In this example, we will characterize the performance of a nanobeam photonic crystal (PC) electro-optic modulator using CHARGE (electrical simulation) and FDTD (optical simulation).
Theory
In a photonic crystal (PC) structure, changes to the effective index of the cladding will result in a shift in the resonant frequency, which can be used to modulate an optical signal [1]. This change in effective index is driven by modulation of the electric field within the resonant cavity. To simulate the electric field established in the dielectric between two doped silicon arms of the nanobeam, a self-consistent simulation of the charge and electrostatic potential is performed using CHARGE. To calculate the effect of the change in the electric field on the transmission of the PC, an FDTD simulation will be run. A structure group in FDTD will use the DC electric field data from CHARGE to calculate the corresponding changes in the real and imaginary parts of refractive index of the material based on the formulation presented in [1].
$$\Delta n=\frac{\gamma_{33}}{2} n_{d i}^{3} E$$
where E is the magnitude of the electric field and the dielectric (an electro-optically active polymer) has baseline refractive index ndi = 1.6. The EO coefficient γ33 is 150pm/V [1].
Simulation Setup
Electrical Simulation
The PC structure in the waveguide modulator can be set up in CHARGE using a construction group. Using a construction group enables the parameterization of the design, including the grating period, slot width, etc. Open the nanobeam_pc_eomod.ldev project in CHARGE. The dimensions of the waveguide and PC reflect the structures in the referenced paper by Biao Qui, et al. [1]. The component is fabricated in an SOI process, where a thin layer of silicon is formed epitaxially on a buried oxide. The silicon is then etched back to pattern the waveguide structures. The peak doping concentrations and dimensions of the doping profile in the referenced paper are used to define the contact and background doping profile in the waveguide. Analytic models are used to construct the doping profile, however it can also be imported from a process simulation result. The top surface is passivated with an electro-optic (EO) polymer, which also fills the holes and slot in the PC.
In the referenced device, a cavity is formed at the centre of the PC. A silicon waveguide is etched with a central slot and a periodic array of holes. The hole radius and spacing is tapered towards the cavity, as illustrated in the figure below. The dashed line represents the symmetry line at y=0. The first five holes extending to the right form the taper. In the layout, +y is chosen as the direction of optical propagation. Holes with the regular periodicity and radius extend to the length of the active area (16um).
The PC design is parameterized by a set of properties associated with the "pc" structure group in the CHARGE layout. These properties can be modified to adjust the layout of the holes and slot within the waveguide.
In the CHARGE, two simulations are run: one for the cavity and one for a single regular hole in the PC. While the entire component could be simulated at once, this approach is more efficient, relying on the assumption that the component exhibits mirror symmetry at the simulation region boundaries in the y-direction. The electric field information is recorded by the electric field monitors, and stored in a file that will be loaded in the optical simulation. The electric field monitors are also used to calculate the capacitance of the PC by evaluating the differential charge at each bias. A mesh override region has been added in the slot to further refine the mesh in that region, such that the electric field can be accurately sampled.
Optical Simulation
The layout for the optical simulation mirrors that of the electrical simulation. Open the nanobeam_pc_eomod.fsp file in FDTD. The electrical simulation calculated the electric field in response to the applied voltage. To represent a material whose index varies spatially in response the distribution of that electric field, (n,k) import material objects are used. These objects are constructed using a structure group that imports the electric field data calculated with CHARGE and performs an interpolation to the rectilinear grid in FDTD. Two objects are used to represent the two (symmetric) sides of the taper, and the field profile for a regular hole is repeated several times to define the remainder of the PC.
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Note: the electrical simulation must first be run in CHARGE to generate the data required to construct the waveguide with a spatially varying index in FDTD. Alternatively, the provided data files, nanobeam_pc_cavity_field.mat and nanobeam_pc_grating_field.mat, can downloaded into your working directory. |
The constructed "field dependent index material" object with spatially varying index is rectangular, and is defined for the extents of the waveguide. To correctly represent EO polymer, a construction group (similar to the one used to define the PC in CHARGE) is used to define the surrounding silicon of the waveguide. This material is given a higher priority by setting its mesh order before that of the material with the spatially varying index. As a result, the silicon will cut away the excess material from the EO polymer, such that the EO polymer will only be defined for the volume within the holes and slot. A background dielectric with the same base refractive index is used to fill the remainder of the volume.
To excite modes in the PC, dipoles are added to the FDTD simulation volume. A time monitor is used to record the electric field at the centre of the cavity and determine the wavelength of the resonance as well as the quality factor. Note that mesh overrides are used to ensure the correct mesh period such that the uniform holes in the grating are discretized in the same way. For more information, users are encouraged to review the tutorial on the photonic crystal cavity for a discussion of simulation techniques required for accurate results.
Results and Discussion
Electrical Bandwidth
Load the nanobeam_pc_eomod.ldev project in CHARGE. The intrinsic bandwidth of the modulator is calculated by treating the component as a series RC circuit. This is illustrated in the schematic below.
The resistance is dominated by the low conductance of the thin silicon slab connecting the waveguide to the adjacent contacts. An important design tradeoff is made between modal confinement (thin silicon) and low loss (thicker silicon). The resistance of the slab can be simulated by adding a "test" contact that connects to the waveguide on the same side of the slot as one of the two contacts, such that a continuous circuit is formed through the silicon (the "anode_test" contact in the project is added for this purpose). Align the solver region with this cross-section, set the solver geometry to 2D y-normal, and run the simulation. The following script commands can be used to reposition the simulation region programmatically.
switchtolayout;
setnamed('CHARGE simulation region', 'dimension','2D Y-Normal');
setnamed('CHARGE simulation region', 'y', 8.5e-6);
A 2D simulation is run at two bias points (0V and 0.1V). The simulation results are normalize to a length of 1cm, and the conductance per unit length can be found as the ratio of the current to voltage. The following script commands will calculate the resistivity, then normalize to the overall device length (14.8um)
test_contact = getresult('CHARGE','anode_test');
I_test = pinch(test_contact.I);
R_test = 0.1/I_test(2); # ohm-cm
Rslab = R_test/14.8e-4; # ohm
The physical structure of the PC resonator is that of two doped semiconductor regions separated by a dielectric, forming a semiconductor-insulator-semiconductor (SIS) capacitor. The capacitance can be calculated as a function of voltage by determining the change in net charge stored on one side of the structure when the voltage is perturbed. By running two simulations at bias voltage V and V+ΔV, the capacitance can be estimated as
$$C_{n, p} \approx \frac{Q_{n, p}(V+\Delta V)-Q_{n, p}(V)}{\Delta V}$$
The script file nanobeam_pc_run_cv.lsf can be used to set-up and run a DC sweep over the range from 0-0.5V with a perturbation of ΔV=25mV at each step. The net charge is calculated using a electric field monitor, which will integrate the electric flux through the surface (Gauss's law). Two sweeps are performed: one for the half-cavity, and one for a single hole in the grating. The script file will combine both results and perform the capacitance calculation. The capacitance is nearly constant as a function of applied bias, as expected for a SIS capacitor.
The bandwidth can then be estimated from the product of the net resistance and capacitance,
$$f_{3 d B} \approx \frac{1}{2 \pi R C}$$
The bandwidth from this calculation is estimated as approximately 120GHz.
Electrostatic Field Distribution
To determine the perturbation to the refractive index of the polymer within the cavity and grating of the PC, the electric field is required. This quantity is calculated in a typical CHARGE simulation, and can be recorded for the waveguide region using an electric field monitor. Using the same project as for the bandwidth calculation, change the DC sweep type for the anode contact from "value" to "range." This will change the simulation to sweep over biases from 0 to 0.5V without perturbing the voltage at each bias point. The following script command will also change the contact mode.
select('CHARGE::boundary conditions::anode');
set('sweep type','range');
Set the solver region to span the tapered region of the PC by setting the properties y min = 0um and y max = 2.251um. Change the solver region geometry to 3D. Enable the monitors wg_cavity and wg_grating. Run the simulation for the taper and cavity first. The electric field monitor wg_cavity will automatically save the electric field data to file. When the simulation is complete, the electric field in the taper area can be visualized from the data recorded by the cavity_field_xsec monitor. The magnitude of E is shown in the plot below at V = 0.2V. Note that the electric field is concentrated in the narrow slot regions, but becomes much weaker in the wider holes.
Next, shift the simulation region to span one hole in the grating: set properties y min = 3.52um and y max = 3.943um. Run the simulation. In this simulation, the electric field monitor wg_grating will save the electric field data to a second file.
Electro-optic Modulator Response
Load the nanobeam_pc_eomod.fsp project in FDTD. In this layout, (n,k) import objects are constructed using the electric field data from the electrical simulation according to the formula presented in the first section. This object is restricted to the PC grating and cavity and is composed of three adjacent structures: one representing the taper and cavity, and two others that repeat the result for the single hole on either side of the taper to represent the grating. Outside of that region, an idealized index material is used to represent the background EO polymer, with an index of 1.6. An index monitor has been included to visualize the index profile. The selected bias point for simulation can be changed by setting the "index_V" property of the "field dependent index material" structure group, which contains the (n,k) import objects constructed from the electric field data. There are six bias points ranging from 0 to 0.5V. At 0.2V (index_V = 3), the index in the vicinity of the cavity is shown in the figure below. Note that the scale of the colorbar is restricted to values around the base index of the EO polymer (the index of the surrounding silicon is approximately 3.48).
In the optimizations and sweeps toolbox, a sweep named "voltage" has been included which will vary the bias point used to determine the field-dependent index of the EO polymer. Using a modified high Q analysis object from the object library, the Q factor and resonant peak of the cavity will be recorded. Running this sweep will simulate the response of the PC as a function of voltage, and generate the following figures. At 0V, the resonant frequency is approximately 186THz, with a Q factor of approximately 5300. Therefore, the FWHM is estimated as 35GHz, which requires a wavelength shift of 0.30nm. From the plot of the wavelength of resonance vs. applied voltage, a slope of 0.9nm/V is obtained, such that a bias of approximately 0.33V would be required to shift the spectrum by the FWHM. For a 14.8um device, this corresponds to a Vπ-L value of 0.5mV-cm. These results are comparable to the values obtained in the referenced publication [1].
(a)
(b)
Modulation response for the photonic crystal electro-optic modulator, showing variation in (a) the quality factor and (b) the wavelength of resonance as a function of the applied voltage.
Reference
- Biao Qi, et al., "Analysis of Electrooptic Modulator With 1-D Slotted Photonic Crystal Nanobeam Cavity," IEEE Photonics Tech. Lett., 23, 992 (2011) [doi]