Abstract

We theoretically investigate the properties of vector and scalar phonon–polariton cnoidal waves and spatial solitons propagating in a boundless dielectric medium. We obtain analytically the expressions for the envelopes of linearly and circularly polarized nonlinear polariton waves in self-focusing and self-defocusing media. The expressions of spatial solitons and cnoidal waves describe one and several flat flows of polaritons for the linearly polarized wave, respectively. The equation for a right or left circularly polarized polariton scalar wave has a soliton solution and a cnoidal wave solution. The polariton vector wave with right and left polariton spiralities has an analytical solution too. Also we examine the linearly and circularly polarized polariton wave instability in the nonlinear dielectric medium.

© 2013 Optical Society of America

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  1. C. Kittel, Quantum Theory of Solids (Wiley, 1963).
  2. O. Madelung, Festkörpertheorie, I, II (Springer-Verlag, 1972).
  3. I. V. Dzedolik, “One-dimensional controllable photonic crystal,” J. Opt. Soc. Am. B 24, 2741–2745 (2007).
    [CrossRef]
  4. I. V. Dzedolik and J. P. Mikulska, “Polariton spectrum control in dielectric medium,” in Proceedings 2nd IEEE International Workshop on Thz Radiation (TERA) (IEEE, 2010), pp. 288–290.
  5. D. N. Klyshko, Quantum and Nonlinear Optics (Nauka, 1980) (in Russian).
  6. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).
  7. I. V. Dzedolik, Polaritons in Optical Fibers and Dielectric Resonators (DIP, 2007) (in Russian).
  8. I. V. Dzedolik, “Period variation of polariton waves in optical fiber,” J. Opt. Pure Appl. Opt. 11, 094012 (2009).
    [CrossRef]
  9. I. V. Dzedolik and S. N. Lapayeva, “Mass of polariton in different dielectric media,” J. Opt. 13, 015204 (2011).
    [CrossRef]
  10. I. V. Dzedolik and O. S. Karakchieva, “Polaritons in nonlinear medium: generation, propagation and interaction,” in Proceedings of International Workshop on Nonlinear Photonics (NLP) (IEEE, 2011), pp. 1–3.
  11. N. Bloembergen, Nonlinear Optics (W. A. Benjamin, 1965).
  12. A. P. Sukhorukov, Nonlinear Wave Interactions in Optics and Radiophysics (Nauka, 1988) (in Russian).
  13. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  14. R. W. Boyd, Nonlinear Optics (Academic, 2003).
  15. J. Xu, V. Shandarov, M. Wesner, and D. Kip, “Observation of two-dimensional spatial solitons in iron-doped barium-calcium titanate crystals,” Phys. Status Solidi. A, Appl. Res. 189, R4–R5 (2002).
    [CrossRef]
  16. I. V. Dzedolik, “Transformation of sinusoidal electromagnetic and polarization waves into cnoidal waves in an optical fibre,” Ukr. J. Phys. Opt. 9, 226–235 (2008).
    [CrossRef]
  17. S. Ouyang and Q. Guo, “Dark and gray spatial optical solitons in Kerr-type nonlocal media,” Opt. Express 17, 5170–5175 (2009).
    [CrossRef]
  18. S. Zhong, C. Huang, C. Li, and L. Dong, “Surface defect kink solitons,” Opt. Commun. 285, 3674–3678 (2012).
    [CrossRef]
  19. D. Buccoliero and A. S. Desyatnikov, “Quasi-periodic transformations of nonlocal spatial solitons,” Opt. Express 17, 9608–9613 (2009).
    [CrossRef]
  20. J. D. Jackson, Classical Electrodynamics (Wiley, 1962).
  21. G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).
  22. S. Santran, L. Cathoni, T. Cardinal, E. Farginc, G. Le Flem, C. Rouyera, and L. Sarger, “Precise and absolute measurements of the complex third-order optical susceptibility,” Proc. SPIE 4106, 349 (2000).
  23. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–26 (2007).

2012 (1)

S. Zhong, C. Huang, C. Li, and L. Dong, “Surface defect kink solitons,” Opt. Commun. 285, 3674–3678 (2012).
[CrossRef]

2011 (1)

I. V. Dzedolik and S. N. Lapayeva, “Mass of polariton in different dielectric media,” J. Opt. 13, 015204 (2011).
[CrossRef]

2009 (3)

2008 (1)

I. V. Dzedolik, “Transformation of sinusoidal electromagnetic and polarization waves into cnoidal waves in an optical fibre,” Ukr. J. Phys. Opt. 9, 226–235 (2008).
[CrossRef]

2007 (2)

I. V. Dzedolik, “One-dimensional controllable photonic crystal,” J. Opt. Soc. Am. B 24, 2741–2745 (2007).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–26 (2007).

2002 (1)

J. Xu, V. Shandarov, M. Wesner, and D. Kip, “Observation of two-dimensional spatial solitons in iron-doped barium-calcium titanate crystals,” Phys. Status Solidi. A, Appl. Res. 189, R4–R5 (2002).
[CrossRef]

2000 (1)

S. Santran, L. Cathoni, T. Cardinal, E. Farginc, G. Le Flem, C. Rouyera, and L. Sarger, “Precise and absolute measurements of the complex third-order optical susceptibility,” Proc. SPIE 4106, 349 (2000).

Agrawal, G. P.

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

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (W. A. Benjamin, 1965).

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Buccoliero, D.

Cardinal, T.

S. Santran, L. Cathoni, T. Cardinal, E. Farginc, G. Le Flem, C. Rouyera, and L. Sarger, “Precise and absolute measurements of the complex third-order optical susceptibility,” Proc. SPIE 4106, 349 (2000).

Cathoni, L.

S. Santran, L. Cathoni, T. Cardinal, E. Farginc, G. Le Flem, C. Rouyera, and L. Sarger, “Precise and absolute measurements of the complex third-order optical susceptibility,” Proc. SPIE 4106, 349 (2000).

Desyatnikov, A. S.

Dong, L.

S. Zhong, C. Huang, C. Li, and L. Dong, “Surface defect kink solitons,” Opt. Commun. 285, 3674–3678 (2012).
[CrossRef]

Dzedolik, I. V.

I. V. Dzedolik and S. N. Lapayeva, “Mass of polariton in different dielectric media,” J. Opt. 13, 015204 (2011).
[CrossRef]

I. V. Dzedolik, “Period variation of polariton waves in optical fiber,” J. Opt. Pure Appl. Opt. 11, 094012 (2009).
[CrossRef]

I. V. Dzedolik, “Transformation of sinusoidal electromagnetic and polarization waves into cnoidal waves in an optical fibre,” Ukr. J. Phys. Opt. 9, 226–235 (2008).
[CrossRef]

I. V. Dzedolik, “One-dimensional controllable photonic crystal,” J. Opt. Soc. Am. B 24, 2741–2745 (2007).
[CrossRef]

I. V. Dzedolik and J. P. Mikulska, “Polariton spectrum control in dielectric medium,” in Proceedings 2nd IEEE International Workshop on Thz Radiation (TERA) (IEEE, 2010), pp. 288–290.

I. V. Dzedolik, Polaritons in Optical Fibers and Dielectric Resonators (DIP, 2007) (in Russian).

I. V. Dzedolik and O. S. Karakchieva, “Polaritons in nonlinear medium: generation, propagation and interaction,” in Proceedings of International Workshop on Nonlinear Photonics (NLP) (IEEE, 2011), pp. 1–3.

Farginc, E.

S. Santran, L. Cathoni, T. Cardinal, E. Farginc, G. Le Flem, C. Rouyera, and L. Sarger, “Precise and absolute measurements of the complex third-order optical susceptibility,” Proc. SPIE 4106, 349 (2000).

Guo, Q.

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–26 (2007).

Huang, C.

S. Zhong, C. Huang, C. Li, and L. Dong, “Surface defect kink solitons,” Opt. Commun. 285, 3674–3678 (2012).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1962).

Karakchieva, O. S.

I. V. Dzedolik and O. S. Karakchieva, “Polaritons in nonlinear medium: generation, propagation and interaction,” in Proceedings of International Workshop on Nonlinear Photonics (NLP) (IEEE, 2011), pp. 1–3.

Kip, D.

J. Xu, V. Shandarov, M. Wesner, and D. Kip, “Observation of two-dimensional spatial solitons in iron-doped barium-calcium titanate crystals,” Phys. Status Solidi. A, Appl. Res. 189, R4–R5 (2002).
[CrossRef]

Kittel, C.

C. Kittel, Quantum Theory of Solids (Wiley, 1963).

Kivshar, Y. S.

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

Klyshko, D. N.

D. N. Klyshko, Quantum and Nonlinear Optics (Nauka, 1980) (in Russian).

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).

Lapayeva, S. N.

I. V. Dzedolik and S. N. Lapayeva, “Mass of polariton in different dielectric media,” J. Opt. 13, 015204 (2011).
[CrossRef]

Le Flem, G.

S. Santran, L. Cathoni, T. Cardinal, E. Farginc, G. Le Flem, C. Rouyera, and L. Sarger, “Precise and absolute measurements of the complex third-order optical susceptibility,” Proc. SPIE 4106, 349 (2000).

Li, C.

S. Zhong, C. Huang, C. Li, and L. Dong, “Surface defect kink solitons,” Opt. Commun. 285, 3674–3678 (2012).
[CrossRef]

Madelung, O.

O. Madelung, Festkörpertheorie, I, II (Springer-Verlag, 1972).

Mikulska, J. P.

I. V. Dzedolik and J. P. Mikulska, “Polariton spectrum control in dielectric medium,” in Proceedings 2nd IEEE International Workshop on Thz Radiation (TERA) (IEEE, 2010), pp. 288–290.

Ouyang, S.

Rouyera, C.

S. Santran, L. Cathoni, T. Cardinal, E. Farginc, G. Le Flem, C. Rouyera, and L. Sarger, “Precise and absolute measurements of the complex third-order optical susceptibility,” Proc. SPIE 4106, 349 (2000).

Said, A. A.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–26 (2007).

Santran, S.

S. Santran, L. Cathoni, T. Cardinal, E. Farginc, G. Le Flem, C. Rouyera, and L. Sarger, “Precise and absolute measurements of the complex third-order optical susceptibility,” Proc. SPIE 4106, 349 (2000).

Sarger, L.

S. Santran, L. Cathoni, T. Cardinal, E. Farginc, G. Le Flem, C. Rouyera, and L. Sarger, “Precise and absolute measurements of the complex third-order optical susceptibility,” Proc. SPIE 4106, 349 (2000).

Shandarov, V.

J. Xu, V. Shandarov, M. Wesner, and D. Kip, “Observation of two-dimensional spatial solitons in iron-doped barium-calcium titanate crystals,” Phys. Status Solidi. A, Appl. Res. 189, R4–R5 (2002).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–26 (2007).

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

Sukhorukov, A. P.

A. P. Sukhorukov, Nonlinear Wave Interactions in Optics and Radiophysics (Nauka, 1988) (in Russian).

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–26 (2007).

Wei, T.-H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–26 (2007).

Wesner, M.

J. Xu, V. Shandarov, M. Wesner, and D. Kip, “Observation of two-dimensional spatial solitons in iron-doped barium-calcium titanate crystals,” Phys. Status Solidi. A, Appl. Res. 189, R4–R5 (2002).
[CrossRef]

Xu, J.

J. Xu, V. Shandarov, M. Wesner, and D. Kip, “Observation of two-dimensional spatial solitons in iron-doped barium-calcium titanate crystals,” Phys. Status Solidi. A, Appl. Res. 189, R4–R5 (2002).
[CrossRef]

Zhong, S.

S. Zhong, C. Huang, C. Li, and L. Dong, “Surface defect kink solitons,” Opt. Commun. 285, 3674–3678 (2012).
[CrossRef]

IEEE LEOS Newsletter (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–26 (2007).

J. Opt. (1)

I. V. Dzedolik and S. N. Lapayeva, “Mass of polariton in different dielectric media,” J. Opt. 13, 015204 (2011).
[CrossRef]

J. Opt. Pure Appl. Opt. (1)

I. V. Dzedolik, “Period variation of polariton waves in optical fiber,” J. Opt. Pure Appl. Opt. 11, 094012 (2009).
[CrossRef]

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

Opt. Commun. (1)

S. Zhong, C. Huang, C. Li, and L. Dong, “Surface defect kink solitons,” Opt. Commun. 285, 3674–3678 (2012).
[CrossRef]

Opt. Express (2)

Phys. Status Solidi. A, Appl. Res. (1)

J. Xu, V. Shandarov, M. Wesner, and D. Kip, “Observation of two-dimensional spatial solitons in iron-doped barium-calcium titanate crystals,” Phys. Status Solidi. A, Appl. Res. 189, R4–R5 (2002).
[CrossRef]

Proc. SPIE (1)

S. Santran, L. Cathoni, T. Cardinal, E. Farginc, G. Le Flem, C. Rouyera, and L. Sarger, “Precise and absolute measurements of the complex third-order optical susceptibility,” Proc. SPIE 4106, 349 (2000).

Ukr. J. Phys. Opt. (1)

I. V. Dzedolik, “Transformation of sinusoidal electromagnetic and polarization waves into cnoidal waves in an optical fibre,” Ukr. J. Phys. Opt. 9, 226–235 (2008).
[CrossRef]

Other (13)

C. Kittel, Quantum Theory of Solids (Wiley, 1963).

O. Madelung, Festkörpertheorie, I, II (Springer-Verlag, 1972).

J. D. Jackson, Classical Electrodynamics (Wiley, 1962).

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).

I. V. Dzedolik and J. P. Mikulska, “Polariton spectrum control in dielectric medium,” in Proceedings 2nd IEEE International Workshop on Thz Radiation (TERA) (IEEE, 2010), pp. 288–290.

D. N. Klyshko, Quantum and Nonlinear Optics (Nauka, 1980) (in Russian).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

I. V. Dzedolik, Polaritons in Optical Fibers and Dielectric Resonators (DIP, 2007) (in Russian).

I. V. Dzedolik and O. S. Karakchieva, “Polaritons in nonlinear medium: generation, propagation and interaction,” in Proceedings of International Workshop on Nonlinear Photonics (NLP) (IEEE, 2011), pp. 1–3.

N. Bloembergen, Nonlinear Optics (W. A. Benjamin, 1965).

A. P. Sukhorukov, Nonlinear Wave Interactions in Optics and Radiophysics (Nauka, 1988) (in Russian).

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

R. W. Boyd, Nonlinear Optics (Academic, 2003).

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Figures (7)

Fig. 1.
Fig. 1.

Bound of charges by the electromagnetic field.

Fig. 2.
Fig. 2.

Potential energy of an ion (dimensionless units).

Fig. 3.
Fig. 3.

Polariton wave with the envelope ex(y): (a) one flat flow and (b) several flat flows (dimensionless units).

Fig. 4.
Fig. 4.

Square grid of polariton flows at the scalar polariton wave with the envelope Ree¯(ξ¯), where ξ¯=x+iy (dimensionless units).

Fig. 5.
Fig. 5.

Row of polariton flows at the vector polariton wave with the envelope Ree(ξ¯¯,η¯¯) at α1>0 and α3>0, where ξ¯¯=|α1|(x+iy) and η¯¯=|α1|(xiy) (dimensionless units).

Fig. 6.
Fig. 6.

Row of polariton flows at the vector polariton wave with the envelope Ree(ξ¯¯,η¯¯) at α1=α¯1 and α3>0 (dimensionless units).

Fig. 7.
Fig. 7.

Row of polariton flows at the vector polariton wave with the envelope Ree(ξ¯¯,η¯¯) at α1=α¯1 and α3<0 (dimensionless units).

Equations (32)

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

meffd2Rdt2+meffΓdRdt+RUR=eeff(E+1cdRdt×B),
md2rdt2+mΓdrdt+rUr=e(E+1cdrdt×B),
×B=c1(E˙+4πP˙),×E=c1B˙,
P=χ1Eaexp(iωt)+χ20EaEa+χ22EaEaexp(i2ωt)+χ31Ea2Eaexp(iωt)+χ33Ea2Eaexp(i3ωt),
χ1=14π(ωe2ω˜12+ωI2Ω˜12),χ20=14π(eα2rωe2mω02((ω02ω2)2+ω2Γ2)eeffα2RωI2meffΩ2((Ω2ω2)2+ω2Γ2)),χ22=14π(eα2rωe2m(ω˜12)2ω˜22eeffα2RωI2meff(Ω˜12)2Ω˜22),χ31=14π(e2α3rωe2m2(ω˜12)3(ω˜12)*+eeff2α3RωI2meff2(Ω˜12)3(Ω˜12)*),χ33=14π(e2α3rωe2m2(ω˜12)3ω˜32+eeff2α3RωI2meff2(Ω˜12)3Ω˜32)
P=χ1Eaexp(iωt)+χ31Ea2Eaexp(iωt),
2E+(E)+c2E¨=c24πP¨,
2Exy2+2Exz22Eyxy+ω2c2(1+4πχ1)Ex+4πω2χ31c2(|Ex|2+|Ey|2)Ex=0,2Eyx2+2Eyz22Exxy+ω2c2(1+4πχ1)Ey+4πω2χ31c2(|Ex|2+|Ey|2)Ey=0,
i2kE˜xz+2E˜xy22E˜yxy+α3(|E˜x|2+|E˜y|2)E˜x=0,i2kE˜yz+2E˜yx22E˜xxy+α3(|E˜x|2+|E˜y|2)E˜y=0,
2exy22eyxy+α1ex+α3(ex2+ey2)ex=0,2eyx22exxy+α1ey+α3(ex2+ey2)ey=0,
d2exdy2α¯1ex+α3ex3=0.
d2eydx2α¯1ey+α3ey3=0,
E˜x(y,z)=|2α¯1xα3|sch(sch1|e(0)α32α¯1x||α¯1x|y)×exp(iqxz),E˜y=0;
E˜y(x,z)=|2α¯1yα3|sch(sch1|e(0)α32α¯1y||α¯1y|x)×exp(iqyz),E˜x=0,
E˜x(y,z)=e˜0xcn((αx2/4+Cx)1/4α3yK(k˜x),k˜x)×exp(iqxz),E˜y=0;
E˜y(x,z)=e˜0ycn((αy2/4+Cy)1/4α3xK(k˜y),k˜y)×exp(iqyz),E˜x=0,
E˜(r,z)=e˜˜0rsn{[αr/2+(αr2/4+Cr)1/2]1/2|α3|/2r,k˜˜r}cn{[αr/2+(αr2/4+Cr)1/2]1/2|α3|/2r,k˜˜r}×exp(iqrz),E˜r=0,
y(exey)+(α1+α3|e|2)e=0,x(exey)+(α1+α3|e|2)e=0,
2eη2+2(α1+α3|e|2)e=0,
2eξη=0.
(2r2+α)u(e2r2+2err)w=2kewz,(e2r2+2err)w=2kewz(2kz+α)u,
(2r2+α)u=2kewz,(e2r2)w=(2kz+α)u.
κ2+(α/2k)κ+k2(k2α)/4k2=0.
2e+ξ¯2+2e+η¯2+i2eξ¯2i2eη¯222eξ¯η¯(α1+α3e2)e=0,2eξ¯2+2eη¯2+i2e+ξ¯2i2e+η¯222e+ξ¯η¯(α1+α3e2)e+=0.
(1i)2eξ¯2+(1+i)2eη¯2+22eξ¯η¯+(α1+α3e2)e=0.
d2e+dξ¯2+α1(+)e++α3(+)e+3=0,
d2edη¯2+α1()e+α3()e3=0,
(1i)2eξ¯¯2+(1+i)2eη¯¯2+22eξ¯¯η¯¯+(1+α11α3e2)e=0,
e(ξ¯¯,η¯¯)=i|α1||α3|tanh{C1+C2ξ¯¯+i2[C2(i+1)(i1i2C22)1/2]η¯¯}.
e(ξ¯¯,η¯¯)=i|α1||α3|tanh{ξ¯¯+12[i1+i+1]η¯¯}.
e(ξ¯¯,η¯¯)=α¯1|α3|tanh{ξ¯¯+12[i1i13i]η¯¯},
e(ξ¯¯,η¯¯)=iα¯1|α3|tanh{ξ¯¯+12[i1i13i]η¯¯}.

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