Abstract

We performed and studied cascaded third-harmonic generation (THG) in a quasi-periodically poled KTP (QPPKTP) crystal allowing simultaneous phase-matching of the two cascading steps ω + ω → 2ω and 2ω + ω → 3ω. The phase-matching was achieved at the fundamental wavelength λω = 1587 nm when the QPPKTP crystal was heated to 95°C. The energy conversion efficiency reached 40% in the picosecond regime for a fundamental energy of 20 µJ that corresponds to an intensity of 1.5 GW/cm2. It is the highest value of THG efficiency ever reported to the best of our knowledge. The modeling in the case of the depleted pump regime accurately described the experiments.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (2)

2017 (1)

2016 (1)

2014 (1)

2013 (1)

Z. V. Vardeny, A. Nahata, and A. Agrawal, “Optics of photonic quasicrystals,” Nat. Photonics 7(3), 177–187 (2013).
[Crossref]

2010 (2)

H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466(7306), 601–603 (2010).
[Crossref]

A. V. Simakin and G. A. Shafeev, “Initiation of nuclear reactions under laser irradiation of Au nanoparticles in the aqueous solution of Uranium salt,” Appl. Phys. A 101(1), 199–203 (2010).
[Crossref]

2008 (2)

A. Bahabad, R. Lifshitz, N. Voloch, and A. Arie, “Nonlinear photonic quasicrystals for novel optical devices,” Philos. Mag. 88(13-15), 2285–2293 (2008).
[Crossref]

A. Bahabad, A. Ganany-Padowicz, and A. Arie, “Engineering two-dimensional nonlinear photonic quasi-crystals,” Opt. Lett. 33(12), 1386–1388 (2008).
[Crossref]

2007 (2)

2006 (1)

2005 (1)

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic Quasicrystals for Nonlinear Optical Frequency Conversion,” Phys. Rev. Lett. 95(13), 133901 (2005).
[Crossref]

2004 (1)

2003 (2)

S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for KTiOPO4 and KTiOAsO4,” Appl. Opt. 42(33), 6661–6665 (2003).
[Crossref]

V. G. Dmitriev and R. Singh, “Generation of polarization squeezed light in PPNC,” Int. J. Quantum Inform. 01(3), 403–416 (2003).
[Crossref]

2002 (3)

R. P. Schmid, T. Schneider, and J. Reif, “Optical processing on a femtosecond time scale,” Opt. Commun. 207(1-6), 155–160 (2002).
[Crossref]

P. S. Banks, M. D. Feit, and M. D. Perry, “High-intensity third-harmonic generation,” J. Opt. Soc. Am. B 19(1), 102 (2002).
[Crossref]

N. V. Kravtsov, G. D. Laptev, I. I. Naumova, A. A. Novikov, V. V. Firsov, and A. S. Chirkin, “Intracavity quasi-phase matched frequency summing in a laser based on a periodically poled active nonlinear Nd:Mg:LiNbO3crystal,” Quantum Electron. 32(10), 923–924 (2002).
[Crossref]

2001 (1)

K. Fradkin-Kashi, A. Arie, P Urenski, and G. Rosenman, “Multiple Nonlinear Optical Interactions with Arbitrary Wave Vector Differences,” Phys. Rev. Lett. 88(2), 023903 (2001).
[Crossref]

1999 (1)

B. Boulanger, J. P. Fève, P. Delarue, I. Rousseau, and G. Marnier, “Cubic optical nonlinearity of KTiOPO4,” J. Phys. B: At., Mol. Opt. Phys. 32(2), 475–488 (1999).
[Crossref]

1997 (1)

S. N. Zhu, Y. Y. Zhu, and N. Ben Ming, “Quasi–Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science 278(5339), 843–846 (1997).
[Crossref]

1994 (1)

1991 (1)

K. Kato, “Parametric Oscillation at 3.2 µm in KTP Pumped at 1.064 µm,” IEEE J. Quantum Electron. 27(5), 1137–1140 (1991).
[Crossref]

1988 (1)

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillator pumped at 355 nm by a singleaxialmode pulsed source,” Appl. Phys. Lett. 53(21), 2014–2016 (1988).
[Crossref]

1981 (1)

N. G. De Brujin, “Algebraic theory of Penrose’s non-periodic tilings of the plane,” Indag. Math. (Proc.) 84(1), 39–52 (1981).
[Crossref]

1965 (1)

J. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14(24), 973–976 (1965).
[Crossref]

1961 (1)

P. Franken, A. E. Hills, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7(4), 118–119 (1961).
[Crossref]

Agrawal, A.

Z. V. Vardeny, A. Nahata, and A. Agrawal, “Optics of photonic quasicrystals,” Nat. Photonics 7(3), 177–187 (2013).
[Crossref]

Arie, A.

Bahabad, A.

A. Bahabad, R. Lifshitz, N. Voloch, and A. Arie, “Nonlinear photonic quasicrystals for novel optical devices,” Philos. Mag. 88(13-15), 2285–2293 (2008).
[Crossref]

A. Bahabad, A. Ganany-Padowicz, and A. Arie, “Engineering two-dimensional nonlinear photonic quasi-crystals,” Opt. Lett. 33(12), 1386–1388 (2008).
[Crossref]

A. Bahabad, N. Voloch, A. Arie, and R. Lifshitz, “Experimental confirmation of the general solution to the multiple-phase-matching problem,” J. Opt. Soc. Am. B 24(8), 1916 (2007).
[Crossref]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic Quasicrystals for Nonlinear Optical Frequency Conversion,” Phys. Rev. Lett. 95(13), 133901 (2005).
[Crossref]

Banks, P. S.

Bassignot, F.

Belinsky, A. V

A. V Belinsky and R. Singh, “Simultaneous nonlinear conversion of light in periodically poled crystals,” Quantum Electron. 48(7), 611–614 (2018).
[Crossref]

Ben Ming, N.

S. N. Zhu, Y. Y. Zhu, and N. Ben Ming, “Quasi–Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science 278(5339), 843–846 (1997).
[Crossref]

Bonnin, C.

Boulanger, B.

Boutou, V.

Byer, R. L.

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillator pumped at 355 nm by a singleaxialmode pulsed source,” Appl. Phys. Lett. 53(21), 2014–2016 (1988).
[Crossref]

Cabirol, X.

Chauvet, M.

Chirkin, A. S.

N. V. Kravtsov, G. D. Laptev, I. I. Naumova, A. A. Novikov, V. V. Firsov, and A. S. Chirkin, “Intracavity quasi-phase matched frequency summing in a laser based on a periodically poled active nonlinear Nd:Mg:LiNbO3crystal,” Quantum Electron. 32(10), 923–924 (2002).
[Crossref]

De Brujin, N. G.

N. G. De Brujin, “Algebraic theory of Penrose’s non-periodic tilings of the plane,” Indag. Math. (Proc.) 84(1), 39–52 (1981).
[Crossref]

Delarue, P.

B. Boulanger, J. P. Fève, P. Delarue, I. Rousseau, and G. Marnier, “Cubic optical nonlinearity of KTiOPO4,” J. Phys. B: At., Mol. Opt. Phys. 32(2), 475–488 (1999).
[Crossref]

Dmitriev, V. G.

V. G. Dmitriev and R. Singh, “Generation of polarization squeezed light in PPNC,” Int. J. Quantum Inform. 01(3), 403–416 (2003).
[Crossref]

Douady, J.

Eckardt, R. C.

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillator pumped at 355 nm by a singleaxialmode pulsed source,” Appl. Phys. Lett. 53(21), 2014–2016 (1988).
[Crossref]

Emanueli, S.

Fan, Y. X.

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillator pumped at 355 nm by a singleaxialmode pulsed source,” Appl. Phys. Lett. 53(21), 2014–2016 (1988).
[Crossref]

Fedrizzi, A.

H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466(7306), 601–603 (2010).
[Crossref]

Feit, M. D.

Félix, C.

Fève, J. P.

Firsov, V. V.

N. V. Kravtsov, G. D. Laptev, I. I. Naumova, A. A. Novikov, V. V. Firsov, and A. S. Chirkin, “Intracavity quasi-phase matched frequency summing in a laser based on a periodically poled active nonlinear Nd:Mg:LiNbO3crystal,” Quantum Electron. 32(10), 923–924 (2002).
[Crossref]

Fradkin-Kashi, K.

K. Fradkin-Kashi, A. Arie, P Urenski, and G. Rosenman, “Multiple Nonlinear Optical Interactions with Arbitrary Wave Vector Differences,” Phys. Rev. Lett. 88(2), 023903 (2001).
[Crossref]

Franken, P.

P. Franken, A. E. Hills, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7(4), 118–119 (1961).
[Crossref]

Ganany-Padowicz, A.

Giordmaine, J. A.

J. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14(24), 973–976 (1965).
[Crossref]

Gravier, F.

F. Gravier and B. Boulanger, “Third order frequency generation in TiO2 rutile and KTiOPO4,” Opt. Mater. 30(1), 33–36 (2007).
[Crossref]

Hamel, D. R.

H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466(7306), 601–603 (2010).
[Crossref]

Hills, A. E.

P. Franken, A. E. Hills, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7(4), 118–119 (1961).
[Crossref]

Hübel, H.

H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466(7306), 601–603 (2010).
[Crossref]

Jennewein, T.

H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466(7306), 601–603 (2010).
[Crossref]

Kato, K.

K. Kato, “Parametric Oscillation at 3.2 µm in KTP Pumped at 1.064 µm,” IEEE J. Quantum Electron. 27(5), 1137–1140 (1991).
[Crossref]

Kravtsov, N. V.

N. V. Kravtsov, G. D. Laptev, I. I. Naumova, A. A. Novikov, V. V. Firsov, and A. S. Chirkin, “Intracavity quasi-phase matched frequency summing in a laser based on a periodically poled active nonlinear Nd:Mg:LiNbO3crystal,” Quantum Electron. 32(10), 923–924 (2002).
[Crossref]

Laptev, G. D.

N. V. Kravtsov, G. D. Laptev, I. I. Naumova, A. A. Novikov, V. V. Firsov, and A. S. Chirkin, “Intracavity quasi-phase matched frequency summing in a laser based on a periodically poled active nonlinear Nd:Mg:LiNbO3crystal,” Quantum Electron. 32(10), 923–924 (2002).
[Crossref]

Leshem, A.

Lifshitz, R.

A. Bahabad, R. Lifshitz, N. Voloch, and A. Arie, “Nonlinear photonic quasicrystals for novel optical devices,” Philos. Mag. 88(13-15), 2285–2293 (2008).
[Crossref]

A. Bahabad, N. Voloch, A. Arie, and R. Lifshitz, “Experimental confirmation of the general solution to the multiple-phase-matching problem,” J. Opt. Soc. Am. B 24(8), 1916 (2007).
[Crossref]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic Quasicrystals for Nonlinear Optical Frequency Conversion,” Phys. Rev. Lett. 95(13), 133901 (2005).
[Crossref]

Lobino, M.

Lupinski, D.

Marangoni, M.

Marnier, G.

Ménaert, B.

Miller, R. C.

J. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14(24), 973–976 (1965).
[Crossref]

Nahata, A.

Z. V. Vardeny, A. Nahata, and A. Agrawal, “Optics of photonic quasicrystals,” Nat. Photonics 7(3), 177–187 (2013).
[Crossref]

Naumova, I. I.

N. V. Kravtsov, G. D. Laptev, I. I. Naumova, A. A. Novikov, V. V. Firsov, and A. S. Chirkin, “Intracavity quasi-phase matched frequency summing in a laser based on a periodically poled active nonlinear Nd:Mg:LiNbO3crystal,” Quantum Electron. 32(10), 923–924 (2002).
[Crossref]

Nolting, J.

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillator pumped at 355 nm by a singleaxialmode pulsed source,” Appl. Phys. Lett. 53(21), 2014–2016 (1988).
[Crossref]

Novikov, A. A.

N. V. Kravtsov, G. D. Laptev, I. I. Naumova, A. A. Novikov, V. V. Firsov, and A. S. Chirkin, “Intracavity quasi-phase matched frequency summing in a laser based on a periodically poled active nonlinear Nd:Mg:LiNbO3crystal,” Quantum Electron. 32(10), 923–924 (2002).
[Crossref]

Perry, M. D.

Peters, C. W.

P. Franken, A. E. Hills, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7(4), 118–119 (1961).
[Crossref]

Qi, H.

Ramelow, S.

H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466(7306), 601–603 (2010).
[Crossref]

Ramponi, R.

Reif, J.

R. P. Schmid, T. Schneider, and J. Reif, “Optical processing on a femtosecond time scale,” Opt. Commun. 207(1-6), 155–160 (2002).
[Crossref]

Resch, K. J.

H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466(7306), 601–603 (2010).
[Crossref]

Rosenman, G.

K. Fradkin-Kashi, A. Arie, P Urenski, and G. Rosenman, “Multiple Nonlinear Optical Interactions with Arbitrary Wave Vector Differences,” Phys. Rev. Lett. 88(2), 023903 (2001).
[Crossref]

Rousseau, I.

B. Boulanger, J. P. Fève, P. Delarue, I. Rousseau, and G. Marnier, “Cubic optical nonlinearity of KTiOPO4,” J. Phys. B: At., Mol. Opt. Phys. 32(2), 475–488 (1999).
[Crossref]

Schmid, R. P.

R. P. Schmid, T. Schneider, and J. Reif, “Optical processing on a femtosecond time scale,” Opt. Commun. 207(1-6), 155–160 (2002).
[Crossref]

Schneider, T.

R. P. Schmid, T. Schneider, and J. Reif, “Optical processing on a femtosecond time scale,” Opt. Commun. 207(1-6), 155–160 (2002).
[Crossref]

Shafeev, G. A.

A. V. Simakin and G. A. Shafeev, “Initiation of nuclear reactions under laser irradiation of Au nanoparticles in the aqueous solution of Uranium salt,” Appl. Phys. A 101(1), 199–203 (2010).
[Crossref]

Shiloh, R.

Simakin, A. V.

A. V. Simakin and G. A. Shafeev, “Initiation of nuclear reactions under laser irradiation of Au nanoparticles in the aqueous solution of Uranium salt,” Appl. Phys. A 101(1), 199–203 (2010).
[Crossref]

Singh, R.

A. V Belinsky and R. Singh, “Simultaneous nonlinear conversion of light in periodically poled crystals,” Quantum Electron. 48(7), 611–614 (2018).
[Crossref]

V. G. Dmitriev and R. Singh, “Generation of polarization squeezed light in PPNC,” Int. J. Quantum Inform. 01(3), 403–416 (2003).
[Crossref]

Sun, X.

Trajtenberg-Mills, S.

Urenski, P

K. Fradkin-Kashi, A. Arie, P Urenski, and G. Rosenman, “Multiple Nonlinear Optical Interactions with Arbitrary Wave Vector Differences,” Phys. Rev. Lett. 88(2), 023903 (2001).
[Crossref]

Vardeny, Z. V.

Z. V. Vardeny, A. Nahata, and A. Agrawal, “Optics of photonic quasicrystals,” Nat. Photonics 7(3), 177–187 (2013).
[Crossref]

Vernay, A.

Villeval, P.

Voloch, N.

A. Bahabad, R. Lifshitz, N. Voloch, and A. Arie, “Nonlinear photonic quasicrystals for novel optical devices,” Philos. Mag. 88(13-15), 2285–2293 (2008).
[Crossref]

A. Bahabad, N. Voloch, A. Arie, and R. Lifshitz, “Experimental confirmation of the general solution to the multiple-phase-matching problem,” J. Opt. Soc. Am. B 24(8), 1916 (2007).
[Crossref]

Wallenstein, R.

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillator pumped at 355 nm by a singleaxialmode pulsed source,” Appl. Phys. Lett. 53(21), 2014–2016 (1988).
[Crossref]

Wang, Z.

Weinreich, G.

P. Franken, A. E. Hills, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. 7(4), 118–119 (1961).
[Crossref]

Xu, X.

Yu, F.

Zhao, X.

Zhu, S. N.

S. N. Zhu, Y. Y. Zhu, and N. Ben Ming, “Quasi–Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science 278(5339), 843–846 (1997).
[Crossref]

Zhu, Y. Y.

S. N. Zhu, Y. Y. Zhu, and N. Ben Ming, “Quasi–Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science 278(5339), 843–846 (1997).
[Crossref]

Appl. Opt. (1)

Appl. Phys. A (1)

A. V. Simakin and G. A. Shafeev, “Initiation of nuclear reactions under laser irradiation of Au nanoparticles in the aqueous solution of Uranium salt,” Appl. Phys. A 101(1), 199–203 (2010).
[Crossref]

Appl. Phys. Lett. (1)

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillator pumped at 355 nm by a singleaxialmode pulsed source,” Appl. Phys. Lett. 53(21), 2014–2016 (1988).
[Crossref]

IEEE J. Quantum Electron. (1)

K. Kato, “Parametric Oscillation at 3.2 µm in KTP Pumped at 1.064 µm,” IEEE J. Quantum Electron. 27(5), 1137–1140 (1991).
[Crossref]

Indag. Math. (Proc.) (1)

N. G. De Brujin, “Algebraic theory of Penrose’s non-periodic tilings of the plane,” Indag. Math. (Proc.) 84(1), 39–52 (1981).
[Crossref]

Int. J. Quantum Inform. (1)

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Figures (3)

Fig. 1.
Fig. 1. (a) A schematic picture of the cascaded processes; (x,y,z) refers to the dielectric frame ; kω,2ω,3ω stands for the wave vectors and $\varDelta {k_{SHG,THG}}$ for the phase-mismatches, with no metric matching between the vector sizes and the pattern in the background. (b) Microscopy picture of the QPPKTP crystal exhibiting the poling pattern; the yellow scale bar is 28 µm long. (c) Far field diffraction pattern from the crystal. In orange are the peaks that correspond to the peaks used for the design. (d) Fourier Transform of the nonlinear modulation pattern, exhibiting two peaks at the mismatch frequencies, i.e. $\varDelta k_{SHG}^{} = 0.24441\mu {m^{ - 1}}$ for the SHG and $\varDelta k_{THG}^{} = 0.22031\mu {m^{ - 1}}$ for the THG.
Fig. 2.
Fig. 2. (a) Normalized intensities of the second-harmonic (SH) wave and third-harmonic (TH) waves as a function of the QPPKTP temperature when the waves propagate along the x-axis, the quantities (y) and (z) referring to the polarization directions of the interacting waves; the dots correspond to experimental data and the dashed lines to fits. (b) Normalized intensities of the SH and TH waves as a function of the polarization angle α defined in the right inset. The fundamental intensity was lower than 0.5 GW.cm-2, hence pump depletion is negligible here.
Fig. 3.
Fig. 3. SHG (a) and THG (b) energy conversion efficiencies as a function of the total incident fundamental intensity. The dots correspond to experimental data and the continuous lines stand for the interpolations.

Tables (1)

Tables Icon

Table 1. Wavelengths λ ω i where ω i stands for the frequencies ω , 2 ω or 3 ω , pulse durations τ ω i , principal refractive indices n y ( λ ω / 3 ω ) and n y ( λ ω / 3 ω ) [26], beam wait radii w ω i , Fresnel transmissions T ω i y , z and nonlinear coefficients χ i j ( 2 ) used for the numerical integration of Eqs. (1) [27].

Equations (6)

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{ | E ω z ( X ) | X = π λ ω n z ( λ ω ) χ 33 ( 2 ) ( ω = 2 ω ω ) F S H G | E 2 ω z ( X ) | . | E ω z ( X ) | | E ω y ( X ) | X = π λ ω n y ( λ ω ) χ 24 ( 2 ) ( ω = 3 ω 2 ω ) F T H G | E 3 ω y ( X ) | . | E 2 ω z ( X ) | | E 2 ω z ( X ) | X = + π λ 2 ω n z ( λ 2 ω ) χ 33 ( 2 ) ( 2 ω = ω + ω ) F S H G | E ω z ( X ) | . | E ω z ( X ) | π λ 2 ω n z ( λ 2 ω ) χ 24 ( 2 ) ( 2 ω = 3 ω ω ) F T H G | E 3 ω y ( X ) | . | E ω y ( X ) | | E 3 ω y ( X ) | X = + π λ 3 ω n y ( λ 3 ω ) χ 24 ( 2 ) ( 3 ω = ω + 2 ω ) F T H G | E ω y ( X ) | . | E 2 ω z ( X ) | .
{ | E ω y ( X = 0 ) | = μ 0 c n y ( λ ω ) κ ω T ω y cos 2 ( α ) ε ω t o t ( X = 0 ) | E ω z ( X = 0 ) | = μ 0 c n z ( λ ω ) κ ω T ω z sin 2 ( α ) ε ω t o t ( X = 0 )
{ T ω y , z = 4 n y , z ( λ ω ) [ 1 + n y , z ( λ ω ) ] 2 κ ω = ( π 2 ) 3 / 2 τ ω 2 w ω 2 .
{ ε 2 ω ( X = L ) = κ 2 ω T 2 ω z n z ( λ 2 ω ) μ 0 c | E 2 ω z ( X = L ) | 2 ε 3 ω ( X = L ) = κ 3 ω T 3 ω y n y ( λ 3 ω ) μ 0 c | E 3 ω y ( X = L ) | 2
{ T 2 ω z = 4 n z ( λ 2 ω ) [ 1 + n z ( λ 2 ω ) ] 2 κ 2 ω = ( π 2 ) 3 / 2 τ 2 ω 2 w 2 ω 2 T 3 ω y = 4 n y ( λ 3 ω ) [ 1 + n y ( λ 3 ω ) ] 2 κ 3 ω = ( π 2 ) 3 / 2 τ 3 ω 2 w 3 ω 2
{ w 2 ω = w ω 2 , τ 2 ω = τ ω 2 w 3 ω = w ω 3 , τ 3 ω = τ ω 3 .

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