Abstract

Results are presented from a prototype phase-shifting interferometer capable of measuring both the real and the imaginary part of the complex index of refraction and the surface profile of a test surface. The three parameters of interest are extracted from the measured data by maximum-likelihood estimation theory. The performance of the system is quantitatively assessed with Cramer–Rao lower bounds. The results are shown to be strongly dependent on the quantization of the interferograms from the 8-bit CCD camera, the incident electric field amplitude, and the relative amplitude and phase difference of each polarized component through each arm of the interferometer.

© 2002 Optical Society of America

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References

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  1. E. W. Rogala, H. H. Barrett, “A phase-shifting interferometer/ellipsometer capable of measuring the complex index of refraction and the surface profile of a test surface,” J. Opt. Soc. Am. A 15, 538–548 (1998).
    [CrossRef]
  2. E. W. Rogala, H. H. Barrett, “Assessing and optimizing the performance of a phase-shifting interferometer/ellipsometer capable of measuring the complex index of refraction and the surface profile of a test surface,” J. Opt. Soc. Am. A 15, 1670–1685 (1998).
    [CrossRef]
  3. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
    [CrossRef]
  4. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1988), Vol. 26, pp. 349–393.
    [CrossRef]
  5. K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. SPIE680, 19–29 (1986).
    [CrossRef]
  6. D. Apostle, P. C. Logofatu, V. Damian, A. Dobrolu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. 35, 1288–1291 (1996).
    [CrossRef]
  7. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [CrossRef] [PubMed]
  8. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]
  9. E. Rogala, “Task-based assessment of a proposed phase-shifting interferometer/ellipsometer,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1999).
  10. H. Cramer, Mathematical Methods of Statistics (Princeton University, Princeton, N.J., 1946).
  11. C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).
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    [CrossRef]
  14. D. Dugue, “Application des proprietes de la limite au sens de calcul des probabilities a l’etude des diverses questions d’estimation,” Ecol. Poly. 3, 305–372 (1937).
  15. H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 1 (Wiley, New York, 1968).
  16. E. W. Barankin, “Locally best unbiased estimators,” Ann. Math. Stat. 20, 477–501 (1949).
    [CrossRef]
  17. A. Bhattacharyya, “On some analogues of the amount of information and their use in statistical estimation,” Sankhya 8, 1–15, 201–218, 315–328 (1946).
  18. C. Bungay, “VASE™ characterization of unknown substrate,” in Measurements Report for Erik Rogala (J. A. Woollam, Lincoln, Neb., 1999).
  19. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [CrossRef]
  20. E. W. Rogala, H. H. Barrett, “Phase-shifting interferometry and maximum-likelihood estimation theory,” Appl. Opt. 36, 8871–8876 (1997).
    [CrossRef]
  21. E. W. Rogala, H. H. Barrett, “Phase-shifting interferometry and maximum-likelihood estimation theory. II: A generalized solution,” Appl. Opt. 37, 7253–7258 (1998).
    [CrossRef]

1998

1997

1996

D. Apostle, P. C. Logofatu, V. Damian, A. Dobrolu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. 35, 1288–1291 (1996).
[CrossRef]

1995

1994

1984

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

1982

1949

E. W. Barankin, “Locally best unbiased estimators,” Ann. Math. Stat. 20, 477–501 (1949).
[CrossRef]

1946

A. Bhattacharyya, “On some analogues of the amount of information and their use in statistical estimation,” Sankhya 8, 1–15, 201–218, 315–328 (1946).

1945

C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).

1942

A. C. Aitkin, H. Silverstone, “On the estimation of statistical parameters,” Proc. R. Soc. Edinburg Section A Math. 61, 186–194 (1942).

1937

D. Dugue, “Application des proprietes de la limite au sens de calcul des probabilities a l’etude des diverses questions d’estimation,” Ecol. Poly. 3, 305–372 (1937).

1925

R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 700–725 (1925).
[CrossRef]

Aitkin, A. C.

A. C. Aitkin, H. Silverstone, “On the estimation of statistical parameters,” Proc. R. Soc. Edinburg Section A Math. 61, 186–194 (1942).

Apostle, D.

D. Apostle, P. C. Logofatu, V. Damian, A. Dobrolu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. 35, 1288–1291 (1996).
[CrossRef]

Barankin, E. W.

E. W. Barankin, “Locally best unbiased estimators,” Ann. Math. Stat. 20, 477–501 (1949).
[CrossRef]

Barrett, H. H.

Bhattacharyya, A.

A. Bhattacharyya, “On some analogues of the amount of information and their use in statistical estimation,” Sankhya 8, 1–15, 201–218, 315–328 (1946).

Bungay, C.

C. Bungay, “VASE™ characterization of unknown substrate,” in Measurements Report for Erik Rogala (J. A. Woollam, Lincoln, Neb., 1999).

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton University, Princeton, N.J., 1946).

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1988), Vol. 26, pp. 349–393.
[CrossRef]

K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. SPIE680, 19–29 (1986).
[CrossRef]

Damian, V.

D. Apostle, P. C. Logofatu, V. Damian, A. Dobrolu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. 35, 1288–1291 (1996).
[CrossRef]

Dobrolu, A.

D. Apostle, P. C. Logofatu, V. Damian, A. Dobrolu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. 35, 1288–1291 (1996).
[CrossRef]

Dugue, D.

D. Dugue, “Application des proprietes de la limite au sens de calcul des probabilities a l’etude des diverses questions d’estimation,” Ecol. Poly. 3, 305–372 (1937).

Fisher, R. A.

R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 700–725 (1925).
[CrossRef]

Ghiglia, D. C.

Greivenkamp, J. E.

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

Logofatu, P. C.

D. Apostle, P. C. Logofatu, V. Damian, A. Dobrolu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. 35, 1288–1291 (1996).
[CrossRef]

Morgan, C. J.

Rao, C. R.

C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).

Rathjen, C.

Rogala, E.

E. Rogala, “Task-based assessment of a proposed phase-shifting interferometer/ellipsometer,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1999).

Rogala, E. W.

Romero, L. A.

Silverstone, H.

A. C. Aitkin, H. Silverstone, “On the estimation of statistical parameters,” Proc. R. Soc. Edinburg Section A Math. 61, 186–194 (1942).

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 1 (Wiley, New York, 1968).

Ann. Math. Stat.

E. W. Barankin, “Locally best unbiased estimators,” Ann. Math. Stat. 20, 477–501 (1949).
[CrossRef]

Appl. Opt.

Bull. Calcutta Math. Soc.

C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).

Ecol. Poly.

D. Dugue, “Application des proprietes de la limite au sens de calcul des probabilities a l’etude des diverses questions d’estimation,” Ecol. Poly. 3, 305–372 (1937).

J. Opt. Soc. Am. A

Opt. Eng.

D. Apostle, P. C. Logofatu, V. Damian, A. Dobrolu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. 35, 1288–1291 (1996).
[CrossRef]

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

Opt. Lett.

Proc. Cambridge Philos. Soc.

R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 700–725 (1925).
[CrossRef]

Proc. R. Soc. Edinburg Section A Math.

A. C. Aitkin, H. Silverstone, “On the estimation of statistical parameters,” Proc. R. Soc. Edinburg Section A Math. 61, 186–194 (1942).

Sankhya

A. Bhattacharyya, “On some analogues of the amount of information and their use in statistical estimation,” Sankhya 8, 1–15, 201–218, 315–328 (1946).

Other

C. Bungay, “VASE™ characterization of unknown substrate,” in Measurements Report for Erik Rogala (J. A. Woollam, Lincoln, Neb., 1999).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1988), Vol. 26, pp. 349–393.
[CrossRef]

K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. SPIE680, 19–29 (1986).
[CrossRef]

E. Rogala, “Task-based assessment of a proposed phase-shifting interferometer/ellipsometer,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1999).

H. Cramer, Mathematical Methods of Statistics (Princeton University, Princeton, N.J., 1946).

H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 1 (Wiley, New York, 1968).

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Figures (17)

Fig. 1
Fig. 1

Schematic of the proposed Mach–Zehnder PSI. PZT, piezoelectric transducer.

Fig. 2
Fig. 2

Schematic of the fundamental building block of the Mach–Zehnder design illustrating the Scheimpflug condition and telecentricity in object and image space.

Fig. 3
Fig. 3

Relationship between the test arm tilt θ and the angle of incidence ϕ from an arbitrary point in the pupil, characterized by a radial coordinate r and an angular coordinate α.

Fig. 4
Fig. 4

GaAs reflectance for 632.8-nm light as a function of incident angle for both TE- and TM-polarized light.

Fig. 5
Fig. 5

TE and TM results at pixel (390,270) for phase offset investigation and measurement. PZT, piezoelectric transducer.

Fig. 6
Fig. 6

Representative complete fringe image. The underlying circular fringe pattern is a result of focus error.

Fig. 7
Fig. 7

Four TE fringe images: (a) 0°, (b) 90°, (c) 180°, (d) 270°.

Fig. 8
Fig. 8

Four (scaled) TM fringe images: (a) 0°, (b) 90°, (c) 180°, (d) 270°.

Fig. 9
Fig. 9

Reconstruction of n for the system parameters 44, 0.35, and 22.7°.

Fig. 10
Fig. 10

Reconstruction of k for the system parameters 44, 0.35, and 22.7°.

Fig. 11
Fig. 11

Unwrapped h reconstruction for the system parameters 44, 0.35, and 22.7°.

Fig. 12
Fig. 12

Reconstruction of h with the TE measurements and the conventional four-step reconstruction algorithm.

Fig. 13
Fig. 13

Reconstruction of h with the TM measurements and the conventional four-step reconstruction algorithm.

Fig. 14
Fig. 14

Relationship between the estimate results and the system parameters phase offset and scaleTM for levelTE = 43: (a) ML, (b) ML, (c) ĥ ML.

Fig. 15
Fig. 15

Histogram results of the camera output at pixel (290,270).

Fig. 16
Fig. 16

Scatterplot projections of Monte Carlo simulation results and Lorentz analysis results for a GaAs, F/12.5 system: 8 bit.

Fig. 17
Fig. 17

Scatterplot projections onto Lorentz analyzer results for the GaAs, F/12.5 system: scaleTM = 1.0.

Tables (6)

Tables Icon

Table 1 Reconstruction Results for the 501 Monte Carlo Data Trials: Mean Values

Tables Icon

Table 2 Reconstruction Results for the 501 Data Trials: Standard Deviation

Tables Icon

Table 3 Cramer–Rao Lower Bounds for the 501 Data Trial Analysis

Tables Icon

Table 4 Effects of Quantization on the Estimate of n, k, and h in the No-Noise Case for LevelTE = 50, ScaleTM = 0.35, and Phase Offset = 22.7°

Tables Icon

Table 5 Monte Carlo Results for 10,000 Trials and 1% Noise: Mean

Tables Icon

Table 6 Monte Carlo Results for 10,000 Trials and 1% Noise: Standard Deviation

Equations (18)

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

Ījmn, k, h=E0ref expimπ/2iλfrminrefrmaxrefθminrefθmaxref ×ρ˜jrefr, θrdrdθ+E0test expi2πh/λiλfrmintestrmaxtestθmintestθmaxtest ×ρ˜jtestr, θrdrdθ2.
ϕ=θ±tan-1r/f.
nˆ=0xˆ+-sin θyˆ+-cos θzˆ,
ν=r sin αxˆ+r cos αyˆ+-fzˆ.
nˆ · ν=nˆνcos ϕ.
cos ϕ=f cos θ-r cos α sin θr2+f21/2.
Ījmn, k, h=π expimπ/2iλf E00R ρ˜jrefrrdr+π expi2πh/λiλf E00R ρ˜jtestrrdr2.
ρ˜TEϕ=cosϕ-n+ikcosϕcosϕ+n+ikcosϕ,
ρ˜TMϕ=n+ikcosϕ-cosϕn+ikcosϕ+cosϕ,
cosϕ=+1-sinϕn+ik21/2.
Ijm=Ījmn, k, h+ηjm,
lnpI|θ=j=01m=03ln12π σN- 12σN2×Ijm-Ījmθ2,
θilnpI|θθ=θˆMLI=1σN2j=01m=03Ijm-ĪjmθθiĪjmθθ=θˆMLI=0.
Φn, k, h=-p I|θ=-12π σNexp-12σN2j=01m=03×Ijm-Ījmn, k, h2,
σθˆi2=θˆiI-θi2 J-1ii,
Jig=- d8IlnpI|θθilnpI|θθgpI|θ.
Jig=1σN2j=01m=03ĪjmθiĪjmθg,i, g=n, k, and h.
Δϕsystem=1.375 nm1 step360°632.8 nm 27 steps=21.12°.

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