Abstract

Linear regression, combined with search over nonlinear parameters, is useful in fitting Sellmeier formulas to dispersion data. A special advantage accrues in fitting several sets of data to one formula: linear regression permits systematic normalization to a common absolute value. Rational procedures are discussed also for weighting of data sets of unequal precision. The fitting of common formulas to six sets of data on helium, for wavelengths from 0.09 to 2 μm, illustrates the various procedures.

© 1983 Optical Society of America

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References

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  1. P. W. Langhoff, M. Karplus, J. Opt. Soc. Am. 59, 863 (1969).
    [CrossRef]
  2. K. T. Tang, J. Chem. Phys. 55, 1064 (1971).
    [CrossRef]
  3. P. W. Langhoff, J. Chem. Phys. 57, 2604 (1972).
    [CrossRef]
  4. O. N. Stavroudis, L. E. Sutton, J. Opt. Soc. Am. 51, 368 (1961).
    [CrossRef]
  5. L. E. Sutton, O. N. Stavroudis, J. Opt. Soc. Am. 51, 901 (1961).
    [CrossRef]
  6. M. J. D. Powell, “Minimization of Functions of Several Variables,” in Numerical Analysis: An Introduction, J. Walsh, Ed. (Thompson Book Co., Washington, D.C., 1967), p. 143.
  7. P. J. Leonard, At. Data Nucl. Tables 14, 22 (1974).
    [CrossRef]
  8. J. Koch, Art. Mat., Astron. Fys. 9, No. 6, 1 (1913).
  9. C. Cuthbertson, M. Cuthbertson, Proc. R. Soc. London Ser. A 135, 40 (1932).
    [CrossRef]
  10. C. R. Mansfield, “Measurement of the Dispersion of Helium and Neon,” Doctoral Thesis, U. Idaho (1969).
  11. C. R. Mansfield, E. R. Peck, J. Opt. Soc. Am. 59, 199 (1969).
    [CrossRef]
  12. P. L. Smith, M. C. E. Huber, W. H. Parkinson, Phys. Rev. A 13, 1422 (1976).
    [CrossRef]
  13. M. C. E. Huber, G. Tondello, J. Opt. Soc. Am. 64, 390 (1974).
    [CrossRef]

1976

P. L. Smith, M. C. E. Huber, W. H. Parkinson, Phys. Rev. A 13, 1422 (1976).
[CrossRef]

1974

1972

P. W. Langhoff, J. Chem. Phys. 57, 2604 (1972).
[CrossRef]

1971

K. T. Tang, J. Chem. Phys. 55, 1064 (1971).
[CrossRef]

1969

1961

1932

C. Cuthbertson, M. Cuthbertson, Proc. R. Soc. London Ser. A 135, 40 (1932).
[CrossRef]

1913

J. Koch, Art. Mat., Astron. Fys. 9, No. 6, 1 (1913).

Cuthbertson, C.

C. Cuthbertson, M. Cuthbertson, Proc. R. Soc. London Ser. A 135, 40 (1932).
[CrossRef]

Cuthbertson, M.

C. Cuthbertson, M. Cuthbertson, Proc. R. Soc. London Ser. A 135, 40 (1932).
[CrossRef]

Huber, M. C. E.

P. L. Smith, M. C. E. Huber, W. H. Parkinson, Phys. Rev. A 13, 1422 (1976).
[CrossRef]

M. C. E. Huber, G. Tondello, J. Opt. Soc. Am. 64, 390 (1974).
[CrossRef]

Karplus, M.

Koch, J.

J. Koch, Art. Mat., Astron. Fys. 9, No. 6, 1 (1913).

Langhoff, P. W.

Leonard, P. J.

P. J. Leonard, At. Data Nucl. Tables 14, 22 (1974).
[CrossRef]

Mansfield, C. R.

C. R. Mansfield, E. R. Peck, J. Opt. Soc. Am. 59, 199 (1969).
[CrossRef]

C. R. Mansfield, “Measurement of the Dispersion of Helium and Neon,” Doctoral Thesis, U. Idaho (1969).

Parkinson, W. H.

P. L. Smith, M. C. E. Huber, W. H. Parkinson, Phys. Rev. A 13, 1422 (1976).
[CrossRef]

Peck, E. R.

Powell, M. J. D.

M. J. D. Powell, “Minimization of Functions of Several Variables,” in Numerical Analysis: An Introduction, J. Walsh, Ed. (Thompson Book Co., Washington, D.C., 1967), p. 143.

Smith, P. L.

P. L. Smith, M. C. E. Huber, W. H. Parkinson, Phys. Rev. A 13, 1422 (1976).
[CrossRef]

Stavroudis, O. N.

Sutton, L. E.

Tang, K. T.

K. T. Tang, J. Chem. Phys. 55, 1064 (1971).
[CrossRef]

Tondello, G.

Art. Mat., Astron. Fys.

J. Koch, Art. Mat., Astron. Fys. 9, No. 6, 1 (1913).

At. Data Nucl. Tables

P. J. Leonard, At. Data Nucl. Tables 14, 22 (1974).
[CrossRef]

J. Chem. Phys.

K. T. Tang, J. Chem. Phys. 55, 1064 (1971).
[CrossRef]

P. W. Langhoff, J. Chem. Phys. 57, 2604 (1972).
[CrossRef]

J. Opt. Soc. Am.

Phys. Rev. A

P. L. Smith, M. C. E. Huber, W. H. Parkinson, Phys. Rev. A 13, 1422 (1976).
[CrossRef]

Proc. R. Soc. London Ser. A

C. Cuthbertson, M. Cuthbertson, Proc. R. Soc. London Ser. A 135, 40 (1932).
[CrossRef]

Other

C. R. Mansfield, “Measurement of the Dispersion of Helium and Neon,” Doctoral Thesis, U. Idaho (1969).

M. J. D. Powell, “Minimization of Functions of Several Variables,” in Numerical Analysis: An Introduction, J. Walsh, Ed. (Thompson Book Co., Washington, D.C., 1967), p. 143.

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Figures (4)

Fig. 1
Fig. 1

Deviations of 106 (n − 1) from Eq. (42): linear plot.

Fig. 2
Fig. 2

Deviations of 106 (n − 1) from Eq. (42): cube root plot.

Fig. 3
Fig. 3

Relative deviations of data from Eq. (42). Each deviation is divided by the rms deviation of its own data set from an individual curve fit.

Fig. 4
Fig. 4

Maximum positive and negative spread of 106 (n − 1) computed from the eight common curve fits about their mean: cube root plot.

Tables (2)

Tables Icon

Table I Characteristics of Data Sets

Tables Icon

Table II Characteristics of the Fit of Eq. (42) and Variations Among the Eight Common Fits

Equations (52)

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3 ( n 2 1 ) 2 ( n 2 + 2 ) .
Y = j = 0 P C j X j ,
Y = D + j = 1 N A j B j X .
S = r = 1 M [ y ( r ) Y ( r ) ] 2 .
q = S / M .
Y = j = j p c j x j .
k = j p G j k c k = V j ( j = j p ) ,
G j k = r x j ( r ) x k ( r ) ; V j = r y ( r ) x j ( r ) .
S y = r y ( r ) 2 ,
S = S y j = j p c j V j .
L = V 0 k = j p G 0 k c k .
1 B X = 1 B ( 1 + X B X ) .
Z = c 0 + c 1 x 1 + c 2 x 2 .
H = A 2 B 1 + A 1 B 2 A 1 + A 2 = B 1 / A 1 + B 2 / A 2 1 / A 1 + 1 / A 2 .
x 1 = X 1 X / H ; x 2 = X 2 1 X / H .
Z = c 0 + c 1 x 1 + c 2 x 2 + c 3 x 3 .
H = A 1 B 2 B 3 + A 2 B 3 B 1 + A 3 B 1 B 2 A 1 ( B 2 + B 3 ) + A 2 ( B 3 + B 1 ) + A 3 ( B 1 + B 2 ) ,
K = A 1 ( B 2 + B 3 ) + A 2 ( B 3 + B 1 ) + A 3 ( B 1 + B 2 ) A 1 + A 2 + A 3 .
B j B k B j + B k ; j , k = 1 , 2 , 3 ; j k .
x 1 = X 1 ( X / H ) ( 1 X / K ) ; x 2 = X x 1 ; x 3 = X x 2 .
T j k ( l ) = r ( l ) x j ( r ) x k ( r ) ; T j 0 ( l ) = r ( l ) x j ( r ) ; T y j ( l ) = r ( l ) y ( r ) x j ( r ) ; T y 0 ( l ) = r ( l ) y ( r ) ; T y y ( l ) = r ( l ) y 2 ( r ) .
G j k = l = 0 t 1 W ( l ) T j k ( l ) ; V j = t = 0 t 1 W ( l ) F ( l ) T y j ( l ) ; S y = l = 0 t 1 W ( l ) F 2 ( l ) T y y ( l ) .
L ( l ) = F ( l ) T y 0 ( l ) k = j p c k T k 0 ( l ) .
L = l = 0 t 1 W ( l ) L ( l ) .
D L ( m ) = l = 0 t 1 a m l f ( l ) ;
D S = l = 0 t 1 m = 0 t 1 b l m f ( l ) f ( m ) + l = 0 t 1 b l f ( l ) ,
l = 1 t 1 a m l f ( l ) = L ( m ) , m = 1 ( t 1 ) .
D S / f ( u ) = 0 , u = 1 ( t 1 ) .
l = 1 t 1 ( b l u + b u l ) f ( l ) = b u , u = 1 ( t 1 ) .
D c k = l = 1 t 1 g k l f ( l ) .
k = j p G j k g k l = W ( l ) T y j ( l ) , j = j p .
a m l = I ̂ m l T y 0 ( l ) k = j p g k m T k 0 ( l ) ,
b m l = I ̂ m l W ( l ) T y y ( l ) k = j p g k m W ( l ) T y k ( l ) ,
b l = 2 W ( l ) F ( l ) T y y ( l ) k = j p [ c k W ( l ) T y k ( l ) + g k l V k ] .
R ( l ) = D ( l ) / [ y ( l ) M ( l ) ] .
R ( l , 0 ) = R ( l ) 2 + R ( 0 ) 2 .
S = l = 0 t 1 W ( l ) M ( l ) q ( l ) 2 .
D S = l = 0 t 1 W ( l ) M ( l ) q ( l ) 2 D [ q ( l ) 2 ] q ( l ) 2 .
W ( l ) = 1 M 1 q 0 ( l ) 2 , M = l = 0 t 1 M ( l ) .
S = 1 M l = 0 t 1 M ( l ) q ( l ) 2 q 0 ( l ) 2 .
W ( l ) = 1 t M ( l ) 1 q 0 ( l ) 2 .
S = 1 t l = 0 t 1 q ( l ) 2 q 0 ( l ) 2 .
Y = Y w l = 0 t 1 w ( l ) F ( l ) , w = l = 0 t 1 w ( l ) .
10 6 ( n 1 ) = 7.3123 + 9279.7 / ( 339.82 X ) .
10 6 ( n 1 ) = 5976.9 / ( 302.6 X ) + 14606.4 / ( 982.4 X ) .
n 1 = 34.8920 × 10 6
P 1 = [ ( c 0 / H ) c 1 ] / ( 2 c 2 ) ; P 2 = c 0 / c 2 .
P 2 2 P 1 B + B 2 = 0 ,
B = P 1 ± P 1 2 P 2 .
A 1 = B 1 H c 2 H ( B 1 B 2 ) ; A 2 = H B 2 c 2 H ( B 1 B 2 ) .
P 3 = c 0 c 3 , P 2 = P 3 H + c 1 c 3 P 1 = P 3 H K c 2 c 3 .
A j = Q 3 B j Q 2 + B j 2 Q 1 ( B k B j ) ( B l B j ) ,

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