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Journal articles on the topic 'Sommerfeld integrals'

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Published: 4 June 2021
Last updated: 5 February 2022

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1

Chew, W. C. "Sommerfeld integrals for left-handed materials." Microwave and Optical Technology Letters 42, no. 5 (2004): 369–73. http://dx.doi.org/10.1002/mop.20307.

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2

Shanin, A. V., and A. I. Korolkov. "Sommerfeld-type integrals for discrete diffraction problems." Wave Motion 97 (September 2020): 102606. http://dx.doi.org/10.1016/j.wavemoti.2020.102606.

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3

Singh, Surendra, and Ritu Singh. "Computation of Sommerfeld integrals using tanh transformation." Microwave and Optical Technology Letters 37, no. 3 (March 19, 2003): 177–80. http://dx.doi.org/10.1002/mop.10860.

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4

Dvorak, Steven L. "Numerical computation of 2D Sommerfeld integrals—Decomposition of the angular integral." Journal of Computational Physics 94, no. 1 (May 1991): 253–54. http://dx.doi.org/10.1016/0021-9991(91)90156-f.

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5

Dvorak, Steven L., and Edward F. Kuester. "Numerical computation of 2D sommerfeld integrals— Deccomposition of the angular integral." Journal of Computational Physics 98, no. 2 (February 1992): 199–216. http://dx.doi.org/10.1016/0021-9991(92)90138-o.

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6

Dvorak, Steven L., and Edward F. Kuester. "Numerical computation of 2D Sommerfeld integrals—decomposition of the angular integral." Journal of Computational Physics 98, no. 1 (January 1992): 178. http://dx.doi.org/10.1016/0021-9991(92)90184-z.

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7

Petrović, Vladimir V., and Antonije R. Djordjević. "General singularity extraction technique for reflected Sommerfeld integrals." AEU - International Journal of Electronics and Communications 61, no. 8 (September 2007): 504–8. http://dx.doi.org/10.1016/j.aeue.2006.08.006.

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8

Hubral, Peter, and Martin Tygel. "Transient response from a planar acoustic interface by a new point‐source decomposition into plane waves." GEOPHYSICS 50, no. 5 (May 1985): 766–74. http://dx.doi.org/10.1190/1.1441951.

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For the wave field of a point source in full space there currently exist two classical decompositions into plane waves. The wave field can be decomposed into either (a) homogeneous and horizontally propagating (vertically attenuated) inhomogeneous plane waves by using the so‐called Sommerfeld‐Weyl integral, or (b) upgoing and downgoing homogeneous plane waves only using the Whittaker integral. Transient representations of both integrals exist. We propose a new decomposition integral that has a greater flexibility than both classical decompositions. Solutions for the point‐source reflection/transmission response from a planar interface, if based on the Sommerfeld‐Weyl integral, have for instance inherently an infinite integration limit. With the new formula, by which the wave field of a transient point source is decomposed into upgoing and downgoing homogeneous as well as horizontally propagating inhomogeneous transient plane waves, the point‐source response is directly obtained in the form of an integral with a finite integration limit. It can also be interpreted in terms of certain plane waves by which the point source is simulated in a new manner. For that matter, solutions based on the new integral readily reveal the “evanescent” or “nonray” character of the point source. The new formula may be considered an extension of the Sommerfeld‐Weyl or Whittaker integral. It can be used to compute reflection/transmission responses in a compact form in situations where the Sommerfeld‐Weyl integral was hitherto considered fundamental.
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9

Petrovic, Vladimir, Aleksandra Krneta, and Branko Kolundzija. "Singularity extraction for reflected Sommerfeld integrals over multilayered media." Telfor Journal 6, no. 2 (2014): 137–41. http://dx.doi.org/10.5937/telfor1402137p.

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10

Annaert, G. "Evaluation of Sommerfeld integrals using Chebyshev decomposition (antenna analysis)." IEEE Transactions on Antennas and Propagation 41, no. 2 (1993): 159–64. http://dx.doi.org/10.1109/8.214606.

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11

Jackson, D., and N. Alexopoulos. "An asymptotic extraction technique for evaluating Sommerfeld-type integrals." IEEE Transactions on Antennas and Propagation 34, no. 12 (December 1986): 1467–70. http://dx.doi.org/10.1109/tap.1986.1143774.

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12

Tygel, Martin, and Peter Hubral. "Transient analytic point‐source response of a layered acoustic medium: Part II." GEOPHYSICS 50, no. 9 (September 1985): 1478–87. http://dx.doi.org/10.1190/1.1442015.

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In part I (this issue) analytic methods are described for propagating directly in the time domain the full wave field of transient plane waves at arbitrary incidence angles through parallel acoustic homogeneous layers. The theory of part I is used here to compute transient point‐source responses by way of integrating the transient plane‐wave responses over various (real and nonreal) incidence angles. At first the complete transient point‐source reflection (transmission) responses are formulated with the help of the transient Sommerfeld‐Weyl integral that was developed previously by us. This leads to a transient solution in form of an infinite integral over incidence angles involving an (angle‐dependent) integrand that is a time‐convolution between a transient reflectivity (transmissivity) function and an inverse transient analytic square root operator. The transient Sommerfeld‐Weyl integral solution is then proven to be causal. This causality provides the starting point for formulating exact transient point‐source responses in terms of integrals over a finite range of incidence angles only.
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13

Ma, Junjie. "Fast and high-precision calculation of earth return mutual impedance between conductors over a multilayered soil." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 37, no. 3 (May 8, 2018): 1214–27. http://dx.doi.org/10.1108/compel-09-2017-0408.

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Purpose Solutions for the earth return mutual impedance play an important role in analyzing couplings of multi-conductor systems. Generally, the mutual impedance is approximated by Pollaczek integrals. The purpose of this paper is devising fast algorithms for calculation of this kind of improper integrals and its applications. Design/methodology/approach According to singular points, the Pollaczek integral is divided into two parts: the finite integral and the infinite integral. The finite part is computed by combining an efficient Levin method, which is implemented with a Chebyshev differential matrix. By transforming the integration path, the tail integral is calculated with help of a transformed Clenshaw–Curtis quadrature rule. Findings Numerical tests show that this new method is robust to high oscillation and nearly singularities. Thus, it is suitable for evaluating Pollaczek integrals. Furthermore, compared with existing method, the presented algorithm gives high-order approaches for the earth return mutual impedance between conductors over a multilayered soil with wide ranges of parameters. Originality/value An efficient truncation strategy is proposed to accelerate numerical calculation of Pollaczek integral. Compared with existing algorithms, this method is easier to be applied to computation of similar improper integrals, such as Sommerfeld integral.
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14

Alcalá Ochoa, Noé. "A unifying approach for the vectorial Rayleigh–Sommerfeld diffraction integrals." Optics Communications 448 (October 2019): 104–10. http://dx.doi.org/10.1016/j.optcom.2019.05.021.

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15

Drachman, B., M. Cloud, and D. P. Nyquist. "Accurate evaluation of Sommerfeld integrals using the fast Fourier transform." IEEE Transactions on Antennas and Propagation 37, no. 3 (March 1989): 403–6. http://dx.doi.org/10.1109/8.18740.

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16

Wu, Bi-Yi, and Xin-Qing Sheng. "On the Asymptotics of Sommerfeld Integrals Over an Impedance Plane." IEEE Transactions on Antennas and Propagation 68, no. 4 (April 2020): 3318–22. http://dx.doi.org/10.1109/tap.2019.2943332.

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17

Ung, Bora, and Yunlong Sheng. "Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited." Optics Express 16, no. 12 (June 4, 2008): 9073. http://dx.doi.org/10.1364/oe.16.009073.

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18

Bruno, O. P., and C. Pérez-Arancibia. "Windowed Green function method for the Helmholtz equation in the presence of multiply layered media." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2202 (June 2017): 20170161. http://dx.doi.org/10.1098/rspa.2017.0161.

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This paper presents a new methodology for the solution of problems of two- and three-dimensional acoustic scattering (and, in particular, two-dimensional electromagnetic scattering) by obstacles and defects in the presence of an arbitrary number of penetrable layers. Relying on the use of certain slow-rise windowing functions, the proposed windowed Green function approach efficiently evaluates oscillatory integrals over unbounded domains, with high accuracy, without recourse to the highly expensive Sommerfeld integrals that have typically been used to account for the effect of underlying planar multilayer structures. The proposed methodology, whose theoretical basis was presented in the recent contribution (Bruno et al. 2016 SIAM J. Appl. Math. 76 , 1871–1898. ( doi:10.1137/15M1033782 )), is fast, accurate, flexible and easy to implement. Our numerical experiments demonstrate that the numerical errors resulting from the proposed approach decrease faster than any negative power of the window size. In a number of examples considered in this paper, the proposed method is up to thousands of times faster, for a given accuracy, than corresponding methods based on the use of Sommerfeld integrals.
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19

Parise, Mauro. "On the Electromagnetic Field of an Overhead Line Current Source." Electronics 9, no. 12 (November 27, 2020): 2009. http://dx.doi.org/10.3390/electronics9122009.

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This work presents an analytical series-form solution for the time-harmonic electromagnetic (EM) field components produced by an overhead current line source. The solution arises from casting the integral term of the complete representation for the generated axial electric field into a form where the non-analytic part of the integrand is expanded into a power series of the vertical propagation coefficient in the air space. This makes it possible to express the electric field as a sum of derivatives of the Sommerfeld integral describing the primary field, whose explicit form is known. As a result, the electric field is given as a sum of cylindrical Hankel functions, with coefficients depending on the position of the field point relative to the line source and its ideal image. Analogous explicit expressions for the magnetic field components are obtained by applying Faraday’s law. The results from numerical simulations show that the derived analytical solution offers advantages in terms of time cost with respect to conventional numerical schemes used for computing Sommerfeld-type integrals.
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20

Ma, Junjie, and Shuhuang Xiang. "High-order fast integration for earth-return impedance between underground and overhead conductors in Matlab." COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 33, no. 5 (August 26, 2014): 1809–18. http://dx.doi.org/10.1108/compel-12-2013-0416.

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Purpose – The earth-return mutual impedances between underground and overhead conductors can be expressed by Pollaczek integrals. Many efforts have been exerted to calculating this kind of integrals. However, most of numerical methods turn out to be time-consuming as integrands become highly oscillatory and strongly singular. Therefore, efficient algorithms should be devised. The paper aims to discuss these issues. Design/methodology/approach – The paper separates the singularity from the whole integral and couple with the singularity and oscillation, respectively. A sinh transformation is applied for the finite part and complex integration method is used to calculate the tail. Findings – Numerical experiments show that the given method shares the property that the stronger the singularity and the higher the oscillation, the better the accuracy of the calculation. Originality/value – The sinh transformation is first proposed to calculate Pollaczek integrals. This efficient algorithm can be used to evaluate mutual impedances between conductors. Also, it provides a new aspect of the research on fast calculation of Pollaczek integrals and Sommerfeld integrals.
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21

Michalski, K. A., and C. M. Butler. "Evaluation of Sommerfeld integrals arising in the ground stake antenna problem." IEE Proceedings H Microwaves, Antennas and Propagation 134, no. 1 (1987): 93. http://dx.doi.org/10.1049/ip-h-2.1987.0018.

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22

Fukushima, Toshio. "Analytical computation of generalized Fermi–Dirac integrals by truncated Sommerfeld expansions." Applied Mathematics and Computation 234 (May 2014): 417–33. http://dx.doi.org/10.1016/j.amc.2014.02.053.

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23

Dvorak, Steven L., and Edward F. Kuester. "Numerical computation of 2D sommerfeld integrals— A novel asymptotic extraction technique." Journal of Computational Physics 98, no. 2 (February 1992): 217–30. http://dx.doi.org/10.1016/0021-9991(92)90139-p.

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24

Dvorak, Steven L., and Edward F. Kuester. "Numerical computation of 2D Sommerfeld integrals—A novel asymptotic extraction technique." Journal of Computational Physics 98, no. 1 (January 1992): 178. http://dx.doi.org/10.1016/0021-9991(92)90185-2.

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25

Cui, T. J., and W. C. Chew. "Efficient Evaluation of Sommerfeld Integrals for Tm Wave Scattering By Buried Objects." Journal of Electromagnetic Waves and Applications 12, no. 5 (January 1998): 607–57. http://dx.doi.org/10.1163/156939398x00160.

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26

Tsai, Ming-Ju, Chinglung Chen, and Nicolaos G. Alexopoulos. "Sommerfeld Integrals in Modeling Interconnects and Microstrip Elements in Multi-Layered Media." Electromagnetics 18, no. 3 (May 1998): 267–88. http://dx.doi.org/10.1080/02726349808908587.

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27

Ge, Yuehe, and Karu P. Esselle. "A fast and general complex image method for evaluating the Sommerfeld integrals." Microwave and Optical Technology Letters 30, no. 1 (2001): 24–26. http://dx.doi.org/10.1002/mop.1209.

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28

MOSIG, JUAN R., and KRZYSZTOF A. MICHALSKI. "Sommerfeld Integrals and Their Relation to the Development of Planar Microwave Devices." IEEE Journal of Microwaves 1, no. 1 (2021): 470–80. http://dx.doi.org/10.1109/jmw.2020.3032399.

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29

Ochoa, Noé Alcalá. "Alternative approach to evaluate the Rayleigh-Sommerfeld diffraction integrals using tilted spherical waves." Optics Express 25, no. 10 (May 12, 2017): 12008. http://dx.doi.org/10.1364/oe.25.012008.

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30

Arnoldus, Henk F. "Numerical evaluation of Sommerfeld-type integrals for reflection and transmission of dipole radiation." Computer Physics Communications 257 (December 2020): 107510. http://dx.doi.org/10.1016/j.cpc.2020.107510.

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31

Jiang, Mei Ping, Ning Ning Yu, Yi Jin, Xing Fang Jiang, and Bin Tang. "Far-Field Properties of Vectorial Cosh-Squared-Gaussian Beams beyond the Paraxial Approximation." Applied Mechanics and Materials 143-144 (December 2011): 204–10. http://dx.doi.org/10.4028/www.scientific.net/amm.143-144.204.

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The concept of vectorial nonparaxial cosh-squared-Gaussian (ChsG) beams is introduced and their free space propagation equations are derived analytically by using the vectorial Rayleigh-Sommerfeld diffraction integrals, and far-field expressions as well as paraxial propagation formula are obtained in free space. Numerical calculations and analysis show that, the f-parameter and decentered parameter plays an important role in determining the far-field properties of the vectorial nonparaxial ChsG beams.
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32

Ioannidi, K., Ch Christakis, S. Sautbekov, P. Frangos, and S. K. Atanov. "The Radiation Problem from a Vertical Hertzian Dipole Antenna above Flat and Lossy Ground: Novel Formulation in the Spectral Domain with Closed-Form Analytical Solution in the High Frequency Regime." International Journal of Antennas and Propagation 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/989348.

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We consider the problem of radiation from a vertical short (Hertzian) dipole above flat lossy ground, which represents the well-known “Sommerfeld radiation problem” in the literature. The problem is formulated in a novel spectral domain approach, and by inverse three-dimensional Fourier transformation the expressions for the received electric and magnetic (EM) field in the physical space are derived as one-dimensional integrals over the radial component of wavevector, in cylindrical coordinates. This formulation appears to have inherent advantages over the classical formulation by Sommerfeld, performed in the spatial domain, since it avoids the use of the so-called Hertz potential and its subsequent differentiation for the calculation of the received EM field. Subsequent use of the stationary phase method in the high frequency regime yields closed-form analytical solutions for the received EM field vectors, which coincide with the corresponding reflected EM field originating from the image point. In this way, we conclude that the so-called “space wave” in the literature represents the total solution of the Sommerfeld problem in the high frequency regime, in which case the surface wave can be ignored. Finally, numerical results are presented, in comparison with corresponding numerical results based on Norton’s solution of the problem.
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33

Wu, M., R. G. Olsen, and S. W. Plate. "Wideband Approximate Solutions for the Sommerfeld Integrals Arising in the Wire over Earth Problem." Journal of Electromagnetic Waves and Applications 4, no. 6 (January 1, 1990): 479–504. http://dx.doi.org/10.1163/156939390x00159.

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34

Kurup, Dhanesh G. "Spatial Domain Green's Functions of Layered Media Using a New Method for Sommerfeld Integrals." IEEE Microwave and Wireless Components Letters 22, no. 4 (April 2012): 161–63. http://dx.doi.org/10.1109/lmwc.2012.2188020.

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35

Kaifas, T. N. "Direct Rational Function Fitting Method for Accurate Evaluation of Sommerfeld Integrals in Stratified Media." IEEE Transactions on Antennas and Propagation 60, no. 1 (January 2012): 282–91. http://dx.doi.org/10.1109/tap.2011.2167915.

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36

Esina, Anna I., and Andrei I. Shafarevich. "SEMICLASSICAL ASYMPTOTICS OF EIGENVALUES FOR NON-SELFADJOINT OPERATORS AND QUANTIZATION CONDITIONS ON RIEMANN SURFACES." Acta Polytechnica 54, no. 2 (April 30, 2014): 101–5. http://dx.doi.org/10.14311/ap.2014.54.0101.

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This paper reports a study of the semiclassical asymptotic behavior of the eigenvalues of some nonself-adjoint operators that are important for applications. These operators are the Schrödinger operator with complex periodic potential and the operator of induction. It turns out that the asymptotics of the spectrum can be calculated using the quantization conditions. These can be represented as the condition that the integrals of a holomorphic form over the cycles on the corresponding complex Lagrangian manifold, which is a Riemann surface of constant energy, are integers. In contrast to the real case (the Bohr–Sommerfeld–Maslov formulas), in order to calculate a chosen spectral series, it is sufficient to assume that the integral over only one of the cycles takes integer values, and different cycles determine different parts of the spectrum.
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37

Bourgiotis, Sotiris, Panayiotis Frangos, Seil Sautbekov, and Mustakhim Pshikov. "The Evaluation of an Asymptotic Solution to the Sommerfeld Radiation Problem Using an Efficient Method for the Calculation of Sommerfeld Integrals in the Spectral Domain." Electronics 10, no. 11 (June 2, 2021): 1339. http://dx.doi.org/10.3390/electronics10111339.

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A recently developed high-frequency asymptotic solution for the famous “Sommerfeld radiation problem” is revisited. The solution is based on an analysis performed in the spectral domain, through which a compact asymptotic formula describes the behavior of the EM field, which emanates from a vertical Hertzian radiating dipole, located above flat, lossy ground. The paper is divided into two parts. We first demonstrate an efficient technique for the accurate numerical calculation of the well-known Sommerfeld integrals. The results are compared against alternative calculation approaches and validated with the corresponding Norton figures for the surface wave. In the second part, we introduce the asymptotic solution and investigate its performance; we compare the solution with the accurate numerical evaluation for the received EM field and with a more basic asymptotic solution to the given problem, obtained via the application of the Stationary Phase Method. Simulations for various frequencies, distances, altitudes, and ground characteristics are illustrated and inferences for the applicability of the solution are made. Finally, special cases leading to analytical field expressions close as well as far from the interface are examined.
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38

Beldi, M., and A. Maghrebi. "Some New Results for the Study of Acoustic Radiation within a Uniform Subsonic Flow Using Boundary Integral Method." Advanced Materials Research 488-489 (March 2012): 383–95. http://dx.doi.org/10.4028/www.scientific.net/amr.488-489.383.

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In this paper, a reformulation of the Helmholtz integral equation for tridimesional acoustic radiation in a uniform subsonic flow is presented. An extension of the Sommerfeld radiation condition, for a free space in a uniform movement, makes possible the determination of the convected Green function, the elementary solution of the convected Helmholtz equation. The gradients of this convected Green function are, so, analyzed. Using these results, an integral representation for the acoustic pressure is established. This representation has the advantage of expressing itself in terms of new surface operators, which simplify the numerical study. For the case of a regular surface, the evaluation of the free term associated with the singular integrals shows that it is independent of the Mach number of the uniform flow. A physical interpretation of this result is offered. A numerical approximation method of the integral representation is developed. Furthermore, for a given mesh, an acoustic discretization criterion in a uniform flow is proposed. Finally, numerical examples are provided in order to validate the integral formula.
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39

Seong-Ook Park and C. A. Balanis. "Analytical technique to evaluate the asymptotic part of the impedance matrix of Sommerfeld-type integrals." IEEE Transactions on Antennas and Propagation 45, no. 5 (May 1997): 798–805. http://dx.doi.org/10.1109/8.575625.

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40

Tie Jun Cui and Weng Cho Chew. "Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects." IEEE Transactions on Geoscience and Remote Sensing 37, no. 2 (March 1999): 887–900. http://dx.doi.org/10.1109/36.752208.

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41

Alcalá Ochoa, Noé. "One-dimensional integrals to calculate the two-dimensional Rayleigh–Sommerfeld diffraction integrals for non-rotationally symmetric functions and general polarizing illuminating fields." Optics Communications 437 (April 2019): 264–70. http://dx.doi.org/10.1016/j.optcom.2018.12.079.

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42

Kang, Xiao Ping, Chang Wei Li, Zhong He, and Lu Mei Zhao. "The Far Properties of Nonparaxial Cosine-Squared Gaussian Beams." Applied Mechanics and Materials 738-739 (March 2015): 440–43. http://dx.doi.org/10.4028/www.scientific.net/amm.738-739.440.

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Based on the Rayleigh-Sommerfeld diffraction integrals, the closed-form expression of nonparaxial cosine-squared Gaussian beams in free space is derived. The power in the bucket (PIB) of nonparaxial regime is introduced to characterize the beam quality of nonparaxial beams in the far field. It is found that the intensity and PIB of nonparaxial cosine-squared Gaussian beams depends on f parameter, decentered parameter, and the PIB is additionally depends on the bucket’s size taken. For a given bucket’s size, the smaller decentered parameter and f parameter result in the better power focusability in the far field and the better beam quality.
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43

Guo, Lan-Wei, Yongpin Chen, Jun Hu, and Joshua Le-Wei Li. "P-FFT Accelerated EFIE with a Novel Diagonal Perturbed ILUT Preconditioner for Electromagnetic Scattering by Conducting Objects in Half Space." International Journal of Antennas and Propagation 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/813273.

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A highly efficient and robust scheme is proposed for analyzing electromagnetic scattering from electrically large arbitrary shaped conductors in a half space. This scheme is based on the electric field integral equation (EFIE) with a half-space Green’s function. The precorrected fast Fourier transform (p-FFT) is first extended to a half space for general three-dimensional scattering problems. A novel enhanced dual threshold incomplete LU factorization (ILUT) is then constructed as an effective preconditioner to improve the convergence of the half-space EFIE. Inspired by the idea of the improved electric field integral operator (IEFIO), the geometrical-optics current/principle value term of the magnetic field integral equation is used as a physical perturbation to stabilize the traditional ILUT perconditioning matrix. The high accuracy of EFIE is maintained, yet good calculating efficiency comparable to the combined field integral equation (CFIE) can be achieved. Furthermore, this approach can be applied to arbitrary geometrical structures including open surfaces and requires no extra types of Sommerfeld integrals needed in the half-space CFIE. Numerical examples are presented to demonstrate the high performance of the proposed solver among several other approaches in typical half-space problems.
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44

Bridges, G., O. Aboul-Atta, and L. Shafai. "Solution of discrete modes for wave propagation along multiple conductor structures above a dissipative Earth." Canadian Journal of Physics 66, no. 5 (May 1, 1988): 428–38. http://dx.doi.org/10.1139/p88-071.

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The discrete propagating modes for an N-conductor transmission line located above a dissipative Earth are determined. The boundary-value problem is solved using a Fourier transform approach that results in a set of Sommerfeld-type integrals, whose accurate evaluation is shown to become imperative at higher frequencies. Because the integrand of one of the integrals exhibits a pair of first-order poles, a significantly different solution to the customary transmission line dispersion equations is obtained, and additional discrete modal solutions become mathematically feasible in the neighbourhood of the poles. An analytical expression is presented for the general evaluation of the multiple-conductor transmission line problem. A parametric study of the discrete propagating modes for typical single- and two-conductor systems is made and discussed. The validity of a simplified quasi-transverse-electromagnetic approach as well as a small-argument expansion are also presented.
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45

Hochman, A., and Y. Leviatan. "A Numerical Methodology for Efficient Evaluation of 2D Sommerfeld Integrals in the Dielectric Half-Space Problem." IEEE Transactions on Antennas and Propagation 58, no. 2 (February 2010): 413–31. http://dx.doi.org/10.1109/tap.2009.2037761.

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46

Reddy, V. S., and R. Garg. "Efficient analytical evaluation of the asymptotic part of Sommerfeld type reaction integrals in microstrip/slot structures." IEE Proceedings - Microwaves, Antennas and Propagation 147, no. 1 (2000): 1. http://dx.doi.org/10.1049/ip-map:20000036.

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47

Askarpour, Amir Nader, and Reza Faraji-Dana. "Complex Time Representation of the Sommerfeld-Type Integrals and Its Application to 1-D Periodic Structures." IEEE Transactions on Antennas and Propagation 60, no. 9 (September 2012): 4306–15. http://dx.doi.org/10.1109/tap.2012.2207041.

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48

Spowart, Michael P., and Edward F. Kuester. "A novel asymptotic extraction technique for the efficient evaluation of a class of double Sommerfeld integrals." Journal of Computational and Applied Mathematics 197, no. 2 (December 2006): 597–611. http://dx.doi.org/10.1016/j.cam.2005.11.020.

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49

Egel, Amos, Siegfried W. Kettlitz, and Uli Lemmer. "Efficient evaluation of Sommerfeld integrals for the optical simulation of many scattering particles in planarly layered media." Journal of the Optical Society of America A 33, no. 4 (March 22, 2016): 698. http://dx.doi.org/10.1364/josaa.33.000698.

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50

Boon, James K. "Electromagnetic volume scattering in the half-space: a moment-method analysis without Sommerfeld integrals or their approximations." Waves in Random Media 14, no. 3 (July 2004): 199–216. http://dx.doi.org/10.1088/0959-7174/14/3/001.

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