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Unformatted text preview: Photonic band structure in periodic dielectric structures Mustafa Muhammad Department of Physics University of Cincinnati Cincinnati, Ohio 45221 December 4, 2001 Abstract Recent experiments have found the existence of photon bands in periodic dielectric structures analogous to the electron bands in the solid. One-dimensional crystals are considered here. 1 Introduction It is well known that the electron forms energy bands in periodic crystals. the deviation from the free-particle dispersion may be thought to be caused by the coherent interference of scattering of electrons from individual atoms. This leads to the formation of gaps and other characteristic aspects in the electron band structure. Analogously, any particle would coherently scatter and form energy bands in a medium that provides a periodic scattering potential with a length scale comparable to the wavelength of the particle. Specifically this should be true for the propagation of classical electromagnetic (EM) waves in periodic dielectric structures. 2 Electromagnetism in Mixed Dielectric Media All of macroscopic electromagnetism, including the propagation of light in a photonic crystal, is governed by the four macroscopic Maxwell equations. In cgs units, they are, .B = 0 (1) E + 1 c B t = 0 (2) .D = 4 (3) H- 1 c D t = 4 c J (4) where (respectively) E and H are the macroscopic electric and magnetic fields, D and B are the displacement and magnetic induction fields, and and J are the free charges and currents. We will restrict ourselves to propagation within a mixed dielectric medium, a composite regions of homogeneous dielectric material, with no free charges and currents. With this type of medium in mind, in which light propagates but there are no sources of light, we can set = J = 0. We have D ( r ) = ( r ) E ( r ). However, for most dielectric materials of interest, the magnetic permeability is very close to unity and we may set B = H . With all of these assumptions in place, the Maxwell equations(1-4) become .H ( r, t ) = 0 (5) E ( r, t ) + 1 c H ( r, t ) t = 0 (6) .E ( r, t ) = 0 (7) H ( r, t )- ( r ) c E ( r, t ) t = 0 . (8) We emply the familier trick of using a complex-valued field for mathematical convenience, re- membering to take the real part to obtain the physical fields. This allowa us to write a harmonic mode as a certain field pattern times a complex exponential: H ( r, t ) = H ( r ) e it (9) E ( r, t ) = E ( r ) e it (10) To find equations for mode profiles of a given frequency, we insert the above equations into (5-8). The two divergence equations give the simple conditions: .H ( r ) = .D ( r ) = 0 (11) These equations have a simple physical interpretation. There are no point sources or sinks of displacement and magnetic fields in the medium. Alternatively, the field configurations are built up of electromagnetic waves that are transverse. We can focus on the other two Maxwell equations asof electromagnetic waves that are transverse....
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