Minimal Waveguide Example

In this example we show how to calculate the stationary paraxial field for a slab and cylindircal waveguide using PyPropagate.



In [1]:

    
from pypropagate import *
%matplotlib inline

Setting up the propagators

We begin by creating the settings for the propagator. We set a simulation box of size $1 \mu \text{m} \times 1 \mu \text{m} \times 0.5 \text{mm}$ and $1000 \cdot 1000 \cdot 1000$ voxels and set initial and boundary conditions for a monochromatic plane wave with $12 \text{keV}$ photon energy. Note that we could also set the $x_\text{min}$, $x_\text{max}$, $N_x$, ... boundaries individually.



In [2]:

    
settings = presets.settings.create_paraxial_wave_equation_settings()
settings.simulation_box.set((0.25*units.um,0.25*units.um,0.25*units.mm),(1000,1000,1000))
presets.boundaries.set_plane_wave_initial_conditions(settings)
settings.wave_equation.set_energy(12*units.keV)

Our waveguide will consist of a vacuum core with a Titanium cladding. We can automatically lookup the refraction index for a compund material by providing chemical formula.



In [3]:

    
nVa = 1
nGe = presets.medium.create_material('Ge',settings)

For a slab waveguide we will use a one dimensional and for a circular waveguide a two dimensional propagator. Since the one dimensional propagator assumes the values for $y = 0$ are valid everywhere we can define both waveguides with one formula. As the waveguide radius we choose $24\text{nm}$.



In [4]:

    
s = settings.symbols
waveguide_radius = 25*units.nm
settings.wave_equation.n = pc.piecewise((nVa,pc.sqrt(s.x**2+s.y**2) <= waveguide_radius),(nGe,True))
settings.get_numeric(s.n)









    Out[4]:





$$\begin{cases} 1 & \text{if } \sqrt{{x}^{2}+{y}^{2}} \le \frac{\mathrm{m}}{4 \cdot 10^{7}} \,   \\ \frac{9999935658021387}{10^{16}} \,  -\frac{3560064055767287\, i }{5 \cdot 10^{21}} \,   & \text{otherwise }  \end{cases}$$

Slab Waveguide

We now calculate the stationary solution for the slab waveguide using the one dimensional finite differences propagator.



In [5]:

    
propagator = propagators.FiniteDifferences2D(settings)



In [6]:

    
field = propagator.run_slice()[-2*waveguide_radius:2*waveguide_radius]
plot(field,figsize = (13,5));









    



propagating:|███████████████████████| 999/999 [0.2s < 0(0)s]]

Cylindrical Waveguide

With the same settings we can calculate the solution for a cylindrical waveguide by using a cylindrically symmetrical finite differences propagator.



In [7]:

    
propagator = propagators.FiniteDifferencesCS(settings)



In [8]:

    
field = propagator.run_slice()[-2*waveguide_radius:2*waveguide_radius]
plot(field,figsize = (13,5));









    



propagating:|███████████████████████| 999/999 [0.3s < 0(0)s]]