INTERCONNECT as a Laser Design Platform

Introduction

The Traveling Wave Laser Model (TWLM) in Ansys Lumerical INTERCONNECT™ is a physical 1-D laser model that can be used for laser design. Compared to other laser compact models in INTERCONNECT, such as the Continuous Wave Laser (CWL) and Directly Modulated Laser (DML), the TWLM relies less on fitting and more on first principles, allowing it to be used for laser design, similar to traditional 2D/3D physical models.

The TWLM utilizes a 1D traveling wave rate equation model to accurately model the behavior of an edge emitting laser. This model couples electrical, optical, gain, and thermal effects, and naturally outputs important figures of merit including LI curves, power gain, linewidth, relative intensity noise, and others. The TWLM, either used standalone or combined with other INTERCONNECT elements, can be used to model a variety of systems such as Fabry-Perot lasers, Distributed Bragg Reflector (DBR) lasers, microring lasers, Distributed FeedBack (DFB) lasers, and semiconductor optical amplifiers.

This article introduces the theory of the TWLM and provides resources detailing important model parameters. For a comparison of different INTERCONNECT laser models, see Hierarchy of laser models in Lumerical INTERCONNECT and Hierarchy of Methods for Laser Design and Compact Modeling in a Photonic Integrated Circuit Simulator.

Theory of Traveling Wave Laser Model

Basic Structure

The TWLM implemented in INTERCONNECT is based on a model known as the Transmission Line Laser Model [1]. It can be used to model a variety of edge-emitting laser systems with the only requirements being that the system must have a waveguide-liked geometry, and the emitted wave propagates parallel to the gain layer. This model cannot be used to model system where the wave propagates perpendicular to the gain layer, such as a vertical cavity surface emitting laser.

Inherently, the TWLM is a 1-D model that captures the spatial behaviour of physical quantities such as charge and photon density along the wave propagation direction, with discrete segments of length $\Delta L$. While the discretization is only 1-D, the TWLM can also capture important physical phenomena in the lateral direction normal to the gain layer, such as carrier capture and escape from the gain layer, or thermionic leakage from the gain layer due to self-heating. These interactions will be introduced in more detail in the subsections below as well as in pages dedicated to specific physics of the TWLM.

A generic schematic diagram for the gain element is shown below. The gain medium, also known as the active layer, is surrounded by separate confinement layers (SCLs), also known as separate confinement heterostructures (SCHs). In a fabricated device, more than one well and confinement layers can be present.

In the schematic, $l_{\text{act}}$, $d_{\text{act}}$, and $w_{\text{act}}$ are the length, depth, and width of the entire gain element, respectively.

When defining the gain medium thickness via the “active region thickness” parameter in the TWLM element in INTERCONNECT, you can choose to either define it only as a quantum well, or a quantum well with additional barrier layers. In either case, the mode confinement factor, which captures the mode overlap with the active layer, must be calculated consistently with the defined thickness.

Discretization and Governing Physics

Each segment in the discretized structure has an electrical and optical element, self-consistently coupled as described in the “Carrier-Photon Interactions” section below. Each segment in the structure is also coupled to nearby segments to model propagation of the optical mode envelope as well as longitudinal carrier diffusion, naturally capturing the effects of longitudinal spatial hole burning (LSHB). A general schematic of coupling between segments and components within a segment is shown in the diagram below.

Carrier-Photon Interactions

Carrier Transport and Recombination

Injected charge carriers in the TWLM model diffuse across the barrier layers until they are captured by the active region quantum well. Within the quantum well, electron-hole pairs can be formed by absorption of photon, and they can recombine radiatively or non-radiatively, emitting a photon in the radiative processes of spontaneous and stimulated emissions.

In the schematic below, processes included in the TWLM are described conceptually. Depending on settings, certain processes, such as capture and escape of carriers from the gain layer, may not be modeled.

Spatially, the interactions of photons and carriers can be thought using the band structure diagram below. This diagram is in the thickness $(x)$ direction, and depicts the qualitative band structure of the generic gain element schematic from the “Basic Structure” section above.

In the band diagram above, each region of the gain element schematic in the “Basic Structure” section has the same color as those in the schematic (blue for active layer, dark grey for barrier, and light grey for SCH). In the TWLM, only a single quantum well is modeled. If the actual device has multiple wells, the cumulative thickness of the well and barriers should be inputted into their respective fields.

In the model, injected carrier from the contact diffuses across the barrier region and recombines in the active layer quantum well after being captured by it. During this process, some carriers can also escape from the well. This process is modeled by carrier capture and escape rates, which represents the sum of carrier diffusion time across barriers and the capture/escape time from the active layer. These values are specified in the TWLM element. A more detailed description of how to estimate these rates can be found in the Knowledge Base article on capture and escape rates.

The SCH layer effects can also be turned off by specifying the value of “enable SCH” to “false” in the parameters. If the SCH layer is turned off, the total carrier and escape rates are not included, and it is assumed that carriers are instantaneously injected from the contacts. This assumption usually valid for steady-state simulations, such as extracting the LI curve. However, carrier dynamics are important for bandwidth simulations.

Inside the well, carrier recombination is modeled by non-radiative, spontaneous, and stimulated recombination processes. In addition, carrier generation via photon absorption is also modeled. These processes are set via recombination parameters in the TWLM element. Further information on these parameters can be found in the Knowledge Base article on carrier recombination in the TWLM.

Electrical and thermal effects from the SCH is controlled by the SCH properties, while optical effects of the SCH is only controlled by the Mode Confinement Factor property.

Rate Equation Model

Analytically, interactions between the photon and carriers in in the active layer can be described conceptually by the laser rate equations [2].

While the implemented TWLM model in INTERCONNECT is more advanced than the model shown here, with details discussed in the section below, this simple rate equation model can be used to qualitatively understand how carriers and photons interact in the model.

$$
\frac{dN}{dt} = \frac{I}{qV_a} - \frac{N}{\tau_N} - \Gamma Gp
$$

$$
\frac{dp}{dt} = \left(\Gamma G - \frac{1}{\tau_p}\right)p + \beta\frac{N}{\tau_{\text{rad}}}
$$

where $n$ is the carrier density, $p$ is the photon density, $I$ is the injected current, $q$ is the elementary charge, $V_a$ is the volume of the active region, $\tau_n$ is the carrier lifetime, $\tau_p$ is the photon lifetime, $\tau_{\text{rad}}$ is the radiative carrier lifetime, $\Gamma$ is the mode confinement factor, $G$ is the gain factor, and $\beta$ is the spontaneous emission factor.

The first equation describes the rate of change of charge carriers in the gain medium, which is the amount of injected charge carriers (first term), less the depletion of charge carriers from recombination processes, with the second term representing all processes except recombination from stimulated emission, and the third term representing carrier loss from stimulated emission.

The second equation describes the rate of change of photons in the cavity, which is the sum of the stimulated (first term) and spontaneous (third) emission, less the loss of photons from exiting the cavity and absorption, described by $\tau_p$ [2] [3].

INTERCONNECT TWLM Model

As stated above, the TWLM implementation in INTERCONNECT is more advanced than the simple rate equation model.

Several key differences versus the simple rate equation model are as follows:

The TWLM is inherently a 1-D model in the longitudinal direction for both the photons and carriers, allowing it to capture important spatially dependent effects such as LHSB.
Different types of cavities can be modeled from solving the field propagation equation. In the TWLM, the propagation of the mode envelope is simulated by propagating a plane wave through the chain of segments, which naturally incorporates effects such as amplification, phase shift, or loss, as the wave moves through the gain layer towards emitting facet.
Carrier diffusion is solved in the longitudinal direction, incorporated into the rate equation for the carriers. This diffusion constant can be set under the “Waveguide/Gain Properties” parameter field of the model.
Carrier lifetime, $\tau_N$, is a polynomial expression that depends on the carrier density, capturing effects of common recombination mechanisms such as Shockley-Read-Hall (SRH), spontaneous, and Auger recombination.
- Further information on how the lifetime is modeled and associated parameters can be found in the Knowledge Base page on recombination.
Photon lifetime, $\tau_p$, is calculated from first principles using mirror loss and absorption through solution of the field propagation equation.
Self heating effects are included via self-consistent coupling with heat equations.
Thermionic leakage of carriers from the active layer are included.
External feedback effects are included due to having bidirectional optical ports.

Self Heating

Physics Coupling

Self-heating effects of the laser can also be modeled in the TWLM. In this model, the heat sources inside the laser come from Joule heating due to current injection, non-radiative recombination effects inside the active layer, carrier capture and escape energy transitions, and heating from absorption. Temperature elevation from these sources can then affect the electrical and optical equations that ultimately governs laser behavior. A schematic diagram of how heat sources couple into different physics component is shown below, depicting the variables exchanged between each physics component.

Note: Heating effects from facet surface recombination is included in non-radiative heating for elements on the surface.

Spatial Distribution and Thermal Impedances

Each segment within the TWLM structure has its own temperature, coupled with each other and with the surrounding environment with the thermal resistances $R_G$ and $R_{GA}$, respectively representing the thermal resistance of the active layer and the thermal resistance between the active layer and the ambient for each segment. A schematic showing this coupling can be found below, the facet elements have an addition heat source from facet surface recombination.

The extraction of the ambient thermal resistance, $R_{GA}$, is detailed in a separate Knowledge Base article.

Thermionic Leakage

When self heating effects are enabled, carriers can additionally leak from the active layer via thermionic leakage, which is essentially a temperature dependent model of carrier escape. This leakage follows the model in [4], with parameters controlled by parameters in “Waveguide/SCH/Thermionic leakage Properties.”

Overall, the total rate of loss of carriers from the quantum well can be expressed as follows:

$$
w\frac{dN}{dt} = -\left[\frac{2}{\tau_B}\right]wN
$$

$$
\tau_B = \frac{1}{w}\left[\frac{k_BT}{2\pi m^\ast}\right]^\frac{1}{2}\exp{\left(-\frac{E_B}{k_BT}\right)}
$$

where $w$ is the width of the quantum well, $E_B$ is the barrier height of the quantum well, $m^\ast$ is the quantum well effective mass, and $N$ is the carrier density as described in the rate equation.

In the TWLM, it is possible to model the carrier loss from the gain layer as a two-step process when the separate confinement structure (SCH) is included. In this case, the same carrier loss equations from above are used with two barrier heights and two effective masses, specified by “quantum barrier height”, “SCH barrier height”, “quantum well effective mass”, and “quantum barrier effective mass” parameters. The transport between the two wells is included in the capture and escape time settings of the SCH. In the case where multiple quantum wells are present in the active layer, the cumulative width of the well should be used.

In this case, the following schematic band diagram can be used to conceptually visualize the situation, with the “Barrier” layer corresponding to the dark grey layer in the diagram in the “Basic Structure” section, and “SCH” layer corresponding to the light grey layer in the same diagram.

TWLM Configuration Pages

Important configurations regarding the TWLM are discussed in the pages below, which will help you configure your model.

TWLM Application Gallery Examples

Examples demonstrating interoperability between the TWLM, MODE/FEEM, and MQW solvers to calculate and import the mode confinement factor and the gain and spontaneous emission spectra. These examples also provide benchmarks against 2D/3D laser solvers.

Examples demonstrating the usage of the three main TWLM internal cavity types: Fabry-Perot (internal grating model turned off), Distribute Feedback Bragg (internal grating model turned on), and multisection (different combinations of the previous two).

Examples demonstrating how to connect passive photonic elements in INTERCONNECT to the TWLM as a gain element to simulate lasers with external cavities such as DBR lasers and lasers with external microring reflectors.

Examples demonstrating post-processing steps to obtain additional figures of merit, such as the bandwidth and relative intensity noise (RIN).

References

A.J. Lowery, “New dynamic semiconductor laser model based on the transmission-line modelling method,” IEE Proceedings J (Optoelectronics), vol. 134, no. 5, pp. 281-289, Oct. 1987, doi: 10.1049/ip-j.1987.0047.
“Laser diode rate equations,” Wikipedia.com. Accessed: Feb. 3, 2025. [Online] Available: https://en.wikipedia.org/wiki/Laser_diode_rate_equations.
L.Coldren, S.Corzine, and M.Mashanovitch, Diode Lasers and Photonic Integrated Circuits, 1st Edition, Wiley, 1995.
H. Schneider and K.Klitzing, “Thermionic emission and gaussian transport of holes in GaAs/Al_xGa_1-xAs multiple-quantum-well structure,” Phys. Rev. B, vol. 38, no. 9, pp.6160-6165, Sep. 1988, doi: 10.1103/PhysRevB.38.6160.