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 polarizations 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).
1. Introduction Ey (x; y; z; t) = Ey (x) expfi !t i z g;
Here p (x) = " (x) for TE-modes, p(x) = (x) for TN-modes, V (x) = k02n2(x), n2(x) = "(x) (x), V (x) is a piecewise continuous function (continuous in each of the layer), k2 = k02 2 is the spectral parameter, and T E (x) = Ey(x), T M (x) = Hy(x).
Description of the Method
This article describes how to find waveguide modes in the case of potential discontinuity. The problem of describing a complete set of modes in an ordinary plane waveguide 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
"s, "c [
Here s;c = k0q 2 ns2;c = 2 pVs;c k2.
In the waveguide layer, we seek the solution in the form of an expansion in the
Chebyshev polynomials of the first kind [
(x + 1)g ; + (x) = C+ exp f +(x
1)g : f (k; x) =
N X Cj (k) Tj (x): j=0 k n k n
In this relation we substitute the x1; x2; : : :; xN 1. In points x0; xN we require the following conditions: s (k; x0) = f (k; x0) ;
's (k; x0) = 'f (k; x0) f (k; xN ) = c (k; xN ) ;
'f (k; xN ) = 'c (k; xN ) :
Polynomial of degree n is denoted Tn (x), and is given by the trigonometric
formula [
Tn (x) = cos(n arccos (x)): 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 [
T n0 (x) = n sin (n acos (x)) ;
x2 ( 1; 1) ; T n0 (x) = n2xn+1; x = 1 or x = 1: (2) (3) 1) xn+2
2 b a 2Tj0 (x)
N j=0 The second derivative is given by the following formulas:
T n00 (x) =
10. On the interval (xN ; 1) the solution satisfies the asymptotic condition (x)x !! 10. We write down boundary conditions of the third kind with the help of Chebyshev polynomials:
X Cj (k) Tj (x0) = Сs e s(x0 x0) = 1 ; X Cj (k) Tj0 (x0) = sСs; X Cj (k) Tj (xN ) = Сc e c(xN xN ) = 1 ; X Cj (k) Tj0 (xN ) = cСc: Denote fj1 = Tj0 (x0) sTj (x0) and fj2 = Tj0 (xN ) + cTj (xN ).
From the boundary conditions we obtain the following system: Solve this system by Cramer: 8 >>>>f01C0 (k) + f N1 CN (k) = > > < > >>>>f02C0 (k) + f N2 CN (k) = > :
N 1 X fj2Cj(k); j=1 N 1 X fj2Cj(k): j=1 det = f01f N2 8 >>>>C0 (k) = > > < > >>>CN (k) = > > :
If we substitute the coefficients (5) into (4), then we will obtain the following system 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) ; Mi0j = V (xi) T0 (xl)
T000 (xi) MiNj = V (xi) TN (xi)
T N00 (xi) f N2 fj1 ; We compute the eigenvectors C~ (k) and eigenvalues k2 for the system: M (k) C~ (k) = k2B(k)C~ (k): (6) value: 3. The Initial Approximation
Since the problem has the capacity gap, it is necessary to choose the initial approximation. 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 ! 1. In this case the spectral parameter s;c, too, tend to infinity, which gives new relations:
N j=0 X Tj (x0) Cj (k) = 0;
X Tj (xN ) Cj (k) = 0: N j=0 As a result, fj1 ^fj2 take the form ffj1 = Tj (x0)^ffj2 = Tj(xN ). We get the following
Mij = V (xi) Tj (xi) Tj00 (xi) ; Bij = Tj (xi) : Mi0j =
V (xi) T0 (xl) T000 (xi) MiNj = V (xi) TN (xi) T N00 (xi) ( TN (x0) Tj (xN ) + TN (xN ) Tj (x0)) ( TN (x0) T0 (xN ) + TN (xN ) T0 (x0)) ( T0 (x0) Tj (xN ) + T0 (xN ) Tj (x0)) ( TN (x0) T0 (xN ) + TN (xN ) T0 (x0)) ; ; Bi0j = T0 (xi) ( TN (x0) T0 (xN ) + TN (xN ) T0 (x0)) ( TN (x0) Tj (xN ) + TN (xN ) Tj (x0)) ; BiNj = TN (xi) ( TN (x0) T0 (xN ) + TN (xN ) T0 (x0)) ( T0 (x0) Tj (xN ) + T0 (xN ) Tj (x0)) :
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.
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.