Abstract

Real-time monitoring and control of the growth of plasma-deposited gradient-index structures by multiwavelength phase-modulated ellipsometry are presented. An efficient method for estimating the optical parameters based on a least-squares fitting on the most recent recorded measurements is described. This method is used for real-time monitoring of the outerlayer refractive index and rate of deposition during deposition of inhomogeneous transparent films. An accurate integral expansion of the reflection coefficient, giving a continuous optical model describing inhomogeneous films, is used in the real-time modeling of the deposited structure. A variety of increasing complexities of the optical model is studied within inversion algorithms. Furthermore several ways of defining the optical parameters to be estimated are discussed. Inversion of simulated growth trajectories by using the described algorithms show what kind of information is available from the various approximations and in what conditions they are useful. Since real-time measurements during growth involves statistical noise and systematic errors, it becomes necessary to stabilize the fit. Various stabilizing functionals are discussed and implemented to solve this problem. Several plasma-deposited silicon oxynitride gradient-index structures where both rate of deposition and the refractive index are varied continuously are inverted in real time. As an example of application, a successful real-time control of the growth of a linear gradient index is demonstrated by using inversion algorithms. Inversion algorithms are extremely fast, with calculation times from less than a second (for the lowest-order approximation) to ∼3 s.

© 1998 Optical Society of America

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  1. H. Fabricius, “Gradient-index filters: designing filters with steep skirts, high reflection, and quintic matching layers,” Appl. Opt. 31, 5191–5196 (1992).
    [CrossRef] [PubMed]
  2. W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8, 584–586 (1983).
    [CrossRef] [PubMed]
  3. A. Dobrowolski, “Optical properties of films and coatings,” in Handbook of Optics, M. Bass, ed., 2nd ed. (McGraw-Hill, New York, 1994).
  4. V. Nguyen Van, A. Brunet-Bruneau, S. Fisson, J. M. Frigerio, G. Vuye, Y. Wang, F. Abelés, M. Berger, P. Chaton, “Determination of refractive-index profiles by a combination of visible and infrared ellipsometry measurements,” Appl. Opt. 35, 5540–5544 (1996).
    [CrossRef] [PubMed]
  5. A. V. Tikhonravov, M. K. Trubetskov, “Program package for the ellipsometry of inhomogeneous layers,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 167–178 (1993).
    [CrossRef]
  6. A. V. Tikhonravov, M. K. Trubetskov, J. Hrdina, J. Sobota, “Characterization of quasirugate filters using ellipsometric measurements,” Thin Solid Films 277, 83–89 (1996).
    [CrossRef]
  7. P. G. Snyder, Y. Xiong, A. Woollam, G. A. Al-Jumaily, F. J. Gagliardi, “Graded refractive index silicon oxynitride thin film characterized by spectroscopic ellipsometry,” J. Vac. Sci. Technol. A 10, 1462–1466 (1992).
    [CrossRef]
  8. C. K. Carniglia, “Ellipsometric calculation for nonabsorbing thin films with linear refractive-index gradients,” J. Opt. Soc. Am. A 7, 848–855 (1990).
    [CrossRef]
  9. M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
    [CrossRef]
  10. M. Kildemo, S. Deniau, P. Bulkin, B. Drevillon, “Real time control of the growth of silicon alloy multilayers by multiwavelength ellipsometry,” Thin Solid Films 290-291, 46–50 (1996).
    [CrossRef]
  11. I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
    [CrossRef]
  12. W. M. Duncan, S. A. Henck, J. W. Kuehne, L. M. Loewenstein, S. Maung, “High-speed spectral ellipsometry for in situ diagnostics and process control,” J. Vac. Sci. Technol. B 12, 2779–2784 (1994).
    [CrossRef]
  13. M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
    [CrossRef]
  14. M. Kildemo, P. Bulkin, B. Drévillon, O. Hunderi, “Real time control by multiwavelength ellipsometry of plasma deposited multilayers on glass using incoherent reflection model,” Appl. Opt. 36, 6352–6359 (1997).
    [CrossRef]
  15. M. Kildemo, O. Hunderi, B. Drévillon, “Approximation of the reflection coefficient for rapid real time calculation of inhomogenous films,” J. Opt. Soc. Am. A 14, 931–939 (1997).
    [CrossRef]
  16. G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech. Signal. Process. 37, 2024–2036 (1989).
    [CrossRef]
  17. M. Bertero, C. de Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–212.
    [CrossRef]
  18. A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Winston-Wiley, New York, 1977).
  19. J. H. Kaiser, “Regularization in ellipsometry,” Appl. Phys. B 45, 1–5 (1988).
    [CrossRef]
  20. P. V. Bulkin, P. L. Swart, B. M. Lacquet, “Fourier-transform design and electron cyclotron resonance-PECVD of lossy graded-index optical coatings,” Appl. Opt. 35, 4413–4419 (1996).
    [CrossRef] [PubMed]
  21. B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: In situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mat. 27, 1–87 (1993).
    [CrossRef]
  22. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).
  23. D. E. Aspnes, “Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements,” J. Opt. Soc. Am. A 10, 974–983 (1993).
    [CrossRef]
  24. S. Kim, R. W. Collins, “Optical characterization of continuous compositional gradients in thin films by real time spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 3010–3012 (1995).
    [CrossRef]
  25. F. K. Urban, M. F. Tabet, “Development of artificial neural networks for in situ ellipsometry of films growing on unknown substrates,” J. Vac. Sci. Technol. A 12, 1952–1956 (1994).
    [CrossRef]
  26. D. A. G. Bruggeman, “Berechnung verschiedener physikalisher konstanten von heterogenen substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
  27. J. C. Maxwell Garnett, “Colors in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
    [CrossRef]
  28. F. Abeles, “Optical properties of inhomogeneous films,” Natl. Bur. Stand. (U.S.) Misc. Publ. 256, 41–58 (1964).

1997 (2)

1996 (6)

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

P. V. Bulkin, P. L. Swart, B. M. Lacquet, “Fourier-transform design and electron cyclotron resonance-PECVD of lossy graded-index optical coatings,” Appl. Opt. 35, 4413–4419 (1996).
[CrossRef] [PubMed]

V. Nguyen Van, A. Brunet-Bruneau, S. Fisson, J. M. Frigerio, G. Vuye, Y. Wang, F. Abelés, M. Berger, P. Chaton, “Determination of refractive-index profiles by a combination of visible and infrared ellipsometry measurements,” Appl. Opt. 35, 5540–5544 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, J. Hrdina, J. Sobota, “Characterization of quasirugate filters using ellipsometric measurements,” Thin Solid Films 277, 83–89 (1996).
[CrossRef]

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

M. Kildemo, S. Deniau, P. Bulkin, B. Drevillon, “Real time control of the growth of silicon alloy multilayers by multiwavelength ellipsometry,” Thin Solid Films 290-291, 46–50 (1996).
[CrossRef]

1995 (1)

S. Kim, R. W. Collins, “Optical characterization of continuous compositional gradients in thin films by real time spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 3010–3012 (1995).
[CrossRef]

1994 (2)

F. K. Urban, M. F. Tabet, “Development of artificial neural networks for in situ ellipsometry of films growing on unknown substrates,” J. Vac. Sci. Technol. A 12, 1952–1956 (1994).
[CrossRef]

W. M. Duncan, S. A. Henck, J. W. Kuehne, L. M. Loewenstein, S. Maung, “High-speed spectral ellipsometry for in situ diagnostics and process control,” J. Vac. Sci. Technol. B 12, 2779–2784 (1994).
[CrossRef]

1993 (3)

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: In situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mat. 27, 1–87 (1993).
[CrossRef]

D. E. Aspnes, “Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements,” J. Opt. Soc. Am. A 10, 974–983 (1993).
[CrossRef]

I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
[CrossRef]

1992 (2)

H. Fabricius, “Gradient-index filters: designing filters with steep skirts, high reflection, and quintic matching layers,” Appl. Opt. 31, 5191–5196 (1992).
[CrossRef] [PubMed]

P. G. Snyder, Y. Xiong, A. Woollam, G. A. Al-Jumaily, F. J. Gagliardi, “Graded refractive index silicon oxynitride thin film characterized by spectroscopic ellipsometry,” J. Vac. Sci. Technol. A 10, 1462–1466 (1992).
[CrossRef]

1990 (1)

1989 (1)

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech. Signal. Process. 37, 2024–2036 (1989).
[CrossRef]

1988 (1)

J. H. Kaiser, “Regularization in ellipsometry,” Appl. Phys. B 45, 1–5 (1988).
[CrossRef]

1983 (1)

1964 (1)

F. Abeles, “Optical properties of inhomogeneous films,” Natl. Bur. Stand. (U.S.) Misc. Publ. 256, 41–58 (1964).

1935 (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalisher konstanten von heterogenen substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

1904 (1)

J. C. Maxwell Garnett, “Colors in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
[CrossRef]

Abeles, F.

F. Abeles, “Optical properties of inhomogeneous films,” Natl. Bur. Stand. (U.S.) Misc. Publ. 256, 41–58 (1964).

Abelés, F.

Al-Jumaily, G. A.

P. G. Snyder, Y. Xiong, A. Woollam, G. A. Al-Jumaily, F. J. Gagliardi, “Graded refractive index silicon oxynitride thin film characterized by spectroscopic ellipsometry,” J. Vac. Sci. Technol. A 10, 1462–1466 (1992).
[CrossRef]

Arsenin, V.

A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Winston-Wiley, New York, 1977).

Aspnes, D. E.

Berger, M.

Bertero, M.

M. Bertero, C. de Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–212.
[CrossRef]

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechnung verschiedener physikalisher konstanten von heterogenen substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

Brunet-Bruneau, A.

Bulkin, P.

M. Kildemo, P. Bulkin, B. Drévillon, O. Hunderi, “Real time control by multiwavelength ellipsometry of plasma deposited multilayers on glass using incoherent reflection model,” Appl. Opt. 36, 6352–6359 (1997).
[CrossRef]

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

M. Kildemo, S. Deniau, P. Bulkin, B. Drevillon, “Real time control of the growth of silicon alloy multilayers by multiwavelength ellipsometry,” Thin Solid Films 290-291, 46–50 (1996).
[CrossRef]

Bulkin, P. V.

Carniglia, C. K.

Chaton, P.

Collins, R. W.

S. Kim, R. W. Collins, “Optical characterization of continuous compositional gradients in thin films by real time spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 3010–3012 (1995).
[CrossRef]

Dagenais, M.

I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
[CrossRef]

de Mol, C.

M. Bertero, C. de Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–212.
[CrossRef]

Demoment, G.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech. Signal. Process. 37, 2024–2036 (1989).
[CrossRef]

Deniau, S.

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

M. Kildemo, S. Deniau, P. Bulkin, B. Drevillon, “Real time control of the growth of silicon alloy multilayers by multiwavelength ellipsometry,” Thin Solid Films 290-291, 46–50 (1996).
[CrossRef]

Dobrowolski, A.

A. Dobrowolski, “Optical properties of films and coatings,” in Handbook of Optics, M. Bass, ed., 2nd ed. (McGraw-Hill, New York, 1994).

Dottelis, J. B.

I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
[CrossRef]

Drevillon, B.

M. Kildemo, S. Deniau, P. Bulkin, B. Drevillon, “Real time control of the growth of silicon alloy multilayers by multiwavelength ellipsometry,” Thin Solid Films 290-291, 46–50 (1996).
[CrossRef]

Drévillon, B.

M. Kildemo, O. Hunderi, B. Drévillon, “Approximation of the reflection coefficient for rapid real time calculation of inhomogenous films,” J. Opt. Soc. Am. A 14, 931–939 (1997).
[CrossRef]

M. Kildemo, P. Bulkin, B. Drévillon, O. Hunderi, “Real time control by multiwavelength ellipsometry of plasma deposited multilayers on glass using incoherent reflection model,” Appl. Opt. 36, 6352–6359 (1997).
[CrossRef]

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: In situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mat. 27, 1–87 (1993).
[CrossRef]

Duncan, W. M.

W. M. Duncan, S. A. Henck, J. W. Kuehne, L. M. Loewenstein, S. Maung, “High-speed spectral ellipsometry for in situ diagnostics and process control,” J. Vac. Sci. Technol. B 12, 2779–2784 (1994).
[CrossRef]

Fabricius, H.

Fisson, S.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).

Frigerio, J. M.

Gagliardi, F. J.

P. G. Snyder, Y. Xiong, A. Woollam, G. A. Al-Jumaily, F. J. Gagliardi, “Graded refractive index silicon oxynitride thin film characterized by spectroscopic ellipsometry,” J. Vac. Sci. Technol. A 10, 1462–1466 (1992).
[CrossRef]

Henck, S. A.

W. M. Duncan, S. A. Henck, J. W. Kuehne, L. M. Loewenstein, S. Maung, “High-speed spectral ellipsometry for in situ diagnostics and process control,” J. Vac. Sci. Technol. B 12, 2779–2784 (1994).
[CrossRef]

Hrdina, J.

A. V. Tikhonravov, M. K. Trubetskov, J. Hrdina, J. Sobota, “Characterization of quasirugate filters using ellipsometric measurements,” Thin Solid Films 277, 83–89 (1996).
[CrossRef]

Hunderi, O.

Kaiser, J. H.

J. H. Kaiser, “Regularization in ellipsometry,” Appl. Phys. B 45, 1–5 (1988).
[CrossRef]

Kildemo, M.

M. Kildemo, O. Hunderi, B. Drévillon, “Approximation of the reflection coefficient for rapid real time calculation of inhomogenous films,” J. Opt. Soc. Am. A 14, 931–939 (1997).
[CrossRef]

M. Kildemo, P. Bulkin, B. Drévillon, O. Hunderi, “Real time control by multiwavelength ellipsometry of plasma deposited multilayers on glass using incoherent reflection model,” Appl. Opt. 36, 6352–6359 (1997).
[CrossRef]

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

M. Kildemo, S. Deniau, P. Bulkin, B. Drevillon, “Real time control of the growth of silicon alloy multilayers by multiwavelength ellipsometry,” Thin Solid Films 290-291, 46–50 (1996).
[CrossRef]

Kim, S.

S. Kim, R. W. Collins, “Optical characterization of continuous compositional gradients in thin films by real time spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 3010–3012 (1995).
[CrossRef]

Kuehne, J. W.

W. M. Duncan, S. A. Henck, J. W. Kuehne, L. M. Loewenstein, S. Maung, “High-speed spectral ellipsometry for in situ diagnostics and process control,” J. Vac. Sci. Technol. B 12, 2779–2784 (1994).
[CrossRef]

Lacquet, B. M.

Loewenstein, L. M.

W. M. Duncan, S. A. Henck, J. W. Kuehne, L. M. Loewenstein, S. Maung, “High-speed spectral ellipsometry for in situ diagnostics and process control,” J. Vac. Sci. Technol. B 12, 2779–2784 (1994).
[CrossRef]

Maung, S.

W. M. Duncan, S. A. Henck, J. W. Kuehne, L. M. Loewenstein, S. Maung, “High-speed spectral ellipsometry for in situ diagnostics and process control,” J. Vac. Sci. Technol. B 12, 2779–2784 (1994).
[CrossRef]

Maxwell Garnett, J. C.

J. C. Maxwell Garnett, “Colors in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
[CrossRef]

Nguyen Van, V.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).

Snyder, P. G.

P. G. Snyder, Y. Xiong, A. Woollam, G. A. Al-Jumaily, F. J. Gagliardi, “Graded refractive index silicon oxynitride thin film characterized by spectroscopic ellipsometry,” J. Vac. Sci. Technol. A 10, 1462–1466 (1992).
[CrossRef]

Sobota, J.

A. V. Tikhonravov, M. K. Trubetskov, J. Hrdina, J. Sobota, “Characterization of quasirugate filters using ellipsometric measurements,” Thin Solid Films 277, 83–89 (1996).
[CrossRef]

Southwell, W. H.

Swart, P. L.

Tabet, M. F.

F. K. Urban, M. F. Tabet, “Development of artificial neural networks for in situ ellipsometry of films growing on unknown substrates,” J. Vac. Sci. Technol. A 12, 1952–1956 (1994).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).

Tikhonov, A.

A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Winston-Wiley, New York, 1977).

Tikhonravov, A. V.

A. V. Tikhonravov, M. K. Trubetskov, J. Hrdina, J. Sobota, “Characterization of quasirugate filters using ellipsometric measurements,” Thin Solid Films 277, 83–89 (1996).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, “Program package for the ellipsometry of inhomogeneous layers,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 167–178 (1993).
[CrossRef]

Trubetskov, M. K.

A. V. Tikhonravov, M. K. Trubetskov, J. Hrdina, J. Sobota, “Characterization of quasirugate filters using ellipsometric measurements,” Thin Solid Films 277, 83–89 (1996).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, “Program package for the ellipsometry of inhomogeneous layers,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 167–178 (1993).
[CrossRef]

Urban, F. K.

F. K. Urban, M. F. Tabet, “Development of artificial neural networks for in situ ellipsometry of films growing on unknown substrates,” J. Vac. Sci. Technol. A 12, 1952–1956 (1994).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).

Viano, G. A.

M. Bertero, C. de Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–212.
[CrossRef]

Vuye, G.

Wang, Y.

Woollam, A.

P. G. Snyder, Y. Xiong, A. Woollam, G. A. Al-Jumaily, F. J. Gagliardi, “Graded refractive index silicon oxynitride thin film characterized by spectroscopic ellipsometry,” J. Vac. Sci. Technol. A 10, 1462–1466 (1992).
[CrossRef]

Wu, I. F.

I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
[CrossRef]

Xiong, Y.

P. G. Snyder, Y. Xiong, A. Woollam, G. A. Al-Jumaily, F. J. Gagliardi, “Graded refractive index silicon oxynitride thin film characterized by spectroscopic ellipsometry,” J. Vac. Sci. Technol. A 10, 1462–1466 (1992).
[CrossRef]

Ann. Phys. (Leipzig) (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalisher konstanten von heterogenen substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

Appl. Opt. (4)

Appl. Phys. B (1)

J. H. Kaiser, “Regularization in ellipsometry,” Appl. Phys. B 45, 1–5 (1988).
[CrossRef]

Appl. Phys. Lett. (2)

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

S. Kim, R. W. Collins, “Optical characterization of continuous compositional gradients in thin films by real time spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 3010–3012 (1995).
[CrossRef]

IEEE Trans. Acoust. Speech. Signal. Process. (1)

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech. Signal. Process. 37, 2024–2036 (1989).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Vac. Sci. Technol. A (3)

F. K. Urban, M. F. Tabet, “Development of artificial neural networks for in situ ellipsometry of films growing on unknown substrates,” J. Vac. Sci. Technol. A 12, 1952–1956 (1994).
[CrossRef]

P. G. Snyder, Y. Xiong, A. Woollam, G. A. Al-Jumaily, F. J. Gagliardi, “Graded refractive index silicon oxynitride thin film characterized by spectroscopic ellipsometry,” J. Vac. Sci. Technol. A 10, 1462–1466 (1992).
[CrossRef]

I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
[CrossRef]

J. Vac. Sci. Technol. B (1)

W. M. Duncan, S. A. Henck, J. W. Kuehne, L. M. Loewenstein, S. Maung, “High-speed spectral ellipsometry for in situ diagnostics and process control,” J. Vac. Sci. Technol. B 12, 2779–2784 (1994).
[CrossRef]

Natl. Bur. Stand. (U.S.) Misc. Publ. (1)

F. Abeles, “Optical properties of inhomogeneous films,” Natl. Bur. Stand. (U.S.) Misc. Publ. 256, 41–58 (1964).

Opt. Lett. (1)

Philos. Trans. R. Soc. London (1)

J. C. Maxwell Garnett, “Colors in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
[CrossRef]

Prog. Cryst. Growth Charact. Mat. (1)

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: In situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mat. 27, 1–87 (1993).
[CrossRef]

Rev. Sci. Instrum. (1)

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

Thin Solid Films (2)

M. Kildemo, S. Deniau, P. Bulkin, B. Drevillon, “Real time control of the growth of silicon alloy multilayers by multiwavelength ellipsometry,” Thin Solid Films 290-291, 46–50 (1996).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, J. Hrdina, J. Sobota, “Characterization of quasirugate filters using ellipsometric measurements,” Thin Solid Films 277, 83–89 (1996).
[CrossRef]

Other (5)

A. Dobrowolski, “Optical properties of films and coatings,” in Handbook of Optics, M. Bass, ed., 2nd ed. (McGraw-Hill, New York, 1994).

A. V. Tikhonravov, M. K. Trubetskov, “Program package for the ellipsometry of inhomogeneous layers,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 167–178 (1993).
[CrossRef]

M. Bertero, C. de Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–212.
[CrossRef]

A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Winston-Wiley, New York, 1977).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).

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Figures (11)

Fig. 1
Fig. 1

Kinetic window with the most recent measurement at t j and the oldest measurements at t i+1. The dotted box illustrates the window sliding from t i+1 to t i+2, with the acquisition of new measurements at t j+1.

Fig. 2
Fig. 2

Inversion of a simulated linear gradient-index profile from 1.7 to 1.5 in 400 nm (B = 0.5 μm-1), by using the WKBJ algorithm. (Top) fitted refractive indices n N and n f and the nominal profile. (Bottom) physical thickness and optical thickness β as defined in Eq. (4) together with the nominal thicknesses.

Fig. 3
Fig. 3

Inversion of a simulated linear gradient-index profile at 2.15 eV, from 1.7 to 1.5 in 400 nm at B = 0.5 μm-1, by using the Bruggeman EMT form of the WKBJ algorithm. The simulated profile for the three wavelengths, 1.8, 2.2, 2.6 eV, is calculated by using the Bruggeman EMT assuming two components, SiO2 and Si3N4. (Top) fitted refractive indices n N obtained from the fitted composition f N and n f obtained from f f and the nominal refractive-index profile at 2.6 eV. (Bottom) physical and optical thicknesses β as defined in Eq. (4) together with nominal thicknesses.

Fig. 4
Fig. 4

Inversion of a simulated linear gradient-index profile from 1.7 to 1.5 in 200 nm (B = 1 μm-1) on glass by using the single-integral algorithm. (Top) fitted parameters, B =dn/dy and n N , together with the resulting refractive-index profile n(t). (Bottom) fitted rate of deposition r d on the right axis and the thickness and optical thickness as defined in Eq. (4) on the left axis.

Fig. 5
Fig. 5

Inversion of a simulated linear gradient-index profile from 2.5 to 1.5 in 200 nm (B = 5 μm-1) on c-Si by using the single-integral algorithm. (Top) fitted parameters, B = dn/dy and n N , together with the resulting refractive-index profile n(t). (Bottom) fitted rate of deposition r d on the right axis and the thickness and optical thickness as defined in Eq. (4) on the left axis.

Fig. 6
Fig. 6

Real-time monitoring of thickness (solid curve) during deposition of SiO2 with a linearly time-varying deposition rate proportional to the SiH4 flow (dotted line).

Fig. 7
Fig. 7

Real-time monitoring during the growth of a gradient-index layer by using the WKBJ and the single-integral algorithm. The refractive index varies nominally linearly between 1.7 and 1.5 in 100 nm. (Top) fitted refractive index found by the single-integral and the WKBJ algorithms. (Bottom) thickness and optical thickness as defined in Eq. (4), found by the single-integral and the WKBJ algorithm.

Fig. 8
Fig. 8

Real-time monitoring of the growth of a gradient-index layer, where both the refractive index and the rate of deposition vary continuously during the deposition. (Top) fitted refractive index and rate of deposition found by applying the WKBJ, the single-integral method, and spectroscopic PME. (Bottom) physical and optical thickness found by the single integral and the WKBJ algorithm.

Fig. 9
Fig. 9

Nominal or target refractive-index profile and the refractive index found by WKBJ in real time and the refractive index obtained by the single-integral algorithm, plotted versus the fitted thickness used in real-time control. The flow rate of N2O is controlled as a function of the fitted thickness, as shown in the figure.

Fig. 10
Fig. 10

Physical thickness applied for real-time control and the optical thickness as defined in Eq. (4), obtained in real time by using the WKBJ algorithm. The physical and optical thicknesses obtained from the single-integral algorithm are shown by solid curves.

Fig. 11
Fig. 11

Spectroscopic ellipsometry after the controlled deposition of the Inhomogeneous layer. The target spectra of (Ψ, Δ) and the measured (Ψ, Δ) after controlled deposition are shown.

Equations (26)

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I s = sin 2 ψ sin Δ = 2   Im r s r p * r s r s * + r p r p * , I c = sin 2 ψ cos Δ = 2   Re r s r p * r s r s * + r p r p * .
r ˆ i = r i + r ˆ i + 1 exp 2 i β i 1 + r i r ˆ i + 1 exp 2 i β i ,
β i = 2 π λ   d i ε i - ε a sin 2   φ 0 1 / 2 ,
β i = 2 π λ   β i .
r ˆ = r 0 , 1 + I 1 + r 0 , 1 r N I 2 exp i 2 β d + r 0 , 1 DI 1 + r N DI 2 exp i 2 β d + TI 1 + r 0 , 1 r N TI 2 exp i 2 β d + + r N exp i 2 β d 1 + r 0 , 1 I 1 + r N I 2 exp i 2 β d + DI 1 + r 0 , 1 r N DI 2 exp i 2 β d + r 0 , 1 TI 1 + r N TI 2 exp i 2 β d + + r 0 , 1 r N exp i 2 β d ,
β z = 2 π λ 0 z ε ζ - ε a sin 2   φ 0 1 / 2 d ζ
I 1 = 0 d 1 2 η z d η z d z exp 2 i β z d z
η = n   cos   φ = ε - ε a sin 2   φ o 1 / 2 n cos   φ = ε ε - ε a sin 2   φ o 1 / 2 s   polarization p   polarization .
r ˆ = r 0 , 1 + I 1 + r 0 , 1 r N I 2 exp i 2 β d + r N exp i 2 β d 1 + r 0 , 1 I 1 + r N I 2 exp i 2 β d + r 0 , 1 r N exp i 2 β d ,
r = r 0 , 1 + r N exp i 2 β d 1 + r 0 , 1 r N exp i 2 β d .
ρ 2 = k = 1 3 r = i + 1 j I s x ¯ i , θ ¯ i + 1 , t r , λ k - I smeas t r , λ k 2 δ k , r 2 + I c x ¯ i , θ ¯ i + 1 , t r , λ k - I cmeas t r , λ k 2 δ k , r 2 ,
d t = 0 t   r d τ d τ .
y j = y i + dy j , i ,
dy j , i = r d Δ T s j - i
y i + 1 = y i + r d Δ T s .
β t = 0 t   r d τ n 2 - ε a 2 sin 2 φ 0 1 / 2 d τ .
n y = n N + 0 y   B ξ d ξ ,
B = dn dy .
n j = n N + IB i + Bdy j , i ,
I j = I i + d η j , i 2 η j exp i 2 d β j , i ,
dn eff dz = 1 2 ε eff d ε eff df v df v dz .
M α = ρ 2 + α Ω θ ¯ ,
Ω =   d 2 n dz 2 2 =   dB dz 2 d z .
Ω = ln n N - b n N 0 - b 2 ,
Ω = ln n N - b n N 0 - b - n N - b n N 0 - b - 1 + 1 2 n N - b n N 0 - b - 1 2 2 .
λ 2 π η 3 n dn dz     1 .

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