**1. Introduction**

366 Optical Communication

10(2) 329-340.

Wiley; 2004.

2008.

[17] Ma, Yiran; Yang, Qi; Tang, Y.; Chen, Simin & Shieh, W. 1 Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth

[18] Bigo, S. Multiterabit DWDM terrestrial transmission with bandwidth-limiting optical filtering. IEEE Journal of Selected Topics in Quantum Electronics, March/April 2004;

[19] Rao, R.M.& A.S. Bopardikar, A. S. Wavelet Transforms. Introduction to Theory and

[20] Daubechies, I Ten Lectures on Wavelets. Philadelphia, Pennsylvania: Society for

[21] Sarkar, K.T.; Salazar-Palma, M.; Wicks, M.C. Wavelet Applications in Engineering

[22] Moreolo, M. S.; Cincotti, G. and Neri A. Synthesis of optical wavelet filters, IEEE

[23] Bulakci, Ö.; Schuster, M.; Bunge, C.-A.; Spinkler, B.; Hanik, N. Wavelet Transform Based Optical OFDM. In: Optical Fiber Communication (OFC), collocated National Fiber Optic Engineers Conference, 2009 Conference on (OFC/NFOEC), 2009, pp. 1-3. [24] Reed, G. T. and Knights, A. P. Silicon photonics. An introduction. Chichester, England:

[25] Reed, G. T., editor. Silicon photonics. The state of the art. Chichester, England: Wiley;

[26] Reed, G. T.; Mashanovich, G.; Gardes, F.Y, & Thomson, D.J. Silicon optical modulators,

[27] Marris-Morini, D. et al. Recent progress in high-speed silicon-based optical modulators.

[28] Marris-Morini, D. et al. Optical modulation by carrier depletion in a silicon PIN diode,

[29] G. Cincotti, G. Full optical encoders/decoders for photonic IP routers, Journal of

[30] Moreolo, M. S. and Cincotti, G. Compact low-loss planar architectures for all-optical wavelet signal processing, In: Transparent Optical Networks, 2005, Proceedings of 2005

[31] Densmore, A. et al. Compact and low power thermo-optic switch using folded silicon

[32] Harjanne, M.; Kapulainen, M.; Aalto, T. and Heimala, P. Sub-μs switching time in silicon-on-insulator Mach-Zehnder thermooptic switch. IEEE Photonics Technology

[33] Hillerkuss, D. et al. Single source optical OFDM transmitter and optical FFT receiver demonstrated at line rates of 5.4 and 10.8 Tbit/s, In: Optical Fiber Communication (OFC), collocated National Fiber Optic Engineers Conference, 2010 Conference on

[34] Hillerkuss, D. et al. 26 Tbit s-1 line-rate super-channel transmission utilizing all-optical

[35] Okamoto, K. Fundamentals of Optical Waveguides. Academic Press, San-Diego, USA:

fast Fourier transform processing, Nature Photonics, June 2011; 5(6) 364-371.

access, Optics Express, May 2009; 17(11) 9421-9427.

Industrial and Applied Mathematics; 2006.

Nature Photonics. August 2010; 4, 518-526.

Letters, September 2004; 16(9) 2039-2041.

(OFC/NFOEC), 2010, pp. 1-3.

Academic Press; 2000.

Proceedings of the IEEE, July 2009; 97(7) 1199-1215.

Optics Express, October 2006; 14(22) 10838-10843.

7th International Conference, 2005, vol. 1, pp. 319- 322.

waveguides, Optics Express, June 2009; 17(13) 10457-10465.

Lightwave Technology, vol. 22, no. 2, (February 2004) 337- 342.

Electromagnetics. Boston: Artech House; 2002.

Applications. Reading, Massachusetts: Addison-Wesley; 1998.

Photonics Technology Letters. July 2004, 16 (7) 1679-1681.

Vortices are fundamental objects which appear in many branches of physics [1] such as optics [2, 3] and Bose-Einstein condensates [4]. In nonlinear optics, vortex solitons are associated with the phase dislocations (or phase singularities) carried by the nondiffracting optical beams [5], and share many common properties with the vortices observed in other systems, e.g., superfluids and Bose-Einstein condensates [6, 7]. In a homogeneous medium, stable vortex solitons were proposed to exist in the so-called cubic-quintic or other similar nonlinear media, for example, combination of *χ*(2) and *χ*(3) nonlinear media, based on competing self-focusing and self-defocusing nonlinearities [8–11]. However, the experimental realization of vortex solitons in such media is hard, as the requirement of very high energy flow of light usually excites other higher-order nonlinearities, which may be dominant and suppress the occurrence of competing nonlinearities.

Successful alternatives are confined systems, such as graded-index optical fibers [12], nonlinear photonic crystals with defects [13], linear and nonlinear optical lattices [14–19], or optical lattices with defects [20], where the azimuthal instability of vortices can be suppressed by the corresponding confining potentials. Stable vortices with charges lower than two are possible within certain ranges of lattice (transverse refractive index modulation) parameters [14, 17, 21–28]. Different types of vortex solitons, such as discrete vortices [21], vortex-ring "discrete" solitons [28], and second-band vortices [22] were observed in experiments. For a review of early works, see [5, 29, 30] and references therein.

Thus far, dynamics of higher-charged vortex solitons are still poorly understood. The optical settings allowing stable higher-charged vortex solitons are rare. Main efforts in bulk or lattice-modulated nonlinear media were devoted to the analysis of vortex solitons with charges less than or equal to two. In the following sections, three different schemes for the realization of vortex solitons with higher charges will be addressed.

In section 2, the dynamics of vortex solitons in a radial lattice with a lower-index defect covering several lattice rings is revealed. The defect scale can be utilized to control the energy flow of vortices. Vortex solitons with various charges are stable in a region near the upper

©2012 Dong and Li, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Dong and Li, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

cutoffs of propagation constant. Although higher-charged vortices at higher energy flow suffer oscillatory instability, they can survive very long distances without visible distortions. Vortex solitons at lower or moderate energy flow are completely stable under appropriate conditions. The variation of topological charges slightly influences the existence and stability domains of vortex solitons. This property provides an effective way for the experimental realization of vortex solitons with higher charges in an optical setting with fixed parameters.

In section 3, the existence, stability and propagation dynamics of vortex solitons in a defocusing Kerr medium with an imprinted azimuthally modulated Bessel lattice are investigated. Since the special amplitude distribution of vortex soliton resembles to the azimuthons stated in [31] and the phase distribution is also a staircase function of the polar angle, such vortex solitons can also be termed as "azimuthons". The azimuthal refractive index modulation admits stable vortex solitons with lower or higher topological charges. The "stability rule" of azimuthons in defocusing cubic media is exactly opposite to that of vortex solitons in focusing media with the same transverse refractive index modulation. It is the first example of stable azimuthons in local nonlinear media.

In section 4, the stability of vortex solitons supported by a circular waveguide array with out-of-phase modulation of linear and nonlinear refractive indices is studied. The out-of-phase competition between two effects substantially modifies the stability properties of vortex solitons. Vortex solitons undergo remarkable power-dependent shape transformations. They expand or shrink radially with the propagation constant, depending on the phase difference between the neighboring lobes. In particular, we revealed that increasing waveguide number of circular array can stabilize vortex solitons with higher topological charges.
