Abstract

Presented here is a modification in the adjustment for refractive-index nonlinear regression through a Sellmeier expansion. It is free of the deficiencies inherent in the earlier nonlinear method and notably improves the adjustment without excessively complicating the mathematical procedures. The method is applied to CdS with the object of comparing the results with those obtained previously. In this manner, the best adjustment has been found for CdS dispersion. It has also allowed a physical explanation of some adjustment parameters.

© 1985 Optical Society of America

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References

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  1. B. J. Pernick, “Nonlinear Regression Analysis for the Sellmeier Dispersion Equation of CdS,” Appl. Opt. 22, 1133 (1983).
    [CrossRef] [PubMed]
  2. F. Vilches, J. M. Guerra, M. S. Gomez, “Nonlinear Regression Analysis of a Sellmeier Equation with Various Resonances: Best Fit of CdS Dispersion,” Appl. Opt. 23, 2044 (1984).
    [CrossRef] [PubMed]
  3. T. M. Bieniewski, S. J. Czyzak, “Refractive Indexes of Single Hexagonal ZnS and CdS Crystals,” J. Opt. Soc. Am. 53, 496 (1963).
    [CrossRef]
  4. S. Chattergee, B. Price, Regression Analysis by Example (Wiley, New York, 1977).
  5. D. G. Thomas, J. J. Hopfield, “Exciton Spectrum of Cadmium Sulfide,” Phys. Rev. 116, 573 (1959).
    [CrossRef]

1984 (1)

1983 (1)

1963 (1)

1959 (1)

D. G. Thomas, J. J. Hopfield, “Exciton Spectrum of Cadmium Sulfide,” Phys. Rev. 116, 573 (1959).
[CrossRef]

Bieniewski, T. M.

Chattergee, S.

S. Chattergee, B. Price, Regression Analysis by Example (Wiley, New York, 1977).

Czyzak, S. J.

Gomez, M. S.

Guerra, J. M.

Hopfield, J. J.

D. G. Thomas, J. J. Hopfield, “Exciton Spectrum of Cadmium Sulfide,” Phys. Rev. 116, 573 (1959).
[CrossRef]

Pernick, B. J.

Price, B.

S. Chattergee, B. Price, Regression Analysis by Example (Wiley, New York, 1977).

Thomas, D. G.

D. G. Thomas, J. J. Hopfield, “Exciton Spectrum of Cadmium Sulfide,” Phys. Rev. 116, 573 (1959).
[CrossRef]

Vilches, F.

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Figures (2)

Fig. 1
Fig. 1

Residuals diagram (difference between experimental and fit values) corresponding to (a) Pernick adjustment,1 (b) our previous fit,2 and (c) fit weighting with 1/xi (see text). The circles indicate the ordinary index and crosses the extraordinary index.

Fig. 2
Fig. 2

Residuals diagram for fit obtained by the iterative procedure described in the text. Circles indicate the ordinary index, crosses the extraordinary index. The two broken lines indicate experimental data round-off level.

Tables (1)

Tables Icon

Table I Experimental Values of Ordinary and Extraordinary Refractive Indices of CdS, According to Ref. 3, and Those from the Residuals Corresponding to the Adjustment Expressions (19) and (20)

Equations (22)

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y = b 0 + k = 1 N b k x d k ,
S L = i = 1 n ( y i b 0 k = 1 N b k x i d k ) 2 = i = 1 n z i 2 .
S = i = 1 n k = 1 N ( x i d k ) 2 ( y i b o k = 1 N b k x i d k ) 2 = i = 1 n z i 2 .
S = i = 1 n [ k = 0 N ( β 2 k + 1 y i + β 2 k ) x i k ] 2 ,
β 2 N + 1 = 1 ; β 2 N = b o ; β 2 ( N α ) + 1 = ( 1 ) α + 1 k α > k 2 > k 1 = 1 d k 1 d k 2 d k α , β 2 ( N α ) = β 2 N β 2 ( N α ) + 1 + ( 1 ) α k = 1 N b k ( ( k α 1 > k 2 > k k 1 d k 1 d k 2 d k α 1 ) α = 1 , 2 , N
k = 1 N ( x i d k ) 2
S = i = 1 n ω i 2 z i 2 .
β = A 1 C ,
a i j = R = 1 n ω R 2 x ( i + j ) / 2 if i and j are even , a i j = R = 1 n ω R 2 y R 2 x | ( i + j ) / 2 | 1 if i and j are odd , a i j = R = 1 n ω R 2 y R x R ( i + j 1 ) / 2 otherwise , i , j = 0 , 1 , 2 , , 2 N ,
C j = R = 1 n ω R 2 y R x R N + ( j / 2 ) if j is even , C j = R = 1 n ω R 2 y R 2 x R N + | ( i j ) / 2 | otherwise , j = 0 , 1 , 2 , , 2 N .
S b o = 0 ; S b k = 0 ; S d k = 0 ; k = 1 , 2 , N .
S L b o = i = 1 n z i 2 b o = 0 , S L b k = i = 1 n z i 2 b k = 0 , S L d k = i = 1 n z i 2 b k = 0 , k = 1 , 2 , N
S b o = i = 1 n ω i 2 [ j = 1 N ( x i d j ) 2 ] z i 2 b o , S b k = i = 1 n ω i 2 [ j = 1 N ( x i d j ) 2 ] z i 2 b k , S d k = i = 1 n ω i 2 [ j = 1 N ( x i d j ) 2 ] z i 2 d k 2 i = 1 n ω i 2 z i 2 ( x i d k ) [ j k N ( x i d j ) 2 ] .
ω i = j = 1 N ( x i d j ) 1 ,
S L b o S b o , S L b k S b k , S L d k S d k + R k , k = 1 , 2 , N ,
R k 2 i = 1 n z i 2 x i d k ; z i = y i b o k = 1 n b k x i d k .
S b o = 0 ; S b k = 0 ; S d k = R k ; k = 1 , 2 , N ,
S β m = M m ; m = 0 , 1 , , 2 N ,
M m = k = 1 N S d k d k β m = k = 1 N R k d k β m .
β = A 1 ( C + M ) ,
n 2 = 5.1792 + 0.23504 λ 2 0.083591 + 0.036927 λ 2 0.23504
n 2 = 5.2599 + 0.20865 λ 2 0.10799 + 0.027527 λ 2 0.23305

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