Abstract

When subjected to random background vibrations, a standard, circular-aperture Michelson interferometer with a dynamic alignment servo system has a misalignment angle that is a random function of time. Here we derive formulas for the loss in performance when the misalignment angle is modeled as a stationary stochastic time series and show how these formulas are simplified when the power spectrum is band-limited white noise.

© 1997 Optical Society of America

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References

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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. J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979) p. 237.
  4. D. R. Hearn, Fourier Transform Interferometry, (Lincoln Laboratory, MIT, Cambridge, Mass., 1995), pp. 12–18.
  5. B. P. Lathi, An Introduction to Random Signals and Communication Theory (International Textbook, Scranton, Pa., 1968), pp. 182–185.
  6. A. Papoulis, Probability, Random Variables, and Stochastic Processes3rd ed. (McGraw–Hill, New York, 1991), pp. 139–140.

Bold, D. R.

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).

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).

Chamberlain, J.

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

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).

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, 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).

Hearn, D. R.

D. R. Hearn, Fourier Transform Interferometry, (Lincoln Laboratory, MIT, Cambridge, Mass., 1995), pp. 12–18.

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).

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).

Lathi, B. P.

B. P. Lathi, An Introduction to Random Signals and Communication Theory (International Textbook, Scranton, Pa., 1968), pp. 182–185.

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).

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, 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).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes3rd ed. (McGraw–Hill, New York, 1991), pp. 139–140.

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).

Ryan-Howard, D. 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).

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).

Other (6)

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).

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

D. R. Hearn, Fourier Transform Interferometry, (Lincoln Laboratory, MIT, Cambridge, Mass., 1995), pp. 12–18.

B. P. Lathi, An Introduction to Random Signals and Communication Theory (International Textbook, Scranton, Pa., 1968), pp. 182–185.

A. Papoulis, Probability, Random Variables, and Stochastic Processes3rd ed. (McGraw–Hill, New York, 1991), pp. 139–140.

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Figures (4)

Fig. 1
Fig. 1

Angle ε(x) is the misalignment angle of the moving mirror with respect to the virtual image of the fixed mirror at an OPD of x (in which the moving mirror has moved x/2 from its ZPD position).

Fig. 2
Fig. 2

Graph of the normalized noise-to-signal ratio for a unit continuum spectrum [see relation (25d)] converging to a limiting curve independent of σ M for wave numbers greater than 1000 cm-1.

Fig. 3
Fig. 3

Parabolic FWHM is a good approximation of the exact FWHM of the sinc function’s main lobe.

Fig. 4
Fig. 4

Graph of the expected variation in resolution for r2〈ε〉2 = 1 × 10-9 cm2, 2 × 10-9 cm2, and 3 × 10-9 cm2 [see relation (42f)].

Equations (83)

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Ix=0Beffσ2J14πσrε4πσrεcos2πσxdσ,
Beffσ=1/2 A0ΩBσηστ0σRσ,
2J14πσrε4πσrε1-Aσ2ε2 for A=2π2r2,
Ix=0Beffσcos2πσx1-Aσ2εx2dσ.
Ix=-Sσexp2πiσx1-Aσ2εx2dσ,
Sσ=1/2 Beffσ so that S-σ=Sσ.
Seσ=-LL Ixexp-2πiσxdx
Seσ=-LLdx exp-2πiσx -dσSσ×exp2πiσx1-Aσ2εx2.
S0σ=-dxΠL, xexp-2πiσx×-dσSσexp2πiσx,
ΠL, x=1for x  L0for x > L.
S0σ=Sσ*2L sinc2πσL,
fσ*gσ  - fσgσ-σdσ.
ESeσ=-LLdx exp-2πiσx -dσ×1-Aσ2Eεx2Sσexp2πiσx=Sσ*2L sinc2πσL-AEε2×σ2Sσ*2L sinc2πσL,
ESeσ=ESeσ*=ESeσ* or EImSeσ=0.
EIx=-dσ1-Aσ2Eε2Sσexp2πiσx
ESeσ-ESeσ2=ESeσ2-ESeσ2.
ΔSeσ=Seσ-ESeσ.
EΔSeσ2+ESeσ2=S0σ2-2AS0σEε2×σ2Sσ*2L sinc2πσL+A2-LLdx-LLdx -dσσ2Sσ×-dσσ2SσEεx2εx2×exp-2πixσ-σexp2πixσ-σ.
Eεx2εx2=Rε2x-x+Eε22,
Eεx2-Eε2εx2-Eε2=Rε2x-x.
Sε2σ=-Rε2xexp-2πiσxdx,
Rε2x=-Sε2σexp2πiσxdσ,
Sε2-σ=Sε2σ.
EΔSeσ2+ESeσ2=S0σ2-2AEε2S0σ×σ2Sσ*2L sinc2πσL+A2Eε22σ2Sσ*2L sinc2πσL2+A2-LLdx-LLdx-dσσ2Sσ×-dσσ2Sσexp-2πixσ-σ×exp2πixσ-σ×-dσSε2σexp2πiσx-x.
EΔSeσ2=A2-LLdx -LLdxI2xI2x×-dσSε2σexp-2πiσ×x-xexp2πiσx-x,
S2Lσ=-LLdxI2xexp-2πiσx=2L sinc2πσL*σ2Sσ.
EΔSeσ2=A2Sε2σ*S2Lσ2=A2Sε2σ*σ2Sσ*2L sinc2πσL2
EΔSeσ2A2Sε2σ*σ2Sσ2,
Sε2σ=Sε0ΠσM, σ.
Sε2σ=Sε0Πβu-1, σ,
σM=β/u.
5 cm-1  σM  350 cm-1.
Rε20=Eε4-Eε22,
2σMSε0=Rε20,
Sε0=Eε4-Eε222σM.
EΔSeσ2A2Eε4-Eε222σM×ΠσM, σ*σ4Sσ2.
EReΔSeσ2=EImΔSeσ2,
EΔSeσ2=EReΔSeσ2+EImΔSeσ2=2EReΔSeσ2.
EReΔSeσ21/2=1/2 EΔSeσ21/2.
EReΔSeσ21/2=A/2Sε2σ*σ2Sσ*2L sinc2πσL21/2,
EReΔSeσ21/2A/2σMEε4-Eε221/2×ΠσM, σ*σ4Sσ21/2.
EReΔSeσ21/22Aε2/πσM×ΠσM, σ*σ4Sσ21/2.
EReΔSeσ21/2Sσ=12Aε2/π2/5σM4+10σM2σ2+5σ41/2.
EReΔSeσ21/2Sσ=142πr2ε2σ2.
EReΔSeσ21/2Sσ=1Aε2
NEdN=2 2ASε2σ*σ2Sσ*2L sinc2πσL21/2A0Ωτ0σησRσ
NEdN16πr2ε2ΠσM, σ*σ4Sσ21/2σMA0Ωτ0σησRσ.
heσ, σ0=-LLdx exp-2πiσ-σ0x1-Aσ02εx2.
Eheσ, σ0=1-Aσ02Eε22L sinc2πLσ-σ0.
sinc2πLσ-σ01-2/3 π2L2σ-σ02.
1/2=1-2/3 π2L2σw-σ02
FWHM2σw-σ0=3/πL.
gσ0=2/σ2heσ, σ0|σ=σ0
heσ, σ0heσ0, σ0+1/2σ-σ02gσ0.
gσ0=-4π2-LLdxx21-Aσ02εx2,
Eheσ, σ0Eheσ0, σ0+1/2σ-σ02Egσ0=2L1-Aσ02Eε2-4π2L3/3×σ-σ021-Aσ02Eε2,
Egσ0=-8π2L3/31-Aσ02Eε2.
heσw, σ0=1/2 heσ0, σ0=heσ0, σ0+1/2σw-σ02gσ0
2σw-σ0=2-LLdx1-Aσ02εx2-gσ01/222L1-Aσ02Eε2-gσ01/2,
-LLdxεx22LEε2.
Δσw-σ0Eσw-σ012Egσ0-Egσ021/2Egσ0.
Egσ0-Egσ02=Egσ02-Egσ02,
Egσ02=16π42L3/321-2Aσ02Eε2+16π4A2σ042L3/32Eε22+16π4A2σ04-LL x2dx×-LL x2dxRε2x-x.
Egσ02-Egσ02=16π4A2σ04×-LLx2dx -LL x2dxRε2x-x.
-LLx2dx -LL x2dxRε2x-x=-dσSε2σ-LL x2dx cos2πσx2=L24π4-dσSε2σ2π2σ2L2-1πLσ3×sin2πσL+2 cos2πσLσ22Eε4-Eε222σM×-σMσMdσ2π2σ2L2-12π3σ3×sin2πσL+L cos2πσLπ2σ22=Eε4-Eε222σMWσ-σMσM,
Wσ=1-2π2σ2L22-4π2σ2L240π6σ5 cos4πσL+L10π5σ41-π2σ2L2sin4πσL-140π6σ5-L42π2σ+2L55πSi4πσL.
Siz=0zsint/tdt.
Δσw-σ0Eσw-σ03Aσ02ε2GσMLπ1-4Aσ02πε2,
Gz=12z1-2π2z22-4π2z220π6z5×cos4πz+15π5z41-π2z2sin4πz-120π6z5-1π2z+45πSi4πz1/2.
Gz231-4π2z2/51/2 for 0  z  1 with 0 < Gz  23 for all z  0.
Δσw-σ0Eσw-σ0 < 4πr2σ02ε21-8πr2σ02ε2,
1/2σM  L/10 or σML5.
Gz1/5z,
σMLΔσw-σ0Eσw-σ06πr2σ02ε251-8πr2σ02ε2.
ε=ϕx2+ϕy21/2,
pε=εs2 exp-ε22s2,  ε0,
Eε=0ε2s2dε exp-ε22s2=sπ2,
Eε2=0ε3s2dε exp-ε22s2=2s2,
Eε4=0ε5s2dε exp-ε22s2=8s4,
Eε4-Eε22=4s4.
Eε2=4/π ε2,
Eε4=32/π2 ε4,
Eε4-Eε22=16/π2 ε4.

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