The Maxwell equations have a fairly simple form. However, finding solutions to Maxwell's equations is an extremely dificult task. Therefore, various simplifying approaches are often used in optics. One such simplifying approach is to use the approximation of geometric optics. The approximation of geometric optics is constructed with the assumption that the wavelengths are small (short-wavelength approximation). The basis of geometric optics is the eikonal equation. The eikonal equation can be obtained from the wave equation (Helmholtz equation). Thus, the eikonal equation relates the wave and geometric optics. In fact, the eikonal equation is a quasi-classical approximation (the Wentzel-Kramers-Brillouin method) of wave optics. This paper shows the application of geometric methods of electrodynamics to the calculation of optical devices, such as lenses Maxwell and Luneburg. The eikonal equation, which was transformed to the ODE system by the method of characteristics, is considered. The resulting system is written for the case of Maxwell and Luneburg lenses and solved by standard numerical methods. Describes the implementation details and images of the trajectories of rays and fronts of the waves.
Copyright © 2018 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. Samouylov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the VIII Conference “Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”, Moscow, Russia, 20-Apr-2018, published at http://ceur-ws.org
In this article, we consider the approach to transform the eikonal equations to the ODE system. In the first part we briefly formulate the problem and describe Luneburg and Maxwell lens. The second part describes a software implementation that allows to visualize the trajectory of rays and wave fronts.
The eikonal equation can be obtained from Maxwell’s equations written for the
regions free of currents and charges and under the condition of a time-changing harmonic
electromagnetic field in a nonconducting isotropic medium [
r ∈ R3,
x ∈ ⊂ R3.
where r = (, , ) is radius-vector, (r) is boundary condition, (r) is refractive index
of the medium. The function (r) is real scalar function with a physical meaning of
time. It is also often called as the eikonal function [
For visualization of lens modeling results, we will consider their projection on the plane. In this case, the eikonal equation is reduced to a two-dimensional form: ⎧ (︂ (, ) )︂ 2 ⎪ ⎨
+ ︂( (, ) )︂ 2
⎪ ⎩ (, ) = (, ), = 2(, ),
(, ) ∈ R2, (, ) ∈ ⊂ R2.
Using the characteristic method, the eikonal equation can be transformed into an ODE system that can be solved by standard numerical methods.
2.2.
By using the method of characteristics [
The following relation 21 + 22 = 1 is imposed on constants 1 and 2. Thus we may take: 1 = cos( ) and 2 = sin( ). Initial conditions give a mathematical description of the source of the rays. For example, to model a point source, we need to fix the initial
In polar coordinates, the eikonal equation has the following form: coordinates 0, 0 and change the angle , which will set the angle of the beam exit from the source-point. To simulate the radiating surface, on the contrary, it is necessary to fix the angle
and change the coordinates 0 and 0.
The parameter has a physical meaning of the signal passing time from the point (0, 0) to the point (, ).
︂( (, ) )︂ 2
1 (︂ (, ) )︂ 2 + 2 = 2(), and corresponding system of ODEs will have the form: = , = , ⎧ d ⎪ ⎪⎪⎪ d ⎪ ⎪⎪⎪ d ⎪ ⎪ ⎪⎨ d
d ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ ⎩ d ⎪ d = ⎪ = −
+
The Luneburg lens [
√︂ ⎧ ⎨0 ⎩0, 2 − where (, ) = √︀( − 0)2 + ( −
√ to an arbitrary point in the (, ) plane. The formula implies that the coeficient continuously varies from 0
2 to 0 starting from the center of the lens and ending with its boundary. The refractive index of the medium outside the lens is constant and is equal to 0. Usually 0 is equal to 1.
To solve the eikonal equation by the method of characteristics, it is necessary to ifnd partial derivatives of the function
(, ). For the Luneburg lens case the partial 0)2 is the distance from the center of the lens derivatives are:
Outside lens region the derivatives are equal to 0.
The Maxwell’s fish eye lens [
Maxwell’s fish eye lens ⎧ ⎪ (, ) = ⎨ 1 + ⎪⎩0, 0 3.
= −
We carry on numerical modeling for lenses with a radius = 1, the refractive
index of the external medium 0 = 1, the center of the lens was placed in the point
(0, 0) = (2, 0), the boundary region was set as the rectangle min = 0, max = 5,
min = − 1.5 and max = 1.5. The point source was placed on the lens boundary
at (0, 0) = (0, 0). 50 values of the parameter have bee taken from the interval
[− / 2 + / 100, / 2 − / 100], which allowed to simulate rays trajectories from a point
source within an angle slightly smaller than 180∘ . The parameter was changed within
the [
Each parameter value sets new initial conditions for the ODE system. The process of numerical simulation consists in multiple solution of this system for diferent initial conditions. The numerical solution of the ODE system for a particular initial condition gives us a set of points ( , ), = 1, . . . , approximating the trajectories of a particular beam. After performing calculations for all the selected initial conditions, we obtain a set of rays. To visualize the rays, it is enough to depict each of the obtained numerical solutions. The result of the simulation may be seen in 1 and 2.
To visualize the wave fronts with the resulting numerical data it is necessary to carry out additional manipulations. From each numerical solution, we must select points ( , ) that correspond to a specific point in time .
The use of a numerical method with a fixed step gives an advantage, since each numerical solution will be obtained for the same uniform grid 0 < 1 < . . . < < . . . < .
3.2.
For the simulation of rays transmission through the lens we used Julia language [
The main program files are located in the src directory. Files of Julia source code have jl extension. Note that the Julia language encourages usage of Unicode variable so we may use various characters of utf-8 encoding, which are often found in mathematical formulas. So, we used Greek letters and symbols of partial diferentiation . – eikonal.jl — the right part of the system of diferential equations, for all lenses it has the same form, difering only in functions n(x,y) and n(x,y). – maxwell.jl — refractive index n(x,y) and its derivatives n (x,y) for the Maxwell lens. – luneburg.jl — similar to the previous item, but for Luneberg lens. – parameters.jl — common parameters for all lenses: lens radius, refractive index of the external medium, coordinates of the center of the lens, area boundaries for Cartesian and polar coordinates, etc. – RK.jl — a separate function that implements the Runge-Kutta method.
Outside the directory src are scripts that trigger calculations. The script cartesian1.jl performs calculations for the point source, the script cartesian2.jl for parallel rays, and the script polar.jl for polar coordinates. Depending on the argument, the files corresponding to the desired lens are loaded. The calculation process for all lenses is the same. The script plot.py is written in Python and is used to visualize the trajectories of rays and fronts of the waves.
Consider, for example, the script cartesian1.jl, briefly explaining the Julia syntax. At the very beginning of the script the variables and functions are selectively imported from files. include("src/RK.jl") include("src/parameters.jl") if length(ARGS) > 0
ARGS[
ARGS[
include("src/maxwell.jl") end include("src/eikonal.jl") Julia is a JIT-compiled language, so it considers the speed of a compiled language with the flexibility of an interpreted one. In this case, it allows us to include the necessary code during the program execution and we get versions of the n(x,y) and n(x,y) functions for Maxwell and Luneberg lens depending on which command-line argument is passed when the program is run. The operator && allows you write conditional execution in the short form. So, the code from the maxwell.jl file will be enabled only if the first conditional expression is true (if maxwell argument is passed).
Then follows the process of calculation. As described above, we have to solve the ODE system for the set of initial values. Each solution gives the coordinates (, ) of the corresponding ray. The calculation results for each ray are stored in a separate file. isdir(dir) ? true : mkdir(dir) # x_0, y_0, p1_0, p2_0 xn_0 = Vector{Float64}(4) i = 0 # The point source. Rays are radiated at an angle alpha for alpha in linspace(alpha_min, alpha_max, 50) i = i + 1 file = open("./$(dir)/data$i.txt", "w") xn_0[1:2] = [x_0, y_0] xn_0[3:4] = [cos(alpha)*n(x_0, y_0), sin(alpha)*n(x_0, y_0)] tn, xn = RK.RKp6n1(eikonal, xn_0, h, t_start, t_stop) for (t, x, y) in zip(tn, xn[:, 1], xn[:, 2])
write(file, "$t,$x,$y\n") end
Since the solution of the ODE system gives the trajectory of a single ray, visualization of rays trajectories is a simple task. The solutions for each system are saved as three columns of data: , and . To visualize the ray it is suficient to read columns and and and plot them.
A more complex task is to visualize the wave fronts. Geometrically, the wave front is formed from points of the beam corresponding to the same time. Thus, to get all the points of the front, it is necessary to consider the coordinates and corresponding to the same time from each file with ODE solutions. Since we used the Runge–Kutta method with a constant step, the same time in diferent files corresponds to the same line number. The following code fragment shows how a list containing the points of the wave fronts is formed. # The lines_num variable contains the number of lines in the file levels = [[] for i in range(lines_num)] # Sequentially open all files for reading for fname in fnames: with open(fname) as f: for li, line in enumerate(f.readlines()):
levels[li].append([float(s) for s in line.strip(’\n’).split(’,’)]) levels = np.array(levels) As a result, we form the nested list, each element containing all the points of the corresponding front.
The paper presents the description of the numerical solution of the eikonal equation for the case of Luneburg and Maxwell lenses. The results are visualized as trajectories of rays passing through lenses and as fronts of electromagnetic waves.
The work is partially supported by Russian Foundation for Basic Research (RFBR) grants No 16-07-00556. Also the publication was prepared with the support of the “RUDN University Program 5-100”.