We present in this paper a spectrally accurate numerical method for computing the spherical/vector spherical harmonic expansion of a function/vector field with given (elemental) nodal values on a spherical surface. Built upon suitable analytic formulas for dealing with the involved highly oscillatory integrands, the method is robust for high mode expansions. We apply the numerical method to the simulation of three-dimensional acoustic and electromagnetic multiple scattering problems. Various numerical evidences show that the high accuracy can be achieved within reasonable computational time. This also paves the way for spectral-element discretization of 3D scattering problems reduced by spherical transparent boundary conditions based on the Dirichlet-to-Neumann map.
The analytical expansion of linearly, azimuthally, and radially polarized rigorous beam-type solutions of Maxwell's equations into vector spherical harmonics (VSHs) is presented. We report on the dominance of higher order multipoles in highly focused radially and azimuthally polarized beams compared to linearly polarized beams under similar conditions. Furthermore, we theoretically investigate a scenario in which highly focused azimuthally and radially polarized beams interact with a linear polarizer placed in the focal plane and expand the resulting fields into VSHs. The generalized Mie theory is used afterwards to investigate the scattering of the studied beams off a spherical gold nanoparticle.
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Until recently there has been nothing in the geomagnetic literature giving the Gauss coefficients (equivalent to magnetic multipole moments) for the magnetic scalar potential produced outside a finite-sized region of electric current. Nor has there been an expression for the corresponding magnetic vector potential. This paper presents a simple expression for the Gauss coefficients in terms of a volume integral over the current, and also a series expansion of the vector potential in terms of these coefficients. We show how our result is related to the classical expressions for the scalar potential given by a spherical current sheet, and to the results of the recent papers by Engels and Olsen (1998), Stump and Pollack (1998) and Kazantsev (1999).
In considering the main geomagnetic field outside the Earth, most workers specify the field by its scalar potential expanded in terms of spherical harmonics, and the corresponding Gauss coefficients, which are scaled versions of the classical multipole moments. There is a similar approach in electrostatics, and many classical texts show how to calculate these moments by integrating over the electric charges (or the equivalent magnetic monopoles) that are the source of the field. However the source of the main geomagnetic field is not monopoles but electric currents, and there did not appear to be in the literature general expressions for calculating the moments (higher than the dipole) by integrating over the current system. Nor were there readily available expressions for the equivalent vector potential distribution. The present paper derives explicit expressions for the Gauss coefficients as integrals over an arbitrary current distribution. It also presents the vector potential analogue of the scalar spherical harmonics, and relates the moments in the two approaches. It compares our results with those of previous workers in a consistent notation.
Outside the source region the same technique of expanding as a series can be applied, though of course we now have to use vector algebra. For example, Backus et al. (1996) expanded using the spherical harmonic approach, and by comparing the expressions for the resulting radial field terms with those given by the scalar potential (16), derived a general expression for the in terms of integrals involving derivatives of J(s)
We saw in Section 1.2 that a particular external scalar potential came from that part of the charge/pole source distribution in a thin spherical shell that was proportional to . Because of the orthogonality of the , the total moment could be obtained by (in effect) performing a spherical harmonic analysis of the charge/pole source distribution within each shell, then weighting by when integrating over radius to give the total moment. What is now described is the equivalent for an arbitrary current distribution. The middle term of (20) shows that (when integrated over the source region) any radial component of current produces no external field, so in our spherical shell we need consider only the tangential components of current. As our basis functions we use the dimensionless surface vector harmonics Kazantsev (1999) called ,
(Note that, by analogy with the moments we used for the magnetic scalar potential, we use for the corresponding moments for the vector potential; there should not be any confusion with the superscript m used to denote spherical harmonic order.) For a given the field curl given by this vector potential approach, must have the same field geometry as the field given by a scalar potential approach; the two fields can differ only by a constant factor. It is straightforward to show that
Unlike Engels and Olsen (1998) and Kazantsev (1999), the paper by Stump and Pollack (1998), was restricted to the calculation of the field produced outside the current distribution on a single surface, and they apparently did not know of the Chapman and Bartels (1940) current function approach. As in the other two papers they expanded the current density at radius s in the form (their vector spherical harmonics had the opposite sign)
giving the external vector potential in terms of the Gauss coefficients and the surface vector harmonics . (Of course when using vector potential, possible gauge transformations mean that in a particular situation different approaches might give different A(r); however the difference will simply be the gradient of a scalar function, and will not affect the resultant field.)
The spherical harmonics are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present. Some care must be taken in identifying the notational convention being used. In this entry, is taken as the polar (colatitudinal) coordinate with , and as the azimuthal (longitudinal) coordinate with . This is the convention normally used in physics, as described by Arfken (1985) and the Wolfram Language (in mathematical literature, usually denotes the longitudinal coordinate and the colatitudinal coordinate). Spherical harmonics are implemented in the Wolfram Language as SphericalHarmonicY[l, m, theta, phi].
(Arfken 1985, p. 681). Here, denotes the complex conjugate and is the Kronecker delta. Sometimes (e.g., Arfken 1985), the Condon-Shortley phase is prepended to the definition of the spherical harmonics.
In this paper we present the first practical method for importance sampling functions represented as spherical harmonics (SH). Given a spherical probability density function (PDF) represented as a vector of SH coefficients, our method warps an input point set to match the target PDF using hierarchical sample warping. Our approach is efficient and produces high quality sample distributions. As a by-product of the sampling procedure we produce a multi-resolution representation of the density function as either a spherical mip-map or Haar wavelet. By exploiting this implicit conversion we can extend the method to distribute samples according to the product of an SH function with a spherical mip-map or Haar wavelet. This generalization has immediate applicability in rendering, e.g., importance sampling the product of a BRDF and an environment map where the lighting is stored as a single high-resolution wavelet and the BRDF is represented in spherical harmonics. Since spherical harmonics can be efficiently rotated, this product can be computed on-the-fly even if the BRDF is stored in local-space. Our sampling approach generates over 6 million samples per second while significantly reducing precomputation time and storage requirements compared to previous techniques.
Abstract:Linearizations of the spherical harmonic discrete ordinate method (SHDOM) by means of a forward and a forward-adjoint approach are presented. Essentially, SHDOM is specialized for derivative calculations and radiative transfer problems involving the delta-M approximation, the TMS correction, and the adaptive grid splitting, while practical formulas for computing the derivatives in the spherical harmonics space are derived. The accuracies and efficiencies of the proposed methods are analyzed for several test problems.Keywords: 3D radiative trasnfer; SHDOM; Jacobian; linearization; adjoint radiative transfer
is the solid angle subtended by each sample, [4] and definitions for (the spherical harmonic basis functions) can be found in Ramamoorthi and Hanrahan 2001b, Green 2003, or any spherical harmonics reference.
While a detailed discussion of spherical harmonics is outside the scope of this chapter, the References section lists a number of excellent articles that provide in-depth explanations (and other applications) of spherical harmonics. In order to use spherical harmonics for computing irradiance maps, all we need to know are some basic properties:
The last property (known as the convolution theorem in signal-processing parlance) is an extremely powerful one, and the reason why we can use spherical harmonics to efficiently generate irradiance maps: Given two functions that we want to convolve (the Lambertian BRDF and a sampled lighting environment, for example), we can compute the convolved result by separately projecting each function into an lth-order spherical harmonic representation, and then multiplying and summing the resulting (l + 1)2 coefficients. This is more efficient than brute-force convolution because for diffuse lighting (and low-exponent specular), the coefficients (Equation 10-3) rapidly approach 0 as l increases, so we can generate very accurate approximations to the true diffuse convolution with a very small number of terms. In fact, Ramamoorthi and Hanrahan (2001a) demonstrate that l = 2 (9 coefficients) is all that is required for diffuse lighting, and l = is sufficient for specular lighting (where s is the Phong exponent). This reduces the runtime complexity of the example convolution to just 9 x 64 x 64 x 6 221 thousand operations per function (442 thousand total, 1 for the lighting environment, and 1 for the reflection function), compared to the 151 million operations using the brute-force technique. 2ff7e9595c
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