MIcro machining of materials with high power ultra-short-pulsed lasers is becoming a preferred technique to obtain cleaner surface characteristics. Due to the short duration of the pulse, there is insufficient time to establish the thermal equilibrium. Consequently, ablation does not pass through the melting phase. Instead, it proceeds mainly with direct removal of the material at the molecular level. To fully benefit from these properties, a high quality beam profile is required. However, during processing the optical wave front suffers distortions while passing through the medium such as air. Passage through the medium causes the beam to self-focus and the gas breaks down, thus generating plasma, which distorts the geometrical and energy profiles of the beam. This phenomenon offsets the advantages of the procedure to a certain extent. For these reasons, processing is usually conducted in vacuum with associated inconvenience and expense. As a step towards improvement over the technique, we develop a numerical scheme to determine the beam profile in air medium. The profile of the beam is then used to determine the shape of the processed surface by a geometrical method developed recently. The calculated surface profile is compared with the experimental observations with good agreement. This provides a method to develop an understanding of the interactions of the laser beam, air and the material.
The dielectric distribution and polarizability of certain materials, e.g., the liquid dielectrics, change in response to the external electromagnetic field. Since their photonic properties can be adjusted by controlling the applied field, these materials can be used to construct tunable photonic band gap crystals. Due to recent advances in tunable photonic bandgap materials technology, it has become necessary to determine the properties of the propagating fields accurately. Numerical methods currently in use are quite cumbersome and place limits on the accuracy of the solutions. A numerical scheme is developed here by expressing the solution in the framework of the Feynman path integral formulation of quantum mechanics. The formulation describes the evolution of the solution in terms of a propagator, which can be determined by the method of fast Fourier transforms. The resulting numerical scheme is more efficient and reliable than other similar methods.
A photonic band gap is determined by its boundaries, which are frequently computed by the Rayleigh-Ritz method, with the plane wave or the finite element basis functions. This method produces a sequence of upper bounds. Since there are no error estimates available on these approximations, the extent of the band gap is not accurately determined, particularly as this method is also known to suffer from a poor rate of convergence for the cases of interest. We adopt the method of intermediate problems to develop a procedure to calculate the lower bounds to the photonic band gap edges. The lower and the upper bounds supplement each other to determine a band gap with arbitrary accuracy, which is essential for designing the photonic band gap material. Computation of the lower bounds requires only slightly more effort than the upper bounds to produce the approximations with comparable accuracy. An alternative method to determine upper bounds is also developed in the process.
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