# Modulation Response of Semiconductor Lasers

This is continuation from the previous tutorial - ** noise characteristics of semiconductor lasers**.

One of the important advantages of semiconductor lasers is that they can be directly modulated; i.e., one can readily obtain short optical pulses useful for optical communications by modulating the device current.

The modulation. response of semiconductor lasers have been studied from the early days.

Because of an intrinsic resonance of the device it was found that the response peaks at the relaxation-oscillation frequency \(\Omega_\text{R}\), and the modulation efficiency sharply drops for modulation frequencies \(\omega_\text{m}\) greater than \(\Omega_\text{R}\).

Experimentally, however, the resonance peak is often less pronounced than the one predicted by the single-mode rate equations. Mechanisms such as spontaneous emission, carrier diffusion, and spectral-hole burning have been proposed to account for the suppression of the relaxation-oscillation peak.

The initial small-signal analysis has been extended to include the nonlinear effects that become important under large-signal modulation. Considerable attention has recently been paid to increase the modulation bandwidth of semiconductor lasers.

A characteristic feature of semiconductor lasers is that intensity or amplitude modulation (AM) leads simultaneously to phase or frequency modulation (FM). The interdependence between AM and FM under direct current modulation is governed by the line-width enhancement factor \(\beta_\text{c}\) encountered in the noise characteristics of semiconductor lasers tutorial.

It has its origin in the index change that invariably occurs when the optical gain changes in response to variations in the carrier population. As a result of simultaneously occurring AM and FM, the mode frequency of a directly modulated semiconductor laser shifts periodically during each modulation cycle.

This phenomenon, called frequency chirping, has received considerable attention recently as it is often a limiting factor in the performance of 1.55-μm optical communication systems. On the other hand, the same physical phenomenon can be used for direct frequency modulation, a useful technique for coherent optical communication systems.

## Small-Signal Analysis

A basic understanding of the modulation characteristics can be developed by using the single-mode rate equations (6-5-1) to (6-5-3). Without the noise sources they become

\[\tag{6-6-1}\dot{P}=(G-\gamma)P+R_\text{sp}\]

\[\tag{6-6-2}\dot{N}=I/q-\gamma_\text{e}N-GP\]

\[\tag{6-6-3}\dot{\phi}=-(\omega_0-\omega_\text{th})+\frac{1}{2}\beta_\text{c}(G-\gamma)\]

Under direct-current modulation, the device current \(I\) is time-dependent and consists of two parts:

\[\tag{6-6-4}I(t)=I_\text{b}+I_\text{m}(t)\]

In the absence of modulation, the laser operates continuously at the bias level \(I_\text{b}\). The corresponding steady-state values \(P\), \(N\), and \(\phi\) are obtained by setting all time derivatives to zero in Equations (6-6-1) to (6-6-3).

The effect of modulation current \(I_\text{m}(t)\) is to introduce deviations \(\delta{P}(t)\), \(\delta{N}(6)\), and \(\delta{\phi}(t)\), which vary periodically at the modulation frequency \(\omega_\text{m}\).

In the small-signal analysis \(I_\text{m}(t)\) is assumed to be small enough such that the deviation from the steady state remains small at all times [\(\delta{P}(t)\ll{P}\), \(\delta{N}(t)\ll{N}\), and \(\delta{\phi}(t)\ll\phi\)]. A criterion for its validity is generally expressed using the depth of modulation defined as

\[\tag{6-6-5}m=\frac{(\delta{P})_\text{max}}{P}=\frac{[I_\text{m}(t)]_\text{max}}{I_\text{b}-I_\text{th}}\]

The small-signal analysis is valid only if \(m\ll1\). Assuming its validity, the rate equations for \(\delta{P}\), \(\delta{N}\), and \(\delta{\phi}\) can be linearized using Equations (6-6-1) to (6-6-4). We then obtain

\[\tag{6-6-6}\delta\dot{P}=-\Gamma_\text{P}\delta{P}+(G_\text{N}P)\delta{N}\]

\[\tag{6-6-7}\delta\dot{N}=-\Gamma_\text{N}\delta{N}-G\delta{P}+I_\text{m}(t)/q\]

\[\tag{6-6-8}\delta\dot{\phi}=\frac{1}{2}\beta_\text{c}G_\text{N}\delta{N}\]

These equations are similar to Equations (6-5-8) to (6-5-10) and have been simplified using \(\partial{R_\text{sp}}/\partial{N}\ll{G_\text{N}}P\) and \(G_\text{P}P\ll{G}\).

They are readily solved in the frequency domain using Fourier analysis. The solutions are

\[\tag{6-6-9}\delta\tilde{P}(\omega)=\frac{G_\text{N}P\tilde{I}_\text{m}(\omega)/q}{(\Omega_\text{R}+\omega-\text{i}\Gamma_\text{R})(\Omega_\text{R}-\omega+\text{i}\Gamma_\text{R})}\]

\[\tag{6-6-10}\delta\tilde{N}(\omega)=\frac{(\Gamma_\text{P}+\text{i}\omega)(\tilde{I}_\text{m}(\omega)/q)}{(\Omega_\text{R}+\omega-\text{i}\Gamma_\text{R})(\Omega_\text{R}-\omega+\text{i}\Gamma_\text{R})}\]

\[\tag{6-6-11}\delta\tilde{\phi}(\omega)=\frac{\beta_\text{c}}{2\text{i}\omega}[G_\text{N}\delta\tilde{N}(\omega)]\]

where

\[\tag{6-6-12}\tilde{I}_\text{m}(\omega)=\displaystyle\int\limits_{-\infty}^{\infty}I_\text{m}(t)\exp(-\text{i}\omega{t})\text{d}t\]

is the Fourier transform of the modulation current and a similar relation exists for \(\delta\tilde{P}\), \(\delta\tilde{N}\), and \(\delta\tilde{\phi}\).

The decay rate \(\Gamma_\text{R}\) and the frequency \(\Omega_\text{R}\) of the relaxation oscillations are given by Equations (6-4-21) and (6-4-23) [refer to the transient response of semiconductor lasers tutorial].

Equations (6-6-9) to (6-6-11) shows that the modulation current changes the photon and carrier populations inside the active region, which in turn affect the optical phase. However, because of intrinsic laser resonance, the modulation response is frequency-dependent.

## Intensity Modulation

The preceding small-signal analysis can be used for an arbitrary form of \(I_\text{m}(t)\). The results are particularly simple for the case of sinusoidal modulation, which is often used experimentally as well. Using

\[\tag{6-6-13}I_\text{m}(t)=I_\text{P}\sin(\omega_\text{m}t)\]

in Equation (6-6-12), we obtain

\[\tag{6-6-14}\tilde{I}_\text{m}(\omega)=-\text{i}\pi{I_\text{P}}[\delta(\omega-\omega_\text{m})-\delta(\omega+\omega_\text{m})]\]

where the modulation frequency \(\nu_\text{m}=\omega_\text{m}/2\pi\) and \(I_\text{P}\) is the peak value of the modulation current.

Using Equation (6-6-14) in Equation (6-6-9) and taking the inverse Fourier transform, we obtain the modulated power

\[\tag{6-6-15}\delta{P}(t)=\delta{P_0}\sin(\omega_\text{m}t+\theta_\text{P})\]

where

\[\tag{6-6-16}\delta{P_0}=\frac{G_\text{N}PI_\text{P}/q}{[(\omega_\text{m}^2-\Omega_\text{R}^2-\Gamma_\text{R}^2)^2+4\omega_\text{m}^2\Gamma_\text{R}^2]^{1/2}}\]

and

\[\tag{6-6-17}\theta_\text{P}=\tan^{-1}\left(\frac{2\Gamma_\text{R}\omega_\text{m}}{\omega_\text{m}^2-\Omega_\text{R}^2-\Gamma_\text{R}^2}\right)\]

We find that the photon population (or equivalently the output power) varies sinusoidally with a phase shift \(\theta_\text{P}\) and with the peak amplitude \(\delta{P_0}\). Both \(\delta{P_0}\) and \(\theta_\text{P}\) vary with the modulation frequency \(\omega_\text{m}\).

Figure 6-16 shows the frequency dependence of the modulation response for several values of \(\Gamma_\text{R}\) and \(\Omega_\text{R}\).

The response is relatively flat when \(\omega_\text{m}\ll\Omega_\text{R}\), peaks in the vicinity of \(\Omega_\text{R}\), and then drops sharply for \(\omega_\text{m}\gt\Omega_\text{R}\), indicating that the laser is no longer able to respond at such high modulation speeds.

The peak height decreases with an increase in \(\Gamma_\text{R}\). Equation (6-6-16) can be used to obtain the peak height given by

\[\tag{6-6-18}\frac{\delta{P_0}(\omega_\text{m}=\Omega_\text{R})}{\delta{P_0}(\omega_\text{m}=0)}\approx\frac{\Omega_\text{R}}{2\Gamma_\text{R}}=\frac{\pi\nu_\text{R}}{\Gamma_\text{R}}\]

where we assumed that \(\Gamma_\text{R}\ll\Omega_\text{R}\), and

\[\nu_\text{R}=\Omega_\text{R}/2\pi\]

The peak value of the modulated power in the flat region (\(\omega_\text{m}\ll\Omega_\text{R}\)) can also be obtained using Equation (6-6-16) and is given by the remarkably simple expression

\[\tag{6-6-19}\delta{P_0}=\tau_\text{P}(I_\text{P}/q)\]

where we used Equation (6-4-23) [refer to the transient response of semiconductor lasers tutorial] and \(G\approx\tau_\text{P}^{-1}\). This equation shows that the modulation response is mainly governed by the photon lifetime \(\tau_\text{P}\).

From a practical viewpoint the quantity of interest is the modulation bandwidth \(\nu_\text{B}\), which indicates the frequency range over which the laser responds to the current modulation. It is usually defined as the frequency at which the modulation response has dropped by 3 dB relative to its low-frequency or dc value.

Figure 6-16 suggests that under ideal conditions, \(\nu_\text{B}\) exceeds the relaxation-oscillation frequency, and \(\nu_\text{R}\) provides a reasonable estimate of it.

In practice, however, \(\nu_\text{B}\) may be significantly lower than \(\nu_\text{R}\) if the electrical parasitics associated with a specific device structure lead to a premature roll-off in the modulation response.

This decrease in the response occurs when an increasing fraction of the applied modulation current is bypassed outside the active region with an increase in the modulation frequency.

The parasitic roll-off is of concern for some buried-heterostructure lasers employing a current-blocking junction and requires design improvements so as to reduce the parasitic capacitance.

Figure 6-17 shows the effect of electrical parasitics on the modulation response of a 1.3-μm etched-mesa buried-heterostructure (EMBH) laser (\(I_\text{th}\approx30\text{ mA}\)) for three bias currents.

Even thought the relaxation-oscillation frequency increases with an increase in the bias current, the modulation bandwidth \(\nu_\text{B}\ll\nu_\text{R}\) and actually decreases with a higher biasing of the laser.

To model the effect of electrical parasitics, circuit models that incorporate all the relevant features of the rate equations have been developed. The continuous curves in Figure 6-17 were obtained using one such circuit model and appears to explain the data reasonably well.

As one may expect, the parasitics are structure-dependent and vary from one laser structure to another. In general, weakly index-guided lasers such as those using the ridge waveguide structure suffer less from the parasitic problem.

Recently considerable attention has been paid to increasing the modulation bandwidth of semiconductor lasers. At room temperature (\(T=20^\circ\text{C}\)) the modulation bandwidth of 24 GHz under CW operation was obtained in 1990 for a 1.3-μm InGaAsP laser.

A comparison of Figure 6-16 and 6-17 shows that, apart from the parasitic effects, the height of the observed relaxation-oscillation peak is considerably suppressed in Figure 6-17.

This suggest that the decay rate \(\Gamma_\text{R}\) is larger compared to the value estimated in Table 6-2 [refer to the transient response of semiconductor lasers tutorial] and that it increases with an increase in the bias power.

If we use Equations (6-4-16) and (6-4-17) [refer to the transient response of semiconductor lasers tutorial] to obtain \(\Gamma_\text{R}=(\Gamma_\text{N}+\Gamma_\text{P})/2\), we find that \(\Gamma_\text{N}\) is carrier-dominated and varies little with power.

By contrast, the dominant contribution to \(\Gamma_\text{P}\) comes from the power-dependent gain-suppression term \(G_\text{P}P\). Several mechanisms such as carrier diffusion, carrier heating, and spectral-hole burning may contribute to gain suppression and can be accounted for by increasing the absolute value of \(G_\text{P}\).

Sinusoidal current modulation, although often used, does not cover the practical case where rectangular pulses of duration \(T\) are used to transit the information at a bit rate of \(B=T^{-1}\). If we allow for the finite rise and fall times, a realistic description of a single pulse is provided by

\[\tag{6-6-20}I_\text{m}(t)=I_\text{P}(H(t)[1-\exp(-t/\tau)]-H(t-\tau)\{1-\exp[-(t-T)/\tau]\})\]

where the Heaviside step function \(H(t)=1\) for \(t\ge0\), \(H(t)=0\) for \(t\lt0\), and \(I_\text{P}\) is the peak value of the modulation current.

If we define the rise time \(\tau_\text{r}\) as the time during which the current changes from 10 to 90% of its peak value, \(\tau_\text{r}\) is related to the exponential decay time \(\tau\) by \(\tau_\text{r}=(\ln9)\tau\approx2.2\tau\). For simplicity we have assumed that the rise and fall times are equal. A typical value of \(\tau_\text{r}\) is ~ 100 ps.

Using Equations (6-6-20) and (6-6-12), the Fourier components of the modulation currents are

\[\tag{6-6-21}\tilde{I}_\text{m}(\omega)=I_\text{P}[1-\exp(-\text{i}\omega\tau)]\left(\frac{1}{\text{i}\omega}-\frac{1}{\text{i}\omega+1/\tau}\right)\]

The term proportional to \((\text{i}\omega+1/\tau)^{-1}\) does not contribute when \(\tau=0\) and is responsible for the effects arising from the finite rise and fall times associated with the modulation circuitry.

The modulation response is readily obtained by substituting \(\tilde{I}_\text{m}(\omega)\) from Equation (6-6-21) into Equation (6-6-9) and taking the inverse Fourier transform. The frequency integration can be carried out using the method of contour integration. The result is

\[\tag{6-6-22}\begin{align}\delta{P}(t)=\frac{G_\text{N}PI_\text{P}}{q}&\{H(t)[V(t,0)-V(t,\tau^{-1})]\\&-H(t-T)[V(t-T,0)-V(t-T,\tau^{-1})]\}\end{align}\]

where

\[\tag{6-6-23}V(t,\eta)=\frac{\exp(-\eta{t})}{[\Omega_\text{R}^2+(\Gamma_\text{R}-\eta)^2]}+\frac{\exp(-\Gamma{t})\sin(\Omega_\text{R}t+\theta_\text{P})}{\Omega_\text{R}[\Omega_\text{R}^2+(\Gamma_\text{R}-\eta)^2]^{1/2}}\]

and the phase shift \(\theta_\text{P}\) is given by

\[\tag{6-6-24}\theta_\text{P}=-\tan^{-1}\left(\frac{\Gamma-\eta}{\Omega_\text{R}}\right)\]

Clearly the response to the square-wave modulation has qualitatively different features when compared with the case of sinusoidal modulation.

The physical origin of the four terms in Equation (6-6-22) is self-evident. The terms proportional to \(H(t)\) and the terms proportional to \(H(t-T)\) correspond respectively to the leading and the trailing edges of the current pulse. In each case the term in the form of \(V(t,\tau^{-1})\) represents the effect of a finite rise time and vanishes when \(\tau=0\).

Equation (6-6-23) shows that under square-wave modulation the optical pulse exhibits damped oscillations during turn-on and turn-off whose frequency and damping time are governed by relaxation oscillations. Furthermore, these oscillations are also affected by the current-pulse rise and fall time.

In comparing the results of sinusoidal and square-wave modulations, it should be noted that for an on-off sequence of pulses of duration \(T\), the effective modulation frequency \(\nu_\text{m}=(2T)^{-1}\).

Equation (6-6-16) and (6-6-22) shows that the modulation response of a semiconductor laser is strongly affected by the frequency \(\Omega_\text{R}\) and the damping rate \(\Gamma_\text{R}\) of relaxation oscillations.

The derivation of \(\Omega_\text{R}\) and \(\Gamma_\text{R}\) in the transient response of semiconductor lasers tutorial is based on a truncated Taylor-series expansion of \(G(N,P)\) [see Equation (6-4-18) in the transient response of semiconductor lasers tutorial].

For a more accurate description of modulation response at high powers one should use the exact functional form of \(G(N,P)\). A simple approach assumes, in analogy with a two-level atomic system, that \(G(N,P)\) is given by

\[\tag{6-6-25}G(N,P)=\frac{G_\text{N}(N-N_0)}{1+P/P_\text{s}}\]

where \(P_\text{s}\) is the saturation photon number.

In general, semiconductor lasers cannot be modeled as a two-level system. For a single-mode semiconductor laser the density-matrix equations can be solved approximately to yield the following expression:

\[\tag{6-6-26}G(N,P)=\frac{G_\text{N}(N-N_0)}{\sqrt{1+P/P_\text{s}}}\]

An advantage of the density-matrix approach is that it provides an expression that relates \(P_\text{s}\) to the fundamental material parameters through the relation

\[\tag{6-6-27}P_\text{s}=\frac{\epsilon_0\bar{\mu}\mu_\text{g}V\hbar}{\Gamma\omega_0d^2\tau_\text{in}(\tau_\text{c}+\tau_\text{v})}\]

where \(\epsilon_0\) is the vacuum permittivity, \(d\) is the dipole moment, \(\tau_\text{in}\) is the dipole relaxation time, and \(\tau_\text{c}\) and \(\tau_\text{v}\) are the carrier scattering time for the conduction and valence bands respectively (\(\tau_\text{in}\approx0.1\text{ ps}\), \(\tau_\text{c}=0.2-0.3\text{ ps}\), \(\tau_\text{v}=0.05-0.1\text{ ps}\)).

One can perform the small-signal analysis by using \(G(N,P)\) from Equation (6-6-26) in Equations (6-6-1) and (6-6-2). The modulation response for the case of sinusoidal current modulation is still given by an equation similar to Equation (6-6-16) but with the following expressions for \(\Omega_\text{R}\) and \(\Gamma_\text{R}\):

\[\tag{6-6-28}\Omega_\text{R}=\left[G_\text{L}G_\text{N}P\frac{1+p/2}{(1+p)^2}-\frac{1}{4}(\Gamma_\text{P}-\Gamma_\text{N})^2\right]^{1/2}\]

and

\[\tag{6-6-29}\Gamma_\text{R}=(\Gamma_\text{N}+\Gamma_\text{P})/2\]

\[\tag{6-6-30}\Gamma_\text{N}=\gamma_\text{e}+N\frac{\partial\gamma_\text{e}}{\partial{N}}+\frac{G_\text{N}P}{(1+p)^{1/2}}\]

\[\tag{6-6-31}\Gamma_\text{P}=\frac{R_\text{sp}}{P}+\frac{G_\text{L}}{2}\frac{P}{(1+p)^{3/2}}\]

where \(G_\text{L}=G_\text{N}(N-N_0)\) is the linear part of the gain and \(p\) is a dimensionless parameter defined as \(p=P/P_\text{s}\).

At low operating powers such that \(p\ll1\), \(\Omega_\text{R}\) varies as \(\sqrt{P}\), and Equation (6-6-28) reduces to the result given in Equation (6-4-23) [refer to the transient response of semiconductor lasers tutorial].

At high powers, \(\Omega_\text{R}\) begins to saturate. The damping rate \(\Gamma_\text{R}\) peaks near \(p=2\) and then decreases with a further increase in \(p\).

The quantity of interest from a practical standpoint is not the relaxation-oscillation frequency but the 3-dB bandwidth \(\nu_\text{B}\), defined as the modulation frequency at which the modulation response drops by a factor of 2 from its zero-frequency value. By using Equation (6-6-16), \(\nu_\text{B}\) is found to be related to the relaxation-oscillation parameters by the relation

\[\tag{6-6-32}\nu_\text{B}=\frac{1}{2\pi}\{\Omega_\text{R}^2-\Gamma_\text{R}^2+2[\Omega_\text{R}^2(\Omega_\text{R}^2+\Gamma_\text{R}^2)+\Gamma_\text{R}^4]^{1/2}\}^{1/2}\]

Figure 6-18 shows the variation of \(\nu_\text{B}\) as a function of \(p\) for several values of \(G_\text{N}P_\text{s}\) with \(\tau_\text{p}=1\text{ ps}\).

The modulation bandwidth \(\nu_\text{B}\) saturates at high powers, and the limiting value depends on the value of \(G_\text{N}P_\text{s}\). The limiting value is obtained from (6-6-32) and is given by

\[\tag{6-6-33}\nu_\text{B, max}=(3G_\text{N}P_\text{s}/8\pi^2\tau_\text{p})^{1/2}\]

where \(\Gamma_\text{R}\ll\Omega_\text{R}\) was assumed.

Equations (6-6-33) can be written in terms of the device and material parameters by using \(G_\text{N}=\Gamma{v}_\text{g}a/V\) and \(P_\text{s}\) from Equation (6-6-27). The result is

\[\tag{6-6-34}\nu_\text{B, max}=\sqrt{\frac{3\epsilon_0\hbar\bar{\mu}a}{16\pi^3d^2\tau_\text{p}\tau_\text{in}(\tau_\text{c}+\tau_\text{v})}}\]

The most crucial parameter is the differential gain coefficient \(a\) that depends on the density of states and can be enhanced by using a quantum-well structure.

If we use \(a=2\times10^{-16}\text{ cm}^2\) and typical parameter values for 1.55-μm InGaAsP laser, \(\nu_\text{B, max}\) is found to be about 32 GHz. For a quantum-well laser \(a\) is enhanced by a factor of nearly 2, and \(\nu_\text{B, max}\) should increase by \(\sqrt{2}\) if other parameters remain the same.

The main conclusion is that the nonlinear gain provides a fundamental mechanism that limits the modulation response of semiconductor lasers.

## Frequency Chirping

We have seen that modulation of the device current leads to laser-power modulation that is useful for transmitting information as a sequence of optical pulses.

However, modulation of the current also affects the optical frequency. This is seen clearly in Equation (6-6-8), which emphasizes that carriers influence the optical phase.

If the steady-state frequency \(\nu_0=\omega_0/2\pi\), this frequency shifts by an amount

\[\tag{6-6-35}\delta\nu(t)=\delta\dot{\phi}(t)/2\pi\]

during modulation.

The phenomenon of the dynamic shift of the lasing frequency is referred to as ** frequency chirping** [or wavelength chirping, since \(\delta\lambda=(-\lambda^2/c)\delta\nu\)].

An expression for the chirp \(\delta\nu(t)\) can be obtained using the Fourier relation

\[\tag{6-6-36}\delta\dot{\phi}(t)=\frac{1}{2\pi}\displaystyle\int\limits_{-\infty}^\infty\text{i}\omega\tilde{\phi}(\omega)\exp(\text{i}\omega{t})\text{d}\omega\]

Using Equations (6-6-11), (6-6-35), and (6-6-36), we obtain

\[\tag{6-6-37}\delta\nu(t)=\frac{\beta_\text{c}G_\text{N}}{8\pi^2q}\displaystyle\int\limits_{-\infty}^\infty\frac{[\Gamma_\text{P}+\text{i}\omega]\tilde{I}_\text{m}(\omega)\exp(\text{i}\omega{t})\text{d}\omega}{(\Omega_\text{R}+\omega-\text{i}\Gamma_\text{R})(\Omega_\text{R}-\omega+\text{i}\Gamma_\text{R})}\]

We note that the chirp is directly proportional to the line-width enhancement factor \(\beta_\text{c}\) and has its origin in the carrier-induced index change that accompanies any gain change in semiconductor lasers.

Equation (6-6-37) can be used to obtain the frequency chirp \(\delta\nu(t)\) for an arbitrary time dependence of the modulation current \(I_\text{m}(t)\).

For the case of sinusoidal modulation, \(I_\text{m}(t)\) is given by Equation (6-6-13). The use of Equation (6-6-14) in Equation (6-6-37) then leads to

\[\tag{6-6-38}\delta\nu(t)=\delta\nu_0\sin(\omega_\text{m}t+\theta_\text{c})\]

where

\[\tag{6-6-39}\delta\nu_0=\frac{\beta_\text{c}I_\text{P}G_\text{N}}{4\pi{q}}\left(\frac{\omega_\text{m}^2+\Gamma_\text{P}^2}{(\omega_\text{m}^2-\Omega_\text{R}^2-\Gamma_\text{R}^2)^2+(2\omega_\text{m}\Gamma_\text{R})^2}\right)^{1/2}\]

and

\[\tag{6-6-40}\theta_\text{c}=\tan^{-1}\left(\frac{\omega_\text{m}}{\Gamma_\text{P}}\right)+\tan^{-1}\left(\frac{2\Gamma_\text{R}\omega_\text{m}}{\omega_\text{m}^2-\Omega_\text{R}^2-\Gamma_\text{R}^2}\right)\]

Equation (6-6-38) shows that the mode frequency shifts periodically in response to current modulation and that \(\delta\nu_0\) is the maximum shift. Similar to the power response, \(\delta\nu_0\) peaks at the relaxation-oscillation frequency.

If we use the parameter values given in Tables 6-1 [refer to the rate equations for semiconductor lasers tutorial] and 6-2 [refer to the transient response of semiconductor lasers tutorial] for a typical buried-heterostructure laser, \(\delta\nu_0\approx2\text{ GHz/mA}\) when \(\omega_\text{m}=\Omega_\text{R}/2\). At a given \(\omega_\text{m}\), however, the chirp decreases with an increase in the bias level.

The dependence of the frequency chirp on various parameters under sinusoidal modulation has been investigated, and the experimental results in the small-signal regime are in agreement with the rate-equation analysis.

The case of square-wave modulation can be treated using Equation (6-6-21) in Equation (6-6-37) and is useful to understand the effects of the shape of the current pulse on frequency chirping.

The phenomenon of frequency chirping under current modulation has attracted considerable attention. It manifests as a broadening of the CW line width associated with a single longitudinal mode when the time-averaged optical spectrum is recorded using a spectrometer.

Figure 6-19 shows the experimentally observed spectra of a 1.3-μm InGaAsP laser under sinusoidal modulation, when \(\omega_\text{m}=100\text{ MHz}\), for several values of the modulation current \(I_\text{P}\).

In the absence of modulation (the narrowest spectrum in Figure 6-19) the observed line width is limited by the spectrometer resolution (~ 0.02 nm), and the actual CW line width is typically less than ~ 100 MHz.

Under modulation, the spectrum develops an asymmetric double-peaked profile whose width is in the gigahertz range and increases with an increase in the amplitude \(I_\text{P}\) of the modulation current.

The calculated chirp \(\delta\nu_0\) is related to the width of the dynamically broadened optical spectrum; according to Equation (6-6-39), \(\delta\nu_0\) should vary linearly with \(I_\text{P}\).

Figure 6-20 shows the measured chirp as a function of \(I_\text{P}\) for several laser structures and verifies the predicted linear increase.

We note from Figure 6-20 that the chirp depends on the device structure. To account for the structure-dependent variation, consider the limit where \(\omega_\text{m}\ll\Omega_\text{R}\) in Equation (6-6-39).

If we use \(\Gamma_\text{R}\ll\Omega_\text{R}\), and \(\Omega_\text{R}^2\approx{GG_\text{N}P}\) from Equation (6-4-23) [refer to the transient response of semiconductor lasers tutorial], we obtain the simplified expression

\[\tag{6-6-41}\delta\nu_0=\frac{\beta_\text{c}I_\text{P}}{4\pi{q}GP}(\omega_\text{m}^2+\Gamma_\text{P}^2)^{1/2}\]

The largest chirp in Figure 6-20 occurs for a gain-guided device. This can be understood by noting that \(R_\text{sp}\), and hence \(\Gamma_\text{P}\) [see Equation (6-4-16) in the transient response of semiconductor lasers tutorial], is generally larger for a gain-guided laser.

Using \(G\approx\tau_\text{P}^{-1}\), the chirp at a given bias power \(P\) depends mainly on two device parameters \(\beta_\text{c}\) and \(\tau_\text{P}\).

From Equation (6-2-16) [refer to the rate equations for semiconductor lasers tutorial] the photon lifetime \(\tau_\text{P}\) depends on the internal loss that may vary somewhat with structure. In particular, \(\tau_\text{P}\) may be shorter for ridge waveguide lasers if \(\alpha_\text{int}\) is larger because of diffraction loss.

The line-width enhancement factor \(\beta_\text{c}\) may also vary from device to device because it is strongly wavelength-dependent.

Such variations of \(\beta_\text{c}\) and \(\tau_\text{P}\) can account for the measured chirp differences in index-guided lasers.

It should be noted, however, that in Figure 6-20 the comparison is being made at low modulation frequencies (\(\nu_\text{m}=100\text{ MHz}\)) that are well below the relaxation-oscillation frequency.

At high modulation frequencies in the gigahertz range, weakly index-guided lasers such as the ridge waveguide laser generally show stronger chirping simply because they have lower values of \(\Omega_\text{R}\) at the same bias power.

The frequency chirp also varies with the device wavelength. Figure 6-20 shows the data obtained for 1.3-μm InGaAsP lasers. Similar measurements for 1.55-μm lasers show that the chirp is larger typically by about a factor of 2 when the comparison is made at a constant value of the modulation current and the bias power. This may be attributed in part to the line-width enhancement factor \(\beta_\text{c}\), which may be larger at 1.55-μm.

The frequency chirping of 1.55-μm lasers at high bit rates is of major concern since, in combination with fiber dispersion, chirping degrades system performance considerably.

Several methods have been proposed to reduce the extent of frequency chirping.

One scheme uses injection locking of the modulating laser by a CW semiconductor laser. The external injection provides a mode-pulling mechanism, reducing the dynamic frequency shift or the chirp of the modulating laser.

Another scheme requires a careful tailoring of the current-pulse shape. A small current step in the leading edge of the current pulse is applied to reduce the amplitude of relaxation oscillations, thus reducing the chirp significantly.

These chirp-reduction schemes are particularly useful for single-frequency lasers.

We have seen that the chirp arises from periodic oscillations of the carrier and photon populations in response to modulation of the current. Clearly the power spectrum would be different at different times during one modulation cycle.

Time-resolved power spectra show that the laser frequency briefly shifts upwards during turn-on (leading edge of the current pulse) and downwards during turn-off (trailing edge of the current pulse).

A theoretical description of the time-resolved or the time-averaged spectra such as those shown in Figure 6-19 requires a generalization of the phase noise discussed in the noise characteristics of semiconductor lasers tutorial to include the effect of current modulation. Such a generalization is readily carried out in the small-signal regime.

The time-averaged power spectrum can be calculated using Equation (6-5-43) [refer to the noise characteristics of semiconductor lasers tutorial] and includes the effect of frequency chirping as well as the effect of phase diffusion due to spontaneous emission.

Figure 6-21 shows the calculated power spectra for four modulation currents at a modulation frequency of 1 GHz.

The central mode corresponds to the single longitudinal mode supported by the laser in the absence of modulation.

The effect of modulation is to generate sidebands at multiples of the modulation frequency on both sides of the optical line. These are the well-known FM sidebands and arise since the current modulation leads to both AM and FM.

When the power spectrum is recorded using a spectrometer (see Figure 6-19), the FM sidebands are often unresolved due to a limited resolution and the observed spectrum corresponds to the envelope of the calculated power spectra.

Figure 6-21 shows that the envelope is asymmetric and double-peaked, which agrees with experimental observations. The asymmetry results from the simultaneously occurring AM and FM under current modulation.

The chirp-induced fine structure of the FM sidebands can be observed using an FP interferometer or by increasing the modulation frequency so that their separation exceeds the spectrometer resolution.

## Large-Signal Modulation

The small-signal analysis of the modulation response, although useful for predicting the parameter dependence, is not necessarily valid under practical conditions.

For use in optical communication systems, semiconductor lasers are generally biased below or close to threshold and the modulation depth \(m\) defined by Equation (6-6-5) does not satisfy the validity criterion that \(m\ll1\).

The nonlinear effects then play an important role, and a numerical analysis of the rate equations becomes a necessity. In this section we briefly describe the new qualitative features that may arise under large-signal modulation.

Numerical simulations indicate that, when the modulation depth increases, the power does not vary sinusoidally under sinusoidal modulation. Figure 6-22 shows the optical pulses for several modulation depths obtained for a 1.3-μm buried-heterostructure laser modulated sinusoidally at 1.73 GHz.

The power response is not sinusoidal even at \(m=0.35\). Under large-signal modulation the optical response takes the form of a narrow pulse followed by "ringing," which is related to damped relaxation oscillations.

The main pulse corresponds to the first peak of the relaxation oscillations. As the modulation depth increases, the subsequent peaks are strongly suppressed and the output takes the form of a short optical pulse.

In this way large-signal modulation can be used to generate short optical pulses. This technique is referred to as gain switching and is discussed in the section below.

For lightwave-system applications semiconductor lasers are used to convert a digital electrical bit stream (containing "1" and "0" bits in random order) into an optical replica with as little distortion as possible. The laser is biased close to threshold so as not to turn off the laser completely during "0" bits.

Figure 6-23 shows the optical pulse obtained by integrating the rate equations numerically when a 500-ps rectangular current pulse is applied at \(t=0\) and the laser is biased 10% above threshold.

The optical pulse has finite rise and fall times and exhibits features related to relaxation oscillations. Even though the optical pulse is far from being an exact replica of the electrical pulse, it can be used to transmit information over optical fibers.

Similar to the case of small-signal modulation, power variations in the light output are accompanied by phase variations, resulting in considerable frequency chirping. The dashed curve in Figure 6-23 shows the frequency chirp imposed on the pulse through current modulation.

The laser frequency shifts toward the high-frequency (blue) side near the leading edge, and then shifts back toward the red side near the trailing edge. Such a frequency chirp is harmful for optical communication systems since it leads to considerable broadening of optical pulses during their propagation in the anomalous-dispersion regime of optical fibers.

The physical origin of chirp is related to the amplitude-phase coupling in semiconductor lasers governed by the line-width enhancement factor \(\beta_\text{c}\).

Semiconductor lasers with low values of \(\beta_\text{c}\) are desirable for lightwave-system applications. This parameter is generally lower for multiquantum-well (MQW) lasers. It is reduced further in strained MQW semiconductor lasers.

Considerable attention has been paid to obtaining an analytic expression for the frequency chirp \(\delta\nu_0\) that is valid for large-signal modulation.

Using Equations (6-6-16) and (6-6-39), the chirp for the case of small-signal modulation can be related to the modulation depth (\(m=\delta{P}_0/P\)) by

\[\tag{6-6-42}\delta\nu_0=\frac{1}{2}\beta_\text{c}\nu_\text{m}m\]

where we assumed that \(\omega_\text{m}=2\pi\nu_\text{m}\gg\Gamma_\text{P}\).

Equation (6-6-42) shows that the chirp varies linearly with the modulation depth \(m\) as long as \(m\ll1\). This expression is not applicable when \(m\ge1\).

One way to obtain the instantaneous frequency shift during large-signal modulation is to use the phase equation (6-6-3) with \(\omega_0=\omega_\text{th}\) and note that

\[\tag{6-6-43}\delta\nu(t)=\frac{\dot{\phi}(t)}{2\pi}=\frac{\beta_\text{c}}{4\pi}(G-\gamma)\]

If we now use Equation (6-6-1) to eliminate \(G-\gamma\) and add the contribution of the gain nonlinearities, we obtain

\[\tag{6-6-44}\delta\nu(t)=\frac{\beta_\text{c}}{4\pi}\left(\frac{1}{P}\frac{\partial{P}}{\partial{t}}-\frac{R_\text{sp}}{P}+G_\text{P}P\right)\]

where the last two terms correspond to the dc shift (related to the frequency offset between the on and off power levels during modulation) arising from spontaneous emission and gain suppression.

The first terms corresponds to the dynamic frequency shift or the transient chirp.

Both the transient and dc chirps have been observed experimentally through time-resolved spectral measurements.

Since \(\delta\nu_0\) represents the maximum transient frequency shift, for the case of large-signal modulation it is given by

\[\tag{6-6-45}\delta\nu_0=\frac{\beta_\text{c}}{4\pi}\left(\frac{1}{P}\frac{\partial{P}}{\partial{t}}\right)_\text{max}\]

Equation (6-6-45) can be used to estimate the frequency chirp \(\delta\nu_0\) for specific pulse profiles. Even though the exact value depends on details of the pulse shape, an order-of-magnitude estimate of \(\delta\nu_0\) can be obtained using a Gaussian pulse shape \(\exp(-t^2/T^2)\) and gives \(\delta\nu_0\approx\beta_\text{c}/2\pi{T}\).

In contrast to Equation (6-6-42), \(\delta\nu_0\) becomes independent of the modulation depth under large-signal modulation (\(m\ge1\)).

It has been suggested that the ratio of the frequency chirp to the modulated power is a useful measure of the modulation response of semiconductor lasers. This ratio varies linearly with the modulation frequency.

## Ultrashort Pulse Generation

Direct current modulation of semiconductor lasers generally produces optical pulses whose width is about the same as the applied current pulse.

This technique can produce optical pulses as short as 100 ps by modulating the laser at 10 Gb/s, but it becomes increasingly difficult to obtain shorter pulses simply because current pulses shorter than 100 ps are not readily available.

Many applications of semiconductor lasers require ultrashort optical pulses of width a few picoseconds or shorter. An example is provided by the fifth generation of fiber-optic communication systems based on the concept of optical solitons.

These systems require semiconductor lasers capable of emitting optical pulses of duration 10-20 ps at high repetition rates in the range of 2-10 GHz. The techniques of gain switching and mode locking have been used to generate ultrashort optical pulses.

**Gain Switching**

Gain switching, as the name suggests, consists of switching the optical gain rapidly from a low value to a high value so that the semiconductor laser, biased well below threshold, suddenly finds itself well above threshold.

After an initial turn-on delay [refer to the transient response of semiconductor lasers tutorial] the optical field builds up rapidly during its first relaxation-oscillation cycle, and drives the laser below threshold through the process of gain saturation.

If duration of the current pulse is chosen such that the current is turned off before formation of the second relaxation-oscillation peak, the output consists of a short optical pulse of 10-20 ps duration even though the current pulse may be longer than 100 ps.

By applying the current pulse repetitively, a picosecond pulse train can be generated at a repetition rate as high as a few gigahertz.

Gain switching was studied extensively during the 1980s. Starting in 1988 it was used to demonstrate soliton propagation in optical fibers.

The main qualitative features of gain switching can be understood by solving the single-mode rate equations numerically. Figure 6-24 shows the pulse shape and the frequency chirp for the case in which the laser is biased below threshold (\(I_\text{b}/I_\text{th}=0.8\)) and switched 3 times above threshold by applying a current pulse of 200-ps duration.

The effect of finite rise and fall times associated with the current pulse is included by using a super-Gaussian pulse shape such that

\[\tag{6-6-46}I(t)=I_\text{b}+I_\text{p}\exp[-(t/T_\text{p})^{2T_\text{p}/T_\text{r}}]\]

where the rise time \(T_\text{r}\) was chosen to be 10% of the pulse duration \(T_\text{p}\).

The optical pulse is about 30-ps wide under such conditions.

Numerical simulations show that the pulse width depends sensitively on many operating parameters such as the bias level \(I_\text{b}\), the peak value \(I_\text{p}\), the current-pulse width \(T_\text{p}\), the rise time \(T_\text{r}\), and the photon lifetime \(\tau_\text{p}\).

The last parameter changes with the cavity length and causes the pulse width to depend on the laser length. Pulse widths shorter than 10 ps can be realized by optimizing the operating parameters.

The experimental results support the rate-equation model well. Optical pulses as short as 15 ps were generated in 1981 by using a 60-μm-long semiconductor laser. In a later experiment, the pulse width was reduce to 6.7 ps for a 100-μm laser by optimizing other device parameters.

The use of multiquantum-well (MQW) active region helps to reduce the pulse width even further. In a GaAs MQW laser of 155-μm length the pulse width was reduce to below 1.8 ps.

The physical reason behind pulse shortening for MQW lasers is related to the enhanced value of the differential gain constant \(a\) that leads to relatively large values of the parameter \(G_\text{N}\) in Equation (6-4-26) [refer to the transient response of semiconductor lasers tutorial].

A shortcoming of the gain-switching method is that optical pulses are considerably chirped because of the amplitude-phase coupling governed by the line-width enhancement factor \(\beta_\text{c}\).

This shortcoming, however, can be used to advantage if the pulse is propagated through an optical fiber of optimized length chosen such that it experiences normal (positive) group-velocity dispersion.

The frequency chirp imposed on the pulse during gain switching manifests through a frequency shift of the leading and trailing edges toward blue and red sides, respectively (see Figure 6-24). Since red-shifted frequency components travel faster than blue-shifted components in the normal-dispersion regime, the trailing edge is advanced and the leading edge is retarded during propagation inside the fiber, and the output pulse is compressed.

With suitable optimization, this technique can compress the pulse by a factor of 4-5. Indeed, pulses shorter than 4 ps have been obtained by compressing 15-20 ps pulses commonly obtained by the technique of gain switching.

Compressed optical pulses are also nearly chirp-free and can be used in the design of soliton communication systems operating near 1.5 μm.

The gain-switching technique in combination with a suitable pulse-compression technique can be used to produce pulses as short as 1-2 ps.

In one experiment 17-ps optical pulses were obtained by gain switching a 1.3-μm semiconductor laser. They were compressed to about 2.5 ps by launching them in a 500-m-long dispersion-shifted fiber exhibiting normal dispersion of 32 ps/(km-nm).

The compressed pulse was amplified by using semiconductor laser amplifiers which broadened it to about 5.5 ps. However, the amplified pulse could be compressed to 1.26 ps by passing git through a 1.1-km fiber exhibiting anomalous dispersion at 1.3 μm.

The physical mechanism behind the second state of pulse compression is self-phase modulation occurring inside the amplifier as a result of gain saturation.

Clearly gain switching is a useful technique to generate picosecond pulses from a semiconductor laser.

**Mode Locking**

The mode-locking technique is well known in laser literature and is commonly used to obtain ultrashort optical pulses (as short as 11 fs) by using solid-state and dye lasers.

The basic idea behind mode locking is quite simple.

A large number of longitudinal modes can be excited simultaneously in lasers with a broad gain spectrum. In the absence of a definite phase relationship among different longitudinal modes, the laser operates continuously simply because the interference effects due to mode beating are washed out.

However, if the phase difference between neighboring modes can be forced to lock to a fixed value, beating among the longitudinal modes forces the laser to emit short pulses at a repetition rate equal to the mode spacing (one pulse on every round trip inside the laser cavity).

The gain bandwidth sets the ultimate limit on the pulse width since the pulse spectrum cannot be broader than the gain spectrum.

In practice, pulse width depends not only on the gain bandwidth, but also on many parameters associated with the specific mode-locking technique. Two commonly used mode-locking techniques are known as active and passive depending on whether an external signal at the pulse-repetition rate is applied or not.

Since the gain bandwidth of semiconductor lasers is quite large (about 10 THz) pulses shorter than 100 fs are possible in principle. For this reason, mode locking in semiconductor lasers has been extensively studied by using active, passive , or hybrid techniques.

Active mode-locking of semiconductor lasers requires modulation of the applied current at a frequency equal to the longitudinal-mode spacing. However, because of relatively small cavity length (0.2-0.5 mm) the required frequency (~ 100 GHz) is so large that current sources are not readily available.

A commonly used technique to reduce the modulation frequency is to extend the laser cavity by placing an external mirror or a grating next to an antireflection-coated laser facet.

Such external-cavity semiconductor lasers can be actively mode-locked relatively easily and provide pulse trains at repetition rates of a few GHz depending on the cavity length.

The pulse width can vary over a wide range with typical values in the range 10-50 ps. With a proper design, pulses shorter than 1 ps can be obtained, but they are often accompanied by one or more trailing pulses whose spacing is governed by the round-trip time in the semiconductor-laser cavity. Their presence is related to the residual reflectivity of the laser facet facing the external cavity; reflectivities as small as \(10^{-16}\) are needed to eliminate these trailing satellite pulses.

Passive mode locking generally requires a fast saturable absorber. Subpicosecond optical pulses by this technique were obtained as early as 1981. An MQW device whose carrier lifetime is artificially shortened through creation of defects (e.g. by proton bombardment) is sometimes used as a saturable absorber.

The use of an MQW saturable absorber in the external-cavity configuration together with a grating-pair pulse compressor has provided 0.46-ps mode-locked pulses at 300-MHz repetition rate. The laser current was also modulated at the pulse-repetition rate. Such a scheme is often referred to as hybrid mode locking since it makes use of both active and passive techniques.

For lightwave-system applications it is desirable to use a monolithically integrated version of the external-cavity semiconductor laser. Considerable progress has been made in this direction. Mode-locked pulses as short as 1.2 ps were obtained by using hybrid mode locking of a monolithic device.

The colliding-pulse mode-locking (CPM) technique, similar to that used for dye lasers, has also been implemented in a monolithic implementation of the ring-cavity geometry. It produced in 1990 1.4-ps pulses at a repetition rate of 32.6 GHz. Further refinements of this technique have generated subpicosecond pulses at a repetition rate as high as 300 GHz by using passive mode locking.

Mode-locked semiconductor lasers are commonly used as a pulse source for soliton communication systems. They are likely to find many other applications as a compact source of ultrashort optical pulses at a repetition rate that is much higher than that achieved in solid-state and dye lasers.

The next tutorial discusses about the ** effects of external optical feedback on semiconductor lasers**.