Abstract

We propose a 3D imaging technique based on the combination of full-field swept-source optical coherence microscopy (FF-SSOCM) with low spatial coherence illumination and a special numerical processing that allows for numerically focused coherent-noise-free imaging without mechanical scanning in longitudinal or transversal directions. We show, both theoretically and experimentally, that the blurring effects arising in FF-SSOCM due to defocus can be corrected by appropriate numerical processing even when low spatial coherence illumination is used. A FF-SSOCM system was built for testing the performance of this technique. Coherent-noise-free imaging of a sample with longitudinal extent exceeding the optical depth of field is demonstrated without displacement of the sample or any optical element.

© 2014 Optical Society of America

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2013

A. A. Grebenyuk and V. P. Ryabukho, “Numerical reconstruction of volumetric image in swept-source interference microscopy,” AIP Conf. Proc. 1537, 147–154 (2013).
[CrossRef]

V. P. Ryabukho, D. V. Lyakin, A. A. Grebenyuk, and S. S. Klykov, “Wiener–Khintchin theorem for spatial coherence of optical wave field,” J. Opt. 15, 025405 (2013).
[CrossRef]

2012

M. D. Bayleyegn, H. Makhlouf, C. Crotti, K. Plamann, and A. Dubois, “Ultrahigh resolution spectral-domain optical coherence tomography at 1.3  μm using a broadband superluminescent diode light source,” Opt. Commun. 285, 5564–5569 (2012).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Theoretical model of volumetric objects imaging in a microscope,” Proc. SPIE 8430, 84301B (2012).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Coherence effects of thick objects imaging in interference microscopy,” Proc. SPIE 8427, 84271M (2012).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Numerical correction of coherence gate in full-field swept-source interference microscopy,” Opt. Lett. 37, 2529–2531 (2012).
[CrossRef]

2011

2010

2008

2007

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3, 129–134 (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]

2006

2005

2004

2003

2002

1999

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).
[CrossRef]

1998

1995

J. Rosen and A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[CrossRef]

1994

1991

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

1982

D. K. Hamilton and C. J. R. Sheppard, “A confocal interference microscope,” Optica Acta 29, 1573–1577 (1982).
[CrossRef]

Abdulhalim, I.

A. Safrani and I. Abdulhalim, “Spatial coherence effect on layer thickness determination in narrowband full-field optical coherence tomography,” Appl. Opt. 50, 3021–3027 (2011).
[CrossRef]

I. Abdulhalim, “Competence between spatial and temporal coherence in full field optical coherence tomography and interference microscopy,” J. Opt. A 8, 952–958 (2006).
[CrossRef]

Aguirre, A. D.

A. D. Aguirre and J. G. Fujimoto, “Optical coherence microscopy,” in Optical Coherence Tomography, W. Drexler and J. G. Fujimoto, eds. (Springer, 2008), pp. 505–542.

Arthaber, H.

Bachmann, A. H.

Bayleyegn, M. D.

M. D. Bayleyegn, H. Makhlouf, C. Crotti, K. Plamann, and A. Dubois, “Ultrahigh resolution spectral-domain optical coherence tomography at 1.3  μm using a broadband superluminescent diode light source,” Opt. Commun. 285, 5564–5569 (2012).
[CrossRef]

Beaurepaire, E.

Benattar, L.

Blanchot, L.

Boccara, A. C.

Boccara, C.

A. Dubois, G. Moneron, and C. Boccara, “Thermal-light full-field optical coherence tomography in the 1.2  μm wavelength region,” Opt. Commun. 266, 738–743 (2006).
[CrossRef]

Bonin, T.

Boppart, S. A.

Bourquin, S.

Brzezinski, M.

Carney, P. S.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Charriere, F.

Chen, Z.

Choma, M. A.

Colomb, T.

Crotti, C.

M. D. Bayleyegn, H. Makhlouf, C. Crotti, K. Plamann, and A. Dubois, “Ultrahigh resolution spectral-domain optical coherence tomography at 1.3  μm using a broadband superluminescent diode light source,” Opt. Commun. 285, 5564–5569 (2012).
[CrossRef]

Cuche, E.

de Groot, P.

de Lega, X. C.

De Martino, A.

Depeursinge, C.

Ding, Z.

Drévillon, B.

Drexler, W.

B. Povazay, A. Unterhuber, B. Hermann, H. Sattmann, H. Arthaber, and W. Drexler, “Full-field time-encoded frequency-domain optical coherence tomography,” Opt. Express 14, 7661–7669 (2006).
[CrossRef]

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).
[CrossRef]

Dubois, A.

Fercher, A. F.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).
[CrossRef]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Fujimoto, J. G.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

A. D. Aguirre and J. G. Fujimoto, “Optical coherence microscopy,” in Optical Coherence Tomography, W. Drexler and J. G. Fujimoto, eds. (Springer, 2008), pp. 505–542.

Georges, P.

Grebenyuk, A. A.

V. P. Ryabukho, D. V. Lyakin, A. A. Grebenyuk, and S. S. Klykov, “Wiener–Khintchin theorem for spatial coherence of optical wave field,” J. Opt. 15, 025405 (2013).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Numerical reconstruction of volumetric image in swept-source interference microscopy,” AIP Conf. Proc. 1537, 147–154 (2013).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Numerical correction of coherence gate in full-field swept-source interference microscopy,” Opt. Lett. 37, 2529–2531 (2012).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Theoretical model of volumetric objects imaging in a microscope,” Proc. SPIE 8430, 84301B (2012).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Coherence effects of thick objects imaging in interference microscopy,” Proc. SPIE 8427, 84271M (2012).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Numerical reconstruction of 3D image in Fourier domain confocal optical coherence microscopy,” in Proceedings of the International Conference on Advanced Laser Technologies (ALT) (Bern Open, 2013), pp. 1–5.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Hamilton, D. K.

D. K. Hamilton and C. J. R. Sheppard, “A confocal interference microscope,” Optica Acta 29, 1573–1577 (1982).
[CrossRef]

Hee, M. R.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Hermann, B.

Herminjard, S.

Hillmann, D.

Hitzenberger, C. K.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).
[CrossRef]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Huttmann, G.

Itzkan, I.

M. D. Modell, V. Ryabukho, D. Lyakin, V. Lychagov, E. Vitkin, I. Itzkan, and L. T. Perelman, “Autocorrelation low coherence interferometry,” Opt. Commun. 281, 1991–1996 (2008).
[CrossRef]

Izatt, J. A.

Karamata, B.

Kim, M. K.

Klykov, S. S.

V. P. Ryabukho, D. V. Lyakin, A. A. Grebenyuk, and S. S. Klykov, “Wiener–Khintchin theorem for spatial coherence of optical wave field,” J. Opt. 15, 025405 (2013).
[CrossRef]

Koch, P.

Kuhn, J.

Lambelet, P.

Lasser, T.

Laubscher, M.

Laude, B.

Lebec, M.

Lee, K. S.

Leitgeb, R. A.

Leutenegger, M.

Lexer, F.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).
[CrossRef]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Lobachev, M.

Luhrs, C.

Lyakin, D.

Lyakin, D. V.

V. P. Ryabukho, D. V. Lyakin, A. A. Grebenyuk, and S. S. Klykov, “Wiener–Khintchin theorem for spatial coherence of optical wave field,” J. Opt. 15, 025405 (2013).
[CrossRef]

Lychagov, V.

M. D. Modell, V. Ryabukho, D. Lyakin, V. Lychagov, E. Vitkin, I. Itzkan, and L. T. Perelman, “Autocorrelation low coherence interferometry,” Opt. Commun. 281, 1991–1996 (2008).
[CrossRef]

Makhlouf, H.

M. D. Bayleyegn, H. Makhlouf, C. Crotti, K. Plamann, and A. Dubois, “Ultrahigh resolution spectral-domain optical coherence tomography at 1.3  μm using a broadband superluminescent diode light source,” Opt. Commun. 285, 5564–5569 (2012).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Section 2.4.

Marks, D. L.

Marquet, P.

Meemon, P.

Modell, M. D.

M. D. Modell, V. Ryabukho, D. Lyakin, V. Lychagov, E. Vitkin, I. Itzkan, and L. T. Perelman, “Autocorrelation low coherence interferometry,” Opt. Commun. 281, 1991–1996 (2008).
[CrossRef]

Molebny, S.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).
[CrossRef]

Moneron, G.

A. Dubois, G. Moneron, and C. Boccara, “Thermal-light full-field optical coherence tomography in the 1.2  μm wavelength region,” Opt. Commun. 266, 738–743 (2006).
[CrossRef]

Montfort, F.

Moreau, J.

Murali, S.

Nelson, J. S.

Owen, G. M.

Perelman, L. T.

M. D. Modell, V. Ryabukho, D. Lyakin, V. Lychagov, E. Vitkin, I. Itzkan, and L. T. Perelman, “Autocorrelation low coherence interferometry,” Opt. Commun. 281, 1991–1996 (2008).
[CrossRef]

Plamann, K.

M. D. Bayleyegn, H. Makhlouf, C. Crotti, K. Plamann, and A. Dubois, “Ultrahigh resolution spectral-domain optical coherence tomography at 1.3  μm using a broadband superluminescent diode light source,” Opt. Commun. 285, 5564–5569 (2012).
[CrossRef]

Povazay, B.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Ralston, T. S.

Ren, H.

Rolland, J. P.

Rosen, J.

J. Rosen and A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[CrossRef]

Ryabukho, V.

Ryabukho, V. P.

V. P. Ryabukho, D. V. Lyakin, A. A. Grebenyuk, and S. S. Klykov, “Wiener–Khintchin theorem for spatial coherence of optical wave field,” J. Opt. 15, 025405 (2013).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Numerical reconstruction of volumetric image in swept-source interference microscopy,” AIP Conf. Proc. 1537, 147–154 (2013).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Numerical correction of coherence gate in full-field swept-source interference microscopy,” Opt. Lett. 37, 2529–2531 (2012).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Theoretical model of volumetric objects imaging in a microscope,” Proc. SPIE 8430, 84301B (2012).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Coherence effects of thick objects imaging in interference microscopy,” Proc. SPIE 8427, 84271M (2012).
[CrossRef]

A. A. Grebenyuk and V. P. Ryabukho, “Numerical reconstruction of 3D image in Fourier domain confocal optical coherence microscopy,” in Proceedings of the International Conference on Advanced Laser Technologies (ALT) (Bern Open, 2013), pp. 1–5.

Sacchet, D.

Safrani, A.

Saint-Jalmes, H.

Sarunic, M. V.

Sattmann, H.

B. Povazay, A. Unterhuber, B. Hermann, H. Sattmann, H. Arthaber, and W. Drexler, “Full-field time-encoded frequency-domain optical coherence tomography,” Opt. Express 14, 7661–7669 (2006).
[CrossRef]

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).
[CrossRef]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Schwartz, L.

Sheppard, C. J. R.

D. K. Hamilton and C. J. R. Sheppard, “A confocal interference microscope,” Optica Acta 29, 1573–1577 (1982).
[CrossRef]

Steinmann, L.

Sticker, M.

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of the FF-SSOCM setup. Imaging system: OP—principal planes of the microscope objectives; OA—aperture planes of the objectives; TP—principal planes of the tube lens; BS, cube beamsplitter; (x,y), registration plane, where a matrix photodetector (CCD) is placed. Illumination system: light from the swept-source is guided via an optical fiber and directed by a lens system, L1, onto the rotating diffuser, RD; AD, diaphragm of variable radius, located right after the diffuser and determining the illumination aperture size; lenses, L2 and L3, image AD onto the aperture planes, OA; FD, field diaphragm of variable radius, which is optically conjugated by lens, L3, and the objectives’ lenses with the front focal planes of the objectives.

Fig. 2.
Fig. 2.

Absolute value (a), (b) and phase (c), (d) of the Ξa(ω;kx,0) distribution versus the reduced spatial frequency kx and the defocus value Δz; (a), (c), NA=0.1, NAi0; (b), (d), NA=0.1, NAi=0.05. A frequency ω=1.2×1015Hz (λ=1.57μm) was considered. Note that we introduced here the normalized spatial frequency kx=M×kx/k where M is the magnification of the imaging system.

Fig. 3.
Fig. 3.

(a) and (b) 3D graph represents the absolute value of the ΓPSF(ω;xS,0) distribution over xS versus defocus value Δz; the 2D projected plot corresponds to the evolution of the absolute value of ΓPSF(ω;0,0) with respect to the defocus value. (a), without numerical correction; (b), with numerical correction according to Eq. (16a). (c) and (d) absolute value of the ΓPSF(ω;xS,0) distribution over xS after normalization to unity maximum for comparison of the widths for three defocus values: Δz=0μm (red dash line), Δz=300μm (blue point-dash line), Δz=600μm (black continuous line), (c), without numerical correction; (d), with numerical correction according to Eq. (16a). For all images NA=0.1, NAi=0.05, ω=1.2×1015Hz.

Fig. 4.
Fig. 4.

Comparison of FF-SSOCM images obtained without the focalization algorithm for two mechanically focused position (a), (b) and with numerical focalization (c), (d) by using the algorithm for spatially coherent illumination (c) or the proposed algorithm (d). Positions of cuts (see Fig. 5) along the xS axis are indicated in the zoomed windows. The field of view is 700μm×870μm.

Fig. 5.
Fig. 5.

Distributions of the absolute value of the reconstructed coherence function along the xS axis in the lower circuit images (see Fig. 4) with mechanical focalization on the lower circuit (black line with filled dots) and mechanical focalization on the upper circuit (all other data points): without numerical focusing (orange line with white dots), with numerical focusing for spatially coherent illumination (red line with white rectangles) and with numerical focusing by the proposed technique (green line with filled rectangles). The distributions for the case of focalization on the upper circuit are magnified 4 times in comparison to the distributions for focalization on the lower circuit. Unlike the algorithm for spatially coherent illumination, the proposed algorithm for arbitrary state of spatial coherence clearly reveals the dark thin scratch, visible in the optically focused image.

Equations (25)

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I(ω;x,y)=IS(ω;x,y)+IR(ω;x,y)+Γ(ω;x,y)+Γ*(ω;x,y),
2πδ(ωω)IS(ω;x,y)=VS(ω;x,y)VS*(ω;x,y),
2πδ(ωω)IR(ω;x,y)=VR(ω;x,y)VR*(ω;x,y),
2πδ(ωω)Γ(ω;x,y)=VS(ω;x,y)VR*(ω;x,y),
VR(ω;x,y)μ(ω;x,y)dx0dy0V0(ω;x0,y0)ΦR2(ω;kx0f,ky0f)rR(ω;kx0f,ky0f)×A(ω;x0,y0)exp[ik(x0x+y0y)/fL],
μ(ω;x,y)=iλfLexp{ik[2f(1+n02)+L+fL]}exp[iπλfL(1LfL)(x2+y2)],
ΦR(ω;kx,ky)=exp[i(zR|f|)(k2n02kx2ky2)1/2],
VS(ω;x,y)μ(ω;x,y)(λf)2dxSdySrS(ω;xS,yS)×A(ω;x3,y3)T2(ω;kx3f,ky3f)exp{ik[x3(xfL+xSf)+y3(yfL+ySf)]}dx3dy3×V0(ω;x0,y0)T1(ω;kx0f,ky0f)exp[ik(x0xS+y0yS)/f]dx0dy0,
T1(ω;kx,ky)=j=1Ntj1,j(ω;kx,ky)j=0Nexp[iΔzj(k2nj2kx2ky2)1/2],
T2(ω;kx,ky)=j=1Ntj,j1(ω;kx,ky)j=0Nexp[iΔzj(k2nj2kx2ky2)1/2],
V0(ω;x0,y0)V0*(ω;x0,y0)=2πδ(ωω)I0(ω;x0,y0)δ(x0x0)δ(y0y0).
Γ(ω;x,y)μ0(ω)dxSdySrS(ω;xS,yS)×A2(ω;x3,y3)exp{ik[x3(xfL+xSf)+y3(yfL+ySf)]}dx3dy3×A1(ω;x0,y0)exp{ik[x0(xfL+xSf)+y0(yfL+ySf)]}dx0dy0,
μ0(ω)=|μ(ω;x,y)|2(λf)2=λ4f2fL2,
A1(ω;x0,y0)=I0(ω;x0,y0)A*(ω;x0,y0)rR*(ω;kx0f,ky0f)×T1(ω;kx0f,ky0f)[ΦR*(ω;kx0f,ky0f)]2,
A2(ω;x3,y3)=A(ω;x3,y3)T2(ω;kx3f,ky3f).
ΓL(ω;x,y)exp[2ikΔL]Γ(ω;x,y),
Γ˜L(ω;kx,ky)=ΓL(ω;x,y)exp[i(kxx+kyy)]dxdy.
Γ˜L(ω;kx,ky)μ0(ω)M2[j=1Ntj1,j(ω)tj,j1(ω)]r˜S(ω;Mkx,Mky)Ξ(ω;kx,ky),
Ξ(ω;kx,ky)=exp[2ikΔL]dxSdySexp[iM(kxxS+kyyS)]×A(ω;x3,y3)exp[ikj=0NΔzjnj2x32+y32f2]exp[ikf(x3xS+y3yS)]dx3dy3×Ai(ω;x0,y0)exp{ik[(zS2zR+|f|)n02x02+y02f2+j=1NΔzjnj2x02+y02f2]}×exp[ikf(x0xS+y0yS)]dx0dy0,
Ai(ω;x0,y0)=I0(ω;x0,y0)A*(ω;x0,y0)rR*(ω;kx0f,ky0f),
Ξ(ω;kx,ky)=Ξt(ω)Ξa(ω;kx,ky),
Ξt(ω)=exp{2ik[ΔL+n0(zSzR)+j=1NnjΔzj]},
Ξa(ω;kx,ky)=Ξ(ω;kx,ky)Ξt*(ω).
Ψ(ω;kx,ky)={Ξ*(ω;kx,ky)/|Ξ(ω;kx,ky)|,if|Ξ(ω;kx,ky)|>0,0,if|Ξ(ω;kx,ky)|=0.
Ψa(ω;kx,ky)={Ξa*(ω;kx,ky)/|Ξa(ω;kx,ky)|,if|Ξa(ω;kx,ky)|>0,0,if|Ξa(ω;kx,ky)|=0,

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