Overview
We extend the traveling wave laser model (TWLM) in INTERCONNECT to accurately simulate the directly modulated laser bandwidth and create custom script functions to postprocess the timedomain simulation results and calculate the RF response and 3dB bandwidth.
The carrier capture/escape rate model [1] is implemented to enable more accurate modeling of carrier transport effects across the separate confinement heterostructure. These transport effects are important to accurately model the laser modulation bandwidth.
Two equivalent simulation methods are performed and compared:
 Small signal model transfer function informed by the steady state results. The small signal model is implemented in script and corresponds to the full timedomain model implemented in TWLM under small signal approximation. Since TWLM is a timedomain laser solver, to get the steady state results we do a transient simulation with a DC source until it reaches steady state.
 Transient simulation with a delayed small signal step excitation superimposed on the DC source. The bandwidth is then calculated by filtering and Fourier transforming the transient output power for times t >= small signal step source delay time.
This example requires 2020a R7 software version.
Run and results
The simulations are based on our FabryPerot getting started laser example. Below we give the simulation steps for both bandwidth simulation methods.
METHOD I. Small signal model

Upload files [[modulation_bandwidth_small_signal.lsf]], [[small_signal_bandwidth.lsf]], and [[TWLMexampleLaserFP300umCF353THzR3R3nel100_small_signal.icp]] to your working folder. File [[small_signal_bandwidth.lsf]] contains the custom function that implements the small signal model.

Open and run [[modulation_bandwidth_small_signal.lsf]] in INTERCONNECT. This will run [[TWLMexampleLaserFP300umCF353THzR3R3nel100_small_signal.icp]] and calculate the RF response and bandwidth.

Observe the plotted figure showing the RF response and the 3dB cutoff line.
Small signal model workflow:
 Transient simulation with DC source
 Extract steady state parameters
 Calculate small signal model transfer function
 Extract RF bandwidth
METHOD II. Transient simulation with superimposed delayed small signal source
 Upload files [[modulation_bandwidth_transient.lsf]], [[transient_bandwidth.lsf]], [[nuderiv.lsf]], and [[TWLMexampleLaserFP300umCF353THzR3R3nel100_transient.icp]] to your working folder. File [[transient_bandwidth.lsf]] contains the custom function for postprocessing the time domain output power to calculate the RF response.
 Open and run [[modulation_bandwidth_transient.lsf]]. This will run TWLMexampleLaserFP300umCF353THzR3R3nel100_transient.icp and calculate the RF response and bandwidth.
 Observe the plotted figure showing the RF response and the 3dB cutoff line.
Transient simulation with superimposed delayed small signal source workflow:
 Transient simulation with DC source + delayed small signal step source
 Postprocess the time domain output power to obtain RF response
 Extract RF bandwidth
The comparison of bandwidths from small signal model and transient simulation with superimposed delayed small signal source is given in the figure below. They show a very good agreement verifying the model.
Important model settings
For typical laser options important for accurate simulations, please refer to our many existing laser examples. Here, we focus on the options necessary for an accurate bandwidth simulation. For more details about the ways to derive the carrier capture/escape rates refer to ref. [1].
Both methods:
 enable SCH (TWLM) – this option enables the carrier capture/escape rate model.
 well carrier capture rate (TWLM) – The rate at which the carriers are captured by quantum wells. This includes the diffusion transport time across the total barrier thickness and the actual capture time into the quantum well once the carrier reaches the quantum well boundary (the latter is usually much smaller than the former).
 well carrier escape rate (TWLM) – The rate at which the carriers escape the quantum wells.
 total well thickness (TWLM) – The total cumulative thickness of all quantum wells. Usually the same as active region thickness option.
 total barrier thickness (TWLM) – The total cumulative thickness of all the barriers in the multiquantum well stack contained between the two separate confinement heterostructure barriers.
 time window (Root element) – total simulation duration. For method I, it should be sufficiently large for the simulation with a DC source to reach the steady state. For method II, it should be sufficiently larger than the delay of the superimposed small signal step to ensure the second steady state is reached after adding small signal to the DC source.
[[NOTE:]] In TWLM, multiquantum well active regions are treated as single quantum well active region with the quantum well and left/right barrier thicknesses equal to the respective total cumulative thicknesses. 
Method I:
 maxFreq (ss_bandwidth function from small_signal_bandwidth.lsf) – same as previous
 numPoints (ss_bandwidth function from small_signal_bandwidth.lsf)  number of frequency points in the RF response.
 run diagnostic , longitudinal profiles downsample factor , record carrier density profile , record photon density profile , record recombination rates profile (TWLM, method I only) – set these options to generate the required steady state result for the small signal model transfer function.
Method II:
 spontaneous emission factor 1 (TWLM, method II only) – this factor represents the portion of the total spontaneous emission that is coupled into the waveguide and contributes to the optical power. In transient simulation with superimposed delayed small signal source, this factor is set to a very small value, big enough to act as a spark for lasing, but small enough not to pollute the optical power with noise and make the RF response extraction difficult. The modulation bandwidth does not depend significantly on the spontaneous emission [1], so this is a good approximation.
 box_filter_window (rf_bandwidth_postprocess function from transient_bandwidth.lsf)  time window for box filtering. Current example uses 25 ps, which should be enough for most cases.
 maxFreq (rf_bandwidth_postprocess function from transient_bandwidth.lsf)  maximum frequency for the RF response.
 delay (STEP, method II only) – when to turn on the superimposed small step source. This time should be chosen so that the initial large signal transient simulation with a DC source reached steady state.
Updating the model with your parameters
In addition to the parameters listed in Important model settings section, other typical laser parameters that have to be modified to create your own device simulation can be found in our other laser examples.
Taking the model further
 Different laser topologies – For calculating bandwidth of lasers other then FabryPerot please refer to this example and combine it with our other examples illustrating DFB and external cavity laser simulations that can be found here https://support.lumerical.com/hc/enus#appanchor 2 if you filter by application Laser and Gain.
 User defined gain and spontaneous emission  Instead of using Lorentzian shapes the user can import gain and spontaneous emission profiles from simulation (e.g. Ansys Lumerical MQW gain solver) or measurement. Only method II can be used with imported profiles currently, since small signal model currently assumes Lorentzian profile.
References
[1] R. Nagarajan, M. Ishikawa, T. Fukushima, R. S. Geels and J. E. Bowers, “High Speed QuantumWell Lasers and Carrier Transport Effects,” IEEE JOURNAL OF QUANTUM ELECTRONICS, vol. 28, no. 10, p. 1990, 1992.