Abstract

A new data reduction method is presented for single-wavelength ellipsometry. A genetic algorithm is applied to ellipsometric data to find the best fit. The sample consists of a single absorbing layer on a semi-infinite substrate. The genetic algorithm has good convergence and is applicable to many different problems, including those with different independent measurements and situations with more than two angles of incidence. Results are similar to those obtained by other inversion techniques.

© 2000 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1996).
  2. F. L. McCrackin, J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” in “Ellipsometry in the measurement of surfaces and films,” Natl Bur. Stand. Misc. Publ. 256, 61–82 (1964).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. G. E. Jellison, “Use of the biased estimator in the interpretation of spectroscopic ellipsometry data,” Appl. Opt. 30, 3354–3360 (1991).
    [CrossRef] [PubMed]
  8. S. Bosch, F. Monzonı́s, E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54–58 (1996).
    [CrossRef]
  9. J. H. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, Ann Arbor, Mich., 1975).
  10. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1989).
  11. T. Eisenhammer, M. Lazarov, M. Leutbecher, U. Schöffel, R. Sizmann, “Optimization of interference filters with genetic algorithms applied to silver-based heat mirrors,” Appl. Opt. 32, 6310–6315 (1993).
    [CrossRef] [PubMed]
  12. L. Davis, Handbook of Genetic Algorithms (Van Nostrand Reinhold, New York, 1991).
  13. A. H. Wright, “Genetic algorithms for real parameter optimization,” in Foundation of Genetic Algorithms, G. J. E. Rawlins, ed. (Morgan Kaufmann, San Mateo, Calif., 1991), pp. 205–218.
  14. Z. Michalewicz, Genetic Algorithms (Springer-Verlag, New York, 1992).
  15. S. Bosch, J. Perez, A. Canillas, “Numerical algorithm for spectroscopic ellipsometry of thick transparent films,” Appl. Opt. 37, 1177–1179 (1998).
    [CrossRef]

1998 (1)

1996 (1)

S. Bosch, F. Monzonı́s, E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54–58 (1996).
[CrossRef]

1994 (1)

1993 (2)

1991 (1)

1988 (1)

T. Easwarakhanthan, C. Michel, S. Ravelet, “Numerical method for the ellipsometric determination of optical constants and thickness of thin films with microcomputers,” Surf. Sci. 197, 339–345 (1988).
[CrossRef]

1972 (1)

1964 (1)

F. L. McCrackin, J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” in “Ellipsometry in the measurement of surfaces and films,” Natl Bur. Stand. Misc. Publ. 256, 61–82 (1964).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1996).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1996).

Bosch, S.

S. Bosch, J. Perez, A. Canillas, “Numerical algorithm for spectroscopic ellipsometry of thick transparent films,” Appl. Opt. 37, 1177–1179 (1998).
[CrossRef]

S. Bosch, F. Monzonı́s, E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54–58 (1996).
[CrossRef]

Boyanov, M. I.

Canillas, A.

Colson, J. P.

F. L. McCrackin, J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” in “Ellipsometry in the measurement of surfaces and films,” Natl Bur. Stand. Misc. Publ. 256, 61–82 (1964).

Davis, L.

L. Davis, Handbook of Genetic Algorithms (Van Nostrand Reinhold, New York, 1991).

Drolet, J.-P.

Easwarakhanthan, T.

T. Easwarakhanthan, C. Michel, S. Ravelet, “Numerical method for the ellipsometric determination of optical constants and thickness of thin films with microcomputers,” Surf. Sci. 197, 339–345 (1988).
[CrossRef]

Eisenhammer, T.

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

Holland, J. H.

J. H. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, Ann Arbor, Mich., 1975).

Jellison, G. E.

Lazarov, M.

Leblanc, R.

Leutbecher, M.

Masetti, E.

S. Bosch, F. Monzonı́s, E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54–58 (1996).
[CrossRef]

McCrackin, F. L.

F. L. McCrackin, J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” in “Ellipsometry in the measurement of surfaces and films,” Natl Bur. Stand. Misc. Publ. 256, 61–82 (1964).

Michalewicz, Z.

Z. Michalewicz, Genetic Algorithms (Springer-Verlag, New York, 1992).

Michel, C.

T. Easwarakhanthan, C. Michel, S. Ravelet, “Numerical method for the ellipsometric determination of optical constants and thickness of thin films with microcomputers,” Surf. Sci. 197, 339–345 (1988).
[CrossRef]

Monzoni´s, F.

S. Bosch, F. Monzonı́s, E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54–58 (1996).
[CrossRef]

Perez, J.

Ravelet, S.

T. Easwarakhanthan, C. Michel, S. Ravelet, “Numerical method for the ellipsometric determination of optical constants and thickness of thin films with microcomputers,” Surf. Sci. 197, 339–345 (1988).
[CrossRef]

Reinberg, A. R.

Russev, S. C.

Schöffel, U.

Sizmann, R.

Urban, F. K.

Wright, A. H.

A. H. Wright, “Genetic algorithms for real parameter optimization,” in Foundation of Genetic Algorithms, G. J. E. Rawlins, ed. (Morgan Kaufmann, San Mateo, Calif., 1991), pp. 205–218.

Appl. Opt. (5)

J. Opt. Soc. Am. A (1)

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

F. L. McCrackin, J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” in “Ellipsometry in the measurement of surfaces and films,” Natl Bur. Stand. Misc. Publ. 256, 61–82 (1964).

Surf. Sci. (1)

T. Easwarakhanthan, C. Michel, S. Ravelet, “Numerical method for the ellipsometric determination of optical constants and thickness of thin films with microcomputers,” Surf. Sci. 197, 339–345 (1988).
[CrossRef]

Thin Solid Films (1)

S. Bosch, F. Monzonı́s, E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54–58 (1996).
[CrossRef]

Other (6)

J. H. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, Ann Arbor, Mich., 1975).

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

L. Davis, Handbook of Genetic Algorithms (Van Nostrand Reinhold, New York, 1991).

A. H. Wright, “Genetic algorithms for real parameter optimization,” in Foundation of Genetic Algorithms, G. J. E. Rawlins, ed. (Morgan Kaufmann, San Mateo, Calif., 1991), pp. 205–218.

Z. Michalewicz, Genetic Algorithms (Springer-Verlag, New York, 1992).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1996).

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Figures (1)

Fig. 1
Fig. 1

Evolution of the objective function as the generation increases for three different trials (Δm,ψm from Table 3). Note that convergence is achieved after approximately 20 generations.

Tables (7)

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Table 1 Optimized Parameters for the GA

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Table 2 Computed Values of Δ and ψ for a Sample of TiO2a on Glassb for Four Incidence Anglesc

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Table 3 Computed Values of Δ and ψ (from Table 2) with Added Random Noise of 0.02° and 0.01° Standard Deviation, Respectivelya

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Table 4 Computed Values of Δ and ψ (from Table 2) with Added Random Noise of 0.5° and 0.25° Standard Deviation, Respectivelya

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Table 5 Data Obtained by Adding to the Data of Table 2 Gaussian Noise with Different Standard Deviationsa

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Table 6 Data Generated for a Film of n=2.2, k=0.22, and d=10 nm on a Substrate of ns=4.05 and ks=0.028 with λ=546.1 nm

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Table 7 Data Generated for a Film of n=2.2, k=0.22, and d=10 nm on a Substrate of ns=4.05 and ks=0.028 with λ=546.1 nm

Equations (9)

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

ρ=rp/rs=tan(ψ)exp(iΔ),
Rp/Rs=tan ψ exp(iΔ),
χ2=1Nm=1NΔm-ΔcΔ2+ψm-ψcψ2,
PRj=RWj1ηj=1ηRWj.
0.5p1+0.5p2,
1.5p1-0.5p2,
-0.5p1+1.5p2,
Vi=Vi+δ(UB-Vi)(heads)Vi-δ(Vi-LB)(tails),
δ(y)=y[r(1-t/T)B]

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