Laser diode rate equations

The semiconductor laser multimode rate equations relate photon and carrier (electron) numbers or densities, to device parameters such as carrier and photon lifetime, optical gain, injection current and other material parameters.

From these equations, it is possible to derive a set of steady state and small signal equations to help in further understanding the static and dynamic characteristics of semiconductor lasers.

Note that these equations can be cast in many different forms, this is one example of them.

Contents

1 The Multimode Rate Equations 1.1 Explanation of Terms 2 Gain Compression 3 Spectral Shift

The Multimode Rate Equations

<math>\frac{dN}{dt} = \frac{I}{e.V} - \frac{N}{\tau_n} - \sum_{\mu=1}^{\mu=M}G_\mu P_\mu<math>

<math>\frac{dP_\mu}{dt} = \Gamma(G_\mu - \frac{1}{\tau_p})P_\mu + \beta_\mu \frac{N}{\tau_n}<math>

where:

N and P are carrier and photon densities respectively. I is the injected current and e the electronic charge.τ_n and τ_p are carrier and photon lifetimes respectively. M is the number of modes modelled. V is the active region volume which can be omitted if the equations are to be interpreted in terms of carrier and photon numbers as opposed to densities. Γ is the mode confinement factor which describes how much of the mode is confined in the active region. μ is the mode number.β_mu is the spontaneous emission factor.

G_μ is the gain of the μ^th mode and 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: α is the gain coefficient and ε is the gain compression factor (see below). λ_mu is the wavelength of the μ^th mode, δλ_g 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 λ₀ is the centre wavelength for N = N_th and k is the spectral shift constant (see below). N_th is the carrier density at threshold and is given by

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

where N_tr is the carrier density at transparency.

β_μ is given by

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

where

β₀ is the spontaneous emission factor, λ_s is the centre wavelength for spontaneous emission and δλ_s is the spontaneous emission FWHM. Finally, λ_mu is the wavelength of the μ^th mode and is given by

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

where δλ is the mode spacing.

Explanation of Terms

<math>\frac{I}{e.V}<math>

is the increase in carriers due to current injection

<math>-\frac{N}{\tau_n}<math>

is the decrease in carriers due to spontaneous emission

<math>-\sum_{\mu=1}^{\mu=M}G_\mu P_\mu<math>

is the decrease in carriers due to stimulated emission

<math>\Gamma G_\mu P_\mu<math>

is the increase in photons due to stimulated emission

<math>-\Gamma\frac{P_\mu}{\tau_p}<math>

is the decrease in photons as a result of cavity losses

<math>\beta_\mu \frac{N}{\tau_n}<math>

is the increase in carriers as a result of spontaneous emission.

Gain Compression

The gain term, G, 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 I₀ is the injected current and I_th is the lasing threshold current.