TWLM Built-in Gain Shape

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, and \(\Delta L\) is the length of the segment.

Carrier Density Dependency

In TWLM, the Lorentzian gain model assumes a peak gain that has a linear relationship with carrier density around the transparency carrier density \(N_{tr}\), and with a slope given by the gain coefficient \(\alpha_p\). Both of these parameters can be set under the “Waveguide/Gain Properties” section of the TWLM element. At higher photon densities (or optical powers), the gain may drop off due to nonlinear gain saturation. This effect is modeled by the parameters gain compression factor \(\epsilon\) and the gain compression factor type. Overall, the peak gain is calculated using the following formula:

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

where \(S\) 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.

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 with center frequencies and Q's being made to vary dynamically as a linear function of carrier density within each spatial element.

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.

See Also

INTERCONNECT as a Laser Design Platform,  Laser TW – INTERCONNECT Element