Abstract

We demonstrate super-resolution imaging with background fluorescence rejection by interferometric temporal focusing microscopy, in which temporal focusing is combined with structured illumination. The lateral resolution and the optical sectioning capability are simultaneously improved by factors of 1.6 and 1.4, respectively, compared to conventional temporal focusing microscopy. Fluorescent beads (200 nm diameter) that are difficult to distinguish from the background fluorescence in conventional temporal focusing microscopy, are clearly visualized by interferometric temporal focusing microscopy.

© 2013 Optical Society of America

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2013 (3)

2012 (4)

2011 (2)

2010 (2)

2009 (4)

2008 (3)

2007 (1)

2006 (2)

2005 (3)

2003 (5)

R. Heintzmann, “Saturated patterned excitation microscopy with two-dimensional excitation patterns,” Micron 34(6-7), 283–291 (2003).
[CrossRef] [PubMed]

W. R. Zipfel, R. M. Williams, W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[CrossRef] [PubMed]

P. Theer, M. T. Hasan, W. Denk, “Two-photon imaging to a depth of 1000 microm in living brains by use of a Ti:Al2O3 regenerative amplifier,” Opt. Lett. 28(12), 1022–1024 (2003).
[CrossRef] [PubMed]

L. Sacconi, E. Froner, R. Antolini, M. R. Taghizadeh, A. Choudhury, F. S. Pavone, “Multiphoton multifocal microscopy exploiting a diffractive optical element,” Opt. Lett. 28(20), 1918–1920 (2003).
[CrossRef] [PubMed]

H. Mashiko, A. Suda, K. Midorikawa, “All-reflective interferometric autocorrelator for the measurement of ultra-short optical pulses,” Appl. Phys. B 76(5), 525–530 (2003).
[CrossRef]

2002 (1)

2001 (1)

T. Nielsen, M. Fricke, D. Hellweg, P. Andresen, “High efficiency beam splitter for multifocal multiphoton microscopy,” J. Microsc. 201(3), 368–376 (2001).
[CrossRef] [PubMed]

2000 (5)

A. Egner, S. W. Hell, “Time multiplexing and parallelization in multifocal multiphoton microscopy,” J. Opt. Soc. Am. A 17(7), 1192–1201 (2000).
[CrossRef] [PubMed]

D. N. Fittinghoff, P. W. Wiseman, J. Squier, “Widefield multiphoton and temporally decorrelated multifocal multiphoton microscopy,” Opt. Express 7(8), 273–279 (2000).
[CrossRef] [PubMed]

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[CrossRef] [PubMed]

J. T. Frohn, H. F. Knapp, A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A. 97(13), 7232–7236 (2000).
[CrossRef] [PubMed]

K. König, “Multiphoton microscopy in life sciences,” J. Microsc. 200(2), 83–104 (2000).
[CrossRef] [PubMed]

1998 (2)

A. H. Buist, M. Muller, J. Squier, G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192(2), 217–226 (1998).
[CrossRef]

J. Bewersdorf, R. Pick, S. W. Hell, “Multifocal multiphoton microscopy,” Opt. Lett. 23(9), 655–657 (1998).
[CrossRef] [PubMed]

1997 (1)

1990 (1)

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Andresen, P.

T. Nielsen, M. Fricke, D. Hellweg, P. Andresen, “High efficiency beam splitter for multifocal multiphoton microscopy,” J. Microsc. 201(3), 368–376 (2001).
[CrossRef] [PubMed]

Antolini, R.

Aubé, B.

Bahlmann, K.

Beck, M.

Betzig, E.

N. Ji, D. E. Milkie, E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[CrossRef] [PubMed]

Bewersdorf, J.

Brakenhoff, G. J.

A. H. Buist, M. Muller, J. Squier, G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192(2), 217–226 (1998).
[CrossRef]

Buehler, C.

Buist, A. H.

A. H. Buist, M. Muller, J. Squier, G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192(2), 217–226 (1998).
[CrossRef]

Chang, B.-J.

Chang, C.-Y.

Chang, N.-S.

Chang, Y.-C.

Chen, N.

Chen, S.-J.

Chen, Z.

Cheng, L.-C.

Chiang, S.-Y.

Cho, K.-C.

Choi, H.

Chou, L.-J.

Choudhury, A.

Chu, K. K.

Clark, C. G.

N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, C. Xu, “In vivo three-photon microscopy of subcortical structures within an intact mouse brain,” Nat. Photonics 7(3), 205–209 (2013).
[CrossRef]

Côté, D.

Cremer, C.

Davidson, M. W.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Denk, W.

Dong, C. Y.

Durst, M.

Durst, M. E.

Egner, A.

Fantini, S.

Fiolka, R.

Fittinghoff, D. N.

Fricke, M.

T. Nielsen, M. Fricke, D. Hellweg, P. Andresen, “High efficiency beam splitter for multifocal multiphoton microscopy,” J. Microsc. 201(3), 368–376 (2001).
[CrossRef] [PubMed]

Frohn, J. T.

J. T. Frohn, H. F. Knapp, A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A. 97(13), 7232–7236 (2000).
[CrossRef] [PubMed]

Froner, E.

Furukawa, Y.

Y. Nabekawa, T. Shimizu, Y. Furukawa, E. J. Takahashi, K. Midorikawa, “Interferometry of attosecond pulse trains in the extreme ultraviolet wavelength region,” Phys. Rev. Lett. 102(21), 213904 (2009).
[CrossRef] [PubMed]

Gustafsson, M. G. L.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[CrossRef] [PubMed]

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102(37), 13081–13086 (2005).
[CrossRef] [PubMed]

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[CrossRef] [PubMed]

Hallacoglu, B.

Hasan, M. T.

Hashimoto, H.

Heffer, E. L.

Heintzmann, R.

Hell, S. W.

Hellweg, D.

T. Nielsen, M. Fricke, D. Hellweg, P. Andresen, “High efficiency beam splitter for multifocal multiphoton microscopy,” J. Microsc. 201(3), 368–376 (2001).
[CrossRef] [PubMed]

Horton, N. G.

N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, C. Xu, “In vivo three-photon microscopy of subcortical structures within an intact mouse brain,” Nat. Photonics 7(3), 205–209 (2013).
[CrossRef]

D. Kobat, N. G. Horton, C. Xu, “In vivo two-photon microscopy to 1.6-mm depth in mouse cortex,” J. Biomed. Opt. 16(10), 106014 (2011).
[CrossRef] [PubMed]

Isobe, K.

Ji, N.

N. Ji, D. E. Milkie, E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[CrossRef] [PubMed]

Johansson, G. A.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Jovin, T. M.

Juskaitis, R.

Kamps-Hughes, N.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Kannari, F.

Kawano, H.

Kim, K. H.

Knapp, H. F.

J. T. Frohn, H. F. Knapp, A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A. 97(13), 7232–7236 (2000).
[CrossRef] [PubMed]

Kobat, D.

N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, C. Xu, “In vivo three-photon microscopy of subcortical structures within an intact mouse brain,” Nat. Photonics 7(3), 205–209 (2013).
[CrossRef]

D. Kobat, N. G. Horton, C. Xu, “In vivo two-photon microscopy to 1.6-mm depth in mouse cortex,” J. Biomed. Opt. 16(10), 106014 (2011).
[CrossRef] [PubMed]

D. Kobat, M. E. Durst, N. Nishimura, A. W. Wong, C. B. Schaffer, C. Xu, “Deep tissue multiphoton microscopy using longer wavelength excitation,” Opt. Express 17(16), 13354–13364 (2009).
[CrossRef] [PubMed]

König, K.

K. König, “Multiphoton microscopy in life sciences,” J. Microsc. 200(2), 83–104 (2000).
[CrossRef] [PubMed]

Koninck, P. D.

Kumagai, A.

Lee, W.-C. A.

Leray, A.

Lim, D.

Lin, C.-Y.

Macklin, J. J.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Mashiko, H.

H. Mashiko, A. Suda, K. Midorikawa, “All-reflective interferometric autocorrelator for the measurement of ultra-short optical pulses,” Appl. Phys. B 76(5), 525–530 (2003).
[CrossRef]

Mertz, J.

Midorikawa, K.

Milkie, D. E.

N. Ji, D. E. Milkie, E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[CrossRef] [PubMed]

Min, W.

Miyawaki, A.

Mizuno, H.

Muller, M.

A. H. Buist, M. Muller, J. Squier, G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192(2), 217–226 (1998).
[CrossRef]

Nabekawa, Y.

Y. Nabekawa, T. Shimizu, Y. Furukawa, E. J. Takahashi, K. Midorikawa, “Interferometry of attosecond pulse trains in the extreme ultraviolet wavelength region,” Phys. Rev. Lett. 102(21), 213904 (2009).
[CrossRef] [PubMed]

Nedivi, E.

Neil, M. A. A.

Nielsen, T.

T. Nielsen, M. Fricke, D. Hellweg, P. Andresen, “High efficiency beam splitter for multifocal multiphoton microscopy,” J. Microsc. 201(3), 368–376 (2001).
[CrossRef] [PubMed]

Nishimura, N.

Oron, D.

Pagès, S.

Pavone, F. S.

Pick, R.

Ragan, T.

Rego, E. H.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Sacconi, L.

Schaffer, C. B.

N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, C. Xu, “In vivo three-photon microscopy of subcortical structures within an intact mouse brain,” Nat. Photonics 7(3), 205–209 (2013).
[CrossRef]

D. Kobat, M. E. Durst, N. Nishimura, A. W. Wong, C. B. Schaffer, C. Xu, “Deep tissue multiphoton microscopy using longer wavelength excitation,” Opt. Express 17(16), 13354–13364 (2009).
[CrossRef] [PubMed]

Shao, L.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[CrossRef] [PubMed]

Sheppard, C. J. R.

Shimizu, T.

Y. Nabekawa, T. Shimizu, Y. Furukawa, E. J. Takahashi, K. Midorikawa, “Interferometry of attosecond pulse trains in the extreme ultraviolet wavelength region,” Phys. Rev. Lett. 102(21), 213904 (2009).
[CrossRef] [PubMed]

Silberberg, Y.

So, P. T. C.

Squier, J.

D. N. Fittinghoff, P. W. Wiseman, J. Squier, “Widefield multiphoton and temporally decorrelated multifocal multiphoton microscopy,” Opt. Express 7(8), 273–279 (2000).
[CrossRef] [PubMed]

A. H. Buist, M. Muller, J. Squier, G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192(2), 217–226 (1998).
[CrossRef]

Stemmer, A.

R. Fiolka, M. Beck, A. Stemmer, “Structured illumination in total internal reflection fluorescence microscopy using a spatial light modulator,” Opt. Lett. 33(14), 1629–1631 (2008).
[CrossRef] [PubMed]

J. T. Frohn, H. F. Knapp, A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A. 97(13), 7232–7236 (2000).
[CrossRef] [PubMed]

Straub, A. A.

Strickler, J. H.

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Suda, A.

Taghizadeh, M. R.

Takahashi, E. J.

Y. Nabekawa, T. Shimizu, Y. Furukawa, E. J. Takahashi, K. Midorikawa, “Interferometry of attosecond pulse trains in the extreme ultraviolet wavelength region,” Phys. Rev. Lett. 102(21), 213904 (2009).
[CrossRef] [PubMed]

Takeda, T.

Tal, E.

Theer, P.

Therrien, O. D.

van Howe, J.

Wang, K.

N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, C. Xu, “In vivo three-photon microscopy of subcortical structures within an intact mouse brain,” Nat. Photonics 7(3), 205–209 (2013).
[CrossRef]

Webb, W. W.

W. R. Zipfel, R. M. Williams, W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Wei, L.

Williams, R. M.

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Wilson, T.

Winoto, L.

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
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Wise, F. W.

N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, C. Xu, “In vivo three-photon microscopy of subcortical structures within an intact mouse brain,” Nat. Photonics 7(3), 205–209 (2013).
[CrossRef]

Wiseman, P. W.

Wong, A. W.

Wong, C.-H.

Xu, C.

Yen, W.-C.

Yew, E. Y. S.

Zhu, G.

Zhu, X.

Zipfel, W.

Zipfel, W. R.

W. R. Zipfel, R. M. Williams, W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
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Appl. Phys. B (1)

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[CrossRef]

Biomed. Opt. Express (5)

J. Biomed. Opt. (1)

D. Kobat, N. G. Horton, C. Xu, “In vivo two-photon microscopy to 1.6-mm depth in mouse cortex,” J. Biomed. Opt. 16(10), 106014 (2011).
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Micron (1)

R. Heintzmann, “Saturated patterned excitation microscopy with two-dimensional excitation patterns,” Micron 34(6-7), 283–291 (2003).
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Nat. Biotechnol. (1)

W. R. Zipfel, R. M. Williams, W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[CrossRef] [PubMed]

Nat. Methods (1)

N. Ji, D. E. Milkie, E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
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Nat. Photonics (1)

N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, C. Xu, “In vivo three-photon microscopy of subcortical structures within an intact mouse brain,” Nat. Photonics 7(3), 205–209 (2013).
[CrossRef]

Opt. Express (10)

G. Zhu, J. van Howe, M. Durst, W. Zipfel, C. Xu, “Simultaneous spatial and temporal focusing of femtosecond pulses,” Opt. Express 13(6), 2153–2159 (2005).
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N. Chen, C.-H. Wong, C. J. R. Sheppard, “Focal modulation microscopy,” Opt. Express 16(23), 18764–18769 (2008).
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A. Leray, J. Mertz, “Rejection of two-photon fluorescence background in thick tissue by differential aberration imaging,” Opt. Express 14(22), 10565–10573 (2006).
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Science (1)

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Figures (8)

Fig. 1
Fig. 1

Schematics for the ITF technique.

Fig. 2
Fig. 2

Flowchart of reconstruction algorithm for SR-ITF microscopy. FFT and * denote the fast Fourier transform and the complex conjugate, respectively.

Fig. 3
Fig. 3

ITF microscope setup. OPO: optical parametric oscillator, PS: periscope to rotate the spatial beam profile, CCL: concave cylindrical lens, DM: dichroic mirror, OB: objective lens, SPF: short-pass filter, BPF: band-pass filter.

Fig. 4
Fig. 4

(a) 1D signal distribution (solid line) of TF microscopy (black), NSR-ITF microscopy (red), and SR-ITF microscopies composed of the all frequency components (blue), and ± k0 and ± 2k0 components (green) along the axial (z) direction near the interface between the cover slip and the Rhodamine B solution. Dotted lines represent the first derivatives of the 1D signal distributions. (b) 1D signal distribution of a 200-nm fluorescent bead along the lateral (y) direction, which was acquired by TF microscopy (black), NSR-ITF microscopy (red), and SR-ITF microscopies composed of the all frequency components (blue), ± k0 and ± 2k0 components (green), and ± 2k0 components (pink).

Fig. 5
Fig. 5

TPEF images of the fluorescent beads obtained by (left) TF microscopy, (the second from the left) NSR-ITF microscopy, and SR-ITF microscopies composed of (the third from the left) the all frequency components and (right) ± k0 and ± 2k0 components only.

Fig. 6
Fig. 6

TPEF images of the 200-nm fluorescent beads obtained by (left) NSR-ITF microscopy, and SR-ITF microscopies composed of (the second from the left) ± k0 and ± 2k0 components, (the third from the left) 25% ± k0 and ± 2k0 components, and (right) ± 2k0 components.

Fig. 7
Fig. 7

Normalized signal profiles of NSR-ITF microscopy (red), and SR-ITF microscopies composed of ± k0 and ± 2k0 components (green), 25% ± k0 and ± 2k0 components (orange), and ± 2k0 components (pink) along the lateral direction indicated by (a) blue, (b) green, and (c) red arrows in Fig. 6.

Fig. 8
Fig. 8

(a) TPEF images of the fluorescent beads obtained by TF microscopy, NSR microscopy, and SR-ITF microscopies composed of the all frequency components, and ± k0 and ± 2k0. (b-e) Normalized signal profiles of TF microscopy (black), NSR-ITF microscopy (red), and SR-ITF microscopies composed of the all frequency components (blue), and ± k0 and ± 2k0 components (green) along the lateral (x and y) direction indicated by (b) blue, (c) green, (d) pink, and (e) red arrows in (a).

Equations (10)

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S m ( r ) = { C ( r ) I ex 2 ( r , ϕ m ) } h ( r ) ,
I ex 2 ( r , ϕ m ) = [ I 0 ( r ) { 1 + α cos ( k 0 y + ϕ m ) } ] 2 = I 0 2 ( r ) [ 1 + α 2 2 + α { e i ( k 0 y + ϕ m ) + e i ( k 0 y + ϕ m ) } + α 2 4 { e i 2 ( k 0 y + ϕ m ) + e i 2 ( k 0 y + ϕ m ) } ] ,
F l ( r ) = m = 0 4 S m ( r ) e i l m ϕ s { C ( r ) I 0 2 ( r ) e i l ( k 0 y + ϕ 0 ) } h ( r ) = { D ( r ) e i l ( k 0 y + ϕ 0 ) } h ( r ) ,
F ITF ( r ) = { | F 1 ( r ) | 2 + | F 2 ( r ) | 2 } 1 / 2 = { F 1 ( r ) F 1 ( r ) + F 2 ( r ) F 2 ( r ) } 1 / 2 .
F ITF ( r ) = { m = 1 4 ( S 0 S m ) 2 + m = 2 4 ( S 1 S m ) 2 + m = 3 4 ( S 2 S m ) 2 + ( S 3 S 4 ) 2 } 1 / 2 .
F ˜ l ( k ) D ˜ ( k l k 0 ) h ˜ ( k ) e i l ϕ 0 ,
F ˜ SR ITF ( k ) = l = 2 2 a l F ˜ l ( k + l k 0 ) e i l ϕ 0 = D ˜ ( k ) l = 2 2 b l h ˜ ( k + l k 0 ) ,
F SR ITF ( r ) = l = 2 2 a l F l ( r ) e i l ( k 0 y + ϕ 0 ) .
X C k ( k ) = | F ˜ 1 ( k ' ) F ˜ 2 ( k ' k ) d k ' | .
X C ϕ ( r ) = Re { F 1 ( r ' ) e i ( k 0 y ' + ϕ 0 ) } | F 1 ( r ' r ) | d r ' ,

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