# Semiconductor Laser Characteristics

This is a continuation from the previous tutorial - **semiconductor lasers**.

Similarly to an LED, a semiconductor laser is also a junction diode, which has the general electrical characteristics discussed in the semiconductor junctions tutorial with its voltage-current characteristics shown in Figure 12-12.

The difference between a laser and an LED is that the active layer of a laser has to be pumped sufficiently to reach the condition in (13-35) [refer to the optical gain in semiconductors tutorial] for an optical gain.

When a junction diode is forward biased with a voltage \(V\), the splitting of its Fermi levels is given by (12-93) [refer to the semiconductor junctions tutorial], which is valid for both homojunctions and heterojunctions.

In the active region, \(E_\text{Fc}=E_\text{Fn}\) and \(E_\text{Fv}=E_\text{Fp}\). Therefore, we find that to have a positive optical gain coefficient in the active region, a diode has to be forward biased at a voltage larger than the bandgap of its active layer:

\[\tag{13-108}eV=E_\text{Fc}-E_\text{Fv}\gt{h\nu}\gt{E}_\text{g}\]

This condition only specifies the forward voltage required for the active region in a junction diode to reach transparency. To reach the laser threshold, a laser diode still has to be biased somewhat higher to reach a gain that is sufficiently large for overcoming the losses in the laser cavity. The bias voltage of a semiconductor laser remains quite constant when the laser oscillates above threshold because the carrier density is clamped at its threshold value when the injection current is increased above the laser threshold. In contrast, an LED is normally biased at a lower voltage around \(V\ge{h}\nu/e\).

For most applications, it is desired that a semiconductor laser oscillate in a single transverse mode and a single longitudinal mode. Many practical lasers indeed have such a desirable characteristic. For a single-mode semiconductor laser with a uniformly distributed carrier density in a thin active layer of a thickness \(d\), the temporal characteristics of its carrier density \(N\) and its intracavity photon density \(S\) can be described by the following coupled rate equations:

\[\tag{13-109}\frac{\text{d}N}{\text{d}t}=\frac{J}{ed}-\frac{N}{\tau_\text{s}}-\mathrm{g}S\]

\[\tag{13-110}\frac{\text{d}S}{\text{d}t}=-\gamma_\text{c}S+\Gamma{\mathrm{g}S}\]

where \(e\) is the electronic charge, \(\tau_\text{s}\) is the spontaneous carrier lifetime, and \(\gamma_\text{c}\) is the cavity decay rate.

The current density \(J\) in the active region of a junction area \(\mathcal{A}\) is related to the injection current \(I\) through the relation given in (13-71) [refer to the light-emitting diodes (LEDs) tutorial].

The overlap factor \(\Gamma\) appears in the last term of (13-110) because only that fraction of the laser mode intensity overlaps with the gain region to receive stimulated amplification.

According to (11-68) [refer to the laser power tutorial] and (13-40) [refer to the optical gain in semiconductors tutorial], the gain parameter \(\mathrm{g}\) (per second) of a semiconductor laser is related to the gain coefficient \(g\) (per meter) of the semiconductor gain medium by

\[\tag{13-111}\mathrm{g}=\frac{c}{n}g=\frac{c\sigma}{n}(N-N_\text{tr})\]

for an ordinary DH laser, and by

\[\tag{13-112}\mathrm{g}=\frac{c}{n}g=\frac{c\sigma}{n}N_\text{tr}\ln\frac{N}{N_\text{tr}}\]

for a QW laser.

**Laser Threshold**

The threshold characteristics of a laser and its characteristics in steady-state oscillation above threshold can be obtained by considering the steady-state solutions of (13-109) and (13-110) for \(\text{d}N/\text{d}t=\text{d}S/\text{d}t=0\).

From (13-110), we find that the threshold condition for a semiconductor laser is

\[\tag{13-113}\Gamma\mathrm{g}_\text{th}=\gamma_\text{c}\]

which leads to the following ** threshold carrier density**:

\[\tag{13-114}N_\text{th}=N_\text{tr}+\frac{n\gamma_\text{c}}{\Gamma{c}\sigma}=N_\text{tr}+\frac{g_\text{th}}{\sigma}\]

for an ordinary DH laser and

\[\tag{13-115}N_\text{th}=N_\text{tr}\exp\left(\frac{n\gamma_\text{c}}{\Gamma{c\sigma}N_\text{tr}}\right)=N_\text{tr}\exp\left(\frac{g_\text{th}}{\sigma{N_\text{tr}}}\right)\]

for a QW laser.

From (13-109), we find that the ** threshold current density** is \(J_\text{th}=N_\text{th}ed/\tau_\text{s}\) because \(S=0\) right at the laser threshold. Using this result and the relation between \(J\) and \(I\) in (13-71) [refer to the light-emitting diodes (LEDs) tutorial] with a carrier injection efficiency \(\eta_\text{inj}\), we find the following

**:**

*threshold injection current*\[\tag{13-116}I_\text{th}=\frac{eN_\text{th}}{\eta_\text{inj}\tau_\text{s}}\mathcal{V}_\text{active}\]

*The threshold current of a semiconductor laser is linearly proportional to the threshold carrier density and the volume of its active region.*

There is a limit in decreasing the threshold carrier density to reduce the threshold current because \(N_\text{th}\gt{N}_\text{tr}\) and \(N_\text{tr}\) is an intrinsic property of a semiconductor gain medium.

Reducing \(N_\text{th}\) by increasing the value of \(\Gamma\) does not lead to a lower value for \(I_\text{th}\) because both \(\Gamma\) and \(I_\text{th}\) are proportional to \(\mathcal{V}_\text{active}\). It is only practical to reduce the value of \(\gamma_\text{c}\) as much as possible in order to make \(N_\text{th}\) as close to its limit of \(N_\text{tr}\) as possible.

Therefore, the value of \(N_\text{th}\) does not vary much among properly optimized stripe-geometry lasers and VCSELs.

For a VCSEL, the threshold current can be reduced by reducing its transverse dimension without changing its cavity length because \(\mathcal{V}_\text{active}=\mathcal{A}M_\text{QW}d_\text{QW}\) is independent of the cavity length of a QW VCSEL.

In contrast, for a stripe-geometry laser, reduction of its threshold current is limited by its cavity length because \(\mathcal{V}_\text{active}=\mathcal{A}d=lwd\).

Because the junction area of a VCSEL can be easily made two to three orders of magnitude smaller than that of a typical stripe-geometry laser, the threshold current of a VCSEL can be two to three orders of magnitude less than that of a stripe-geometry laser. The threshold current of a VCSEL can be as low as 1 μA.

*Example 13-18*

A GaAs QW VCSEL like the one described in Example 13-17 [refer to the semiconductor lasers tutorial] has a carrier injection efficiency of \(\eta_\text{inj}=70\%\) and a cross-sectional diameter of 5 μm.

(a) Use the data in Examples 13-7 [refer to the semiconductor junction structures tutorial] and (13-17) [refer to the semiconductor lasers tutorial] to find its threshold carrier density.

(b) Carrier recombination in an efficient laser is almost purely radiative. Take the radiative recombination coefficient of \(B=1.77\times10^{-16}\text{ m}^3\text{ s}^{-1}\) found in Example 13-5 [refer to the spontaneous emission in semiconductor tutorial] for GaAs to find the carrier lifetime at the threshold carrier density.

(c) Find the threshold current for this VCSEL.

From Examples 13-17 [refer to the semiconductor lasers tutorial], we have the following parameters for this VCSEL:

\(g_\text{th}=816\text{ cm}^{-1}=8.16\times10^4\text{ m}^{-1}\), \(M_\text{QW}=3\), and \(d_\text{QW}=8\text{ nm}\).

From Example 13-7(b) [refer to the semiconductor junction structures tutorial], we have

\(\sigma=2.2\times10^{-19}\text{ m}^{-2}\), \(N_\text{tr}=1.45\times10^{24}\text{ m}^{-3}\), and \(\sigma{N}_\text{tr}=3.19\times10^5\text{ m}^{-1}\) for a GaAs quantum well of \(d_\text{QW}=8\text{ nm}\).

**(a)**

By using (13-115) for a QW laser, we find that

\[N_\text{th}=N_\text{tr}\exp\left(\frac{g_\text{th}}{\sigma{N}_\text{tr}}\right)=1.45\times10^{24}\times\exp\left(\frac{8.16\times10^4}{3.19\times10^5}\right)\text{ m}^{-3}=1.87\times10^{24}\text{ m}^{-3}\]

For the purpose of comparison, we use (13-114) to find that \(N_\text{th}=1.77\times10^{24}\text{ m}^{-3}\). This value is very close to the value of \(N_\text{th}=1.87\times10^{24}\text{ m}^{-3}\) found by using the relation in (13-115) for a QW laser. Because \(N_\text{th}-N_\text{tr}\ll{N}_\text{tr}\) in this situation, (13-114) is a fairly good approximation to (13-115).

**(b)**

For \(N=N_\text{th}\gg{n}_0,p_0\) in the situation of purely radiative recombination, we find from (13-8) [refer to the radiative recombination tutorial] that

\[\tau_\text{s}=\tau_\text{rad}=\frac{1}{BN_\text{th}}=\frac{1}{1.77\times10^{-16}\times1.87\times10^{24}}\text{ s}=3.02\text{ ns}\]

**(c)**

With a diameter of 5 μm, the active volume of the three quantum wells is

\[\mathcal{V}_\text{active}=\mathcal{A}M_\text{QW}d_\text{QW}=\pi\times\left(\frac{5\times10^{-6}}{2}\right)^2\times3\times8\times10^{-9}\text{ m}^3=4.71\times10^{-19}\text{ m}^3\]

Therefore, we find the following threshold current for this VCSEL:

\[I_\text{th}=\frac{eN_\text{th}}{\eta_\text{inj}\tau_\text{s}}\mathcal{V}_\text{active}=\frac{1.6\times10^{-19}\times1.87\times10^{24}}{0.7\times3.02\times10^{-9}}\times4.71\times10^{-19}\text{ A}=66.7\text{ μA}\]

As expected, the threshold current of this VCSEL is pretty low.

**Laser Power**

In steady-state oscillation above threshold with an injection current of \(I\gt{I}_\text{th}\), the carrier density and the gain are clamped at their respective threshold values, \(N=N_\text{th}\) and \(\mathrm{g}=\mathrm{g}_\text{th}\) given above, while the intracavity photon density builds up for \(S\ne0\). Most of the concepts developed in the laser power tutorial for laser power characteristics are directly applicable to semiconductor lasers.

Taking the relation in (13-111) for an ordinary DH laser, it can be shown that the threshold gain parameter of a semiconductor laser also has the form of (11-72) [refer to the laser power tutorial] as follows:

\[\tag{13-117}\mathrm{g}_\text{th}=\frac{c\sigma}{n}(N_\text{th}-N_\text{tr})=\frac{c\sigma}{n}\frac{N_\text{inj}-N_\text{tr}}{1+S/S_\text{sat}}\]

where \(N_\text{inj}=J\tau_\text{s}/ed\) is the injected carrier density and \(S_\text{sat}\) is the saturation photon density that has the form of (11-74) [refer to the laser power tutorial]:

\[\tag{13-118}S_\text{sat}=\frac{n}{c\tau_\text{s}\sigma}\]

Comparing (13-118) with (11-74), we find that the spontaneous carrier recombination lifetime \(\tau_\text{s}\) of a semiconductor has exactly the same function as the saturation lifetime of an atomic or molecular system in defining the saturation intensity of a gain medium and the saturation photon density of a laser.

Therefore, the spontaneous carrier recombination lifetime is also the saturation lifetime of a semiconductor, as mentioned following (12-56) [refer to the carrier recombination tutorial] where it is defined.

For a semiconductor laser, the dimensionless pumping ratio \(r\), which is defined in (11-76) [refer to the laser power tutorial], is conveniently expressed in terms of the pump current because the bias voltage of a semiconductor laser remains quite constant:

\[\tag{13-119}r=\frac{I-I_\text{tr}}{I_\text{th}-I_\text{tr}}\]

Using the steady-state solution of (13-109) for \(S\) and the relations in (13-118) and (13-119), the output power of a semiconductor laser can be expressed in the form of (11-78) and (11-87) [refer to the laser power tutorial] as

\[\tag{13-120}P_\text{out}=(r-1)\mathcal{V}_\text{mode}S_\text{sat}h\nu\gamma_\text{out}=\frac{I-I_\text{th}}{I_\text{th}-I_\text{tr}}P_\text{out}^\text{sat}\]

where \(P_\text{out}^\text{sat}=\mathcal{V}_\text{mode}S_\text{sat}h\nu\gamma_\text{out}\) as defined in (11-82) [refer to the laser power tutorial].

This relation is obtained for an ordinary DH laser. A similar, but more complicated, relation can be obtained for a QW laser with the same definitions for \(S_\text{sat}\) and \(r\) in (13-118) and (13-119), respectively.

Alternatively, by applying the relation \(\mathrm{g}_\text{th}=\gamma_\text{c}/\Gamma\) from (13-113) and the relation \(N=J\tau_\text{s}/ed\) directly to the steady-state solution of \(S\) from (13-109), the output power of a semiconductor laser can be expressed as

\[\tag{13-121}P_\text{out}=\eta_\text{inj}\frac{\gamma_\text{out}}{\gamma_\text{c}}\frac{h\nu}{e}(I-I_\text{th})\]

For a laser, we have

\[\tag{13-122}\gamma_\text{out}=\frac{c}{n}\alpha_\text{out}\qquad\text{and}\qquad\gamma_\text{c}=\frac{c}{n}\Gamma{g_\text{th}}\]

These relations in (13-121) and (13-122) are generally applicable to all semiconductor lasers, including DH and QW lasers. All of the following discussions are also generally applicable to all semiconductor lasers.

It can be seen from (13-121) that in an ideal situation, the output power of a semiconductor laser above threshold increases linearly with injection current. This characteristic is indeed observed in most semiconductor lasers over a large range of operating conditions.

In (13-121), both \(\eta_\text{inj}\) and \(I_\text{th}\) are temperature dependent. In general, \(\eta_\text{inj}\) decreases but \(I_\text{th}\) increases as the temperature increases.

In addition, at high injection levels, \(\eta_\text{inj}\) normally becomes current dependent and decreases with increasing current for a given device, resulting in nonlinearities in the \(L-I\) characteristics of a laser. Figure 13-38(a) shows the power-current characteristics of a typical single-mode semiconductor laser.

For a multimode laser, the competition and coexistence of multiple modes can lead to the nonlinearities and kinks, shown in Figure 13-38(b), that are often observed in its \(L-I\) characteristics.

After optimizing the structure of a laser to reduce power losses by maximizing the values of \(\eta_\text{inj}\) and \(\gamma_\text{out}/\gamma_\text{c}\) and by minimizing the value of \(I_\text{th}\), the output power available from a semiconductor laser depends solely on the current that can be injected into the laser.

As discussed in the semiconductor junctions tutorial, there is a limit to the current density \(J\) that can be injected into a junction diode. Further limitation on \(J\) comes from the limitation on the carrier density \(N\) in the active region before high-order nonradiative processes dominate the recombination process.

Because of these limitations on \(J\) and because \(I=J\mathcal{A}/\eta_\text{inj}\), the junction area of a laser sets a limit on the output power that the laser can possibly deliver.

As the junction area of a VCSEL is made very small to reduce its threshold current, it also limits the VCSEL to a low output power in comparison to a stripe-geometry laser that has a much larger junction area than a VCSEL.

**Example 13-19**

Find the output power of the GaAs QW VCSEL considered in Examples 13-17 [refer to the semiconductor lasers tutorial] and 13-18 when it is operated with an injection current at twice the threshold level.

From Example 13-17 [refer to the semiconductor lasers tutorial], we have the following parameters for this VCSEL:

\(\lambda=850\text{ nm}\), \(\Gamma=20\%\), \(g_\text{th}=816\text{ cm}^{-1}\), and \(\alpha_\text{out}=145.2\text{ cm}^{-1}\). We then have

\[\frac{h\nu}{e}=\frac{1239.8}{850}\text{ V}=1.459\text{ V}\]

and, from (13-122),

\[\frac{\gamma_\text{out}}{\gamma_\text{c}}=\frac{\alpha_\text{out}}{\Gamma{g_\text{th}}}=\frac{145.2}{0.2\times816}=0.89\]

The device has \(\eta_\text{inj}=0.7\) and \(I_\text{th}=66.7\text{ μA}\) found in Example 13-18. Thus, when it operates at \(I=2I_\text{th}=133.4\text{ μA}\), the output power

\[\begin{align}P_\text{out}&=\eta_\text{inj}\frac{\gamma_\text{out}}{\gamma_\text{c}}\frac{h\nu}{e}(I-I_\text{th})\\&=0.7\times0.89\times1.459\times(133.4-66.7)\times10^{-6}\text{ W}=60.7\text{ μW}\end{align}\]

Because the injection current is still very low for this output power, much higher output powers can be obtained at higher injection levels.

**Laser Efficiency**

Because the pump power is \(P_\text{p}=VI\) in the case of current injection, the power conversion efficiency of a semiconductor laser is

\[\tag{13-123}\eta_\text{c}=\frac{P_\text{out}}{VI}=\eta_\text{inj}\frac{\gamma_\text{out}}{\gamma_\text{c}}\frac{h\nu}{eV}\left(1-\frac{I_\text{th}}{I}\right)=\eta_\text{e}\frac{h\nu}{eV}\left(1-\frac{I_\text{th}}{I}\right)\]

where \(\eta_\text{e}\) is the external quantum efficiency defined in (13-125) below.

The slope efficiency of a semiconductor laser operating above threshold is

\[\tag{13-124}\eta_\text{s}=\frac{\text{d}P_\text{out}}{V\text{d}I}=\eta_\text{inj}\frac{\gamma_\text{out}}{\gamma_\text{c}}\frac{h\nu}{eV}=\eta_\text{e}\frac{h\nu}{eV}\]

The external quantum efficiency of a semiconductor laser operating above threshold is

\[\tag{13-125}\eta_\text{e}=\frac{P_\text{out}/h\nu}{I/e-I_\text{th}/e}=\eta_\text{inj}\frac{\gamma_\text{out}}{\gamma_\text{c}}\]

Above threshold, the voltage across the junction of a laser diode remains fairly constant because the carrier density is clamped at its threshold value of \(N_\text{th}\). Therefore, if all of the bias voltage drops across the junction, the slope efficiency of the laser is a constant that is independent of the injection level.

In reality, however, there is always some series resistance, which may be added intentionally to protect a laser diode or caused by parasitic effects, in a laser diode. Thus, the bias voltage is always increased by the series resistance as \(V=V_\text{j}+IR_\text{s}\), where \(V_\text{j}\) is the junction voltage.

Clearly, this increase in the bias voltage will reduce both the slope efficiency and the power conversion efficiency of the laser as the injection current increases. Such characteristics are schematically shown in Figure 13-38.

As the additional voltage drop across the resistance is unimportant when \(IR_\text{s}\ll{V}_\text{j}\), it is clear that the efficiencies of a laser can be improved by reducing the series resistance and the laser threshold.

Comparing (13-124) and (13-125), we find that \(\eta_\text{e}\gt\eta_\text{s}\) for a semiconductor laser because \(eV\gt{h}\nu\). In addition, if we identify the photon extraction efficiency of a semiconductor laser as

\[\tag{13-126}\eta_\text{t}=\frac{\gamma_\text{out}}{\gamma_\text{c}}\]

We can express \(\eta_\text{e}\) as

\[\tag{13-127}\eta_\text{e}=\eta_\text{inj}\eta_\text{t}\]

Comparing this result to \(\eta_\text{e}\) of an LED in (13-67) [refer to the light-emitting diodes (LEDs) tutorial], we find that the internal quantum efficiency \(\eta_\text{i}\) of a semiconductor laser is

\[\tag{13-128}\eta_\text{i}=1\]

In practice, \(\eta_\text{i}\) of a semiconductor laser is not exactly 100% but is often higher than 90%. Such a high internal quantum efficiency for a semiconductor laser reflects the fact that almost all of the injected carriers recombine radiatively through the stimulated recombination process when a laser oscillates above threshold.

We see from the above discussions that a semiconductor laser typically has a very high external quantum efficiency, as well as a very high slope efficiency. They can be as high as 80-90% if internal losses and diffraction losses are minimized to make \(\gamma_\text{out}\approx\gamma_\text{c}\) while the injection efficiency \(\eta_\text{inj}\) is maximized.

The power conversion efficiency, however, is normally much lower because of the existence of a laser threshold. A typical laser has a power conversion efficiency of 10-20%. Some lasers have power conversion efficiencies as high as 50%. Clearly, it is important to reduce the laser threshold as much as possible.

**Example 13-20**

Find the various efficiencies of the GaAs QW VCSEL considered in the preceding examples if it has a bias voltage of \(V=2.2\text{ V}\) when operating at an injection level twice the threshold.

From the example 13-19, we find the following external quantum efficiency for this VCSEL:

\[\eta_\text{e}=\eta_\text{inj}\frac{\gamma_\text{out}}{\gamma_\text{c}}=0.7\times0.89=62.3\%\]

The photon extraction efficiency

\[\eta_\text{t}=\frac{\gamma_\text{out}}{\gamma_\text{c}}=89\%\]

The photon extraction efficiency is much higher than the external quantum efficiency because the device suffers \(30\%\) loss of the injected carriers with an injection efficiency of \(\eta_\text{inj}=70\%\). This is where efficiency improvement can be targeted for this device.

The power conversion efficiency of this device operating at \(I=2I_\text{th}\) with a bias voltage of \(V=2.2\text{ V}\) is

\[\eta_\text{c}=\eta_\text{e}\frac{h\nu}{eV}\left(1-\frac{I_\text{th}}{I}\right)=62.3\%\times\frac{1.459}{2.2}\times\left(1-\frac{1}{2}\right)=20.7\%\]

The slope efficiency

\[\eta_\text{s}=\eta_\text{e}\frac{h\nu}{eV}=62.3\%\times\frac{1.459}{2.2}=41.3\%\]

We see that \(\eta_\text{s}\lt\eta_\text{e}\) because \(h\nu\lt{eV}\). This reduction of efficiency cannot be avoided because a bias voltage of 2.2 V, which is significantly higher than the photon energy, is required for the laser to reach the desired level of population inversion.

Of course, any series resistance that can further increase the bias voltage will further reduce the slope efficiency and the power conversion efficiency of the laser.

The power conversion efficiency is only half that of the slope efficiency because the laser is operated at twice its threshold. At a given injection current, \(\eta_\text{c}\) can be increased by reducing the laser threshold. For a laser of a given threshold, \(\eta_\text{c}\) can be increased by operating the laser at a level high above its threshold.

**Laser Spectrum**

A semiconductor laser has the general spectral characteristics of a laser, which are very different from those of an LED. The basic difference between a semiconductor laser and other classes of lasers, such as fiber lasers, is that a semiconductor laser has a very short cavity and a high optical gain. As a result, a semiconductor laser has a larger longitudinal mode spacing and a larger linewidth than most other lasers.

As discussed in the semiconductor lasers tutorial, a VCSEL normally oscillates in a single longitudinal mode because of its large mode spacing.

For a semiconductor laser that has a horizontal or folded cavity, the cavity length is typically in the range of 200 - 500 μm with a corresponding longitudinal mode spacing in the range of 100 - 200 GHz.

Because the gain bandwidth of a semiconductor is typically in the range of 10 - 20 THz and can be as large as 40 THz for a highly pumped QW laser, a multimode semiconductor laser easily oscillates in 10 - 20 longitudinal modes.

The linewidth of each longitudinal mode is typically on the order of 10 MHz, but can be as narrow as 1 MHz or as broad as 100 MHz. The linewidth narrows, but the number of oscillating modes increases, as the laser is injected at a current level high above its threshold.

Figure 13-39(a) shows a typical spectrum of a multimode semiconductor laser.

In many applications, a laser oscillating in a single frequency is desired. There are many different approaches to making a semiconductor laser oscillate in a single longitudinal frequency. Some of the most important and practical concepts are already discussed in the semiconductor lasers tutorial.

They include the use of a very short cavity, as is the case of a VCSEL, and the use of a frequency-selective grating, as is the cases of the DBR laser, the phase-shifted DFB laser, and the GCSEL.

For these single-frequency lasers, the linewidth is still in the typical range of 1 - 100 MHz as mentioned above. It is possible to obtain single-frequency output with a linewidth on the order of 100 kHz or less by injection locking with a narrow-linewidth, single-frequency master laser source or by using a highly frequency-selective external grating as one optical-feedback element.

Figure 13-39(b) shows the spectrum of a single-frequency semiconductor laser. Tuning of the laser frequency, in some cases over a large range close to the entire gain bandwidth of the laser, is possible.

**Modulation Characteristics**

A semiconductor laser can be directly current modulated like an LED. Unlike an LED, however, the modulation speed of a semiconductor laser is not limited by the spontaneous carrier lifetime \(\tau_\text{s}\) in the active region of the laser.

The difference is caused by the fact that there is strong coupling between the carrier density and the intracavity laser field. The effective carrier lifetime in an oscillating laser is much shorter than the spontaneous lifetime because of stimulated recombination in a laser.

The modulation speed of a semiconductor laser is primarily determined by the intracavity photon lifetime and the effective carrier lifetime. Because both the photon lifetime and the effective carrier lifetime of a semiconductor laser are generally much shorter than the spontaneous carrier lifetime, a semiconductor laser has a higher modulation speed than an LED.

Because the stimulated recombination rate increases with the intracavity photon density, the modulation speed of a semiconductor laser increases with laser power.

When a laser is in steady-state oscillation at a bias point with a DC current of \(I_0\gt{I}_\text{th}\) in the absence of modulation, the laser gain and the carreir density are both clamped at their respective threshold values of \(\mathrm{g}_\text{th}\) and \(N_\text{th}\), but the photon density has a value of \(S_0\) corresponding to the laser output power \(P_0\) at the bias point.

Under the dynamical perturbation of a modulation signal, the gain can deviate from \(\mathrm{g}_\text{th}\) due to the variations in the carrier and photon densities caused by the external perturbation. The dependence of the gain parameter on the carrier and photon densities can be expressed as

\[\tag{13-129}\mathrm{g}=\mathrm{g}_\text{th}+\mathrm{g}_\text{n}(N-N_\text{th})+\mathrm{g}_\text{p}(S-S_0)\]

where \(\mathrm{g}_\text{n}=c\sigma/n\) is the ** differential gain parameter** characterizing the dependence of the gain parameter on the carrier density as seen in (13-111) and \(\mathrm{g}_\text{p}\) is the

**characterizing the effect of**

*nonlinear gain parameter***due to the saturation of gain by intracavity photons.**

*gain compression*It has been found empirically that both \(\mathrm{g}_\text{n}\) and \(\mathrm{g}_\text{p}\) stay quite constant over large ranges of carrier density and photon density in a given laser. For most practical purposes, they can be treated as constants over the operating range of a laser. These parameters are normally measured experimentally though they can also be calculated theoretically. Note that \(\mathrm{g}_\text{n}\gt0\) but \(\mathrm{g}_\text{p}\lt0\).

It is convenient to define a ** differential carrier relaxation rate**, \(\gamma_\text{n}\), and a

**, \(\gamma_\text{p}\), as**

*nonlinear carrier relaxation rate*\[\tag{13-130}\gamma_\text{n}=\mathrm{g}_\text{n}S_0,\qquad\gamma_\text{p}=-\Gamma\mathrm{g}_\text{p}S_0\]

In addition, we have the cavity decay rate, \(\gamma_\text{c}=1/\tau_\text{c}\), and the spontaneous carrier relaxation rate, \(\gamma_\text{s}=1/\tau_\text{s}\).

These four relaxation rates, together with the linewidth enhancement factor, \(b\), defined in (13-61) [refer to the lateral structures of semiconductor junctions tutorial], are the intrinsic dynamical parameters of a semiconductor laser that completely determine the dynamical behavior of the laser.

The current-modulation characteristics of a laser, however, are independent of the linewidth enhancement factor but are determined only by the four rate parameters. Note that, for a given laser, \(\gamma_\text{c}\) and \(\gamma_\text{s}\) are constants that are independent of laser power, but \(\gamma_\text{n}\) and \(\gamma_\text{p}\) are linearly proportional to laser power.

Because a semiconductor laser has a threshold, the modulation index \(m\) for a laser that is biased at a DC current \(I_0\gt{I}_\text{th}\) and is modulated at a frequency of \(\Omega=2\pi{f}\) is defined as

\[\tag{13-131}I(t)=I_0+I_1(t)=I_\text{th}+(I_0-I_\text{th})(1+m\cos\Omega{t})\]

which is different from that defined in (13-73) [refer to the light-emitting diodes (LEDs) tutorial] for an LED.

In the regime of linear response, the output power of the laser can be expressed in the same form as that in (13-74) [refer to the light-emitting diodes (LEDs) tutorial]:

\[\tag{13-132}P(t)=P_0+P_1(t)=P_0[1+|r|\cos(\Omega{t}-\varphi)]\]

For small-signal modulation with \(m\ll1\), the complex response function of a laser is

\[\tag{13-133}r(\Omega)=|r|\text{e}^{\text{i}\varphi}=-\frac{m\gamma_\text{c}\gamma_\text{n}}{\Omega^2-\Omega_\text{r}^2+\text{i}\Omega\gamma_\text{r}}\]

where \(\Omega_\text{r}\) is the ** relaxation resonance frequency** and \(\gamma_\text{r}\) is the

**for the relaxation oscillation of the coupling between the carriers and the intracavity laser field in the semiconductor laser. They are related to the intrinsic dynamical parameters of the laser through**

*total carrier relaxation rate*\[\tag{13-134}\Omega_\text{r}^2=4\pi^2f_\text{r}^2=\gamma_\text{c}\gamma_\text{n}+\gamma_\text{s}\gamma_\text{p}\]

and

\[\tag{13-135}\gamma_\text{r}=\gamma_\text{s}+\gamma_\text{n}+\gamma_\text{p}\]

Clearly, \(\Omega_\text{r}\) and \(f_\text{r}\) are proportional to the square root of the laser power, whereas \(\gamma_\text{r}\) is linearly dependent on, but not proportional to, the laser power.

The relation between the relaxation resonance frequency and the carrier relaxation rate is often characterized by a ** \(K\) factor** defined as

\[\tag{13-136}K=\frac{\gamma_\text{r}-\gamma_\text{s}}{f_\text{r}^2}\]

The modulation power spectrum of a semiconductor laser is

\[\tag{13-137}R(f)=|r(f)|^2=\frac{m^2\gamma_\text{c}^2\gamma_\text{n}^2}{16\pi^4(f^2-f_\text{r}^2)^2+4\pi^2{f^2}\gamma_\text{r}^2}\]

As shown in Figure 13-40, this spectrum has a resonance peak at

\[\tag{13-138}f_\text{pk}=\left(f_\text{r}^2-\frac{\gamma_\text{r}^2}{8\pi^2}\right)^{1/2}\]

and a 3-dB modulation bandwidth

\[\tag{13-139}f_\text{3dB}=(1+\sqrt{2})^{1/2}\left(f_\text{r}^2-\frac{\gamma_\text{r}^2}{8\sqrt{2}\pi^2}\right)^{1/2}\approx1.554f_\text{pk}\]

Because \(f_\text{r}\gg\gamma_\text{r}/2\pi\) for most lasers and because \(f_\text{r}\propto{P}_0^{1/2}\), the modulation bandwidth of a semiconductor laser increases with laser power and scales roughly as \(f_\text{3dB}\propto{P}_0^{1/2}\).

An intrinsic modulation bandwidth on the order of a few gigahertz is common for a semiconductor laser. A high-speed semiconductor laser can have a bandwidth larger than 20 GHz.

Because the intrinsic modulation bandwidth of a semiconductor laser is significantly higher than that of an LED, it is very important to reduce the parasitic effects from electrical contacts and packaging for high-frequency modulation of a semiconductor laser.

**Example 13-21**

The GaAs QW VCSEL considered in the preceding examples can be modulated at very high frequencies because it has a very short cavity and is a QW laser, two important factors that lead to the high speed of the device. The data given or found in the preceding examples are sufficient to find all parameters for the modulation characteristics of the laser, with the exception of the parameter \(\mathrm{g}_\text{p}\). Here we simply take \(\mathrm{g}_\text{p}=-\mathrm{g}_\text{n}\).

(a) Find the values of \(\gamma_\text{s}\) and \(\gamma_\text{c}\). What are the corresponding carrier lifetime and photon lifetime?

(b) Find the values of \(\mathrm{g}_\text{n}\) and \(\mathrm{g}_\text{p}\). Then find the values of \(\gamma_\text{n}\) and \(\gamma_\text{p}\).

(c) Find the values of \(f_\text{r}\) and \(\gamma_\text{r}\). What is the value of the \(K\) factor of this device?

(d) Find the resonance peak of the modulation spectrum. What is the 3-dB modulation bandwidth of this VCSEL?

**(a)**

We have \(\tau_\text{s}=3.02\text{ ns}\) found in Example 13-18. Thus

\[\gamma_\text{s}=\frac{1}{\tau_\text{s}}=\frac{1}{3.02\times10^{-9}}\text{ s}^{-1}=3.31\times10^8\text{ s}^{-1}\]

From Example 13-17 [refer to the semiconductor lasers tutorial], we have \(n=3.52\), \(\Gamma=0.2\), and \(g_\text{th}=8.16\times10^4\text{ m}^{-1}\). Thus

\[\gamma_\text{c}=\frac{c}{n}\Gamma{g}_\text{th}=\frac{3\times10^8}{3.52}\times0.2\times8.16\times10^4\text{ s}^{-1}=1.39\times10^{12}\text{ s}^{-1}\]

We already have \(\tau_\text{s}=3.02\text{ ns}\) for the carrier lifetime. The photon lifetime

\[\tau_\text{c}=\frac{1}{\gamma_\text{c}}=\frac{1}{1.39\times10^{12}}\text{ s}=719\text{ fs}\]

This laser has a very small photon lifetime because of its very short cavity.

**(b)**

From Example 13-7 [refer to the semiconductor junction structures tutorial], we find that for GaAs quantum wells, \(\sigma=2.2\times10^{-19}\text{ m}^2\), thus, we find

\[\mathrm{g}_\text{n}=\frac{c\sigma}{n}=\frac{3\times10^8\times2.2\times10^{-19}}{3.52}\text{ m}^3\text{ s}^{-1}=1.875\times10^{-11}\text{ m}^3\text{ s}^{-1}\]

Based on the assumption we have made, \(\mathrm{g}_\text{p}=-\mathrm{g}_\text{n}=-1.875\times10^{-11}\text{ m}^3\text{ s}^{-1}\).

To find the values of \(\gamma_\text{n}\) and \(\gamma_\text{p}\), we need to find the intracavity photon density \(S_0\) at the operating point. We have \(P_\text{out}=60.6\text{ μW}\) and \(h\nu=1.459\text{ eV}\), both found in Example 13-19. To find \(S_0\), we also need the following two parameters:

\[\mathcal{V}_\text{mode}\approx{\mathcal{A}l}=\pi\times\left(\frac{5\times10^{-6}}{2}\right)^2\times241.5\times10^{-9}\text{ m}^3=4.74\times10^{-18}\text{ m}^3\]

for \(l=241.5\text{ nm}\) found in Example 13-17 [refer to the semiconductor lasers tutorial], and, from Example 13-20,

\[\gamma_\text{out}=\eta_\text{t}\gamma_\text{c}=89\%\times1.39\times10^{12}\text{ s}^{-1}=1.24\times10^{12}\text{ s}^{-1}\]

Then, the intracavity photon density can be found as

\[\begin{align}S_0&=\frac{P_\text{out}}{\mathcal{V}_\text{mode}h\nu\gamma_\text{out}}\\&=\frac{60.6\times10^{-6}}{4.74\times10^{-18}\times1.459\times1.6\times10^{-19}\times1.24\times10^{12}}\text{ m}^{-3}\\&=4.42\times10^{19}\text{ m}^{-3}\end{align}\]

We then find that

\[\gamma_\text{n}=\mathrm{g}_\text{n}S_0=1.875\times10^{-11}\times4.42\times10^{19}\text{ s}^{-1}=8.29\times10^8\text{ s}^{-1}\]

and

\[\gamma_\text{p}=-\Gamma\mathrm{g}_\text{p}S_0=0.2\times1.875\times10^{-11}\times4.42\times10^{19}\text{ s}^{-1}=1.66\times10^8\text{ s}^{-1}\]

**(c)**

We now find that

\[\begin{align}f_\text{r}&=\frac{(\gamma_\text{c}\gamma_\text{n}+\gamma_\text{s}\gamma_\text{p})^{1/2}}{2\pi}\\&=\frac{(1.39\times10^{12}\times8.29\times10^8+3.31\times10^8\times1.66\times10^8)^{1/2}}{2\pi}\text{ Hz}\\&=5.403\text{ GHz}\end{align}\]

and

\[\gamma_\text{r}=\gamma_\text{s}+\gamma_\text{n}+\gamma_\text{p}=(3.31\times10^8+8.29\times10^8+1.66\times10^8)\text{ s}^{-1}=1.33\times10^9\text{ s}^{-1}\]

The value of the \(K\) factor is

\[K=\frac{\gamma_\text{r}-\gamma_\text{s}}{f_\text{r}^2}=\frac{1.33\times10^9-3.31\times10^8}{(5.403\times10^9)^2}\text{ s}=34.2\text{ ps}\]

**(d)**

The resonance peak of the modulation spectrum is

\[f_\text{pk}=\left(f_\text{r}^2-\frac{\gamma_\text{r}^2}{8\pi^2}\right)^{1/2}=\left(5.403^2-\frac{1.33^2}{8\pi^2}\right)^{1/2}\text{ GHz}=5.401\text{ GHz}\]

We see that \(f_\text{pk}\) is very close to \(f_\text{r}\) but slightly lower. We now find the following 3-dB modulation bandwidth of this VCSEL:

\[\begin{align}f_\text{3dB}&=(1+\sqrt{2})^{1/2}\left(f_\text{r}^2-\frac{\gamma_\text{r}^2}{8\sqrt{2}\pi^2}\right)^{1/2}\\&=(1+\sqrt{2})^{1/2}\left(5.403^2-\frac{1.33^2}{8\sqrt{2}\pi^2}\right)^{1/2}\text{ GHz}\\&=8.39\text{ GHz}\end{align}\]

This VCSEL indeed has a large modulation bandwidth, as expected.

The next tutorial covers the topic of **photodetector noise**.