Abstract

We propose a novel method of measuring the group refractive index of biological tissues at the micrometer scale. The technique utilizes a broadband confocal microscope embedded into a Mach–Zehnder interferometer, with which spectral interferograms are measured as the sample is translated through the focus of the beam. The method does not require phase unwrapping and is insensitive to vibrations in the sample and reference arms. High measurement stability is achieved because a single spectral interferogram contains all the information necessary to compute the optical path delay of the beam transmitted through the sample. Included are a physical framework defining the forward problem, linear solutions to the inverse problem, and simulated images of biologically relevant phantoms.

© 2008 Optical Society of America

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A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12, 051403 (2007).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

J. Fingler, D. Schwartz, C. Yang, and S. E. Fraser, “Mobility and transverse flow visualization using phase variance contrast with spectral optical coherence tomography,” Opt. Express 15, 12636-12653 (2007).
[CrossRef] [PubMed]

2006 (2)

Y.-P. Wang, D.-N. Wang, W. Jin, J.-P. Chen, X.-W. Li, and J.-H. Zhou, “Reflectometry measuring refractive index and thickness of polymer samples simultaneously,” J. Mod. Opt. 53, 1845-1851 (2006).
[CrossRef]

A. M. Zysk, E. J. Chaney, and S. A. Boppart, “Refractive index of carcinogen-induced rat mammary tumours,” Phys. Med. Biol. 51, 2165-2177 (2006).
[CrossRef] [PubMed]

2005 (2)

2004 (2)

Z. Liu, X. Dong, Q. Chen, C. Yin, Y. Xu, and Y. Zheng, “Nondestructive measurement of an optical fiber refractive-index profile by a transmitted-light differential interference contact microscope,” Appl. Opt. 43, 1485-1492 (2004).
[CrossRef] [PubMed]

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

2003 (3)

2002 (2)

S. Singh, “Refractive index measurement and its applications,” Phys. Scr. 65, 167-180 (2002).
[CrossRef]

I. K. Ilev, R. W. Waynant, K. R. Byrnes, and J. J. Anders, “Dual confocal fiber-optic method for measurement of refractive index and thickness of optically transparent media,” Opt. Lett. 27, 1603-1605 (2002).
[CrossRef]

2001 (1)

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

1995 (3)

1994 (1)

D. F. Aldridge, “Linearization of the eikonal equation,” Geophysics 59, 1631-1632 (1994).
[CrossRef]

1990 (1)

H. Takubo, “Refractive index as a measure for saturation and supersaturation in crystal growth of water-soluble substances,” J. Cryst. Growth 104, 239-244 (1990).
[CrossRef]

Aalders, M. C. G.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Aldridge, D. F.

R. Snieder and D. F. Aldridge, “Perturbation theory for travel times,” J. Acoust. Soc. Am. 98, 1565-1569 (1995).
[CrossRef]

D. F. Aldridge, “Linearization of the eikonal equation,” Geophysics 59, 1631-1632 (1994).
[CrossRef]

Alexandrov, S. A.

Anders, J. J.

I. K. Ilev, R. W. Waynant, K. R. Byrnes, and J. J. Anders, “Dual confocal fiber-optic method for measurement of refractive index and thickness of optically transparent media,” Opt. Lett. 27, 1603-1605 (2002).
[CrossRef]

Armstrong, J. J.

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

Badizadegan, K.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Boppart, S. A.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12, 051403 (2007).
[CrossRef] [PubMed]

A. M. Zysk, E. J. Chaney, and S. A. Boppart, “Refractive index of carcinogen-induced rat mammary tumours,” Phys. Med. Biol. 51, 2165-2177 (2006).
[CrossRef] [PubMed]

A. M. Zysk, J. J. Reynolds, D. L. Marks, P. S. Carney, and S. A. Boppart, “Projected index computed tomography,” Opt. Lett. 28, 701-703 (2003).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1980).

Bouma, B. E.

Brezinski, M. E.

Byrnes, K. R.

I. K. Ilev, R. W. Waynant, K. R. Byrnes, and J. J. Anders, “Dual confocal fiber-optic method for measurement of refractive index and thickness of optically transparent media,” Opt. Lett. 27, 1603-1605 (2002).
[CrossRef]

Carney, P. S.

Cense, B.

Chaney, E. J.

A. M. Zysk, E. J. Chaney, and S. A. Boppart, “Refractive index of carcinogen-induced rat mammary tumours,” Phys. Med. Biol. 51, 2165-2177 (2006).
[CrossRef] [PubMed]

Chen, J.-P.

Y.-P. Wang, D.-N. Wang, W. Jin, J.-P. Chen, X.-W. Li, and J.-H. Zhou, “Reflectometry measuring refractive index and thickness of polymer samples simultaneously,” J. Mod. Opt. 53, 1845-1851 (2006).
[CrossRef]

Chen, Q.

Cheriaux, G.

Choi, W.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Dasari, R. R.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

de Boer, J. F.

Decraemer, W. F.

J. J. J. Dirckx, L. C. Kuypers, and W. F. Decraemer, “Refractive index of tissue measured with confocal microscopy,” J. Biomed. Opt. 10, 044014 (2005).
[CrossRef]

Dirckx, J. J. J.

J. J. J. Dirckx, L. C. Kuypers, and W. F. Decraemer, “Refractive index of tissue measured with confocal microscopy,” J. Biomed. Opt. 10, 044014 (2005).
[CrossRef]

Dong, X.

Faber, D. J.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Fang-Yen, C.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Feld, M. S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Fingler, J.

Fraser, S. E.

Fujimoto, J. G.

Gbur, G.

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1996).

Hee, M. R.

Hillman, T. R.

Hooper, B. A.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Ilev, I. K.

I. K. Ilev, R. W. Waynant, K. R. Byrnes, and J. J. Anders, “Dual confocal fiber-optic method for measurement of refractive index and thickness of optically transparent media,” Opt. Lett. 27, 1603-1605 (2002).
[CrossRef]

Jin, W.

Y.-P. Wang, D.-N. Wang, W. Jin, J.-P. Chen, X.-W. Li, and J.-H. Zhou, “Reflectometry measuring refractive index and thickness of polymer samples simultaneously,” J. Mod. Opt. 53, 1845-1851 (2006).
[CrossRef]

Joffre, M.

Kuypers, L. C.

J. J. J. Dirckx, L. C. Kuypers, and W. F. Decraemer, “Refractive index of tissue measured with confocal microscopy,” J. Biomed. Opt. 10, 044014 (2005).
[CrossRef]

Lepetit, L.

Li, X.-W.

Y.-P. Wang, D.-N. Wang, W. Jin, J.-P. Chen, X.-W. Li, and J.-H. Zhou, “Reflectometry measuring refractive index and thickness of polymer samples simultaneously,” J. Mod. Opt. 53, 1845-1851 (2006).
[CrossRef]

Liu, Z.

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Marks, D. L.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12, 051403 (2007).
[CrossRef] [PubMed]

A. M. Zysk, J. J. Reynolds, D. L. Marks, P. S. Carney, and S. A. Boppart, “Projected index computed tomography,” Opt. Lett. 28, 701-703 (2003).
[CrossRef] [PubMed]

Mik, E. G.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Mujat, M.

Nguyen, F. T.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12, 051403 (2007).
[CrossRef] [PubMed]

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Oldenburg, A. L.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12, 051403 (2007).
[CrossRef] [PubMed]

Park, B. H.

Pierce, M. C.

Reynolds, J. J.

Sampson, D. D.

Schwartz, D.

Silva, K. K. M. B. D.

Singh, S.

S. Singh, “Refractive index measurement and its applications,” Phys. Scr. 65, 167-180 (2002).
[CrossRef]

Snieder, R.

R. Snieder and D. F. Aldridge, “Perturbation theory for travel times,” J. Acoust. Soc. Am. 98, 1565-1569 (1995).
[CrossRef]

Southern, J. F.

Takubo, H.

H. Takubo, “Refractive index as a measure for saturation and supersaturation in crystal growth of water-soluble substances,” J. Cryst. Growth 104, 239-244 (1990).
[CrossRef]

Tearney, G. J.

Tsuzuki, T.

van Gemert, M. J. C.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

van Leeuwen, T. G.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1996).

Wang, D.-N.

Y.-P. Wang, D.-N. Wang, W. Jin, J.-P. Chen, X.-W. Li, and J.-H. Zhou, “Reflectometry measuring refractive index and thickness of polymer samples simultaneously,” J. Mod. Opt. 53, 1845-1851 (2006).
[CrossRef]

Wang, Y.-P.

Y.-P. Wang, D.-N. Wang, W. Jin, J.-P. Chen, X.-W. Li, and J.-H. Zhou, “Reflectometry measuring refractive index and thickness of polymer samples simultaneously,” J. Mod. Opt. 53, 1845-1851 (2006).
[CrossRef]

Waynant, R. W.

I. K. Ilev, R. W. Waynant, K. R. Byrnes, and J. J. Anders, “Dual confocal fiber-optic method for measurement of refractive index and thickness of optically transparent media,” Opt. Lett. 27, 1603-1605 (2002).
[CrossRef]

Wolf, E.

Xu, Y.

Yang, C.

Yin, C.

Yun, S.-H.

Zheng, Y.

Zhou, J.-H.

Y.-P. Wang, D.-N. Wang, W. Jin, J.-P. Chen, X.-W. Li, and J.-H. Zhou, “Reflectometry measuring refractive index and thickness of polymer samples simultaneously,” J. Mod. Opt. 53, 1845-1851 (2006).
[CrossRef]

Zvyagin, A. V.

Zysk, A. M.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12, 051403 (2007).
[CrossRef] [PubMed]

A. M. Zysk, E. J. Chaney, and S. A. Boppart, “Refractive index of carcinogen-induced rat mammary tumours,” Phys. Med. Biol. 51, 2165-2177 (2006).
[CrossRef] [PubMed]

A. M. Zysk, J. J. Reynolds, D. L. Marks, P. S. Carney, and S. A. Boppart, “Projected index computed tomography,” Opt. Lett. 28, 701-703 (2003).
[CrossRef] [PubMed]

Appl. Opt. (1)

Geophysics (1)

D. F. Aldridge, “Linearization of the eikonal equation,” Geophysics 59, 1631-1632 (1994).
[CrossRef]

Inverse Probl. (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

J. Acoust. Soc. Am. (1)

R. Snieder and D. F. Aldridge, “Perturbation theory for travel times,” J. Acoust. Soc. Am. 98, 1565-1569 (1995).
[CrossRef]

J. Biomed. Opt. (2)

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12, 051403 (2007).
[CrossRef] [PubMed]

J. J. J. Dirckx, L. C. Kuypers, and W. F. Decraemer, “Refractive index of tissue measured with confocal microscopy,” J. Biomed. Opt. 10, 044014 (2005).
[CrossRef]

J. Cryst. Growth (1)

H. Takubo, “Refractive index as a measure for saturation and supersaturation in crystal growth of water-soluble substances,” J. Cryst. Growth 104, 239-244 (1990).
[CrossRef]

J. Mod. Opt. (1)

Y.-P. Wang, D.-N. Wang, W. Jin, J.-P. Chen, X.-W. Li, and J.-H. Zhou, “Reflectometry measuring refractive index and thickness of polymer samples simultaneously,” J. Mod. Opt. 53, 1845-1851 (2006).
[CrossRef]

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

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

Nat. Methods (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (4)

Phys. Med. Biol. (1)

A. M. Zysk, E. J. Chaney, and S. A. Boppart, “Refractive index of carcinogen-induced rat mammary tumours,” Phys. Med. Biol. 51, 2165-2177 (2006).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Phys. Scr. (1)

S. Singh, “Refractive index measurement and its applications,” Phys. Scr. 65, 167-180 (2002).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1980).

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1996).

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Figures (3)

Fig. 1
Fig. 1

Diagram of group refractive index confocal microscopy instrument.

Fig. 2
Fig. 2

(a), (b), Images of the same slice through the center plane of the simulated phantom: (a) real part of the refractive index, (b) imaginary part. (c), (d), Images of the data acquired by the confocal instrument of these planes: (c) total number of radians of retardance between the two measured frequencies recorded at each beam position, (d) total attenuation difference in nats between the two measured frequencies recorded at each beam position. (e) Real part of the weighted least-squares reconstruction of the index profile, (f) imaginary part.

Fig. 3
Fig. 3

(a) 2-D projection along one transverse axis of the 3-D transfer function F ̃ ( Q ; k ) of the simulated confocal microscope. The function is radially symmetric about the Q = 0 axis. (b) 2-D projection along one transverse axis of the 3-D point-spread function of the simulated confocal microscope. The point-spread function is likewise radially symmetric. Both density plots use relative units.

Equations (33)

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

2 u ( r ) + k 2 [ n 0 ( r ) + ϵ n ( r r 0 ) ] 2 u ( r ) = 0 ,
u 0 ( r ; k ) ϕ ( r , r 0 ; k ) = 2 k 2 V d 3 r g ( r , r ; k ) u 0 ( r ; k ) n 0 ( r ) n ( r r 0 ) ,
u 0 ( r ; k ) = ( 2 π ) 2 k 2 d 2 q exp [ i ( q r + k z ( q ) z ) ] B ̃ ( q k ) ,
g ( r , r ; k ) = i k 2 π z = 0 d 2 r P ( r r ; k ) d 2 q k z ( q ) 1 exp [ i ( q ( r r ) + k z ( q ) ( z z ) ) ] ,
g ( r , r ; k ) = i k ( 2 π ) 3 z = 0 d 2 r d 2 q exp [ i q ( r r ) ] P ̃ ( q k ) d 2 q k z ( q ) 1 exp [ i ( q ( r r ) + k z ( q ) ( z z ) ) ] .
g ( r , r ; k ) = i k 2 π d 2 q k z ( q ) 1 P ̃ ( q k ) exp [ i ( q ( r r ) k z ( q ) z ) ] .
u 0 ( r ; k ) ϕ ( r , r 0 ; k ) = 2 i k ( 2 π ) 3 V d 3 r n 0 ( r ) n ( r r 0 ) d 2 q exp [ i ( q r + k z ( q ) z ) ] B ̃ ( q k ) d 2 q k z ( q ) 1 P ̃ ( q k ) exp [ i ( q ( r r ) k z ( q ) z ) ] .
u 0 ( 0 ; k ) ϕ ̃ ( Q ; k ) = 2 i k ( 2 π ) 3 d 3 r 0 V d 3 r n 0 ( r ) n ( r r 0 ) exp ( i Q r 0 ) d 2 q exp [ i ( q r + k z ( q ) z ) ] B ̃ ( q k ) d 2 q k z ( q ) 1 P ̃ ( q k ) exp [ i ( q r k z ( q ) z ) ] .
u 0 ( 0 ; k ) ϕ ̃ ( Q ; k ) = 2 i k ( 2 π ) 3 n ̃ ( Q ) d 2 q B ̃ ( q k ) d 2 q k z ( q ) 1 P ̃ ( q k ) V d 2 r d z n 0 ( r ) exp [ i ( Q r + Q z z ) ] exp [ i ( q r + k z ( q ) z ) ] exp [ i ( q r k z ( q ) z ) ] ,
u 0 ( 0 ; k ) ϕ ̃ ( Q ; k ) = 2 i k n b ( 2 π ) 1 n ̃ ( Q ) d 2 q B ̃ ( q k ) d 2 q k z ( q ) 1 P ̃ ( q k ) V d z δ ( 2 ) ( Q + q q ) exp ( i Q z z ) exp [ i k z ( q ) z ] exp [ i k z ( q ) z ] .
u 0 ( 0 ; k ) ϕ ̃ ( Q ; k ) = 2 i k n b ( 2 π ) 1 n ̃ ( Q ) d 2 q k z ( q ) 1 P ̃ ( q k ) B ̃ ( q Q k ) V d z exp ( i Q z z ) exp [ i k z ( q Q ) z ] exp [ i k z ( q ) z ] .
u 0 ( 0 ; k ) ϕ ̃ ( Q ; k ) = 2 i k n b n ̃ ( Q ) d 2 q k z ( q ) 1 P ̃ ( q k ) B ̃ ( q Q k ) δ [ Q z + k z ( q Q ) k z ( q ) ] .
F ̃ ( Q ; k ) = 2 i n b d 2 q k z ( q ) 1 P ̃ ( q k ) B ̃ ( q Q k ) δ [ Q z + k z ( q Q ) k z ( q ) ]
I ( r 0 ; k , τ ) = u r exp ( i k c τ ) + u 0 ( 0 ; k ) exp [ ϕ ( 0 , r 0 ; k ) ] 2 .
ϕ ( 0 , r 0 ; k ) = log { [ u r * u 0 ( 0 ; k ) ] 1 [ 1 i 4 I ( r 0 ; k , 0 ) 1 + i 4 I ( r 0 ; k , π c k ) + i 2 I ( r 0 ; k , π 2 c k ) ] } .
d ( u 0 ϕ ̃ ) d k = n ̃ F ̃ + k d n ̃ d k F ̃ + k n ̃ d F ̃ d k .
d ( u 0 ϕ ̃ ) d k = F ̃ ( Q ; k ) [ n ̃ + k d n ̃ d k ] = F ̃ ( Q ; k ) n ̃ g ( Q ) .
I ( r 0 ; k , τ ) = u r 2 + u 0 ( 0 ; k ) exp [ ϕ ( 0 , r 0 ; k ) ] 2 + 2 Re { u r * u 0 ( 0 ; k ) exp [ ϕ ( 0 , r 0 ; k ) i k c τ ] } .
u 0 ϕ = d ( u 0 ϕ ) d k = u 0 d ϕ d k + d u 0 d k ϕ .
ϕ = d ϕ d k ϕ ( 0 , r 0 ; k + Δ k ) ϕ ( 0 , r 0 ; k ) Δ k = log u s ( 0 , r 0 ; k + Δ k ) u 0 ( 0 ; k + Δ k ) log u s ( 0 , r 0 ; k ) u 0 ( 0 ; k ) Δ k = 1 Δ k log u s ( 0 , r 0 ; k + Δ k ) u s ( 0 , r 0 ; k ) u 0 ( 0 ; k ) u 0 ( 0 ; k + Δ k ) = 1 Δ k log u s ( 0 , r 0 ; k + Δ k ) u s ( 0 , r 0 ; k ) .
ϕ = i = 1 N 1 1 k i + 1 k i log u s ( 0 , r 0 ; k i + 1 ) u s ( 0 , r 0 ; k i ) .
u 0 ϕ = u 0 ( 0 ; k ) ϕ ( r 0 ; k ) = Fn g = V d 3 r n g ( r ) F ( r 0 r ; k ) ,
with F ( r ; k ) = ( 2 π ) 3 d 3 Q exp ( i r Q ) F ̃ ( Q ; k ) ,
n g ( r ) = ( 2 π ) 3 d 3 Q exp ( i r Q ) n ̃ g ( Q ) .
n g + = arg min n g u 0 ϕ Fn g 2 + γ n g 2 = arg min n g V d 3 r 0 u 0 ( 0 ; k ) ϕ ( r 0 ; k ) V d 3 r n g ( r ) F ( r 0 r ; k ) 2 + γ V d 3 r n g ( r ) 2 .
n ̃ g + ( Q ; k ) = u 0 ( 0 ; k ) ϕ ̃ ( Q ; k ) F ̃ * ( Q ; k ) F ̃ ( Q ; k ) 2 + γ .
n g + = arg min n g W ( u 0 ϕ Fn g ) 2 + γ n g 2 = arg min n g V d 3 r 0 W ( r 0 ) 2 u 0 ( 0 ; k ) ϕ ( 0 , r 0 ; k ) V d 3 r n g ( r ) F ( r 0 r ; k ) 2 + γ V d 3 r n g ( r ) 2 .
F ̃ ( Q ; k ) = 2 i n b π 2 π 2 d θ 0 d s s k z ( q ) 1 P ̃ ( q k ) B ̃ ( q Q k ) δ [ f ( s ) ] , with f ( s ) = Q z + k 2 ( Q 2 ) 2 s 2 Q s cos θ k 2 ( Q 2 ) 2 s 2 + Q s cos θ .
s = ± Q z 4 k 2 Q 2 Q z 2 2 Q z 2 + Q 2 cos 2 θ .
F ̃ ( Q ; k ) = 2 i n b π 2 π 2 d θ s k z ( q ) 1 P ̃ ( q k ) B ̃ ( q Q k ) d f d s 1 ,
with s = Q z 4 k 2 Q 2 Q z 2 2 Q z 2 + Q 2 cos 2 θ ,
q = Q 2 Q Q s cos θ Q Q s sin θ , Q { Q , z ̂ } = 0 ,
d f d s = 2 s Q cos θ 2 k 2 ( Q 2 ) 2 s 2 + Q s cos θ 2 s + Q cos θ 2 k 2 ( Q 2 ) 2 s 2 Q s cos θ .

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