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The addition of orbital angular momentum has been previously shown to stabilise beams of elliptic cross-section. In this article the evolution of such elliptical beams is explored through the use of an approximate methodology based on modulation theory. An approximate method is used as the equations that govern the optical system have no known exact solitary wave solution. This study brings to light two distinct phases in the evolution of a beam carrying orbital angular momentum. The two phases are determined by the shedding of radiation in the form of mass loss and angular momentum loss. The first phase is dominated by the shedding of angular momentum loss through spiral waves. The second phase is dominated by diffractive radiation loss which drives the elliptical solitary wave to a steady state. In addition to modulation theory, the “chirp” variational method is also used to study this evolution. Due to the significant role radiation loss plays in the evolution of an elliptical solitary wave, an attempt is made to couple radiation loss to the chirp variational method. This attempt furthers understanding as to why radiation loss cannot be coupled to the chirp method. The basic reason for this is that there is no consistent manner to match the chirp trial function to the generated radiating waves which is uniformly valid in time. Finally, full numerical solutions of the governing equations are compared with solutions obtained using the various variational approximations, with the best agreement achieved with modulation theory due to its ability to include both mass and angular momentum losses to shed diffractive radiation. [ABSTRACT FROM AUTHOR]

The equations governing optical solitary waves in nonlinear nematic liquid crystals are investigated in both (1+1) and (2+1) dimensions. An isolated exact solitary wave solution is found in (1+1) dimensions and an isolated, exact, radially symmetric solitary wave solution is found in (2+1) dimensions. These exact solutions are used to elucidate what is meant by a nematic liquid crystal to have a nonlocal response and the full role of this nonlocal response in the stability of (2+1) dimensional solitary waves. General, approximate solitary wave solutions in (1+1) and (2+1) dimensions are found using variational methods and they are found to be in excellent agreement with the full numerical solutions. These variational solutions predict that a minimum optical power is required for a solitary wave to exist in (2+1) dimensions, as confirmed by a careful examination of the numerical scheme and its solutions. Finally, nematic liquid crystals subjected to two different external electric fields can support the same solitary wave, exhibiting a new type of bistability. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the interaction of three optical solitary waves propagating with angular momentum in bulk nematic liquid crystals. The resulting cluster of solitary waves, or nematicons, is shown to orbit about its common centre of “mass”. An elongated isosceles triangle configuration is derived, this solution being the equivalent of the Lagrange solution of Newtonian gravitation. This triangle solution is found to be stable owing to diffractive radiation. A modulation theory explains the existence of the triangle solution as due to the non-monotonicity of an effective potential for the interaction of the solitary waves. This modulation theory also gives good agreement with numerical solutions for the trajectories of the nematicons in the three colours. Finally, it is shown that a cut-off in the shed diffractive radiation prevents the break-up of the triangle due to radiative losses. [Copyright &y& Elsevier]

Abstract: The interaction of two symmetric solitary waves, termed nematicons, in a liquid crystal is considered in the limit of nonlocal response of the liquid crystal. This nonlocal limit is the applicable limit for most experimentally available liquid crystals. In this nonlocal limit, two separate cases for the initial separation of the nematicons are considered, these being large and small separation. Both spinning and nonspinning nematicons are considered. It is found that in the case of large initial separation, the nematicons can form a spinning or nonspinning bound state with a finite steady separation, this being called a nematicon dipole, when they are out of phase. On the other hand, well separated, nonspinning, in-phase nematicons attract and merge, while well separated, spinning, in-phase nematicons can either form a bound state or merge into a single nematicon. In the limit of small initial separation, the nematicons rapidly merge when they are in phase. Modulation equations describing the nematicon interaction are derived via suitable trial functions in an averaged Lagrangian. These modulation equations are further modified to include the effect of the diffractive radiation shed as the nematicons evolve. Finally the modulation equations are approximated in order to investigate the various interaction regimes. Good to excellent agreement is found between their solutions and full numerical solutions of the nematicon equations. [Copyright &y& Elsevier]

The work of Kavitha et al. [1] uses Hirota's method to derive N solitary wave solutions of the equations governing the propagation of coherent light through a nematic liquid crystal. However, Hirota's method implicitly assumes that the governing equations possess an inverse scattering solution, which is not the case for the equations governing light propagation in nematic liquid crystals. The solutions derived in [1] are then invalid. Furthermore, these solutions do not satisfy the assumed boundary conditions and have a complex angle of rotation of the nematic molecules, which is unphysical. [ABSTRACT FROM AUTHOR]