Steady-State Characteristics of Semiconductor Lasers
This is a continuation from the previous tutorial - rate equations of semiconductor lasers.
The rate equations of the previous tutorial can be used to obtain the steady-state response of a semiconductor laser at a fixed value of the current \(I\).
The steady-state solution is obtained by setting all time derivatives to zero and is applicable for continuous-wave (CW) operation after transients have died out. The steady-state or CW solution is also applicable under pulsed operation provided the duration of the current pulse is much longer than the laser response time, which is a few nanoseconds.
Two steady-state features are of general interest in characterizing laser performance, namely the light-current (\(L-I\)) curve and the longitudinal-mode spectrum.
The \(L-I\) curve shows how the output power varies with the device current. The longitudinal-mode spectrum indicates how the power is distributed among various modes at a given current.
Light-Current Curve
Although in general one should consider the multimode rate equations, a simple, physical description of the laser threshold and the resulting \(L-I\) curve can be based on the single-mode rate equations.
From Equation (6-2-14) [refer to the rate equations of semiconductor lasers tutorial] the photon number \(P\) is given by
\[\tag{6-3-1}P=R_\text{sp}/(\gamma-G)\]
This equation stresses that a laser can be viewed as a regenerative noise amplifier. Spontaneously emitted photons provide the noise input that is amplified in the presence of gain provided by the injected charge carriers.
The threshold condition (\(G=\gamma\)) obtained in the threshold condition and longitudinal modes of semiconductor lasers tutorial ignored the contribution of spontaneous emission.
Equation (6-3-1) shows that the required gain \(G\) is slightly below the cavity-loss level and approaches its asymptotic value \(\gamma\) as the power output increases.
To obtain the \(L-I\) curve, we substitute Equation (6-3-1) into Equation (6-2-20) [refer to the rate equations of semiconductor lasers tutorial]. This leads to the relation
\[\tag{6-3-2}\gamma_\text{e}(N)N+R_\text{sp}(N)\frac{G}{\gamma-G}=\frac{I}{q}\]
that can be used to obtain \(N\) for a given \(I\) if the functional dependence of \(G(N)\) is known.
Note that \(G\) corresponds to the peak value of the gain since the longitudinal mode closest to the gain peak reaches threshold first.
The simple model discussed in the gain and stimulated emission for semiconductor lasers tutorial used a linear dependence of \(G\) on \(N\). Using Equations (2-4-3) [refer to the gain and stimulated emission for semiconductor lasers tutorial] and (6-2-15) [refer to the rate equations of semiconductor lasers tutorial], we obtain
\[\tag{6-3-3}G(N)=\Gamma{v_\text{g}}a(N/V-n_0)\]
The use of Equations (6-2-21), (6-2-22) [refer to the rate equations for semiconductor lasers tutorial], and (6-3-3) in Equation (6-3-2) leads to a fourth-degree polynomial in \(N\) that can be used to obtain \(N\) as a function of the device current \(I\).
The photon number \(P\) is then obtained using Equation (6-3-1). The output power is related to \(P\) linearly as given by Equation (6-2-27) [refer to the rate equations for semiconductor lasers tutorial].
Figure 6-1 shows the variation of the output power with the device current \(I\) for a 1.3-μm InGaAsP laser using the parameter values shown in Table 6-1 [refer to the rate equations for semiconductor lasers tutorial].
Different curves correspond to different values of the spontaneous-emission factor \(\beta_\text{sp}\).
We note that \(P\) starts to increase rapidly as the laser threshold is reached. The sudden change in the slope of the \(L-I\) curve in the vicinity of the threshold indicates that stimulated emission has taken over the spontaneous emission and is accompanied by other characteristic features such as narrowing of the spectral width.
In the threshold vicinity, \(P\) changes by orders of magnitude in a narrow current range. However, the sharpness of the laser threshold depends on \(\beta_\text{sp}\).
The variation of the carrier population \(N\) with \(I\) in Figure 6-2 shows that \(N\) is nearly constant in the threshold region and does not increase significantly with a further increase in \(I\).
As is evident from Equation (6-3-1), a small change in \(N\), or equivalently in the gain \(G\), can nonetheless produce a large change in \(P\) when \(G\) is close to \(\gamma\). The limiting value of \(N\) is referred to as the threshold carrier population \(N_\text{th}\) and corresponds to the condition where \(G=\gamma\).
From a practical viewpoint the quantity of interest is the threshold current \(I_\text{th}\). In the presence of spontaneous emission, however, the threshold is not sharply defined.
Figure 6-1 shows that the transition from the nonlasing to the lasing state becomes softer with an increase in the amount of spontaneous emission. It is customary to define \(I_\text{th}\) in the limiting case of \(\beta_\text{sp}=0\). This case was discussed in the emission characteristics of semiconductor lasers tutorial.
From Equation (6-3-2) with \(R_\text{sp}=0\), we obtain
\[\tag{6-3-4}I_\text{th}=q\gamma_\text{e}(N_\text{th})N_\text{th}\]
where \(N_\text{th}\) is obtained using Equation (6-3-3) after setting \(G(N_\text{th})\) equal to \(\gamma\).
For parameter values given in Table 6-1 [refer to the rate equations for semiconductor lasers tutorial], \(I_\text{th}\approx16\text{ mA}\).
It should be stressed that Equation (6-3-4) expresses the current that passes through the active region. The actual device current is generally higher because of current leakage outside the active region, which invariably occurs under realistic conditions [refer to the strongly index-guided lasers tutorial].
The \(L-I\) curves shown in Figure 6-1 model reasonably well an index-guided device. In the presence of gain guiding, carrier diffusion becomes important and the \(L-I\) curve under CW operation is generally obtained numerically.
The rate-equation model neglects the axial variation of mode intensities. The axial effects can be included using an FP-type approach.
Note also that the \(L-I\) curve is strongly temperature-dependent. In obtaining Figure 6-1, the parameter values corresponding to room temperature (\(T=300\text{ K}\)) were used.
The temperature dependence of \(L-I\) curves was discussed in the temperature dependence of threshold current of semiconductor lasers tutorial.
Longitudinal-Mode Spectrum
The power spectrum of a semiconductor laser obtained using a spectrometer shows the presence of several longitudinal modes whose relative powers vary with the current \(I\).
The multimode nature of semiconductor lasers has been of concern from the early days. It is believed that spontaneous emission plays an important role in determining the spectral characteristics of a semiconductor laser.
We use the multimode rate equations to obtain such spectral features as the number of longitudinal modes and their relative intensities. In its simple form, the analysis is applicable to a strongly index-guided device.
To solve the multimode rate equations (6-2-25) and (6-2-26) [refer to the rate equations for semiconductor lasers tutorial], we need to know the gain spectrum \(G_m\) where \(G_m=G(\omega_m)\). The gain spectrum is either obtained experimentally or calculated numerically [see Figure 3-7 in the radiative recombination in semiconductors tutorial]; its functional form is not generally available.
A simple approximation is that the gain decreases quadratically from its peak value and assumes that
\[\tag{6-3-5}G(\omega)=G_0\left[1-\left(\frac{\omega-\omega_0}{\Delta\omega_\text{g}}\right)^2\right]\]
where \(\omega_0\) is the frequency at which the gain takes its maximum value \(G_0\), and \(\Delta\omega_\text{g}\) is the frequency spread over which the gain is nonzero on either side of the gain peak (see Figure 6-3).
We assume that the central (most intense) mode is located at the gain-peak frequency \(\omega_0\). The other longitudinal modes have frequencies
\[\tag{6-3-6}\omega_m=\omega_0+m\Delta\omega_\text{L}\]
where \(\Delta\omega_\text{L}\) is the longitudinal-mode spacing [see Equation (2-3-14) in the threshold condition and longitudinal modes of semiconductor lasers tutorial] and the integer \(m\) varies from \(-M\) to \(+M\).
The value of \(M\) is determined by the largest integer contained in the ratio
\[\tag{6-3-7}M\approx\frac{\Delta\omega_\text{g}}{\Delta\omega_\text{L}}\]
where \(2M+1\) represents the number of longitudinal modes for which the gain is positive.
Using Equations (6-3-5) to (6-3-7), the mode gain is approximated by
\[\tag{6-3-8}G_m=G_0[1-(m/M)^2]\]
The photon number \(P_m\) can be obtained using Equation (6-2-25) [refer to the rate equations for semiconductor lasers tutorial] and is given by
\[\tag{6-3-9}P_m=\frac{R_\text{sp}(\omega_m)}{\gamma-G_m}\]
where we have assumed that all modes have the same loss \(\gamma=\tau_\text{p}^{-1}\).
One can now follow a procedure similar to that for the single-mode case. The substitution of Equations (6-3-8) and (6-3-9) into Equation (6-2-26) [refer to the rate equations for semiconductor lasers tutorial] leads to an implicit relation that can be solved to obtain the steady-state carrier number \(N\). This in turn determines the mode gain \(G_m\) and the steady-state photon population \(P_m\) for each value of the current \(I\).
Figure 6-4 shows the results of such a numerical calculation using a total of 19 longitudinal modes (\(M=9\)) for a 1.3-μm InGaAsP laser.
The output power in the central or main mode (\(m=0\)) and in side modes (the four adjacent modes on both sides of the central mode) is shown as a function of the device current.
Below or near threshold, the power increases in all of the longitudinal modes. However, the side-mode power saturates in the above-threshold regime, whereas the main-mode power keeps increasing in a manner similar to the single-mode case as shown in Figure 6-1.
The power level at which a given side mode saturates depends on the spontaneous-emission factor \(\beta_\text{sp}\) and increases with an increase in \(\beta_\text{sp}\).
The power degeneracy for positive and negative values of \(m\) implies that the longitudinal-mode spectrum is symmetric with respect to the main mode, which is a consequence of our assumption that the gain spectrum (6-3-5) is symmetric. In practice, both the gain spectrum and the longitudinal-mode spectrum show some asymmetry.
To get some physical insight into the multimode behavior of semiconductor lasers, we note from our previous discussion for the single-mode case that the peak gain \(G_0\) asymptotically approaches \(\gamma\) and that \(\gamma-G_0\) is inversely proportional to the main-mode power.
If we substitute \(m=0\) and use
\[\tag{6-3-10}G_0=\gamma(1-\delta)\]
in Equation (6-3-9), we find that
\[\tag{6-3-11}\delta=\frac{R_\text{sp}(\omega_0)}{\gamma{P_0}}\]
Using Equations (6-3-8) and (6-3-10) in (6-3-9), the photon number in the \(m\)th side mode is given by
\[\tag{6-3-12}P_m\approx\frac{R_\text{sp}(\omega_m)}{\gamma}\left(\frac{1}{\delta+(m/M)^2}\right)\]
This equation shows that the power distribution among various side modes is approximately Lorentzian and that the side-mode power is half the main-mode power when \(m=\delta^{1/2}M\).
The width (FWHM) of the spectral envelope is thus given by
\[\tag{6-3-13}\Delta\omega_\text{s}=2m(\Delta\omega_\text{L})=\delta^{1/2}(2\Delta\omega_\text{g})\]
where we have used Equation (6-3-7).
Since \(\delta\) decreases with an increase in the main-mode power, the spectral width \(\Delta\omega_\text{s}\) decreases continuously in the above-threshold regime.
A measure of the spectral purity of a semiconductor laser is the mode suppression ratio (MSR), defined as the ratio of the main-mode power to the power of the most intense side mode,
\[\tag{6-3-14}\text{MSR}=\frac{P_0}{P_1}=1+\frac{1}{\delta{M^2}}\]
where we used Equation (6-3-12) and assumed that \(R_\text{sp}(\omega_0)\approx{R_\text{sp}}(\omega_1)\).
Clearly the MSR increases with an increase in the main-mode power \(P_0\), which reduces \(\delta\). For a given \(\delta\), the MSR can be improved by decreasing \(M\). This suggests that short-cavity semiconductor lasers have an improved MSR, since \(M\) can be made smaller by increasing the longitudinal-mode spacing.
Using Equations (6-3-7) and (6-3-11) in Equation (6-3-14), the explicit dependence of the MSR on the laser parameters is given by
\[\tag{6-3-15}\text{MSR}=1+\frac{P_0}{\tau_\text{p}R_\text{sp}}\left(\frac{\Delta\omega_\text{L}}{\Delta\omega_\text{g}}\right)^2\]
where \(\tau_\text{p}=\gamma^{-1}\) is the photon lifetime.
This equation shows that the MSR increases linearly with the main-mode power. The term single-mode operation implies a large value of MSR. However, the exact value of the MSR above which the laser qualities as being single mode is a matter of definition. A value of 20 for the MSR is often used for this purpose.
The total power emitted by a multimode laser can be obtained by summing over all the modes. If we use Equation (6-2-27) [refer to the rate equations of semiconductor lasers tutorial] to relate the power emitted per facet to the intracavity photon number, we obtain
\[\tag{6-3-16}P^\text{out}\approx\frac{1}{2}\hbar\omega_0v_\text{g}\alpha_\text{m}\sum_mP_m\]
We substitute \(P_m\) from Equation (6-3-12). The summation can be performed if we replace the finite sum by an infinite sum. This does not introduce significant errors since \(P_m\) is almost zero for large \(m\).
The total intracavity photon population is given by
\[\tag{6-3-17}P_\text{T}=\sum_mP_m=\frac{R_\text{sp}(\omega_0)}{\gamma\delta^{1/2}}\pi{M}\coth(\pi{M}\delta^{1/2})\]
where we have assumed that \(R_\text{sp}(\omega_m)\) can be replaced by \(R_\text{sp}(\omega_0)\). This assumption is justified since the spontaneous-emission spectrum is much wider than the longitudinal-mode spectrum.
The previous analysis shows that the multimode characteristics of a semiconductor laser can be described in terms of two dimensionless parameters \(M\) and \(\delta\).
The number \(2M+1\) corresponds to the total number of longitudinal modes that fit within the gain spectrum and experience gain.
The parameter \(\delta\), defined by Equation (6-3-10), is a measure of how closely the peak gain approaches the total cavity loss (see Figure 6-3) and decreases with an increase in the laser power.
The numerical value of \(\delta\) depends on the spontaneous-emission rate \(R_\text{sp}\) and is sensitive to the numerical value of the spontaneous-emission factor \(\beta_\text{sp}\) [see Equations (6-2-22) and (6-2-23) in the rate equations of semiconductor lasers tutorial].
For index-guided lasers, \(\beta_\text{sp}\le10^{-4}\), and at a power level of a few milliwatts, \(\delta\approx10^{-4}\). Using a typical value (\(M=10\)) for a 250-μm-long laser, Equation (6-3-14) predicts that an MSR of about 100 can be achieved under CW operation.
Figure 6-5 shows the calculated longitudinal-mode spectra at various power levels for a 1.3-μm InGaAsP laser. At a power level of 5 mW, most of the power is carried out by the main mode.
Gain-guided lasers, by contrast, exhibit longitudinal mode spectra that are strikingly different from those shown in Figure 6-5. By way of an example, Figure 6-6 shows the experimentally measured mode spectra of an AlGaAs gain-guided laser.
Many modes are present and \(\text{MSR}\approx2\) even at a power level of 11.7 mW. From Equation (6-3-14) we note that this would be the case if \(\delta\ge10^{-2}\). Such values of \(\delta\) are consistent with \(\beta_\text{sp}\approx10^{-3}\).
It has been suggested that an order-of-magnitude increase in the spontaneous-emission factor \(\beta_\text{sp}\) is due to the enhancement factor \(K\) in Equation (6-2-23) [refer to the rate equations of semiconductor lasers tutorial].
The effect of the lateral waveguiding mechanism on the longitudinal-mode spectrum is not very well understood.
In the previous analysis the multimode nature of semiconductor lasers is attributed to spontaneous emission. The semiconductor laser behaves as a regenerative noise amplifier. It amplifies all modes for which the roundtrip gain is positive.
Once the threshold is reached, the gain is approximately clamped and the power in the side modes saturates.
In this model, the MSR increases continuously with an increase in the laser power, as is evident from Equation (6-3-15). In practice, however, with an increase in the laser power several other phenomena, such as spatial and spectral hole burning, start to influence the longitudinal-mode behavior.
Spatial-hole burning is a result of the standing-wave nature of the optical mode and is known to lead to multimode oscillation. However, its relative importance depends on carrier diffusion, which is likely to wash out axial inhomogeneities in the carrier density.
The effectiveness of the diffusion process decreases with an increase in the laser power since the stimulated recombination time becomes much shorter relative to the carrier-diffusion time.
As carriers are depleted faster at the crest of the standing wave, the main-mode gain decreases. As a result the side-mode power starts to increase when the main-mode power exceeds a critical value.
Spectral-hole burning is related to the broadening mechanism of the gain profile. In semiconductor lasers the gain profile is nearly homogeneously broadened because the intraband carrier scattering time is extremely short (~ 100 fs).
At high laser powers the stimulated recombination time becomes small enough (~ 1 ps) that the peak gain decreases because of partial spectral-hole burning. Consequently other longitudinal modes start to grow, and the range of single-mode operation is inherently limited.
The power-dependent or the nonlinear part of the gain has been used to explain the experimental results wherein the power distribution among the longitudinal modes is asymmetric and the dominant mode shifts toward longer wavelengths with an increase in device current.
Power-dependent changes in the mode gain are referred to as the nonlinear gain and have been studied extensively by using a density-matrix approach commonly used in laser theory. We shall see in later tutorials that the nonlinear gain especially affects the dynamic response of semiconductor lasers.
A comprehensive analysis of the multimode characteristics of a semiconductor laser is extremely involved. Together with such mechanisms as spatial- and spectral-hole burning, whose main effect is to destabilize the main mode, there are other nonlinear processes that tend to stabilize the main mode.
One such nonlinear process results from the beating of lasing and nonlasing modes, which modulates the carrier population at the beat frequency corresponding to the longitudinal-mode spacing (typically ~ 100 GHz).
Since the mode gain and the refractive index both depend on the carrier population, they are also modulated at the beat frequency, an effect referred to as the creation of dynamic gain and index gratings.
Bragg diffraction from these gratings couples the longitudinal modes and affects their steady-state power distribution through a well-known nonlinear phenomenon called four-wave mixing. This mechanism has been used to explain extraordinarily large MSR values observed in some index-guided AlGaAs lasers. It can also explain shifting of the main mode toward longer wavelengths with an increase in the current.
The multimode rate equations (6-2-25) and (6-2-26) [refer to the rate equations for semiconductor lasers tutorial] are able to model reasonably well the observed features in the longitudinal-mode spectra of semiconductor lasers. However, their use neglects axial variations of the mode intensities. Recently attempts have been made to include the axial effects.
In general, the rate of spontaneous emission \(R_\text{sp}\) increases with a decrease in the facet reflectivity \(R_\text{m}\). For a semiconductor laser whose \(R_\text{m}=0.32\), the increase is about \(10\%\).
The next tutorial discusses about the transient response of semiconductor lasers.