Abstract

A novel interferometer based upon a conventional phase-shifting design is further investigated. This interferometer is capable of measuring both the real and imaginary parts of the complex index of refraction and the surface profile of a test surface. Maximum-likelihood estimation theory is shown to be an effective means of extracting the three parameters of interest from the measured data. Cramér–Rao lower bounds are introduced as a means of quantitatively assessing the performance of the system. Furthermore, it is shown that as the design parameters are optimized, the results approach the theoretical performance limit. We conclude by developing the underlying theory behind the relationship of the complex-index-of-refraction estimates to the surface-profile estimate.

© 1998 Optical Society of America

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References

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  1. Y. Liu, C. W. See, M. G. Somekh, “Common path interferometric microellipsometry,” Proc. SPIE 2782, 635–645 (1996).
    [CrossRef]
  2. R. D. Holmes, C. W. See, M. G. Somekh, “Scanning microellipsometry for extraction of true topography,” Electron. Lett. 31, 358–359 (1995).
    [CrossRef]
  3. C. W. See, R. K. Appel, M. G. Somekh, “Scanning differential optical profilometer for simultaneous measurement of amplitude and phase variation,” Appl. Phys. Lett. 53, 10–12 (1988).
    [CrossRef]
  4. Y. Lin, C. Chou, K. Chang, “Real-time interferometric ellipsometry with optical heterodyne and phase lock-in techniques,” Appl. Opt. 29, 5159–5162 (1990).
    [CrossRef] [PubMed]
  5. S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
    [CrossRef]
  6. A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 1301–1303 (1992).
    [CrossRef]
  7. N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. patent4,999,014 (March12, 1991).
  8. H. F. Hazebroek, A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
    [CrossRef]
  9. T. Mishima, K. C. Kao, “Detection of thickness uniformity of film layers in semiconductor devices by spatially resolved ellipsometry,” Opt. Eng. 21, 1074–1078 (1982).
    [CrossRef]
  10. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  11. M. Pluta, Advanced Light Microscopy (Elsevier, New York, 1989).
  12. E. W. Rogala, H. H. Barrett, “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]
  13. E. W. Rogala, H. H. Barrett, “Phase-shifting interferometry and maximum-likelihood estimation theory,” Appl. Opt. 36, 8871–8876 (1997).
    [CrossRef]
  14. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
    [CrossRef]
  15. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 349–393.
  16. K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. SPIE680, 19–29 (1986).
    [CrossRef]
  17. D. Apostol, P. C. Logofatu, V. Damian, A. Dobrolu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. (Bellingham) 35, 1288–1291 (1996).
    [CrossRef]
  18. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Eng. 7, 368–370 (1982).
  19. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]
  20. H. Cramer, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946).
  21. C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).
  22. A. C. Aitkin, H. Silverstone, “On the estimation of statistical parameters,” Proc. R. Soc. Edinburgh Sect. A 61, 186–194 (1942).
  23. R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 700–725 (1925).
    [CrossRef]
  24. D. Dugue, “Application des propriétes de la limite au sens de calcul des probabilities a l’étude des diverses questions d’estimation,” Ecol. Poly. 3, 305–372 (1937).
  25. H. L. Van Trees, Detection, Estimation Theory, and Linear Modulation Theory, Part 1 (Wiley, New York, 1968).
  26. E. W. Barankin, “Locally best unbiased estimators,” Ann. Math. Stat. 20, 477–501 (1949).
    [CrossRef]
  27. 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).
  28. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

1998

1997

1996

Y. Liu, C. W. See, M. G. Somekh, “Common path interferometric microellipsometry,” Proc. SPIE 2782, 635–645 (1996).
[CrossRef]

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

1995

R. D. Holmes, C. W. See, M. G. Somekh, “Scanning microellipsometry for extraction of true topography,” Electron. Lett. 31, 358–359 (1995).
[CrossRef]

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
[CrossRef]

1992

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 1301–1303 (1992).
[CrossRef]

1990

1988

C. W. See, R. K. Appel, M. G. Somekh, “Scanning differential optical profilometer for simultaneous measurement of amplitude and phase variation,” Appl. Phys. Lett. 53, 10–12 (1988).
[CrossRef]

1984

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

1982

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Eng. 7, 368–370 (1982).

T. Mishima, K. C. Kao, “Detection of thickness uniformity of film layers in semiconductor devices by spatially resolved ellipsometry,” Opt. Eng. 21, 1074–1078 (1982).
[CrossRef]

1973

H. F. Hazebroek, A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

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. Edinburgh Sect. A 61, 186–194 (1942).

1937

D. Dugue, “Application des propriétes de la limite au sens de calcul des probabilities a l’étude 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. Edinburgh Sect. A 61, 186–194 (1942).

Apostol, D.

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

Appel, R. K.

C. W. See, R. K. Appel, M. G. Somekh, “Scanning differential optical profilometer for simultaneous measurement of amplitude and phase variation,” Appl. Phys. Lett. 53, 10–12 (1988).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Barankin, E. W.

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

Barrett, H. H.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

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).

Chang, K.

Chou, C.

Cramer, H.

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

Creath, K.

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

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

Damian, V.

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

Dobrolu, A.

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

Dugue, D.

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

Fanton, J. T.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 1301–1303 (1992).
[CrossRef]

Fisher, R. A.

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

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

Gold, N.

N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. patent4,999,014 (March12, 1991).

Greivenkamp, J. E.

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

Hazebroek, H. F.

H. F. Hazebroek, A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

Holmes, R. D.

R. D. Holmes, C. W. See, M. G. Somekh, “Scanning microellipsometry for extraction of true topography,” Electron. Lett. 31, 358–359 (1995).
[CrossRef]

Holscher, A. A.

H. F. Hazebroek, A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

Juskaitis, R.

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

Kao, K. C.

T. Mishima, K. C. Kao, “Detection of thickness uniformity of film layers in semiconductor devices by spatially resolved ellipsometry,” Opt. Eng. 21, 1074–1078 (1982).
[CrossRef]

Kelso, S. M.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 1301–1303 (1992).
[CrossRef]

Lin, Y.

Liu, Y.

Y. Liu, C. W. See, M. G. Somekh, “Common path interferometric microellipsometry,” Proc. SPIE 2782, 635–645 (1996).
[CrossRef]

Logofatu, P. C.

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

Mishima, T.

T. Mishima, K. C. Kao, “Detection of thickness uniformity of film layers in semiconductor devices by spatially resolved ellipsometry,” Opt. Eng. 21, 1074–1078 (1982).
[CrossRef]

Morgan, C. J.

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Eng. 7, 368–370 (1982).

Opsal, J.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 1301–1303 (1992).
[CrossRef]

N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. patent4,999,014 (March12, 1991).

Pluta, M.

M. Pluta, Advanced Light Microscopy (Elsevier, New York, 1989).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

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. W.

Rosencwaig, A.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 1301–1303 (1992).
[CrossRef]

N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. patent4,999,014 (March12, 1991).

See, C. W.

Y. Liu, C. W. See, M. G. Somekh, “Common path interferometric microellipsometry,” Proc. SPIE 2782, 635–645 (1996).
[CrossRef]

R. D. Holmes, C. W. See, M. G. Somekh, “Scanning microellipsometry for extraction of true topography,” Electron. Lett. 31, 358–359 (1995).
[CrossRef]

C. W. See, R. K. Appel, M. G. Somekh, “Scanning differential optical profilometer for simultaneous measurement of amplitude and phase variation,” Appl. Phys. Lett. 53, 10–12 (1988).
[CrossRef]

Shatalin, S. V.

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

Silverstone, H.

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

Somekh, M. G.

Y. Liu, C. W. See, M. G. Somekh, “Common path interferometric microellipsometry,” Proc. SPIE 2782, 635–645 (1996).
[CrossRef]

R. D. Holmes, C. W. See, M. G. Somekh, “Scanning microellipsometry for extraction of true topography,” Electron. Lett. 31, 358–359 (1995).
[CrossRef]

C. W. See, R. K. Appel, M. G. Somekh, “Scanning differential optical profilometer for simultaneous measurement of amplitude and phase variation,” Appl. Phys. Lett. 53, 10–12 (1988).
[CrossRef]

Tan, J. B.

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

Van Trees, H. L.

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

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

Willenborg, D.

N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. patent4,999,014 (March12, 1991).

Willenborg, D. L.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 1301–1303 (1992).
[CrossRef]

Wilson, T.

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

Ann. Math. Stat.

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

Appl. Opt.

Appl. Phys. Lett.

A. Rosencwaig, J. Opsal, D. L. Willenborg, S. M. Kelso, J. T. Fanton, “Beam profile reflectometry: a new technique for dielectric film measurements,” Appl. Phys. Lett. 60, 1301–1303 (1992).
[CrossRef]

C. W. See, R. K. Appel, M. G. Somekh, “Scanning differential optical profilometer for simultaneous measurement of amplitude and phase variation,” Appl. Phys. Lett. 53, 10–12 (1988).
[CrossRef]

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 propriétes de la limite au sens de calcul des probabilities a l’étude des diverses questions d’estimation,” Ecol. Poly. 3, 305–372 (1937).

Electron. Lett.

R. D. Holmes, C. W. See, M. G. Somekh, “Scanning microellipsometry for extraction of true topography,” Electron. Lett. 31, 358–359 (1995).
[CrossRef]

J. Microsc.

S. V. Shatalin, R. Juskaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin-film structures,” J. Microsc. 179, pt. 3, 241–252 (1995).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. E

H. F. Hazebroek, A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

Opt. Eng.

T. Mishima, K. C. Kao, “Detection of thickness uniformity of film layers in semiconductor devices by spatially resolved ellipsometry,” Opt. Eng. 21, 1074–1078 (1982).
[CrossRef]

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Eng. 7, 368–370 (1982).

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

Opt. Eng. (Bellingham)

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

Proc. Cambridge Philos. Soc.

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

Proc. R. Soc. Edinburgh Sect. A

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

Proc. SPIE

Y. Liu, C. W. See, M. G. Somekh, “Common path interferometric microellipsometry,” Proc. SPIE 2782, 635–645 (1996).
[CrossRef]

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

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 274–328.

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

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

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

M. Pluta, Advanced Light Microscopy (Elsevier, New York, 1989).

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

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

N. Gold, D. Willenborg, J. Opsal, A. Rosencwaig, “Method and apparatus for measuring thickness of thin films,” U.S. patent4,999,014 (March12, 1991).

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Figures (28)

Fig. 1
Fig. 1

Twyman–Green interferometer configuration under consideration. Only a single point on the test surface is under consideration here.

Fig. 2
Fig. 2

Polarization pupil mask. The transmitting regions are defined by rmin, rmax, and α0.

Fig. 3
Fig. 3

Relationship between pupil coordinate and angle of incidence onto the test surface.

Fig. 4
Fig. 4

Scatterplot of the 20,000 ML estimates for the F/1.0 simulation, where the minimum radius of the pupil was 10 mm and the maximum was 20 mm. No tilt of the test surface was employed here.

Fig. 5
Fig. 5

Histograms showing the distribution of the ML estimates for the F/1.0 simulation, where the minimum radius of the pupil was 10 mm and the maximum was 20 mm. No tilt of the test surface was employed here. (a) nˆML(I), (b) kˆML(I), (c) hˆML(I).

Fig. 6
Fig. 6

Same as Fig. 4, except that the minimum radius of the pupil was 24 mm and the maximum was 25 mm.

Fig. 7
Fig. 7

Same as Fig. 5, except that the minimum radius of the pupil was 24 mm and the maximum was 25 mm.

Fig. 8
Fig. 8

Scatterplot of the 20,000 ML estimates for the F/2.8 simulation, where the minimum radius of the pupil was 0 mm and the maximum was 25 mm. No tilt of the test surface was employed here.

Fig. 9
Fig. 9

Histograms showing the distribution of the ML estimates for the F/2.8 simulation, where the minimum radius of the pupil was 0 mm and the maximum was 25 mm. No tilt of the test surface was employed here. (a) nˆML(I), (b) kˆML(I), (c) hˆML(I).

Fig. 10
Fig. 10

Proposed design for a PSI incorporating tilt into the interferometer design.

Fig. 11
Fig. 11

Relationship between the test arm tilt θ and the angle of incidence ϕ from an arbitrary point in the pupil.

Fig. 12
Fig. 12

Scatterplot of the 20,000 ML estimates for the F/1.0 simulation, where the test surface was tilted 45° with respect to the optical axis.

Fig. 13
Fig. 13

Histograms showing the distribution of the ML estimates for the F/1.0 simulation, where the test surface was tilted 45° with respect to the optical axis: (a) nˆML(I), (b) kˆML(I), (c) hˆML(I).

Fig. 14
Fig. 14

Same as Fig. 12, but for the F/11.2 simulation.

Fig. 15
Fig. 15

Same as Fig. 13, but for the F/11.2 simulation.

Fig. 16
Fig. 16

Relationship between theoretical Cramér–Rao lower bound on the estimation of the parameter n and test surface tilt for F/1.0, F/2.8, F/5.6, and F/11.2 systems.

Fig. 17
Fig. 17

Same as Fig. 16, but for the parameter k.

Fig. 18
Fig. 18

Same as Fig. 16, but for the parameter h.

Fig. 19
Fig. 19

Same as Fig. 16, but for the parameter h in the h-alone estimation case.

Fig. 20
Fig. 20

Monte Carlo simulation results. The crosses denote the wide-open pupil mask design (r=[0 mm, 25 mm]). The open circles denote the restricted mask design (r=[24 mm, 25 mm]). (a) σnˆML(I), (b) σkˆML(I), (c) σhˆML(I).

Fig. 21
Fig. 21

Magnitude of the complex reflectivity for gold at 589.3-nm wavelength as a function of incident angle.

Fig. 22
Fig. 22

Phase associated with reflection for gold at 589.3-nm wavelength as a function of incident angle.

Fig. 23
Fig. 23

Estimators nˆML(I) and kˆML(I) as a function of the test surface parameters and hˆML(I).

Fig. 24
Fig. 24

Overlay of Monte Carlo results and derived analytical relationships for the F/1.0, 0° tilt design, where rmin=0 mm and rmax=25 mm.

Fig. 25
Fig. 25

Same as Fig. 24, but for rmin=24 mm.

Fig. 26
Fig. 26

Overlay of Monte Carlo results and derived analytical relationships for the F/2.8, 0° tilt design, where rmin=24 mm and rmax=25 mm.

Fig. 27
Fig. 27

Overlay of Monte Carlo results and derived analytical relationships for the F/1.0, 45° tilt design, where rmin=0 mm and rmax=25 mm.

Fig. 28
Fig. 28

Same as Fig. 27, but for the F/11.2 system.

Tables (7)

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Table 1 Monte Carlo Results and Cramér–Rao Lower Bounds for the Initial PSI Configuration with F/1.0 Cone Incident at 0° Tilt

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Table 2 Monte Carlo Results and Cramér–Rao Lower Bounds for the Modified PSI Configuration with F/1.0 Cone Incident at 0° Tilt with rpupil=[10 mm, 20 mm]

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Table 3 Monte Carlo Results and Cramér–Rao Lower Bounds for the Modified PSI Configuration with F/1.0 Cone Incident at 0° Tilt with rpupil=[24 mm, 25 mm]

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Table 4 Monte Carlo Results and Cramér–Rao Lower Bounds for the Modified PSI Configuration with F/2.8 Cone Incident at 0° Tilt

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Table 5 Monte Carlo Results and Cramér–Rao Lower Bounds for the Modified PSI Configuration with F/2.8 Cone Incident at 0° Tilt with rpupil=[24 mm, 25 mm]

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Table 6 Monte Carlo (MC) Results and Cramér–Rao (CR) Lower Bounds for the Modified PSI Configuration Incorporating Tilt and the F/1.0 System

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Table 7 Monte Carlo (MC) Results and Cramér–Rao (CR) Lower Bounds for the F/1.0, F/2.8, F/5.6, and F/11.2 Systems at 45° Tilt

Equations (48)

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I¯jm(n, k, h)
=exp[i(mπ/2)]iλfE0(ref)4α0(ref)rmin(ref)rmax(ref)ρ˜j(ref)(r)r dr+expi4πhλiλfE0(test)4α0(test)rmax(test)rmax(test)ρ˜j(test)(r)r dr2.
ϕ=tan-1(r/f ).
Ijm=I¯jm(n, k, h)+ηjm,
ln[p(I|θ)]
=j=01m=03ln12πσN-12σN2[Ijm-I¯jm(θ)]2,
θiln[p(Iθ)]θ=θˆML(I)=1σN2j=01m=03[Ijm-I¯jm(θ)]×θi[I¯jm(θ)]θ=θˆML(I)=0.
σi2=[aˆi(I)-θi]2[J-l]ii,
Jig=-d8I ln[p(I|θ)]θi ln[p(I|θ)]θgp(I|θ).
Jig=1σN2j=01m=03I¯jmθiI¯jmθg,i, g=n, k, h.
Φ(n, k, h)
=-p(I|θ)=-12πσN×exp-12σN2j=01m=03[Ijm-I¯jm(n, k, h)]2.
σN=1%(Iref+Itest).
nˆ=0xˆ+(-sin θ)yˆ+(-cos θ)zˆ,
ν=(r sin α)xˆ+(r cos α)yˆ+(-f )zˆ.
nˆ·ν=|nˆ||ν|cos ϕ.
cos ϕ=f cos θ-r cos α sin θr2+f2.
Ijm(n, k, h)=ρR2IR+ρT2IT+2ρRρTIRIT×cosmπ2+4πh0λ+ΔφRT+ηjm.
Ijm(nˆ, kˆ, hˆ)=ρR2IR+ρˆT2IT+2ρRρˆTIRIT×cosmπ2+4πhˆλ+ΔφˆRT.
ρˆT=ρT,
mπ2+4πhˆλ+ΔφˆRT=mπ2+4πh0λ+ΔφRT.
φˆT=φT+4π(h0-hˆ)λ.
ρˆ=+Rˆ2+Iˆ2=ρT,
φˆT=tan-1IˆRˆ=φT+4π(h0-hˆ)λ.
Iˆ=Rˆ tanφT+4π(h0-hˆ)λ,
Rˆ=±ρT21+tan2φT+4πλ(h0-hˆ).
T=tanφT+4π(h0-hˆ)λ,
γ=ρT21+T21/2.
Rˆ=±γ,
Iˆ=±γT.
ρ˜=rminrmaxαminαmaxp˜(r, α)rdrdα,
ρ˜TE=1-N1+N,
ρ˜TM=N-1N+1,
ρ˜TE=1-(n2+k2)(1+n)2+k2+i -2k(1+n)2+k2,
ρ˜TM=-1+(n2+k2)(1+n)2+k2+i 2k(1+n)2+k2.
-2kˆ(1+nˆ)2+kˆ2=-γT.
2kˆ=γT(1+nˆ)2+γTkˆ2,
(1+nˆ)2=2kˆγT-kˆ2.
1-(nˆ2+kˆ2)(1+nˆ)2+kˆ2=-γ,
(1-nˆ2)-kˆ2=-γ(1+nˆ)2-γkˆ2.
1-nˆ2=(1-nˆ)(1+nˆ)=-(-1+nˆ)(1+nˆ)=-(1+nˆ-2)(1+nˆ)=-(1+nˆ)2+2(1+nˆ).
-(1+nˆ)2+γ(1+nˆ)2+2(1+nˆ)-kˆ2+γkˆ2=0.
±2kˆγT-kˆ2=kˆT1γ-1.
kˆ2(1-γ)2(γT)2+1-kˆ 2γT=0.
kˆ=0,
kˆ=2γT(1-γ)2+(γT)2.
nˆ=2(1-γ)(1-γ)2+(γT)2-1,kˆ=2γT(1-γ)2+(γT)2,
γ=ρT21+T21/2,T=tanφT+4π(h0-hˆ)λ.

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