Abstract

We present dynamics of spatial solitons propagating through a PT symmetric optical lattice with a longitudinal potential barrier. We find that a spatial soliton evolves a transverse drift motion after transmitting through the lattice barrier. The gain/loss coefficient of the PT symmetric potential barrier plays an essential role on such soliton dynamics. The bending angle of solitons depends on the lattice parameters including the modulation frequency, incident position, potential depth and the barrier length. Besides, solitons tend to gain a certain amount of energy from the barrier, which can also be tuned by barrier parameters.

© 2015 Optical Society of America

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References

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    [Crossref]
  2. C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Amer. J. Phys. 71, 1095 (2003).
    [Crossref]
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  4. K. G. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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2012 (6)

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

Y. He and D. Mihalache, “Spatial solitons in parity-time-symmetric mixed linear-nonlinear optical lattices: recent theoretical results,” Rom. Rep. Phys. 64, 1243 (2012).

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85, 043826 (2012).
[Crossref]

I. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Breathers in PT-symmetric optical couplers,” Phys. Rev. A 86, 053809 (2012).
[Crossref]

Y. He, D. Mihalache, X. Zhu, L. Guo, and Y. V. Kartashov, “Stable surface solitons in truncated complex potentials,” Opt. Lett. 37, 2526 (2012).
[Crossref] [PubMed]

2011 (3)

2010 (5)

2008 (3)

R. Yang and X. Wu, “Spatial soliton tunneling, compression and splitting,” Opt. Express 16, 17759 (2008).
[Crossref] [PubMed]

Z. Musslimani, K. Makris, R. El-Ganainy, and D. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

2006 (1)

2005 (2)

2004 (1)

2003 (3)

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710–712 (2003).
[Crossref] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[Crossref] [PubMed]

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Amer. J. Phys. 71, 1095 (2003).
[Crossref]

2001 (1)

M. J. Ablowitz and Z. H. Musslimani, “Discrete diffraction managed spatial solitons,” Phys. Rev. Lett. 87, 254102 (2001).
[Crossref] [PubMed]

2000 (1)

H. Eisenberg, Y. Silberberg, R. Morandotti, and J. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863 (2000).
[Crossref] [PubMed]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Ablowitz, M. J.

M. J. Ablowitz and Z. H. Musslimani, “Discrete diffraction managed spatial solitons,” Phys. Rev. Lett. 87, 254102 (2001).
[Crossref] [PubMed]

Aitchison, J.

H. Eisenberg, Y. Silberberg, R. Morandotti, and J. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863 (2000).
[Crossref] [PubMed]

Barashenkov, I.

I. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Breathers in PT-symmetric optical couplers,” Phys. Rev. A 86, 053809 (2012).
[Crossref]

Bender, C. M.

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Amer. J. Phys. 71, 1095 (2003).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Amer. J. Phys. 71, 1095 (2003).
[Crossref]

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[Crossref] [PubMed]

Chen, Z.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

Christodoulides, D.

Z. Musslimani, K. Makris, R. El-Ganainy, and D. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Christodoulides, D. N.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Y. V. Kartashov, L. Torner, and D. N. Christodoulides, “Soliton dragging by dynamic optical lattices,” Opt. Lett. 30, 1378 (2005).
[Crossref] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[Crossref] [PubMed]

Dmitriev, S. V.

I. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Breathers in PT-symmetric optical couplers,” Phys. Rev. A 86, 053809 (2012).
[Crossref]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, “Binary parity-time-symmetric nonlinear lattices with balanced gain and loss,” Opt. Lett. 35, 2976 (2010).
[Crossref] [PubMed]

Driben, R.

Efremidis, N. K.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[Crossref] [PubMed]

Eisenberg, H.

H. Eisenberg, Y. Silberberg, R. Morandotti, and J. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863 (2000).
[Crossref] [PubMed]

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Z. Musslimani, K. Makris, R. El-Ganainy, and D. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

Fleischer, J. W.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[Crossref] [PubMed]

Garanovich, I.

Guo, L.

Guo, Z.

Guo, Z. Y.

He, Y.

Y. He, D. Mihalache, X. Zhu, L. Guo, and Y. V. Kartashov, “Stable surface solitons in truncated complex potentials,” Opt. Lett. 37, 2526 (2012).
[Crossref] [PubMed]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

Y. He and D. Mihalache, “Spatial solitons in parity-time-symmetric mixed linear-nonlinear optical lattices: recent theoretical results,” Rom. Rep. Phys. 64, 1243 (2012).

Hu, S.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85, 043826 (2012).
[Crossref]

Hu, W.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85, 043826 (2012).
[Crossref]

Jiang, X.

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011).
[Crossref]

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Amer. J. Phys. 71, 1095 (2003).
[Crossref]

Kartashov, Y. V.

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Kivshar, Y.

Kivshar, Y. S.

I. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Breathers in PT-symmetric optical couplers,” Phys. Rev. A 86, 053809 (2012).
[Crossref]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, “Binary parity-time-symmetric nonlinear lattices with balanced gain and loss,” Opt. Lett. 35, 2976 (2010).
[Crossref] [PubMed]

Krolikowski, W.

Li, H.

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011).
[Crossref]

Liu, J.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

Liu, S.

Liu, S. T.

Lu, D.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85, 043826 (2012).
[Crossref]

Ma, X.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85, 043826 (2012).
[Crossref]

Makris, K.

Z. Musslimani, K. Makris, R. El-Ganainy, and D. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Malomed, B. A.

Mihalache, D.

Y. He, D. Mihalache, X. Zhu, L. Guo, and Y. V. Kartashov, “Stable surface solitons in truncated complex potentials,” Opt. Lett. 37, 2526 (2012).
[Crossref] [PubMed]

Y. He and D. Mihalache, “Spatial solitons in parity-time-symmetric mixed linear-nonlinear optical lattices: recent theoretical results,” Rom. Rep. Phys. 64, 1243 (2012).

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

Morandotti, R.

H. Eisenberg, Y. Silberberg, R. Morandotti, and J. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863 (2000).
[Crossref] [PubMed]

Musslimani, Z.

Z. Musslimani, K. Makris, R. El-Ganainy, and D. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

M. J. Ablowitz and Z. H. Musslimani, “Discrete diffraction managed spatial solitons,” Phys. Rev. Lett. 87, 254102 (2001).
[Crossref] [PubMed]

Neshev, D.

Ostrovskaya, E.

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[Crossref] [PubMed]

Shi, Z.

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011).
[Crossref]

Silberberg, Y.

H. Eisenberg, Y. Silberberg, R. Morandotti, and J. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863 (2000).
[Crossref] [PubMed]

Suchkov, S. V.

I. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Breathers in PT-symmetric optical couplers,” Phys. Rev. A 86, 053809 (2012).
[Crossref]

Sukhorukov, A.

Sukhorukov, A. A.

I. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Breathers in PT-symmetric optical couplers,” Phys. Rev. A 86, 053809 (2012).
[Crossref]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, “Binary parity-time-symmetric nonlinear lattices with balanced gain and loss,” Opt. Lett. 35, 2976 (2010).
[Crossref] [PubMed]

Torner, L.

Vysloukh, V. A.

Wang, H.

Wang, J.

Wu, X.

Yang, R.

Zheng, Y.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85, 043826 (2012).
[Crossref]

Zhou, K. Y.

Zhu, X.

Y. He, D. Mihalache, X. Zhu, L. Guo, and Y. V. Kartashov, “Stable surface solitons in truncated complex potentials,” Opt. Lett. 37, 2526 (2012).
[Crossref] [PubMed]

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011).
[Crossref]

Amer. J. Phys. (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Amer. J. Phys. 71, 1095 (2003).
[Crossref]

J. Opt. Soc. Am. B (1)

Nat. Phys. (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Opt. Express (3)

Opt. Lett. (8)

Phys. Rev. A (6)

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85, 043826 (2012).
[Crossref]

I. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Breathers in PT-symmetric optical couplers,” Phys. Rev. A 86, 053809 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Schematic Profile of a PT symmetric lattice potential. barrier described by Eq. (2)
Fig. 2
Fig. 2 Propagation of spatial soliton through lattice barriers with typical transverse modulation frequencies (a) Ωx = 2.4, (b) 2.8 and (c) 4; (d–f) Beam profiles of solitons at z = 0, zb, and 2zb accordingly. Other parameters of the barrier are set as ω0 = 1, p = 6 and δ = 1.
Fig. 3
Fig. 3 (a) Definition of bending angle; (b) Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on modulation frequency.
Fig. 4
Fig. 4 (a) Dependence of soliton bending angle(blue line) and emergent beam energy (red line) on the gain/loss coefficients; (b) and (c) are solitons propagating through lattice barriers with ω0 = 1 and 0, respectively. other lattice parameters are Ωx = 2 and p = 3, and δ = 1. (d) and (e) are beam profiles corresponding to the cases in (b) and (c).
Fig. 5
Fig. 5 Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on the incident position; (b–c) are soliton propagation through the lattice barrier at x0 = 0.4T and 0.3T respectively, other parameters are Ωx = 2.4, ω0 = 1, A = 1, p = 6, δ = 1; (d) and (e) are the beam profiles corresponds to the cases of (b) and (c), respectively.
Fig. 6
Fig. 6 Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on the potential depth; (b–c) are soliton propagation through the lattice barrier at p = 2 and 4 respectively, other parameters are Ωx = 2.4, ω0 = 1, A = 1, x0 = 0, δ = 1; (d) and (e) are the beam profiles corresponds to the cases of (b) and (c), respectively.
Fig. 7
Fig. 7 Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on slope rate δ of the barrier; (b–c) are soliton propagation through the lattice barrier at δ = 2 and δ = 4, respectively, other parameters are Ωx = 2.4, ω0 = 1, p = 6, A = 1, x0 = 0.;(d) and (e) are the beam profiles corresponds to the cases of (b) and (c), respectively.

Equations (4)

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i q z = 1 2 2 q 2 x q | q | 2 p R ( x , z ) q ,
R ( x , z ) = { V ( x ) exp [ δ ( z z b ) ] z b z 2 z b . V ( x ) exp [ δ ( z z b ) ] 0 z z b ,
q ( x , 0 ) = A sech [ A ( x x 0 ) ] ,
P 0 = + | q ( x , 0 ) | 2 d x = 2 A .

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