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Discrete & Continuous Models & Applied Computational Science 2022, 30 (4) 330-341
ISSN 2658-7149 (online), 2658-4670 (print) http://journals-rudn-ru/miph
Research article
UDC 517.958:621.372.8
PACS 07.05.Tp, 02.60.Pn, 02.70.Bf
DOI: 10.22363/2658-4670-2022-30-4-330-341
On a dispersion curve of a waveguide filled with inhomogeneous substance
Oleg K. Kroytor1, Mikhail D. Malykh1'2
1 Peoples' Friendship University of Russia (RUDN University), 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation 2 Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear Research, 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation
(received: October 26, 2022; revised: December 8, 2022; accepted: December 19, 2022)
Abstract. The paper discusses the relationship between the modes traveling along the axis of the waveguide and the standing modes of a cylindrical resonator, and shows how this relationship can be explored using the Sage computer algebra system. In this paper, we study this connection and, on its basis, describe a new method for constructing the dispersion curve of a waveguide with an optically inhomogeneous filling. The aim of our work was to find out what computer algebra systems can give when calculating the points of the waveguide dispersion curve. Our method for constructing the dispersion curve of a waveguide with optically inhomogeneous filling differs from those proposed earlier in that it reduces this problem to calculating the eigenvalues of a self-adjoint matrix, i.e., a well-studied problem. The use of a self-adjoint matrix eliminates the occurrence of artifacts associated with the appearance of a small imaginary addition to the eigenvalues. We have composed a program in the Sage computer algebra system that implements this method for a rectangular waveguide with rectangular inserts and tested it on SLE modes. The obtained results showed that the program successfully copes with the calculation of the points of the dispersion curve corresponding to the hybrid modes of the waveguide, and the points found fit the analytical curve with graphical accuracy even when with a small number of basis elements taken into account.
Key words and phrases: waveguide, Maxwell's equations, normal modes, partial radiation conditions
1. Introduction
In classical electrodynamics, there are two related spectral problems — the problem of the normal modes of the waveguide and the problem of eigenmodes of the resonator. Recall their formulations.
© Kroytor O.K., Malykh M.D., 2022
This work is licensed under a Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by-nc/4-0/legalcode
Let G be a regular domain in R3, below referred to as a resonator. A nontrivial field of the form
E = E(x,y,z)eiwt, H = H(x, y, z)eiwt,
satisfying the system of homogeneous Maxwell equations and boundary conditions
nxE = 0, n-H = 0,
is called the eigenmode, and the corresponding value of the positive parameter u is called the eigenfrequency, instead of which the wave number k = u/c is usually used. To find the eigenfrequencies, it is necessary to solve the eigenvalue problem
rot^ = —ik^H, rotH = iktE with the boundary conditions
nxE = 0, n-H = 0.
This problem, like the scalar one, is written as an eigenvalue problem for a completely continuous self-adjoint operator, and therefore it has a discrete spectrum, which can be found approximately using the Ritz method [1, p. 181]. To theoretically substantiate this statement, special Sobolev spaces are introduced, the embedding theorems for which, unfortunately, have so far been proved under the assumption of a smooth boundary [2, § 6.1].
Let S be a regular domain in R2, let us call the cylinder S xR a waveguide. Assume that the axis Oz of the used Cartesian coordinate system is directed along the axis of the cylinder. A non-trivial field of the form
E = E(x, y)eiwt-i^z, H = H(x, y)eiwt-i^z,
satisfying the system of homogeneous Maxwell equations and boundary conditions
nxE = 0, n-H = 0,
is called the normal waveguide mode [3]. In this case, the parameter ft = ^/c is called the phase constant. To find it at a fixed frequency, it is required to solve the eigenvalue problem
rot^ = —ik^H, rotH = iktE with the boundary conditions
nxE = 0, n-H = 0.
Here, rot means a differential operator in which differentiation with respect to z is replaced with multiplication by —¿7. The points of the k^y plane, at which this problem has a nontrivial solution, form a certain curve called the waveguide dispersion curve.
The spectral problem for waveguide modes does not belong to any studied type. In the case when the filling of the waveguide is homogeneous, the
complete system of waveguide modes can be composed of two types of modes: transverse magnetic (Hz = 0) and transverse electric (Ez = 0) ones. The Borgnis theory of functions makes it possible to reduce the study of such modes to the study of the Laplace operator spectrum. In this case, the dispersion curve turns out to be the union of a countable number of hyperbolas [3, 4].
Traditionally, this problem is written as an eigenvalue problem with respect to three field components. The choice of the three field components from the six components of the vectors E and H can be done in different ways, which leads to different formulations of the problem. A. N. Bogolyubov and T. V. Edakina [5, 6] and Frank Schmidt [7, 8] used the components of vector H, in the papers by E. Lezar and D. Davidson [9] vector E was used, A. L. Delitsyn [10-13] formulated the problem in terms of Hx,Hy,EZ. Normal modes of an axially symmetric waveguide with a dielectric core were considered by N. A. Novoselova, S. B. Raevsky and A. A. Titarenko [14], as well as by A. L. Delitsyn and S. I. Kruglov [15]. Potentials can be used instead of fields, for example, four scalar functions, as proposed in Refs. [16-18]. Finally, with considerable efforts, it is possible to reduce the spectral problem to the study of the spectrum of a self-adjoint quadratic pencil [19].
Traditionally, the spectral problem of the theory of waveguides was considered as an eigenvalue problem with respect to the parameter 7, and the wave number was considered given. This approach is justified, since the problem of waveguide diffraction considers the incidence of a monochromatic wave, which partially passes through and is partially reflected from the inhomogene-ity; in this case, transmitted and reflected waves arise, travelling from the inhomogeneity, but having the same frequency as the incident wave. On the other hand, to construct a dispersion curve, it is quite unnecessary to to calculate its points at a sequence of frequency values. It is possible to search for its points at fixed values of 7. What is of importance here is only the convenience of solving the problem.
It is obvious from physical considerations that there should be a simple relationship between the modes traveling along the waveguide axis and the standing modes of the cylindrical resonator. In this paper, we investigate this relationship and, based on it, describe a new method for constructing the dispersion curve of a waveguide with an optically inhomogeneous filling.
2. Relation between travelling and standing modes
We managed to express the relation between the travelling and standing modes by two theorems.
Theorem 1. If the waveguide S x R has a normal mode E = E(x, y)eiut-i^z, H = H(x, y)eiut-i^z
at certain values of then the resonator S x [0,-ku/^] has an eigenmode at the same u, j.
Theorem 2. If the resonator S x [0, L] has an eigenmode
E = E(x,y,z)e™t, H = H(x,y,z)e™t
whose components have continuous derivatives at z = 0, L, then it also has an eigenmode
Ex,y = Ex,y(x, y) sin , Ez = Ez(x, y) cos ^-e™*1,
Hx,y = Hx,y(x, y) cos —e^1, Hz = Hz(x, y) sin—e1^,
for some natural value n, and the waveguide S x R has a normal mode at the same frequency u and j = ■nn/L.
Proven theorems allow us to reduce the solution of the spectral problem of the theory of waveguides to the solution of the spectral problem of the theory of cylindrical resonators.
3. Eigenmodes of resonators
Let G be a resonator filled with a substance characterized, generally speaking, by variable e and constant In this case, it is convenient to exclude
E from the system of Maxwell equations and write down the system of second-order equations
rot1 rotH = k2 pH, (1)
to which the boundary conditions should be added
H-n = 0, rotH xn = 0 (2)
Every non-trivial solution H of the problem (1), (2) has an eigenmode
E = -1 rotHelujt, H = He'lujt ike
of the resonator G. Let us multiply (1) by the test vector F, integrate over G and apply the integration by parts:
JJJ rotF* ■ rotH*^- - k2ii JJJ F* ■ Hd.xd.ydz = 0. (3)
G G
The closure of the set of vectors F £ C 1(G) satisfying the condition
div F = 0 in G and F ■ n = 0 on its boundary, in the norm generated by the scalar product
(F, H) = JJJ rotF* ■ rotHdxd^dZ + ^ JJJ F* ■ Hdxdydz,
G G
is a Hilbert space, which we will denote as fy(G). The generalized eigenmode of the resonator G is the nonzero vector H £ fy(G) that satisfies the identity (3) for any vector F £ fy(G).
It follows from the embedding theorem [2] that there exists a completely continuous self-adjoint operator A such that
^JJJf* ■ Hdxdydz = JJJ rotF* ■ rot(AH)^^-.
G G
Therefore, the relation (3) can be written as H = k2AH.
Hence, the eigenfrequencies of the resonator form an infinitely large sequence, and the eigenmodes corresponding to them form an orthonormal basis in the space fy(G).
The eigenvalues of a self-adjoint, completely continuous operator can be found using the Ritz method, it is only necessary to choose the basis of the space in a proper way.
4. Dispersion curve of a waveguide
Let us consider the points of intersection of the waveguide dispersion curve SxK with the straight line 7 = const. Each such point corresponds to the normal mode of the waveguide, and, by virtue of theorem 1, to the eigenmode of the resonator S x [0,^/^{]. The eigenfrequencies of this resonator form an infinite monotonic sequence ,u2,.... By virtue of theorem 2, these points correspond to modes from which one can construct normal waveguide modes. From here a theorem immediately follows.
Theorem 3. The points of intersection of the waveguide dispersion curve S x R with the straight line j = const form a countable set of points, whose set of abscissas coincides with the set of the resonator eigenfrequencies S x [0,k/^(}.
Thus, in order to construct a dispersion curve, it is necessary to solve the self-adjoint problem of natural vibrations of a cylindrical resonator G using the Ritz method. The choice of a basis is complicated by the fact that the
elements of the space must satisfy the condition divF = 0.
To construct such a basis, we took the eigenfunctions of the Dirichlet problem on the section of a cylinder
A2 ^ + a2 $ = 0, = 0
and eigenfunctions of the Neumann problem on the section of a cylinder
a2t + p4 = 0, ^
= 0.
dS
We will take orthonormal systems and with respect to L2 (S). With their help, we construct the basis formed by TM fields
( dyK (x,y) \
H =
-dx (x,y)
\ 0 J
cos s^z, n,m G N,
and another basis for the TE fields
H =
(x,y) cos s-yz^ 7dy(x,y) cos s^z V ft2(x,y) sin s^z J
n £ m £ Z, m ^ 0,
where 7 = n/L, and s takes integer values, but for our case when e is independent of z, it is sufficient to take s = 1.
Truncating the infinite system to the first N basis functions, we reduce the problem of finding the wave numbers (3) to an algebraic eigenvalue problem
AH = k2 BH. (4)
If we agree to write the first N TM waves, and then the first N TE waves, then the matrices of this system will have a block form
A= I Al1, Al2 i, B = A21, A22/
We composed the matrix elements as integrals over a cylinder and took those integrals that could be calculated explicitly for t(x,y). The matrix B turned out to be diagonal, and
B11 = diag (if), B22 = diag (^f W + & ))
Therefore, the generalized eigenvalue problem (4) is reduced to the standard eigenvalue problem:
DH = k2 H.
The elements of the symmetric matrix D are defined as double integrals:
72 ff dxdy ff dxdy
II Wn ^^m^ + ^n^m 11 ,
' an am J J J J
S S
n,m = 1,... ,N1,
d =
7
72
0n dxdy dxy t^
n = 1,...,Ni, m = Ni + 1,...Ni + N2,
d,r, ,rr, = P2m + 12 VM+f U ■ dXdy
fin ft.
,
n,m = N1 + 1,.,N1 + N2.
5. Calculation of the points of the waveguide dispersion
curve in the Sage system
As is usually the case when applying the Ritz method, the matrix elements are integrals, which in the general case are calculated approximately. To avoid this problem in the first tests, we considered a special case when all integrals are calculated in elementary functions.
Consider a rectangular waveguide S = Lx with uniform filling e0 = 1 and ^ = 1 (see figure 1). Inside it we place a rectangular insert S1 with constant filling e1 and ^ = 1. In this case, 0n and are products of sines and cosines, and the matrix elements are integrals of such products over rectangular regions.
y
Lv
£0 = 1
Ly
2
Figure 1. Waveguide cross section
We compiled a program (https://github.com/malykhmd) in the Sage computer algebra system that calculates these integrals in symbolic form, composes the D matrix for any given numerical value of 7, and calculates its eigenvalues using the standard linalg library function.
To test this program, we considered a waveguide in which the insert occupies the bottom half (see figure 1). In this case, two families of normal modes can be written analytically: SLE and SLH modes [4]. This example is interesting because these modes are hybrid and cannot be found by methods that ignore this effect.
Figures 2, 3, and 4 demonstrate the results of analytical and numerical calculations performed using our program. It is clearly seen that for lower modes, the points found by our program fit the analytical curve with graphical accuracy even when a very small number of basis elements are taken into account (three for each direction). This is not hindered by the proximity of the neighboring arc due to the close values of e0 and e1. The accuracy of the calculations decreases with the growth of e1, but when using 10 modes the calculated points fit the analytical curve even at e1 =2 (see figure 4).
The obtained results showed that the program successfully copes with the calculation of the points of the dispersion curve corresponding to the hybrid modes of the waveguide, and the points found fit the analytical curve with graphical accuracy even when with a small number of basis elements taken into account.
e-
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 k
(a) e-L = 1.1
e-
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 k
(b) = 1.5
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 k
(c) £l =2
Figure 2. Dispersion curve of a waveguide with an insert at Lx = 1, Ly = 2. The dots indicate the points of the dispersion curve found numerically (3 modes were taken in each of the directions), the solid lines are arcs of the dispersion curve corresponding
to the SLE modes
(a) e-L = 1.1
(b) = 1.5
= 2
Figure 3. Dispersion curve of a waveguide with an insert at Lx = 1, L = 2. The dots indicate the points of the dispersion curve found numerically (6 modes are taken in each of the directions), the solid lines are arcs of the dispersion curve corresponding
to the SLE modes
6. Conclusion
Our approach to constructing the dispersion curve of a waveguide with an optically inhomogeneous filling differs from those proposed earlier in that it reduces the problem to calculating eigenvalues of a self-adjoint matrix, i.e., a well-studied problem. The use of a self-adjoint matrix eliminates the occurrence of artifacts associated with the appearance of a small imaginary addition to the eigenvalues.
Figure 4. Dispersion curve of a waveguide with an insert at Lx = 1, Ly = 2. The dots indicate the points of the dispersion curve found numerically (10 modes were taken in each of the directions), thesolid lines are arcs of the dispersion curve corresponding
to the SLE modes
We implemented this approach for an example of a rectangular waveguide with rectangular inserts in the Sage computer algebra system and tested it using SLE modes. At the same time, it was shown that our program perfectly copes with calculating the points of the dispersion curve corresponding to the hybrid modes of the waveguide.
Acknowledgments
This work is supported by the Russian Science Foundation (grant no. 20-11-20257).
References
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For citation:
O. K. Kroytor, M. D. Malykh, On a dispersion curve of a waveguide filled with inhomogeneous substance, Discrete and Continuous Models and Applied Computational Science 30 (4) (2022) 330-341. DOI: 10.22363/2658-46702022-30-4-330-341.
Information about the authors:
Kroytor, Oleg K. — Senior lecturer of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University) (e-mail: kroytor-ok@rudn.ru, phone: +7(495)9550927, ORCID: https://orcid.org/0000-0002-5691-7331)
Malykh, Mikhail D. — Doctor of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University) (e-mail: malykh-md@rudn.ru, phone: +7(495)9550927, ORCID: https://orcid.org/0000-0001-6541-6603, ResearcherID: P-8123-2016, Scopus Author ID: 6602318510)
УДК 517.958:621.372.8
PACS 07.05.Tp, 02.60.Pn, 02.70.Bf
DOI: 10.22363/2658-4670-2022-30-4-330-341
О о о
дисперсионном кривои волновода, заполненного неоднородным веществом
О. К. Кройтор1, М. Д. Малых1'2
1 Российский университет дружбы народов, ул. Миклухо-Маклая, д. 6, Москва, 117198, Россия 2 Лаборатория информационных технологий им. М. Г. Мещерякова,
Объединённый институт ядерных исследований, ул. Жолио-Кюри, д. 6, Дубна, Московская область, 141980, Россия
Аннотация. В статье рассматривается связь между модами, бегущими вдоль оси волновода, и стоячими модами цилиндрического резонатора. Показывается, как данная связь может быть исследована с помощью системы компьютерной алгебры Sage. В работе мы исследуем эту связь и на её основе описываем новый метод построения дисперсионной кривой волновода с оптически неоднородным заполнением. Целью нашей работы было выяснить, что могут дать системы компьютерной алгебры при вычислении (точек) дисперсионной кривой волновода. Метод построения дисперсионной кривой волновода с оптически неоднородным заполнением, предложенный нами, отличается от предложенных ранее тем, что сводит эту задачу к вычислению собственных значений самосопряжённой матрицы, то есть к задаче, хорошо изученной. Использование самосопряжённой матрицы исключает возникновение артефактов, связанных с появлением малой мнимой добавки у собственных значений. Мы составили программу в системе компьютерной алгебры Sage, в которой реализован этот метод для волновода прямоугольного сечения с прямоугольными вставками, и протестировали её на SLE-модах. Полученные результаты показали, что программа успешно справляется с вычислением точек дисперсионной кривой, отвечающих гибридным модам волновода, и найденные точки с графической точностью ложатся на аналитическую кривую даже при небольшом числе учитываемых базисных элементов.
Ключевые слова: волновод, уравнения Максвелла, нормальные моды, парциальные условия излучения
