Abstract

The maximum entropy model (MEM) and Kramers–Kronig (K-K) analysis were compared with the aim of phase retrieval from reflectance. The object was to test two different phase-retrieval methods when reflectance is known at a finite frequency range and data fitting is not performed beyond the finite frequency band. In addition, it was assumed that the phase is known only at one or two anchor points. As an example, we study the terahertz reflection spectrum related to a semiconductor and an optical spectrum of potassium chloride. It is shown that the MEM resolves the complex refractive index of a medium, in the vicinity of initial and final points of the spectra, better than singly and doubly subtractive K-K relations. Both methods give only satisfactory results in the event of one anchor point, but in the case of two anchor points, the MEM is better than doubly subtractive K-K. It is proposed that the MEM should be used instead of K-K analysis, for a priori information of phase at two anchor points, for the purpose of resolving the complex refractive index of a medium from reflectance with high accuracy.

© 2006 Optical Society of America

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  1. V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005).
  2. R. K. Ahrenkiel, "Modified Kramers-Kronig analysis of optical spectra," J. Opt. Soc. Am. 61, 1651-1655 (1971).
    [CrossRef]
  3. K. F. Palmer, M. Z. Williams, and B. A. Budde, "Multiply subtractive Kramers-Kronig analysis of optical data," Appl. Opt. 37, 2660-2672 (1998).
    [CrossRef]
  4. K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, "Dispersion relations and phase retrieval in optical spectroscopy," in Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37.
    [CrossRef]
  5. K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).
  6. K.-E. Peiponen and E. M. Vartiainen, "Dispersion theory of the reflectivity of s-polarized and p-polarized light," J. Opt. Soc. Am. B 23, 114-119 (2006).
    [CrossRef]
  7. J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
    [CrossRef]
  8. E. M. Vartiainen, J. J. Saarinen, and K.-E. Peiponen, "Method for extracting the complex dielectric function of nanoparticles in a water matrix from surface-plasmon resonance data," J. Opt. Soc. Am. B 22, 1173-1178 (2005).
    [CrossRef]
  9. E. M. Vartiainen and K.-E. Peiponen, "Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum," Phys. Rev. B 50, 1941-1944 (1994).
    [CrossRef]
  10. K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, "Dispersion theory and phase retrieval of meromorphic total susceptibility," J. Phys:Condens. Matter 9, 8937-8943 (1997).
    [CrossRef]
  11. V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, "Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations," Phys. Rev. B 72, 1251071 (2005).
    [CrossRef]
  12. E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, "Numerical phase correction method for terahertz time-domain reflection spectroscopy," J. Appl. Phys. 96, 4171-4175 (2004).
    [CrossRef]
  13. K.-E. Peiponen, E. Gornov, Y. Svirko, Y. Ino, M. Kuwata-Gonokami, and V. Lucarini, "Testing the validity of terahertz reflection spectra by dispersion relations," Phys. Rev. B 72, 2451091 (2005).
    [CrossRef]
  14. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).
  15. K.-E. Peiponen and E. M. Vartiainen, "Kramers-Kronig relations in optical data inversion," Phys. Rev. B 44, 8301-8303 (1991).
    [CrossRef]

2006 (1)

2005 (3)

E. M. Vartiainen, J. J. Saarinen, and K.-E. Peiponen, "Method for extracting the complex dielectric function of nanoparticles in a water matrix from surface-plasmon resonance data," J. Opt. Soc. Am. B 22, 1173-1178 (2005).
[CrossRef]

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, "Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations," Phys. Rev. B 72, 1251071 (2005).
[CrossRef]

K.-E. Peiponen, E. Gornov, Y. Svirko, Y. Ino, M. Kuwata-Gonokami, and V. Lucarini, "Testing the validity of terahertz reflection spectra by dispersion relations," Phys. Rev. B 72, 2451091 (2005).
[CrossRef]

2004 (1)

E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, "Numerical phase correction method for terahertz time-domain reflection spectroscopy," J. Appl. Phys. 96, 4171-4175 (2004).
[CrossRef]

2003 (1)

J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
[CrossRef]

1998 (1)

1997 (1)

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, "Dispersion theory and phase retrieval of meromorphic total susceptibility," J. Phys:Condens. Matter 9, 8937-8943 (1997).
[CrossRef]

1994 (1)

E. M. Vartiainen and K.-E. Peiponen, "Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum," Phys. Rev. B 50, 1941-1944 (1994).
[CrossRef]

1991 (1)

K.-E. Peiponen and E. M. Vartiainen, "Kramers-Kronig relations in optical data inversion," Phys. Rev. B 44, 8301-8303 (1991).
[CrossRef]

1971 (1)

Ahrenkiel, R. K.

Asakura, T.

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, "Dispersion theory and phase retrieval of meromorphic total susceptibility," J. Phys:Condens. Matter 9, 8937-8943 (1997).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, "Dispersion relations and phase retrieval in optical spectroscopy," in Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37.
[CrossRef]

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

Budde, B. A.

Gornov, E.

K.-E. Peiponen, E. Gornov, Y. Svirko, Y. Ino, M. Kuwata-Gonokami, and V. Lucarini, "Testing the validity of terahertz reflection spectra by dispersion relations," Phys. Rev. B 72, 2451091 (2005).
[CrossRef]

Ino, Y.

K.-E. Peiponen, E. Gornov, Y. Svirko, Y. Ino, M. Kuwata-Gonokami, and V. Lucarini, "Testing the validity of terahertz reflection spectra by dispersion relations," Phys. Rev. B 72, 2451091 (2005).
[CrossRef]

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, "Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations," Phys. Rev. B 72, 1251071 (2005).
[CrossRef]

E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, "Numerical phase correction method for terahertz time-domain reflection spectroscopy," J. Appl. Phys. 96, 4171-4175 (2004).
[CrossRef]

Kuwata-Gonokami, M.

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, "Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations," Phys. Rev. B 72, 1251071 (2005).
[CrossRef]

K.-E. Peiponen, E. Gornov, Y. Svirko, Y. Ino, M. Kuwata-Gonokami, and V. Lucarini, "Testing the validity of terahertz reflection spectra by dispersion relations," Phys. Rev. B 72, 2451091 (2005).
[CrossRef]

E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, "Numerical phase correction method for terahertz time-domain reflection spectroscopy," J. Appl. Phys. 96, 4171-4175 (2004).
[CrossRef]

Lucarini, V.

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, "Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations," Phys. Rev. B 72, 1251071 (2005).
[CrossRef]

K.-E. Peiponen, E. Gornov, Y. Svirko, Y. Ino, M. Kuwata-Gonokami, and V. Lucarini, "Testing the validity of terahertz reflection spectra by dispersion relations," Phys. Rev. B 72, 2451091 (2005).
[CrossRef]

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

Palik, E. D.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

Palmer, K. F.

Peiponen, K.-E.

K.-E. Peiponen and E. M. Vartiainen, "Dispersion theory of the reflectivity of s-polarized and p-polarized light," J. Opt. Soc. Am. B 23, 114-119 (2006).
[CrossRef]

K.-E. Peiponen, E. Gornov, Y. Svirko, Y. Ino, M. Kuwata-Gonokami, and V. Lucarini, "Testing the validity of terahertz reflection spectra by dispersion relations," Phys. Rev. B 72, 2451091 (2005).
[CrossRef]

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, "Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations," Phys. Rev. B 72, 1251071 (2005).
[CrossRef]

E. M. Vartiainen, J. J. Saarinen, and K.-E. Peiponen, "Method for extracting the complex dielectric function of nanoparticles in a water matrix from surface-plasmon resonance data," J. Opt. Soc. Am. B 22, 1173-1178 (2005).
[CrossRef]

E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, "Numerical phase correction method for terahertz time-domain reflection spectroscopy," J. Appl. Phys. 96, 4171-4175 (2004).
[CrossRef]

J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, "Dispersion theory and phase retrieval of meromorphic total susceptibility," J. Phys:Condens. Matter 9, 8937-8943 (1997).
[CrossRef]

E. M. Vartiainen and K.-E. Peiponen, "Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum," Phys. Rev. B 50, 1941-1944 (1994).
[CrossRef]

K.-E. Peiponen and E. M. Vartiainen, "Kramers-Kronig relations in optical data inversion," Phys. Rev. B 44, 8301-8303 (1991).
[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, 2005).

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, "Dispersion relations and phase retrieval in optical spectroscopy," in Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37.
[CrossRef]

Saarinen, J. J.

E. M. Vartiainen, J. J. Saarinen, and K.-E. Peiponen, "Method for extracting the complex dielectric function of nanoparticles in a water matrix from surface-plasmon resonance data," J. Opt. Soc. Am. B 22, 1173-1178 (2005).
[CrossRef]

J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
[CrossRef]

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

Shimano, R.

E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, "Numerical phase correction method for terahertz time-domain reflection spectroscopy," J. Appl. Phys. 96, 4171-4175 (2004).
[CrossRef]

Svirko, Y.

K.-E. Peiponen, E. Gornov, Y. Svirko, Y. Ino, M. Kuwata-Gonokami, and V. Lucarini, "Testing the validity of terahertz reflection spectra by dispersion relations," Phys. Rev. B 72, 2451091 (2005).
[CrossRef]

Svirko, Y. P.

E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, "Numerical phase correction method for terahertz time-domain reflection spectroscopy," J. Appl. Phys. 96, 4171-4175 (2004).
[CrossRef]

Vartiainen, E. M.

K.-E. Peiponen and E. M. Vartiainen, "Dispersion theory of the reflectivity of s-polarized and p-polarized light," J. Opt. Soc. Am. B 23, 114-119 (2006).
[CrossRef]

E. M. Vartiainen, J. J. Saarinen, and K.-E. Peiponen, "Method for extracting the complex dielectric function of nanoparticles in a water matrix from surface-plasmon resonance data," J. Opt. Soc. Am. B 22, 1173-1178 (2005).
[CrossRef]

E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, "Numerical phase correction method for terahertz time-domain reflection spectroscopy," J. Appl. Phys. 96, 4171-4175 (2004).
[CrossRef]

J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, "Dispersion theory and phase retrieval of meromorphic total susceptibility," J. Phys:Condens. Matter 9, 8937-8943 (1997).
[CrossRef]

E. M. Vartiainen and K.-E. Peiponen, "Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum," Phys. Rev. B 50, 1941-1944 (1994).
[CrossRef]

K.-E. Peiponen and E. M. Vartiainen, "Kramers-Kronig relations in optical data inversion," Phys. Rev. B 44, 8301-8303 (1991).
[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, 2005).

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, "Dispersion relations and phase retrieval in optical spectroscopy," in Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37.
[CrossRef]

Williams, M. Z.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
[CrossRef]

J. Appl. Phys. (1)

E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, "Numerical phase correction method for terahertz time-domain reflection spectroscopy," J. Appl. Phys. 96, 4171-4175 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

J. Phys:Condens. Matter (1)

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, "Dispersion theory and phase retrieval of meromorphic total susceptibility," J. Phys:Condens. Matter 9, 8937-8943 (1997).
[CrossRef]

Phys. Rev. B (4)

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, "Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations," Phys. Rev. B 72, 1251071 (2005).
[CrossRef]

E. M. Vartiainen and K.-E. Peiponen, "Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum," Phys. Rev. B 50, 1941-1944 (1994).
[CrossRef]

K.-E. Peiponen and E. M. Vartiainen, "Kramers-Kronig relations in optical data inversion," Phys. Rev. B 44, 8301-8303 (1991).
[CrossRef]

K.-E. Peiponen, E. Gornov, Y. Svirko, Y. Ino, M. Kuwata-Gonokami, and V. Lucarini, "Testing the validity of terahertz reflection spectra by dispersion relations," Phys. Rev. B 72, 2451091 (2005).
[CrossRef]

Other (4)

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

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

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, "Dispersion relations and phase retrieval in optical spectroscopy," in Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37.
[CrossRef]

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

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Figures (6)

Fig. 1
Fig. 1

Reflectance (solid curve) and phase (dotted curve) for n-type, undoped (100) InAs wafers obtained from the Drude permittivity for metal. Γ = 3.3 × 10 12 s 1 , ω p 2 = 130 × 10 24 s 2 , and ε b = 15.6 .

Fig. 2
Fig. 2

(a) Real refractive index and (b) extinction coefficient calculated in the case of one anchor point 11 , 6 × 10 12 l / s , indicated by an arrow. MEM gives a better match than SSKK with the exact curves.

Fig. 3
Fig. 3

(a) Real refractive index and (b) extinction coefficient obtained with the MEM and DSKK for two anchor points 6, 7 and 14 , 8 × 10 12 l / s , which are indicated by two arrows. The MEM provides an excellent match with exact curves.

Fig. 4
Fig. 4

Reflectance and phase of KCl.

Fig. 5
Fig. 5

(a) Real refractive index (b) and extinction coefficient obtained with the MEM and SSKK in the case of one anchor point, 24   eV , indicated by an arrow. SSKK gives a better estimate than the MEM but all the curves depart significantly from the exact ones.

Fig. 6
Fig. 6

(a) Real refractive index and (b) extinction coefficient of KCl in the case of two anchor points 13, 1 and 31, 3   eV , indicated by two arrows. The MEM gives almost a perfect match with the exact curves.

Equations (16)

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R = | N 1 N + 1 | 2 .
φ ( ω ) ω = φ ( ω 1 ) ω 1 2 π ( ω 2 ω 1 2 ) × P 0 log | r ( ω ) | ( ω 2 ω 2 ) ( ω 2 ω 1 2 ) d ω ,
φ ( ω ) ω = ( ω 2 ω 2 2 ) ( ω 1 2 ω 2 2 ) φ ( ω 1 ) ω 1 + ( ω 2 ω 1 2 ) ( ω 2 2 ω 1 2 ) φ ( ω 2 ) ω 2 2 π ( ω 2 ω 2 2 ) ( ω 2 ω 1 2 ) × P 0 log | r ( ω ) | ( ω 2 ω 2 ) ( ω 2 ω 1 2 ) ( ω 2 ω 2 2 ) d ω .
ω q = { ( ω b 2 ω a 2 ) cos [ ( 2 q + 1 ) π 2 Q ] + ( ω b 2 + ω a 2 ) 2 } 1 / 2 ,
R ( ν ) = | r ( ν ) | 2 | b 0 1 + m = 1 M b m exp ( 2 π i m ν ) | 2 ,
ν = ω ω a ω b ω a .
p = 1 M b p C ( m p ) = { | b 0 | 2 , m = 0 0 , m = 1 ,   , M ,
C ( m ) = 0 1 R ( ν ) exp [ i 2 π m ν ] d ν ,
C ( m ) = N 1 n = 0 N 1 R ( ν n ) exp [ i 2 π m n / N ] .
r ( ν ) | b o | exp { i [ arg { r ( ν ) } ψ ( ν ) ] } 1 + m = 1 M b m * exp ( 2 π i m ν ) = | b o | exp [ i ϕ ( ν ) ] 1 + m = 1 M b m * exp ( 2 π i m ν ) ,
ϕ ( ν ) = l = 0 L B ν l ,
( 1 ν o ν o L 1 ν 1 L 1 ν L ν L L ) ( B o B L ) = ( ϕ ( ν o ) ϕ ( ν L ) ) .
R ( ν ) = { R ( ν a ) , 0 ν < w K ( ω a ) R ( ν ) , w K ( ω a ) ν w K ( ω b ) R ( ν b ) , w K ( ω b ) < ν 1 ,
w K ( ω ) = 1 2 K + 1 ( ω ω a ω b ω a + K ) ,
ν = w K ( ω ) w K ( ω a ) w K ( ω b ) w K ( ω a ) ,
ε ( ω ) = ε b ( 1 ω p 2 ω 2 + i Γ ω ) ,

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