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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 4 — Feb. 21, 2005
  • pp: 1072–1078
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Nodal solitons and the nonlinear breaking of discrete symmetry

Albert Ferrando, Mario Zacarés, Pedro Andreés, Pedro Fernández de Córdoba, and Juan A. Monsoriu  »View Author Affiliations


Optics Express, Vol. 13, Issue 4, pp. 1072-1078 (2005)
http://dx.doi.org/10.1364/OPEX.13.001072


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Abstract

We present a new type of soliton solutions in nonlinear photonic systems with discrete point-symmetry. These solitons have their origin in a novel mechanism of breaking of discrete symmetry by the presence of nonlinearities. These so-called nodal solitons are characterized by nodal lines determined by the discrete symmetry of the system. Our physical realization of such a system is a 2D nonlinear photonic crystal fiber owning 𝓒 symmetry.

© 2005 Optical Society of America

1. Introduction

Symmetry is one of the most powerful and elegant concepts in physics. The use of group theory provides an extraordinary mathematical tool to classify solutions according to the symmetries of the physical system. In recent years, the increasing interest in physical systems owning 2D discrete symmetries, such as 2D nonlinear photonic crystals or Bose-Einstein condensates in 2D periodic potentials, raises the question of utilizing group theory as an analysis tool. Certainly, this approach is a standard in solid state physics. Its use in the topic of photonic crystals is less extended and it has been traditionally confined to the classification of linear modes [1

1. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).

]. Its generalization to the nonlinear case has become of great interest after the recent experimental observation of fundamental and vortex solitons in optically-induced 2D nonlinear photonic crystals [2

2. J.W. Fleischer, M. Segev, N. K. Efremedis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef] [PubMed]

, 3

3. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. G. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004). [CrossRef] [PubMed]

, 4

4. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004). [CrossRef] [PubMed]

]. In fact, attempts to apply group theory to the analysis of this type of solutions permitted the analytical prediction of the angular dependence of vortex solitons in 2D photonic crystals [5

5. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Vortex solitons in photonic crystal fibers,” Opt. Express 12, 817–822 (2004). [CrossRef] [PubMed]

]. Following this approach, in this paper we will use group theory as a general framework to analyze the role played by nonlinearities in the realization of discrete symmetry. We will see that discrete symmetry is realized differently when nonlinearities are present and that a new phenomenon of discrete-symmetry breaking occurs. The physical outcome of this general process is the generation of a new type of solitons with lesser symmetry than that of the original system.

2. Group self-consistency condition

So that, we start by analyzing the general problem of finding stationary solutions -Φ(x,y,z)=ϕ (x,y) expiβ z-of a nonlinear operator of the form:

(L0+LNL(Φ))Φ=2Φz2,
(1)

where L 0 is a linear operator (depending on the transverse coordinates xt=(x,y) only) invariant under a 2D discrete point-symmetry group G and LNL is a nonlinear operator depending locally on the modulus of the ϕ field. We are interested then in solving the following nonlinear eigenvalue problem:

(L0+LNL(ϕ))ϕ=β2ϕ.
(2)

A solution ϕs of Eq. (2) has to satisfy the so-called self-consistency condition, namely, ϕs has to appear as an eigenmode of the operator generated by itself, L(ϕs )≡L 0+LNL (|ϕs |). From a symmetry point of view, the self-consistency condition implies that if ϕs belongs to some representation of a finite group G′, then the entire operator L(ϕs ) has to be invariant under the same group, [L(ϕs ),G′]=0 (otherwise, L(ϕs ) would not contain in its spectrum the representation where ϕs lies on). We call this property the group self-consistency condition.

The simplest attempt to find solutions of Eq. (2) satisfying the group self-consistency condition is trying functions that enjoy the full symmetry of the linear operator; i.e., functions that are invariant under G. Functions belonging to the fundamental representation of G satisfy this property [6

6. M. Hamermesh, Group theory and its application to physical problems, Addison-Wesley series in physics, 1st ed. (Addison-Wesley, Reading, Massachusetts, 1964).

]: ϕfundg fundfund, ∀gG. Group self-consistency is satisfied because g -1 LNLg=LNL (|ϕfundg|)=LNL (|ϕ fund|), ∀gG;i.e., [LNL,G]=0 and, thus, [L(ϕ fund),G]=0. Solutions that satisfy this property are called fundamental solitons and they have been found in different systems of the type described by Eq. (2). A less obvious choice is the selection of functions belonging to higher-order representations of the same symmetry group G of the linear system. For 2D point-symmetry groups, these higher-order representations can be either non-degenerated (one-dimensional) or doubly-degenerated (two-dimensional) [6

6. M. Hamermesh, Group theory and its application to physical problems, Addison-Wesley series in physics, 1st ed. (Addison-Wesley, Reading, Massachusetts, 1964).

]. Vortex-antivortex solutions, appearing always as conjugated pairs (ϕv, ϕ*ν ), belong to two-dimensional representations of G [5

5. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Vortex solitons in photonic crystal fibers,” Opt. Express 12, 817–822 (2004). [CrossRef] [PubMed]

].We note that the modulus of a vortex solution is a group invariant (this is a general property also fulfilled by the modulus of functions belonging to one-dimensional representations of G). Since |ϕνG |=|ϕv |, then g-1LNLg=LNL (|ϕvg |)=LNL (|ϕv |), ∀gG and, consequently, [LNL,G]=0 and [L(ϕv ),G]=0. It is apparent that vortex solitons also fulfill the group self-consistency condition.

Fig. 1. Two nodal solitons for l=1 (Λ=23µm, a=8µm, γ=0.006 and wavelength λ=1064nm): (a)–(b) amplitude and phase, respectively, of the S nodal soliton; (c)–(d) amplitude and phase, respectively, of the A nodal soliton. Inset: schematic transverse representation of a PCF.

3. Group self-consistency theorem and nonlinear breaking of discrete symmmetry

4. Nodal solitons in photonic crystal fibers

Our specific physical system is a triangular photonic crystal fiber (PCF), although similar results can be expected in other 2D photonic crystals. We study the propagation of the electric component of a monochromatic electromagnetic field (at fixed polarization: E=ϕ u,|u|=1). PCF’s are thin silica fibers possessing a regular array of holes extending the entire fiber length and characterized by the hole radius a and the spatial period Λ of the photonic crystal cladding (see inset in Fig. 1). When silica nonlinearity is not neglected, a PCF is a particular case of a 2D nonlinear photonic crystal with a defect (where guidance occurs). In this case, L 0=t2+k02n02 (x,y), where ∇t is the transverse gradient operator, k 0 is the vacuum wave number, and n 0 is the refractive-index profile function (n0=n (silica) in silica and n 0=1 in air). The nonlinear term is LNL=k02γΔ(x,y)|ϕ|2, where Δ is the distribution function of nonlinear material (Δ=1 in silica and Δ=0 in air) and γ is a dimensionless nonlinear coupling constant, γ≡3χ(silica)(3) P/(2ε0cn (silica) A 0) (P is the total power and A 0 is an area parameter: A 0=π(Λ/2)2).

The symmetry group of a triangular PCF is 𝓒6v , i.e., [L0,𝓒6v ]=0. This group is constituted by discrete π/3-rotations (Rπ /3) plus specular reflections with respect to the x and y axes:

θRxθ

and

θRyπθ

, in polar coordinates. Solutions with the PCF 𝓒6v -symmetry have been previously found in the form of fundamental and vortex solitons [7

7. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Spatial soliton formation in photonic crystal fibers,” Opt. Express 11, 452–459 (2003). [CrossRef] [PubMed]

, 5

5. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Vortex solitons in photonic crystal fibers,” Opt. Express 12, 817–822 (2004). [CrossRef] [PubMed]

]. We will focus now on new solutions belonging to the subgroup 𝓒2v of 𝓒6v , formed by Rπ,Rx and Ry . Thus, we study the particular symmetry breaking pattern G=𝓒6vG′=𝓒2v . We can explicitly construct functions belonging to the four non-degenerated representations of 𝓒2v out of functions in the two-dimensional representations of 𝓒6v . The latter functions come in conjugated pairs (ϕl, ϕ*l ) (l=1,2), whose angular dependence is fixed by symmetry: ϕl=rleilθϕls (r,θ)exp[iϕlp (r,θ)], where ϕs (r,θ) is a scalar function, characterized by ϕs (r,θ+π/3)=ϕs (r,θ) and ϕs (r,-θ)=ϕs (r,π-θ)=ϕs (r,θ), and ϕp (r,θ) is a pseudoescalar function characterized by ϕp (r,θ+π/3)=ϕp (r,θ) and ϕp (r,-θ)=ϕp (r,π-θ)=-ϕp (r,θ). Let us consider the two following linear combinations (l=1, 2): 1/√2[ϕl ±ϕ*l ]). By writing the angular dependence of ϕl, the new functions adopt the form:

ϕδl(r,θ)=2rϕls(r,θ)cos[lθ+ϕlp(r,θ)+δ],l=1,2.
(3)

Solutions of Eq. (2) of the form given by Eq. (3) are indeed found. They are characterized by nodal lines determined by symmetry through the implicit equation cos[lθ+φlp (r,θ)+δ]=0 (l=1, 2). For this reason, we call them nodal solitons. The solution with δ=0 corresponds to the symmetric (S) 𝓒6v combination ϕSl ≡1/√2(ϕl+ϕ*l ) and that with δ=π/2 to the antisymmetric (A) one ϕAl≡i/√2(ϕl-ϕ*l ) (l=1, 2). Note that the φl function is not a solution of the Eq. (2) because the superposition principle does not hold. This function can only be approximated by a vortex solution in the linear regime (γ≈0). Although the complete structure of nodal lines could be rather intrincate, S and A nodal solitons are characterized by principal nodal lines: a single principal line for l=1 solitons and two orthogonal principal lines for l=2. In our simulations, we have found this four different types of nodal solitons. Nevertheless, we will show here results corresponding to the S and A nodal soliton solutions with l=1 only. In Fig. 1 we show the amplitude and phase of S and A nodal solitons, respectively. As predicted by the nodal line condition, the l=1, S nodal soliton presents a single vertical nodal line, whereas for the A soliton this line is horizontal.

At this point, it is interesting to remark that the so-called “dipole lattice solitons” found in perfectly periodic 𝓒4v lattices [8

8. Z. H. Musslimani and J. Yang, “Self-trapping of light in a two-dimensional photonic lattice,” J. Opt. Soc. Am. B 21, 973–981 (2004). [CrossRef]

] can be analogously explained in this framework. They are nothing but nodal solitons associated to the breaking of the original 𝓒4v symmetry of the photonic crystal into its subgroup 𝓒2 . They correspond to the symmetry-breaking pattern 𝓒4v→𝓒2 and, consequently they belong to the symmetric (S) and antisymmetric (A) representations of the 𝓒2 group. The numerically-found amplitudes and phases of “dipole lattice solitons” are exactly those predicted by group theory.

5. The role of symmetry. Soliton spectrum and stability

Fig. 2. Effective index of a soliton solution, n sol vs the nonlinear coupling γ for symmetric (dotted line) and antisymmetric (dashed line) solitons and vortex and antivortex solitons with l=1 (solid line).
Fig. 3. Evolution of a diagonal perturbation of a S nodal soliton showing asymptotic stability. (749 KB)

In order to check the stability of nodal solitons we need to solve the evolution Eq. (1) after perturbing the solution: ϕ′S,A =ϕS,A+δϕ. The stability analysis follows that of vortex solitons in Ref. [5

5. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Vortex solitons in photonic crystal fibers,” Opt. Express 12, 817–822 (2004). [CrossRef] [PubMed]

]. In there we introduced the concepts of diagonal and non-diagonal perturbations. For one-dimensional representations, a diagonal perturbation is defined as that that preserves the representation in which the solution lies on. In the present case—in which ϕS and ϕA belongs to the B 1 and B 2 one-dimensional representations of 𝓒2v , respectively—, this definition implies that ϕ′SB 1 and ϕ′AB 2. Explicit examples of such perturbations are scaled solutions: ϕ′S,A =(1+ε)ϕS,A≠0. Evolution yields numerical evidence that nodal solitons are stable under such perturbations, as shown in Fig. 3 in which a diagonal perturbation (scaled solution) is applied. However, non-diagonal perturbations, taking the perturbed solution out of its original representation, provide instabilities. These instabilities are of the oscillatory type, as shown in Fig. 4, and they can be understood as a simultaneous oscillation among modes belonging to all the representations of 𝓒2v . This instability pattern, however, shows no trace of pseudo-soliton collapse nor of transverse ejection of pseudo-solitons typical of Kerr nonlinearities in homogeneous media treated in the paraxial approximation. This particular behavior was first observed in vortices in PCF’s [5

5. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Vortex solitons in photonic crystal fibers,” Opt. Express 12, 817–822 (2004). [CrossRef] [PubMed]

]. Since the self-focussing instability seems to be rooted in the paraxial approximation [9

9. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?,” Opt. Lett. 18, 411–413 (1993). [CrossRef] [PubMed]

], a plausible explanation of its absence is the non-paraxial nature of evolution in this case. Absence of ejection can be qualitative understood by the inhibition of transverse radiation induced by the photonic crystal cladding.

Fig. 4. Non-diagonal perturbation of a S nodal soliton. In this case, an oscillatory instability occurs. (576 KB)

An interesting interpretation of nodal solitons is as interacting pseudo-solitons. It can be proven that a S nodal soliton can be written as ϕS0 (x+x0,y)-ϕ0 (x-x0,y), ϕ0 being a localized function in the fundamental representation of 𝓒6v . In the case that ϕ 0 is a sufficiently localized function (large nonlinear coupling γ or strong lattice index contrast), ϕ 0 can be approximated by a fundamental soliton solution. Then, a nodal soliton can be envisaged as a pair of weakly interacting pseudo-solitons. Like in an homogeneous medium this interaction is repulsive [10

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

]. In our case, nor γ nor the index contrast are necessarely large, consequently, the soliton-soliton interaction cannot longer be considered weak. However, the group theory antisymmetric decomposition in terms of localized solutions in the fundamental representation of 𝓒6v remains valid. In this way, the concept of nodal soliton generalizes the idea of interacting pseudo-solitons into a regime of strong particle coupling (intermediate γ and index contrast). In the weak soliton-interaction regime, new soliton solutions have been recently found that can also be interpreted in the context of group theory reported in this paper [11

11. T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett 93, 063901 (2004). [CrossRef] [PubMed]

].

We are thankful to H. Michinel for useful discussions. This work was financially supported by the Plan Nacional I+D+I (grant TIC2002-04527-C02-02), Ministerio de Ciencia y Tecnología (Spain) and FEDER funds. Authors also acknowledge the financial support from the Generalitat Valenciana, Spain (grants Grupos03/227 and GV04B-390). M. Z. gratefully acknowledges Fundación Ramón Areces grant.

References and links

1.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).

2.

J.W. Fleischer, M. Segev, N. K. Efremedis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef] [PubMed]

3.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. G. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004). [CrossRef] [PubMed]

4.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004). [CrossRef] [PubMed]

5.

A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Vortex solitons in photonic crystal fibers,” Opt. Express 12, 817–822 (2004). [CrossRef] [PubMed]

6.

M. Hamermesh, Group theory and its application to physical problems, Addison-Wesley series in physics, 1st ed. (Addison-Wesley, Reading, Massachusetts, 1964).

7.

A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Spatial soliton formation in photonic crystal fibers,” Opt. Express 11, 452–459 (2003). [CrossRef] [PubMed]

8.

Z. H. Musslimani and J. Yang, “Self-trapping of light in a two-dimensional photonic lattice,” J. Opt. Soc. Am. B 21, 973–981 (2004). [CrossRef]

9.

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?,” Opt. Lett. 18, 411–413 (1993). [CrossRef] [PubMed]

10.

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

11.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett 93, 063901 (2004). [CrossRef] [PubMed]

OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(190.4370) Nonlinear optics : Nonlinear optics, fibers

ToC Category:
Research Papers

History
Original Manuscript: December 1, 2004
Revised Manuscript: December 1, 2004
Published: February 21, 2005

Citation
Albert Ferrando, Mario Zacarés, Pedro Andreés, Pedro Fernández de Córdoba, and Juan Monsoriu, "Nodal solitons and the nonlinear breaking of discrete symmetry," Opt. Express 13, 1072-1078 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-4-1072


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References

  1. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001)
  2. J.W. Fleischer,M. Segev, N. K. Efremedis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003) [CrossRef] [PubMed]
  3. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. G. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004) [CrossRef] [PubMed]
  4. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004) [CrossRef] [PubMed]
  5. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Vortex solitons in photonic crystal fibers,” Opt. Express 12, 817–822 (2004). [CrossRef] [PubMed]
  6. M. Hamermesh, Group theory and its application to physical problems, Addison-Wesley series in physics, 1st ed. (Addison-Wesley, Reading, Massachusetts, 1964).
  7. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Spatial soliton formation in photonic crystal fibers,” Opt. Express 11, 452–459 (2003). [CrossRef] [PubMed]
  8. Z. H. Musslimani and J. Yang, “Self-trapping of light in a two-dimensional photonic lattice,” J. Opt. Soc. Am. B 21, 973–981 (2004) [CrossRef]
  9. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?,” Opt. Lett. 18, 411–413 (1993). [CrossRef] [PubMed]
  10. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518 (1999) [CrossRef] [PubMed]
  11. T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett 93, 063901 (2004) [CrossRef] [PubMed]

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