Abstract

The truncated Kramers–Kronig transform, used widely in the analysis of optical data, is recast into a form that avoids the need to evaluate a Cauchy principal-value integral. A specialized Gaussian quadrature involving the weight function logex1 is employed. This approach yields accurate results for functions that lead to kernels with relatively rapid decay, which covers the cases most commonly encountered in optical data analysis. An application to the reststrahlen region of the GaAs spectrum is made.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).
  2. D. E. Aspnes, "The accurate determination of optical properties by ellipsometry," in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 89-112.
  3. V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).
  4. V. Lucarini and K.-E. Peiponen, "Verification of generalized Kramers-Kronig relations and sum rules on experimental data of third harmonic generation susceptibility on polymer," J. Chem. Phys. 119, 620-627 (2003).
    [CrossRef]
  5. V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities," Opt. Commun. 218, 409-414 (2003).
    [CrossRef]
  6. A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, 1966).
  7. J. E. Bertie and S. L. Zhang, "Infrared intensities of liquids. IX. The Kramers-Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms," Can. J. Chem. 70, 520-531 (1992).
    [CrossRef]
  8. V. I. Krylov and A. A. Pal'cev, "Numerical integration of functions having logarithmic and power singularities," Vestsi Akad. Navuk BSSR, Ser. Fiz.-Tekh. Navuk 14-23 (1963).
  9. B. Danloy, "Numerical construction of Gaussian quadrature formulas for ∫01(−Logx)∙xα∙f(x)∙dx and ∫01Em(x)∙f(x)∙dx," Math. Comput. 27, 861-869 (1973).
  10. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN 77, 2nd ed. (Cambridge U. Press, 1992).
    [PubMed]
  11. R. A. Sack and A. F. Donovan, "An algorithm for Gaussian quadrature given modified moments," Numer. Math. 18, 465-478 (1972).
    [CrossRef]
  12. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).
  13. E. D. Palik, "Gallium arsenide (GaAs)," in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), p. 429.
  14. F. W. King, "Efficient numerical approach to the evaluation of Kramers-Kronig transforms," J. Opt. Soc. Am. B 19, 2427-2436 (2002).
    [CrossRef]

2003 (2)

V. Lucarini and K.-E. Peiponen, "Verification of generalized Kramers-Kronig relations and sum rules on experimental data of third harmonic generation susceptibility on polymer," J. Chem. Phys. 119, 620-627 (2003).
[CrossRef]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities," Opt. Commun. 218, 409-414 (2003).
[CrossRef]

2002 (1)

1992 (1)

J. E. Bertie and S. L. Zhang, "Infrared intensities of liquids. IX. The Kramers-Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms," Can. J. Chem. 70, 520-531 (1992).
[CrossRef]

1973 (1)

B. Danloy, "Numerical construction of Gaussian quadrature formulas for ∫01(−Logx)∙xα∙f(x)∙dx and ∫01Em(x)∙f(x)∙dx," Math. Comput. 27, 861-869 (1973).

1972 (1)

R. A. Sack and A. F. Donovan, "An algorithm for Gaussian quadrature given modified moments," Numer. Math. 18, 465-478 (1972).
[CrossRef]

1963 (1)

V. I. Krylov and A. A. Pal'cev, "Numerical integration of functions having logarithmic and power singularities," Vestsi Akad. Navuk BSSR, Ser. Fiz.-Tekh. Navuk 14-23 (1963).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Asakura, T.

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

Aspnes, D. E.

D. E. Aspnes, "The accurate determination of optical properties by ellipsometry," in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 89-112.

Bertie, J. E.

J. E. Bertie and S. L. Zhang, "Infrared intensities of liquids. IX. The Kramers-Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms," Can. J. Chem. 70, 520-531 (1992).
[CrossRef]

Danloy, B.

B. Danloy, "Numerical construction of Gaussian quadrature formulas for ∫01(−Logx)∙xα∙f(x)∙dx and ∫01Em(x)∙f(x)∙dx," Math. Comput. 27, 861-869 (1973).

Donovan, A. F.

R. A. Sack and A. F. Donovan, "An algorithm for Gaussian quadrature given modified moments," Numer. Math. 18, 465-478 (1972).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN 77, 2nd ed. (Cambridge U. Press, 1992).
[PubMed]

King, F. W.

Krylov, V. I.

V. I. Krylov and A. A. Pal'cev, "Numerical integration of functions having logarithmic and power singularities," Vestsi Akad. Navuk BSSR, Ser. Fiz.-Tekh. Navuk 14-23 (1963).

Lucarini, V.

V. Lucarini and K.-E. Peiponen, "Verification of generalized Kramers-Kronig relations and sum rules on experimental data of third harmonic generation susceptibility on polymer," J. Chem. Phys. 119, 620-627 (2003).
[CrossRef]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities," Opt. Commun. 218, 409-414 (2003).
[CrossRef]

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

Pal'cev, A. A.

V. I. Krylov and A. A. Pal'cev, "Numerical integration of functions having logarithmic and power singularities," Vestsi Akad. Navuk BSSR, Ser. Fiz.-Tekh. Navuk 14-23 (1963).

Palik, E. D.

E. D. Palik, "Gallium arsenide (GaAs)," in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), p. 429.

Peiponen, K.-E.

V. Lucarini and K.-E. Peiponen, "Verification of generalized Kramers-Kronig relations and sum rules on experimental data of third harmonic generation susceptibility on polymer," J. Chem. Phys. 119, 620-627 (2003).
[CrossRef]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities," Opt. Commun. 218, 409-414 (2003).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN 77, 2nd ed. (Cambridge U. Press, 1992).
[PubMed]

Saarinen, J. J.

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities," Opt. Commun. 218, 409-414 (2003).
[CrossRef]

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

Sack, R. A.

R. A. Sack and A. F. Donovan, "An algorithm for Gaussian quadrature given modified moments," Numer. Math. 18, 465-478 (1972).
[CrossRef]

Secrest, D.

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, 1966).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Stroud, A. H.

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, 1966).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN 77, 2nd ed. (Cambridge U. Press, 1992).
[PubMed]

Vartiainen, E. M.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN 77, 2nd ed. (Cambridge U. Press, 1992).
[PubMed]

Zhang, S. L.

J. E. Bertie and S. L. Zhang, "Infrared intensities of liquids. IX. The Kramers-Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms," Can. J. Chem. 70, 520-531 (1992).
[CrossRef]

Can. J. Chem. (1)

J. E. Bertie and S. L. Zhang, "Infrared intensities of liquids. IX. The Kramers-Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms," Can. J. Chem. 70, 520-531 (1992).
[CrossRef]

J. Chem. Phys. (1)

V. Lucarini and K.-E. Peiponen, "Verification of generalized Kramers-Kronig relations and sum rules on experimental data of third harmonic generation susceptibility on polymer," J. Chem. Phys. 119, 620-627 (2003).
[CrossRef]

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

Math. Comput. (1)

B. Danloy, "Numerical construction of Gaussian quadrature formulas for ∫01(−Logx)∙xα∙f(x)∙dx and ∫01Em(x)∙f(x)∙dx," Math. Comput. 27, 861-869 (1973).

Numer. Math. (1)

R. A. Sack and A. F. Donovan, "An algorithm for Gaussian quadrature given modified moments," Numer. Math. 18, 465-478 (1972).
[CrossRef]

Opt. Commun. (1)

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities," Opt. Commun. 218, 409-414 (2003).
[CrossRef]

Vestsi Akad. Navuk BSSR, Ser. Fiz.-Tekh. Navuk (1)

V. I. Krylov and A. A. Pal'cev, "Numerical integration of functions having logarithmic and power singularities," Vestsi Akad. Navuk BSSR, Ser. Fiz.-Tekh. Navuk 14-23 (1963).

Other (7)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN 77, 2nd ed. (Cambridge U. Press, 1992).
[PubMed]

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, 1966).

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

D. E. Aspnes, "The accurate determination of optical properties by ellipsometry," in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 89-112.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

E. D. Palik, "Gallium arsenide (GaAs)," in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), p. 429.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Numerical quadrature result for ε r ( ω ) ε . The values of ε r ( ω ) ε from the exact result, Eq. (27), are represented by the long-dashed curve. The input ε i ( ω ) for the numerical quadrature is represented by the short-dashed curve. The solid curve is the numerical quadrature result superimposed on the exact result for the truncated Kramers–Kronig transform from Eq. (34).

Fig. 2
Fig. 2

Numerical quadrature result for ε i ( ω ) . The values of ε i ( ω ) from the exact result, Eq. (28), are represented by the short-dashed curve. The input ε r ( ω ) ε for the numerical quadrature is represented by the long-dashed curve. The solid curve is the numerical quadrature result superimposed on the exact result for the truncated Kramers–Kronig transform from Eq. (38).

Tables (2)

Tables Icon

Table 1 Comparison of Numerical Quadrature Values for the Finite Hilbert Transform versus Exact Evaluation a

Tables Icon

Table 2 Abscissas and Weights for a Gaussian Quadrature with a Weight Function of log e x 1 for N ¯ = 60

Equations (38)

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

K { O } ( ω ) = 2 π P 0 ω O ( ω ) d ω ω 2 ω 2 ,
k ω 1 , ω 2 ( ω ) K T { h } ( ω ) = 1 π P ω 1 ω 2 h ( ω ) d ω ω 2 ω 2 ,
k ( ω ) = 1 2 ω π P ω 1 ω 2 h ( ω ) d ω ω ω + 1 2 ω π ω 1 ω 2 h ( ω ) d ω ω + ω .
ω = 1 2 ( ω 1 + ω 2 ) + 1 2 ( ω 2 ω 1 ) x ;
k 2 ( ω ) = ( ω 2 ω 1 ) 2 ω π 1 1 h ( 1 2 [ ω 1 + ω 2 ] + 1 2 [ ω 2 ω 1 ] x ) d x 2 ω + ω 1 + ω 2 + ( ω 2 ω 1 ) x .
O ( ω , x ) = ( ω 2 ω 1 ) h ( 1 2 [ ω 1 + ω 2 ] + 1 2 [ ω 2 ω 1 ] x ) 2 ω π { 2 ω + ω 1 + ω 2 + ( ω 2 ω 1 ) x } ;
k 2 ( ω ) = 1 1 O ( ω , x ) d x .
k 2 ( ω ) i = 1 N O ( ω , x i ) w i ,
k 1 ( ω ) = 1 2 ω π P ω 1 ω 2 h ( ω ) d ω ω ω = 1 2 ω π P 1 1 h ( 1 2 [ ω 1 + ω 2 ] + 1 2 [ ω 2 ω 1 ] x ) d x ( 2 ω ω 1 ω 2 ) ( ω 2 ω 1 ) 1 x .
x 0 = ( 2 ω ω 1 ω 2 ) ( ω 2 ω 1 ) 1 ,
f ( x ) = h ( 1 2 [ ω 1 + ω 2 ] + 1 2 [ ω 2 ω 1 ] x ) ,
K ( x 0 ) = { ω 1 + ω 2 + x 0 ( ω 2 ω 1 ) } k 1 ( 1 2 [ ω 1 + ω 2 ] + 1 2 [ ω 2 ω 1 ] x ) ,
K ( x 0 ) = 1 π P 1 1 f ( x ) d x x 0 x ,
f ( x ε ) f ( x + ε ) C ε m ,
K ( x 0 ) = 1 π lim ε 0 { 1 x 0 ε f ( x ) d x x 0 x + x 0 + ε 1 f ( x ) d x x 0 x } = 1 π lim ε 0 { ε 1 + x 0 f ( x 0 s ) d s s ε 1 x 0 f ( x 0 + s ) d s s } ,
K ( x 0 ) = 1 π lim ε 0 ε 1 { f ( x 0 s ) f ( x 0 + s ) } d s s + 1 π { 1 1 + x 0 f ( x 0 s ) d s s + 1 x 0 1 f ( x 0 + s ) d s s } .
lim ε 0 ε 1 { f ( x 0 s ) f ( x 0 + s ) } d s s = lim ε 0 ε 1 { f ( x 0 s ) f ( x 0 + s ) } d log e s d s d s = lim ε 0 { [ f ( x 0 ε ) f ( x 0 + ε ) ] log e ε ε 1 log e s { f ( x 0 s ) f ( x 0 + s ) } d s } = 0 1 log e s 1 { f ( x 0 s ) f ( x 0 + s ) } d s ,
K ( x 0 ) = 1 π 0 1 log e x 1 { f ( x 0 x ) f ( x 0 + x ) } d x + x 0 π 1 1 { f ( x 0 2 [ 1 x ] 1 ) x 0 ( x + 1 ) + 2 f ( x 0 2 [ 1 x ] + 1 ) x 0 ( x + 1 ) 2 } d x .
g 1 ( x 0 , x ) = 1 π { f ( x 0 x ) f ( x 0 + x ) } ,
g 2 ( x 0 , x ) = x 0 π { f ( x 0 2 [ 1 x ] 1 ) x 0 ( x + 1 ) + 2 f ( x 0 2 [ 1 x ] + 1 ) x 0 ( x + 1 ) 2 } ;
K ( x 0 ) i = 1 N ¯ g 1 ( x 0 , x ¯ i ) w ¯ i + i = 1 N g 2 ( x 0 , x i ) w i ,
Ci ( x ) = x cos y d y y ;
Si ( x ) = 0 x sin y d y y ;
E n ( z ) = 1 e z y d y y n for n = 0 , 1 , 2 , , with Re z > 0 ;
Shi ( z ) = 0 z sinh t d t t .
ε ( ω ) ε = ε ( ω L 2 - ω T 2 ) ω T 2 ω 2 i Γ ω ;
ε r ( ω ) ε = ε ( ω L 2 - ω T 2 ) ( ω T 2 ω 2 ) ( ω T 2 ω 2 ) 2 + Γ 2 ω 2 ,
ε i ( ω ) = ε ( ω L 2 - ω T 2 ) Γ ω ( ω T 2 ω 2 ) 2 + Γ 2 ω 2 .
ε i ( ω ) = 2 ω π P 0 [ ε r ( ω ) ε ] d ω ω 2 ω 2 ,
ε r ( ω ) ε = 2 π P 0 ω ε i ( ω ) d ω ω 2 ω 2 .
ε r ( ω ) ε = 2 π P ω 1 ω 2 ω ε i ( ω ) d ω ω 2 ω 2 ,
a 2 = ω 2 ω , a 1 = ω 1 ω , a = ω T ω ,
A = ω L 2 ω T 2 , b = Γ 2 ω , c = a 2 b 2 ,
ε r ( ω ) ε = A ε 2 ω 2 c π { b ( 1 ( c + 1 ) 2 + b 2 1 ( c 1 ) 2 + b 2 ) log e [ ( 1 + a 2 ) ( 1 a 1 ) ( a 2 1 ) ( a 1 + 1 ) ] + { ( c 1 ) ( c 1 ) 2 + b 2 + ( c + 1 ) ( c + 1 ) 2 + b 2 } { tan 1 [ a 2 c b ] + tan 1 [ a 2 + c b ] tan 1 [ a 1 c b ] tan 1 [ a 1 + c b ] } + b 2 { 1 ( c 1 ) 2 + b 2 + 1 ( c + 1 ) 2 + b 2 } { log e [ ( b 2 + ( a 2 + c ) 2 ) ( b 2 + ( a 1 + c ) 2 ) ] log e [ ( b 2 + ( a 2 c ) 2 ) ( b 2 + ( a 1 c ) 2 ) ] } } .
ε r ( ω ) ε = A ε 2 ω 2 c { ( c 1 ) ( c 1 ) 2 + b 2 + ( c + 1 ) ( c + 1 ) 2 + b 2 } .
ε r ( ω ) ε = A ε ω 2 ( a 2 1 ) ( a 2 1 ) 2 + 4 b 2 ,
ε i ( ω ) = 2 ω π P ω 1 ω 2 [ ε r ( ω ) ε ] d ω ω 2 ω 2 ,
ε i ( ω ) = A ε 2 ω 2 c π { ( c + 1 ( c + 1 ) 2 + b 2 + c 1 ( c 1 ) 2 + b 2 ) log e [ ( 1 + a 2 ) ( 1 a 1 ) ( a 2 1 ) ( a 1 + 1 ) ] ] + b { 1 ( c 1 ) 2 + b 2 1 ( c + 1 ) 2 + b 2 } { tan 1 [ a 2 + c b ] tan 1 [ a 1 + c b ] ] + tan 1 [ a 2 c b ] tan 1 [ a 1 c b ] } + 1 2 { c + 1 ( c + 1 ) 2 + b 2 c 1 ( c 1 ) 2 + b 2 } { log e [ ( b 2 + ( a 2 + c ) 2 ) ( b 2 + ( a 1 + c ) 2 ) ] log e [ ( b 2 + ( a 2 c ) 2 ) ( b 2 + ( a 1 c ) 2 ) ] } } .

Metrics