![]() ![]() Moreover, the new initialization of the iterative solver extends the practical applicability of the DDA to size parameter equal to 250, 160, and 140 for m = 1.1, 1.2, and 1.3, respectively. The acceleration becomes more pronounced with increasing size, while the WKBr becomes clearly superior to the WKB. It allowed us to accelerate the DDA solution without compromising the final accuracy. Further, we tested the WKB and WKBr as an initial guess in the iterative solution of the linear system in the framework of the discrete dipole approximation (DDA). While keeping the same order of errors with varying (m − 1), they significantly reduce the errors for moderate values of m (from 1.1 to 1.3). These versions additionally account for Fresnel transmission and wavefront focusing for rays entering the scatterer, combination of intersecting rays, and the vanishing of the electric field in the shadow region. We extensively studied several versions WKBr for the case of a homogeneous sphere. Thus, the resulting accuracy is determined by (m − 1) uniformly for all particle sizes, in contrast to the original WKB, which accuracy deteriorates with increasing size. For large particles the WKBr is equivalent to the geometric optics in the limit of relative refractive index m approaching 1. The new method, named WKBr, additionally takes into account refraction (rotation) of the incident rays at the particle-medium interface. We improved the Wentzel-Kramers-Brillouin (WKB) approximation for calculating the electric field inside a scatterer. For practical calculations the À π=2-phase shift is applied each time the Jacobian changes its. At a focal point, this amounts to a phase-shift of À π, depending on the degeneracy of the eigenvalues of the Jacobian DðτÞ ¼ det Q ðτÞ according to. At the caustics, a phase- shift of À π=2 needs to be added to the phase for each focal line that is passed. This is explained in greater detail by Berry or again in the book by Kravtsov. This also leads to the possibility of caustics, which are curves and surfaces on which the intensity of the electro- magnetic field becomes infinite according to geometrical optics. As the cross-section decreases, the amplitude of the electromagnetic oscillation increases correspond- ingly. 4 where the cross-section of the ray segment decreases as the initial wavefront converges to the focal lines 1 and 2. (2) quantifies the influence of the change in cross-sectional area of the ray segment on the amplitude of the electric field vector. The figure also shows the electric field vector EðτÞ as two scalar quantities E r and E b through its projection onto the normal ^ r and binormal vector ^ b associated to the vector ^ k. This model of the electromagnetic field, combining the ray equa- tion (1) and the field equation (2), is illustrated in Fig. The matrix Q is the so-called wavefront curvature matrix and E 0 is the electric field vector at the starting point x 0 of the ray segment. The phase φ ¼ kτ depends linearly on the distance τ along the ray segment starting from x 0 and takes the wavenumber k ¼ 2π=λ, with λ being the wavelength, as a proportionality constant. the time-dependence expðiωtÞ for time t and fre- quency ω has been suppressed.
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