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Exact soliton solutions of the nonlinearHelmholtz equation: communication

P. Chamorro-Posada

Area de Teorıa de la Senal y Comunicaciones, Departamento de Ingenierıa Electrica, Electronica, Computadores ySistemas, Universidad de Oviedo Edificio Departamental 4, Campus de Viesques, 33204-Gijon, Spain

G. S. McDonald

Photonics and Nonlinear Science Group, Joule Physics Laboratory, School of Sciences, University of Salford,Salford M5 4WT, UK

G. H. C. New

Quantum Optics and Laser Science Group, The Blackett Laboratory, Imperial College of Science, Technology,and Medicine, Prince Consort Road, London SW7 2BW, UK

Received March 9, 2001

Exact analytical soliton solutions of the nonlinear Helmholtz equation are reported. A lucid generalization ofparaxial soliton theory that incorporates nonparaxial effects is found. © 2002 Optical Society of America

OCIS codes: 190.3270, 190.5530, 240.4350.

1. INTRODUCTION

Investigations of the propagation of optical beams inKerr-type media are usually based on the nonlinearSchrodinger equation (NSE), which is particularly ap-pealing because of its analytical properties. The maindrawback of this approach comes from the limitations im-posed by the paraxial approximation that is implicit inthe NSE. To overcome these difficulties we developedtechniques based on a nonparaxial NSE (NNSE) that isderived without the use of the paraxial approximation;exact analytical solutions were derived for bright two-dimensional solitons, and their properties were thor-oughly analyzed.1 Further results were presented in re-cent publications2,3 and have been widely disseminated atconferences.4–7

In a recent paper,8 analytical solutions that correspondto soliton solutions of the nonlinear Helmholtz equationwere presented. The bright and dark solutions forN 1 1 dimensions were reported as new, and the authorsthen concentrated attention on the highly physical two-dimensional solutions. In this Communication we ad-dress key questions that were not fully answered in Ref.8.

2. RESULTS

The two-dimensional nonlinear Helmholtz equation,

0740-3224/2002/051216-02$15.00 ©

]2E

]z2 1]2E

]x2 1 k2E 1 guEu2E 5 0, (1)

where E(x, z) is the electric field, k 5 n0v/c, and g5 2n0n2v2/c2, is transformed into the NNSE:

k]2u

]z2 1 i]u

]z1

1

2

]2u

]j2 1 uuu2u 5 0, (2)

when it is written in terms of the field envelopeA(x, z), where E(x, z) 5 A(x, z)exp( jkz) and the nor-malizations z 5 z/LD , j 5 A2x/w0 , and u(j, z)5 (kn2LD /n0)1/2A(j, z) have been employed. v0 is areference beam width, LD 5 kv0

2/2 is the correspondingdiffraction length, and k 5 1/(kv0)2 is a measure of thedegree to which an input beam that propagates along thez axis deviates from being paraxial. In the paraxiallimit, when the first term in Eq. (2) is neglected the NSEis recovered. A detailed account of the physical proper-ties of the NNSE and its solutions can be found in Refs.1–7. In what follows, we focus only on those results thatare relevant to the study presented in Ref. 8.

The fundamental bright-soliton solution of the nonlin-ear Helmholtz equation, in terms of the normalizations ofthe NNSE, can be written as a lucid generalization of aparaxial soliton:

2002 Optical Society of America

Chamorro-Posada et al. Vol. 19, No. 5 /May 2002/J. Opt. Soc. Am. B 1217

u~j, z! 5 h sechF h~j 1 Vz!

A1 1 2kV2G3 expF iS 1 1 2kh2

1 1 2kV2D 1/2S 2Vj 1z

2kD G

3 expS 2iz

2kD , (3)

where h, V, and k are the amplitude, transverse velocity,and nonparaxial parameters, respectively. When non-paraxial effects are negligible, i.e., when k → 0, kh2

→ 0, and kV2 → 0, one recovers the well-known paraxialsoliton solution. The width of the nonparaxial soliton isgiven by j0 5 (1 1 2kV2)1/2/h. The soliton’s area isthus proportional to j0h 5 (1 1 2kV2)1/2 and depends onboth the transverse velocity and the actual size of thebeam (through k). The latter property reflects the factthat the solutions of the NNSE are invariant under trans-formations

j 5j8 1 Vz8

~1 1 2kV2!1/2 , z 522kVj8 1 z8

~1 1 2kV2!1/2 , (4)

u~j, z! 5 expXiH Vj8

~1 1 2kV2!1/2

11

2kF1 2

1

~1 1 2kV2!1/2Gz8J Cu8~j8, z8!,

(5)

from which the well-known Galilean transformation in-variance of the NSE is recovered in the appropriateparaxial limit. The transformations of Eqs. (4) and (5)correspond to a rotation of the solution in the original (un-scaled) coordinate system by an angle u, given by sec u5 (1 1 2kV2)1/2 (Refs. 1 and 2); this allows the analysis tobe extended to common nonparaxial situations in whichoff-axis soliton beams are considered. It has further beenshown2 that these transformations can be used to predictthe long-term evolution of nonparaxial solitons when theyare used in conjunction with analytical paraxialtechniques.9

With respect to the conservation properties of theNNSE, we have generalized the first conserved quantityof the NSE, *2`

1`uu(j, z)u2dj 5 C, where C is a constant,to the framework of the NNSE. This quantity has a cor-respondence to the conservation of an energy flow that isgiven by

E2`

1`F 1

2k1

]f~j, z!

]zG uu~j, z!u2dj 5 C8, (6)

where C8 is another constant. This exact result general-izes a previously published (approximate) expression forwhich only fast on-axis phase variations were taken intoaccount.10,11 The NSE conservation law is recoveredfrom Eq. (6) in the paraxial limit.

Considering the effect that exact solutions of the NSEhave had on a wide range of physical systems in whichsolitonlike solutions exert a dominant influence, the re-sults given above, along with those of Refs. 1–8, are likelyto underpin a broad class of exciting new research topicsin nonlinear optics in which higher-order effects and re-lated geometries are considered.

P. Chamorro-Posada’s e-mail address is pedro@tsc.uniovi.es.

REFERENCES1. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New,

‘‘Non-paraxial solitons,’’ J. Mod. Opt. 45, 1111–1121 (1998).2. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New,

‘‘Propagation properties of non-paraxial spatial solitons,’’ J.Mod. Opt. 47, 1877–1886 (2000).

3. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New,‘‘Non-paraxial beam propagation methods,’’ Opt. Commun.192, 1–12 (2001).

4. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New,‘‘Nonparaxial solitons,’’ presented at the UK National Con-ference of Quantum Electronics, Cardiff, UK, 1997.

5. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New,‘‘Exact analytical nonparaxial optical solitons,’’ presented atthe 1998 International Conference on Quantum Electron-ics, San Francisco, Calif.

6. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New,‘‘Nonparaxial solitons: accurate numerical methods andnew results,’’ presented at the 1998 European Conferenceon Quantum Electronics, Glasgow, UK.

7. P. Chamorro-Posada, G. S. McDonald, G. H. C. New, J.Noon, and S. Chavez-Cerda, ‘‘Fundamental properties of(1 1 1)D and (2 1 1)D nonparaxial optical solitons,’’ pre-sented at the UK National Conference of Quantum Elec-tronics, Manchester, UK, 1999.

8. T. A. Laine and A. T. Friberg, ‘‘Self-guided waves and exactsolutions of the nonlinear Helmholtz equation,’’ J. Opt. Soc.Am. B 17, 751–757 (2000).

9. J. Satsuma and N. Yajima, ‘‘Initial value problems of one-dimensional self-modulation of nonlinear waves in disper-sive media,’’ Suppl. Prog. Theor. Phys. 55, 284–306 (1974).

10. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, ‘‘Doesthe nonlinear Schrodinger equation correctly describe beampropagation?’’ Opt. Lett. 18, 411–413 (1993).

11. J. M. Soto-Crespo and N. Akhmediev, ‘‘Description of theself-focusing and collapse effects by a modified nonlinearSchrodinger equation,’’ Opt. Commun. 101, 223–230 (1993).