Based on the symmetry considerations, the general form of the magnetic field vector of a TE-polarized mode and the electric field vector of a TM-polarized mode expanded into plane wave vector [??] with respect to the 2D reciprocal lattice vector [??], labeled with a Bloch wave
number [k.sub.y], which is given by .
For two-dimensional (2D) interfaces, since the electron transport properties depend not only on the incident electron energy across the interfaces but also on the 2D momentum parallel to the interfaces, the computational load for calculating the Green's functions at each incident energy using a 2D Bloch wave
vector has become severely demanding.
The dispersion relation for TM waves in the infinite periodic structure, which relates the frequency [omega], the longitudinal wave number [k.sub.x] and the Bloch wave
number [bar.k], can be written as 
Due to the inherent loss in InSb, it is found that the photonic band structure of an SDPC should be complex, i.e., a complex Bloch wave
number should exist in both the PBGs and passbands.
In this equation K is the Bloch wave
vector that takes the values of the reduced Brillouin zone, 0 [less than or equal to] K [less than or equal to] [pi] / a; ([m.sub.11], [m.sub.12]; [m.sub.21], [m.sub.22]) is the transfer matrix of the unit cell.
The Bloch wave
vector K can be determined by the half trace of the translational matrix in Eq.
where K is the Bloch wave
vector, and the amplitude is a periodic function of spatial periodicity, i.e., [E.sub.K] (z + [LAMBDA]) = [E.sub.K] (z).
This features resemble those appearing in the band structure associated to two-dimensional photonic crystals where the dispersion relation depends on a Bloch wave
vector with two components.
In sections on acoustic waves in sonic crystals, elastic waves in phononic crystals, and wave phenomena in phononic crystals, he considers such topics as scalar waves in periodic media, sonic crystals, phononic crystals for surface and plate waves, coupling acoustic and elastic waves in phononic crystals, evanescent Bloch waves
, and spatial and temporal dispersion.
where [[omega].sub.R] = 1/ [square root of ([L'.sub.R][C'.sub.R])], [[omega].sub.E] = 1/ [square root of ([L'.sub.L][C'.sub.R])] and d and [beta] are a length of the unit cell and a phase constant for Bloch waves
Theory of dynamic scattering and phase contrast formation is now well developed for multislice and Bloch waves
Among his topics are from classical bodies to microscopic particles, electrons in crystals and Bloch waves
in crystals, the tight-binding model and embedded-atom potentials, transition metals, high-temperature creep, and modeling kinetic processes.