Abstract

A model is developed for optical coherence tomography and interferometric synthetic aperture microscopy (ISAM) systems employing full-field frequency-scanned illumination with partial spatial coherence. This model is used to derive efficient ISAM inverse scattering algorithms that give diffraction-limited resolution in regions typically regarded as out of focus. Partial spatial coherence of the source is shown to have the advantage of mitigating multiple-scattering effects that can otherwise produce significant artifacts in full-field coherent imaging.

© 2009 Optical Society of America

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2008 (2)

B. J. Davis, D. L. Marks, T. S. Ralston, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy: computed imaging for scanned coherent microscopy,” Sensors 8, 3903-3931 (2008).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Real-time interferometric synthetic aperture microscopy,” Opt. Express 16, 2555-2569 (2008).
[CrossRef] [PubMed]

2007 (4)

2006 (4)

2005 (8)

Y. Watanabe, Y. Hayasaka, M. Sato, and N. Tanno, “Full-field optical coherence tomography by achromatic phase shifting with a rotating polarizer,” Appl. Opt. 44, 1387-1392 (2005).
[CrossRef] [PubMed]

G. Moneron, A.-C. Bocarra, and A. Dubois, “Stroboscopic ultrahigh-resolution full-field optical coherence tomography,” Opt. Lett. 30, 1351-1353 (2005).
[CrossRef] [PubMed]

B. Karamata, M. Laubscher, M. Leutenegger, S. Bourquin, and T. Lasser, “Multiple scattering in optical coherence tomography. I. Investigation and modeling,” J. Opt. Soc. Am. A 22, 1369-1379 (2005).
[CrossRef]

B. Karamata, M. Leutenegger, M. Laubscher, S. Bourquin, and T. Lasser, “Multiple scattering in optical coherence tomography. II. Experimental and theoretical investigation of cross talk in wide-field optical coherence tomography,” J. Opt. Soc. Am. A 22, 1380-1388 (2005).
[CrossRef]

K. Grieve, A. Dubois, M. Simonutti, M. Paques, J. Sahel, J.-F. Le Gargasson, and C. Bocarra, “In vivo anterior segment imaging in the rat eye with high speed white light full-field optical coherence tomography,” Opt. Express 13, 6286-6295 (2005).
[CrossRef] [PubMed]

P. Blazkiewicz, M. Gourlay, J. R. Tucker, A. D. Rakic, and A. V. Zvyagin, “Signal-to-noise ratio study of full-field Fourier-domain optical coherence tomography,” Appl. Opt. 34, 7722-7729 (2005).
[CrossRef]

K. Grieve, G. Moneron, A. Dubois, J.-F. Le Gargasson, and C. Boccara, “Ultrahigh resolution ex vivo ocular imaging using ultrashort acquisition time en face optical coherence tomography,” J. Opt. A, Pure Appl. Opt. 7, 368-373 (2005).
[CrossRef]

A. V. Zvyagin, P. Blazkiewicz, and J. Vintrou, “Image reconstruction in full-field Fourier-domain optical coherence tomography,” J. Opt. A, Pure Appl. Opt. 7, 350-356 (2005).
[CrossRef]

2004 (6)

2003 (5)

2002 (2)

2000 (1)

1999 (1)

1998 (1)

1997 (1)

1995 (1)

1959 (1)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

Akiba, M.

Arthaber, H.

Beaurepaire, E.

Benattar, L.

Blazkiewicz, P.

A. V. Zvyagin, P. Blazkiewicz, and J. Vintrou, “Image reconstruction in full-field Fourier-domain optical coherence tomography,” J. Opt. A, Pure Appl. Opt. 7, 350-356 (2005).
[CrossRef]

P. Blazkiewicz, M. Gourlay, J. R. Tucker, A. D. Rakic, and A. V. Zvyagin, “Signal-to-noise ratio study of full-field Fourier-domain optical coherence tomography,” Appl. Opt. 34, 7722-7729 (2005).
[CrossRef]

Bocarra, A.-C.

Bocarra, C.

Boccara, A.-C.

Boccara, C.

K. Grieve, G. Moneron, A. Dubois, J.-F. Le Gargasson, and C. Boccara, “Ultrahigh resolution ex vivo ocular imaging using ultrashort acquisition time en face optical coherence tomography,” J. Opt. A, Pure Appl. Opt. 7, 368-373 (2005).
[CrossRef]

A. K. Grieve, G. Moneron, R. Lecaque, L. Vabre, and C. Boccara, “Ultrahigh-resolution full-field optical coherence tomography,” Appl. Opt. 43, 2874-2883 (2004).
[CrossRef] [PubMed]

Bonner, R. F.

Boppart, S. A.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Real-time interferometric synthetic aperture microscopy,” Opt. Express 16, 2555-2569 (2008).
[CrossRef] [PubMed]

B. J. Davis, D. L. Marks, T. S. Ralston, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy: computed imaging for scanned coherent microscopy,” Sensors 8, 3903-3931 (2008).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 5, 129-134 (2007).
[CrossRef]

B. J. Davis, S. C. Schlachter, D. L. Marks, T. S. Ralston, S. A. Boppart, and P. S. Carney, “Nonparaxial vector-field modeling of optical coherence tomography and interferometric synthetic aperture microscopy,” J. Opt. Soc. Am. A 24, 2527-2542 (2007).
[CrossRef]

B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Autocorrelation artifacts in optical coherence tomography and interferometric synthetic aperture microscopy,” Opt. Lett. 32, 1441-1443 (2007).
[CrossRef] [PubMed]

D. L. Marks, T. S. Ralston, S. A. Boppart, and P. S. Carney, “Inverse scattering for frequency-scanned full-field optical coherence tomography,” J. Opt. Soc. Am. A 24, 1034-1041 (2007).
[CrossRef]

D. L. Marks, T. S. Ralston, P. S. Carney, and S. A. Boppart, “Inverse scattering for rotationally scanned optical coherence tomography,” J. Opt. Soc. Am. A 23, 2433-2439 (2006).
[CrossRef]

T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett. 31, 3585-3587 (2006).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23, 1027-1037 (2006).
[CrossRef]

D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, “A digital algorithm for dispersion correction in optical coherence tomography for homogeneous and stratified media,” Appl. Opt. 42, 204-217 (2003).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1980), Chap. 10.5.2, pp. 524-526.

Bourquin, S.

Carney, P. S.

B. J. Davis, D. L. Marks, T. S. Ralston, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy: computed imaging for scanned coherent microscopy,” Sensors 8, 3903-3931 (2008).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Real-time interferometric synthetic aperture microscopy,” Opt. Express 16, 2555-2569 (2008).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 5, 129-134 (2007).
[CrossRef]

B. J. Davis, S. C. Schlachter, D. L. Marks, T. S. Ralston, S. A. Boppart, and P. S. Carney, “Nonparaxial vector-field modeling of optical coherence tomography and interferometric synthetic aperture microscopy,” J. Opt. Soc. Am. A 24, 2527-2542 (2007).
[CrossRef]

D. L. Marks, T. S. Ralston, S. A. Boppart, and P. S. Carney, “Inverse scattering for frequency-scanned full-field optical coherence tomography,” J. Opt. Soc. Am. A 24, 1034-1041 (2007).
[CrossRef]

B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Autocorrelation artifacts in optical coherence tomography and interferometric synthetic aperture microscopy,” Opt. Lett. 32, 1441-1443 (2007).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett. 31, 3585-3587 (2006).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23, 1027-1037 (2006).
[CrossRef]

D. L. Marks, T. S. Ralston, P. S. Carney, and S. A. Boppart, “Inverse scattering for rotationally scanned optical coherence tomography,” J. Opt. Soc. Am. A 23, 2433-2439 (2006).
[CrossRef]

Chan, K. P.

Chen, Z.

Chinn, S. R.

Choma, M. A.

Davis, B. J.

de Boer, J. F.

De Martino, A.

Drevillon, B.

Drexler, W.

Dubois, A.

Dubois, F.

Fercher, A. F.

Fujimoto, J. G.

Goodman, J.

J. Goodman, Statistical Optics (Wiley, 1985), Chap. 5.3.3, pp. 193-195.

Gourlay, M.

P. Blazkiewicz, M. Gourlay, J. R. Tucker, A. D. Rakic, and A. V. Zvyagin, “Signal-to-noise ratio study of full-field Fourier-domain optical coherence tomography,” Appl. Opt. 34, 7722-7729 (2005).
[CrossRef]

Grieve, A. K.

Grieve, K.

K. Grieve, A. Dubois, M. Simonutti, M. Paques, J. Sahel, J.-F. Le Gargasson, and C. Bocarra, “In vivo anterior segment imaging in the rat eye with high speed white light full-field optical coherence tomography,” Opt. Express 13, 6286-6295 (2005).
[CrossRef] [PubMed]

K. Grieve, G. Moneron, A. Dubois, J.-F. Le Gargasson, and C. Boccara, “Ultrahigh resolution ex vivo ocular imaging using ultrashort acquisition time en face optical coherence tomography,” J. Opt. A, Pure Appl. Opt. 7, 368-373 (2005).
[CrossRef]

A. Dubois, G. Moneron, K. Grieve, and A.-C. Boccara, “Three-dimensional cellular-level imaging using full-field optical coherence tomography,” Phys. Med. Biol. 49, 1227-1234 (2004).
[CrossRef] [PubMed]

Hayasaka, Y.

Hermann, B.

Hitzenberger, C. K.

Istasse, E.

Izatt, J. A.

Joannes, L.

Karamata, B.

Lambelet, P.

Lasser, T.

Laubscher, M.

Laude, B.

Le Gargasson, J.-F.

K. Grieve, G. Moneron, A. Dubois, J.-F. Le Gargasson, and C. Boccara, “Ultrahigh resolution ex vivo ocular imaging using ultrashort acquisition time en face optical coherence tomography,” J. Opt. A, Pure Appl. Opt. 7, 368-373 (2005).
[CrossRef]

K. Grieve, A. Dubois, M. Simonutti, M. Paques, J. Sahel, J.-F. Le Gargasson, and C. Bocarra, “In vivo anterior segment imaging in the rat eye with high speed white light full-field optical coherence tomography,” Opt. Express 13, 6286-6295 (2005).
[CrossRef] [PubMed]

Lecaque, R.

Legros, J.-C.

Leitgeb, R.

Leutenegger, M.

Lorlette, V.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Marks, D. L.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Real-time interferometric synthetic aperture microscopy,” Opt. Express 16, 2555-2569 (2008).
[CrossRef] [PubMed]

B. J. Davis, D. L. Marks, T. S. Ralston, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy: computed imaging for scanned coherent microscopy,” Sensors 8, 3903-3931 (2008).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 5, 129-134 (2007).
[CrossRef]

B. J. Davis, S. C. Schlachter, D. L. Marks, T. S. Ralston, S. A. Boppart, and P. S. Carney, “Nonparaxial vector-field modeling of optical coherence tomography and interferometric synthetic aperture microscopy,” J. Opt. Soc. Am. A 24, 2527-2542 (2007).
[CrossRef]

B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Autocorrelation artifacts in optical coherence tomography and interferometric synthetic aperture microscopy,” Opt. Lett. 32, 1441-1443 (2007).
[CrossRef] [PubMed]

D. L. Marks, T. S. Ralston, S. A. Boppart, and P. S. Carney, “Inverse scattering for frequency-scanned full-field optical coherence tomography,” J. Opt. Soc. Am. A 24, 1034-1041 (2007).
[CrossRef]

D. L. Marks, T. S. Ralston, P. S. Carney, and S. A. Boppart, “Inverse scattering for rotationally scanned optical coherence tomography,” J. Opt. Soc. Am. A 23, 2433-2439 (2006).
[CrossRef]

T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett. 31, 3585-3587 (2006).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23, 1027-1037 (2006).
[CrossRef]

D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, “A digital algorithm for dispersion correction in optical coherence tomography for homogeneous and stratified media,” Appl. Opt. 42, 204-217 (2003).
[CrossRef] [PubMed]

Minetti, C.

Moneron, G.

G. Moneron, A.-C. Bocarra, and A. Dubois, “Stroboscopic ultrahigh-resolution full-field optical coherence tomography,” Opt. Lett. 30, 1351-1353 (2005).
[CrossRef] [PubMed]

K. Grieve, G. Moneron, A. Dubois, J.-F. Le Gargasson, and C. Boccara, “Ultrahigh resolution ex vivo ocular imaging using ultrashort acquisition time en face optical coherence tomography,” J. Opt. A, Pure Appl. Opt. 7, 368-373 (2005).
[CrossRef]

A. K. Grieve, G. Moneron, R. Lecaque, L. Vabre, and C. Boccara, “Ultrahigh-resolution full-field optical coherence tomography,” Appl. Opt. 43, 2874-2883 (2004).
[CrossRef] [PubMed]

A. Dubois, G. Moneron, K. Grieve, and A.-C. Boccara, “Three-dimensional cellular-level imaging using full-field optical coherence tomography,” Phys. Med. Biol. 49, 1227-1234 (2004).
[CrossRef] [PubMed]

Monnom, O.

Moreau, J.

Nelson, J. S.

Oldenburg, A. L.

Paques, M.

Potton, R. J.

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717-754 (2004).
[CrossRef]

Povazay, B.

Rakic, A. D.

P. Blazkiewicz, M. Gourlay, J. R. Tucker, A. D. Rakic, and A. V. Zvyagin, “Signal-to-noise ratio study of full-field Fourier-domain optical coherence tomography,” Appl. Opt. 34, 7722-7729 (2005).
[CrossRef]

Ralston, T. S.

B. J. Davis, D. L. Marks, T. S. Ralston, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy: computed imaging for scanned coherent microscopy,” Sensors 8, 3903-3931 (2008).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Real-time interferometric synthetic aperture microscopy,” Opt. Express 16, 2555-2569 (2008).
[CrossRef] [PubMed]

D. L. Marks, T. S. Ralston, S. A. Boppart, and P. S. Carney, “Inverse scattering for frequency-scanned full-field optical coherence tomography,” J. Opt. Soc. Am. A 24, 1034-1041 (2007).
[CrossRef]

B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Autocorrelation artifacts in optical coherence tomography and interferometric synthetic aperture microscopy,” Opt. Lett. 32, 1441-1443 (2007).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 5, 129-134 (2007).
[CrossRef]

B. J. Davis, S. C. Schlachter, D. L. Marks, T. S. Ralston, S. A. Boppart, and P. S. Carney, “Nonparaxial vector-field modeling of optical coherence tomography and interferometric synthetic aperture microscopy,” J. Opt. Soc. Am. A 24, 2527-2542 (2007).
[CrossRef]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23, 1027-1037 (2006).
[CrossRef]

T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett. 31, 3585-3587 (2006).
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K. Grieve, G. Moneron, A. Dubois, J.-F. Le Gargasson, and C. Boccara, “Ultrahigh resolution ex vivo ocular imaging using ultrashort acquisition time en face optical coherence tomography,” J. Opt. A, Pure Appl. Opt. 7, 368-373 (2005).
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Opt. Commun. (1)

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Figures (4)

Fig. 1
Fig. 1

Diagram of full-field OCT instrument with a source of adjustable partial coherence.

Fig. 2
Fig. 2

Contours of the mapping β ( q ; k ) for various coherence parameters χ. The transverse widths of the contours are representative of the bandwidth for a system with α = 1.57 ( NA = 0.9 ) . This bandwidth is determined by C ( q ; k ) [Eq. (32)]. Note that β ( q ; k ) is a function on the two-dimensional q plane, but varies only with the magnitude q, as plotted here.

Fig. 3
Fig. 3

An illustration of multiple scattering—in this case second-order scattering. Light from the source plane is focused into the sample, scatters twice, and is focused onto the detector. The reference-arm optics image the source plane onto the detector, so in this simplified diagram the source and detector planes are colocated.

Fig. 4
Fig. 4

(a), (c), (e), (g) OCT and (b), (d), (f), (h) ISAM images of an object consisting of three point scatterers at ( 20 , 0 , 0 ) λ , ( 0 , 0 , 15 ) λ , and ( 20 , 0 , 30 ) λ . The source spatial coherence is varied as described by the parameter χ, the numerical aperture of the objective is 0.2 and the focal plane is at z = 0 . The coherence lengths are (a), (b) ( χ = 0 ) , (c), (d) 25 λ ( χ = 0.002 ) , (e), (f) 11 λ ( χ = 0.01 ) , and (g), (h) 1.1 λ ( χ = 0.5 ) . The images are formed from data consisting of first- and second-order scattering effects. A projection over the y axis of the three-dimensional image magnitudes is taken to produce the two-dimensional images displayed.

Equations (44)

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I ( r , ω ) = E * ( r , ω ) E ( r , ω ) ,
I ( ρ , k ) = E r ( ρ , k ) + E s ( ρ , k ) 2 = E r ( ρ , k ) 2 + 2 Re { E r * ( ρ , k ) E s ( ρ , k ) } + E s ( ρ , k ) 2 ,
S ( ρ , k ) = E r * ( ρ , k ) E s ( ρ , k ) .
S ̂ ( ρ , Δ z ) = E ̂ r * ( ρ , z Δ z ) E ̂ s ( ρ , z ) .
S ̂ ( ρ , Δ z ) = 0 , Δ z < 0 .
E i ( r , k ) = d 2 ρ E 0 ( ρ , k ) G i 0 ( r , ρ ; z , k ) ,
E s ( ρ , k ) = d 3 r [ k 2 η ( r ) E i ( r , k ) ] G 0 i ( ρ , r ; z , k ) = k 2 d 3 r d 2 ρ E 0 ( ρ , k ) G i 0 ( r , ρ ; z , k ) G 0 i ( ρ , r ; z , k ) η ( r ) ,
G 0 i ( ρ , r ; z , k ) = G i 0 ( r , ρ ; z , k ) .
G i 0 ( r , ρ ; z , k ) = g ( r ρ M ; z , k ) ,
E s ( ρ , k ) = k 2 d 3 r d 2 ρ E 0 ( ρ , k ) g ( r ρ M ; z , k ) g ( r ρ M ; z , k ) η ( r ) .
S ( ρ , k ) = E r * ( ρ , k ) E s ( ρ , k ) E 0 * ( ρ , k ) E s ( ρ , k ) .
S ( ρ , k ) = k 2 M 2 d 3 r d 2 ρ W ( ρ M , ρ M ; k ) g ( r ρ M ; z , k ) g ( r ρ M ; z , k ) η ( r ) ,
W ( ρ , ρ ; k ) = M 2 E 0 * ( M ρ , k ) , E 0 ( M ρ , k ) .
W ( ρ , ρ ; k ) = A ( k ) b ( ρ ρ ; k ) .
S ( ρ , k ) = k 2 A ( k ) d z d 2 r [ b g ] ( r ρ M ; z , k ) g ( r ρ M ; z , k ) η ( r ; z ) .
S ( ρ , k ) = d z [ h η ] ( ρ M ; z , k ) ,
h ( r ; z , k ) = k 2 A ( k ) [ b g ] ( r ; z , k ) g ( r ; z , k ) .
g ( r ; z , k ) = i 2 π d 2 q G ( q k ) k z ( q ) e i [ q r + k z ( q ) z ] ,
k z ( q ) = k 2 q 2 ,
g ̃ ( q ; z , k ) = 2 π i G ( q k ) k z ( q ) e i k z ( q ) z .
h ̃ ( q ; z , k ) = 4 π 2 k 2 A ( k ) d 2 q B ( q ; k ) G ( q k ) k z ( q ) G [ ( q q ) k ] k z ( q q ) e i [ k z ( q ) + k z ( q q ) ] z ,
S ̃ ( q ; k ) = 4 π 2 k 2 M 2 A ( k ) d z d 2 q B ( q ; k ) G ( q k ) G [ ( M q q ) k ] k z ( q ) k z ( M q q ) η ̃ ( M q ; z ) e i [ k z ( q ) + k z ( M q q ) ] z .
S ̃ ( q ; k ) = 4 π 2 k 2 M 2 A ( k ) d 2 q B ( q ; k ) G ( q k ) G [ ( M q q ) k ] k z ( q ) k z ( M q q ) × η ͌ { M q ; [ k z ( q ) + k z ( M q q ) ] } ,
h ̃ ( q ; z , k ) = 16 π 4 k 2 A ( k ) G ( 0 ) k G [ q k ] k z ( q ) e i [ k + k z ( q ) ] z .
S ̃ ( q ; k ) = 16 π 4 k M 2 A ( k ) G ( 0 ) G ( M q k ) k z ( M q ) η ͌ { M q ; [ k + k z ( M q ) ] } .
h ̃ ( q ; z , k ) = 4 π 2 k 2 A ( k ) d 2 q G ( q k ) k z ( q ) G [ ( q q ) k ] k z ( q q ) e i [ k z ( q ) + k z ( q q ) ] z .
G ( q k ) = exp ( α 2 q 2 2 k 2 ) ,
B ( q ; k ) = 1 χ α 2 2 π exp ( 1 χ χ α 2 q 2 2 k 2 ) .
h ̃ ( q ; z , k ) 4 π 2 k 2 A ( k ) e i [ k z ( p ) + k z ( q p ) ] z k z ( p ) k z ( q p ) d 2 q B ( q ; k ) G ( q k ) G ( q q k ) .
h ̃ ( q ; z , k ) 4 π 2 k 2 A ( k ) exp { i [ k z ( q χ 1 + χ ) + k z ( q 1 + χ ) ] z } k z ( q χ 1 + χ ) k z ( q 1 + χ ) k 2 1 + χ exp ( α 2 q 2 2 k 2 ( 1 + χ ) ) .
S ( q ; k ) = C ( q ; k ) η ̃ ̃ ( M q ; β ( q ; k ) ) ,
C ( q ; k ) = 4 π 2 k 4 M 2 A ( k ) ( 1 + χ ) k z ( M q χ 1 + χ ) k z ( M q 1 + χ ) exp ( α 2 M q 2 2 k 2 ( 1 + χ ) ) ,
β ( q ; k ) = k z ( M q χ 1 + χ ) k z ( M q 1 + χ ) .
h ̃ ( q ; z , k ) 4 π 2 k 2 A ( k ) π i k z e i 2 k z ( q 2 ) z B ( q 2 ; k ) G ( q 2 k ) G ( q 2 k ) .
S ( q ; k ) = C ( q ; k ) η ̃ ̃ ( M q ; β ( q ; k ) ) ,
η ̃ ̃ ( q , β ) = d z η ̃ ( q , z ) k z e i β z ,
β ( q ; k ) = 2 k z ( M q 2 ) ,
C ( q ; k ) = i 2 π 2 k 2 M 2 α 2 A ( k ) χ exp ( α 2 M ( 1 + χ ) q 2 8 k 2 χ ) .
z t = 1 + χ 2 χ α 2 k = 1 + χ 2 χ λ π N A 2 .
S + ( q , k ) = C + ( q ; k ) S ( q , k ) .
C + ( q ; k ) = C * ( q ; k ) C ( q ; k ) 2 + γ 2 ,
E s ( ρ , k ) = k 4 d 3 r d 3 r d 2 ρ E 0 ( ρ , k ) G i 0 ( r , ρ ; z , k ) η ( r ) G f ( r , r ; k ) η ( r ) G 0 i ( ρ , r ; z , k ) .
G f ( r , r ; k ) = exp ( i k r r ) r r .
S ( ρ , k ) = k 4 M 2 d 3 r d 3 r d 2 ρ W ( ρ M , ρ M ; k ) × g ( r ρ M ; z , k ) exp ( i k r r ) r r g ( r ρ M ; z , k ) η ( r ) η ( r ) .

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