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MODELING OF WAVEGUIDE WITH AN ANISOTROPIC MEDIUM
At present, optoelectronics and integrated optical technologies are developing rapidly, which requires new research in the field of wave optical technologies, including the use of optical transmission systems [1]. This can be explained by the development of optical technologies for processing and transmitting information, the development of photonic integrated circuits and optical memory [2,3]. The advantages of the optical converter and data processing technologies are high data transfer rate, low cost, structurally small size, etc. The further development of fiber-optic transmission systems is associated with the creation of complete optical photon networks and optical communication lines. In such systems, the transmission, reception and processing of signals without the use of electronic devices and electronic processes will occur at the full level of photons. It is known that in anisotropic media, electromagnetic waves propagate differently than in isotropic media [4]. In an anisotropic medium, the optical properties depend on the direction of light propagation, and therefore, the orientation of the optical axis of the waveguide should affect the propagation conditions of natural waves. New optical solutions require the creation of media with special optical properties. As you know, integrated optical components are manufactured in very complex processes, for example, in the process of ion implantation. In order for the devices to be created to function as designed by the developer, a detailed analysis of the waveguide propagation characteristics is necessary, as well as the development of a simple set of calculation tools for production. This requires an adequate mathematical model of the interaction of electromagnetic waves with matter, built on Maxwell's equations.
In this work, we obtained a mathematical model for calculating the eigen modes of a planar anisotropic waveguide for an arbitrary tilt of the optical axis in the plane of incidence. New mathematical models of the dispersion equations of a waveguide with an anisotropic medium for TE- and TM-waves are obtained in this article. The asymptotic behavior of dispersion curves for a TM-wave is studied as a function of the angle of orientation of the optical axis in the wave propagation plane.The mathematical model of the interaction of an electromagnetic field with anisotropic materials in planar anisotropic waveguides, proposed in the work, allows one to find, with controlled accuracy, the distribution of the electromagnetic field in an arbitrarily anisotropic and arbitrarily inho-mogeneous material bounded by a conducting surface. Using numerical methods, the dispersion equations are solved. It was revealed that the position of the dispersion curves depends on the angle for the TM-wave.
Information about author:
Mehman Huseyn Hasanov, Candidate of Technical Sciences, PhD, Department of "Telecommunication systems and information security", Azerbaijan Technical University, Baku, Azerbaijan
Для цитирования:
Гасанов М.Г. Моделирование волновода с анизотропной средой // T-Comm: Телекоммуникации и транспорт. 2019. Том 13. №12. С. 62-68.
For citation:
Hasanov M.H. (2019). Modeling of waveguide with an anisotropic medium. T-Comm, vol. 13, no.12, рр. 62-68.
DOI 10.24411/2072-8735-2018-10335
Mehman Huseyn Hasanov,
Azerbaijan Technical University, Baku, [email protected]
Keywords: anisotropic medium, optical waveguides, waveguide modes, three-layer homogeneous waveguide.
T
Introduction
Optical switches, new high-capacity optical cross-switches, optical amplifiers with an optical feeder with remote power supply, optical pulse generators, optical passive elements: optical channel dividers, couplers, optical valves, chromatic dispersion compensators, optical filters, multiplexers and demultiplexers are used optica! processors [2, 3]. (t is also important in full optical photon networks to study the variability of certain parameters during the transition from the same isotropic medium, for example, from fiber, to another anisotropic medium i.e., to the optical signal switch, or when transmitting optica! signals to optical processing devices.
Over the past decades, many analytical and numerical methods for calculating waveguide modes have been developed [5, 6]. The beam propagation method was applied in [7] for an anisotropic thin waveguide. A three-layer system was considered. In the ease of arbitrary orientation of the optical axis of the anisotropic waveguide, cross-polarization and wave interaction will occur in each layer. To describe this phenomenon, a new 4x4 matrix algebra was proposed in [8], which combines the 2x2 matrix method and the Jones matrix method. It was used to study the propagation of a plane wave in an arbitrary anisotropic medium. For each layer, a propagation matrix was recorded; it was found that the form of the matrix depends on the orientation of the optical tensor relative to the axes of the waveguide. The procedure based on the formalism of 4x4 Juh matrices was developed in J9j].
An asymptotic solution was found for a set of very thin layers much thicker than the wavelength. A method for selecting the zero elements of the characteristic matrix was proposed in [10]. It was used for a multilayer planar optical waveguide with a given refractive index profile. The propagation constants of the waveguide were obtained from the conditions of the vanishing of the elements of the transfer matrix. The method was applied to waveguides with losses and anisotropy. In 111], the transfer matrix method was proposed. This is the standard 2x2 matrix method used for optics of thin films. It was applied to a planar multilayer waveguide. Expressions are obtained for calculating the field and energy profiles in each layer. The method applied to an absorbing multilayer waveguide and to the reflection of a plane wave was proposed in [12]; it is a modification of the method of characteristic matrices adapted for a planar waveguide 112], A detailed theoretical analysis of various types of optical waveguides was proposed in [13]. The conditions for the existence of waves arc studied in detail, dispersion equations are obtained for calculating the dependence of the refractive index for the first two modes on the relative thickness of the waveguide, the interval for the upper limit of the "change in the refractive index" is indicated.
Modifications of the finite difference method for the time and frequency domain, as well as the finite element method, are often used to calculate and analyze waveguide modes. An original algorithm was proposed in [14]. A new full-vector finite-difference discretization for waveguides with transverse anisotropy was proposed in [14]. Unlike previous solutions, the method allows solving problems for arbitrary orientation of the optical axis with respect to wave propagation. The finite element method was used to calculate the modes at different propagation angles in [16] and [17]. Numerical methods make it possible to cany out solutions for eigen modes for waveguides with various
optical properties and geometry. However, their common drawbacks are the appearance of non-phystea! dependencies, as well as the difficulty in analyzing the solution.
Statement of the problem
In the present work, based on the equations of classical electrodynamics, the solutions are stitched together at the boundaries of a three-layer homogeneous waveguide with an internal anisotropic medium using the Cauchy matrix. The solutions of the dispersion equations for the TE- and TM-waves propagating in the anisotropic waveguide are calculated, and the asymptotic behavior of the dispersion curves is obtained as a function of the angle of the optical axis of the waveguide.
Consider the propagation of an electromagnetic wave in an anisotropic plane layer. It is presented in Fig. 1. The coating medium "1" and the substrate "3" have lower values of dielectric constant than the values of EQ,seof the second medium, and ensure the propagation of the electromagnetic wave within the layer "2" in the XOZ plane due to the phenomenon of total internal reflection. The XOZ wave propagation plane shown in the figure will be called the plane of incidence. Let the optical axis of the anisotropic medium of the waveguide be oriented in the plane of incidcnce at an angle <p to the axis OX.
/ P
\A /\ 1 S mm
O'd
Figure 1. Uniform anisotropic planar waveguide: OO'- the optical axis of the anisotropic medium "2" is located in the XOZ plane at an angle (f> to the axis OX
When the specified position of the optical axis, the dielectric constant of the uniaxial anisotropic medium "2" has the form:
£ =
£t, COS' $£>-I-£(, sirr ——sin2$i
S
0;fo;0
-sin cos2 + sin" (p
(1)
Consider the propagation of two types of waves in medium "2", The first wave will be an ordinary wave in the crystal "2"; this is a TE-wave. The vectors of electric and magnetic fields for it have the form:
Ete = (0; Ey ;0), HTE=(Hv-fi-H__).
(2a) (2b)
The second wave in the "2" crystal is extraordinary; it is a TM-wave. The vector of electric field strength for a given wave lies in the plane of incidence and has components Ex, Ezand
vector H is perpendiculyar to it, therefore it has only component
'y ■
Em = (Ex;0:E_),
(3a) (3b)
We substitute the vectors (2) and (3) into the Maxwell equations and take into account that the conditions are satisfied for all components of the fields Fj :
—Ft ~ ik0aFr oz
(4a)
(4b)
a ~ yjc|(j| sin 0] ~ ^63^3 sin 63 = ^Eyyji. sin tljg = ne sin Oj|v| = const.
(5)
Where ne = yjen cos2 (<9 -<p) + se sin2 (6 - (p) is the refractive index for the TM-wave in layer "2" [13J.
Solution of Maxwell equations
Substitution of vectors (2) and (3) into Maxwell's equations, taking into account constraints (4) and (5), allows us to obtain two systems consisting of two ordinary differential equations. The first system of ordinary differential equations describes the propagation of TE-waves in a "2'" medium:
—E = ikufjH ax
(6)
The second system of two ordinary differentia) equations is responsible for the propagation of the TM-wave in medium "2":
±H =-ikoa^-Hv-ikJ e„-£"s" ax e„
E.,
— E,= ~ikti dx
M'
a
2\
"xx J
Hv - ik0a^Er.
(7)
To solve the systems of ordinary differential equations (6) and (7), fundamental solution matrices were found, and then
Cauchy matrices for TE- and TM-waves in the anisotropic layer weve obtained. And also matrix solutions were obtained for a homogeneous anisotropic layer. The "stitching" of solutions at the boundaries "1-2" when x = 0, and "2-3" when x = d, for the components of the TE-wave fields has the form:
Ey(t!) KHXd)
"¡1 n\2
<TE\<TEi
V 21 22 7
Ey( 0) Hz( 0).
(8)
The coefficients of the Cauchy matrix in equation (8) have the form:
jsm^dyje^fi-à2}
Parameter a depends on the angle of incidence of the wave at the x = 0 and x = d interfaces and on the optical properties of the waveguide. The values of the angles of incidence 0], 63 in
the media "1" and "3", as well as the values of the angles dTE
and Bm in the layer "2" are associated with the optical
properties of materials by Snell's law:
- or
(8a) (8b)
(8v)
Matrix "stitching" of a solution for a TM-wave at the boundaries of layer "2" is written as:
rHy{df
f ,JTM)iTh )\
"11 ] 1
JxmjjM) V — 1 22
EA 0)
/
(9)
The Cauchy matrix Np(d,0)in formula (8) for the TM-wave has the form:
JTM) _ „(TM) _
-ik^ad—
'¡I
= n\2 = e
cos
Va (sxjt-cf)^
knd
-iknad-^
n[™>«=ie
kod
i
4(exxM - a )
(9a)
(9b)
-ik0ad— I
sin
77 \
k0d
. (9v)
The value of A — £ir£.. — £y.€.x = SqS* does not depend
on the orientation of the optical axis in the XOZ plane. Formulas (8) and (9) make it possible to calculate the localization of the fields of an electromagnetic wave in a waveguide. From formulas (9a)-(9v) it can be seen that the eigenvalues of the system of ordinary differential equations (5) fake the values:
(10)
The second term in formula (9) explains the appearance of a phase factor in the coefficients of matrix (9). Cross 1inking (8) and (9) was obtained by solving systems of ordinary differential equations using a matrix. As a rule, in the calculations, the
Conclusion
In this work, we obtained a mathematical model for calculating the eigen modes of a planar anisotropic waveguide for an arbitrary tilt of the optical axis in the plane of incidence. The asymptotic behavior of dispersion curves for a TM-wave is studied as a function of the angle of orientation of the optical axis in the wave propagation plane. The results can be used in the development of various types of planar anisotropic waveguides, which allow you to effectively control the radiation pattern. The mathematical model of the interaction of an electromagnetic field with anisotropic materials in planar anisotropic waveguides, proposed in the work, allows one to find, with controlled accuracy, the distribution of the electromagnetic field in an arbitrarily anisotropic and arbitrarily inhomogeneous material bounded by a conducting surface.
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( I л
МОДЕЛИРОВАНИЕ ВОЛНОВОДА С АНИЗОТРОПНОЙ СРЕДОЙ
Гасанов Мехман Гусейн оглы, Азербайджанский Технический Университет, г. Баку, Азербайджан,
Аннотация
В данной работе получена математическая модель для расчета собственных мод плоского анизотропного волновода для произвольного наклона оптической оси в плоскости падения. Разработаны математические модели дисперсионных уравнений волновода с анизотропной средой для TE- и TM-волн. Используя численные методы, решаются дисперсионные уравнения. Выявлено, что положение дисперсионных кривых зависит от угла для TM-волны. Асимптотическое поведение дисперсионных кривых для TM-волны исследовано в зависимости от угла ориентации оптической оси в плоскости распространения волны. Полученные результаты могут быть использованы при разработке различных типов плоских анизотропных волноводов, которые позволяют эффективно контролировать диаграмму направленности. Предложенная в работе математическая модель взаимодействия электромагнитного поля с анизотропными материалами в плоских анизотропных волноводах позволяет с контролируемой точностью находить распределение электромагнитного поля в произвольно анизотропном и сколь угодно неоднородном материале, ограниченном проводимостью. поверхность.
Ключевые слова: анизотропная среда, оптические волноводы, волноводных мод, трехслойный однородной волновод. Литература
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Информация об авторе:
Мехман Гусейн оглы Гасанов, к.т.н., доцент, кафедры " Tелекоммуникационные системы и информационные безопасности", Азербайджанский Технический Университет, г. Баку, Азербайджан
7ТТ
