In the Traveling Wave Laser Model (TWLM), the spontaneous emission factor, \(\beta\), captures how much of the spontaneously emitted photons is coupled into the lasing mode. This article describes how this factor can be calculated. In the TWLM element, this parameter can be set under “Waveguide/Mode 1 Properties/spontaneous emission factor 1.” For more information on how this term couples into the governing physics of the TWLM, see INTERCONNECT as a Laser Design Platform.
For more information on the radiative carrier lifetime \(\tau_n\), which affects overall spontaneous recombination, see the Knowledge Base article on Carrier Recombination in TWLM. For more information on the spectrum of spontaneous emission, see the Knowledge Base article on Spontaneous Emission Spectrum Model in the TWLM.
There are multiple ways to estimate this spontaneous emission factor; two commonly used methods are described below. For typical materials, this value is between \(10^{-4}\) and \(10^{-5}\).
Semiclassical Estimation
One way of calculating the spontaneous emission factor depends on the assumption that spontaneous emission is uniformly distributed amongst all cavity modes of the laser. The ratio of spontaneous emission into a single mode is inversely proportional to the number of supported cavity modes. In this case, the following expression is used to calculate \(\beta\)
$$
\beta = \frac{\Gamma \lambda^4}{8\pi V n^2 n_g \Delta\lambda_{sp}}
$$
where \(\Gamma\) is the mode confinement factor, \(\lambda\) is the center frequency of operation, \(V\) is the volume of the cavity, \(n\) is the effective refractive index, \(n_g\) is the group index, and \(\Delta\lambda_{sp}\) is the bandwidth of the spontaneous emission.
For more information with regards to the physics of the formula above, please see [1, Ch. A4.3].
Quantum-Mechanical Estimation
Another way to calculate the spontaneous emission factor uses a quantum mechanical approach, which is more general. In this case, the following expression can be used for \(\beta\)
$$
\beta = \frac{\Gamma v_g g n_{sp}}{\eta_i \eta_r \frac{I}{q}}
$$
where \(\Gamma\) is the mode confinement factor, \(v_g\) is the group velocity, \(g\) is the optical gain, \(n_{sp}\) is the population inversion factor, \(\eta_i\) is the internal quantum efficiency, \(\eta_r\) represents the efficiency of radiative vs. non-radiative recombination, \(I\) is the injection current, and \(q\) is the elementary charge.
In the formula above, even though the spontaneous emission factor is current-dependent, a constant approximation of this factor is correct for near and above-threshold analysis of lasers [1, Ch. 2, Ch. 5].
For more information with regards to the physics and derivation of the formula above, please see [1, Ch. 4.4.3].
References
- L.Coldren, S.Corzine, and M.Mashanovitch, Diode Lasers and Photonic Integrated Circuits, 1st Edition, Wiley, 1995.
See Also
INTERCONNECT as a Laser Design Platform, Laser TW – INTERCONNECT Element