Abstract

An optimization-based strategy is introduced for suppressing errors due to vibration in phase-shifting-interferometry algorithms. A norm-square integral criterion of the error as a function of vibration frequency is used as the basis of the optimization procedure. Analytical results are obtained for certain classes of problems, and numerical algorithms are used when these are not available. It is also shown that the effect of vibration-induced errors in the computation of a time-averaged phase estimate diminishes as the measurements are averaged. Simulations are used to validate the analysis and demonstrate the overall efficacy of the approach. Generalizations to multiple objective optimization problems are briefly discussed.

© 2002 Optical Society of America

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References

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  1. R. Danner, S. Unwin, “Space interferometry mission, taking the measure of the universe,” (Jet Propulsion Laboratory, Pasadena, Calif., 1999).
  2. P. J. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]
  3. P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Vibration-induced phase errors in high-speed phase-shifting speckle-pattern interferometry,” Appl. Opt. 40, 2117–2125 (2001).
    [CrossRef]
  4. P. J. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [CrossRef]
  5. J. Schwider, R. Burrow, K.-E. Elsner, J. Grzana, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  6. B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
    [CrossRef]
  7. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase shifting-algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  8. Y. Surrel, “Phase-stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [CrossRef] [PubMed]
  9. M. Milman, S. Basinger, “White light fringe estimation: algorithms, error sources and mitigation strategies,” in Proceedings of the 2001 IEEE Aerospace Conference (Institute of Electrical and Electronics Engineers, New York, 2001), Paper 6.0811.
  10. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York (1974).
  11. K. L. Chung, A Course in Probability Theory, 2nd ed. (Academic, New York, 1974).
  12. E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).

2001 (1)

1995 (3)

1993 (1)

1992 (1)

1983 (1)

Basinger, S.

M. Milman, S. Basinger, “White light fringe estimation: algorithms, error sources and mitigation strategies,” in Proceedings of the 2001 IEEE Aerospace Conference (Institute of Electrical and Electronics Engineers, New York, 2001), Paper 6.0811.

Burrow, R.

Chung, K. L.

K. L. Chung, A Course in Probability Theory, 2nd ed. (Academic, New York, 1974).

Coddington, E. A.

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).

Coggrave, C. R.

Danner, R.

R. Danner, S. Unwin, “Space interferometry mission, taking the measure of the universe,” (Jet Propulsion Laboratory, Pasadena, Calif., 1999).

de Groot, P. J.

Elsner, K.-E.

Grzana, J.

Huntley, J. M.

Kaufmann, G. H.

Larkin, K. G.

Levinson, N.

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).

Merkel, K.

Milman, M.

M. Milman, S. Basinger, “White light fringe estimation: algorithms, error sources and mitigation strategies,” in Proceedings of the 2001 IEEE Aerospace Conference (Institute of Electrical and Electronics Engineers, New York, 2001), Paper 6.0811.

Oreb, B. F.

Rudin, W.

W. Rudin, Real and Complex Analysis, McGraw-Hill, New York (1974).

Ruiz, P. D.

Schwider, J.

Shen, Y.

Spolaczyk, R.

Surrel, Y.

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Y. Surrel, “Phase-stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[CrossRef] [PubMed]

Unwin, S.

R. Danner, S. Unwin, “Space interferometry mission, taking the measure of the universe,” (Jet Propulsion Laboratory, Pasadena, Calif., 1999).

Zhao, B.

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Other (5)

M. Milman, S. Basinger, “White light fringe estimation: algorithms, error sources and mitigation strategies,” in Proceedings of the 2001 IEEE Aerospace Conference (Institute of Electrical and Electronics Engineers, New York, 2001), Paper 6.0811.

W. Rudin, Real and Complex Analysis, McGraw-Hill, New York (1974).

K. L. Chung, A Course in Probability Theory, 2nd ed. (Academic, New York, 1974).

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).

R. Danner, S. Unwin, “Space interferometry mission, taking the measure of the universe,” (Jet Propulsion Laboratory, Pasadena, Calif., 1999).

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Figures (10)

Fig. 1
Fig. 1

Linearization error.

Fig. 2
Fig. 2

Response of 8-bin least-squares algorithm to vibration disturbance.

Fig. 3
Fig. 3

Response of four different 8-bin algorithms to vibration disturbance.

Fig. 4
Fig. 4

Response of 8- and 32-bin least-squares and optimal algorithms to vibration disturbance.

Fig. 5
Fig. 5

Response of 8-bin algorithms with unmatched modulation length and wavelength.

Fig. 6
Fig. 6

Response of 32-bin algorithms with unmatched modulation length and wavelength.

Fig. 7
Fig. 7

Comparison of variance between estimators.

Fig. 8
Fig. 8

Time-averaged response to sinusoidal disturbance.

Fig. 9
Fig. 9

Time-averaged response to exponentially correlated disturbance.

Fig. 10
Fig. 10

Monte Carlo simulations of exponentially correlated disturbance.

Equations (138)

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yi=ui-Δ/2ui+Δ/2I0{1+V cos[u+ϕ(u)]}du,
yi=I0[Δ+2V sin(Δ/2)cos(ui+ϕ)].
y1yN=Δcos(u1)-sin(u1)Δcos(uN)-sin(uN)I0I0V˜ cos(ϕ)I0V˜ sin(ϕ).
y=Ax.
yi=ui-Δ/2ui+Δ/2I0{1+V cos[u+ϕ¯+δϕ(u)]}du=I0[Δ+2V sin(Δ/2)cos(ui+ϕ¯)]-I0V cos(ϕ¯)×ui-Δ/2ui+Δ/2 sin(u)δϕ(u)du-I0V sin(ϕ¯)×ui-Δ/2ui+Δ/2 cos(u)δϕ(u)du.
B=12 sin(Δ/2)0ui-Δ/2ui+Δ/2 sin(u)δϕ(u)duui-Δ/2ui+Δ/2 cos(u)δϕ(u)du.
y=Ax-Bx.
xˆ=Ky
KA=I3×3(I3×3=3×3identitymatrix)
x=K(y+Bx),
e=KBx.
ϕ(x)=tan-1(x3/x2),
ϕ(x)-ϕ(xˆ)=ϕ(x)(x-xˆ)=ϕ(x)e=-x3x22+x32e2+x2x22+x32e3=-sin(ϕ¯)e2I0V˜+cos(ϕ¯)e3I0V˜.
e2=I0V˜2 sin(Δ/2)j=1N k2jcos(ϕ¯)ui-Δ/2ui+Δ/2 sin(u)δϕ(u)du+sin(ϕ¯)ui-Δ/2ui+Δ/2 cos(u)δϕ(u)du,
e3=I0V˜2 sin(Δ/2)j=1N k3jcos(ϕ¯)uj-Δ/2uj+Δ/2 sin(u)δϕ(u)du+sin(ϕ¯)ujΔ/2uj+Δ/2 cos(u)δϕ(u)du.
ϕ(x)-ϕ(xˆ)
12 sin(Δ/2)k3jcos2(ϕ¯)uj-Δ/2uj+Δ/2 sin(u)δϕ(u)du+sin(2ϕ¯)/2uj-Δ/2uj+Δ/2 cos(u)δϕ(u)du-k2jsin(2ϕ¯)/2uj-Δ/2uj+Δ/2 sin(u)δϕ(u)du+sin2(ϕ¯)uj-Δ/2uj+Δ/2 cos(u)δϕ(u)du.
ϕ(x)-ϕ(xˆ)
12 sin(Δ/2)k3juj-Δ/2uj+Δ/2 sin(u)δϕ(u)du.
k2j=2Ncos(uj),k3j=-2Nsin(uj).
δϕ(u)=cos(ωu)-c(ω),δϕ(u)=sin(ωu)-s(ω),
c(ω)=12π02π cos(ωx)dx,
s(ω)=12π02π sin(ωx)dx
E(iω)=12 sin(Δ/2)k3juj-Δ/2uj+Δ/2 sin(u)[cos(ωu)-c(ω)+i sin(ωu)-is(ω)]du.
minαΩ-Ω+|E(iω)|2dω,subjecttoαA=ez,
uj-Δ/2uj+Δ/2 sin(u)du=2 sin(Δ/2)sin(uj),
αjuj-Δ/2uj+Δ/2 sin(u)du=2 sin(Δ/2)
E(iω)=αjuj-Δ/2uj+Δ/2 sin(u)cos(ωu)du+2 sin(Δ/2)c(ω)
+iαjuj-Δ/2uj+Δ/2 sin(u)sin(ωu)du+2 sin(Δ/2)s(ω).
|E(iω)|2=αT(ω)αT+αh(ω)+4 sin2(Δ/2)[s2(ω)+c2(ω)],
Tij(ω)=ui-Δ/2ui+Δ/2uj-Δ/2uj+Δ/2 sin(u)sin(v)[sin(ωu)sin(ωv)+cos(ωu)cos(ωv)]dudvdω,
hj(ω)=4 sin(Δ/2)uj-Δ/2uj+Δ/2 sin(u)[s(ω)sin(ωu)+c(ω)cos(ωu)]du.
Tij=Ω-Ω+Tij(ω)dω
hj=Ω-Ω+hj(ω)dω.
minα αTαT+αhsubjecttoαA=ez.
L(α, λ)=αTαT+αh+(αA-ez)λ,
αT=-T-1h+T-1A(ATT-1A)-1[ezT+ATT-1h].
Ω-Ω+Tij(ω)dω=Ω-Ω+ui-Δ/2ui+Δ/2uj-Δ/2uj+Δ/2 sin(u)sin(v)×[sin(ωu)sin(ωv)+cos(ωu)cos(ωv)]dudvdω=ui-Δ/2ui+Δ/2uj-Δ/2uj+Δ/2 sin(u)sin(v)×sin Ω+(u-v)-sin Ω-(u-v)(u-v)×dudv,
Ω-Ω+ sin(ωu)sin(ωv)dω
=sin[Ω+(u-v)]2(u-v)-sin[Ω+(u+v)]2(u+v)-sin[Ω-(u-v)]2(u-v)-sin[Ω-(u+v)]2(u+v),
Ω-Ω+ cos(ωu)cos(ωv)dω
=sin[Ω+(u-v)]2(u-v)+sin[Ω+(u+v)]2(u+v)-sin[Ω-(u-v)]2(u-v)+sin[Ω-(u+v)]2(u+v).
limΩsin (Ωu)u=πδ,
Tii=π2[Δ-sin(Δ)cos(2ui)].
hj=4 sinΔ20uj-Δ/2uj+Δ/2 sin(u)[s(ω)sin(ωu)+c(ω)cos(ωu)]dudω
=4 sinΔ2uj-Δ/2uj+Δ/2 sin(u)0[s(ω)sin(ωu)+c(ω)cos(ωu)]dωdu.
s(ω)=12πωγ[1-cos(2πωγ)]
c(ω)=12πωγsin(2πωγ).
0[s(ω)sin(ωu)+c(ω)cos(ωu)]dω
=0cos(ωu)sin(2πωγ)+sin(ωu)[1-cos(2πωγ)]2πωγdω=0sin[ω(2πγ-u)]2πωγdω+0sin(ωu)2πωγdω.
0sin(αx)xdx=π2
0[s(ω)sin(ωu)+c(ω)cos(ωu)]dω=1γ,
u[0, 2π).
αT=T-1A(ATT-1A)-1ezT.
M33=i=1NTii-1 sin2(ui)
=1πi=1N1-cos(2ui)Δ-sin(Δ)cos(2ui)
=1πΔi=1N1-cos(2ui)1-sin(Δ)cos(2ui)/Δ.
αj=-sin(uj)TjjM33,
αj-2 sin(uj)N[1-cos(2uj)].
ϕ(x)-ϕ(xˆ)1202π cos(2u)δϕ(u).
0|E(iω)|2dω=14 sin2(Δ/2)βTβT+0[c2(ω)+s2(ω)]dω>0.
βTβT=2πN2i=1N[Δ-sin(Δ)cos(2ui)]sin2(ui)
=2π2N2+πNsin(Δ)2
3π2N2.
limN14 sin2(Δ/2)βTβT=34.
limN14 sin2(Δ/2)αTαT=12.
ϕ(x)-ϕ(xˆ)=02πzN(u)δϕ(u)du,
zN=αj sin(u)2 sin(Δ/2)foru[(j-1)Δ, jΔ].
limN02πzN(u)δϕ(u)du=02πδϕ(u)du=0,
E(|x-xˆ|2)=trace(KQKT),
K=k1k2α,
mink1,k2 trace(KQKT)subjecttotheconstraints
k1A=ex,k2A=ey,
trace(KQKT)=k1Qk1T+k2Qk2T+αQαT,
k1T=Q-1A(ATQ-1A)-1ex,
k2T=Q-1A(ATQ-1A)-1ey.
R(λ)=λQ+(1-λ)T,λ[0, 1].
x¯=1T0Tx(t)dt.
ϕ¯=12πM02πMϕ(u)du.
E0sin(ω)=02πh(u)[sin(ωu)-ϕ¯]du,
ϕ¯=12π02π sin(ωu)du,
h(u)=12 sin(Δ/2)k3j sin(u),u[(j-1)Δ, jΔ]
η=12π02πh(u)du.
E0sin(ω)=02πh(u)sin(ωu)du-ηω[1-cos(2πω)].
Eksin(ω)=02πh(u)sin(2πkω+ωu)du-η02π sin(2πkω+ωu)du
=02πh(u)[sin(2πkω)cos(ωu)+cos(2πkω)sin(ωu)]du
-ηω{sin(2πω)sin(2πkω)
+cos(2πkω)[1-cos(2πω)]}.
CM(ω)=1Mk=0M-1 cos(2πkω),
SM(ω)=1Mk=0M-1 sin(2πkω).
Esin(M, ω)=1Mk=0M-1Eksin(ω)=SM(ω)02πh(u)cos(ωu)du-CM(ω)02πh(u)sin(ωu)du-ηω{sin(2πω)SM(ω)+[1-cos(2πω)]CM(ω)}.
Ecos(M, ω)=1Mk=0M-1Ekcos(ω)=CM(ω)02πh(u)cos(ωu)du-SM(ω)02πh(u)sin(ωu)du-ηω{sin(2πω)CM(ω)-[1-cos(2πω)]SM(ω)}.
z(u)=ϕ(u)-ϕ¯k,u[2πk, 2π(k+1)],
k=0,1,.
|ϕ¯|2du=k|ϕ¯k|2=k2πk2π(k+1)ϕ(u)du22πk2πk2π(k+1|ϕ(u)|2du(byJensenssinequality11)=2π0|ϕ(u)|2du.
EM(ϕ)=1Mk=0M-12kπ2(k+1)πh(u)[ϕ(u)-ϕ¯k]du.
EM(ϕ)=1M02Mπh(u)z(u)du=1M02Mπh(u)-zˆ(ω)exp(iωu)dωdu=-zˆ(ω)1M02Mπh(u)exp(iωu)dudω.
ΨM(ω)=1M02Mπh(u)exp(iωu)dudω.
Re(ΨM)=CM(ω)02πh(u)cos(ωu)du-SM(ω)02πh(u)sin(ωu)du,
Im(ΨM)=SM(ω)02πh(u)cos(ωu)du+CM(ω)02πh(u)sin(ωu)du.
H(ω)=02πh(u)exp(iωu)du.
limM CM,SM=0,
Hc(ω)=02πh(u)cos(ωu)du,
Hs(ω)=02πh(u)sin(ωu)du.
[Re ΨM]22CM2Hc2+2SM2Hs2.
-CM2(ω)Hc2(ω)dω=-2(k-1)π2kπCM2(ω)Hc2(ω)dω
-supω(2(k-1)π,2kπ)Hc2(ω)×2(k-1)π2kπCM2(ω)dω.
1M2- supω(2(k-1)π,2kπ) Hc2(ω).
- supω(2(k-1)π,2kπ) Hc2(ω)<,
-CM2(ω)Hc2(ω)dω=O(1/M2).
-|ΨM2(ω)|dω=O(1/M2).
αjN=sin(ujN)TjjNM33N,
Tjj=π2[ΔN-sin(ΔN)cos(2ujN)],
M33N=1πΔNj=1N1-cos(2ujN)1-sin(ΔN)cos(2ujN)/ΔN,
EN(δϕ)=1πΔN sin(ΔN/2)M33N02πΨN(u)δϕ(u)du,
ΨN(u)=sin(ujN)1-sin(ΔN)cos(2ujN)/ΔNsin(u),
u[(j-1)ΔN, jΔN).
sin(u)=sin(ujN)+cos(u)(u-ujN),
ΨN(u)=sin(ujN)1-sin(ΔN)cos(2ujN)/ΔNsin(u)
=[1-cos(2ujN)]/21-sin(ΔN)cos(2ujN)/ΔN+sin(ujN)cos(u)(u-u)1-sin(ΔN)cos(2ujN)/ΔN.
limN[1-cos(2ujN)]/21-sin(ΔN)cos(2ujN)/ΔN=1/2,
limNsin(ujN)cos(u)(u-u)1-sin(ΔN)cos(2ujN)/ΔN=0.
1-cos(2ujN)1-sin(ΔN)cos(2ujN)/ΔN
=1-cos(2ujN)1-[1+sin(ΔN)/ΔN-1]cos(2ujN)
=11+[ΔN-sin(ΔN)]cos(2ujN)ΔN[1-cos(2ujN)].
g(x)=[ΔN-sin(ΔN)]cos(x)ΔN[1-cos(x)].
g(x)>-ΔN-sin(ΔN)ΔN.
1-cos(2ujN)1-sin(ΔN)cos(2ujN)/ΔN<ΔN/sin(ΔN).
cos(u)(u-ujN)=cos(ujN)γ,
sin(ujN)cos(u)(u-ujN)1-sin(ΔN)cos(2ujN)/ΔN=γ sin(2ujN)/21-sin(ΔN)cos(2ujN)/ΔN.
f(x)=κΔN sin(x)1-sin(ΔN)cos(x)/ΔN.
|δϕ(u)|<k,|δϕ(u)|<k.
02πΨn(u)δϕ(u)<
02πΨnk(u)δϕnk(u).
02πΨnkl(u)δϕnkl(u)du
=02πΨnkl(u)δϕdu+02πΨnkl(u)[δϕnkl(u)-δϕdu]du
02πc0δϕ=0
limN02πΨN(u)δϕ(u)du=0.

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