Abstract

The polarization effects in the diffraction-induced pulse splitting (DIPS) observed under the dynamical Bragg diffraction in the Laue geometry in linear one-dimensional photonic crystals (PCs) are studied theoretically and experimentally. It is demonstrated that the characteristic length of the laser pulse path in a PC, or splitting length, used to describe the temporal pulse splitting, as well as the number of the outgoing femtosecond pulses, are influenced significantly by the polarization of the incident laser pulse. We have observed that the characteristic splitting time in porous quartz PCs for the s-polarized probe pulse is approximately 1.5 times smaller as compared with that measured for the p-polarized radiation. These results are supported by the theoretical description and ensure that the polarization sensitivity of the DIPS effect is due to a large lattice-induced dispersion of the PC. It is also shown that the number of output pulses can be varied from two up to four in both transmission and diffraction directions depending on the polarization of incident femtosecond pulses.

© 2013 Optical Society of America

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2012 (4)

A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction-induced splitting, selective compression, and focusing of pulses,” JETP 115, 56–67 (2012).
[CrossRef]

S. E. Svyahovskiy, V. O. Kompanets, A. I. Maidykovskiy, T. V. Murzina, S. V. Chekalin, V. A. Bushuev, A. A. Skorynin, and B. I. Mantsyzov, “Observation of diffraction-induced laser pulse splitting in a photonic crystal,” Phys. Rev. A 86, 013843 (2012).
[CrossRef]

S. E. Svyakhovskiy, A. I. Maydykovsky, and T. V. Murzina, “Mesoporous silicon photonic structures of thousands of periods,” J. Appl. Phys. 112, 013106 (2012).
[CrossRef]

B. Bruser, I. Staude, G. Freymann, M. Wegener, and U. Pietsch, “Visible light Laue diffraction from woodpile photonic crystals,” Appl. Opt. 51, 6732–6737 (2012).
[CrossRef]

2011 (2)

B. Terhalle, A. Desyatnikov, D. Neshev, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Dynamic diffraction and interband transition in two-dimensional photonic lattices,” Phys. Rev. Lett. 106, 083902 (2011).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

2009 (1)

2008 (2)

V. A. Bushuev and B. I. Mantsyzov, “Linear effect of doubling of the laser pulse repetition rate in the Laue geometry of Bragg diffraction in a photonic crystal,” Bull. Russ. Acad. Sci.: Phys. 72, 30–34 (2008).

S. Savo, E. Di Gennaro, C. Miletto, A. Andreone, P. Dardano, L. Moretti, and V. Mocella, “Pendellosung effect in photonic crystals,” Opt. Express 16, 9097–9105 (2008).
[CrossRef]

2007 (1)

K. Busch, G. Von Freymann, S. Linder, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

2006 (5)

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

A. Balestreri, L. C. Andreani, and M. Agio, “Optical properties and diffraction effects in opal photonic crystals,” Phys. Rev. E 74, 036603 (2006).
[CrossRef]

J. T. Mok, C. M. deSterke, and B. J. Eggleton, “Delay-tunable gap-soliton-based slow-light system,” Opt. Express 14, 11987–11996 (2006).
[CrossRef]

M. Calvo, P. Cheben, O. Martinez-Matos, F. del Monte, and J. A. Rodrigo, “Experimental detection of the optical Pedellosung effect,” Phys. Rev. Lett. 97, 084801 (2006).
[CrossRef]

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

2005 (2)

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef]

H. Altug, and J. Vuckovic, “Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays,” Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

2002 (1)

M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. Qiu, “Low-threshold photonic crystal laser,” Appl. Phys. Lett. 81, 2680–2682 (2002).
[CrossRef]

2001 (1)

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

1997 (1)

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear one-dimensional periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

1977 (1)

Agio, M.

A. Balestreri, L. C. Andreani, and M. Agio, “Optical properties and diffraction effects in opal photonic crystals,” Phys. Rev. E 74, 036603 (2006).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Altug, H.

H. Altug, and J. Vuckovic, “Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays,” Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

Andreani, L. C.

A. Balestreri, L. C. Andreani, and M. Agio, “Optical properties and diffraction effects in opal photonic crystals,” Phys. Rev. E 74, 036603 (2006).
[CrossRef]

Andreone, A.

Balakin, A. V.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Balestreri, A.

A. Balestreri, L. C. Andreani, and M. Agio, “Optical properties and diffraction effects in opal photonic crystals,” Phys. Rev. E 74, 036603 (2006).
[CrossRef]

Belardini, A.

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

Benton, C. J.

Bertolotti, M.

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

Bloemer, M. J.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear one-dimensional periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Bogaerts, W.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef]

Bowden, C. M.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear one-dimensional periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Bruser, B.

Buganov, O.

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

Busch, K.

K. Busch, G. Von Freymann, S. Linder, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Bushuev, V. A.

A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction-induced splitting, selective compression, and focusing of pulses,” JETP 115, 56–67 (2012).
[CrossRef]

S. E. Svyahovskiy, V. O. Kompanets, A. I. Maidykovskiy, T. V. Murzina, S. V. Chekalin, V. A. Bushuev, A. A. Skorynin, and B. I. Mantsyzov, “Observation of diffraction-induced laser pulse splitting in a photonic crystal,” Phys. Rev. A 86, 013843 (2012).
[CrossRef]

V. A. Bushuev and B. I. Mantsyzov, “Linear effect of doubling of the laser pulse repetition rate in the Laue geometry of Bragg diffraction in a photonic crystal,” Bull. Russ. Acad. Sci.: Phys. 72, 30–34 (2008).

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Calvo, M.

M. Calvo, P. Cheben, O. Martinez-Matos, F. del Monte, and J. A. Rodrigo, “Experimental detection of the optical Pedellosung effect,” Phys. Rev. Lett. 97, 084801 (2006).
[CrossRef]

Centini, M.

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

Cheben, P.

M. Calvo, P. Cheben, O. Martinez-Matos, F. del Monte, and J. A. Rodrigo, “Experimental detection of the optical Pedellosung effect,” Phys. Rev. Lett. 97, 084801 (2006).
[CrossRef]

Chekalin, S. V.

S. E. Svyahovskiy, V. O. Kompanets, A. I. Maidykovskiy, T. V. Murzina, S. V. Chekalin, V. A. Bushuev, A. A. Skorynin, and B. I. Mantsyzov, “Observation of diffraction-induced laser pulse splitting in a photonic crystal,” Phys. Rev. A 86, 013843 (2012).
[CrossRef]

Dardano, P.

del Monte, F.

M. Calvo, P. Cheben, O. Martinez-Matos, F. del Monte, and J. A. Rodrigo, “Experimental detection of the optical Pedellosung effect,” Phys. Rev. Lett. 97, 084801 (2006).
[CrossRef]

Denz, C.

B. Terhalle, A. Desyatnikov, D. Neshev, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Dynamic diffraction and interband transition in two-dimensional photonic lattices,” Phys. Rev. Lett. 106, 083902 (2011).
[CrossRef]

deSterke, C. M.

Desyatnikov, A.

B. Terhalle, A. Desyatnikov, D. Neshev, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Dynamic diffraction and interband transition in two-dimensional photonic lattices,” Phys. Rev. Lett. 106, 083902 (2011).
[CrossRef]

Di Gennaro, E.

Dosco, A.

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear one-dimensional periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Eggleton, B. J.

Engelen, R. J.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef]

Fazio, E.

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

Fedotov, A. B.

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

Freymann, G.

Gaponenko, S.

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

Gersen, H.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef]

Gogna, P.

M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. Qiu, “Low-threshold photonic crystal laser,” Appl. Phys. Lett. 81, 2680–2682 (2002).
[CrossRef]

Golovan, L. A.

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

Haus, J. W.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear one-dimensional periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Hong, C.

Ivanov, D. A.

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

Karle, T. J.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef]

Kartashov, Y. V.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

Kashkarov, P. K.

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

Kivshar, Y. S.

B. Terhalle, A. Desyatnikov, D. Neshev, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Dynamic diffraction and interband transition in two-dimensional photonic lattices,” Phys. Rev. Lett. 106, 083902 (2011).
[CrossRef]

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Kompanets, V. O.

S. E. Svyahovskiy, V. O. Kompanets, A. I. Maidykovskiy, T. V. Murzina, S. V. Chekalin, V. A. Bushuev, A. A. Skorynin, and B. I. Mantsyzov, “Observation of diffraction-induced laser pulse splitting in a photonic crystal,” Phys. Rev. A 86, 013843 (2012).
[CrossRef]

Konorov, S. O.

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

Korterik, J. P.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef]

Krauss, T. F.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef]

Krolikowski, W.

B. Terhalle, A. Desyatnikov, D. Neshev, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Dynamic diffraction and interband transition in two-dimensional photonic lattices,” Phys. Rev. Lett. 106, 083902 (2011).
[CrossRef]

Kuipers, L.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef]

Leahu, G.

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

Linder, S.

K. Busch, G. Von Freymann, S. Linder, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Loncar, M.

M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. Qiu, “Low-threshold photonic crystal laser,” Appl. Phys. Lett. 81, 2680–2682 (2002).
[CrossRef]

Maidykovskiy, A. I.

S. E. Svyahovskiy, V. O. Kompanets, A. I. Maidykovskiy, T. V. Murzina, S. V. Chekalin, V. A. Bushuev, A. A. Skorynin, and B. I. Mantsyzov, “Observation of diffraction-induced laser pulse splitting in a photonic crystal,” Phys. Rev. A 86, 013843 (2012).
[CrossRef]

Malomed, B. A.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear one-dimensional periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Mantsyzov, B. I.

S. E. Svyahovskiy, V. O. Kompanets, A. I. Maidykovskiy, T. V. Murzina, S. V. Chekalin, V. A. Bushuev, A. A. Skorynin, and B. I. Mantsyzov, “Observation of diffraction-induced laser pulse splitting in a photonic crystal,” Phys. Rev. A 86, 013843 (2012).
[CrossRef]

A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction-induced splitting, selective compression, and focusing of pulses,” JETP 115, 56–67 (2012).
[CrossRef]

V. A. Bushuev and B. I. Mantsyzov, “Linear effect of doubling of the laser pulse repetition rate in the Laue geometry of Bragg diffraction in a photonic crystal,” Bull. Russ. Acad. Sci.: Phys. 72, 30–34 (2008).

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Martinez-Matos, O.

M. Calvo, P. Cheben, O. Martinez-Matos, F. del Monte, and J. A. Rodrigo, “Experimental detection of the optical Pedellosung effect,” Phys. Rev. Lett. 97, 084801 (2006).
[CrossRef]

Masselin, P.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Maydykovsky, A. I.

S. E. Svyakhovskiy, A. I. Maydykovsky, and T. V. Murzina, “Mesoporous silicon photonic structures of thousands of periods,” J. Appl. Phys. 112, 013106 (2012).
[CrossRef]

Melnikov, V. A.

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

Miletto, C.

Mingaleev, S.

K. Busch, G. Von Freymann, S. Linder, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Mocella, V.

Mok, J. T.

Moretti, L.

Mouret, G.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Murzina, T. V.

S. E. Svyahovskiy, V. O. Kompanets, A. I. Maidykovskiy, T. V. Murzina, S. V. Chekalin, V. A. Bushuev, A. A. Skorynin, and B. I. Mantsyzov, “Observation of diffraction-induced laser pulse splitting in a photonic crystal,” Phys. Rev. A 86, 013843 (2012).
[CrossRef]

S. E. Svyakhovskiy, A. I. Maydykovsky, and T. V. Murzina, “Mesoporous silicon photonic structures of thousands of periods,” J. Appl. Phys. 112, 013106 (2012).
[CrossRef]

Neshev, D.

B. Terhalle, A. Desyatnikov, D. Neshev, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Dynamic diffraction and interband transition in two-dimensional photonic lattices,” Phys. Rev. Lett. 106, 083902 (2011).
[CrossRef]

Ozheredov, I. A.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Petrov, E. V.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Petrov, G. I.

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

Pietsch, U.

Pinsker, Z. G.

Z. G. Pinsker, Dynamical Scattering of X-rays in Crystals, Vol. 3 of Springer Series On Solid-State Science (Springer, 1977).

Qiu, Y.

M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. Qiu, “Low-threshold photonic crystal laser,” Appl. Phys. Lett. 81, 2680–2682 (2002).
[CrossRef]

Rodrigo, J. A.

M. Calvo, P. Cheben, O. Martinez-Matos, F. del Monte, and J. A. Rodrigo, “Experimental detection of the optical Pedellosung effect,” Phys. Rev. Lett. 97, 084801 (2006).
[CrossRef]

Savo, S.

Scalora, M.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear one-dimensional periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Scherer, A.

M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. Qiu, “Low-threshold photonic crystal laser,” Appl. Phys. Lett. 81, 2680–2682 (2002).
[CrossRef]

Shkurinov, A. P.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Sibilia, C.

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

Skorynin, A. A.

S. E. Svyahovskiy, V. O. Kompanets, A. I. Maidykovskiy, T. V. Murzina, S. V. Chekalin, V. A. Bushuev, A. A. Skorynin, and B. I. Mantsyzov, “Observation of diffraction-induced laser pulse splitting in a photonic crystal,” Phys. Rev. A 86, 013843 (2012).
[CrossRef]

A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction-induced splitting, selective compression, and focusing of pulses,” JETP 115, 56–67 (2012).
[CrossRef]

Skryabin, D. V.

Staude, I.

Svyahovskiy, S. E.

S. E. Svyahovskiy, V. O. Kompanets, A. I. Maidykovskiy, T. V. Murzina, S. V. Chekalin, V. A. Bushuev, A. A. Skorynin, and B. I. Mantsyzov, “Observation of diffraction-induced laser pulse splitting in a photonic crystal,” Phys. Rev. A 86, 013843 (2012).
[CrossRef]

Svyakhovskiy, S. E.

S. E. Svyakhovskiy, A. I. Maydykovsky, and T. V. Murzina, “Mesoporous silicon photonic structures of thousands of periods,” J. Appl. Phys. 112, 013106 (2012).
[CrossRef]

Terhalle, B.

B. Terhalle, A. Desyatnikov, D. Neshev, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Dynamic diffraction and interband transition in two-dimensional photonic lattices,” Phys. Rev. Lett. 106, 083902 (2011).
[CrossRef]

Timoshenko, V. Y.

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

Tkeshelashvili, L.

K. Busch, G. Von Freymann, S. Linder, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Torner, L.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

van Hulst, N. F.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef]

Viswanathan, R.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear one-dimensional periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Von Freymann, G.

K. Busch, G. Von Freymann, S. Linder, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Vuckovic, J.

H. Altug, and J. Vuckovic, “Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays,” Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

Wegener, M.

B. Bruser, I. Staude, G. Freymann, M. Wegener, and U. Pietsch, “Visible light Laue diffraction from woodpile photonic crystals,” Appl. Opt. 51, 6732–6737 (2012).
[CrossRef]

K. Busch, G. Von Freymann, S. Linder, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Yakovlev, V. V.

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

Yariv, A.

Yeh, P.

Yoshie, T.

M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. Qiu, “Low-threshold photonic crystal laser,” Appl. Phys. Lett. 81, 2680–2682 (2002).
[CrossRef]

Zheltikov, A. M.

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

Zhukovsky, S.

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

Appl. Opt. (1)

Appl. Phys. Lett. (2)

H. Altug, and J. Vuckovic, “Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays,” Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. Qiu, “Low-threshold photonic crystal laser,” Appl. Phys. Lett. 81, 2680–2682 (2002).
[CrossRef]

Bull. Russ. Acad. Sci.: Phys. (1)

V. A. Bushuev and B. I. Mantsyzov, “Linear effect of doubling of the laser pulse repetition rate in the Laue geometry of Bragg diffraction in a photonic crystal,” Bull. Russ. Acad. Sci.: Phys. 72, 30–34 (2008).

J. Appl. Phys. (1)

S. E. Svyakhovskiy, A. I. Maydykovsky, and T. V. Murzina, “Mesoporous silicon photonic structures of thousands of periods,” J. Appl. Phys. 112, 013106 (2012).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Optoelectron. Adv. Mater. (1)

A. Belardini, O. Buganov, G. Leahu, A. Dosco, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. Gaponenko, “Dynamic response of a coupled-cavities one-dimensional photonic crystal in the femtosecond regime,” J. Optoelectron. Adv. Mater. 8, 2015–2017 (2006).

JETP (1)

A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction-induced splitting, selective compression, and focusing of pulses,” JETP 115, 56–67 (2012).
[CrossRef]

Opt. Express (3)

Phys. Rep. (1)

K. Busch, G. Von Freymann, S. Linder, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Phys. Rev. A (2)

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear one-dimensional periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

S. E. Svyahovskiy, V. O. Kompanets, A. I. Maidykovskiy, T. V. Murzina, S. V. Chekalin, V. A. Bushuev, A. A. Skorynin, and B. I. Mantsyzov, “Observation of diffraction-induced laser pulse splitting in a photonic crystal,” Phys. Rev. A 86, 013843 (2012).
[CrossRef]

Phys. Rev. B (1)

L. A. Golovan, V. A. Melnikov, S. O. Konorov, A. B. Fedotov, V. Y. Timoshenko, A. M. Zheltikov, P. K. Kashkarov, D. A. Ivanov, G. I. Petrov, and V. V. Yakovlev, “Linear and nonlinear optical anisotropy of amorphous oxidized silicon films induced by a network of pores,” Phys. Rev. B 73, 115337 (2006).
[CrossRef]

Phys. Rev. E (2)

A. Balestreri, L. C. Andreani, and M. Agio, “Optical properties and diffraction effects in opal photonic crystals,” Phys. Rev. E 74, 036603 (2006).
[CrossRef]

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Phys. Rev. Lett. (3)

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef]

M. Calvo, P. Cheben, O. Martinez-Matos, F. del Monte, and J. A. Rodrigo, “Experimental detection of the optical Pedellosung effect,” Phys. Rev. Lett. 97, 084801 (2006).
[CrossRef]

B. Terhalle, A. Desyatnikov, D. Neshev, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Dynamic diffraction and interband transition in two-dimensional photonic lattices,” Phys. Rev. Lett. 106, 083902 (2011).
[CrossRef]

Rev. Mod. Phys. (1)

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

Other (2)

Z. G. Pinsker, Dynamical Scattering of X-rays in Crystals, Vol. 3 of Springer Series On Solid-State Science (Springer, 1977).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of the diffraction-induced pulse splitting within 1D PC at the Bragg diffraction in the Laue geometry.

Fig. 2.
Fig. 2.

Dispersion curves qzj(s,p) (α0) for the Borrmann (red lines 1s, 1p) and anti-Borrmann (blue lines 2s, 2p) modes calculated by Eq. (30) for the s- (dashed lines) and p-polarized (solid lines) incident radiation, and for a homogeneous medium with the same value of the average refraction index na (gray line). The refractive indexes of the layers are: n1=1.445 and n2=1.355; d=775nm, d1/d=0.5, λ0=800nm, θB=31°.

Fig. 3.
Fig. 3.

Dependencies of the electric fields |Egj(s,p)| on the detuning α0 for the Borrmann (1) and anti-Borrmann (2) modes for the s- (dashed lines) and p-polarized (solid lines) incident radiation calculated by Eq. (26), and Eqs. (19) and (21), correspondingly. The parameters are the same as in Fig. 2.

Fig. 4.
Fig. 4.

Dependencies of the group velocity projections vz1,2(s,p) Eq. (32) on the detuning α0. The parameters are the same as in Fig. 2.

Fig. 5.
Fig. 5.

Measured (blue dashed lines) and calculated by Eqs. (34) and (35) (red solid lines) intensity autocorrelation functions IAC(τ) of laser pulses passed through the PC in the direction of the transmitted wave at (a) s-polarized incident pulse and (b) p-polarized one for sample I (L=3.8mm). The time dependence of the calculated by Eq. (35) pulse intensities I0(s,p)(t) that provide the best correlation with the experimental data for (c) s-polarized and (d) p-polarized incident pulse radiation. The refractive indexes of the layers are: n1,o=1.455, n2,o=1.345, n1,e=1.440, and n2,e=1.320; pulse duration is τ0=110fs.

Fig. 6.
Fig. 6.

(a), (b) Experimental (dashed lines) and calculated by Eq. (34) for I0(m), Eq. (36) (solid lines) autocorrelation functions IAC(m)(τ) of the laser pulses passed through the PC in the direction of the transmitted wave for the “mixed” polarizations of the incident and outgoing pulses for sample I. The polarization angles are (a) ϕ=64° and (b) 28°. (c), (d) The intensity of the pulses calculated by Eqs. (35) and (36). Parameters are the same as in Fig. 5.

Fig. 7.
Fig. 7.

Experimentally measured (blue dashed lines) and calculated by Eqs. (34) and (35) (red solid lines) intensity autocorrelation functions IAC(τ) of laser pulses passed through the PC in the direction of the transmitted wave for (a) s-polarized incident pulse and (b) for the p-polarized pulses for the sample II with thickness L=2mm. The intensity of the pulses I0(s,p)(t) calculated by Eq. (35) (c) for s-polarized incident pulse and (d) for p-polarized one. The PC parameters used in calculations are: for ordinary wave n1,o=1.445 and n2,o=1.355, for extraordinary n1,e=1.433 and n2,e=1.327; d=775nm, d1/d=0.5, λ0=800nm, θ=θB=31°, τ0=30fs, D=30μm.

Equations (49)

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

Ein(r,t)=einEin(r,t)exp(ik0riω0t),
Ein(x,t)=Ein(kx,ω)exp(ikxxiωt)dkxdω,
Ein(kx,ω)=einEin(k,Ω)=ein14π2Ein(x,t)exp(iKx+iΩt)dxdt.
××E(r,t)+ε(x)c22E(r,t)t2=0,
n(x)=na+Δn(x),
na=(n1d1+n2d2)/d=n2+ξδ,
ΔH(r,t)+εε××H(r,t)ε(x)c22H(r,t)t2=0.
H(r,t)=H0(r,t)+Hh(r,t),E(r,t)=E0(r,t)+Eh(r,t).
Hg(x,z,t)=Hg(K,Ω)exp[i(q0xg)x+iqzziωt]dKdΩ,g=0,h,
ε(x)=χ0+χhexp(ihx)+χhexp(ihx),
χg=1d0dε(x)exp(igx)dx,g=0,h,h,
χ0=na2+δ2(ξξ2),χh=iπ(naδ+δ212ξ2)[1exp(i2πξ)],χh=iπ(naδ+δ212ξ2)[1exp(i2πξ)].
ν(x)=ex[ν0+νhexp(ihx)+νhexp(ihx)],
νg=1d0d1ε(x)ε(x)xexp(igx)dx,g=0,h,h.
νh=ihχh/χ0,νh=ihχh/χ0.
βH0χhHhiνhex×qh×Hh/k2=0,χhH0(βα)Hh+iνhex×q0×H0/k2=0,
β=(qz2k2γ2)/k2,γ=(k2χ0q0x2)1/2/k,α=h(2q0xh)/k2,k=ω/c.
βH0ChχhHh=0,ChχhH0(βα)Hh=0,
β(βα)ChChχhχh=0,
β1,2=(1/2)[α(α2+4ChChχhχh)1/2],
Ch=1hq0x/k2χ0,Ch=1h(hq0x)/k2χ0.
qzj=k(γ2+βj)1/2,j=1,2.
Hhj=RjH0j,
E0j=D0jDhjχh/χ0χ0χ2/χ0,Ehj=DhjD0jχh/χ0χ0χ2/χ0.
Dgj=qgjHgj/k.
E0j=q0jkχ0(1Rjχhχ0q0j2q0xhq0j2)H0j,Ehj=qhjkχ0(1χhRjχ0q0j2q0xhqhj2)Hhj.
Hin+Hr=H01+H02,kz(HinHr)=f1H01+f2H02,R1H01+R2H02=0,
H0j=B0jHin,
Hg(x,z,t)=eyBg(K,Ω,z)Hin(K,Ω)exp(iKxiΩt)dKdΩ×exp[i(k0xg)xiω0t],
Bg(K,Ω,z)=j=1,2Bgjexp(iqzjz),
βH0χh(q0/qh)Hh=0,χh(qh/q0)H0(βα)Hh=0.
βE0χhEh=0,χhE0(βα)Eh=0.
β1,2(s)=(1/2)[α(α2+4χhχh)1/2],qzj(s)=k(γ2+βj(s))1/2,j=1,2.
Ein+Er=E01(s)+E02(s),kz(EinEr)=qz1(s)E01(s)+qz2(s)E02(s),R1(s)E01(s)+R2(s)E02(s)=0.
E0j(s)=B0j(s)Ein,
Eg(s)(x,z,t)=eyBg(s)(K,Ω,z)Ein(K,Ω)exp(iKxiΩt)dKdΩ×exp[i(k0xg)xiω0t],
Bg(s)(K,Ω,z)=j=1,2Bgj(s)exp(iqzj(s)z).
E(s)(x,z,t)=[E1(s)(x,z,t)+E2(s)(x,z,t)]exp(ik0xiω0t),
Ej(s)(x,z,t)=eyBj(s)(K,Ω,z)Ein(K,Ω)exp(iKxiΩt)dKdΩ,j=1,2,Bj(s)(K,Ω,z)=exp(iqzj(s)z)g=0,hBgj(s)exp(igx).
H(p)(x,z,t)=[H1(p)(x,z,t)+H2(p)(x,z,t)]exp(ik0xiω0t),Hj(p)(x,z,t)=eyBj(p)(K,Ω,z)Hin(K,Ω)exp(iKxiΩt)dKdΩ,
Bj(p)(K,Ω,z)=exp(iqzj(p)z)g=0,hBgj(p)exp(igx).
qzj(s,p)2=k2χ0q0x2+hα0h2α02+C(s,p)2χhχhk4,
C(p)2(ω,q0x)=1h2/χ0k2+h2(hq0x)q0x/χ02k4.
vzj(s,p)=cqzj(s,p)k[χ0C(s,p)χhχhk2(2C(s,p)+ωC(s,p)/ω)2h2α02+C(s,p)2χhχhk4]1,
t12(s)=(zχ/cχ01/2)(1sin2θB/2χ0),t12(p)=(zχ/cχ01/2)(1+3sin2θB/2χ0).
LDIPS(p)/LDIPS(s)=(1sin2θB/2χ0)/(1+3sin2θB/2χ0)<1.
IAC(s,p)(τ)=I0(s,p)(t)I0(s,p)(t+τ)dt,
I0(s)(t)=|E0(s)(t)|2,I0(p)(t)=|H0(p)(t)|2,
I0(m)(t,ϕ)=I0(p)(t)cos2(ϕ)+I0(s)(t)sin2(ϕ).

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