Abstract

Published Kramers–Kronig dispersion-analysis methods have been evaluated and a new technique has been developed for determining the optical indices from transmittance data obtained from thin films supported by a transparent substrate. To solve the problem of the treatment of the phase-shift integral in the observed wings, a previously suggested mean-value-theorem method has been adopted. The unique feature of the new method is the reduction of systematic errors introduced by using the k = 0 approximation. These errors are reduced by the use of a Cauchy relation to generate acceptable k values in the observed wings.

© 1972 Optical Society of America

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References

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  1. P. Grant and W. Paul, J. Appl. Phys. 37, 3110 (1966).
    [Crossref]
  2. P. O. Nilsson, Appl. Opt. 7, 435 (1968).
    [Crossref] [PubMed]
  3. H. Verleur, J. Opt. Soc. Am. 58, 1356 (1968).
    [Crossref]
  4. D. Glover and J. Hollenberg, J. Phys. Chem. 73, 880 (1969).
    [Crossref]
  5. P. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
    [Crossref]
  6. S. Maeda, G. Thyagarajan, and P. Schatz, J. Chem. Phys. 39, 3474 (1963).
    [Crossref]
  7. G. Andermann, A. Caron, and D. Dows, J. Opt. Soc. Am. 55, 1210 (1965).
    [Crossref]
  8. K. Kozima, W. Suetaka, and P. Schatz, J. Opt. Soc. Am. 56, 181 (1966).
    [Crossref]
  9. D. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965).
    [Crossref]
  10. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), p. 92.
  11. I. S. Sokolnikoff and E. S. Sokolnikoff, Higher Mathematics for Engineers and Physicists (McGraw–Hill, New York, 1941), pp. 97–101.
  12. G. Andermann and L. Brantley, J. Opt. Soc. Am. 58, 171 (1968).
    [Crossref]
  13. L. Brantley, G. Andermann, and P. Sakamoto, Spectrosc. Letters 4, 47 (1971).
    [Crossref]
  14. J. Neufeld and G. Andermann, Appl. Spectrosc. 25, 597 (1971).
    [Crossref]
  15. G. Andermann and D. Dows, J. Phys. Chem. Solids 28, 1307 (1967).
    [Crossref]

1971 (2)

L. Brantley, G. Andermann, and P. Sakamoto, Spectrosc. Letters 4, 47 (1971).
[Crossref]

J. Neufeld and G. Andermann, Appl. Spectrosc. 25, 597 (1971).
[Crossref]

1969 (1)

D. Glover and J. Hollenberg, J. Phys. Chem. 73, 880 (1969).
[Crossref]

1968 (3)

1967 (1)

G. Andermann and D. Dows, J. Phys. Chem. Solids 28, 1307 (1967).
[Crossref]

1966 (2)

1965 (2)

1963 (2)

P. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
[Crossref]

S. Maeda, G. Thyagarajan, and P. Schatz, J. Chem. Phys. 39, 3474 (1963).
[Crossref]

Andermann, G.

Brantley, L.

L. Brantley, G. Andermann, and P. Sakamoto, Spectrosc. Letters 4, 47 (1971).
[Crossref]

G. Andermann and L. Brantley, J. Opt. Soc. Am. 58, 171 (1968).
[Crossref]

Caron, A.

Dows, D.

G. Andermann and D. Dows, J. Phys. Chem. Solids 28, 1307 (1967).
[Crossref]

G. Andermann, A. Caron, and D. Dows, J. Opt. Soc. Am. 55, 1210 (1965).
[Crossref]

Glover, D.

D. Glover and J. Hollenberg, J. Phys. Chem. 73, 880 (1969).
[Crossref]

Grant, P.

P. Grant and W. Paul, J. Appl. Phys. 37, 3110 (1966).
[Crossref]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), p. 92.

Hollenberg, J.

D. Glover and J. Hollenberg, J. Phys. Chem. 73, 880 (1969).
[Crossref]

Kozima, K.

K. Kozima, W. Suetaka, and P. Schatz, J. Opt. Soc. Am. 56, 181 (1966).
[Crossref]

P. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
[Crossref]

Maeda, S.

S. Maeda, G. Thyagarajan, and P. Schatz, J. Chem. Phys. 39, 3474 (1963).
[Crossref]

P. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
[Crossref]

Neufeld, J.

Nilsson, P. O.

Paul, W.

P. Grant and W. Paul, J. Appl. Phys. 37, 3110 (1966).
[Crossref]

Roessler, D.

D. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965).
[Crossref]

Sakamoto, P.

L. Brantley, G. Andermann, and P. Sakamoto, Spectrosc. Letters 4, 47 (1971).
[Crossref]

Schatz, P.

K. Kozima, W. Suetaka, and P. Schatz, J. Opt. Soc. Am. 56, 181 (1966).
[Crossref]

S. Maeda, G. Thyagarajan, and P. Schatz, J. Chem. Phys. 39, 3474 (1963).
[Crossref]

P. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
[Crossref]

Sokolnikoff, E. S.

I. S. Sokolnikoff and E. S. Sokolnikoff, Higher Mathematics for Engineers and Physicists (McGraw–Hill, New York, 1941), pp. 97–101.

Sokolnikoff, I. S.

I. S. Sokolnikoff and E. S. Sokolnikoff, Higher Mathematics for Engineers and Physicists (McGraw–Hill, New York, 1941), pp. 97–101.

Suetaka, W.

Thyagarajan, G.

S. Maeda, G. Thyagarajan, and P. Schatz, J. Chem. Phys. 39, 3474 (1963).
[Crossref]

Verleur, H.

Appl. Opt. (1)

Appl. Spectrosc. (1)

Brit. J. Appl. Phys. (1)

D. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965).
[Crossref]

J. Appl. Phys. (1)

P. Grant and W. Paul, J. Appl. Phys. 37, 3110 (1966).
[Crossref]

J. Chem. Phys. (2)

P. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963).
[Crossref]

S. Maeda, G. Thyagarajan, and P. Schatz, J. Chem. Phys. 39, 3474 (1963).
[Crossref]

J. Opt. Soc. Am. (4)

J. Phys. Chem. (1)

D. Glover and J. Hollenberg, J. Phys. Chem. 73, 880 (1969).
[Crossref]

J. Phys. Chem. Solids (1)

G. Andermann and D. Dows, J. Phys. Chem. Solids 28, 1307 (1967).
[Crossref]

Spectrosc. Letters (1)

L. Brantley, G. Andermann, and P. Sakamoto, Spectrosc. Letters 4, 47 (1971).
[Crossref]

Other (2)

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), p. 92.

I. S. Sokolnikoff and E. S. Sokolnikoff, Higher Mathematics for Engineers and Physicists (McGraw–Hill, New York, 1941), pp. 97–101.

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Figures (7)

F. 1
F. 1

Representation of thin-film-transmittance measurements where the thin film A is deposited on a transparent substrate B. I0 is the irradiance of the normal incident beam, t ̂ is the amplitude of the light passing through the substrate, and I is the irradiance of the transmitted beam. Rw and Rb represent the reflection at the respective interfaces and d is the thickness of the film.

F. 2
F. 2

Phase-shift-error evaluation using ks values at ν1 and ν2. The solid line represents the synthetic θs curve. The other points represent θ values when ν1 and ν2 are chosen from Table II as follows: Triangles—trial No. 1; open circles—trial No. 5; closed circles—trial No. 8.

F. 3
F. 3

Q(nk) represents the integrated area between the nsks and nk curves for various values of ν1. ν2 was placed symmetrically on the opposite side of the resonant frequency, 484 cm−1. (a) Error in MVT-C method. (b) Error due to the k(ν1) = k(ν2) = 0 approximation.

F. 4
F. 4

Far-ir transmittance spectrum of NaClO3 lattice bands.

F. 5
F. 5

Illustration of the reliability of the MVT-C method for calculating the optical indices n and k. Solid line represents the synthetic values and the dots represent the calculated values.

F. 6
F. 6

Graphical solutions for n corresponding to given values of T and k. (Heavy solid line) k = 0.0; (dash-dot line) k = 0.04; (dashed line) k = 0.08; (light solid line) k = 0.10. Other parameters in the calculation were ν = 310 cm−1, ϕ = 1.5, and d = 1.0.

F. 7
F. 7

Effect of film thickness on the solution for n at a given T. (Solid line) d = 1.40 μm; (dash-dot line) d = 1.00 μm; (dashed line) d = 0.60 μm. The other parameters in the calculation were ν = 310 cm−1, ϕ = 1.5, and k = 0.04.

Tables (4)

Tables Icon

Table I Parameters used in single-band error evaluation.

Tables Icon

Table II Errors in phase-shift calculation by using various values for ν1 and ν2 in the MVT-C method.

Tables Icon

Table III Band parameters for generating a multiband DHO synthetic spectra.

Tables Icon

Table IV Summary of error analysis for multiband spectra. The right two columns are Q(nk) values for the indicated approximations.

Equations (21)

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I = I 0 ϕ | t ̂ | 2 ( 1 R w ) 1 R w R b ,
I 0 = I 0 1 R w 1 + R w .
1 R b R w 1 + R w = 1 R w + R w ( R w R b ) 1 + R w 1 R w .
T = I I 0 = | t ̂ | 2 ϕ 1 R w .
t ̂ = t exp ( i θ )
ln t ̂ = ln t + i θ .
θ ( ν 0 ) = 2 ν 0 π 0 ln t ( ν ) ln t ( ν 0 ) ν 2 ν 0 2 d ν + 2 π ν 0 d .
θ ( ν 0 ) = θ A ( ν 0 ) + θ X ( ν 0 ) + θ B ( ν 0 ) + 2 π ν 0 d ,
θ A ( ν 0 ) = 2 ν 0 π 0 ν a ln t ( ν ) ln t ( ν 0 ) ν 2 ν 0 2 d ν ,
θ X ( ν 0 ) = 2 ν 0 π ν a ν b ln t ( ν ) ln t ( ν 0 ) ν 2 ν 0 2 d ν ,
θ B ( ν 0 ) = 2 ν 0 π ν b ν ln t ( ν ) ln t ( ν 0 ) ν 2 ν 0 2 d ν .
θ A ( ν 0 ) = ( A ( ν 0 ) ln t ( ν ) π ) ln | ν 0 + ν a ν 0 ν a |
θ B ( ν 0 ) = ( B ( ν 0 ) + ln t ( ν 0 ) π ) ln | ν 0 + ν b ν 0 ν b | ,
t = t ( n , k , ϕ , d )
θ = θ ( n , k , ϕ , d ) ,
n 1 = n 0 + t * ( θ θ c ) θ * ( t t c ) t * θ t θ *
k 1 = k 0 + θ ( t t c ) t ( θ θ c ) t * θ t θ * ,
θ ( ν 1 ) = ( A ln t ( ν 1 ) π ) ln | ν 1 + ν a ν 1 ν a | + θ X ( ν 1 ) + ( B + ln t ( ν 1 ) π ) ln | ν 1 + ν b ν 1 ν b | + 2 π ν 1 d
n ( ν ) k ( ν ) = j = 1 N a j ( ν ν j ) 2 + b j 2 ,
Q ( θ ) = ν x ν y [ θ s ( ν ) θ ( ν ) ] 2 d ν ,
Q ( n k ) = ν x ν y [ n s k s ( ν ) n k ( ν ) ] 2 d ν