Abstract

We study the evolution of dark spatial solitons by χ(2)(2) cascaded nonlinearity. The generation and evolution of solitons pairs by the optical branching effect is investigated. It is shown that the number of solitons pairs and the transverse velocity of a given pair depend on the parameters of the input beam. We also investigate the hollows generated in a beam with a uniform background when the two sides of the input beam have various phase difference. It is concluded that dark spatial solitons by χ(2)(2) cascaded nonlinearity have analog performance compared with that in χ(3) defocusing Kerr media.

©2005 Optical Society of America

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  6. M. Segev and G. Stegeman , “ Self-trapping of optical beams: spatial solitons ,” Phys. Today   51 , 42 – 48 ( 1998 ).
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    [Crossref]
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    [Crossref]
  11. A. D. Boardman and K. Xie , “ Spatial bright-dark soliton steering through waveguide coupling ,” Opt. Quantum. Electron.   30 , 783 – 794 ( 1998 ).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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  30. H. Kim , L. Jankovic , and G. I. Stegeman , “ Quadratic spatial solitons in periodically poled KTiOPO4 ,” Opt. Lett.   28 , 640 – 642 ( 2003 ).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  32. S. A. Gredeskul and Yu. S. Kivshar , “ Dark-pulse solitons in nonlinear-optical fibers ,” Phys. Rev. A   41 , 3994 – 4008 ( 1990 ).
    [Crossref] [PubMed]
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    [Crossref]

2005 (1)

G. Bartal , O. Manela , O. Cohen , J. W. Fleischer , and M. Segev , “ Observation of 2nd-band vortex solitons in 2D photonic lattices ,” Phys. Rev. Lett.   95 , 053904 ( 2005 ).
[Crossref] [PubMed]

2004 (4)

D. S. Jiang , S. G. Ouyang , and W. L. She , “ Interaction of photovotaic spatial solitons between dark and dark as well as dark and bright ,” Acta. Phys. Sin.   53 , 3777 – 3785 ( 2004 ).

S. G. Ouyang , X. S. Wang , and W. L. She , “ Interaction between photovotaic spatial solitons with difference wavelengths ,” Acta. Phys. Sin.   53 , 761 – 772 ( 2004 ).

Y. Q. Xie and Q. Guo , “ Interaction between optical spatial solitons in nonlocal Kerr media ,” Acta. Phys. Sin.   53 , 3020 – 3024 ( 2004 )

Y. V. Kartashov , L. Torner , and V. A. Vysloukh , “ Parametric amplification of soliton steering in optical lattices ,” Opt. Lett.   29 , 10 – 13 ( 2004 ).
[Crossref]

2003 (2)

N. Nikolov , D. Neshev , and O. Bang , “ Quadratic solitons as nonlocal solitons ,” Phys. Rev. E   68 , 036614 ( 2003 ).
[Crossref]

H. Kim , L. Jankovic , and G. I. Stegeman , “ Quadratic spatial solitons in periodically poled KTiOPO4 ,” Opt. Lett.   28 , 640 – 642 ( 2003 ).
[Crossref] [PubMed]

2002 (1)

2000 (2)

G. I. Stegeman , D. N. Christodoulides , and M. Segev , “ Optical spatial solitons: historical perspectives ,” Invited Paper, Special Millennium Issue of the IEEE J. Sel Top. Quantum. Electron.   6 , 1419 – 1427 ( 2000 ).
[Crossref]

A. D. Boardman , K. Marinov , D. I. Pushkarov , and A. Shivarova , “ Wave-beam coupling in quadratic nonlinear optical waveguides: effects of nonlinearly induced diffraction ,” Phys. Rev. E.   62 , 2871 – 2877 ( 2000 ).
[Crossref]

1999 (3)

1998 (6)

N. N. Akhmediev , “ Spatial solitons in Kerr and Kerr-like Media ,” Opt. Quantum. Electron.   30 , 535 – 569 ( 1998 ).
[Crossref]

M. Segev , “ Optical spatial solitons ,” Opt. Quantum. Electron.   30 , 503 – 533 ( 1998 ).
[Crossref]

A. D. Boardman and K. Xie , “ Spatial bright-dark soliton steering through waveguide coupling ,” Opt. Quantum. Electron.   30 , 783 – 794 ( 1998 ).
[Crossref]

Y. S. Kivshar and B. Luther-Davies , “ Dark optical solitons: physics and applications ,” Phys. Rep.   298 , 81 – 197 ( 1998 ).
[Crossref]

A. D. Boardman , L. Pavlov , and S. Tanev , Advanced Photonics with Second-order Optical Nonlinear Processes ( Kluwer , 1998 ).

M. Segev and G. Stegeman , “ Self-trapping of optical beams: spatial solitons ,” Phys. Today   51 , 42 – 48 ( 1998 ).
[Crossref]

1997 (2)

C. B. Clausen , O. Bang , and Y. S. Kivshar , “ Spatial solitons and Induced Kerr effects in quasi-phase-matched quadratic media ,” Phys. Rev. Lett.   78 , 4749 – 4752 ( 1997 ).
[Crossref]

P. Vidakovic , D. J. Lovering , and J. A. Levenson , “ Large nonlinear phase shift owing to cascaded χ (2) in quasi-phase-matched bulk LiNbO 3 ,” Opt. Lett.   22 , 277 – 229 ( 1997 ).
[Crossref]

1995 (2)

W. E. Torruellas , Z. Wang , D. J. Hagan , E. W. VanStryland , and G. I. Stegeman , “ Observation of two-dimensional spatial solitary waves in quadratic midium ,” Phys. Rev. Lett.   74 , 5036 – 5039 ( 1995 ).
[Crossref] [PubMed]

L. Torner , C. R. Menyuk , W. E. Torruellas , and G. I. Stegeman , “ Two-dimensional solitons with second-order nonlinearities ,” Opt. Lett.   20 , 13 – 15 ( 1995 ).
[Crossref] [PubMed]

1994 (1)

C. R. Menyuk , R. Schiek , and L. Torner , “ Solitary waves due to χ (2) : χ (2) cascading ,” Opt. Soc. Am. B   11 , 2434 – 2443 ( 1994 ).
[Crossref]

1992 (1)

1990 (3)

1987 (1)

1976 (1)

V. E. Zakharov and S. V. Manakov “ The theory of resonance interaction of wave packets in nonlinear media ,” Sov. Phys. JETP   42 , 842 – 850 ( 1976 ).

1972 (1)

V. E. Zakharov and A. B. Shabat , “ Exact Theory of two-dimensional self-focusing and one dimensional self-modulation of waves in nonlinear media ,” Sov. Phys. JETP   34 , 62 – 69 ( 1972 ).

1967 (1)

L. A. Ostrovskii , “ Self action of light in crystals ,” JETP Lett.   5 , 272 – 275 ( 1967 ).

Agrawal, G. P.

G. P. Agrawal , Nonlinear Fiber Optics ( Academic 1995 ).

Y. S. Kivshar and G. P. Agrawal , Optical Solitons ( Academic , 2003 ).

Aitchison, J. S.

Akhmediev, N. N.

N. N. Akhmediev , “ Spatial solitons in Kerr and Kerr-like Media ,” Opt. Quantum. Electron.   30 , 535 – 569 ( 1998 ).
[Crossref]

Andersen, D. R.

-Andre Belanger, P.

Assanto, G.

Bang, O.

N. Nikolov , D. Neshev , and O. Bang , “ Quadratic solitons as nonlocal solitons ,” Phys. Rev. E   68 , 036614 ( 2003 ).
[Crossref]

C. B. Clausen , O. Bang , and Y. S. Kivshar , “ Spatial solitons and Induced Kerr effects in quasi-phase-matched quadratic media ,” Phys. Rev. Lett.   78 , 4749 – 4752 ( 1997 ).
[Crossref]

Bartal, G.

G. Bartal , O. Manela , O. Cohen , J. W. Fleischer , and M. Segev , “ Observation of 2nd-band vortex solitons in 2D photonic lattices ,” Phys. Rev. Lett.   95 , 053904 ( 2005 ).
[Crossref] [PubMed]

Barthelemy, A.

Boardman, A. D.

A. D. Boardman , K. Marinov , D. I. Pushkarov , and A. Shivarova , “ Wave-beam coupling in quadratic nonlinear optical waveguides: effects of nonlinearly induced diffraction ,” Phys. Rev. E.   62 , 2871 – 2877 ( 2000 ).
[Crossref]

A. D. Boardman , L. Pavlov , and S. Tanev , Advanced Photonics with Second-order Optical Nonlinear Processes ( Kluwer , 1998 ).

A. D. Boardman and K. Xie , “ Spatial bright-dark soliton steering through waveguide coupling ,” Opt. Quantum. Electron.   30 , 783 – 794 ( 1998 ).
[Crossref]

Bourliaguet, B.

Christodoulides, D. N.

G. I. Stegeman , D. N. Christodoulides , and M. Segev , “ Optical spatial solitons: historical perspectives ,” Invited Paper, Special Millennium Issue of the IEEE J. Sel Top. Quantum. Electron.   6 , 1419 – 1427 ( 2000 ).
[Crossref]

Clausen, C. B.

C. B. Clausen and L. Torner , “ Spatial switching of quadratic solitons in engineered quasi-phase-matched structures ,” Opt. Lett.   24 , 7 – 10 ( 1999 ).
[Crossref]

C. B. Clausen , O. Bang , and Y. S. Kivshar , “ Spatial solitons and Induced Kerr effects in quasi-phase-matched quadratic media ,” Phys. Rev. Lett.   78 , 4749 – 4752 ( 1997 ).
[Crossref]

Cohen, O.

G. Bartal , O. Manela , O. Cohen , J. W. Fleischer , and M. Segev , “ Observation of 2nd-band vortex solitons in 2D photonic lattices ,” Phys. Rev. Lett.   95 , 053904 ( 2005 ).
[Crossref] [PubMed]

Couderc, V.

Fleischer, J. W.

G. Bartal , O. Manela , O. Cohen , J. W. Fleischer , and M. Segev , “ Observation of 2nd-band vortex solitons in 2D photonic lattices ,” Phys. Rev. Lett.   95 , 053904 ( 2005 ).
[Crossref] [PubMed]

Gredeskul, S. A.

S. A. Gredeskul and Yu. S. Kivshar , “ Dark-pulse solitons in nonlinear-optical fibers ,” Phys. Rev. A   41 , 3994 – 4008 ( 1990 ).
[Crossref] [PubMed]

Guo, Q.

Y. Q. Xie and Q. Guo , “ Interaction between optical spatial solitons in nonlocal Kerr media ,” Acta. Phys. Sin.   53 , 3020 – 3024 ( 2004 )

Hagan, D. J.

W. E. Torruellas , Z. Wang , D. J. Hagan , E. W. VanStryland , and G. I. Stegeman , “ Observation of two-dimensional spatial solitary waves in quadratic midium ,” Phys. Rev. Lett.   74 , 5036 – 5039 ( 1995 ).
[Crossref] [PubMed]

Hooton, D. E.

Jankovic, L.

Jiang, D. S.

D. S. Jiang , S. G. Ouyang , and W. L. She , “ Interaction of photovotaic spatial solitons between dark and dark as well as dark and bright ,” Acta. Phys. Sin.   53 , 3777 – 3785 ( 2004 ).

Kartashov, Y. V.

Y. V. Kartashov , L. Torner , and V. A. Vysloukh , “ Parametric amplification of soliton steering in optical lattices ,” Opt. Lett.   29 , 10 – 13 ( 2004 ).
[Crossref]

Kim, H.

Kivshar, Y. S.

Y. S. Kivshar and B. Luther-Davies , “ Dark optical solitons: physics and applications ,” Phys. Rep.   298 , 81 – 197 ( 1998 ).
[Crossref]

C. B. Clausen , O. Bang , and Y. S. Kivshar , “ Spatial solitons and Induced Kerr effects in quasi-phase-matched quadratic media ,” Phys. Rev. Lett.   78 , 4749 – 4752 ( 1997 ).
[Crossref]

Y. S. Kivshar and G. P. Agrawal , Optical Solitons ( Academic , 2003 ).

Kivshar, Yu. S.

S. A. Gredeskul and Yu. S. Kivshar , “ Dark-pulse solitons in nonlinear-optical fibers ,” Phys. Rev. A   41 , 3994 – 4008 ( 1990 ).
[Crossref] [PubMed]

Levenson, J. A.

Lovering, D. J.

Luther-Davies, B.

Manakov, S. V.

V. E. Zakharov and S. V. Manakov “ The theory of resonance interaction of wave packets in nonlinear media ,” Sov. Phys. JETP   42 , 842 – 850 ( 1976 ).

Manela, O.

G. Bartal , O. Manela , O. Cohen , J. W. Fleischer , and M. Segev , “ Observation of 2nd-band vortex solitons in 2D photonic lattices ,” Phys. Rev. Lett.   95 , 053904 ( 2005 ).
[Crossref] [PubMed]

Marinov, K.

A. D. Boardman , K. Marinov , D. I. Pushkarov , and A. Shivarova , “ Wave-beam coupling in quadratic nonlinear optical waveguides: effects of nonlinearly induced diffraction ,” Phys. Rev. E.   62 , 2871 – 2877 ( 2000 ).
[Crossref]

Mathieu, Pierre

Menyuk, C. R.

L. Torner , C. R. Menyuk , W. E. Torruellas , and G. I. Stegeman , “ Two-dimensional solitons with second-order nonlinearities ,” Opt. Lett.   20 , 13 – 15 ( 1995 ).
[Crossref] [PubMed]

C. R. Menyuk , R. Schiek , and L. Torner , “ Solitary waves due to χ (2) : χ (2) cascading ,” Opt. Soc. Am. B   11 , 2434 – 2443 ( 1994 ).
[Crossref]

Neshev, D.

N. Nikolov , D. Neshev , and O. Bang , “ Quadratic solitons as nonlocal solitons ,” Phys. Rev. E   68 , 036614 ( 2003 ).
[Crossref]

Nikolov, N.

N. Nikolov , D. Neshev , and O. Bang , “ Quadratic solitons as nonlocal solitons ,” Phys. Rev. E   68 , 036614 ( 2003 ).
[Crossref]

Ostrovskii, L. A.

L. A. Ostrovskii , “ Self action of light in crystals ,” JETP Lett.   5 , 272 – 275 ( 1967 ).

Ouyang, S. G.

D. S. Jiang , S. G. Ouyang , and W. L. She , “ Interaction of photovotaic spatial solitons between dark and dark as well as dark and bright ,” Acta. Phys. Sin.   53 , 3777 – 3785 ( 2004 ).

S. G. Ouyang , X. S. Wang , and W. L. She , “ Interaction between photovotaic spatial solitons with difference wavelengths ,” Acta. Phys. Sin.   53 , 761 – 772 ( 2004 ).

Pavlov, L.

A. D. Boardman , L. Pavlov , and S. Tanev , Advanced Photonics with Second-order Optical Nonlinear Processes ( Kluwer , 1998 ).

Pushkarov, D. I.

A. D. Boardman , K. Marinov , D. I. Pushkarov , and A. Shivarova , “ Wave-beam coupling in quadratic nonlinear optical waveguides: effects of nonlinearly induced diffraction ,” Phys. Rev. E.   62 , 2871 – 2877 ( 2000 ).
[Crossref]

Schiek, R.

C. R. Menyuk , R. Schiek , and L. Torner , “ Solitary waves due to χ (2) : χ (2) cascading ,” Opt. Soc. Am. B   11 , 2434 – 2443 ( 1994 ).
[Crossref]

Segev, M.

G. Bartal , O. Manela , O. Cohen , J. W. Fleischer , and M. Segev , “ Observation of 2nd-band vortex solitons in 2D photonic lattices ,” Phys. Rev. Lett.   95 , 053904 ( 2005 ).
[Crossref] [PubMed]

G. I. Stegeman , D. N. Christodoulides , and M. Segev , “ Optical spatial solitons: historical perspectives ,” Invited Paper, Special Millennium Issue of the IEEE J. Sel Top. Quantum. Electron.   6 , 1419 – 1427 ( 2000 ).
[Crossref]

G. I. Stegeman and M. Segev , “ Optical spatial solitons and their interactions: universality and diversity ,” Science   286 , 1518 – 1523 ( 1999 ).
[Crossref] [PubMed]

M. Segev and G. Stegeman , “ Self-trapping of optical beams: spatial solitons ,” Phys. Today   51 , 42 – 48 ( 1998 ).
[Crossref]

M. Segev , “ Optical spatial solitons ,” Opt. Quantum. Electron.   30 , 503 – 533 ( 1998 ).
[Crossref]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat , “ Exact Theory of two-dimensional self-focusing and one dimensional self-modulation of waves in nonlinear media ,” Sov. Phys. JETP   34 , 62 – 69 ( 1972 ).

She, W. L.

D. S. Jiang , S. G. Ouyang , and W. L. She , “ Interaction of photovotaic spatial solitons between dark and dark as well as dark and bright ,” Acta. Phys. Sin.   53 , 3777 – 3785 ( 2004 ).

S. G. Ouyang , X. S. Wang , and W. L. She , “ Interaction between photovotaic spatial solitons with difference wavelengths ,” Acta. Phys. Sin.   53 , 761 – 772 ( 2004 ).

Shivarova, A.

A. D. Boardman , K. Marinov , D. I. Pushkarov , and A. Shivarova , “ Wave-beam coupling in quadratic nonlinear optical waveguides: effects of nonlinearly induced diffraction ,” Phys. Rev. E.   62 , 2871 – 2877 ( 2000 ).
[Crossref]

Stegeman, G.

M. Segev and G. Stegeman , “ Self-trapping of optical beams: spatial solitons ,” Phys. Today   51 , 42 – 48 ( 1998 ).
[Crossref]

Stegeman, G. I.

H. Kim , L. Jankovic , and G. I. Stegeman , “ Quadratic spatial solitons in periodically poled KTiOPO4 ,” Opt. Lett.   28 , 640 – 642 ( 2003 ).
[Crossref] [PubMed]

G. Assanto and G. I. Stegeman , “ Simple physics of quadratic spatial solitons ,” Opt. Express   10 , 388 – 396 ( 2002 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-9-388 .
[PubMed]

G. I. Stegeman , D. N. Christodoulides , and M. Segev , “ Optical spatial solitons: historical perspectives ,” Invited Paper, Special Millennium Issue of the IEEE J. Sel Top. Quantum. Electron.   6 , 1419 – 1427 ( 2000 ).
[Crossref]

G. I. Stegeman and M. Segev , “ Optical spatial solitons and their interactions: universality and diversity ,” Science   286 , 1518 – 1523 ( 1999 ).
[Crossref] [PubMed]

W. E. Torruellas , Z. Wang , D. J. Hagan , E. W. VanStryland , and G. I. Stegeman , “ Observation of two-dimensional spatial solitary waves in quadratic midium ,” Phys. Rev. Lett.   74 , 5036 – 5039 ( 1995 ).
[Crossref] [PubMed]

L. Torner , C. R. Menyuk , W. E. Torruellas , and G. I. Stegeman , “ Two-dimensional solitons with second-order nonlinearities ,” Opt. Lett.   20 , 13 – 15 ( 1995 ).
[Crossref] [PubMed]

Swartzlander, G. A.

Tanev, S.

A. D. Boardman , L. Pavlov , and S. Tanev , Advanced Photonics with Second-order Optical Nonlinear Processes ( Kluwer , 1998 ).

Torner, L.

Y. V. Kartashov , L. Torner , and V. A. Vysloukh , “ Parametric amplification of soliton steering in optical lattices ,” Opt. Lett.   29 , 10 – 13 ( 2004 ).
[Crossref]

C. B. Clausen and L. Torner , “ Spatial switching of quadratic solitons in engineered quasi-phase-matched structures ,” Opt. Lett.   24 , 7 – 10 ( 1999 ).
[Crossref]

L. Torner , C. R. Menyuk , W. E. Torruellas , and G. I. Stegeman , “ Two-dimensional solitons with second-order nonlinearities ,” Opt. Lett.   20 , 13 – 15 ( 1995 ).
[Crossref] [PubMed]

C. R. Menyuk , R. Schiek , and L. Torner , “ Solitary waves due to χ (2) : χ (2) cascading ,” Opt. Soc. Am. B   11 , 2434 – 2443 ( 1994 ).
[Crossref]

Torruellas, W. E.

W. E. Torruellas , Z. Wang , D. J. Hagan , E. W. VanStryland , and G. I. Stegeman , “ Observation of two-dimensional spatial solitary waves in quadratic midium ,” Phys. Rev. Lett.   74 , 5036 – 5039 ( 1995 ).
[Crossref] [PubMed]

L. Torner , C. R. Menyuk , W. E. Torruellas , and G. I. Stegeman , “ Two-dimensional solitons with second-order nonlinearities ,” Opt. Lett.   20 , 13 – 15 ( 1995 ).
[Crossref] [PubMed]

VanStryland, E. W.

W. E. Torruellas , Z. Wang , D. J. Hagan , E. W. VanStryland , and G. I. Stegeman , “ Observation of two-dimensional spatial solitary waves in quadratic midium ,” Phys. Rev. Lett.   74 , 5036 – 5039 ( 1995 ).
[Crossref] [PubMed]

Vidakovic, P.

Vysloukh, V. A.

Y. V. Kartashov , L. Torner , and V. A. Vysloukh , “ Parametric amplification of soliton steering in optical lattices ,” Opt. Lett.   29 , 10 – 13 ( 2004 ).
[Crossref]

Wang, X. S.

S. G. Ouyang , X. S. Wang , and W. L. She , “ Interaction between photovotaic spatial solitons with difference wavelengths ,” Acta. Phys. Sin.   53 , 761 – 772 ( 2004 ).

Wang, Z.

W. E. Torruellas , Z. Wang , D. J. Hagan , E. W. VanStryland , and G. I. Stegeman , “ Observation of two-dimensional spatial solitary waves in quadratic midium ,” Phys. Rev. Lett.   74 , 5036 – 5039 ( 1995 ).
[Crossref] [PubMed]

Xie, K.

A. D. Boardman and K. Xie , “ Spatial bright-dark soliton steering through waveguide coupling ,” Opt. Quantum. Electron.   30 , 783 – 794 ( 1998 ).
[Crossref]

Xie, Y. Q.

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Figures (5)

Fig. 1.
Fig. 1. (a) Input box-like pulse (partial) (A=6); evolution of box-like dark soliton in cascaded media (A=6 β=-25 ) at diffraction length of n =15(b), 20(c), and 35(d). After about 30 diffraction length, the number of soliton pairs (3 pairs for A=6) does not increase further.
Fig. 2.
Fig. 2. Steady state of optical branching effect with input normalized amplitude of 2(a), 4(b), and 7(c). The required diffraction lengths are 100(a), 80(b), and 60(c), respectively, and numbers of soliton pairs are 1(a), 3(b), and 4(c).
Fig. 3.
Fig. 3. Steady state of optical branching effect with initial width of ω/2 (a), ω(b), 2ω(c), and 4ω(d).
Fig. 4.
Fig. 4. Relation between the y and the displacements of the first-order pairs at 100 diffraction lengths.
Fig. 5.
Fig. 5. Generation of dark solitons in a uniform background. (a) Input uniform background, (b) hollow generated in the center of background with a phase difference of π at two edges of the uniform background, (c) hollow generated at left side of the background when the left side of the hollow has a phase of +π/2 ahead of the right side, (d) hollow generated at the right side of the background when the left side of the hollow has a phase of +3π/2 ahead of the right side.

Equations (5)

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i a 1 ξ + 1 2 2 a 1 + d ( ξ ) a 1 * a 2 exp ( iβξ ) = 0 i a 2 ξ + α 2 2 a 2 i δ 2 a 2 + d ( ξ ) a 1 2 exp ( iβξ ) = 0 ,
{ A = 0 x < a A = 1 x > a .
{ A exp [ ( x + 400 ) 2 400 ] 500 < x < 400 A 400 < x < 2 π A 2 [ cos ( x 2 + π ) + 1 ] 2 π < x < 2 π A 2 π < x < 400 A exp [ ( x 400 ) 2 400 ] 400 < x < 500 .
λ n = v ( 2 n 0 2 n 2 E 0 2 ) 1 2 ,
{ A e ( x + 400 ) 2 / 400 500 < x < 400 A 400 < x < 400 A e ( x 400 ) 2 / 400 400 < x < 500 ,

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