Artificial Second Order Non-Linearity in Photonic Crystals
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Artificial Second Order Non-Linearity in Photonic Crystals. A Feigel, Z. Kotler and B. Sfez Electro-Optics Division, NRC Soreq, 81800 Yavne, Israel ABSTRACT We describe a technique for obtaining effective second order non-linearity χ2 in non centro-symmetric Photonic Crystal made from centro-symmetric materials (e.g. glass, Ge or Si). The effect is based on the electric quadrupole transition, strong electromagnetic mode deformation and different contributions to the volume polarization from different parts of the photonic crystal1. INTRODUCTION Many new application based on different physical phenomena are feasible now with the help of photonic crystals. Possibility to design photonic density of states and spatial electromagnetic modes structure open new ways for creation of materials with extraordinary optical properties. The second order nonlinear materials are highly required both for fundamental research and for industrial applications. Unfortunately there are many constrains that limit the choice of such materials: the value of χ2 should be reasonably high, absorption in required spectrum should be low, high damage threshold is required and finally to possess second order non-linearity the materials has to be non-centro-symmetric. The latter immediately eliminates all amorphous materials (like glass or Silicon) and crystals from 11 of 32 symmetry classes. Hence the technology that allows fabrication of non-linear materials from previously unsuitable centrosymmetric substrates can significantly enlarge the material choice for non-linear optics. A local second order polarization P(2) exists even in centro-symmetric materials due to the higher than dipole electromagnetic transitions2. The asymmetry of the electromagnetic field spatial mode leads to quadrupole transition, while dipole transition is based on the asymmetry of the electron wave function. The second order polarization corresponding to a quadrupole transition is: r r r (1) PQ(2 ) = Q M E∇E where Q is a fourth-order tensor. Generally the volume contribution of eq. (1) polarization vanishes, due to periodicity of electromagnetic mode and gradient dependence of quadrupole transition polarization. However the result in properly designed photonic crystals can be quite different. Integration of eq. (1) over the volume in dielectric/air photonic crystal can be different from zero due to not equal contribution to volume polarization from different parts of the crystal. The polarization of the air regions can be totally neglected due to low electron density. Constructing photonic crystal in such a way that in dielectric part the quadrupole polarization has one sign and in the air the opposite, effective ''structural'' volume polarization can be obtained. The required symmetry breaking is introduced on the macroscale of the photonic crystal unit cell, contrary to atomic scale asymmetry in ordinary non-linear materials. Electromagnetic mode r inside photonic crystal can be highly modulated3, leading to large ∇E term.
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Figure 1: Array of waveguides in Phot
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