TWLM Built-in Lorentzian Gain Options

The Traveling Wave Laser Model (TWLM) in INTERCONNECT supports both using a built-in Lorentzian optical gain shape as well as custom gain shape from gain measurement or simulation. This article will discuss options related to the Lorentzian gain shape. For fitting a custom shape to measured or simulated gain data, see the Knowledge Base article on Gain Fitting.

Theory

As an optical mode traverses a section of gain element, its power grows exponentially. This exponential rate of growth is proportional to the modal confinement factor due to the overlap of the optical mode and the active region, and to the material gain. The modal gain is a product of the mode confinement factor and the material gain. In general, the material gain is frequency, carrier density, and temperature dependent.

In a given segment in the TWLM, this exponential increase is calculated as follows:

$$
\frac{\left|E\left(\Delta L\right)\right|^2}{\left|E\left(0\right)\right|^2} = e^{\Gamma g\left(f, N\right)\Delta L}
$$

where $E$ is the optical mode field amplitude, $g$ is the frequency and carrier density dependent gain explained below, $\Gamma$ is the mode confinement factor, and $\Delta L$ is the length of the segment.

Carrier and Photon Density Dependency

In TWLM, the built-in Lorentzian gain model uses a peak gain that depends on both the carrier and photon densities.

With respect to carrier density, you can select either a linear or logarithmic relationship in the “Waveguide/Gain Properties” section of the TWLM element:

Linear: the peak gain varies linearly with carrier density $N$ above the transparent carrier density $N_{tr}$, with a gain coefficient of $a_p$. This is the default model in the TWLM. In this case
$$g_{peak}\propto a_p (N-N_{tr})$$
Logarithmic: the peak gain varies logarithmically with respect to the carrier density $N$ above the transparent carrier density $N_{tr}$, and is also controlled with a gain coefficient $a_p$. This setting is supported starting in 2025 R2.4.
$$g_{peak}\propto a_p \ln(N/N_{tr})$$

In both models, you can also set the carrier transparent density $N_{tr}$ and the gain coefficient $a_p$ under the "Waveguide/Gain Properties" section of the TWLM element.

Note: The unit for $a_p$ differs depending on the gain carrier density dependence. When the dependence is linear, the unit for $a_p$ is $m^2$; when the dependence is logarithmic, the unit for $a_p$ is $1/m$.

At higher photon densities, corresponding to higher optical powers, the gain may drop off due to nonlinear gain saturation. The built-in gain model models this effect through the parameter gain compression factor ϵ and gain compression factor type. Both options are also under the “Waveguide/Gain Properties” section of the TWLM element.

Overall, the peak gain is calculated using the following formulas:

Linear gain carrier density dependence

$$g_{peak}=\frac{1}{1+\epsilon S} a_p (N-N_{tr})$$

Logarithmic gain carrier density dependence

$$g_{peak}=\frac{1}{1+\epsilon S} a_p \ln(N/N_{tr})$$

In the formulas above, $S$ is the photon density and other parameters are as previously defined.

A reference table for all parameters related to the gain shape can be found in the section below.

In lasers with bulk active layers, where density of states has a square root dependence on energy, gain saturation occurs high above the threshold. In this case, the linear dependence of peak gain on carrier density is acceptable. In lasers with quantum-well based active layers, the step-like density of states causes the gain saturation to occur at a lower carrier density. In these lasers, the logarithmic gain dependence on carrier density is more accurate.

Frequency Dependency

With respect to frequency, the gain is assumed to have a Lorentzian relationship, controlled by a center frequency $f_{0G}$ and quality factor $Q_{0G}$. Both of these variables are assumed to vary linearly with carrier density, controlled by differential gain center frequency and quality factor, $\alpha_{fG}$ and $\alpha_{QG}$, reference carrier density $N_{Gref}$, and gain shape center frequency and quality factor, $f_{0G}$ and $Q_{0G}$. All parameters stated here can also be set under the “Waveguide/Gain Properties” section of the TWLM element. Overall, the frequency dependence of the gain is calculated using the following formulas for a given segment:

$$
g\left(f, N\right) = g_{peak} L\left(f_{cG}, Q_G; f\right)
$$

$$
f_{cG}\left(N\right) = f_{0G} + \alpha_{fG}\left(N - N_{Gref}\right)
$$

$$
Q_G\left(N\right) = Q_{0G} + \alpha_{QG}\left(N - N_{Gref}\right)
$$

where $g_{peak}$ is the carrier density dependent peak gain as defined earlier, $L\left(f_c, Q; f\right)$ is a unity peak Lorentzian function centered at $f_c$ with a quality factor $Q$, and all other variables are as previously defined.

To achieve a constant gain shape, the differential values $\alpha_{fG}$ and $\alpha_{QG}$ can be simply set to zero.

A reference table for all parameters related to the gain shape can be found in the section below.

Example and Parameter Reference

The following figure shows a plot of a family of Lorentzian Gain Shapes, $L(f_{cG},Q_G;f)$, with center frequencies and Q's being made to vary, while setting $a_{fG}$ and $a_{QG}$ to zero.

The script and project file used to generate the plot is attached to this article. The script file plotGain.lsf generates the plot below. In the script, the gain shape center frequency, set with “Waveguide/Gain Properties/Gain Shape Center Frequency”, is set identically to the TWLM element simulation frequency, set with “Standard/Frequency”. This is important for accuracy considerations, which is further discussed below.

The full parameters used to represent both carrier density dependency and frequency dependency is listed in the table below. All parameters are adjusted from the “Waveguide/Gain Properties” section of the TWLM element.

Carrier Density Dependency
Parameter Name	Symbol
Transparency Carrier Density	$N_{tr}$
Gain Coefficient	$a_p$
Gain Compression Factor	$\epsilon$
Gain Compression Factor Type	N/A
Frequency Dependency
Parameter Name	Symbol
Gain Center Frequency	$f_{0G}$
Differential Gain Center Frequency	$\alpha_{fG}$
Quality Factor	$Q_{0G}$
Differential Quality Factor	$\alpha_{QG}$
Reference Carrier Density	$N_{Gref}$

Accuracy Considerations

Center Frequency

The gain shape center frequency, which is adjusted using the “Waveguide/Gain Properties/Gain Shape Center Frequency” property is set independently from the TWLM simulation center frequency, which is adjusted using the “Standard/Frequency” property. For accurate simulation results, you must ensure these frequencies are the same, otherwise, the fitted gain shape used in TWLM for simulation may not match the exact Lorentzian formulated with input shape parameters. The TWLM first fits the spectrum prior to using it in the TWLM model, and this fit is the most accurate around the simulation center frequency.

Fitted and Measured Gain

To check whether the spectrum used in TWLM matches the expected Lorentzian shape, you can examine the “fitted spectrum” and “measured spectrum” quantities in the diagnostic output of the element, which is enabled by the property “Diagnostic Properties/run diagnostic”. The fitted spectrum is the spectrum used as input into the TWLM simulation, and the measured spectrum is the gain shape dictated by the Lorentzian function discussed above. If you find the fitted spectrum to be insufficient in terms of accuracy away from the center frequency, you can increase the sample rate to improve the fit. Increasing the sample rate increases the simulation bandwidth, which provides a wider range of frequencies around the simulation center frequency where the Lorentzian gain fit is accurate.

A plot of the fitted vs. measured frequency is seen below, generated using the same script, plotGain.lsf which is attached to this article.

Using Custom Gain Definition

Alternative to increasing the simulation bandwidth and sampling rate, you can save the theoretical Lorentzian gain to a properly formatted file and fit it using the mczfit script command with the User Defined Gain Fitting workflow. Typically, this allows for more accurate gain fitting with a smaller simulation bandwidth, which improves the simulation speed.

An example script and project file is attached to this article inside the UserDefinedGain folder. The fit is done for a fixed quality factor (Q) and center frequency, considering a variety of carrier densities, and a logarithmic dependence of gain on the carrier density.

The comparison between the generated gain shape and gain spectrum extracted from the diagnostics of the TWLM simulation is shown below, showing good agreement across frequencies and carrier densities. Note that the true gain is plotted, not the gain shape.

Beyond the plot shown, the script also compares the results measured using an ONA and gain spectrum extracted from the diagnostics of the TWLM, using a circuit identical to that of the User Defined Gain Fitting workflow.