Abstract

Two completely independent systematic approaches for designing algorithms are presented. One approach uses recursion rules to generate a new algorithm from an old one, only with an insensitivity to more error sources. The other approach uses a least-squares method to optimize the noise performance of an algorithm while constraining it to a desired set of properties. These properties might include insensitivity to detector nonlinearities as high as a certain power, insensitivity to linearly varying laser power, and insensitivity to some order to the piezoelectric transducer voltage ramp with the wrong slope. A noise figure of merit that is valid for any algorithm is also derived. This is crucial for evaluating algorithms and is what is maximized in the least-squares method. This noise figure of merit is a certain average over the phase because in general the noise sensitivity depends on it. It is valid for both quantization noise and photon noise. The equations that must be satisfied for an algorithm to be insensitive to various error sources are derived. A multivariate Taylor-series expansion in the distortions is used, and the time-varying background and signal amplitudes are expanded in Taylor series in time. Many new algorithms and families of algorithms are derived.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. 26, Chap. 5, pp. 349–383.
    [CrossRef]
  2. J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (May1982).
  3. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995). His values of ε and γ for π/2 algorithms are, respectively, -π/2 and π/16 times the values in this paper.
  4. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  5. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  6. R. Onodera, Y. Ishii, “Phase extraction analysis of laser-diode phase-shifting interferometry that is insensitive to changes in laser power,” J. Opt. Soc. Am. A 13, 139–146 (1996). Their formula for the phase has the form θ = tan-1 (Σ bjIj/Σ aiIi) and their phase step is really -π/2. For +π/2 phase step, let bj → -bj.
  7. C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [CrossRef] [PubMed]
  8. J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
    [CrossRef] [PubMed]
  9. Y. Surrel, “Design of algorithms for phase measurements by the use of phase-shifting,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  10. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digitial wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef]
  11. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2507 (1987).
    [CrossRef] [PubMed]
  12. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]
  13. J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
  14. P. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29, 3271–3279 (1990).
    [CrossRef] [PubMed]
  15. B. Truax, Truax Associates, 189 Olson Drive, Southington, Conn. 06489 (personal communication, 1997).

1996

1995

1992

1990

1987

1984

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

1983

1982

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (May1982).

C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
[CrossRef] [PubMed]

Burow, R.

Creath, K.

J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[CrossRef] [PubMed]

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. 26, Chap. 5, pp. 349–383.
[CrossRef]

de Groot, P.

Eiju, T.

Elssner, K. E.

Freischlad, K.

Grievenkamp, J. E.

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Grzanna, J.

Hariharan, P.

Ishii, Y.

Koliopoulos, C. L.

Larkin, K. G.

Merkel, K.

Morgan, C. J.

Onodera, R.

Oreb, B. F.

Schmit, J.

Schwider, J.

Spolaczyk, R.

Surrel, Y.

Truax, B.

B. Truax, Truax Associates, 189 Olson Drive, Southington, Conn. 06489 (personal communication, 1997).

Wizinowich, P.

Wyant, J. C.

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (May1982).

Appl. Opt.

J. Opt. Soc. Am. A

Laser Focus

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (May1982).

Opt. Eng.

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Opt. Lett.

Other

B. Truax, Truax Associates, 189 Olson Drive, Southington, Conn. 06489 (personal communication, 1997).

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. 26, Chap. 5, pp. 349–383.
[CrossRef]

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 (4)

Fig. 1
Fig. 1

Absolute phase error δθ = arg[G(ν0)] - θ versus ε at θ = 45° for the P = 7, 9, 11, and 13 α = π/2 optimally distortion-insensitive algorithms whose weights are given in Table 4. For small ε, δθ ∝ ε P-3 sin 2θ.

Fig. 2
Fig. 2

Relative phase error δθrel = arg[G(γ)] - θ - arg[G+(γ)] versus θ for γ = 0.05 for the P = 7 and P = 9 α = π/2 optimally distortion-insensitive algorithms whose weights are given in Table 4. The algorithms are sensitive to second-order distortion γ in order int[(P - 3)/2], where int denotes the integer part. Thus the P = 7 and P = 9 algorithms are sensitive to γ in the second and third orders, respectively.

Fig. 3
Fig. 3

Absolute phase error δθ = arg[G(ν0)] - θ versus ε at θ = 45° for the P = 8, 10, 12, and 14 α = π/4 algorithms whose weights are given in Table 6. For small ε δθ ∝ (-ε)(P-6)/2 sin 2θ.

Fig. 4
Fig. 4

Comparison of the average sensitivity to small amplitude vibration for two π/2 algorithms: curve A, algorithm with weights 1 3 4 4 3 1; curve B, Algorithm with weights 1 3+2 5+32 7+42 7+42 5+32 3+2 1. Algorithm B is derived from algorithm A with the application of recursion rules to make it relatively insensitive to a vibration of frequency νvib = ν/2.

Tables (9)

Tables Icon

Table 1 Summary of P = 3 Algorithms for α = 2π/3, π/2, π/3, and π/4a

Tables Icon

Table 2 Summary of Recursion Relations for Creating a New Algorithm with Weights j from an Existing One with Weights wja

Tables Icon

Table 3 Family of Algorithms with Phase Step 2π/3 Generated by j = wj + wj+1 + wj+2a

Tables Icon

Table 4 Family of Algorithms with Phase Step π/2 Generated by j = wj + wj + 1a

Tables Icon

Table 5 Family of Algorithms with Phase Step π/3 Generated by j = wj + wj +1 + wj + 2a

Tables Icon

Table 6 Family of Algorithms with Phase Step π/4 Generated by j = wj + wj+2a

Tables Icon

Table 7 Comparison of Algorithm Generated with Recursion Rules with Algorithm Generated with Least-Squares Methoda

Tables Icon

Table 8 Sums Σj r wj exp(imφj) for Onodera’s6 Algorithm; Phase Step α = π/2a

Tables Icon

Table 9 Sums Σ jrw j exp(imφ j) for the P = 11 α = π/2 Algorithm from Table 4a

Equations (117)

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

Ijx, y=Ax, y+Bx, ycos2πνtj+θx, y,
B expiθ=jrjIj=jRerjIj+i ImrjIj,
θ=tan-1j ImrjIjj RerjIj.
G=jrjIj,
G=G0+G+ expiθ+G- exp-iθ.
rj=wj exp-iφj.
G=jwjIj exp-iφj,
φj=2πνtj.
w¯j=kakwj+k.
G=jwjIj exp-iφj.
Ij=A0+B0 cos2πν0tj+θ,
G=G0+G+ν0exp+iθ+G-ν0exp-iθ,
G0=A0jwj exp-iφj,  G+ν0=12B0jwj exp2πiν0tj-iφj,  G-ν0=12B0jwj exp-2πiν0tj-iφj.
G+=B02jwj,  G0=A0jwj exp-iφj,  G-=B02jwj exp-2iφj.
jwj=positive real,
jwj exp-iφj=0,
jwj exp-2iφj=0.
It=xt+a2xt2+a3xt3+,
xt=At+Bt cos2πν0t+γ22ν0t2+γ33ν0t3++θ+Δθvib,
Δθvib=ωLcεvib cos2πνvibt+θvib.
At=A0+A1t+A2t2+,  Bt=B0+B1t+B2t2+.
ε=ν0ν-1.
G=n=-+Gnν0, γj, aj, Aj, Bj, εvib, νvib, θvibexpinθ,
G0=A0jwj exp-iφj,  G+ν0, γ2, =B02jwj exp2πiν0tj+γ22νtj2+γ32νtj3+-iφj,  G-ν0, γ2, =B02jwj exp-2πiν0tj+γ22νtj2+γ32νtj3+-iφj,
rν0rG-ν0ν0=ν=B0-iαr2jjrwj exp-2iφj.
I+J+K+ν0Iγ2Jγ3KG-ν0, γ2, γ3, ν0=νγ2=γ3==0=K jI+2J+3K+wj exp-2iφj,
K=B02-2πi1α2π1I-2πi2α2π2J×-2πi3α2π3K.
G0A=jwjA0+A1tj+A2tj2+exp-iφj,  G+B=12jwjB0+B1tj+B2tj2+,  G-B=12jwjB0+B1tj+B2tj2+exp-2iφj.
jjrwj exp-2iφj=0.
jjrwj exp-iφj=0.
G=jwjn=1anklnklAn-k-lB2k+l×expik-lθ+φjexp-iφj.
G=m=-N-1N-1KmA, B, anjwj expimφj×expim+1θ.
jwj expimφj=0
G=n=-+Gnεvib, νvib, θvibexpinθ,
It=A+Bcos2πνt+θ+ωLcεvib cos2πνvibt+θvib.
Xt=2πνt+θ,  Yt=ωLεvib/ccos2πνvibt+θvib.
ItA+Bcos2πνt+θ-xsin2πνt+θcosYt.
G+=B2jwj1+ix cosYj,  G0=Ajwj exp-iφj,  G-=B2jwj1-ix cosYjexp-2iφj,
G+=B2jwj+B2jwjix cosYj,  G0=0,  G-=B2jwj-ix cosYjexp-2iφj.
G+=B2jwj+ix2expiθvibG+(+)+exp-iθvibG+(-),  G0=0,  G-=-ix2expiθvibG-(+)+exp-iθvibG-(-),
G+(+)=B2jwj expi2πνvibtj,  G+(-)=B2jwj exp-i2πνvibtj,  G-(+)=B2jwj expi2πνvibtjexp-2iφj,  G-(-)=B2jwj exp-i2πνvibtjexp-2iφj.
G+(+)=G+ν+νvib,  G+(-)=G+ν-νvib,  G-(+)=G-ν-νvib,  G-(-)=G-ν+νvib.
G+=B2jwj+i2ωLεvibcexpiθvibG+ν+νvid+exp-iθvibG+ν-νvib,  G0=0,  G-=-i2ωLεvibcexpiθvibG-ν-νvib+exp-iθvibG-ν+νvib,
G=G++δG+expiθ+G-+δG-exp-iθ,
δθ=ImδG++δG- exp-2iθG+.
δθ=δθ++δθ-,
δθ+=ωLεvib2cG+cosθvibG+ν+νvib+cosθvibG+ν-νvib,  δθ-=-ωLεvib2cG+cos2θ+θvibG-ν+νvib+cos2θ-θvibG-ν-νvib.
δθ2¯=δθ+2¯+δθ-2¯,
δθ2¯=δθ+2¯+δθ-2¯,
δθ+2¯=12ωLεvib2cG+2G+ν+νvib+G+ν-νvib2,  δθ-2¯=12ωLεvib2cG+2G-ν+νvib2+G-ν-νvib2.
w¯j=k=0q-1wj+k,
jw¯j exp-imφj=0,
jw¯j exp-imφj=1-expimqα1-expimαjwj expimφj,
jjrwj exp-2iφj=0,
w¯j=a-wj+a+wj+Δj,
jjR+1wj exp-2iφj=0.
w¯j=exp-iβwj+expiβwj+Δj,
β=π2-Δjα.
w¯j=a-wj-Δj+a0wj+a+wj+Δj,
a0+2a+ cos2Δjα=0.
jjswj exp-iφj=0
jjS+1wj exp-iφj=0.
β=π-Δjα2.
a0+2a+ cosΔjα=0.
Px=j=0P-1rjxj,
px=j=0P-1wjxj.
w¯j=k=0Nakwj+k,
qx=k=0Nakxk,
Qx=k=0Nak exp-ikαxk,
PNx=x+1x-1x+iN+1
jwj1j=positive real,  jwjexp-iαj=0,  jwjexp-2iαj=0.
jrjexp+iαj=positive real,  jrj1j=0,  jrjexp-iαj=0,
px=x-exp-iαx-exp-2iαN+1
Px=x-1x-exp-iαN+1
Px=x-1x+iN+1
Ij=A+B cosθ+φj+δIj,
G=BU expiθ+V2+jδIjuj expiνj-φj.
δθ=2jδIjuj sinνj-φj-V-θBU.
δθ2=2juj21-cosνj-φj-V-θB2U2fIj.
δθ2¯=12π02πδθ2dθ.
δθ2¯=jwj21/2jwj216B2.
δθ2¯=jwj21/2jwj22kAB2.
NF=jwjjwj21/2.
δθ21/2/2π=2.261×10-41-0.04 sin2 θ1/2 waves,
kwk=1,
krkαwk=0,
kzkβwk=0,
krkzkβ=0,
w¯k=wk+βcβzkβ,
kzkβ*zkβ=Nβδββ,
kw¯k2=kwk2+βkcβwk*zkβ+cβ*wkzkβ*+βcβcβ*kzkβzkβ*.
kwk+ββcβzkβzkβ*=kwkzkβ*.
kwk*zk+c*zk2=0,
kwkzk*+czk2=0.
c=-kwkzk*kzk2.
cβ=-kwk+ββcβzkβzkβ*kzkβ2.
cβ=-kwkzkβ*kzkβ2
δθ=argG-argG+-θ.
θ=12 argG-G+±π4+nπ.
δθ=±tan-1G-G+.
δθ=tan-1x sin y1+x cos y,  x=G-G+  y=argG-G+-2θ.
θ=argG0G+±π2+2nπ.
δθ=tan-1x sin y1+x cos y,  x=G0G+  y=argG0G+-θ.
G=m=-N-1N-1KmA, B, anjwj expimφj×expimθexpiθ,
δθm=-N-1m0N-1KmA, B, anK0A, B, an×sinmθ+argKmA, B, anK0A, B, an.
δθG-(ν0)G+(ν0) sin(2θ).
G-γ=12B0jwj exp-πiγν0tj2exp-2iφj,
G-γ12B0jwj-πiγν0tj2rr! exp-2iφj.
010x-11x-1x-21x-2w-1w0w+1=100.
w-1w0w1=-x2/1+x1-x2/1+x*.
w¯j=exp-iβwj+expiβwj+Δj,
β=π-ν0ν+1αΔj2G¯-ν0=0,  β=ν0ν-1αΔj-π2G¯+ν0=0.
G¯=jw¯jIj exp-iφj.
w¯j=a-wj-Δj+a0wj+a+wj+Δj
jw¯j expijxα=0
a0+a+ cosxαΔj=0.
δθ-2¯1/2ωLεvibc

Metrics