Abstract

We apply the multiply subtractive Kramers–Kronig (MSKK) method to the derivative of a medium’s optical transfer function. That is, the phase “difference” or derivative Δθ (instead of the phase) can be evaluated from the measurements of the relative derivative of the intensity Δln[I(ω)]=ΔI(ω)I(ω) with the aid of a few Δθ measurements. As a result, we obtain a method that integrates two different techniques, MSKK and spectral ballistic imaging. We show that the transfer function can be evaluated with great accuracy without the need to measure the phases at all but rather its derivative, which is a much simpler process.

© 2008 Optical Society of America

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References

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  14. 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).
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  15. 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]
  16. E. Granot, S. Sternklar, Y. Ben-Aderet, and D. Schermann, “Quasi-ballistic imaging through a dynamic scattering medium with optical-field averaging using Spectral-Ballistic-Imaging,” Opt. Express 14, 8598-8603 (2006).
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2006 (2)

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]

E. Granot, S. Sternklar, Y. Ben-Aderet, and D. Schermann, “Quasi-ballistic imaging through a dynamic scattering medium with optical-field averaging using Spectral-Ballistic-Imaging,” Opt. Express 14, 8598-8603 (2006).
[CrossRef] [PubMed]

2005 (1)

2003 (2)

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]

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]

1998 (1)

1997 (1)

1995 (1)

1994 (1)

1991 (1)

L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769-771 (1991).
[CrossRef] [PubMed]

1971 (1)

1970 (1)

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

1926 (1)

Appl. Opt. (1)

Appl. Phys. B (1)

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. (1)

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. (2)

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

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

E. Granot and S. Sternklar, “Reconstructing the impulse response of a diffusive medium with the Kramers-Kronig relations,” J. Opt. Soc. Am. B (to be published).

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. B (1)

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

Science (1)

L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769-771 (1991).
[CrossRef] [PubMed]

Other (2)

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).

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Figures (2)

Fig. 1
Fig. 1

Schematic presentation of the diffusive medium. Each scatterer is presented in the model as a delta function change in the refractive index.

Fig. 2
Fig. 2

Reconstruction of the phase difference Δ θ by the DMSKK method (dashed curve) and its comparison to the ordinary KK reconstruction (dashed-dotted curve) and to the theory (solid curve).

Equations (19)

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Re { a ( ω ) } = 2 π P 0 ω Im { a ( ω ) } ω 2 ω 2 d ω ,
Im { a ( ω ) } = 2 ω π P 0 Re { a ( ω ) } ω 2 ω 2 d ω ,
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 ) = I ( ω ) exp [ i θ KK ( ω ) ] , or H KK ( ω ) = H ( 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 2 π ( ω 2 ω 1 2 ) P 0 ln H ( ω ) ( ω 2 ω 2 ) ( ω 2 ω 1 2 ) d ω .
d ln [ H ( ω ) ] d ω = d ln H ( ω ) d ω + i d θ d 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 d ln H ( ϖ ) d ϖ ϖ = ω ( ω 2 ω 2 ) ( ω 2 ω 1 2 ) ( ω 2 ω Q 2 ) d ω
θ ( ω ) ω = j = 1 Q θ ( ω j ) ω j n = 1 j Q ( ω 2 ω n 2 ) ( ω j 2 ω n 2 ) 2 π n = 1 Q ( ω 2 ω n 2 ) P 0 d ln H ( ϖ ) d ϖ ϖ = ω ( ω 2 ω 2 ) n = 1 Q ( ω 2 ω n 2 ) d ω ,
Δ θ ( ω ) ω = j = 1 Q Δ θ ( ω j ) ω j n = 1 j Q ( ω 2 ω n 2 ) ( ω j 2 ω n 2 ) 2 π n = 1 Q ( ω 2 ω n 2 ) m Δ ln H ( ω m ) ( ω m 2 ω 2 ) n = 1 Q ( ω m 2 ω n 2 ) Δ ω ,
ω m = ω L + ( m 1 ) Δ ω
Δ θ ( ω ) ω = Δ θ ( ω 1 ) ω 1 2 π ( ω 2 ω 1 2 ) m Δ ln H ( ω m ) ( ω m 2 ω 2 ) ( ω m 2 ω 1 2 ) Δ ω .
θ ( ω m ) = j = 1 m Δ θ ( ω j ) .
2 x 2 ψ + ( 2 π n ( x ) λ ) 2 ψ = 0 ,
2 x 2 ψ + k 2 ( 1 + 2 Δ n ( x ) n 0 ) ψ = 0 ,
α j = 2 δ n j l j n 0
2 x 2 ψ + k 2 [ 1 + j = 1 N α j δ ( x L j ) ] ψ = 0 .

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