Field projection for a zone plate
Field projection for a zone plate#
This tutorial will show you how to solve for electromagnetic fields far away from your structure using field information stored on a nearby surface.
This field projection technique is very useful for reducing the simulation size needed for structures involving lots of empty space.
As an example, we will simulate a simple zone plate lens with a very thin domain size, and measure the transmitted fields just above the structure. Then, we’ll show how to use field projections to compute the fields at the focal plane above the lens.
For more simulation examples, please visit our examples page. If you are new to the finite-difference time-domain (FDTD) method, we highly recommend going through our FDTD101 tutorials. FDTD simulations can diverge due to various reasons. If you run into any simulation divergence issues, please follow the steps outlined in our troubleshooting guide to resolve it.
# standard python imports import numpy as np import matplotlib.pyplot as plt # tidy3d imports import tidy3d as td import tidy3d.web as web
Below is a rough sketch of the setup of the field projection.
The transmitted near fields are measured just above the metalens on the blue line, and are projected to the focal plane denoted by the red line.
Define Simulation Parameters#
As always, we first need to define our simulation parameters. As a reminder, all length units in
tidy3D are specified in microns.
# 1 nanometer in units of microns (for conversion) nm = 1e-3 # free space central wavelength wavelength = 1.0 # numerical aperture NA = 0.8 # height of lens features height_lens = 200 * nm # space between bottom PML and substrate (-z) # and the space between lens structure and top pml (+z) space_below_sub = 1.5 * wavelength # height of substrate (um) height_sub = wavelength / 2 # side length (xy plane) of entire metalens (um) length_xy = 20 * wavelength # Lens and substrate refractive index n_TiO2 = 2.40 n_SiO2 = 1.46 # define material properties air = td.Medium(permittivity=1.0) SiO2 = td.Medium(permittivity=n_SiO2**2) TiO2 = td.Medium(permittivity=n_TiO2**2)
Next we perform some conversions based on these parameters to define the simulation.
# because the wavelength is in microns, use builtin td.C_0 (um/s) to get frequency in Hz f0 = td.C_0 / wavelength # Define PML layers, for this application we surround the whole structure in PML to isolate the fields boundary_spec = td.BoundarySpec.all_sides(boundary=td.PML()) # domain size in z, note, we're just simulating a thin slice: (space -> substrate -> lens height -> space) length_z = space_below_sub + height_sub + height_lens + space_below_sub # construct simulation size array sim_size = (length_xy, length_xy, length_z)
Now we create the ring metalens programatically
# define substrate substrate = td.Structure( geometry=td.Box( center=[0, 0, -length_z / 2 + space_below_sub + height_sub / 2.0], size=[td.inf, td.inf, height_sub], ), medium=SiO2, ) # focal length focal_length = length_xy / 2 / NA * np.sqrt(1 - NA**2) # location from center for edge of the n-th inner ring, see https://en.wikipedia.org/wiki/Zone_plate def edge(n): return np.sqrt(n * wavelength * focal_length + n**2 * wavelength**2 / 4) # loop through the ring indeces until it's too big and add each to geometry list n = 1 r = edge(n) rings =  while r < 2 * length_xy: # progressively wider cylinders, material alternating between air and TiO2 cylinder = td.Structure( geometry=td.Cylinder( center=[ 0, 0, -length_z / 2 + space_below_sub + height_sub + height_lens / 2, ], axis=2, radius=r, length=height_lens, ), medium=TiO2 if n % 2 == 0 else air, ) rings.append(cylinder) n += 1 r = edge(n) # reverse geometry list so that inner, smaller rings are added last and therefore override larger rings. rings.reverse() geometry = [substrate] + rings
Create a plane wave incident from below the structure
# Bandwidth in Hz fwidth = f0 / 10.0 # Gaussian source offset; the source peak is at time t = offset/fwidth offset = 4.0 # time dependence of source gaussian = td.GaussianPulse(freq0=f0, fwidth=fwidth) source = td.PlaneWave( center=(0, 0, -length_z / 2 + space_below_sub / 2), size=(td.inf, td.inf, 0), source_time=gaussian, direction="+", pol_angle=0.0, ) # Simulation run time run_time = 40 / fwidth
We’ll create a FieldProjectionCartesianMonitor monitor which measures the fields just above the structure, and projects them to a Cartesian plane in the far field a given distance away. We’ll also make a dedicated near-field monitor just to see what the near fields look like.
# place the monitors halfway between top of lens and PML pos_monitor_z = ( -length_z / 2 + space_below_sub + height_sub + height_lens + space_below_sub / 2 ) # set the points on the observation grid at which fields should be projected num_far = 40 xs_far = 4 * wavelength * np.linspace(-0.5, 0.5, num_far) ys_far = 4 * wavelength * np.linspace(-0.5, 0.5, num_far) monitor_far = td.FieldProjectionCartesianMonitor( center=[ 0.0, 0.0, pos_monitor_z, ], # center of the near field surface on which fields are recorded size=[ td.inf, td.inf, 0, ], # size of the near field surface on which fields are recorded normal_dir="+", # normal vector direction of the near field surface on which fields are recorded freqs=[f0], name="farfield", x=xs_far, y=ys_far, proj_axis=2, # direction along which fields are projected proj_distance=focal_length, # signed distance along the projection axis at which the observations grid resides far_field_approx=False, # turn off the geometric far field approximations, which in this case would lead to # inaccurate fields because the distance to the focal plane is of the same order as # the size of the structure and the near field monitor ) monitor_near = td.FieldMonitor( center=[0.0, 0.0, pos_monitor_z], size=[td.inf, td.inf, 0], freqs=[f0], name="nearfield", )
Put everything together and define a simulation object. A nonuniform simulation grid is generated automatically based on a given number of cells per wavelength in each material (10 by default), using the frequencies defined in the sources.
simulation = td.Simulation( size=sim_size, grid_spec=td.GridSpec.auto(min_steps_per_wvl=20), structures=geometry, sources=[source], monitors=[monitor_far, monitor_near], run_time=run_time, boundary_spec=boundary_spec, )
Note: Tidy3D warns us that one of the rings is touching the PML boundary. This warning is intended to prevent people from placing, for example, a substrate next to the PML without extending into it. In our case, the ring is not supposed to be extending into the PML so we can safely ignore the warning.
Let’s take a look and make sure everything is defined properly
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(20, 8)) simulation.plot_eps(x=0, ax=ax1) simulation.plot_eps( z=-length_z / 2 + space_below_sub + height_sub + height_lens / 2, ax=ax2 ) plt.show()
Now we can run the simulation and download the results
import tidy3d.web as web sim_data = web.run( simulation, task_name="zone_plate", path="data/simulation.hdf5", verbose=True )
View task using web UI at webapi.py:190 'https://tidy3d.simulation.cloud/workbench?taskId=fdve- 9982a94e-0039-45c0-8d20-43ce2d6e32a4v1'.
near_field_data = sim_data["nearfield"] fig, (ax1, ax2, ax3) = plt.subplots(1, 3, tight_layout=True, figsize=(15, 3.5)) near_field_data.Ex.real.plot(ax=ax1) near_field_data.Ey.real.plot(ax=ax2) near_field_data.Ez.real.plot(ax=ax3) plt.show()
Getting Far Field Data#
The FieldProjectionCartesianMonitor object ensures that the projected fields are already computed on the server during the simulation run.
The projected fields can then be used to quickly return various quantities such as power and radar cross section.
For this example, we use
FieldProjectionCartesianData.fields_cartesian to get the fields at the previously-set
x,y,z points as an
projected_fields = sim_data[monitor_far.name].fields_cartesian
Now we can plot the near and far fields together
# plot everything f, (axes_near, axes_far) = plt.subplots(2, 3, tight_layout=True, figsize=(10, 5)) def pmesh(xs, ys, array, ax, cmap): im = ax.pcolormesh(xs, ys, array.T, cmap=cmap, shading="auto") return im ax1, ax2, ax3 = axes_near im = near_field_data.Ex.real.plot(ax=ax1) im = near_field_data.Ey.real.plot(ax=ax2) im = near_field_data.Ez.real.plot(ax=ax3) ax1, ax2, ax3 = axes_far im = projected_fields["Ex"].real.plot(ax=ax1) im = projected_fields["Ey"].real.plot(ax=ax2) im = projected_fields["Ez"].real.plot(ax=ax3) plt.show()
We can also use the far field data and plot the field intensity to see the focusing effect.
intensity = 0 for field in ("Ex", "Ey", "Ez"): intensity += np.abs(projected_fields[field].isel(z=0, f=0)) ** 2 _, (ax1, ax2) = plt.subplots(1, 2, figsize=(13, 5)) np.sqrt(intensity).plot(ax=ax2, cmap="magma") intensity.plot(ax=ax1, cmap="magma") ax1.set_title("$|E(x,y)|^2$") ax2.set_title("$|E(x,y)|$") plt.show()