Rate Equations for Semiconductor Lasers
This is a continuation from the previous tutorial - laser arrays.
Since the electromagnetic field inside the laser cavity satisfies Maxwell's equations, the starting point to obtain the field rate equation is the wave equation (2-2-12) [refer to the Maxwell's equations for semiconductor lasers tutorial].
In general, the dynamics of the semiconductor material should be taken into account to obtain the induced polarization \(\pmb{\mathscr{P}}\). However, the material response governed by the intraband scattering processes is relatively fast (~ 0.1 ps) compared to other time scales of interest such as the photon lifetime and the carrier recombination time.
Considerable simplification occurs if we assume that the material response is instantaneous and that Equation (2-2-17) [refer to the Maxwell's equations for semiconductor lasers tutorial] holds even for time-dependent fields. The wave equation then becomes
\[\tag{6-2-1}\pmb{\nabla}^2\pmb{\mathscr{E}}-\frac{1}{c^2}\frac{\partial^2}{\partial{t^2}}(\pmb{\epsilon\mathscr{E}})=0\]
where the loss term related to the medium conductivity has been included in the imaginary part of the dielectric constant \(\pmb{\epsilon}\) [see Equation (2-2-20) in the Maxwell's equations for semiconductor lasers tutorial].
It should be kept in mind that the use of Equation (6-2-1) is justified only for a fast-responding semiconductor. If for a semiconductor material the intraband scattering time becomes comparable to the photon lifetime (~ 1 ps), Equation (2-2-12) [refer to the Maxwell's equations for semiconductor lasers tutorial] should be used with the induced polarization given by Equation (2-4-1) [refer to the gain and stimulated emission for semiconductor lasers tutorial].
The optical field \(\pmb{\mathscr{E}}\) in general may consist of a large number of lateral, transverse, and longitudinal modes oscillating at different frequencies. Furthermore, each mode forms a standing-wave pattern in the axial direction arising from a superposition of the forward and backward running waves.
For simplicity, we assume that the laser structure has been designed to support a single lateral and transverse mode. This is often the case in practical semiconductor lasers.
The optical field \(\pmb{\mathscr{E}}\) can be written as
\[\tag{6-2-2}\pmb{\mathscr{E}}(x,y,z,t)=\frac{1}{2}\hat{\mathbf{X}}\psi(x)\phi(y)\sum_j\sin(k_jz)E_j(t)\text{e}^{-\text{i}\omega_jt}+\text{c}.\text{c}.\]
where \(\psi(x)\) and \(\phi(y)\) are the lateral and transverse field profiles discussed in the waveguide modes in semiconductor lasers tutorial and \(\text{c}.\text{c}.\) stands for complex conjugate.
The sinusoidal variation of the optical field in the \(z\) direction assumes facets with high reflectivity. Even though this assumption is somewhat questionable for a semiconductor laser (facet reflectivity \(\approx32\%\)), its use is essential in order to avoid solving a complicated boundary-value problem and does not introduce significant quantitative errors if the laser is above threshold.
The wave number \(k_j\) is related to the cavity-resonance frequency \(\Omega_j=2\pi\nu_j\) given by Equation (2-3-12) [refer to the threshold condition and longitudinal modes of semiconductor lasers tutorial], i.e.,
\[\tag{6-2-3}k_j=\frac{\mu\Omega_j}{c}=\frac{m_j\pi}{L}\]
where \(L\) is the cavity length and \(m_j\) is an integer. The laser-mode frequency \(\omega_j\) is yet undetermined except that \(\omega_j\) nearly coincides with \(\Omega_j\).
For simplicity of notation, let us first consider the case of a single longitudinal mode and drop the subscript \(j\) in Equation (6-2-2), i.e.,
\[\tag{6-2-4}\pmb{\mathscr{E}}(x,y,z,t)=\frac{1}{2}\hat{\mathbf{X}}\psi(x)\phi(y)\sin(kz)E(t)\text{e}^{-\text{i}\omega{t}}+\text{c}.\text{c}.\]
We substitute \(\pmb{\mathscr{E}}\) from Equation (6-2-4) in Equation (6-2-1), assume that \(E(t)\) varies slowly, multiply by \(\psi(x)\) and \(\phi(y)\), and integrate over the whole range of \(x\) and \(y\). After some simplifications, we obtain
\[\tag{6-2-5}\frac{2\text{i}\omega}{c^2}\left(\langle\boldsymbol{\epsilon}\rangle+\frac{\omega}{2}\frac{\partial\langle\boldsymbol{\epsilon}\rangle}{\partial\omega}\right)\frac{\text{d}E}{\text{d}t}+\left(\frac{\omega^2}{c^2}\langle\boldsymbol{\epsilon}\rangle-k^2\right)E=0\]
where the spatially averaged dielectric constant
\[\tag{6-2-6}\langle\boldsymbol{\epsilon}\rangle=\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\boldsymbol{\epsilon}(x,y)\psi^2(x)\phi^2(y)\text{d}x\text{d}y\]
and \(\psi(x)\) and \(\phi(y)\) are assumed to be properly normalized.
The second term on the left-hand side of Equation (6-2-5) takes into account the dispersive nature of the semiconductor material.
The average in Equation (6-2-6) can be carried out using the analysis of the waveguide modes in semiconductor lasers tutorial. We find that \(\langle\boldsymbol{\epsilon}\rangle\) approximately equals the effective dielectric constant of the waveguide mode and is given by
\[\tag{6-2-7}\langle\boldsymbol{\epsilon}\rangle\approx\bar{\mu}^2+2\Gamma\bar{\mu}\Delta\mu_\text{p}+\text{i}\bar{\mu}\bar{\alpha}/k_0\]
where \(k_0=\omega/c\), \(\Gamma\) is the confinement factor, and \(\Delta\mu_\text{p}\) is the carrier-induced index change.
Furthermore, \(\bar{\mu}\) is the mode index and \(\bar{\alpha}\) is the mode-absorption coefficient given by [refer to Equations (2-5-47) and (2-5-48) in the waveguide modes in semiconductor lasers tutorial]
\[\tag{6-2-8}\bar{\alpha}=-\Gamma{g}+\alpha_\text{int}+\alpha_\text{m}\]
Equation (6-2-8) has been generalized to include the facet loss \(\alpha_\text{m}\) [given by Equation (2-3-10) in the threshold condition and longitudinal modes in semiconductor lasers tutorial].
A rigorous approach would require that \(E\) should be allowed to vary with \(z\) and then to solve the axial-propagation problem with appropriate boundary conditions at the laser facets.
However, considerable simplification occurs if we replace the localized facet loss by an equivalent distributed loss and add it phenomenologically to \(\alpha_\text{int}\).
We substitute (6-2-7) into (6-2-5) and use \(k=\bar{\mu}\Omega/c\) from Equation (6-2-3). Using \((\omega^2-\Omega^2)\approx2\omega(\omega-\Omega)\) and \(\langle\boldsymbol{\epsilon}\rangle\approx\bar{\mu}^2\) on the left-hand side of Equation (6-2-5), we obtain
\[\tag{6-2-9}\frac{\text{d}E}{\text{d}t}=\frac{\text{i}\bar{\mu}}{\mu_\text{g}}(\omega-\Omega)E+\frac{\text{i}\omega}{\mu_\text{g}}(\Gamma\Delta\mu_\text{p}+\text{i}\bar{\alpha}/2k_0)E\]
where \(\mu_\text{g}\) is the group index corresponding to the mode index \(\bar{\mu}\).
It is useful to separate Equation (6-2-9) into its real and imaginary parts using
\[\tag{6-2-10}E=A\exp(-\text{i}\phi)\]
We then obtain the amplitude and phase rate equations
\[\tag{6-2-11}\dot{A}=\frac{1}{2}v_\text{g}[\Gamma{g}-(\alpha_\text{int}+\alpha_\text{m})]A\]
\[\tag{6-2-12}\dot{\phi}=-\frac{\bar{\mu}}{\mu_\text{g}}(\omega-\Omega)-\frac{\omega}{\mu_\text{g}}\Gamma\Delta\mu_\text{p}\]
where a dot represents time derivative, \(v_\text{g}=c/\mu_\text{g}\) and is the group velocity, and we have used Equation (6-2-8) to eliminate \(\bar{\alpha}\).
Equation (6-2-11) suggest that the rate of amplitude growth equals gain minus loss and may have been written directly using heuristic arguments. The advantage of the present approach where it is obtained using Maxwell's equations is that the various approximations made during its derivation have been clearly identified.
Furthermore, the phase equation (6-2-12) follows self-consistently and shows that the carrier-induced index change \(\Delta\mu_\text{p}\) affects the lasing-mode frequency \(\omega\). It will be seen later that this term plays an important role in the discussion of line width and frequency chirping.
It is more convenient to write the amplitude equation (6-2-11) in terms of the photon number \(P\) defined using
\[\tag{6-2-13}P=\frac{\epsilon_0\bar{\mu}\mu_\text{g}}{2\hbar\omega}\displaystyle\int|\pmb{\mathscr{E}}|^2\text{d}V\]
where \(\hbar\omega\) is the photon energy.
Since \(P\propto{A}^2\), we obtain
\[\tag{6-2-14}\dot{P}=(G-\gamma)P+R_\text{sp}\]
where
\[\tag{6-2-15}G=\Gamma{v_\text{g}}g\]
is the net rate of stimulated emission and
\[\tag{6-2-16}\gamma=v_\text{g}(\alpha_\text{m}+\alpha_\text{int})=\tau^{-1}_\text{p}\]
is the photon decay rate that can be used to define the photon lifetime \(\tau_\text{p}\) inside the laser cavity.
The last term \(R_\text{sp}\) in Equation (6-2-14) has been added to take into account the rate at which spontaneously emitted photons are added to the intracavity photon population. The contribution of spontaneous emission to the lasing mode is indispensable and should be included for a correct treatment of laser dynamics.
A rigorous approach would require quantization of the laser field. Within the semiclassical framework, the contribution of spontaneous emission can be included by adding a noise current to Maxwell's equations.
It should be stressed that the photon number \(P\) is used only as a convenient dimensionless measure of the optical intensity \(|E|^2\). Its use is not intended to imply the particulate nature of photons. This distinction is important for the discussion of the intensity noise.
The phase Equation (6-2-12) can be related to the gain if we use the parameter \(\beta_\text{c}\) defined in Equation (2-4-6) [refer to the gain and stimulated emission for semiconductor lasers tutorial] to replace \(\Delta\mu_\text{p}\) by
\[\tag{6-2-17}\Delta\mu_\text{p}=-(\beta_\text{c}/2k_0)\Delta{g}\]
Using Equations (6-2-15) and (6-2-17) and replacing \(\Delta{G}=\Gamma{v}_\text{g}\Delta{g}\) by \(G-\gamma\), Equation (6-2-12) becomes
\[\tag{6-2-18}\dot\phi=-(\omega-\omega_\text{th})+\frac{1}{2}\beta_\text{c}(G-\gamma)\]
This equation shows that when the gain changes from its threshold value, the phase changes as well. Physically, this is so because a gain change is always accompanied by an index change that shifts the longitudinal-mode frequencies.
In Equation (6-2-18) the longitudinal-mode frequency \(\omega_\text{th}\) corresponds to the threshold value of the mode index. By expanding \(\Omega(\omega)\) in the vicinity of \(\omega_\text{th}\), one can show that \((\bar{\mu}/\mu_\text{g})[\omega-\Omega(\omega)]=\omega-\omega_\text{th}\).
The gain \(G\) in Equation (6-2-14) is known in terms of the carrier density \(n\) inside the active layer. The rate equation for \(n\) was considered in the gain and stimulated emission for semiconductor lasers tutorial and show that in general the effect of carrier diffusion should be included.
Its inclusion, however, complicates the analysis considerably. In this tutorial we shall neglect carrier diffusion. The analysis is therefore mainly applicable to an index-guided laser.
We define the number of carriers inside the active layer as
\[\tag{6-2-19}N=\displaystyle\int{n}\text{d}V=nV\]
where we have assumed that \(n\) is approximately constant. \(V=Lwd\) and is the active volume for a laser of length \(L\), width \(w\), and thickness \(d\).
Using Equations (2-4-7) and (2-4-9) [refer to the gain and stimulated emission for semiconductor lasers tutorial], the carrier rate equation is
\[\tag{6-2-20}\dot{N}=I/q-\gamma_\text{e}N-GP\]
where \(I=wLJ\), \(J\) is the current density flowing through the active region (the contribution of leakage current is excluded), and
\[\tag{6-2-21}\gamma_\text{e}=(A_\text{nr}+Bn+Cn^2)=\tau_\text{e}^{-1}\]
is the carrier-recombination rate that can be used to define the spontaneous carrier lifetime \(\tau_\text{e}\).
Both radiative and nonradiative recombination processes contribute to \(\gamma_\text{e}\). The recombination rate \(A_\text{nr}\) is due to mechanisms such as trap or surface recombination. \(B\) is radiative-recombination coefficient, while \(C\) is related to the Auger recombination processes.
The last term in Equation (6-2-20) is due to stimulated recombination and leads to a nonlinear coupling between photons and charge carriers.
To complete the rate-equation description, we need an expression for \(R_\text{sp}\) appearing in Equation (6-2-14). If we assume that a fraction \(\beta_\text{sp}\) of spontaneously emitted photons goes into the lasing mode, \(R_\text{sp}\) is given by
\[\tag{6-2-22}R_\text{sp}=\beta_\text{sp}\eta_\text{sp}\gamma_\text{e}N\]
where \(\eta_\text{sp}=Bn/\gamma_\text{e}\) and is the (internal) spontaneous quantum efficiency showing the fraction of carriers that emit photons through spontaneous recombination.
The parameter \(\beta_\text{sp}\) is referred to as the spontaneous-emission factor and has been discussed extensively in the literature.
Petermann has pointed out that \(\beta_\text{sp}\) depends on the lateral guiding mechanism and is enhanced for gain-guided lasers due to wavefront curvature. For a mode lasing at the wavelength \(\lambda=2\pi{c}/\omega\), his expression for \(\beta_\text{sp}\) is
\[\tag{6-2-23}\beta_\text{sp}=\frac{K\Gamma\lambda^4}{4\pi^2\mu\bar{\mu}\mu_\text{g}V\Delta\lambda_\text{sp}}\]
where \(\mu\) is the bulk-material index and \(\Delta\lambda_\text{sp}\) is the width of the spontaneous-emission spectrum.
The enhancement factor \(K=1\) for index-guided lasers but is larger for gain-guided lasers. However, there is some disagreement over its numerical value.
In practice, \(\beta_\text{sp}\) is treated as a fitting parameter. Its typical numerical value for index-guided InGaAsP lasers is in the range of \(10^{-4}-10^{-5}\).
An alternative expression for \(R_\text{sp}\) can be obtained by using the Einstein relation that relates the gain spectrum to the spontaneous-emission spectrum. In this approach
\[\tag{6-2-24}R_\text{sp}(\omega)=n_\text{sp}(\omega)G(\omega)\]
where
\[n_\text{sp}(\omega)=\left[1-\exp\left(\frac{\hbar\omega-E_\text{f}}{k_\text{B}T}\right)\right]^{-1}\]
is the population-inversion factor and \(E_\text{f}\) is the energy separation between the quasi-Fermi levels. For 1.3-μm InGaAsP lasers, the estimated value of \(n_\text{sp}\) from spectral measurements at room temperature is ~ 1.7.
Equations (6-2-14), (6-2-18), and (6-2-20) are the single-mode rate equations and will be used for following tutorials to discuss the operating characteristics of a semiconductor laser.
For a discussion of multimode phenomena, these equations should be generalized to include the number of possible longitudinal modes for which the gain \(G\) is positive. This number depends on the width of the gain spectrum and the frequency spacing between different longitudinal modes.
The photon rate equation for each mode can be obtained using Equations (6-2-1) and (6-2-2) and following the preceding analysis. If \(P_m\) represents the photon population for the \(m\text{th}\) longitudinal mode oscillating at the frequency \(\omega_m\), the multimode rate equations are
\[\tag{6-2-25}\dot{P}_m=(G_m-\gamma_m)P_m+R_\text{sp}(\omega_m)\]
\[\tag{6-2-26}\dot{N}=I/q-\gamma_\text{e}N-\sum_mG_mP_m\]
where \(G_m=G(\omega_m)\) is the mode gain and \(\gamma_m\) is the mode loss.
In conventional Fabry-Perot semiconductor lasers, \(\gamma_m\) is frequency independent and \(\gamma_m=\gamma=\tau_\text{p}^{-1}\) for all modes. However, frequency-dependent losses can be introduced to enhance mode selectivity.
In Equation (6-2-26) the summation is over all modes participating in the process of stimulated emission.
To simplify the notation, the rate equations have been obtained for the dimensionless variables \(P_m\) and \(N\), which represent the photon and electron populations inside the laser cavity.
The quantity of practical interest is the output power emitted from each facet. As seen in Equation (2-6-7) [refer to the emission characteristics of semiconductor lasers tutorial], it is related linearly to the photon population and is given by
\[\tag{6-2-27}P_m^\text{out}=\frac{1}{2}\hbar\omega{v_\text{g}}\alpha_\text{m}P_m\]
The derivation of Equation (6-2-27) is intuitively obvious if we note that \(v_\text{g}\alpha_\text{m}\) is the rate at which photons of energy \(\hbar\omega\) escape through the two facets. For a 250-μm-long 1.3-μm InGaAsP laser, \(P_m\approx3.88\times10^4\) at a power level of 1 mW.
For ease of reference, Table 6-1 gives typical parameter values for a 1.3-μm buried heterostructure laser. In the case of unequal facet reflectivities the output powers are obtained using Equations (2-6-20) [refer to the emission characteristics of semiconductor lasers tutorial].
The next tutorial discusses about the steady-state characteristics of semiconductor lasers.