Abstract

We apply the multiply subtractive anchoring method for efficient phase spectrum retrieval, which is based on the fast Fourier transform and Lagrange polynomials. Because the polynomials eventually diverge, choosing the optimum anchoring points is crucial. It is demonstrated that, if more than two anchoring points are chosen, the algorithm’s performance can easily deteriorate.

© 2011 Optical Society of America

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  12. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. (Prentice-Hall, 1999).
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    [CrossRef]
  14. K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy. Classical Theory (Springer, 2010).

2010

2009

2008

2006

E. Granot, S. Sternklar, D. Schermann, Y. Ben-Aderet, and M. H. Itzhaq, “200 femtosecond impulse response of a Fabry-Perot etalon with the spectral ballistic imaging technique,” Appl. Phys. B 82, 359–362 (2006).
[CrossRef]

2003

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive generalized Kramers-Kronig relations: application on third harmonic generation susceptibility on polysilane,” J. Chem. Phys. 119, 11095–11098 (2003).
[CrossRef]

E. Granot and S. Sternklar, “Spectral ballistic imaging: a novel technique for viewing through turbid or obstructing media,” J. Opt. Soc. Am. A 20, 1595–1599 (2003).
[CrossRef]

1998

1971

1970

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1, 818–831 (1970).
[CrossRef]

1926

Ahrenkiel, R. K.

Asakura, T.

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy. Classical Theory (Springer, 2010).

Bachrach, R. Z.

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1, 818–831 (1970).
[CrossRef]

Ben-Aderet, Y.

Brown, F. C.

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1, 818–831 (1970).
[CrossRef]

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. (Prentice-Hall, 1999).

Budde, B. A.

Granot, E.

Itzhaq, M. H.

E. Granot, S. Sternklar, D. Schermann, Y. Ben-Aderet, and M. H. Itzhaq, “200 femtosecond impulse response of a Fabry-Perot etalon with the spectral ballistic imaging technique,” Appl. Phys. B 82, 359–362 (2006).
[CrossRef]

Kopeika, N. S.

Kramers, H. A.

H. A. Kramers, Estratto dagli Atti del Congresso Internazionale di Fisici Como (Nicolo Zonichello, 1927).

Kronig, R.

Lucarini, V.

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive generalized Kramers-Kronig relations: application on third harmonic generation susceptibility on polysilane,” J. Chem. Phys. 119, 11095–11098 (2003).
[CrossRef]

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

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. (Prentice-Hall, 1999).

Palmer, K. F.

Peiponen, K.-E.

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive generalized Kramers-Kronig relations: application on third harmonic generation susceptibility on polysilane,” J. Chem. Phys. 119, 11095–11098 (2003).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy. Classical Theory (Springer, 2010).

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

Saarinen, J. J.

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive generalized Kramers-Kronig relations: application on third harmonic generation susceptibility on polysilane,” J. Chem. Phys. 119, 11095–11098 (2003).
[CrossRef]

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

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. (Prentice-Hall, 1999).

Schermann, D.

E. Granot, S. Sternklar, D. Schermann, Y. Ben-Aderet, and M. H. Itzhaq, “200 femtosecond impulse response of a Fabry-Perot etalon with the spectral ballistic imaging technique,” Appl. Phys. B 82, 359–362 (2006).
[CrossRef]

Sternklar, S.

Vartiainen, E. M.

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy. Classical Theory (Springer, 2010).

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

Williams, M. Z.

Appl. Opt.

Appl. Phys. B

E. Granot, S. Sternklar, D. Schermann, Y. Ben-Aderet, and M. H. Itzhaq, “200 femtosecond impulse response of a Fabry-Perot etalon with the spectral ballistic imaging technique,” Appl. Phys. B 82, 359–362 (2006).
[CrossRef]

J. Chem. Phys.

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive generalized Kramers-Kronig relations: application on third harmonic generation susceptibility on polysilane,” J. Chem. Phys. 119, 11095–11098 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Phys. Rev. B

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1, 818–831 (1970).
[CrossRef]

Other

H. A. Kramers, Estratto dagli Atti del Congresso Internazionale di Fisici Como (Nicolo Zonichello, 1927).

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

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. (Prentice-Hall, 1999).

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy. Classical Theory (Springer, 2010).

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Figures (3)

Fig. 1
Fig. 1

Comparison between three different phase spectra (upper figure) and comparison between their spectral derivatives (lower figure): the theory (dotted curve), KK reconstruction without anchoring (dashed curve), and KK with anchoring (solid curve).

Fig. 2
Fig. 2

Same as Fig. 1: comparison between reconstruction with (solid curve) and without (dashed curve) anchoring.

Fig. 3
Fig. 3

Error versus number of anchoring points for three different criteria as explained in the text: dotted curve, minimum error configuration; dashed curve, average error for random configuration; solid curve, uniform spacing configuration.

Equations (22)

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| H ( ω ) | = 2 π P 0 ω θ ( ω ) ω 2 ω 2 d ω ,
θ ( ω ) = 2 ω π P 0 ln | H ( ω ) | ω 2 ω 2 d ω .
θ KK ( ω ) = 2 ω π P ω L ω H ln | H ( ω ) | ω 2 ω 2 d ω or θ KK ( ω ) = ω π P ω L ω H ln [ I ( ω ) ] ω 2 ω 2 d ω ,
H KK ( ω ) = | H ( ω ) | exp [ i θ KK ( ω ) ] or H KK ( ω ) = I ( ω ) exp [ i θ KK ( ω ) ] ,
θ ( ω ) ω = θ ( ω 1 ) ω 1 ( ω 2 ω 2 2 ) ( ω 2 ω 3 2 ) ( ω 2 ω Q 2 ) ( ω 1 2 ω 2 2 ) ( ω 1 2 ω 3 2 ) ( ω 1 2 ω Q 2 ) + θ ( ω j ) ω j ( ω 2 ω 1 2 ) ( ω 2 ω j 1 2 ) ( ω 2 ω j + 1 2 ) ( ω 2 ω Q 2 ) ( ω j 2 ω 1 2 ) ( ω j 2 ω j 1 2 ) ( ω j 2 ω j + 1 2 ) ( ω j 2 ω Q 2 ) + θ ( ω Q ) ω Q ( ω 2 ω 1 2 ) ( ω 2 ω 2 2 ) ( ω 2 ω Q 1 2 ) ( ω Q 2 ω 1 2 ) ( ω Q 2 ω 2 2 ) ( ω Q 2 ω Q 1 2 ) 2 π ( ω 2 ω 1 2 ) ( ω 2 ω 2 2 ) ( ω 2 ω Q 2 ) P 0 ln | H ( ω ) | ( ω 2 ω 2 ) ( ω 2 ω 1 2 ) ( ω 2 ω Q 2 ) d ω
θ ( ω ) ω = θ ( ω 1 ) ω 1 ( ω 2 ω 2 2 ) ( ω 1 2 ω 2 2 ) + θ ( ω 2 ) ω 2 ( ω 2 ω 1 2 ) ( ω 2 2 ω 1 2 ) 2 π ( ω 2 ω 1 2 ) ( ω 2 ω 2 2 ) P 0 ln | H ( ω ) | ( ω 2 ω 2 ) ( ω 2 ω 1 2 ) ( ω 2 ω 2 2 ) d ω .
f k = f 0 + k Δ f ,
Im { H [ k ] } = 2 Re { H [ k ] } η [ k ] 2 N m = 0 N 1 Re { H [ k ] } η [ k m ] ,
η [ k ] { cot ( π k / N ) , k odd 0 , k even .
θ [ k ] = ln { I [ k ] } η [ k ] 1 N m = 0 N 1 ln { I [ k ] } η [ k m ] .
H [ k ] = exp ( FFT [ IFFT { ln | H [ k ] | } u [ k ] ] ) ,
u [ k ] = { 1 k = 0 , N / 2 2 k = 1 , 2 , ( N / 2 ) 1 0 ( N / 2 ) + 1 , N 1
θ [ k ] = arg { H [ k ] } = arg { exp ( FFT [ IFFT { ln | H [ k ] | } u [ k ] ] ) } ,
θ [ k ] = Im { FFT [ IFFT { ln | H [ k ] | } u [ k ] ] } .
α n α [ k n ] for 1 n Q ,
θ A [ k ] = θ [ k ] + 1 Q n = 1 Q ( α n θ [ k n ] ) .
θ A [ k ] = θ [ k ] + n = 1 Q { α n θ [ k n ] } m j = 1 Q k k m k n k m ,
θ A [ k ] = θ [ k ] + { α 1 θ [ k 1 ] } ( k k 2 ) ( k k 3 ) ( k k Q ) ( k 1 k 2 ) ( k 1 k 3 ) ( k 1 k Q ) + { α j θ [ k j ] } ( k k 1 ) ( k k j 1 ) ( k k j + 1 ) ( k k Q ) ( k j k 1 ) ( k j k j 1 ) ( k j k j + 1 ) ( k j k Q ) + { α Q θ [ k Q ] } ( k k 1 ) ( k k 2 ) ( k k Q 1 ) ( k Q k 1 ) ( k Q k 2 ) ( k Q k Q 1 ) ,
θ A [ k ] = θ [ k ] + { α 1 θ [ k 1 ] } k k 2 k 1 k 2 + { α 2 θ [ k 2 ] } k k 1 k 2 k 1 .
Δ θ [ k ] = θ [ k ] θ [ k 1 ] ,
Θ [ k ] θ [ k ] θ theory [ k ] = θ [ k ] α k and Δ Θ [ k ] Δ θ [ k ] Δ θ theory [ k ] = Δ θ [ k ] α k .
Error θ A [ k ] α k = k = 0 N 1 ( θ A [ k ] α k ) 2 .

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