The spontaneous spectrum model in the Traveling Wave Laser Model in INTERCONNECT is handled very similarly to the gain model. Namely, it supports both using a built-in Lorentzian shape as well as an imported custom shape.
Built-In Emission Spectrum Expression
The settings are very similar between the spontaneous emission spectrum and the gain spectrum. To simplify definition, the setting “spontaneous emission from gain” allow you to use the same parameters to define the gain and spontaneous spectrum. If spontaneous emission information is not available for your model, it is recommended to set this property to true. This parameter can be set under the “Waveguide/Spontaneous Emission Properties” section of the TWLM element.
Frequency and Carrier Density Dependency
The following equation and parameters are used to define the spontaneous emission spectrum when “spontaneous emission from gain” is set to false. The spontaneous emission shape, \(E\), is in general given by a unity peak Lorentzian function centered at center frequency \(f_{cE}\) with a quality factor \(Q_E\). The spontaneous emission spectrum has a unit peak, since in TWLM the spontaneous emission rate is determined by the product of the radiative recombination rate and the spontaneous emission factor, set in other TWLM options. This function in general depends on both the frequency \(f\) and the charge carrier density \(n\).
Both the center frequency and the quality factor are assumed to vary linearly with carrier density, controlled by differential gain center frequency and quality factor, \(a_{fE}\) and \(a_{QE}\), reference carrier density \(N_{Eref}\), and gain shape center frequency and quality factor, \(f_{0E}\) and \(Q_{0E}\). All parameters can be set under the “Waveguide/Spontaneous Emission Properties” section of the TWLM element.
Specifically, the following equations are used
$$
E\left(f, N\right) = L(f_{cE}, Q_E; f)
$$
$$
f_{cE}(N) = f_{0E} + a_{fE}(N - N_{Eref})
$$
$$
Q_E\left(N\right) = Q_{0E} + a_{QE}(N - N_{Eref})
$$
where \(L\left(f_c, Q; f\right)\) is a unity peak Lorentzian function centered at \(f_c\) with a quality factor \(Q\), and all other variables are as previously defined.
Custom Emission Spectrum
Similar to gain, a file can be used to fit the spontaneous spectrum from a text file. For more information, see the Knowledge Base article on Gain Fitting for TWLM.
Linewidth Enhancement Factor
The linewidth enhancement factor is responsible for the change in material index with respect to carrier density, which gives rise to laser chirp. This factor is defined as follows
$$
\alpha_H = -\frac{4\pi}{\lambda_0} \frac{\frac{\partial n}{\partial N} \big|_{\lambda_0}}{\frac{\partial g}{\partial N} \big|_{\lambda_0}}
$$
where \(\lambda_0\) is the reference wavelength, as specified in the “Standard” section of the TWLM element, \(n\) is the refractive index, \(g\) is gain, and \(N\) is the charge carrier density. Here, \(\frac{dg}{dN}\) is the gain coefficient in the compact model. If the built-in gain shape is used, the derivative can be evaluated using the definitions for \(g\) given in the Knowledge Base article TWLM Built-in Gain Shape.
See Also
INTERCONNECT as a Laser Design Platform, Laser TW – INTERCONNECT Element