Abstract

Optical sectioning imaging with high spatial resolution deep inside scattering samples such as mammalian brain is of great interest in biological study. Conventional two-photon microscopy deteriorates in focus when light scattering increases. Here we develop an optical sectioning enhanced two-photon technique which incorporates structured illumination into line-scanning spatial-temporal focusing microscopy (LTSIM), and generate patterned illumination via laser intensity modulation synchronized with scanning. LTSIM brings scattering background elimination and in-focus contrast enhancement, and realizes nearly 2-fold increase in spatial resolution to ∼208 nm laterally and ∼0.94 µm axially. In addition, the intensity modulated line-scanning implementation of LTSIM enables fast and flexible generation of structured illumination, permitting adjustable spatial frequency profiles to optimize image contrast. The highly qualified optical sectioning ability of our system is demonstrated on samples including tissue phantom, C. elegans and mouse brain at depths over hundreds of microns.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

2016 (1)

A. Lal, C. Shan, and P. Xi, “Structured illumination microscopy image reconstruction algorithm,” IEEE J. Sel. Top. Quant. 22, 50–63 (2016).
[Crossref]

2015 (1)

2014 (4)

2013 (3)

2010 (4)

E. Papagiakoumou, F. Anselmi, A. Bègue, V. D. Sars, J. GluÜckstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Meth. 7, 848 (2010).
[Crossref]

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Meth. 7, 141 (2010).
[Crossref]

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2010).
[Crossref]

S. Sakadzic, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, and S. A. Vinogradov, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Meth. 7, 755 (2010).
[Crossref]

2009 (3)

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Neuron 32, 435–506 (2009).

P. Kner, B. Chhun, E. Griffis, L. Winoto, and M. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Meth. 6, 339 (2009).
[Crossref]

D. Dèbarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009).
[Crossref] [PubMed]

2008 (2)

M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. 281, 1796–1805 (2008).
[Crossref] [PubMed]

N. Ji, H. Shroff, H. Zhong, and E. Betzig, “Advances in the speed and resolution of light microscopy,” Curr. Opin. Neurobiol. 18, 605 (2008).
[Crossref]

2006 (1)

K. Svoboda and R. Yasuda, “Principles of two-photon excitation microscopy and its applications to neuroscience,” Neuron 50, 823–839 (2006).
[Crossref] [PubMed]

2005 (4)

Anselmi, F.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. D. Sars, J. GluÜckstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Meth. 7, 848 (2010).
[Crossref]

Arai, K.

S. Sakadzic, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, and S. A. Vinogradov, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Meth. 7, 755 (2010).
[Crossref]

Bègue, A.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. D. Sars, J. GluÜckstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Meth. 7, 848 (2010).
[Crossref]

Betzig, E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Meth. 7, 141 (2010).
[Crossref]

N. Ji, H. Shroff, H. Zhong, and E. Betzig, “Advances in the speed and resolution of light microscopy,” Curr. Opin. Neurobiol. 18, 605 (2008).
[Crossref]

Booth, M. J.

Botcherby, E. J.

Brosh, I.

H. Dana, A. Marom, S. Paluch, R. Dvorkin, I. Brosh, and S. Shoham, “Hybrid multiphoton volumetric functional imaging of large-scale bioengineered neuronal networks,” Nat. Commun. 5, 3997 (2014).
[Crossref] [PubMed]

Burns, L. D.

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Neuron 32, 435–506 (2009).

Chen, S. J.

Cheng, L. C.

Chhun, B.

P. Kner, B. Chhun, E. Griffis, L. Winoto, and M. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Meth. 6, 339 (2009).
[Crossref]

chi Chen, S.

Chien, F. C.

Chitnis, A.

Choi, H.

Christensen, R.

Dana, H.

H. Dana, A. Marom, S. Paluch, R. Dvorkin, I. Brosh, and S. Shoham, “Hybrid multiphoton volumetric functional imaging of large-scale bioengineered neuronal networks,” Nat. Commun. 5, 3997 (2014).
[Crossref] [PubMed]

H. Dana, N. Kruger, A. Ellman, and S. Shoham, “Line temporal focusing characteristics in transparent and scattering media,” Opt. Express 21, 5677–5687 (2013).
[Crossref] [PubMed]

Dèbarre, D.

Devor, A.

S. Sakadzic, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, and S. A. Vinogradov, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Meth. 7, 755 (2010).
[Crossref]

Dong, C. Y.

Durst, M.

Durst, M. E.

M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. 281, 1796–1805 (2008).
[Crossref] [PubMed]

Dvorkin, R.

H. Dana, A. Marom, S. Paluch, R. Dvorkin, I. Brosh, and S. Shoham, “Hybrid multiphoton volumetric functional imaging of large-scale bioengineered neuronal networks,” Nat. Commun. 5, 3997 (2014).
[Crossref] [PubMed]

Ellman, A.

Emiliani, V.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. D. Sars, J. GluÜckstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Meth. 7, 848 (2010).
[Crossref]

Fantini, S.

Ghosh, K. K.

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Neuron 32, 435–506 (2009).

GluÜckstad, J.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. D. Sars, J. GluÜckstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Meth. 7, 848 (2010).
[Crossref]

Griffis, E.

P. Kner, B. Chhun, E. Griffis, L. Winoto, and M. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Meth. 6, 339 (2009).
[Crossref]

Groessl, F.

Gustafsson, M.

P. Kner, B. Chhun, E. Griffis, L. Winoto, and M. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Meth. 6, 339 (2009).
[Crossref]

Gustafsson, M. G.

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2010).
[Crossref]

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” P. Natl. Acad. Sci. USA. 102, 13081 (2005).
[Crossref]

Hallacoglu, B.

Haubensak, W. E.

Ho, E. T. W.

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Neuron 32, 435–506 (2009).

Hu, Y. Y.

Ingaramo, M.

Isacoff, E. Y.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. D. Sars, J. GluÜckstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Meth. 7, 848 (2010).
[Crossref]

Ji, N.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Meth. 7, 141 (2010).
[Crossref]

N. Ji, H. Shroff, H. Zhong, and E. Betzig, “Advances in the speed and resolution of light microscopy,” Curr. Opin. Neurobiol. 18, 605 (2008).
[Crossref]

Kner, P.

P. Kner, B. Chhun, E. Griffis, L. Winoto, and M. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Meth. 6, 339 (2009).
[Crossref]

Kruger, N.

Lal, A.

A. Lal, C. Shan, and P. Xi, “Structured illumination microscopy image reconstruction algorithm,” IEEE J. Sel. Top. Quant. 22, 50–63 (2016).
[Crossref]

Li, C.

Lien, C. H.

Lin, C. Y.

Lin, W.

Lo, E. H.

S. Sakadzic, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, and S. A. Vinogradov, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Meth. 7, 755 (2010).
[Crossref]

Mandeville, E. T.

S. Sakadzic, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, and S. A. Vinogradov, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Meth. 7, 755 (2010).
[Crossref]

Marom, A.

H. Dana, A. Marom, S. Paluch, R. Dvorkin, I. Brosh, and S. Shoham, “Hybrid multiphoton volumetric functional imaging of large-scale bioengineered neuronal networks,” Nat. Commun. 5, 3997 (2014).
[Crossref] [PubMed]

Meng, Y.

Milkie, D. E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Meth. 7, 141 (2010).
[Crossref]

Mukamel, E. A.

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Neuron 32, 435–506 (2009).

Na, J.

J. Na, “The practical and fundamental limits of optical imaging in mammalian brains,” Neuron 83, 1242–1245 (2014).
[Crossref]

Nogare, D. D.

Oron, D.

Paluch, S.

H. Dana, A. Marom, S. Paluch, R. Dvorkin, I. Brosh, and S. Shoham, “Hybrid multiphoton volumetric functional imaging of large-scale bioengineered neuronal networks,” Nat. Commun. 5, 3997 (2014).
[Crossref] [PubMed]

Papagiakoumou, E.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. D. Sars, J. GluÜckstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Meth. 7, 848 (2010).
[Crossref]

Patterson, G. H.

Prevedel, R.

Roussakis, E.

S. Sakadzic, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, and S. A. Vinogradov, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Meth. 7, 755 (2010).
[Crossref]

Rupprecht, P.

Ruvinskaya, S.

S. Sakadzic, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, and S. A. Vinogradov, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Meth. 7, 755 (2010).
[Crossref]

Sakadzic, S.

S. Sakadzic, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, and S. A. Vinogradov, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Meth. 7, 755 (2010).
[Crossref]

Sars, V. D.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. D. Sars, J. GluÜckstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Meth. 7, 848 (2010).
[Crossref]

Schnitzer, M. J.

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Neuron 32, 435–506 (2009).

Shan, C.

A. Lal, C. Shan, and P. Xi, “Structured illumination microscopy image reconstruction algorithm,” IEEE J. Sel. Top. Quant. 22, 50–63 (2016).
[Crossref]

Sheppard, C. J. R.

Shoham, S.

H. Dana, A. Marom, S. Paluch, R. Dvorkin, I. Brosh, and S. Shoham, “Hybrid multiphoton volumetric functional imaging of large-scale bioengineered neuronal networks,” Nat. Commun. 5, 3997 (2014).
[Crossref] [PubMed]

H. Dana, N. Kruger, A. Ellman, and S. Shoham, “Line temporal focusing characteristics in transparent and scattering media,” Opt. Express 21, 5677–5687 (2013).
[Crossref] [PubMed]

Shroff, H.

Silberberg, Y.

So, P. T. C.

Srinivas, S.

Srinivasan, V. J.

S. Sakadzic, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, and S. A. Vinogradov, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Meth. 7, 755 (2010).
[Crossref]

Sun, Q. Q.

X. Wang, C. Zhang, G. Szabo, and Q. Q. Sun, “Distribution of camkiiα expression in the brain in vivo, studied by camkiiα-gfp mice,” Brain Res. 1518, 9–25 (2013).
[Crossref] [PubMed]

Svoboda, K.

K. Svoboda and R. Yasuda, “Principles of two-photon excitation microscopy and its applications to neuroscience,” Neuron 50, 823–839 (2006).
[Crossref] [PubMed]

Szabo, G.

X. Wang, C. Zhang, G. Szabo, and Q. Q. Sun, “Distribution of camkiiα expression in the brain in vivo, studied by camkiiα-gfp mice,” Brain Res. 1518, 9–25 (2013).
[Crossref] [PubMed]

Tal, E.

Van, H. J.

Vaziri, A.

Vinogradov, S. A.

S. Sakadzic, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, and S. A. Vinogradov, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Meth. 7, 755 (2010).
[Crossref]

Wang, X.

X. Wang, C. Zhang, G. Szabo, and Q. Q. Sun, “Distribution of camkiiα expression in the brain in vivo, studied by camkiiα-gfp mice,” Brain Res. 1518, 9–25 (2013).
[Crossref] [PubMed]

Watanabe, T.

Wilson, T.

Wilt, B. A.

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Neuron 32, 435–506 (2009).

Winoto, L.

P. Kner, B. Chhun, E. Griffis, L. Winoto, and M. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Meth. 6, 339 (2009).
[Crossref]

Winter, P. W.

Xi, P.

A. Lal, C. Shan, and P. Xi, “Structured illumination microscopy image reconstruction algorithm,” IEEE J. Sel. Top. Quant. 22, 50–63 (2016).
[Crossref]

Xu, C.

Yaseen, M. A.

S. Sakadzic, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, and S. A. Vinogradov, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Meth. 7, 755 (2010).
[Crossref]

Yasuda, R.

K. Svoboda and R. Yasuda, “Principles of two-photon excitation microscopy and its applications to neuroscience,” Neuron 50, 823–839 (2006).
[Crossref] [PubMed]

Yew, E. Y. S.

Yong, D. S.

York, A. G.

Zhang, C.

X. Wang, C. Zhang, G. Szabo, and Q. Q. Sun, “Distribution of camkiiα expression in the brain in vivo, studied by camkiiα-gfp mice,” Brain Res. 1518, 9–25 (2013).
[Crossref] [PubMed]

Zhong, H.

N. Ji, H. Shroff, H. Zhong, and E. Betzig, “Advances in the speed and resolution of light microscopy,” Curr. Opin. Neurobiol. 18, 605 (2008).
[Crossref]

Zhu, G.

M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. 281, 1796–1805 (2008).
[Crossref] [PubMed]

G. Zhu, H. J. Van, M. Durst, W. Zipfel, and C. Xu, “Simultaneous spatial and temporal focusing of femtosecond pulses,” Opt. Express 13, 2153–2159 (2005).
[Crossref] [PubMed]

Zipfel, W.

Biomed. Opt. Express (3)

Brain Res. (1)

X. Wang, C. Zhang, G. Szabo, and Q. Q. Sun, “Distribution of camkiiα expression in the brain in vivo, studied by camkiiα-gfp mice,” Brain Res. 1518, 9–25 (2013).
[Crossref] [PubMed]

Curr. Opin. Neurobiol. (1)

N. Ji, H. Shroff, H. Zhong, and E. Betzig, “Advances in the speed and resolution of light microscopy,” Curr. Opin. Neurobiol. 18, 605 (2008).
[Crossref]

IEEE J. Sel. Top. Quant. (1)

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Figures (7)

Fig. 1
Fig. 1

Line-scanning two-photon structured illumination system design. The ultrashort pulsed laser is modulated in intensity by the acousto-optical modulator and provides a line excitation to the sample. Synchronizing the line-scanning and intensity modulation results in a sinusoidal illumination pattern. The fluorescence image is collected by a camera in epifluorescence microscopy scheme. Temporal chirp induced by the grating is first broadened then compressed to the minimum at the objective focus to improve axial confinement. Symbols: HWP, half wave plate; PBS, polarized beam splitter; M, reflective mirror; TL, tube lens; DM, dichroic mirror; BPF, bandpass filter; SPF, shortpass filter.

Fig. 2
Fig. 2

Evaluation of line-scanning structured illumination. (a) shows the 2D excitation of a fluorescent layer captured by the sCMOS through line-scanning with and without intensity modulation, where N denotes the sinusoidal period number in the FOV. (b) shows magnified subregions of structured patterns with different phase profiles (N = 20). Scale-bar = 10 µm. We measured the intensity fluctation on a selected line of the fluorescent image from (b), as illustrated in (c), and calculated the [1 + m cos(kpr)]2 function fitting of discrete sampling points. (d) presents the Fourier transform of the spatial intensity data. We can clearly see the five spatial frequency components in frequency domain. We magnified the 0, +1 and +2 ordered spatial components in (e) and quantified that kp equals to 0.2 µm−1.

Fig. 3
Fig. 3

Measurement of three-dimensional psf for LTSIM (N = 40). (a) shows lateral FWHMs of LT microscopy, LTSIM and LTSIM after deconvolution. (b) shows axial FWHMs of LT microscopy, LTSIM and LTSIM after deconvolution.

Fig. 4
Fig. 4

Image contrast and resolution enhancement in LTSIM. (a) (b) and (c) are f-actin structures of one same BPAE cell imaged under uniform illuminated LSTF, structure illuminated LTSIM (N = 40) and LTSIM after deconvolution. The insects are magnified views of the squared region, illustrating the imaging performance by LSTF, LTSIM and deconvoluted LTSIM. (d) Intensity along the line-marked region. Scale-bar = 5 µm in (a) (b) and (c) and 2.5 µm in magnified views.

Fig. 5
Fig. 5

Optical section imaging in tissue phantom. (a) shows representative image of fluorescence beads excited at different depths by uniform and structure illuminated (N = 20) two-photon microscopy without deconvolution. Scale-bar = 1 µm. (b) and (c) compare the SNR and SBR of fluorescent signals in two methods. Means and standard deviations are calculated from measurements of three beads at each depth.

Fig. 6
Fig. 6

Optical section imaging of a whole C. elegans in LTSIM. (a) shows the stitched visualization of the worm at single XY slice of 20 µm depth below the coverslip using LTSIM after deconvolution. (b) (d) are higher magnified views of the squared regions in (a) at indicated axial depths using LTSIM and (c) (e) the counterparts using uniform illumination, proving the high qualified optical section imaging of LTSIM. N = 40. Scale-bar = 50 µm in (a) and 5 µm in (b)–(e).

Fig. 7
Fig. 7

Frequency-switchable structured illumination in LTSIM. (a) (b) are fluorescent images captured at separate depths using uniform illumination and SIM with different modulation frequencies from 10 to 80 cycles within the FOV, indicating the reconstruction performance of structure frequencies related to scattering properties. Scale-bar = 10 µm.

Equations (4)

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N = s M H = s M w f c y l t e x p ,
I ( r , ϕ ) = I 0 ( 1 + m cos ( k p r + ϕ ) ) ,
s ( r , ϕ ) = { c ( r ) I ( r , ϕ ) 2 } * h ( r ) = { c ( r ) I 0 2 [ 1 + m 2 2 + m ( e i ( k p r + ϕ ) + e i ( k p r + ϕ ) ) + m 2 4 ( e i 2 ( k p r + ϕ ) + e i 2 ( k p r + ϕ ) ] } * h ( r ) ,
S ( k , ϕ ) = { C ( k ) + m 1 + m 2 [ C ( k k p ) e i ϕ + C ( k + k p ) e i ϕ ] + m 2 4 + 4 m 2 [ C ( k 2 k p ) e i 2 ϕ + C ( k + 2 k p ) e i 2 ϕ ] } × H ( k ) .