## Introduction

We present a laser linewidth simulation method for above-threshold single-mode semiconductor edge emitting laser diodes using the photon number method. This method is based on the modified Schawlow-Townes formula and laser simulation using the travelling wave laser model (TWLM) in INTERCONNECT.

The modified Schawlow-Townes formula for linewidth is given by [1] [2]:

$$

\Delta f=\frac{R}{4 \pi S}\left(1+\alpha_{e f f}^{2}\right)

$$

where \(S\) is the number of photons in the cavity, αeff is the effective linewidth enhancement factor, and \(R\) is the spontaneous emission rate [1/s] given by

$$

R=R_{s p} K_{t r} K_{l}

$$

where \(R_{sp}\) is the spontaneous emission rate [1/s], Ktr and Kl are the correction factors modeling the enhanced spontaneous emission due to the lateral and longitudinal field distribution.

The advantage of the photon number method is that it is very efficient because it is based on the average number of photons in the cavity, for which it is enough to simulate until steady state is reached. This is typically 10-15 nanoseconds. This is much shorter than the simulation time necessary to resolve, for example, 1 MHz spectrum features, which is 1 microsecond.

In this example we reproduce results from [1] for several quarter wave shifted (QWS) DFB lasers using the photon number method.

## Run and results

**Simulation 1:** To reproduce results from table 1 in [1] load file [[photon_number_method_Whiteaway1989_settings.icp]] in INTERCONNECT and open and run script file [[photon_number_method_Whiteaway1989_settings.lsf]]. This will simulate a QWS DFB laser at 5 mW facet power and several different grating coupling strengths from [1]. For other parameters and calculation details please look in the project (icp) and script (lsf) files.

The first two figures below compare the total number of photons in the cavity and the effective mirror loss against the reference. These two quantities are required for the modified Schawlow-Townes formula.

The comparison of linewidth using two equivalent methods of calculating the spontaneous recombination rate Rsp against the reference result from [1] is given in the third figure below. For details about the two equivalent methods of calculating Rsp please refer to section Important model parameters.

**Simulation 2:** To reproduce results from table 2 in [1] load file [[photon_number_method_COST-240_settings.icp]] in INTERCONNECT and open and run script file [[photon_number_method_COST-240_settings.lsf]]. This will simulate a QWS DFB laser with fixed coupling strength and several different above-threshold facet powers from [1]. For other parameters and calculation details please look in the project and script files.

The comparison of linewidth using two equivalent methods of calculating the spontaneous recombination rate against the reference result from [1] is given in the figure below. For details about the two equivalent methods of calculating Rsp please refer to section Important model parameters.

## Important model parameters

**Linewidth correction factor \(K_l\): **This factor, greater or equal to 1, accounts for the enhanced spontaneous emission rate due to the longitudinal field distribution in a DFB laser. It depends on the type of cavity. As mentioned in Introduction, there is also a correction factor due to the lateral field distribution, but it is unity in index-guided lasers [1]. The value of \(K_l\) in this example is taken from [1] and references therein.

**Effective linewidth enhancement factor \(\alpha_{eff}\): **The effective linewidth enhancement factor should be used instead of the material linewidth enhancement factor \(\alpha\) in the modified Schawlow-Townes formula. However, in case of a QWS DFB laser, \(\alpha_{eff}\) is approximately equal to \(\alpha\). The value of \(\alpha_{eff}\) in this example is taken from [1] and references therein.

**Population inversion factor \(n_{sp}\): **The total spontaneous emission rate [1/s] in the cavity, \(R_{sp}\), can be obtained by the following two equivalent methods:

$$R_{s p}=v_{g} g_{t h} n_{s p}$$

$$R_{s p}=v_{g}\left(\alpha_{i}+\alpha_{m}\right) n_{s p}$$

where \(v_g\) is the group velocity, \(g_{th}\) is gain at threshold, \(\alpha_i\) is the waveguide loss, \(\alpha_m\) is the effective mirror loss given by [1]

$$

\alpha_{m}=\frac{P_{1}+P_{2}}{v_{g} h f S}

$$

where \(P_1\) and \(P_2\) are the left and right facet powers, and \(n_{sp}\) is given by (e.g. A6.5 in [3])

$$

n_{s p}=\frac{f_{2}\left(1-f_{1}\right)}{f_{2}-f_{1}}=\frac{1}{\left.1-e^{(}\left(h f_{21}-\Delta E_{F}\right) / k T\right)}

$$

where \(f_1\) and \(f_2\) are the Fermi-Dirac probabilities of occupation of the upper (conduction band) energy level 2 and the lower (valence band) energy level 1, \(f_{21}\) is the transition frequency, and \(\Delta E_{F}=E_{F 2}-E_{F 1}\) is the difference between the electron and hole quasi-Fermi levels. The value of nsp in this example is taken from [1] and references therein.

We can see that the two equations for \(R_{sp}\) are equivalent because gain is equal to total losses at threshold.

The threshold gain can be found by running an LI sweep in INTERCONNECT, identifying the threshold current and extracting the gain for that current. The facet powers and the total number of photons in the cavity \(S\) can also be found from the simulation results in INTERCONNECT. For details please look at the example script files.

## Taking the model further

In the example above, a QWS DFB laser is simulated. For other type of lasers, the parameters listed below may have different values. Also, lasers with external feedback require additional considerations of how the feedback affects the unperturbed linewidth (e.g. section 5.7.2 in [3]).

**Linewidth enhancement factor \(\alpha\): **this is a material property of the gain layer and it is defined in the standard way as

$$

\alpha=-\frac{4 \pi}{\lambda_{0}} \frac{\partial n}{\partial g}

$$

where \(n\) is the refractive index and \(g\) is the material gain. In principle, the value of this parameter can be calculated with mqwindex command, since this command returns both the refractive index perturbation and the gain (or absorption).

**Effective linewidth enhancement factor \(\alpha_{eff}\):**this is the value of \(\alpha\) that should be used in the modified Schawlow-Townes formula. For QWS DFB lasers \(\alpha_{e f f}=\alpha\) , but for different types of lasers \(\alpha_{eff}\) may be different from \(\alpha\). As an initial guess the users may assume \(\alpha_{e f f}=\alpha\). For more details on the value of \(\alpha_{eff}\) please refer to [1] and references therein.

**Linewidth correction factors \(K_{tr}\) and \(K_l\): **these correction factors model the enhancement of the spontaneous emission rate due to the lateral and longitudinal field distribution in the cavity. For index guided lasers \(K_{tr}=1\) , while \(K_l\) depends on the type of cavity. For example, it is close to unity for Fabry-Perot lasers with high reflectivity mirrors, but may be much larger than 1 for DFB lasers. For more details on the value of these parameters please refer to [1] and references therein.

**Population inversion factor \(n_{sp}\): **this factor depends on the material properties of the gain layer, just like material gain g. According to the formula given in section Important model parameters, this parameter depends on the separation of the quasi-Fermi levels in the semiconductor gain layer. For more details on the theoretical background for this factor please refer to, for example, A6.5 in [3].

## References

[1] Y. C. Chan, M. Premaratne and A. J. Lowery, “Semiconductor laser linewidth from the transimssion-line laser model,” IEE Proc.-Optoelectron., vol. 144, no. 4, pp. 246-252, 1997.

[2] J. Wang, N. Schunk and K. Petermann, “Linewidth enhancement for DFB lasers due to longitudinal field dependence in the laser cavity,” Electronics Letters, vol. 23, no. 14, pp. 715-717, 1987.

[3] L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, John Wiley & Sons, Inc., 1995.