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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 5 — Mar. 8, 2004
  • pp: 817–822
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Vortex solitons in photonic crystal fibers

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


Optics Express, Vol. 12, Issue 5, pp. 817-822 (2004)
http://dx.doi.org/10.1364/OPEX.12.000817


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Abstract

We demonstrate the existence of vortex soliton solutions in photonic crystal fibers. We analyze the role played by the photonic crystal fiber defect in the generation of optical vortices. An analytical prediction for the angular dependence of the amplitude and phase of the vortex solution based on group theory is also provided. Furthermore, all the analysis is performed in the non-paraxial regime.

© 2004 Optical Society of America

1. Introduction

Spatial soliton solutions in 1D periodic structures have been theoretically predicted [1

1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794 (1988). [CrossRef] [PubMed]

] and experimentally demonstrated in nonlinear waveguide arrays [2

2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383 (1998). [CrossRef]

]. More recently, similar structures have been observed in optically induced gratings [3

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

, 4

4. D. Neshev, E. Ostrovskaya, Y. S. Kivshar, and W. Krolikowsky, “Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710 (2003). [CrossRef] [PubMed]

] and, for the first time, in 2D photonic lattices [5

5. 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 (2003). [CrossRef] [PubMed]

]. Modeling has been traditionally performed using the discrete nonlinear Schrödinger equation (NLSE)[1

1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794 (1988). [CrossRef] [PubMed]

, 6

6. Y. S. Kivshar, “Self-localization in arrays of defocusing waveguides,” Opt. Lett. 18, 1147 (1993). [CrossRef] [PubMed]

], valid only in the so-called tight-binding approximation. However, the accurate study of nonlinear solutions in optically-induced lattices requires the resolution of the NLSE with a continuous periodic potential model [3

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

, 5

5. 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 (2003). [CrossRef] [PubMed]

]. Discrete optical vortices have been also predicted in the discrete NLSE [7

7. B. A. Malomed and P. G. Keverkidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001). [CrossRef]

] and in continuous models [8

8. J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28, 2094 (2003). [CrossRef] [PubMed]

] and, recently, experimentally observed in optically-induced 2D lattices [9

9. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. Kivshar, H. Martin, and Z. Chen, “Observation of discrete vortex solitons in optically-induced photonic lattices,” arXiv:nlin/0309018 (2003).

].

A different scenario appears in a 2D photonic crystal in the presence of a defect. An example of such a system is a photonic crystal fiber (PCF). A PCF is a silica fiber that possesses a regular array of air-holes extending along the entire fiber length (see Fig. 1(a)). It is an axially invariant 2D photonic crystal with a central defect (the PCF core, where guidance occurs). It has been recently proven that a PCF acting as a 2D nonlinear photonic crystal can support and stabilize fundamental solitons [10

10. 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 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452. [CrossRef] [PubMed]

]. Unlike fundamental solitons embedded in perfectly periodic potentials [5

5. 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 (2003). [CrossRef] [PubMed]

, 8

8. J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28, 2094 (2003). [CrossRef] [PubMed]

], a distinguishing feature of these systems is that no power gap is needed to generate the nonlinear localized solution. In this letter we go a step further and give numerical evidence of the existence of vortex solitons in PCF’s.

2. Description of the method

When one considers that silica can have a nonlinear response, electromagnetic propagation in a PCF is given by the following non-paraxial equation (∇·E≈0):

[t2+k02(n02(x)+n22(x)E2)]E=2Ez2,
(1)

L(ϕ)ϕ=β2ϕL(ϕ)=L0+LNL(ϕ),
(2)

where L 0=t2+k02 n02(x,y) and LNL (|ϕ|)=k02 n22(x,y)|ϕ|2 stand for the linear and nonlinear parts of the differential operator L(|ϕ|), respectively. Due to the geometry of the air-holes distribution, both n 0(x,y) and n 2(x,y) are invariant under the action of the 𝓒 6v point-symmetry group. This group is constituted by discrete π/3-rotations (R π/3) plus specular reflections with respect to the x and y axes: yRxy and xRyx, or, equivalently, θRxθ and θRyπθ, in polar coordinates.

ϕl=rleilθϕls(r,θ)exp[iϕlp(r,θ)]l=1,2,
(3)

where ϕls is a scalar function, that is, a function invariant under finite rotations and specular reflections, whereas ϕlp is a pseudo-scalar function, invariant under rotations but sign-changing under reflections. Mathematically, ϕls (r,θ+π/3)=ϕls (r,θ) and ϕls (r,-θ)=ϕls (r,π-θ)=ϕls (r,θ), whereas ϕlp (r,θ+π/3)=ϕlp (r,θ) but ϕlp (r,-θ)=ϕlp (r,π-θ)=-ϕlp (r,θ). Both ϕls and ϕlp are periodic functions of θ, so they can be expanded in Fourier series in cos(6) and sin(6) (nN). Reflection symmetry forces ϕs (ϕp ) to depend on the cosine (sine) terms only. Thus, ϕls (r,θ)=∑ naln (r)cos(6) and ϕlp (r,θ)=∑ nbln (r) sin(6).

Fig. 1. (a) Schematic representation of a PCF. (b)–(d) Amplitudes of several vortices for increasing values of γ in a PCF with Λ=31µm and a=6µm (λ=1064nm): (b) γ=0,45×10-3; (c) γ=0,95×10-3; (d) γ=1,75×10-3.

Fig. 2. Effective index of a family of vortex solutions as a function of γ (solid line). Same for a family of fundamental solitons (dashed line), as in Ref. [10]. The shadow region corresponds to the conduction band, constituted by Bloch modes, of the 2D photonic cladding with Λ=31µm and a=10µm (λ=1064nm).

3. Results

We have simulated large-scale PCF’s (in order to prevent silica breakdown [10

10. 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 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452. [CrossRef] [PubMed]

]) with a lattice period, or pitch, Λ=31µm, and several air-hole radius at a fixed wavelength λ=1064nm (a convenient quasi-continuum source). We define a dimensionless nonlinear parameter as γ=n2(silica)2 P/A 0 (A 0 is an area parameter: A 0=π (Λ/2)2; P is the total power carried by the solution) and perform calculations for increasing values of γ. We discover that a family of optical vortex solutions of the type given by the ansatz (3) is found. The amplitudes of several solutions (with l=1) for increasing values of γ are represented in Fig. 1(b)–(d). According to group theory (Eq. (3)), these amplitudes have to be scalar functions, thus showing full invariance under 𝓒 6v, and they have to vanish at the origin, as seen in Fig. 1(b)–(d). Besides, they become gradually narrower as the nonlinearity increases. It is interesting to plot the effective index of these solutions (n eff=β/k 0) versus γ, as shown in Fig. 2. The value of neff increases as γ increases, accordingly to the narrowing of solutions depicted in Fig. 1(b)–(d). For comparison, we have also included the n eff(γ) curve of the fundamental soliton family for the same PCF structure, as in Ref. [10

10. 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 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452. [CrossRef] [PubMed]

]. Both lie on the upper forbidden band of the perfectly periodic cladding structure. Here it can be clearly envisaged the role of the central defect. Notice that there is no threshold γ (power) to generate a nonlinear vortex soliton, i.e, there is a continuum of solutions in γ starting from the linear (TE or TM) mode. Unlike in perfectly periodic structures, the defect can eliminate the presence of a threshold power [8

8. J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28, 2094 (2003). [CrossRef] [PubMed]

]. If we consider structures where the linear vortex-like mode is not present [14

14. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A 17, 1333 (2000). [CrossRef]

], a threshold γ is necessary to generate the nonlinear vortex solution in the forbidden band. By playing with the geometric parameters of the PCF, it is possible to tune this threshold value. In all cases, this value is much lesser than in perfectly periodic structures, which can be important for experimental purposes.

Fig. 3. Phase of a vortex with l=1 at r=21µm. We represent both the total phase arg(ϕl ) (solid line) and the group phase arg(ϕl )- (dashed line).

Even more interesting is the phase behavior of these solutions. We show in Fig. 3 a typical phase profile of a PCF vortex calculated at a fixed radius. It shows strict accordance with the group theory prediction for the phase: arg(ϕl )=+ϕlp (r,θ)=+n=1 bln (r) sin(6) and it points out that n≈1. The phase of these vortex solutions differs from that of an ordinary vortex by the presence of the pseudo-scalar function extra term: the group phase. Besides the existence of the standard linear behavior in θ, the group phase provides an additional sinusoidal dependence on the angle (with period determined by the group order: π/3 for order 6) not present in ordinary vortices.

In order to check the stability of a vortex soliton solution ϕl , we have to consider z-dependent perturbations. This implies solving the non-paraxial equation (1) for the perturbed field ϕ′=ϕl +δϕ. In terms of group theory, (ϕl*l) constitute an orthogonal basis of the l representation of 𝓒 6v, so we can consider two types of perturbations around a vortex solution ϕl : diagonal perturbations, in which ϕ′ remains in the one-dimensional subspace spawned by ϕl , that is, 〈ϕ′|ϕ*l〉=0, and non-diagonal, where the perturbation takes ϕ′ out of this subspace, 〈ϕ′|ϕ*l〉=0.

A typical initial condition of the form ϕ′=re exp[-(r/r 0)2] is an example of diagonal perturbation for a vortex (l=1). Another example is a perturbation of the form ϕ′=(1+ε)ϕl ,ε being a small real constant number. In both cases, 〈ϕ′|ϕ*l〉=0. We have simulated the evolution of a perturbation of the latter type launched into the fiber core for different PCF configurations and values of ε (ε~0.025-0.1). In Fig. 4 we present an animation that displays the evolution of the transverse field amplitude along the fiber. The animation shows the transient from the initial amplitude towards an asymptotically stationary profile corresponding to a vortex. This transient is a few centimeters long and has the characteristic form of a damped oscillation. Non-paraxial evolution shows that a vortex is stable under diagonal perturbations. Diagonal perturbations generalize the concept of radial perturbations to the discrete symmetry case. We have also simulated non-diagonal perturbations by introducing random perturbations (|δϕ(x,y)|≤ε, ε~0.05-0.1) that modify both the phase and amplitude of the vortex solution in such a way that 〈ϕ′|ϕ*l〉≠0. Non-diagonal perturbations correspond to azimuthal ones in the radially symmetric case. In this case, an oscillatory instability occurs, as seen in the simulation presented in Fig. 5. This oscillatory instability initially appears as a rotating flux along the vortex ring. After a short evolution, the perturbed vortex breaks into a two strongly interacting single structures of zero vorticity with a cumbersome dynamics, as observed in our simulation. It is remarkable that, unlike vortices in an homogeneous medium [15

15. W. J. Firth and D. V. Skryabin, “Orbital solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450 (1997). [CrossRef]

], these two structures do not fly off tangential to the vortex ring but remain confined in the PCF defect due to the inhibition of transverse flux produced by the PCF cladding. It is also noticeable that, in spite of this instability, no collapse of the structures is detected due to the non-paraxial nature of evolution [16

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

].

Fig. 4. (997 KB) Evolution of the field amplitude in z under a diagonal perturbation (ε=0.095) for a large-scale PCF with Λ=31µm, a=10µm and λ=1064nm. We show the transient from an initial profile towards an asymptotic vortex solution.
Fig. 5. (965 KB) Evolution of the field amplitude in z under a non-diagonal perturbation (ε=0.05) for a large-scale PCF with Λ=31µm, a=10µm and λ=1064nm.

4. Conclusions

Summarizing, we propose a novel way of generating vortex solitons, based on PCF’s, in which the presence of the defect plays a crucial role. Our analytical approach, based on group theory, is general and permits to predict the angular dependence of vortex solutions in a nonlinear system owning a discrete invariance, no matter it is periodic or non-periodic. Finally, the use of a non-paraxial equation permits to analyze new phenomena beyond the paraxial approximation.

We thank 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. M. Zacarés acknowledges Fundación Ramón Areces grant.

References and links

1.

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794 (1988). [CrossRef] [PubMed]

2.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383 (1998). [CrossRef]

3.

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

4.

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

5.

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 (2003). [CrossRef] [PubMed]

6.

Y. S. Kivshar, “Self-localization in arrays of defocusing waveguides,” Opt. Lett. 18, 1147 (1993). [CrossRef] [PubMed]

7.

B. A. Malomed and P. G. Keverkidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001). [CrossRef]

8.

J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28, 2094 (2003). [CrossRef] [PubMed]

9.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. Kivshar, H. Martin, and Z. Chen, “Observation of discrete vortex solitons in optically-induced photonic lattices,” arXiv:nlin/0309018 (2003).

10.

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 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452. [CrossRef] [PubMed]

11.

M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488 (2001). [CrossRef]

12.

M. Hamermesh, Group theory and its application to physical problems, (Addison-Wesley, Reading, Massachusetts, 1964).

13.

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides,” IEEE Trans. Microw. Theory Tech. 23, 421 (1975). [CrossRef]

14.

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A 17, 1333 (2000). [CrossRef]

15.

W. J. Firth and D. V. Skryabin, “Orbital solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450 (1997). [CrossRef]

16.

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?,” Opt. Lett. 18, 411 (1993). [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: February 4, 2004
Revised Manuscript: February 23, 2004
Published: March 8, 2004

Citation
Albert Ferrando, Mario Zacarés, Pedro Fernández de Córdoba, Daniele Binosi, and Juan Monsoriu, "Vortex solitons in photonic crystal fibers," Opt. Express 12, 817-822 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-5-817


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References

  1. D. N. Christodoulides and R. I. Joseph, �??Discrete self-focusing in nonlinear arrays of coupled waveguides,�?? Opt. Lett. 13, 794 (1988). [CrossRef] [PubMed]
  2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, �??Discrete spatial optical solitons in waveguide arrays,�?? Phys. Rev. Lett. 81, 3383 (1998). [CrossRef]
  3. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremedis, and D. N. Christodoulides, �??Observation of discrete solitons in optically induced real time waveguide arrays,�?? Phys. Rev. Lett. 90, 023902 (2003). [CrossRef] [PubMed]
  4. D. Neshev, E. Ostrovskaya, Y. S. Kivshar, and W. Krolikowsky, �??Spatial solitons in optically induced gratings,�?? Opt. Lett. 28, 710 (2003). [CrossRef] [PubMed]
  5. 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 (2003). [CrossRef] [PubMed]
  6. Y. S. Kivshar, �??Self-localization in arrays of defocusing waveguides,�?? Opt. Lett. 18, 1147 (1993). [CrossRef] [PubMed]
  7. B. A. Malomed and P. G. Keverkidis, �??Discrete vortex solitons,�?? Phys. Rev. E 64, 026601 (2001). [CrossRef]
  8. J. Yang and Z. H. Musslimani, �??Fundamental and vortex solitons in a two-dimensional optical lattice,�?? Opt. Lett. 28, 2094 (2003). [CrossRef] [PubMed]
  9. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. Kivshar, H. Martin, and Z. Chen, �??Observation of discrete vortex solitons in optically-induced photonic lattices,�?? arXiv:nlin/0309018 (2003).
  10. 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 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452<a> [CrossRef] [PubMed]
  11. M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, �??Symmetry and degeneracy in microstructured optical fibers,�?? Opt. Lett. 26, 488 (2001). [CrossRef]
  12. M. Hamermesh, Group theory and its application to physical problems, (Addison-Wesley, Reading, Massachusetts, 1964).
  13. P. R. McIsaac, �??Symmetry-induced modal characteristics of uniform waveguides,�?? IEEE Trans. Microw. Theory Tech. 23, 421 (1975). [CrossRef]
  14. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, �??Vector description of higher-order modes in photonic crystal fibers,�?? J. Opt. Soc. Am. A 17, 1333 (2000). [CrossRef]
  15. W. J. Firth and D. V. Skryabin, �??Orbital solitons carrying orbital angular momentum,�?? Phys. Rev. Lett. 79, 2450 (1997). [CrossRef]
  16. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, �??Does the nonlinear Schrödinger equation correctly describe beam propagation?,�?? Opt. Lett. 18, 411 (1993). [CrossRef] [PubMed]

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