=Paper=
{{Paper
|id=Vol-1995/paper-09-964
|storemode=property
|title=
Direct Realization of the Pseudospectral Method of Calculating
Waveguide Mode
|pdfUrl=https://ceur-ws.org/Vol-1995/paper-09-964.pdf
|volume=Vol-1995
|authors=Yaroslav Y. Kuziv,Leonid A. Sevastianov
}}
==
Direct Realization of the Pseudospectral Method of Calculating
Waveguide Mode
==
https://ceur-ws.org/Vol-1995/paper-09-964.pdf
65
Direct Realization of the Pseudospectral Method of Calculating
Waveguide Mode
Yaroslav Y. Kuziv∗ , Leonid A. Sevastianov∗†
∗
Department of Applied Probability and Informatics
Peoples’ Friendship University of Russia (RUDN University)
6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation
†
Bogoliubov Laboratory of Theoretical Physics
Joint Institute for Nuclear Research
6 Joliot-Curie, Dubna, Moscow region, 141980, Russian Federation
Email: kuziw@yandex.ru, sevastianov_la@rudn.university
Often in the applied problems of integrated optics we use regular open gradient (general)
planar waveguides. Waveguides perform conversion, amplification and transmission of light
signals, similar to how it occurs with electrical signals in integrated circuits, but the speed
of information transfer through such devices is much higher. The mathematical model of
light propagation in a waveguide is described by Maxwell’s equations and the corresponding
boundary conditions. The Maxwell’s equations in Cartesian coordinates are separated into
two independent sets for the TE and TM polarizations. Systems for the TE and TM polar-
izations can be transformed into ODEs of the second order. The boundary conditions for
equations are reduced to two pairs of boundary conditions. The problem of finding modes in
regular open gradient planar waveguide is described in terms of an eigenvalue problem (The
generalized eigenvalue problem of two matrices). Numerical simulation of these waveguides
requires modern numerical methods with high efficiency and accuracy.
This article describes the method for finding wave modes for a three-layer waveguide.
The work is partially supported by RFBR grant No’s 15-07-08795, 16-07-00556. Also the
publication was financially supported by the Ministry of Education and Science of the Russian
Federation (the Agreement number 02.A03.21.0008).
Key words and phrases: regular open gradient planar waveguide, waveguide modes,
Chebyshev polynomials, TE and TM polarizations, Maxwell’s equations.
1. Introduction
Optical waveguides are spatially inhomogeneous optical structures that serve to
transmit light energy over sufficiently large distances [1–5]. The regular waveguide
consists of a dielectric waveguide layer (or a few ones) of refractive index nf (or
nf 1 , . . . , nf N ) and the dielectric cladding with smaller refraction indices: ns in the
substrate layer and nc in the cover layer as in Fig. 1.
There are exact and approximate analytical methods for simulation of waveguide
modes in open gradient planar waveguides with selected refractive index in the guiding
layer. In the case of an arbitrary piecewise continuous profile the approximate calcu-
lation of the electromagnetic field of guided modes is possible only by using numerical
methods, implemented on a computer.
We will use Cartesian coordinates associated with the waveguide geometry as in
Fig. 2.
The Maxwell equations can be split into two independent systems of equations.
For each system, there are three types of waveguide modes: guided modes, substrate
radiation modes, and cover radiation modes. Their solutions are, respectively,
Ey (x, y, z, t) = Ey (x) exp{iωt − iβz },
Copyright © 2017 for the individual papers by the papers’ authors. Copying permitted for private and
academic purposes. This volume is published and copyrighted by its editors.
In: K. E. Samouilov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the VII Conference
“Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”,
Moscow, Russia, 24-Apr-2017, published at http://ceur-ws.org
66 ITTMM—2017
Figure 1. A schematic diagram of the potential
Figure 2. Waveguide is formed by media 1–3
Hy (x, y, z, t) = Hy (x) exp{iωt − iβz }.
Both equations for the modes can be written in the more customary form [11–14]
d 1 dψ(k, x)
−p (x) + V (x)ψ(k, x) = k2 ψ(k, x) (1)
dx p (x) dx
Here p (x) = ε (x) for TE-modes, p(x) = µ(x) for TN-modes, V (x) = −k02 n2 (x),
2
n (x) = ε(x)µ(x), V (x) is a piecewise continuous function (continuous in each of the
layer), k2 = −k02 β 2 is the spectral parameter, and ψT E (x) = Ey (x), ψT M (x) = Hy (x).
Kuziv Yaroslav Y., Sevastianov Leonid A. 67
2. Description of the Method
This article describes how to find waveguide modes in the case of potential disconti-
nuity. The problem of describing a complete set of modes in an ordinary plane waveg-
uide is formulated in terms of the eigenvalue problem for a second-order self-adjoint
differential operator (1).
Solutions on the left and on the right are decreasing exponents in the case of real
εs , εc [7]:
ψ− (x) = C− exp {γ− (x + 1)} ,
ψ+ (x) = C+ exp {−γ+ (x − 1)} .
q
Here γs,c = k0 β 2 − n2s,c = 2π
p
λ
Vs,c − k2 .
In the waveguide layer, we seek the solution in the form of an expansion in the
Chebyshev polynomials of the first kind [6, 9]:
N
X
ψf (k, x) = Cj (k) Tj (x). (2)
j=0
In this relation we substitute the x1 , x2 , . . ., xN −1 . In points x0 , xN we require the
following conditions:
ψs (k, x0 ) = ψf (k, x0 ) , ϕs (k, x0 ) = ϕf (k, x0 )
ψf (k, xN ) = ψc (k, xN ) , ϕf (k, xN ) = ϕc (k, xN ) .
Polynomial of degree n is denoted Tn (x), and is given by the trigonometric for-
mula [8, 9, 14]:
Tn (x) = cos(n arccos (x)). (3)
Formula (3) can be replaced by recurrence expressions for Tn (x):
T0 (x) = 1, T1 (x) = x, Tn+1 (x) = 2xTn (x) − Tn−1 (x) n > 1.
The polynomial Tn (x) has n zeros in the interval [−1, 1], and they are located at
the points [8]:
!
π k − 21
x = cos , k = 1, 2, . . ., n.
n
In this same interval there are N + 1 extrema (maxima and minima) [8], located at
πk
x = cos , k = 0, 1, . . ., n,
n
we can also find the derivative of this function. Tn0 (x) derivatives of Chebyshev poly-
nomials approximating the derivative of the function f (x) are calculated using the
following formula [10]:
Tn0 (x) = n· sin (n·acos (x)) , x∈ (−1, 1) ,
Tn0 (x) = n2 xn+1 , x = −1 or x = 1.
68 ITTMM—2017
The second derivative is given by the following formulas:
−n2 cos (nt) n cos (t) sin (nt)
Tn00 (x) = + , x∈ (−1, 1) ,
sin2 (t) sin3 (t)
n3 (n − 1) xn+2
Tn00 (x) = , x = −1 or x = 1.
3
We can get a new eigenvalues and eigenfunctions by expanding the interval (a, b). If
a6= − 1 and b6=1, it is necessary to use the following formula:
2 2
d2 Tj (x)
00 00 2 2
Tf
j (x) = Tj (x) = 2
,
b−a d x b−a
2Tj0 (x) dTj (x) 2
f0 (x) =
T = .
j
b−a dx b − a
We obtain SLAE for undetermined coefficients Cs , C0 , C1 , . . ., CN , Cc .
Let us consider the case of TE modes. The substitution from (2) into (1) leads to
the following relation:
N
X N
X N
X
− Cj (k) Tj00 (xl ) + V (xl ) Cj (k) Tj (xl ) = k2 Cj (k) Tj (xl ) . (4)
j=0 j=0 j=0
On the interval (−∞, x0 ) the solution for equation (4) satisfies the following as-
→
ymptotic condition (x)x → −∞0. On the interval (xN , ∞) the solution satisfies the
→
asymptotic condition (x)x → ∞0. We write down boundary conditions of the third
kind with the help of Chebyshev polynomials:
X X
Cj (k) Tj (x0 ) = Сs eγs (x0 −x0 ) = 1 , Cj (k) Tj0 (x0 ) = γs Сs ,
X X
Cj (k) Tj (xN ) = Сc e−γc (xN −xN ) = 1 , Cj (k) Tj0 (xN ) = −γc Сc .
Denote fj1 = Tj0 (x0 ) − γs Tj (x0 ) and fj2 = Tj0 (xN ) + γc Tj (xN ).
From the boundary conditions we obtain the following system:
NX−1
f01 C0 (k) + fN1
CN (k) = − fj2 Cj (k),
j=1
N
X −1
f02 C0 (k) + fN
2
CN (k) = − fj2 Cj (k).
j=1
Solve this system by Cramer:
det = f01 fN
2 1 2
− fN f0 ,
Kuziv Yaroslav Y., Sevastianov Leonid A. 69
N −1
−1 X
f 1 f 2 − fN
2 1
C0 (k) = fj Cj (k),
det j=1 N j
(5)
N −1
1 X
f 1 f 2 − f02 fj1 Cj (k).
CN (k) =
det j=1 0 j
If we substitute the coefficients (5) into (4), then we will obtain the following sys-
tem of equations for the coefficients Cj (k), j = 1, N − 1 in the form of an eigenvalue
problem:
Mij = V (xi ) Tj (xi ) − Tj00 (xi ) , Bij = Tj (xi ) ,
−1 1 2
Mij0 = V (xi ) T0 (xl ) − T000 (xi ) 2 1
f f − fN fj ,
det N j
00
1
MijN = V (xi ) TN (xi ) − TN f 1 f 2 − f02 fj1 ,
(xi )
det 0 j
0 −1 1 2 2 1
Bij = T0 (xi ) f f − fN fj ,
det N j
N 1
f 1 f 2 − f02 fj1 ,
Bij = TN (xi )
det 0 j
M (k)ij = Mij + Mij0 + MijN ,
0 N
B(k)ij = Bij + Bij + Bij .
~
We compute the eigenvectors C(k) and eigenvalues k2 for the system:
~
M (k) C(k) ~
= k2 B(k)C(k). (6)
Figure 3. The graph of the first eigenfunction of the system (6)
70 ITTMM—2017
3. The Initial Approximation
Since the problem has the capacity gap, it is necessary to choose the initial approx-
imation. As an initial approximation we take the solution of the problem with the
boundary condition independent of the spectral parameter Vs,c . To do this, we let the
height of the potential well to infinity, ie, Vs,c → ∞. In this case the spectral parameter
γs,c , too, tend to infinity, which gives new relations:
N
X N
X
Tj (x0 ) Cj (k) = 0, Tj (xN ) Cj (k) = 0.
j=0 j=0
As a result, fj1 ∧fj2 take the form ff1 f2
j = −Tj (x0 )∧ fj = Tj (xN ). We get the following
value:
Mij = V (xi ) Tj (xi ) − Tj00 (xi ) , Bij = Tj (xi ) .
(−TN (x0 ) Tj (xN ) + TN (xN ) Tj (x0 ))
Mij0 = − V (xi ) T0 (xl ) − T000 (xi ) ,
(−TN (x0 ) T0 (xN ) + TN (xN ) T0 (x0 ))
00 (−T0 (x0 ) Tj (xN ) + T0 (xN ) Tj (x0 ))
MijN = V (xi ) TN (xi ) − TN
(xi ) ,
(−TN (x0 ) T0 (xN ) + TN (xN ) T0 (x0 ))
0 (−TN (x0 ) Tj (xN ) + TN (xN ) Tj (x0 ))
Bij = T0 (xi ) ,
(−TN (x0 ) T0 (xN ) + TN (xN ) T0 (x0 ))
N (−T0 (x0 ) Tj (xN ) + T0 (xN ) Tj (x0 ))
Bij = TN (xi ) .
(−TN (x0 ) T0 (xN ) + TN (xN ) T0 (x0 ))
Figure 4. Graph of initial approximation
The new expressions no longer contain a dependence on the spectral parameter and
correspond to the problem for a closed waveguide with boundary conditions of the first
kind at the potential discontinuity points.
The expressions obtained give approximate values for eigenvalues and eigenvectors.
Kuziv Yaroslav Y., Sevastianov Leonid A. 71
4. Conclusion
The solution of many problems of integrated optics includes a spectral analysis and
spectral synthesis on the basis of a complete system of solutions of differential equations
of second order operator that regulates the guided modes in an open waveguide. In
the simplest case, a regular operator waveguide is essentially self-adjoint and has a
continuous mixed range.
The method described in this paper allows one to find numerical solutions for a
three-layer waveguide. This method can be modified for other types of tasks.
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