Transient Response of Semiconductor Lasers
This is a continuation from the previous tutorial - steady-state characteristics of semiconductor lasers.
The previous tutorial considered device behavior under steady-state conditions. When a semiconductor laser is turned on by changing the current \(I\), a relatively long time (~ 10 ns) elapses before the steady state is reached.
In the transient regime, the power distribution among various longitudinal modes varies periodically as the laser goes through relaxation oscillations. In particular, a semiconductor laser whose CW mode spectrum is predominantly single-mode exhibits poor side-mode suppression under dynamic conditions.
An understanding of the transient response is especially important for optical communication systems where the current is modulated at gigahertz frequencies and the laser never attains a steady state.
The purpose of this tutorial is to use the rate equations to describe the dynamic characteristics of a semiconductor laser.
Dynamic Longitudinal-Mode Spectrum
The longitudinal-mode spectrum under transient conditions is obtained by solving the multimode rate equations (6-2-25) and (6-2-26) [refer to the rate equations for semiconductor lasers tutorial].
The laser current is abruptly increased, say when time \(t=0\), from its initial value \(I_0\) to the final value \(I\), which is greater than \(I_\text{th}\). A numerical approach is generally required because of the nonlinear nature of the rate equations. However, approximate analytic solutions can be obtained under specific conditions and provide physical insight into the dynamic process.
Figure 6-7 shows the temporal evolution of \(N\) and \(P_m\) calculated numerically using parameters corresponding to those of a 1.3-μm InGaAsP laser.
The laser current was increased from \(0\) to \(1.5I_\text{th}\) at \(t=0\). Several features are noteworthy.
We see that the photon population in all modes remains zero for a time period known as the turn-on delay time \(t_\text{d}\), after which it increases rapidly. The turn-on delay is characteristic of any laser and indicates that stimulated emission does not occur until the carrier concentration has reached its threshold value \(N_\text{th}\).
The delay time \(\tau_\text{d}\sim\tau_\text{e}\), where \(\tau_\text{e}\) is the carrier recombination time defined in Equation (6-2-21) [refer to the rate equations for semiconductor lasers tutorial]; typically, \(\tau_\text{e}=2-3\text{ ns}\).
The most important feature of the transient response shown in Figure 6-7 is that the electron and photon populations oscillate before attaining their steady-state values.
These oscillations, referred to as relaxation oscillations, are manifestation of an intrinsic resonance in which energy stored in the system oscillates back and forth between the electron and photon populations.
The natural frequency of relaxation oscillations is in the gigahertz range and plays an important role in determining device response. Relaxation oscillations are considered further in latter half of this tutorial.
It is evident from Figure 6-7 that the transient-mode spectrum is significantly different from the one obtained under CW operation. Figure 6-8 shows the mode spectra under transient and steady-state conditions obtained using Figure 6-7.
The transient spectrum corresponds to the time at which the photon populations first reach their maximum value (the peak of the first relaxation oscillation). The transient spectrum is essentially multimode, while the CW spectrum exhibits an MSR greater than 10.
Relaxation oscillations in Figure 6-7 take several nanoseconds to become sufficiently damped for mode intensities to reach their steady-state values.
However, experiments indicate that relaxation oscillations in most semiconductor lasers damp much faster (typically in < 1 ns). The reason behind this disagreement can be traced to the assumption that the mode gain \(G_m\) in the rate equation (6-2-25) [refer to the rate equations for semiconductor lasers tutorial] is independent of the mode intensities. The nonlinear gain discussed in the steady-state characteristics of semiconductor lasers tutorial leads to a reduction in the magnitude of \(G_m\), which depends on the mode intensities. As discussed in latter half of this tutorial, an important effect of the nonlinear gain is to reduce the damping time of relaxation oscillations.
Turn-On Delay
When the laser is turned on by in creasing the device current from its initial value \(I_0\) to the above-threshold value \(I\) greater than \(I_\text{th}\), stimulated recombination is delayed by \(\tau_\text{d}\), the time during which the carrier population rises to its threshold value (see Figure 6-7).
The delay time is determined by the carrier dynamics alone. Its experimental determination is relatively easy and can be used to extract information about the carrier lifetime \(\tau_\text{e}\).
If we neglect the stimulated-recombination term in Equation (6-2-26) [refer to the rate equations for semiconductor lasers tutorial], the carrier density \(n=N/V\) and satisfies the rate equation
\[\tag{6-4-1}\frac{\text{d}n}{\text{d}t}=\frac{I}{qV}-\gamma_\text{e}(n)n\]
where \(\gamma_\text{e}=\tau_\text{e}^{-1}\) and is given by Equation (6-2-21) [refer to the rate equations for semiconductor lasers tutorial].
Integrating from \(t=0\) to \(t=t_\text{d}\), we obtain
\[\tag{6-4-2}\tau_\text{d}=qV\displaystyle\int\limits_{n_0}^{n_\text{th}}[I-qV\gamma_\text{e}(n)n]^{-1}\text{d}n\]
where \(n_0\) and \(n_\text{th}\) are obtained in terms of the current by solving the steady-state relations
\[\tag{6-4-3a}I_0=qV\gamma_\text{e}(n_0)n_0\]
\[\tag{6-4-3b}I_\text{th}=qV\gamma_\text{e}(n_\text{th})n_\text{th}\]
It is evident that the delay time depends on the functional form of the recombination rate \(\gamma_\text{e}(n)\).
In the presence of Auger recombination, important for long-wavelength semiconductor lasers [refer to the nonradiative recombination in semiconductors tutorial], \(\gamma_\text{e}\) is a quadratic polynomial in \(n\). In this general case a closed-form expression for \(\tau_\text{d}\) is difficult to obtain.
In a simple approximation only the constant term \(\gamma_\text{e}=A_\text{nr}\) is used, and the result for \(\tau_\text{d}\) is
\[\tag{6-4-4}\tau_\text{d}=\frac{1}{A_\text{nr}}\ln\left(\frac{I-I_0}{I-I_\text{th}}\right)\]
If only the linear term (\(\gamma_\text{e}=Bn\)) is used after assuming that radiative recombination dominates, Equation (6-4-2) yields
\[\tag{6-4-5}\tau_\text{d}=\left(\frac{qV}{BI}\right)^{1/2}\left[\tanh^{-1}\left(\frac{I_\text{th}}{I}\right)^{1/2}-\tanh^{-1}\left(\frac{I_0}{I}\right)^{1/2}\right]\]
Dixon and Joyce have obtained an analytic expression for \(\tau_\text{d}\) when \(\gamma_\text{e}=A_\text{nr}+Bn\). As noted by them, it is useful to define a new integration variable
\[\tag{6-4-6}I_\text{R}=qV\gamma_\text{e}(n)n\]
with the physical significance of the recombination current.
Equation (6-4-2) then becomes
\[\tag{6-4-7}\tau_\text{d}=\displaystyle\int\limits_{I_0}^{I_\text{th}}\frac{\tau_\text{e}'(I_\text{R})}{I-I_\text{R}}\text{d}I_\text{R}\]
where \(\tau_\text{e}'\) is the differential recombination time defined as
\[\tag{6-4-8}\text{d}I_\text{R}/\text{d}n=qV/\tau_\text{e}'\]
Using Equation (6-2-21) [refer to the rate equations for semiconductor lasers tutorial] in Equation (6-4-6), one can easily see that
\[\tag{6-4-9}\tau_\text{e}'(n)=(A_\text{nr}+2Bn+3Cn^2)^{-1}\]
Equation (6-4-7) can be used to draw several qualitative conclusions that hold for the general case that includes Auger recombination as well.
If the laser is prebiased close to threshold, \(\tau_\text{e}'\) is approximately constant in the entire range of integration and can be replaced by its threshold value. Similarly, \(I_\text{R}\approx{I}_\text{th}\), and Equation (6-4-7) yields the simple relation
\[\tag{6-4-10}\tau_\text{d}\approx\tau_\text{e}'(n_\text{th})\left(\frac{I_\text{th}-I_0}{I-I_\text{th}}\right)\]
This equation can be used to estimate the differential recombination time \(\tau_\text{e}'\) from the measurement of \(\tau_\text{d}\).
The important point to notice is that depending on the experimental conditions one can obtain either \(\tau_\text{e}\) or \(\tau_\text{e}'\) from the delay-time measurements.
The value of \(\tau_\text{e}(n_\text{th})\) is obtained when \(I_0=0\) and \(I\gg{I}_\text{th}\), so that from Equations (6-4-2) and (6-4-3),
\[\tag{6-4-11}\tau_\text{d}\approx\tau_\text{e}(n_\text{th})\frac{I_\text{th}}{I}\]
To study the effect of Auger recombination on the delay time in the general case, we have evaluated the integral in Equation (6-4-2) numerically using the parameter values shown in Table 6-1 [refer to the rate equations of semiconductor lasers tutorial]. The results for the specific case of no prebias (\(I_0=0\)) are shown in Figure 6-9.
For the sake of comparison, the threshold value of the carrier-recombination time has been kept constant for all three curves and is about 2.5 ns. This was achieved by setting \(\tau_\text{nr}=(A_\text{nr}+Cn_\text{th}^2)^{-1}=5\text{ ns}\) and choosing \(A_\text{nr}\) accordingly for each value of the Auger coefficient \(C\) shown in Figure 6-9.
The main effect of Auger recombination is to reduce the delay time compared to its value expected when nonradiative recombination is independent of the carrier density.
The extent of reduction decreases with an increase in \(I\), and the three curves in Figure 6-9 merge for \(I\gg{I}_\text{th}\) in accordance with Equation (6-4-11).
Relaxation Oscillations
As seen in Figure 6-7, the output of a semiconductor laser exhibits damped periodic oscillations before settling down to its steady-state value. Such relaxation oscillations are due to an intrinsic resonance in the nonlinear laser system.
An expression for their frequency and decay rate can be easily obtained using the small-signal analysis of the single-mode rate equations.
From Equations (6-2-14) and (6-2-20) [refer to the rate equations of semiconductor lasers tutorial], these are
\[\tag{6-4-12}\dot{P}=(G-\gamma)P+R_\text{sp}\]
\[\tag{6-4-13}\dot{N}=I/q-\gamma_\text{e}N-GP\]
where the stimulated-emission rate \(G\) is a function of \(N\).
Even though a linear dependence of \(G\) on \(N\) is often valid, for the moment we leave it unspecified.
Furthermore, we allow \(G\) to vary with \(P\) to allow for the nonlinear-gain effects. The dependence of \(G\) on the power can occur due to several mechanisms such as the spectral-hole burning, carrier heating, and two-photon absorption.
For a given value of the device current \(I\) greater than \(I_\text{th}\), the steady-state solution of each rate equation is readily obtained, as discussed in the first part of the steady-state characteristics of semiconductor lasers tutorial.
In the small-signal analysis the steady-state values \(P\) and \(N\) are perturbed by a small amount \(\delta{P}\) and \(\delta{N}\). The rate equations (6-4-12) and (6-4-13) are linearized by neglecting the quadratic and higher powers of \(\delta{P}\) and \(\delta{N}\). We then obtain
\[\tag{6-4-14}\delta\dot{P}=-\Gamma_\text{P}\delta{P}+(G_\text{N}P+\partial{R_\text{sp}}/\partial{N})\delta{N}\]
\[\tag{6-4-15}\delta\dot{N}=\Gamma_\text{N}\delta{N}-(G+G_\text{P}P)\delta{P}\]
where
\[\tag{6-4-16}\Gamma_\text{P}=R_\text{sp}/P-G_\text{P}P\]
\[\tag{6-4-17}\Gamma_\text{N}=\gamma_\text{e}+N(\partial\gamma_\text{e}/\partial{N})+G_\text{N}P\]
are the decay rates of fluctuations in the photon and carrier populations, respectively.
In obtaining Equations (6-4-14) and (6-4-15), the gain \(G(N,P)\) was expanded in a truncated Taylor series
\[\tag{6-4-18}G(N,P)\approx{G}+G_\text{N}\delta{N}+G_\text{P}\delta{P}\]
where \(G_\text{N}=\partial{G}/\partial{N}\) and \(G_\text{P}=\partial{G}/\partial{P}\).
The gain derivative \(G_\text{P}\) is usually negative because of gain suppression occurring at high powers. Even though the gain reduction is usually less than 1%, its inclusion is important since it contributes significantly to the damping of photon fluctuations as seen by Equation (6-4-16).
Typical values of the parameters appearing in the small-signal analysis are listed in Table 6-2 for a buried heterostructure laser with the device parameters given in Table 6-1 [refer to the rate equations of semiconductor lasers tutorial].
The linear set of Equations (6-4-14) and (6-4-15) can be readily solved. If we assume an exponential time dependence
\[\tag{6-4-19}\begin{cases}\delta{P}(t)=\delta{P_0}\exp(-ht)\\\delta{N}(t)=\delta{N_0}\exp(-ht)\end{cases}\]
where \(\delta{P}_0\) and \(\delta{N}_0\) are the initial values of the perturbation, the decay constant \(h\) is complex, indicating an oscillatory approach to equilibrium, and is given by
\[\tag{6-4-20}h=\Gamma_\text{R}\pm\text{i}\Omega_\text{R}\]
where
\[\tag{6-4-21}\Gamma_\text{R}=\frac{1}{2}(\Gamma_\text{N}+\Gamma_\text{P})\]
is the decay rate of relaxation oscillations and
\[\tag{6-4-22}\Omega_\text{R}=\left[(G+G_\text{P}P)\left(G_\text{N}P+\frac{\partial{R}_\text{sp}}{\partial{N}}\right)-\frac{(\Gamma_\text{N}-\Gamma_\text{P})^2}{4} \right]^{1/2}\]
is the angular frequency of relaxation oscillations.
The expression for \(\Omega_\text{R}\) can be considerably simplified by noting that the combination of \(GG_\text{N}P\) in Equation (6-4-22) dominates by several orders of magnitude and to a good degree of approximation
\[\tag{6-4-23}\Omega_\text{R}\approx(GG_\text{N}P)^{1/2}\]
This is a remarkably simple expression. For practical purposes it is more useful to express it in terms of the device current. From Equation (6-4-13) the steady-state value of \(P\) is given by
\[\tag{6-4-24}P=(I-I_\text{th})/(qG)\]
where \(I_\text{th}=q\gamma_\text{e}N\).
Using this equation an alternative expression for \(\Omega_\text{R}\) is
\[\tag{6-4-25}\Omega_\text{R}=\left[\frac{G_\text{N}(I-I_\text{th})}{q}\right]^{1/2}\]
It is evident that the most important parameter governing the relaxation oscillation frequency is the gain derivative \(G_\text{N}\), which equals \(\partial{G}/\partial{N}\). If we use Equation (6-3-3) [refer to the steady-state characteristics of semiconductor lasers tutorial] and assume the linear dependence of gain on \(N\), then
\[\tag{6-4-26}G_\text{N}=\Gamma{v_\text{g}}a/V\]
where \(a\) is the gain coefficient introduced in Equation (2-4-3) [refer to the gain and stimulated emission of semiconductor lasers tutorial].
Clearly \(\Omega_\text{R}\) increases with a decrease in the active volume \(V\).
The relaxation-oscillation frequency is often expressed in terms of the carrier and photon lifetimes.
If we use Equations (6-3-3) [refer to the steady-state characteristics of semiconductor lasers tutorial], (6-4-25), and (6-2-26) [refer to the rate equations of semiconductor lasers tutorial], together with the equation \(I_\text{th}=q\gamma_\text{e}N\), then \(\Omega_\text{R}\) can also be written as
\[\tag{6-4-27}\Omega_\text{R}=\left[\frac{1+\Gamma{v_\text{g}}an_0\tau_\text{p}}{\tau_\text{e}\tau_\text{p}}\left(\frac{I}{I_\text{th}}-1\right)\right]^{1/2}\]
where \(n_0\) is the transparency value of the carrier density, \(\tau_\text{e}=\gamma_\text{e}^{-1}\), and \(\tau_\text{p}=\gamma^{-1}\).
The term \(\Gamma{v_\text{g}}an_0\tau_\text{p}\) is sometimes neglected in comparison to \(1\). However, typically this term is \(\sim1\) and cannot be neglected. For a laser with parameters given in Table 6-1 [refer to the rate equations of semiconductor lasers tutorial], its magnitude is \(0.9\).
The preceding small-signal analysis shows that as long as \(\Gamma_\text{R}\gt0\), fluctuations from the steady-state exhibit damped relaxation oscillations.
However, if \(\Gamma_\text{R}\) becomes negative, fluctuations grow exponentially and the steady state is no longer stable. From Equation (6-4-16) this can happen only if \(G_\text{P}=\partial{G}/\partial{P}\gt0\), i.e., if the gain increases with power. The laser then exhibits self-pulsing that has been observed in gain-guided AlGaAs lasers [refer to the gain-guided lasers tutorial].
Several possible mechanisms, such as saturable absorption and lateral-hole burning have been proposed; the common feature of these mechanisms is that they all lead to the instability condition \(G_\text{P}\gt0\).
Self-pulsing is rarely observed in index-guided InGaAsP lasers. It is believed that in these devices phenomena such as spectral-hole burning actually reduce the gain with an increase in the output power so that \(G_\text{p}\lt0\). From Equations (6-4-16) and (6-4-21) we find that \(\Gamma_\text{R}\) then increases with power, and the relaxation oscillations are rapidly damped.
The next tutorial discusses about the noise characteristics of semiconductor lasers.