Nonlinear effects and optical bistability in photonic crystals

Nonlinear optics has numerous applications in communications and optical computing, as nonlinear material effects allow for the design of switches, couplers, modulators, and all-optical signal processing systems. Nonlinear optics also has a number of inherent benefits, such as the ability to compensate for dispersion and diffraction effects, as evidenced by temporal and spatial solitons. The idea of combining nonlinear effects with photonic crystals has attracted much attention, since the light guiding and confining properties of the PC structures allow for possible circuit integration. Slow-light enhancement of nonlinear effects in photonic crystal waveguides is also useful when the nonlinear properties of an otherwise desirable material are relatively weak.

The development of a suitable simulation tool for nonlinear optics is crucial if such effects and their applications are ever to be studied. The most common technique is the nonlinear Schrödinger equation, although such approximate techniques are inherently inadequate for describing systems with intricate features over relatively short small lengths, such as photonic crystals. The finite-difference time-domain (FDTD) method of analysis, however, is applicable for a wide range of generalized structures, constrained only by the size of the computational space required for the simulation. Therefore, we have developed our own nonlinear extension to the original linear FDTD method, which is suitable for simulating materials exhibiting anisotropic, χ⁽²⁾, and χ⁽³⁾ nonlinearities. Better than 95% accuracy was calculated for the χ⁽³⁾ code, with similar results expected for the anisotropic and χ⁽²⁾ codes.

In order to verify correct operation of the code, two types of simulations were performed for the well-understood nonlinear optics examples of self-focusing of optical beams in χ⁽³⁾ nonlinear media and second-harmonic generation in χ⁽²⁾ media. In these simulations, a bulk of nonlinear material is excited using a Huygens’ source originating from within a slab waveguide. In the figure below, the field profile is shown for the case of a bulk χ⁽³⁾ material of normalized χ⁽³⁾ = 1 that is excited by a source on the left side propagating from left to right. A qualitative examination of the diagram indicates that the lensing effect of the material focuses an initially divergent beam, causing it to converge by the time is has propagated to the end of the computational domain.

Next, the TM and TE field profiles for a slab of bulk χ⁽²⁾ material are shown the next figure, where the normalized χ⁽²⁾ = 10. The TM field (a) is used to excite the material and the stimulated response shows up in the TE field (b). Again, examining the diagrams qualitatively indicates that a second-harmonic field is generated in the TE field (orthogonal to the TM excitation) and the energy is transferred periodically between the two, as expected. Of course, the energy conversion is not 100%, since there is a mismatch in the group velocities of the TE and TM polarizations due to waveguide dispersion. Also, the null on the axis seen in the second-harmonic field is due to the fact that the transverse magnetic field of the fundamental component has even symmetry and therefore its electric field must vanish on the axis.