Impurity modes from impurity clusters in photonic band structures

Abstract

A Green function calculation for the impurity modes of single impurities and clusters of impurities in a photonic band structure is given. Specifically, a truncated two-dimensional periodic dielectric medium composed from a square lattice array of dielectric rods placed perpendicularly between two parallel perfectly conducting plates is studied. Impurities are introduced into this system substitutionally by replacing one (single impurity case) or a number (cluster impurity case) of the rods in the periodic system by impurity rods. Exact expressions for the impurity mode frequencies are obtained in terms of the Green functions of the pure system. These expressions are then evaluated for the case of impurities with small cross-sectional areas using numerically determined Green functions. Linear frequency-independent, linear frequency-dependent and non-linear impurity dielectric media are treated.

Photonic crystal circuits: A theory for two- and three-dimensional networks

“…We will offer plots illustrating the results from some of these expressions for a particular realization of a twodimensional photonic crystal. 9,11 For the particular twodimensional photonic crystal used in our illustration, 9,11 the plots presented in the text and those the reader may wish to generate for the other systems considered in this paper can be easily produced on a pocket calculator using the equations in the text and the numerical results for the coefficients ␣(r,s) given in the Appendix. Finally, we note here that the difference equations studied in this paper are closely related, in the absence of a Kerr nonlinearity, to those studied in Ref.…”

“…We have discussed this evaluation in detail in the case of impurity clusters of highly symmetric geometry in Ref. 9. This discussion can be directly taken over to the new types of impurities considered above.…”

“…11 to illustrate the waveguide modes of an infinitely long waveguide in a two-dimensional photonic crystal and in Ref. 9 to present results on finite clusters of impurities in photonic crystals.͒ In the circuit examples considered in the text, we take the waveguide channels to be oriented along the set of ͓10͔, ͓01͔ directions of the square lattice. Along these directions ␣(0,0) and ␣(1,0), for the photonic crystal described in the above paragraph, are dominant.…”

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14] These are waveguide structures formed in photonic crystals by the addition of a line of site impurities to the photonic crystal. Electromagnetic waveguide modes then propagate along the line of site impurities.…”

“…In addition, the Green's function method can be easily applied to waveguide channels formed from different dielectric materials or materials that have a number of different interconnecting waveguide channels all formed from different types of dielectric materials. In this approach, Green's functions techniques are applied 3, [9][10][11][12][13][14] to particular types of photonic crystal waveguides for which the equations describing the propagation of light in the waveguide channels reduce to a set of difference equations. These difference equations are treated using standard methods to obtain analytic closed-form expressions for the propagation characteristics of photonic crystal circuits.…”