Modeling of defect modes in photonic crystals using the fictitious source superposition method S. Wilcox, 1 L. C. Botten, 2 R. C. McPhedran, 1 C. G. Poulton, 3 and C. Martijn de Sterke 1 1 Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS) and School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia 2 Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS) and Department of Mathematical Sciences, University of Technology, Sydney, New South Wales 2007, Australia 3 High Frequency and Quantum Electronics Laboratory, University of Karlsruhe, 78128 Karlsruhe, Germany Received 13 December 2004; published 12 May 2005 We present an exact theory for modeling defect modes in two-dimensional photonic crystals having an infinite cladding. The method is based on three key concepts, namely, the use of fictitious sources to modify response fields that allow defects to be introduced, the representation of the defect mode field as a superposi- tion of solutions of quasiperiodic field problems, and the simplification of the two-dimensional superposition to a more efficient, one-dimensional average using Bloch mode methods. We demonstrate the accuracy and efficiency of the method, comparing results obtained using alternative techniques, and then concentrate on its strengths, particularly in handling difficult problems, such as where a mode is highly extended near cutoff, that cannot be dealt with in other ways. DOI: 10.1103/PhysRevE.71.056606 PACS numbers: 42.70.Qs, 42.79.Dj I. INTRODUCTION Photonic crystals 1PCs and photonic crystal fibers 2 PCFs are structures in which the refractive index depends periodically on position. Many of the exciting light propaga- tion properties of these structures arise from the existence of photonic band gaps, frequency intervals in which running wave solutions are not allowed. In this paper, we concern ourselves with structures with a two-dimensional plane of periodicity. In PCs, the light propagates in this plane, and the refractive index in the direction orthogonal to this plane is designed for light confinement in that direction. However, for simplicity these structures are often modeled as being uniform in this direction, so that only a two-dimensional cal- culation is required. In PCFs the refractive index in the di- rection orthogonal to the plane of periodicity genuinely is uniform, but in contrast to two-dimensional PCs, the light propagation is not confined to this plane. Many applications of PCs and PCFs rely on the introduc- tion of defects in an otherwise periodic structure: in PCs, line defects lead to waveguides 3, whereas localized defects give rise to resonators 4,5, while in PCFs, a localized de- fect forms the core of the fiber 6. These applications re- quire a photonic band gap to confine the light to these de- fects. Until now, almost all the modeling of PCs or PCFs with defects has used one of two general methods. The first of these are supercell methods, by which the geometry is repeated periodically 7–9. This turns a nonperiodic struc- ture into one with periodicity, so that methods that have been developed for periodic structures, relying principally on Bloch’s theorem, can be brought to bear. The use of such a procedure is justified when the size of the supercell exceeds the size of any of the features that are to be modeled since, otherwise, artificial overlaps are introduced. The second class of methods to deal with periodic structures with defects model these structures as being of finite extent 10–12of- ten with the use of appropriate absorbing boundary condi- tions to make finite the solution region. Of course, this is well justified since real systems are finite. However, these methods tend not be very efficient since an overarching mathematical framework, such as Bloch’s theorem, is miss- ing. Accordingly, the physical size of the systems that can be modeled is limited. Photonic crystal waveguides may be studied also using methods that fit into neither of the two categories mentioned above. These methods 13 make use of the reflection scat- tering matrix of semi-infinite photonic crystals 14. This matrix R  describes the response of a semi-infinite photonic crystal to an incident plane wave of given frequency, polar- ization, and direction. The response consists of a number of reflected plane waves at angles given by the grating equa- tion, and an infinite number of evanescent orders. The col- umns of R  contain the amplitudes of the reflected waves generated by the respective incoming plane wave orders of unit amplitude. Thus, with this tool, a straight photonic crys- tal waveguide can be modeled as a conventional waveguide: it consists of a uniform medium, sandwiched between two semi-infinite media, the response of which is completely de- termined by R  , and by the separation of the semi-infinite media. In such calculations the modes are found by applying a phase or resonance condition. Calculations of this type, which model a genuinely infinite system, have advantages over conventional methods involving a supercell: they are not only more elegant, but they are also efficient since the calculation of R  involves the Bloch functions of the infinite system. They are also particularly well suited to calculations when the modal fields have a large extent such as occurs close to modal cutoff, since the modeling of these fields would require a very large supercell. By construction, the method described in the previous paragraph applies only to line defects, but not to defects that are localized. In this paper, however, we describe a method for calculating the modes of localized defects in otherwise infinite structures, both in PCs and in PCFs. This fictitious source superposition FSS method shares the advantages listed above: it is theoretically elegant, analytically tractable, PHYSICAL REVIEW E 71, 056606 2005 1539-3755/2005/715/05660611/$23.00 ©2005 The American Physical Society 056606-1