Abstract

A simple method to generate second-harmonic conical waves with tunable cone angles (apex angles) only using a cube uniaxial nonlinear crystal is proposed. The formation mechanism is based on the birefringence phase-matching among a plane fundamental wave, a spherical fundamental wave, and a conical second-harmonic wave. The cone angle in our demonstration of the method could be continuously adjusted from 0 to 30° by just rotating the crystal, which is the largest reported tunable range. Moreover, a tunable range of 0–68° could be achieved by using several crystals with different cut angles. The polarization and the longitudinal distribution of the generated second-harmonic wave are theoretically analyzed. The measured conversion efficiency was around 7%, and it decreased with the increasing of the cone angle.

© 2017 Optical Society of America

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References

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2016 (1)

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Conical refraction: fundamentals and applications,” Laser Photonics Rev. 1, 22 (2016).

2015 (3)

2014 (2)

2013 (3)

2012 (2)

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov Radiation in an Anomalous Dispersive Medium,” Phys. Rev. Lett. 108(22), 223901 (2012).
[Crossref] [PubMed]

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
[Crossref]

2011 (1)

J. Kroupa and M. Fridrichova, “Spontaneous noncollinear second harmonic generation in GUHP,” J. Opt. 13(3), 035204 (2011).
[Crossref]

2010 (1)

S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Y. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. 104(8), 083902 (2010).
[Crossref] [PubMed]

2008 (1)

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

2007 (1)

K. Bastwöste, U. Sander, and M. Imlau, “Conical light scattering in strontium barium niobate crystals related to an intrinsic composition inhomogeneity,” J. Phys. Condens. Matter 19(15), 156225 (2007).
[Crossref]

2006 (1)

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

2004 (1)

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ2 photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

1999 (2)

1990 (1)

Y. H. Ja, “A scattered ring in a natural crystal of tourmaline,” J. Opt. 21(1), 41–43 (1990).
[Crossref]

1981 (1)

1962 (1)

J. A. Giordmaine, “Mixing of light beams in crystals,” Phys. Rev. Lett. 8(1), 19–20 (1962).
[Crossref]

Abdolvand, A.

An, N.

Arie, A.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Y. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. 104(8), 083902 (2010).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Arnold, C. B.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
[Crossref]

Bang, O.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Y. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. 104(8), 083902 (2010).
[Crossref] [PubMed]

Bastwöste, K.

K. Bastwöste, U. Sander, and M. Imlau, “Conical light scattering in strontium barium niobate crystals related to an intrinsic composition inhomogeneity,” J. Phys. Condens. Matter 19(15), 156225 (2007).
[Crossref]

Betzler, K.

K. U. Kasemir and K. Betzler, “Characterization of photorefractive materials by spontaneous noncolinear frequency doubling,” Appl. Phys. B 68(5), 763–766 (1999).
[Crossref]

Chen, X.

Chen, Y. F.

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

Cojocaru, C.

Deng, X.

N. An, Y. Zheng, H. Ren, X. Zhao, X. Deng, and X. Chen, “Normal, degenerated, and anomalous-dispersion-like Cerenkov sum-frequency generation in one nonlinear medium,” Photon. Res. 3(4), 106–109 (2015).
[Crossref]

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Surface phase-matched harmonic enhancement in a bulk anomalous dispersion medium,” Appl. Phys. Lett. 103(2), 021110 (2013).
[Crossref]

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov Radiation in an Anomalous Dispersive Medium,” Phys. Rev. Lett. 108(22), 223901 (2012).
[Crossref] [PubMed]

Duocastella, M.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
[Crossref]

Fischer, R.

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Fridrichova, M.

J. Kroupa and M. Fridrichova, “Spontaneous noncollinear second harmonic generation in GUHP,” J. Opt. 13(3), 035204 (2011).
[Crossref]

Gao, Y.

Gillespie, W. A.

Giordmaine, J. A.

J. A. Giordmaine, “Mixing of light beams in crystals,” Phys. Rev. Lett. 8(1), 19–20 (1962).
[Crossref]

Grant, S. D.

He, J. L.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ2 photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Hong, Q.

Huang, K. F.

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

Imlau, M.

K. Bastwöste, U. Sander, and M. Imlau, “Conical light scattering in strontium barium niobate crystals related to an intrinsic composition inhomogeneity,” J. Phys. Condens. Matter 19(15), 156225 (2007).
[Crossref]

Ja, Y. H.

Y. H. Ja, “A scattered ring in a natural crystal of tourmaline,” J. Opt. 21(1), 41–43 (1990).
[Crossref]

Jarutis, V.

Ji, S. H.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ2 photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Kalinowski, K.

Kalkandjiev, T. K.

Kasemir, K. U.

K. U. Kasemir and K. Betzler, “Characterization of photorefractive materials by spontaneous noncolinear frequency doubling,” Appl. Phys. B 68(5), 763–766 (1999).
[Crossref]

Kivshar, Y. S.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Y. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. 104(8), 083902 (2010).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Krolikowski, W.

V. Roppo, K. Kalinowski, Y. Sheng, W. Krolikowski, C. Cojocaru, and J. Trull, “Unified approach to Cerenkov second harmonic generation,” Opt. Express 21(22), 25715–25726 (2013).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Y. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. 104(8), 083902 (2010).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Kroupa, J.

J. Kroupa and M. Fridrichova, “Spontaneous noncollinear second harmonic generation in GUHP,” J. Opt. 13(3), 035204 (2011).
[Crossref]

Laurell, F.

Lee, S. L.

Li, M. C.

Li, T.

Loiko, Y. V.

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Conical refraction: fundamentals and applications,” Laser Photonics Rev. 1, 22 (2016).

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, J. Trull, C. Cojocaru, and J. Mompart, “Type I and type II second harmonic generation of conically refracted beams,” Opt. Lett. 38(14), 2484–2486 (2013).
[Crossref] [PubMed]

Lu, T. H.

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

Luo, Z.

Ming, N. B.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ2 photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Mompart, J.

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Conical refraction: fundamentals and applications,” Laser Photonics Rev. 1, 22 (2016).

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, J. Trull, C. Cojocaru, and J. Mompart, “Type I and type II second harmonic generation of conically refracted beams,” Opt. Lett. 38(14), 2484–2486 (2013).
[Crossref] [PubMed]

Neshev, D. N.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Y. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. 104(8), 083902 (2010).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Pašiškevicius, V.

Piskarskas, A.

Ren, H.

Roppo, V.

Saltiel, S. M.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Y. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. 104(8), 083902 (2010).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Sander, U.

K. Bastwöste, U. Sander, and M. Imlau, “Conical light scattering in strontium barium niobate crystals related to an intrinsic composition inhomogeneity,” J. Phys. Condens. Matter 19(15), 156225 (2007).
[Crossref]

Sheng, Y.

Smilgevicius, V.

Stabinis, A.

Su, K. W.

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

Sun, J.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ2 photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Tellefsen, J.

Trebino, R.

Trull, J.

Tsai, W. C.

Turpin, A.

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Conical refraction: fundamentals and applications,” Laser Photonics Rev. 1, 22 (2016).

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, J. Trull, C. Cojocaru, and J. Mompart, “Type I and type II second harmonic generation of conically refracted beams,” Opt. Lett. 38(14), 2484–2486 (2013).
[Crossref] [PubMed]

Voloch-Bloch, N.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Y. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. 104(8), 083902 (2010).
[Crossref] [PubMed]

Wang, H. T.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ2 photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Wang, S.

Wang, X.

Wu, S. T.

Xu, P.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ2 photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Yu, X. Q.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ2 photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Zhao, X.

Zheng, Y.

Zhu, R.

Zhu, S. N.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ2 photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Zhu, Y. Y.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ2 photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Zolotovskaya, S. A.

Appl. Opt. (1)

Appl. Phys. B (1)

K. U. Kasemir and K. Betzler, “Characterization of photorefractive materials by spontaneous noncolinear frequency doubling,” Appl. Phys. B 68(5), 763–766 (1999).
[Crossref]

Appl. Phys. Lett. (1)

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Surface phase-matched harmonic enhancement in a bulk anomalous dispersion medium,” Appl. Phys. Lett. 103(2), 021110 (2013).
[Crossref]

J. Opt. (2)

Y. H. Ja, “A scattered ring in a natural crystal of tourmaline,” J. Opt. 21(1), 41–43 (1990).
[Crossref]

J. Kroupa and M. Fridrichova, “Spontaneous noncollinear second harmonic generation in GUHP,” J. Opt. 13(3), 035204 (2011).
[Crossref]

J. Phys. Condens. Matter (1)

K. Bastwöste, U. Sander, and M. Imlau, “Conical light scattering in strontium barium niobate crystals related to an intrinsic composition inhomogeneity,” J. Phys. Condens. Matter 19(15), 156225 (2007).
[Crossref]

Laser Photonics Rev. (2)

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Conical refraction: fundamentals and applications,” Laser Photonics Rev. 1, 22 (2016).

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Photon. Res. (1)

Phys. Rev. Lett. (6)

J. A. Giordmaine, “Mixing of light beams in crystals,” Phys. Rev. Lett. 8(1), 19–20 (1962).
[Crossref]

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ2 photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Y. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. 104(8), 083902 (2010).
[Crossref] [PubMed]

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov Radiation in an Anomalous Dispersive Medium,” Phys. Rev. Lett. 108(22), 223901 (2012).
[Crossref] [PubMed]

Other (1)

R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier Singapore, 2010).

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Figures (8)

Fig. 1
Fig. 1 (a) Experimental setup for generation of SH conical waves. c is the optic axis, ϕ is the angle from the z axis to the optic axis, α is the cone angle (in air) of the SH conical wave, and β is the propagation (or central axis) direction (in air) of the SH conical wave relative to the z axis. (b) Evolution of the observed SH rings on the screen while the BBO crystal was rotated in the clockwise. The ϕ values are about (i) 21.6°, (ii) 22.5°, (iii) 23.5°, (iv) 24.5°, (v) 25.5°, and (vi) 26.5°.
Fig. 2
Fig. 2 (a) Schematic of the beamline inside the crystal for tunable SH conical wave generation. Plane and spherical FH waves are from line- and point-shaped sources, respectively. (b) Position-dependent polarization of the SH conical wave. The inset shows the projection of the polarizations in the normal plane of the SH conical wave. (c) Phase-matching diagram in the x–z plane. (d) Evolution of the phase-matching diagrams while the crystal is rotated in the clockwise around the y axis (c is the optic axis).
Fig. 3
Fig. 3 Phase-matching diagram in the x–y–z space. k1, k1 and k2 are wave vectors of the FH plane wave, FH spherical wave and SH conical wave. θx and θy are orientation angles of k2.
Fig. 4
Fig. 4 Intensity distribution of SH conical waves in two-dimensional space defined by orientation angles θx and θy. The ϕ values are (a) 21.6°, (b) 22.5°, (c) 23.5°, (d) 24.5°, (e) 25.5°, and (f) 26.5°. The orientation angles (θx and θy) are those determined inside the crystal. The interaction length of SHG is 0.5 mm. The spots illustrate the position of the collinear phase-matching SHG of the transmission FH plane wave.
Fig. 5
Fig. 5 Intensity distributions of SH conical waves in the (A, a) θx and (B, b) θy directions with variation of ϕ. (C, c) Cone angles (αx measured in the x–z plane and αy measured in the y–z plane) and the corresponding propagation directions of the SH conical waves (βx and βy) with variation of ϕ. Points in (c) are experiment results. θx, θy, αx, αy, βx, and βy were determined inside the crystal. The interaction length of SHG is 0.5 mm.
Fig. 6
Fig. 6 Longitudinal intensity variation of SH conical waves in the x–z plane. The ϕ values are (a) 21.6°, (b) 22.5°, (c) 23.5°, (d) 21.4°, (e) 21.2°, and (f) 21.0°. The orientation angle θx is that inside the crystal.
Fig. 7
Fig. 7 Vertical intensity distributions of SH waves for interaction lengths of (a) 0.2 mm, (b) 0.4 mm, (c) 0.6 mm, (d) 0.8 mm, (e) 1.0 mm, and (f) 1.2 mm. The ϕ value is 21.4°. The orientation angles θx and θy are those measured inside the crystal.
Fig. 8
Fig. 8 Measured evolution of conversion efficiency and cone angle (inside the crystal) with variation of ϕ.

Equations (6)

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cos k 2 ,c=cos θ y sin θ x sinϕ+cos θ y cos θ x cosϕ.
1 n k2 2 = ( sin k 2 ,c n e ) 2 + ( cos k 2 ,c n o ) 2 ,
Δk= k 1 2+2cos k 1 , k 1 ' k 2 .
A 2 = K g A 1 sin( gL ) e 1 2 iΔkL ,
K= 2i ω 2 2 d eff k 2 c 2 A 1 ',
g= 4 ω 1 2 ω 2 2 d eff 2 | A' | 2 k 1 k 2 c 4 + 1 4 Δ k 2 ,

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