Use of pattern equation method for the analysis of scattering on a thin dielectric cylinder Текст научной статьи по специальности «Физика»

USE OF PATTERN EQUATION METHOD FOR THE ANALYSIS OF SCAT-TERING ON A THIN DIELECTRIC CYLINDER

Dmitry B. Demin,

Moscow Technical University of Communications and Informatics, Moscow, Russia, dbdemin@gmail.com

Andrey I. Kleev,

P.L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Moscow, Russia, kleev@kapitza.ras.ru

Alexander G. Kyurkchyan,

Moscow Technical University of Communications and Informatics; Kotel'nikov Institute of Radio Engineering and Electronics, Fryazino Branch;

Central Research Institute of Communication FSUE, Moscow, Russia,

agkmtuci@yandex.ru

DOI 10.24411/2072-8735-2018-10300

This paper was partially funded by Russian Foundation for Basic Research (project No. l9-02-00654а)

Keywords: Light scattering by small particles, Rayleigh approximation, Pattern Equation Method, electromagnetic scattering, numerical methods in diffraction theory.

Electromagnetic scattering by small particles is an important key problem of the diffraction theory. From the moment of occurrence of the first papers on this subject and up to now, the most widely used mathematical model, applied for solution to a problem of scattering on small objects, is dipole approximation (Rayleigh approximation). This approach is quite detailed for particular cases of scattering on spheres and ellipsoids when solution to an associated electrostatic problem can be obtained explicitly. It should be noted that problem solution in electrostatic approximation in a general case is a complicated problem in itself and labor input for its solution is comparable to the labor input for solution of an initial wave problem. The existing methods for its solution have a range of fundamental limitations. This paper develops methodology based on the use of pattern equation method (PEM) which was initially proposed in 1992. It was clearly demonstrated in a significant number of publications that PEM has important advantages over multiple alternative methods and is quite efficient for solving a wide range of problems. While building up a new approach to the analysis of scattering on small bodies, we used a high convergence of PEM, established in the above papers. Indeed, as was demonstrated by previous works of the authors of the given article, in order to solve a problem of scat-tering on impedance bodies, the typical size of which is comparable to the primary field wavelength, it is sufficient to consider one to three summands in the scattering pattern decomposition, depending on polarization of an incident field. This circumstance allowed obtaining explicit formulas for integrated scattering characteristics, applicable for impedance scatterers of complex shape. This paper develops approximated method of calculation of integrated characteristics of scattering on thin dielectric cylinders, based on the use of PEM. Explicit formulas were obtained for integrated scattering characteristics, which are applied to dielectric cylinders with arbitrary cross section. Applicability of the obtained ratios is analyzed by a range of examples: scattering on an elliptic cylinder and scattering on a cylinder, the cross section of which has a shape of superellipse. As shown by the presented re-sults, the obtained approximated relations are quite accurate in a wide range of problem parameters.

Information about authors:

Dmitry B. Demin, Moscow Technical University of Communications and Informatics, Associate Professor, Cand. Sc., Moscow, Russia Andrey I. Kleev, P.L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Moscow, Russia

Alexander G. Kyurkchyan, Moscow Technical University of Communications and Informatics; Kotel'nikov Institute of Radio Engineering and Electronics, Fryazino Branch; Central Research Institute of Communication FSUE, Head of Chair, Doctor of Science, Moscow, Russia

Для цитирования:

Демин Д.Б., Клеев А.И., Кюркчан А.Г. Использование метода диаграммных уравнений для анализа рассеяния на тонком диэлектрическом цилиндре // T-Comm: Телекоммуникации и транспорт. 2019. Том 13. №8. С. 42-46.

For citation:

Demin D.B., Kleev A.I., Kyurkchyan A.G. (2019). Use of pattern equation method for the analysis of scattering on a thin dielectric cylinder. T-Comm, vol. 13, no.8, pр. 42-46. (in Russian)

Introduction

For the time being, Rayleigh approximation is virtually the only mathematical model used to solve a problem of scattering on small bodies 11 ¡. In well-known monographs \ 2 —4], this approach is quite detailed lor particular cases of scattering on balls and ellipsoids when solution to an associated electrostatic problem can be obtained explicitly. It should be noted that problem solution in electrostatic approximation in a general case is a complicated task. The existing methods for its solution have a range of fundamental limitations (5]. This paper develops an approach based on the use of pattern equation method (PF.M), The approach was proposed in papers [6-9]. it was demonstrated that PEM has important advantages over multiple alternative methods (e.g. see [10]) and is quite efficient for solving a wide range of problems. While building up a new approach to the analysis of scattering on small bodies, we used a high convergence of PEM, established in the above papers. Indeed, as was demonstrated by calculations, in order to solve a problem of scattering on bodies, the typical size of which is comparable to the primary field wavelength, it is sufficient to consider one to three summands in the scattering pattern decomposition, depending on polarization of an incident Held. This circumstance allowed obtaining explicit formulas for integrated scattering characteristics, applicable for scatterers of complex shape.

Basic Relations and Results

Let us illustrate the above-mentioned by several examples. Let an electromagnetic wave having a single magnetic field component which is different from zero - H. be incident on a

magnetodielectric cylinder with directrix S, defined in polar coordinates (r,<p) by a relation

r=p{<p), (1)

/ flit / y

Zyfi f - p{cp) \

U/ yr X

H'

= H

s z

S £ ôn Ôn

(2)

where //''' is electromagnetic field inside the cylinder, H2 - {if + HW) is total magnetic field outside the cylinder, ffV' is the field scattered by the cylinder, e. p are the permittivity and permeability of the cylinder material, djn means differentiation in the direction of outer (regarding the area inside S ) normal Let us consider scattering of a flat wave propagated at angle 0 + 7C to axis X (Fig. I). In this case, incident field is given by

i/f) = &sp[ikr cos(i3 -0)]. (3)

We will use the following representations for fields H^ and //i'1, obtained within the approach stated in [6 — 9]:

J .7/ 2 - rc

7Ï

Hu'\r,(p)-~ jg(<P+ikr co&y/)dy/ ' ^

-JT/2-/1o

/ff =-30

In relations (2) and (3) k = k ep, Jm (x) is Bessel function [12],

g(a)={/{[^-^¿lexptifrcwi«^)!«® <6>

ôn

d_ 5n

is the cylinder scattering pattern and

"Ai

4 ¿ 11 3« du

i?i2)(¿(íV)exp(- im<p')]s Us

:,(7)

wherein //„(x) is Hankel function of second kind [12].

By replacing value /-/ for Hir> in representation (6) and value H\'] for H, in (7) in accordance with boundary conditions (2), we obtain the following system of intcgro-algcbraic equations:

1 Leu^ riti'}

(8)

Fig. I. Problem geometry

and generatrix parallel to axis Oz. We assume that time dependence is proportional to exp(/iyf) Where to = ck. k=2ir X-

while X and c are wavelength and electromagnetic constant correspondingly. The following interface conditions will take place on the border of the scatterer: 1 dH{!] dH.

-ikíñ [/?( $cœ{ a~<¡>)-p[(j>)€m{a-<f)\j„ (¿('V( $)} *

J

x exp [ ikp{ !p) eos ( a ] dtj>

. 2.T -T/2+M cp ,

4,-^+M Í

71 0 -n/2-fa U

]+is[kp(ç)ist*w~

P\<P)

xexp[-/£ ^(^í)cos(^) - in<f^d\// dip wherein

(9)

7TT

T

Y

Full curve presents an exact solution. Circles present an approximated solution given by relations (20) - (24). As demonstrated by the results presented in this figure, the approximated explicit solution has a quite high accuracy in a wide range of problem parameters.

Figure 3 shows dependence of normalized cross section of flat wave scattering on an elliptic dielectric (£ — 2.25,^ = 1) cylinder on ka ■ hi this ease, shape of the cylinder cross section is defined by formula:

(25)

Full curve presents an exact solution. Circles present an approximated solution given by relations (20) - (24), As in the previous example, the approximated explicit solution has a quite high accuracy in a wide range of problem parameters.

2 3

I'"ig. 3. Dependence of normalized cross section of plane wave scattering on an elliptic dielectric (e = 2.25,// = 1) cylinder on ka ■

Full curve presents an exact solution. Circles present an approximated solution given by relations (20) - (24). Curves I - 3 were obtained with q - 2, 4, 8 correspondingly

Conclusion

As shown by the above results, PEM allows obtaining explicit expressions for characteristics of the scattered field, with the accuracy sufficient for practice, which is up to ajX = 0.3 (a is

the typical size of a scatterer). it should be noted as a definite advantage that, contrary to the approach set out in |5, 18 - 20], use of PFM does not imply solving of associated static problems and, particularly, calculation of particle polarizability tensor is not required.

References

1. Landau L.D. and Lifshitz E.M. (1984), Electrodynamics of Continuous Media. Pergamon, Oxford and New York. 460 p.

2. van de Hulst H,C. (1957). Light scattering by small particles. New- York (John Wiley and Sons), London (Chapman and Hall, 470 p.

3. Bohrcn C.F., Huffman D R. {1998). Absorption and Scattering of Light by Small Particles. New York (John Wiley and Sons). 544 p.

4. Mishchenko M.Î., Hovenier J.W., Travis L.D, (2000). Light Scattering by Nonspherical Particles. San Diego: Academic Press. 690 p.

5. Farafonov V.G., Ustimov V.I. (2015). Analysis of the extended boundary condition method: an electrostatic problem for Chebyshev particles. Optics and Spectroscopy. Vol, ! 18. No. 3, pp. 445-459.

6. Kyurkchan A.G. (1992). A new integral equation in the diffraction theory . Soviet Physics-Doklady. vol. 37, no 7, pp. 338-340.

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8. Kyurkchan A.G., Kieev A.I. (1995). Solution of the Problems of Wave Diffraction on Finite Scatterers with the Method of Diagram Equations. Radiotekhnika i elektronika. Vol. 40. No. 6. Pp. C, 897-905.

9. Kyurkchan A.G., Smirnova N.I. (2015). Mathematical Modeling in Diffraction Theory Based on A Priori Information on the Analytic Properties of the Solution. Amsterdam: Elsevier. 280 p.

10. Kleev AJ, A.B, Manenkov A.B. (1986). Adaptive Collocation Method in 2D Diffraction Problems. Radiophysics and Quantum. Vol.29. No. 5. Pp. 557-565.

I I. Dmitriev V.I., Zakharov E.V. (1987). Integral Equations in Boundary Problems of Electrodynamics. Moscow: MSU Publishing House.

12. Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables, Ed. By M. Abramovitz and I.A. Stegun (Dover, New York, 1964) 1046 p.

13. Dentin D.B., Kleev A.I, Kyurkchan A.G. (2016). Use of Pattern F.quation Method for Analysis of Scattering on Small Particles of a Complex Shape. T-Comm. Vol. 10, No. 10, pp. 38-42.

14. Demin D.B., Kleev A.L Kyurkchan A.G. (2017). Modeling of electromagnetic scattering by thin cylinders using Pattern Equation Method. Journal of Quantitative Spectroscopy and Radiative Transfer. Vol. 187, No. I, pp. 287-292.

15. Demin D.B., Kleev A.I, Kyurkchan A.G. (2018). Application of the Pattern Equation Method to the Analysis of Electromagnetic Wave Scattering by a Thin Cylinder of an Arbitrary Cross Section. Journal of Communication Technology and Electronics. Vol. 63, No. 6, pp. 505-512.

16. Demin D.B., Kleev A.I., Kyurkchan A.G. (2017). Solution of Electromagnetic Problems of Diffraction on Small Particles of a Complex Shape Using Pattern Equation Method. T-Comm. Vol. 11, No. 5, pp. 26-32.

I 7. Demin D.B., Kleev A.I, Kyurkchan A.G.(20I9). Construction of the Approximate Solution to the Problems of Diffraction of Electromagnetic Waves by Small Particles with the Use of the Pattern Equation Method. Journal of Communication Technology and Electronics. Vol. 64, No. 1, pp. 13-19.

18. Farafonov V.G. (2000), Light scattering by multilayer ellipsoid in the Rayleigh approximation. Optics and Spectroscopy. Vol. 88. No. 3, pp. 441-443.

19. Farafonov V.G. (2001 ). New recursive solution of the problem of scattering of electromagnetic radiation by multilayer spheroidal particles. Optics and Spectroscopy. Vol. 90. No. 5, pp. 743-752.

20. Posse It B„ Farafonov V.G, IPin V.B., ProkopjevaM.S. (2002). Light scattering by multi-layered ellipsoidal particles in the quasistatic approximation. Measurem. Sci. Techno!. Vol. 13. pp. 256-262.

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ИСПОЛЬЗОВАНИЕ МЕТОДА ДИАГРАММНЫХ УРАВНЕНИЙ ДЛЯ АНАЛИЗА РАССЕЯНИЯ НА ТОНКОМ ДИЭЛЕКТРИЧЕСКОМ ЦИЛИНДРЕ

Демин Дмитрий Борисович, Московский технический университет связи и информатики, Москва, Россия, dbdemin@gmail.com Клеев Андрей Игоревич, Институт физических проблем им. ПЛ.Капицы РАН, Москва, Россия, kleev@kapitza.ras.ru Кюркчан Александр Гаврилович, Московский технический университет связи и информатики; ФИРЭ им. В.А. Котельникова РАН; ФГУП Центральный научно-исследовательский институт связи, Москва, Россия,

agkmtuci@yandex.ru

Аннотация

Рассеяние электромагнитных волн малыми частицами является важной ключевой задачей теории дифракции. С момента появления первых работ, посвященных этой тематике и вплоть до настоящего время наиболее используемой математической моделью, применяемой при решении задачи о рассеянии на малых телах, является дипольное приближение (приближение Рэлея). Данный подход достаточно подробно изложен для частных случаев рассеяния на шарах и эллипсоидах, когда решение вспомогательной электростатической задачи можно получить в явном виде. Отметим, что решение задачи в электростатическом приближении в общем случае, само по себе, является сложной задачей, трудоемкость решения которой сопоставима с трудоемкостью решения исходной волновой задачи. Существующие методы ее решения имеют ряд принципиальных ограничений. В данной работе развита методика, основанная на использовании метода диаграммных уравнений (МДУ), впервые предложенном в 1992 г. В значительном числе публикаций наглядно продемонстрировано, что МДУ обладает важными преимуществами перед многими альтернативными методиками и весьма эффективен при решении широкого класса задач. При построении нового подхода к анализу рассеяния на малых телах нами была использована установленная в указанных выше работах высокая скорость сходимости МДУ. Действительно, как показано в предыдущих работах авторов настоящей статьи, для решения задачи рассеяния на импедансных телах, характерный размер которых сопоставим с длиной волны первичного поля, достаточно учесть, в зависимости от поляризации падающего поля, от одного до трех слагаемых в разложении диаграммы рассеяния. Это обстоятельство дало возможность получить явные формулы для интегральных характеристик рассеяния, применимые для импедансных рассеивателей сложной формы. В данной работе развита приближенная методика расчета интегральных характеристик рассеяния на тонких диэлектрических цилиндрах, основанная на использовании МДУ. Получены явные формулы для интегральных характеристик рассеяния, применимые для диэлектрических цилиндров произвольного поперечного сечения. Применимость полученных соотношений проанализирована на ряде примеров: рассеянии на эллиптическом цилиндре и рассеянии на цилиндре, поперечное сечение которого имеет форму суперэллипса. Как показывают приведенные результаты, полученные приближенные соотношения обладают достаточной точностью в широком диапазоне параметров задачи.

Ключевые слова: рассеяние света на малых частицах, приближение Рэлея, метод диаграммных уравнений, рассеяние электромагнитных волн, численные методы теории дифракции.

Литература

1. Landau L.D. and Lifshitz E.M. Electrodynamics of Continuous Media. Pergamon, Oxford and New York. 1984. 460 p.

2. van de Hulst H.C. Light scattering by small particles. New York (John Wiley and Sons), London (Chapman and Hall). 1957. 470 p.

3. Bohren C.F., Huffman D.R. Absorption and Scattering of Light by Small Particles. New York (John Wiley and Sons). 1998. 544 p.

4. Mishchenko M.I., HovenierJ.W., Travis L.D. Light Scattering by Nonspherical Particles. San Diego: Academic Press. 2000. 690 p.

5. Farafonov V.G., Ustimov V.I. Analysis of the extended boundary condition method: an electrostatic problem for Chebyshev particles. Optics and Spectroscopy. Vol. 118. No. 3. 2015. Pp. 445-459.

6. Кюркчан А.Г. Об одном новом интегральном уравнении в теории дифракции // Доклады Академии наук. 1992. Т. 325. № 2. С. 273-275.

7. Кюркчан А.Г. Об одном методе решения задач дифракции волн на рассеивателях конечных размеров // Доклады Академии наук. 1994. Т. 337. № 6. С. 728-731.

8. Кюркчан А.Г., Клеев А.И. Решение задач дифракции волн на рассеивателях конечных размеров методом диаграммных уравнений // Радиотехника и электроника. 1995. Т. 40. № 6. С. 897 - 905.

9. Kyurkchan A.G., Smirnova N.I. Mathematical Modeling in Diffraction Theory Based on A Priori Information on the Analytic Properties of the Solution. Amsterdam: Elsevier, 2015. 280 p.

10. Клеев А.И., Маненков А.Б. Метод адаптивной коллокации в двумерных задачах дифракции. // Изв. вузов. Радиофизика. 1986. Т. 29. № 5. С. 557-565.

11. Дмитриев В.И., Захаров Е.В. Интегральные уравнения в краевых задачах электродинамики. М.: Изд-во МГУ, 1987.

12. Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables, Ed. By M. Abramovitz and I.A. Stegun (Dover, New York, 1964) 1046 p.

13. Демин Д.Б., Клеев А.И., Кюркчан А.Г. Использование метода диаграммных уравнений для анализа рассеяния на малых частицах сложной формы // T-Comm: Телекоммуникации и транспорт, 2016, том 10, № 10. С. 38-42.

14. Demin D.B., Kleev A.I., Kyurkchan A.G, Modeling of electromagnetic scattering by thin cylinders using Pattern Equation Method // Journal of Quantitative Spectroscopy and Radiative Transfer, 2017, Vol. 187, No. 1, рp. 287-292.

15. Демин Д.Б., Клеев А.И., Кюркчан А.Г. Использование метода диаграммных уравнений для анализа рассеяния электромагнитных волн на тонком цилиндре произвольного поперечного сечения // Радиотехника и электроника, 2018, том 63, № 6. С. 507-514.

16. Демин Д.Б., Клеев А.И., Кюркчан А.Г. Решение электромагнитных задач дифракции на малых частицах сложной формы методом диаграммных уравнений // T-Comm: Телекоммуникации и транспорт, 2017, том 11, № 5. С. 26-32.

17. Демин Д.Б., Клеев А.И., Кюркчан А.Г. Построение приближенного решения задач дифракции электромагнитных волн на малых частицах сложной формы при помощи метода диаграммных уравнений. // Радиотехника и электроника, 2019, том 64, № 1. С. 15-21.

18. Farafonov V.G. Light scattering by multilayer ellipsoid in the Rayleigh approximation // Optics and Spectroscopy. 2000. Vol. 88. No. 3, рp. 441-443.

19. Фарафонов В.Г. Новое рекурсивное решение задачи рассеяния электромагнитного излучения многослойными сфероидальными частицами // Опт. и спектр. 2001. Т. 90. № 5. С. 826-835.

20. Posselt В., Farafonov V.G., Il'in V.B., Prokopjeva M.S. Light scattering by multi-layered ellipsoidal particles in the quasistatic approximation // Measurem. Sci. Technol. 2002. V. 13, рр. 256-262.

Информация об авторах:

Демин Дмитрий Борисович, Московский технический университет связи и информатики, доц., к.ф.-м.н. Москва, Россия Клеев Андрей Игоревич, Институт физических проблем им. ПЛ.Капицы РАН, зам. дир., д.ф.-м.н., Москва, Россия Кюркчан Александр Гаврилович, Московский технический университет связи и информатики; ФИРЭ им. В.А. Котельникова РАН; ФГУП Центральный научно-исследовательский институт связи, зав. каф., д.ф.-м.н., Москва, Россия

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