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Optical Cavities: Free-Space Laser Resonators
Published in Chunlei Guo, Subhash Chandra Singh, Handbook of Laser Technology and Applications, 2021
Iterative computer techniques are commonly used to determine the cavity modes of plane-parallel resonators. In the original paper by Fox and Li [1], iterative solutions to Huygen’s integral in cylindrical symmetry starting with a uniform plane wave were used. It is more common now to use Hankel transform techniques for cylindrical symmetry and Fourier transform techniques for rectangular symmetry [15,16]. A discrete Fourier transform pair such as equations (5.50) and (5.51) is used. The spatial amplitude and phase distribution transmitted through the aperture are transformed into a sum of plane waves propagating at regularly spaced directions with respect to the central direction of propagation. The propagation of the plane waves to the next encounter with the aperture is straightforward, each having a relative phase shift dependent on the direction of propagation. The plane waves are then summed to synthesize the spatial distribution, which is modified by transmission through the aperture. Routines for calculating the Fourier transformations are available [18,19].
A time fractional model of hemodynamic two-phase flow with heat conduction between blood and particles: applications in health science
Published in Waves in Random and Complex Media, 2022
Farhad Ali, Fazli Haq, Naveed Khan, Anees Imtiaz, Ilyas Khan
With the boundary condition in the transformed form: by using the finite Hankel transform of zero-order [63,64] in Equation (22) and applying the boundary condition from Equation (23), we get: where is the finite Hankel transform of zero-order of and are the positive roots of is the Bessel function of the first kind of zero order. Equation (24) can be written in the more suitable form as:
Semi-analytical solution of three-dimensional thermoelastic problem for half-space with gradient coating
Published in Journal of Thermal Stresses, 2018
Roman Kulchytsky-Zhyhailo, Stanisław Jan Matysiak, Adam Stanisław Bajkowski
Assume that the heated region Ω is a circle of radius a and the heat flux q depends only on the r coordinate, where r2 = x2 + y2. Taking into account the relation where Jν(ϖ) is the Bessel function of order ν, the transform may be written in zero-order Hankel transform form:
Theoretical investigation on characteristics of field reactions of saturated ground subjected to vibrations of inclined pile groups
Published in European Journal of Environmental and Civil Engineering, 2022
Wen Liu, Yanlin Zhang, Kang Yao, Li Shi, Dexing Ni
A concentrated force is assumed to act at depth inside the saturated half-space, that is, where and are the unit vectors defining the three coordinate directions; and are the load components along the three directions, respectively; and is the Dirac function. Similarly, the loading vector in Equation (26) can be recast into the frequency–Hankel transform domain as where denotes the number of terms included in the Fourier series; is the order Hankel transform; ‘∼’ denotes variables in the frequency–Hankel transform domain; is the wave number. By satisfying the force equilibrium and displacement compatibility conditions along the plane where the concentrated load is applied, the unknown coefficients defining the fundamental solutions can be determined with additional traction-free and permeable boundary conditions at the surface of the half-space. Consequently, the displacement response can be evaluated for any point of interest in the half-space, that is, the receptance function of the saturated homogeneous half-space is obtained analytically in the transform domain. For detailed derivations of the fundamental solutions, refer to Chen et al.’s study (2007).