Editing Laser diode rate equations

The [[laser diode]] [[rate equation]]s model the electrical and optical performance of a laser diode. This system of [[ordinary differential equation]]s relates the number or density of [[photon]]s and [[charge carrier]]s ([[electron]]s) in the device to the injection [[current (electricity)|current]] and to device and material parameters such as [[carrier lifetime]], photon lifetime, and the [[Semiconductor optical gain|optical gain]].

The rate equations may be solved by [[numerical integration]] to obtain a [[time-domain]] solution, or used to derive a set of [[steady state]] or [[small-signal model|small signal equations]] to help in further understanding the static and dynamic characteristics of [[semiconductor lasers]].

The laser diode rate equations can be formulated with more or less complexity to model different aspects of laser diode behavior with varying accuracy.

==Multimode rate equations==
In the multimode formulation, the rate equations<ref>G. P. Agrawal, "Fiber-Optic Communication Systems", Wiley Interscience, Chap. 3</ref> model a laser with multiple optical [[normal mode|modes]]. This formulation requires one equation for the carrier density, and one equation for the photon density in each of the [[optical cavity]] modes:

:<math>\frac{dN}{dt} = \frac{I}{eV} - \frac{N}{\tau_n} - \sum_{\mu=1}^{\mu=M}\Gamma_\mu G_\mu P_\mu</math>
:<math>\frac{dP_\mu}{dt} = (\Gamma_\mu G_\mu - \frac{1}{\tau_p})P_\mu + \beta_\mu \frac{N}{\tau_r}</math>

where: 
<math>{N}</math> is the carrier density, <math>{P}</math> is the photon density, <math>{I}</math> is the applied current, <math>{e}</math> is the [[elementary charge]], <math>{V}</math> is the volume of the [[Active laser medium|active]] region, <math>{\tau_n}</math> is the carrier lifetime, <math>{G}</math> is the gain coefficient (s<sup>−1</sup>), <math>{\Gamma}</math> is the confinement factor, <math>{\tau_p}</math> is the photon lifetime, <math>{\beta}</math> is the spontaneous emission factor, <math>{\tau_r}</math> is the radiative recombination time constant, <math>{M}</math> is the number of modes modelled, μ is the mode number, and 
subscript <math>{\mu}</math> has been added to <math>{G}</math>, <math>{\Gamma}</math>, and <math>{\beta}</math> to indicate these properties may vary for the different modes.

The first term on the right side of the carrier rate equation is the injected electrons rate (<math>{I/eV}</math>), the second term is the carrier depletion rate due to all recombination processes (described by the decay time <math>{\tau_n}</math>) and the third term is the carrier depletion due to [[Stimulated emission|stimulated recombination]], which is proportional to the photon density and medium gain.

In the photon density rate equation, the first term <math>{\Gamma_\mu G_\mu P_\mu}</math> is the rate at which photon density increases due to stimulated emission (the same term in carrier rate equation, with positive sign and multiplied for the confinement factor <math>{\Gamma}</math>), the second term is the rate at which photons leave the cavity, for internal absorption or exiting the mirrors, expressed via the decay time constant <math>{\tau_p}</math> and the third term is the contribution of spontaneous emission from the carrier radiative recombination into the laser mode.

==The modal gain==
<math>{G_\mu}</math>, the gain of the <math>\mu</math><sup>th</sup> mode, can be modelled by a parabolic dependence of gain
on wavelength as follows:

:<math> G_\mu = \frac{\alpha N [1-(2\frac{\lambda(t)-\lambda_\mu}{\delta\lambda_g})^2] - \alpha N_0}{1 + \epsilon \sum_{\mu=1}^{\mu=M}P_\mu}</math>

where:
<math>{\alpha}</math> is the gain coefficient and ε is the gain compression factor (see below). <math>{\lambda_\mu}</math> is the wavelength of the <math>\mu</math><sup>th</sup> mode, <math>\delta\lambda_g</math> is the full width at half maximum (FWHM) of the gain curve, the centre of which is given by

:<math>\lambda(t)=\lambda_0 + \frac{k(N_{th} - N(t))}{N_{th}}</math>

where <math>\lambda_0</math> is the centre wavelength for <math>{N = N_{th}}</math> and k is the spectral shift constant (see below). <math>N_{th}</math> is the carrier density at threshold and is given by

:<math>N_{th}=N_{tr} + \frac{1}{\alpha\tau_p\Gamma}</math>

where <math>N_{th}</math> is the carrier density at transparency.

<math>\beta_{\mu}</math> is given by
:<math>\beta_\mu=\frac{\beta_0}{1+(2(\lambda_s-\lambda_\mu)/\delta\lambda_s)^2}</math>

where

<math>\beta_{0}</math> is the spontaneous emission factor, <math>\lambda_s</math> is the centre wavelength for spontaneous emission and <math>\delta\lambda_s</math> is the spontaneous emission FWHM. Finally, <math>\lambda_{}</math> is the wavelength of the <math>\mu</math><sup>th</sup> mode and is given by

:<math>\lambda_\mu=\lambda_0 - \mu\delta\lambda + \frac{(n-1)\delta\lambda}{2}</math>

where <math>\delta\lambda</math> is the mode spacing.

==Gain Compression==

The gain term, <math>G</math>, cannot be independent of the high power densities found in
semiconductor laser diodes.  There are several phenomena which cause the gain to
'compress' which are dependent upon optical power.  The two main phenomena are
[[spatial hole burning]] and [[spectral hole burning]].

Spatial hole burning occurs as a result of the standing wave nature of the optical
modes.  Increased lasing power results in decreased carrier diffusion efficiency which
means that the stimulated recombination time becomes shorter relative to the carrier
diffusion time.  Carriers are therefore depleted faster at the crest of the wave causing a
decrease in the modal gain.

Spectral hole burning is related to the gain profile broadening mechanisms such
as short intraband scattering which is related to power density.

To account for gain compression due to the high power densities in semiconductor lasers, the gain equation is modified such that it becomes related to the inverse of the optical power.  Hence, the following term in the denominator of the gain equation :

:<math>1 + \epsilon \sum_{\mu=1}^{\mu=M}P_\mu</math>

==Spectral Shift==

Dynamic wavelength shift in semiconductor lasers occurs as a result of the change
in refractive index in the active region during intensity modulation.  It is possible to
evaluate the shift in wavelength by determining the refractive index change of the active
region as a result of carrier injection.  A complete analysis of spectral shift during direct
modulation found that the refractive index of the active region varies proportionally to carrier density and hence the wavelength varies proportionally to injected current.

Experimentally, a good fit for the shift in wavelength is given by:

:<math>\delta\lambda=k\left(\sqrt{\frac{I_0}{I_{th}}}-1\right)</math>

where <math>I_0</math> is the injected current and <math>I_{th}</math> is the lasing threshold current.

==References==
{{Reflist}}

[[Category:Ordinary differential equations]]
[[Category:Semiconductor lasers]]
{{Semiconductor laser}}