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ABSTRACT. A mathematical model was developed to obtain the population inversion of the atoms in laser field in a laser cavity by considering the electric field in the optical cavity and the atomic states of the medium to be quantized. The master equation of the density operator of the laser field was studied analytically and numerically. Using coherent states, the Fokker-Plank equation for the phase space density for the laser field was solved analytically for the time dependent and steady state situations. The laser field above the threshold can be represented by a randomly phased mixture of coherent states. As the pump parameter increases in the laser process, the phase density becomes narrower, tending toward a delta function, creating a coherence laser field.

ABSTRACT. The photonic energy bands of body centered cubic photonic crystals formed from SiO 2 , GaP, Si, InAs, GaAs, InP, Ge and BaSrTiO 3 dielectric spheres drilled in air and air holes drilled in these dielectric mediums were calculated using the plane wave expansion method. The filling factor for each dielectric material was changed until a complete energy gap was obtained and then the density of states was calculated. There were no complete band gaps for air spheres drilled in these eight dielectric mediums. The lattice constants were determined by using wavelengths in the region 1550 nm -1nm . The variation of the band gap widths with the filling factor and the variation of gap width to midgap frequency ratios with dielectric contrast were investigated. The largest band gap width of 0.021 for normalized frequency was obtained for GaP for the filling factor of 0.0736. The mode filed distributions were obtained by guiding a telecommunication wave with 1.55μm wavelength through a 3 3 3 × × photonic cell formed from GaP spheres in air with a filling factor of 0.0736 for transverse electric and magnetic modes.

The Dirac equation consistent with the principles of quantum mechanics and the special theory of relativity, introduces a set of matrices combined with the wave function of a particle in motion to give rise to the relativistic energy-momentum relation. In this paper a new hypothesis, the wave function of a particle in motion is associated with a pair of complementary waves is proposed. This hypothesis gives rise to the same relativistic energy-momentum relation and achieves results identical to those of Dirac. Additionally, both the energy-time and momentum-position uncertainty relations are derived from the complementary wave interpretation. How the complementary wave interpretation of the Dirac equation is related to the time-arrow and the four-vectors are also presented.

The plane wave expansion method was implemented in modelling and simulating the band structures of two dimensional photonic crystals with square, triangular and honeycomb lattices with circular, square and hexagonal dielectric rods and air holes. Complete band gaps were obtained for square lattice of square GaAs rods and honeycomb lattice of circular and hexagonal GaAs rods as well as triangular lattice of circular and hexagonal air holes in GaAs whereas square lattice of square or circular air holes in a dielectric medium ε = 18 gave complete band gaps. The variation of these band gaps with dielectric contrast and filling factor gave the largest gaps for all configurations for a filling fraction around 0.1.The gap maps presented indicated that TM gaps are more favoured by dielectric rods while TE gaps are favoured by air holes. The geometrical gap maps operating at telecommunication wavelength λ = 1.55 μm showed that a complete band gap can be achieved for triangular lattice with circular and hexagonal air holes in GaAs and for honeycomb lattice of circular GaAs rods.