Statistical Distributions, 2nd ed. (Wiley, New York, 1993), Chaps. 2 and 29, pp. 12, 13, and 114.

Revercomb, H. E.

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (Institute of Electrical and Electronics Engineers, New York, 1987), Chap. 4, p. 60.

Smith, W. L.

Sromovsky, L. A.

Williams, C. S.

Appl. Opt. (4)

Other (4)

M. Evans, N. Hastings, B. Peacock, Statistical Distributions, 2nd ed. (Wiley, New York, 1993), Chaps. 2 and 29, pp. 12, 13, and 114.

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979), Chap. 8, p. 237.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991), Chap. 5, p. 102.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (Institute of Electrical and Electronics Engineers, New York, 1987), Chap. 4, p. 60.

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Figures (4)

Fig. 1
Fig. 1

(a) Signal diagram showing the flow of information from the input scene radiance entering the FTIR spectrometer to the digital signal leaving the FTIR spectrometer. ZPD stands for zero path difference. (b) The z axis (heavy solid arrow) is the correctly aligned normal vector of the mirror surface and the dashed arrow is the misaligned normal vector. The x and y axes show the orientation of the θ̃ x (χ) and θ̃ y (χ) components of the total θ̃(χ) misalignment angle.

Fig. 2
Fig. 2

(a) and (b) Dashed curves represent function f(σ) and the solid curves represent the convolution of function f(σ) with itself—that is, they represent f(σ) * f(σ). (c) When this plot is taken to be a graph of S xx (σ), the dashed line is at S 0 = γ x 2/(2σ M ); when this plot is taken to be a graph of S θ2(σ), the dashed line is at S 0 = 2ϕ2γ x 2 M .

Fig. 3
Fig. 3

(a) Dark solid curve is the spectral radiance of a 320 K blackbody, and the dotted curves are ten measurements of this spectral radiance contaminated by the misalignment noise characterized by the Gaussian noise power spectrum specified in the text. Several of the dotted curves cannot easily be seen because they are too close to the solid curve. (b) The solid curve is the misalignment NEdN predicted by Eqs. (9f) or (9i) for the interferometer measurements in (a) and the crosses mark the calculated standard deviations for 3600 simulated measurements with misalignment noise.

Fig. 4
Fig. 4

(a) Dark solid curves give the true spectral radiance entering the instrument and the light solid curve gives the misalignment NEdN predicted by Eqs. (9f) and (9i). The dotted curves are ten spectral measurements of the spectral radiance contaminated by misalignment noise characterized by the quasi-harmonic noise power spectrum described in the text. Several of the dotted curves cannot be easily seen (except perhaps inside the two ghost-line regions) because they are too close to the dark solid curves. (b) The solid curve is the misalignment NEdN predicted by Eqs. (9f) and (9i) for the interferometer measurements in (a) and the crosses mark the calculated standard deviations for 3600 simulated measurements with misalignment noise.

Equations (131)

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z˜ANχ=- Mσθ˜χZσexp2πiσχdσ,
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|.
Mσθ˜χ=1-aσθ˜χ2,
a=2π2r2.
θ˜χ=θ˜xχ2+θ˜yχ21/2,
Eθ˜χ2=θrms2
Mσθ˜χ=1-aσ2θrms2+aσ2θrms2-θ˜χ2,
Mσθ˜χ=Mσθrms+aσ2ñθ2χ,
ñθ2χ=θrms2-θ˜χ2.
Eñθ2χ=θrms2-Eθ˜χ2=0.
Eñθ2χñθ2χ=Rθ2χ-χ,
Rθ2χ-χ=Rθ2χ-χ,
Rθ2-χ=Rθ2χ.
F±iνtft=- ftexp±2πiνtdt,
Sθ2σ=F-iσχRθ2χ=-Rθ2χexp-2πiσχdχ,
Rθ2χ=FiσχSθ2σ=-Sθ2σexp2πiσχdσ.
z˜ANχ=zAχ+añθ2χWχ,
zAχ=- MσθrmsZσexp2πiσχdσ=FiσχMσθrmsZσ,
Wχ=- σ2Zσexp2πiσχdσ=Fiσχσ2Zσ.
t=χ/u,
z˜ANut=zAut+añθ2utWut.
ht=- Hfexp2πiftdf=FiftHf,
Hf=- htexp-2πiftdt=F-iftht.
ht0 for |t|>T.
fa, b * gb, d=- gb, dfa, b-bdb.
ht * z˜ANut=ht * zAut+aht * ñθ2utWut.
u-1hχu * z˜ANχ=u-1hχu * zAχ+u-1ahχu * ñθ2χWχ.
-LχL.
Πχ, L=1for |χ|L0for |χ|>L
z˜BNχ=Πχ, L · u-1hχu * z˜ANχ =u-1Πχ, Lhχu * zAχ+u-1aΠχ, L×hχu * ñθ2χWχ.
F-iσχz˜BNχ=u-1F-iσχΠχ, Lhχu * zAχ +u-1aF-iσχΠχ, Lhχu * ñθ2χWχ.
F-iσχz˜BNχHuσMσθrmsZσ +aHuσF-iσχΠχ, Lñθ2χWχ.
F-iσχz˜BNχ
EF-iσχz˜BNχHuσMσθrmsZσ+aHuσ-LL Eñθ2χWχ×exp-2πiσχdχ,
EF-iσχz˜BNχ=HuσMσθrmsZσ.
aHuσF-iσχΠχ, Lñθ2χWχ =aHuσF-iσχΠχ, Lñθ2χ * F-iσχWχ,
aHuσF-iσχΠχ, Lñθ2χWχ=aHuσñθ2Lσ * σ2Zσ,
ñθ2Lσ=- Πχ, Lñθ2χexp-2πiσχdχ=-LL ñθ2χexp-2πiσχdχ,
F-iσχWχ=σ2Zσ.
F-iσχz˜BNχHuσMσθrmsZσ+aHuσ×ñθ2Lσ * σ2Zσ.
G1σ=EF-iσχz˜BNχ|for B1=HuσMσθrmsZs1σ+Zbσ,
G2σ=EF-iσχz˜BNχ|for B2=HuσMσθrmsZs2σ+Zbσ,
Zs1,2σ=1/4AdΩscη|σ|R|σ|×τa|σ|τf|σ|B1,2|σ|,
spectral radiancecontaminated bymisalignment noise=HuσMσθrmsZσ+añθ2L×σ * σ2Zσ-G1σ·B2σ-B1σG2σ-G1σ+B1σ.
spectral radiancecontaminated bymisalignment noise=Bscσ+4añθ2Lσ * σ2ZσAdΩscMσθrmsη|σ|R|σ|τa|σ|τf|σ|.
4añθ2Lσ * σ2ZσAdΩscMσθrmsη|σ|R|σ|τa|σ|τf|σ|
spectral radiancecontaminated bymisalignment noise=Bscσ+4aReñθ2Lσ * σ2ZσAdΩscMσθrmsη|σ|R|σ|τa|σ|τf|σ|.
misalignment noise=4aReñθ2Lσ * σ2ZσAdΩscMσθrmsη|σ|R|σ|τa|σ|τf|σ|.
Eñθ2Lσ=-LL Eñθ2χexp-2πiσχdχ=0,
NEdNtilt=4aɈσAdΩscMσθrmsη|σ|R|σ|τa|σ|τf|σ|,
Ɉσ=EReñθ2Lσ * σ2Zσ2.
Ɉσ=14-Sθ2σσ+σ2Zσ+σ+σ-σ2Zσ-σ2dσ.
ΩscAd=ΩFOVπr2,
aAdΩsc=2π2r2ΩFOV · πr2=2πΩFOV,
NEdNtilt=8πɈσΩFOVMσθrmsη|σ|R|σ|τa|σ|τf|σ|.
Eθ˜xχ=ϕ,
Eθ˜yχ=0.
Eθ˜xχ-ϕ21/2=γx,
Eθ˜yχ21/2=γy.
Eθ˜χ2=Eθ˜xχ2+Eθ˜yχ2=γx2+γy2+ϕ2
θrms2=γx2+γy2+ϕ2.
Eθ˜xχ-ϕθ˜xχ-ϕ=Rxxχ-χ,
Eθ˜yχθ˜yχ=Ryyχ-χ.
Sxxσ=F-iσχRxxχ=-Rxxχexp-2πiσχdχ,
Syyσ=F-iσχRyyχ=-Ryyχexp-2πiσχdχ.
Eθ˜xχ-ϕθ˜yχ=Rxyχ-χ,
Sxyσ=F-iσχRxyχ=-Rxyχexp-2πiσχdχ,
Sθ2σ=2Sxxσ * Sxxσ+2Syyσ * Syyσ+4ϕ2Sxxσ+4 ReSxyσ * Sxyσ.
fσ * fσ=- fσ-σfσdσ=- fσ-σfσdσ.
fσ * fσ|σ=0=- fσ2dσ.
Sθ2σ=α exp-σ2/2s2.
Eñθ2χ2=Rθ20=-Sθ2σdσ,
Eñθ2χ2=Eθrms2-θ˜χ22=Eθ˜χ4-θrms4.
Eñθ2χ2=Eθ˜xχ4+θ˜yχ4+2θ˜xχ2θ˜yχ2-θrms4 =Eθ˜xχ4+Eθ˜yχ4+2Eθ˜xχ2Eθ˜yχ2-θrms4.
Eθ˜xχ4=3γx4+6ϕ2γx2+ϕ4, Eθ˜yχ4=3γy4, Eθ˜xχ2Eθ˜yχ2=ϕ2+γx2γy2.
Eñθ2χ2=3γx4+3γy4+2γx2γy2+2ϕ23γx2+γy2+ϕ4-θrms4,
Eñθ2χ2=2γx4+γy4+2ϕ2γx2.
2γx4+γy4+2ϕ2γx2=-Sθ2σdσ.
2γx4+γy4+2ϕ2γx2=α -exp-σ2/2s2dσ=αs2π,
α=1s2πγx4+γy4+2ϕ2γx2.
Sθ2σ=1s2πγx4+γy4+2ϕ2γx2exp-σ2/2s2
γx2=-Sxxσdσ,
Sθ2σ=4ϕ2Sxxσ.
Baσ=Bfσ=0.
τaσ=τfσ=ησ=1
Rσ=1 amp×scm.
F±iσχΠχ, L=- Πχ, Lexp±2πiσχdχ=2L sinc2πσL,
sincx=sinxx.
F-iσχhχu=u - htexp-2πiσutdt=uHuσ
u-1F-iσχΠχ, Lhχu * zAχ =u-1F-iσχΠχ, L * F-iσχhχu×F-iσχzAχ =2L sinc2πσL * HuσF-iσχzAχ.
u-1F-iσχΠχ, Lhχu * zAχ=2L sinc2πσL * HuσMσθrmsZσ.
u-1F-iσχΠχ, Lhχu * zAχHuσMσθrms2L sinc2πσL * Zσ.
2L sinc2πσL * Zσ=2L sinc2πσL* Ad4 η|σ|R|σ|τa|σ|ΩscBsc|σ|τf|σ|+ΩfBf|σ|-ΩaBa|σ|.
2L sinc2πσL * ZσAd4 η|σ|R|σ|τa|σ|ΩfBf|σ|-ΩaBa|σ|+Ωscτf|σ|×2L sinc2πσL * Bsc|σ|.
2L sinc2πσL * ZσAd4 η|σ|R|σ|τa|σ|×ΩfBf|σ|-ΩaBa|σ|+Ωscτf|σ|Bsc|σ| =Zσ,
u-1F-iσχΠχ, Lhχu * zAχHuσMσθrmsZσ.
Πχ, Lhχu * ñθ2χWχ=Πχ, L- ñθ2χWχhχ-χudχ.
Πχ, Lhχu * ñθ2χWχΠχ, Lχ-uTχ+uT ñθ2χWχhχ-χudχ.
Πχ, Lhχu * ñθ2χWχΠχ, L-L+uTL+uT ñθ2χWχhχ-χudχ.
Πχ, Lhχu * ñθ2χWχΠχ, L- Πχ, L+uT ×ñθ2χWχhχ-χudχ,
Πχ, Lhχu * ñθ2χWχΠχ, L×hχu * Πχ, L+uTñθ2χWχ.
F-iσχΠχ, Lhχu * ñθ2χWχF-iσχΠχ, L * F-iσχhχu×F-iσχΠχ, L+uTñθ2χWχ.
F-iσχΠχ, Lhχu * ñθ2χWχu2L sinc2πσL* HuσF-iσχΠχ, L+uT×ñθ2χWχ.
F-iσχΠχ, Lhχu * ñθ2χWχuHuσ2L sinc2πσL* F-iσχΠχ,L+uT×ñθ2χWχ.
F-iσχΠχ, Lhχu * ñθ2χWχuHuσ×F-iσχΠχ, LΠχ, L+uT×ñθ2χWχ.
Πχ, LΠχ, L+uT=Πχ, L,
F-iσχΠχ, Lhχu * ñθ2χWχuHuσF-iσχΠχ, Lñθ2χWχ.
u-1aF-iσχΠχ, Lhχu * ñθ2χWχaHuσF-iσχΠχ, Lñθ2χWχ,
σ2Zσ=- Wχexp±2πiσχdχ=F±iσχWχ.
Wχ=- σ2Zσexp±2πiσχdσ=F±iσχσ2Zσ.
nθ2Lσ=F-iσχΠχ, Lñθ2χ.
nθ2Lσ * σ2Zσ=F-iσχΠχ, Lñθ2χWχ;
Renθ2Lσ * σ2Zσ2=14F-iσχΠχ, Lñθ2χWχ+F-iσχΠχ, Lñθ2χWχ*2=14F-iσχΠχ, Lñθ2χWχ+FiσχΠχ, Lñθ2χWχ2,
Ɉσ=14 EF-iσχΠχ, Lñθ2χWχ2+14 EFiσχΠχ, Lñθ2χWχ2 +12 EFiσχΠχ, Lñθ2χWχ×F-iσχΠχ, Lñθ2χWχ.
Ɉσ=14T1+T1*+2T2,
T1=EF-iσχΠχ, Lñθ2χWχ2,
T2=EFiσχΠχ, Lñθ2χWχ×F-iσχΠχ, Lñθ2χWχ.
T1=-dχΠχ, LWχexp-2πiσχ×-dχΠχ, LWχexp-2πiσχ×Eñθ2χñθ2χ =-dχΠχ, LWχexp-2πiσχ×-dχΠχ, LWχexp-2πiσχ×Rθ2χ-χ,
T1=-dσSθ2σ ×-dχΠχ, LWχexp-2πiχσ-σ×-dχΠχ, LWχexp-2πiχσ+σ.
- Πχ, LWχexp±2πiσχdχ =F±iσχ×Πχ, LWχ =F±iσχΠχ, L * F±iσχWχ.
- Πχ, LWχexp±2πiσχdχ=2L sinc2πσL * σ2Zσ.
- Πχ, LWχexp±2πiσχdχσ22L sinc2πσL * Zσ.
- Πχ, LWχexp±2πiσχdχσ2Zσ,
T1=-Sθ2σσ-σ2Zσ-σ×σ+σ2Zσ+σdσ.
T2=-dχΠχ, LWχexp2πiσχ×-dχΠχ, LWχexp-2πiσχ×Rθ2χ-χ.
Rθ2χ-χ=12Rθ2χ-χ+12Rθ2χ-χ.
T2=12-dσSθ2σ ×-dχΠχ, LWχexp2πiχσ+σ×-dχΠχ, LWχexp-2πiχσ+σ +12-dσSθ2σ ×-dχΠχ, LWχexp2πiχσ-σ×-dχΠχ, LWχexp-2πiχσ-σ,
T2=12-Sθ2σσ+σ2Zσ+σ2dσ+12-Sθ2σσ-σ2Zσ-σ2dσ.
Ɉσ=14-Sθ2σσ+σ2Zσ+σ+σ-σ2Zσ-σ2dσ.

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