Abstract

The Kramers–Kronig (KK) algorithm, useful for retrieving the phase of a spectrum based on the known spectral amplitude, is applied to reconstruct the impulse response of a diffusive medium. It is demonstrated by a simulation of a 1D scattering medium with realistic parameters that its impulse response can be generated from the KK method with high accuracy.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  12. 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]
  13. 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]
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    [CrossRef]
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    [CrossRef]
  20. R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Kramers-Kronig relations for an interferometer," Opt. Commun. , 138, 118-126 (1997).
    [CrossRef]
  21. M. Beck, I. A. Walmsley, and J. D. Kafka, "Group delay measurements of optical components near 800nm," IEEE J. Quantum Electron. 27, 2074-2081 (1991).
    [CrossRef]
  22. V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

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 (2)

2003 (1)

2001 (1)

1998 (1)

1997 (3)

G. W. Milton, D. J. Eyre, and J. V. Mantese, "Finite frequency range Kramers-Kronig relations: bounds on the dispersion," Phys. Rev. Lett. 79, 3062-3065 (1997).
[CrossRef]

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Kramers-Kronig relations for an interferometer," Opt. Commun. , 138, 118-126 (1997).
[CrossRef]

A. Ya. Polishchuk, J. Dolne, F. Liu, and R. R. Alfano,"Average and most-probable photon paths in random media," Opt. Lett. 22, 430-432 (1997).
[CrossRef] [PubMed]

1995 (2)

1994 (1)

1991 (2)

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]

M. Beck, I. A. Walmsley, and J. D. Kafka, "Group delay measurements of optical components near 800nm," IEEE J. Quantum Electron. 27, 2074-2081 (1991).
[CrossRef]

1971 (1)

1956 (1)

J. S. Toll, "Causality and the dispersion relation: logical foundations," Phys. Rev. 104, 1760 (1956).
[CrossRef]

1926 (1)

Ahrenkiel, R. K.

Alfano, R. R.

Beck, M.

M. Beck, I. A. Walmsley, and J. D. Kafka, "Group delay measurements of optical components near 800nm," IEEE J. Quantum Electron. 27, 2074-2081 (1991).
[CrossRef]

Ben-Aderet, Y.

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]

Budde, B. A.

Chance, B.

de Vries, P.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Kramers-Kronig relations for an interferometer," Opt. Commun. , 138, 118-126 (1997).
[CrossRef]

Delpy, D. T.

Dolne, J.

Eyre, D. J.

G. W. Milton, D. J. Eyre, and J. V. Mantese, "Finite frequency range Kramers-Kronig relations: bounds on the dispersion," Phys. Rev. Lett. 79, 3062-3065 (1997).
[CrossRef]

Galland, P.

Granot, E.

Hebden, J. C.

Ho, P. P.

L. Wang, X. Liang, P. Galland, P. P. Ho, and R. R. Alfano, "True scattering coefficients of turbid matter measured by early-time gating," Opt. Lett. 20, 913-915 (1995).
[CrossRef] [PubMed]

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]

Holboke, M. J.

Intes, X.

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]

Kafka, J. D.

M. Beck, I. A. Walmsley, and J. D. Kafka, "Group delay measurements of optical components near 800nm," IEEE J. Quantum Electron. 27, 2074-2081 (1991).
[CrossRef]

Kop, R. H. J.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Kramers-Kronig relations for an interferometer," Opt. Commun. , 138, 118-126 (1997).
[CrossRef]

Kramers, H. A.

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

Kronig, R.

Lagendijk, A.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Kramers-Kronig relations for an interferometer," Opt. Commun. , 138, 118-126 (1997).
[CrossRef]

Liang, X.

Liu, F.

A. Ya. Polishchuk, J. Dolne, F. Liu, and R. R. Alfano,"Average and most-probable photon paths in random media," Opt. Lett. 22, 430-432 (1997).
[CrossRef] [PubMed]

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]

Lucarini, V.

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

Mantese, J. V.

G. W. Milton, D. J. Eyre, and J. V. Mantese, "Finite frequency range Kramers-Kronig relations: bounds on the dispersion," Phys. Rev. Lett. 79, 3062-3065 (1997).
[CrossRef]

Milton, G. W.

G. W. Milton, D. J. Eyre, and J. V. Mantese, "Finite frequency range Kramers-Kronig relations: bounds on the dispersion," Phys. Rev. Lett. 79, 3062-3065 (1997).
[CrossRef]

Ntziachristos, V.

Palmer, K. F.

Peiponen, K.-E.

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

Polishchuk, A. Ya.

Ripoll, J.

Saarinen, J. J.

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

Schermann, D.

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]

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]

Soubret, A.

Sprik, R.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Kramers-Kronig relations for an interferometer," Opt. Commun. , 138, 118-126 (1997).
[CrossRef]

Steinmeyer, G.

Sternklar, S.

Stibenz, G.

Toll, J. S.

J. S. Toll, "Causality and the dispersion relation: logical foundations," Phys. Rev. 104, 1760 (1956).
[CrossRef]

Trebino, R.

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Lasers (Kluwer Academic, 2002).
[CrossRef]

Tuchin, V.

V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE, 2000).

Turner, G. M.

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).

Walmsley, I. A.

M. Beck, I. A. Walmsley, and J. D. Kafka, "Group delay measurements of optical components near 800nm," IEEE J. Quantum Electron. 27, 2074-2081 (1991).
[CrossRef]

Wang, L.

L. Wang, X. Liang, P. Galland, P. P. Ho, and R. R. Alfano, "True scattering coefficients of turbid matter measured by early-time gating," Opt. Lett. 20, 913-915 (1995).
[CrossRef] [PubMed]

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]

Williams, M. Z.

Yodh, A.

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[CrossRef]

Yodh, A. G.

Zacharakis, G.

Zhang, G.

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]

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]

IEEE J. Quantum Electron. (1)

M. Beck, I. A. Walmsley, and J. D. Kafka, "Group delay measurements of optical components near 800nm," IEEE J. Quantum Electron. 27, 2074-2081 (1991).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (1)

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, "Kramers-Kronig relations for an interferometer," Opt. Commun. , 138, 118-126 (1997).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. (1)

J. S. Toll, "Causality and the dispersion relation: logical foundations," Phys. Rev. 104, 1760 (1956).
[CrossRef]

Phys. Rev. Lett. (1)

G. W. Milton, D. J. Eyre, and J. V. Mantese, "Finite frequency range Kramers-Kronig relations: bounds on the dispersion," Phys. Rev. Lett. 79, 3062-3065 (1997).
[CrossRef]

Phys. Today (1)

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[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 (4)

V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE, 2000).

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Lasers (Kluwer Academic, 2002).
[CrossRef]

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 (9)

Fig. 1
Fig. 1

Refractive index as a function of location for a diffusive medium.

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

Impulse response of the medium. The upper plot is the exact reconstruction, while the lower one is the KK reconstruction.

Fig. 4
Fig. 4

Zoom-in of Fig. 3.

Fig. 5
Fig. 5

Same as Fig. 3 but with a spectral width of Δ ω = 2 π × 30 THz .

Fig. 6
Fig. 6

Same as Fig. 4 but with a spectral width of Δ ω = 2 π × 30 THz .

Fig. 7
Fig. 7

Phases with and without the CT. The two upper plots compare the exact phases ϕ (solid curve) and the KK phases ϕ K K (dashed curve) at the two sides of the spectrum. The two lower plots compare ϕ (solid curve) to the KK phases with the CT, i.e., ϕ K K + Δ ϕ K K (dashed curve).

Fig. 8
Fig. 8

In the upper figure, the reconstruction was done with the CT, while in the lower one, the calculations were done without it. In both cases, the solid curve represents the direct calculation, and the dashed curve corresponds to the KK reconstruction.

Fig. 9
Fig. 9

Illustration of the incoming and outgoing waves at the vicinity of a single scatterer.

Equations (31)

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ϕ K K ( ω ) = ω π P 0 d ω ln H ( ω ) ω 2 ω 2 + j arg ( ω ω j ω ω j * ) ,
2 x 2 ψ 2 + ( 2 π n ( x ) λ ) 2 ψ = 0 ,
2 x 2 ψ 2 + ( 2 π λ ) 2 [ n 0 2 + 2 n 0 Δ n ( x ) ] 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 .
ϕ K K ( k ) k π P k min k max d k ln H ( k ) k 2 k 2 ,
H K K ( k ) = H ( k ) exp [ i ϕ K K ( k ) ] ,
ϕ K K ( k ) = k π P k min k max d k ln H ( k ) k 2 k 2 k π ln H ( k ) [ 0 k min d k 1 k 2 k 2 + k max d k 1 k 2 k 2 ] ,
ϕ K K ( k ) = k π P k min k max d k ln H ( k ) k 2 k 2 + Δ ϕ K K ( k ) ,
Δ ϕ K K ( k ) 1 2 π ln H ( k ) ln ( k k min k + k min k max + k k max k )
I ( t ) = k min k max d k H ( k ) exp ( i k c t ) d k 2 ,
I K K ( t ) = k min k max d k H K K ( k ) exp ( i k c t ) d k 2 .
Δ ϕ K K ( k ) = 1 2 π ln H ( k ) [ 4 κ 2 η κ 2 + 16 3 ( 1 + 8 3 η 2 ) κ 3 + O ( κ 4 ) ] ,
τ delay linear d Δ ϕ K K linear ( k ) d ( c k ) = 2 π ln H ( k ) τ .
Δ ϕ K K nonlinear ( k ) 8 3 π ln H ( k ) κ 3 .
τ delay nonlinear d Δ ϕ K K nonlinear ( k ) d ( c k ) < 2 π ln H ( k ) τ ,
v j = ( u j + u j ) ,
ψ ( x j < x < x j + 1 ) = u j + exp ( i k x ) + u j exp ( i k x ) .
ψ ( x = 0 ) = ψ ( x = + 0 ) ,
u j + + u j = u j + 1 + + u j + 1 .
x ψ x = + 0 x ψ x = 0 = α j ψ ( 0 ) ,
i k ( u j + 1 + u j + 1 u j + + u j ) = α ( u j + + u j ) .
v j = ( u j + u j )
v j + 1 = ( u j + 1 + u j + 1 )
v j + 1 = A j v j ,
A j = ( 1 i η j i η j i η j 1 + i η j ) ,
D j = ( exp ( i k a j ) 0 0 exp ( i k a j ) ) .
M j = A j D j M j 1 .
M N = ( m 11 m 12 m 21 m 22 ) ,
H ( k ) = det ( M N ) m 22 ,

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