Rate equations for laser operation
Rate equations for laser operation
Abstract and Keywords
Rate equations for the electron and photon populations provide insight into the variation of light output with current and the highfrequency modulation response of the laser. Above threshold the average photon lifetime exceeds the cavity roundtrip time. The steadystate solution of these equations shows that the gain never becomes exactly equal to the optical loss, as assumed in the traditional description of threshold, the small difference being made up by the spontaneous emission coupled into the lasing mode given by the spontaneous emission factor. Steadystate solutions are also obtained for quantum dot lasers in the random population regime, where quasiequilibrium is not established, by solving electron, photon, and phonon coupled rate equations. Rate equations also give the small signal modulation response of the laser and the resonance frequency, including the influence of carrier transport in quantum well devices. The chapter ends with remarks on what constitutes evidence for laser action.
Keywords: rate equation, photon lifetime, roundtrip time, spontaneous emission factor, quantum dot, random population, resonance frequency, carrier transport
Calculation of the intrinsic threshold current through the gain–current relation is based on a definition of threshold in Section 5.2 in which current due to stimulated emission is neglected by assuming the photon density at threshold is negligible. The cavity enters simply as an optical loss and it is not necessary to know the photon density. This approach gives no information about operation above threshold such as the light output or the differential quantum efficiency. To do this it is necessary to take account of the everincreasing photon density above threshold and include the coupling between carriers and photons by stimulated emission. This is achieved by formulating coupled rate equations for the carrier and photon populations.
In this chapter these rate equations are solved in the steady state to obtain the light–current characteristic. Steadystate solutions are also obtained for quantum dot lasers in the random regime where quasiequilibrium is not established. An important application of these equations is the study of the small signal modulation response of the laser, and the influence of carrier transport in quantum confined devices. The chapter ends with remarks on what constitutes evidence for laser action.
13.1 Formulation of the photon and carrier rate equations
We consider a slab waveguide with a quantum confined gain region (wells or dots) lying in the $(x,y)$ plane and with the optical mode propagating in the y direction, as illustrated in Fig. 13.1. The carrier and photon distributions in the z direction have areal densities n and ${N}_{\text{ph}}$ respectively in the $(x,y)$ plane, the latter obtained by integration over the mode profile. The overlap between them in the z direction is contained within the modal gain G(n), which is a function of carrier density. The carrier scattering rates into the well and within the well are sufficiently fast to maintain a global quasiequilibrium.
Modal gain is the fractional increase in energy in the mode per unit distance (Section 2.1.2); therefore, for a monochromatic beam of photon (p.216) density ${N}_{\text{ph}}$, the increase in photon density per unit distance is $G(n){N}_{\text{ph}}$. The rate of increase per unit time is
where ${v}_{\text{g}}$ is the group velocity.^{1} This is the net stimulated emission rate due to the net effects of stimulated emission and absorption: ${R}_{\text{stim}}^{\text{net}}$ is equal to $[\text{d}{N}_{\text{ph}}/\text{d}t{]}^{\text{net}}$, so
The product ${v}_{\text{g}}G(n)$ has dimensions [T]^{−1} and is the average net stimulated rate per photon, $1/{\mathrm{\tau}}_{\text{ph}}^{\text{stim}}$. Since the gain depends on the carrier density, eqn 13.1 expresses the coupling of photons and carriers.
Carriers
A current density J is driven into the top contact and of this a fraction ${\eta}_{\text{inj}}$ enters the gain medium (Section 5.5.1), as illustrated in Fig. 13.1. The injection efficiency and the fraction of injected current lost by current spreading are not the same below and above threshold (Section 5.5.2) and their relation depends upon the device structure. We therefore formulate the rate equation for the carrier density in terms of the current entering the well, ${J}_{\text{well}}$. Electrons are lost from the active region by recombination by net stimulated emission (eqn 13.2), recombination producing spontaneous emission, ${R}_{\text{spon}}(n)$, and nonradiative recombination, ${R}_{\text{nr}}(n)$. The competition between these rates determines the carrier density in the well represented by the reservoir, illustrated in Fig. 13.2.
The net rate of change of the area density of electrons in the gain region is therefore
(p.217) Below transparency the net stimulated rate is negative and electrons are added to the system by absorption of spontaneous emission.
Photons
Photons are lost from the mode by internal scattering, specified by the internal mode loss ${\mathrm{\alpha}}_{\text{i}}$, and from the end mirrors, specified by the distributed mirror loss ${\mathrm{\alpha}}_{\text{m}}$.^{2} We define a photon lifetimethat represents the average rate at which photons are lost from the mode and which is a property of the cavity (eqn 5.12):
This is the loss in the absence of any amplification, sometimes called a “cold cavity”. When photons are added by amplification their average lifetime in the cavity is increased.
Photons are added to the mode by stimulated emission and by the fraction of spontaneous emission, ${\beta}_{\text{spon}}$, that enters the mode. The rate equation for the photon density is
By writing these equations in terms of densities per unit area and modal gain, the “thickness” of the gain layer and volume of the mode are not required. The overlap of the mode with the gain region is taken into account in the modal gain.
The treatment here considers all the laser emission to be in a single mode. This is a simplification and for a full understanding of the laser (p.218) emission spectrum it is necessary to formulate multimode equations that are themselves coupled through the homogeneous linewidth.
In the following sections we examine the steadystate and small signal modulation solutions of eqns 13.3 and 13.5.
13.2 Steadystate solutions
13.2.1 The steadystate equations
For a given current density the light output in the steady state is calculated by setting the time derivatives of eqns 13.3 and 13.5 to zero. Then eqn 13.5 can be written
and eqn 13.3 becomes
The modal gain G(n) and spontaneous recombination rate ${R}_{\text{spon}}(n)$ are given by calculations described in Chapter 9 (dots) and Chapters 11 and 12 (wells).^{3} An appropriate model must be adopted to calculate the nonradiative recombination rate and usually this is written in terms of a nonradiative lifetime ${\mathrm{\tau}}_{\text{nr}}$ (Section 15.3.1 and eqn E.29):
though a value for ${\mathrm{\tau}}_{\text{nr}}$ may not be known.
Starting with a value for n, G, ${R}_{\text{spon}}$, and ${R}_{\text{nr}}$ are determined and ${N}_{\text{ph}}$ is given by eqn 13.6; eqn 13.7 gives the current density ${J}_{\text{well}}$.
The external light output from one end mirror can be calculated as follows. The total number of photons in the lasing mode is ${N}_{\text{ph}}{L}_{\text{c}}w$, where w is the width of the gain region, and from the definition of photon lifetime (eqn 13.4) the rate at which photons are lost from the mode is ${N}_{\text{ph}}{L}_{\text{c}}w/{\mathrm{\tau}}_{\text{ph}}^{0}={N}_{\text{ph}}{L}_{\text{c}}w{v}_{\text{g}}({\mathrm{\alpha}}_{\text{i}}+{\mathrm{\alpha}}_{\text{m}})$. Of this loss, a fraction ${\mathrm{\alpha}}_{\text{m}}/({\mathrm{\alpha}}_{\text{m}}+{\mathrm{\alpha}}_{\text{i}})$ is lost through the mirrors (eqn 5.24). The power emittedfrom one mirror (assuming ${R}_{1}={R}_{2}$) is therefore
where we have used eqns 13.4 and 5.12. It is assumed that there are no transmission losses in the mirrors.
13.2.2 Light–current curve
Figure 13.3 shows a light–current curve calculated with these equations using typical relations for G(n) and ${R}_{\text{spon}}(n)$ as functions of carrier (p.219) density in a quantum well. We illustrate the general principles with numerical results for a specific example. The cavity loss is 53.2 cm^{−1}, ${\mathrm{\alpha}}_{\text{i}}=5$ cm^{−1}, nonradiative recombination in the well is included.
Extrapolation of this curve to zero output gives a thresholdcurrent density of 410 A cm^{−2} compared with a radiative threshold current density of 250 A cm^{−2} obtained from the input data, giving an internal spontaneous quantum efficiency at threshold of about 0.55 (eqn 5.21).
The external differential efficiency (eqn 5.26) from the slope above threshold is 0.91 and, using the extraction factor (eqn 5.24), the internal differential efficiency ${\eta}_{\text{int}}^{\text{d}}$ is found to be unity, as should occur when the Fermi levels pin above threshold (Section 5.2.4).^{4}
Figure 13.4 shows the calculated emission rates into the lasermode due to spontaneous emission and net induced emission, and their total produces the light output. Initially the net induced rate ${R}_{\text{stim}}^{\text{net}}$ is negative: this is absorption of photons when the population is not inverted. It becomes positive above transparency. The spontaneous emission rate increases with current as the carrier density builds up, then pins just above threshold owing to the rapid increase in photon density causing an increase in the stimulated rate (Section 5.2.4). Figure 13.4 also shows that gain is first produced at a transparency current of about 240 A cm^{−2}, compared with the threshold current density of 410 A cm^{−2}. The extra current generates the gain to overcome the optical loss (Fig. 5.9).^{5}
13.2.3 Spontaneous emission and threshold
The threshold condition derived in Section 5.2.1 is that the threshold gain is equal to the cavity losses (eqn 5.13). However, close scrutiny of the rate equation results shows that the gain never reaches a value exactly equal to the optical loss (Exercise 13.3): if it did so, eqn 13.6 shows (p.220) that the photon density would be infinitely large. At twice the threshold current the gain is 53.156 cm^{−1}, compared with a loss of 53.158 cm^{−1}. The difference is due to the contribution of spontaneous emission to the photons in the mode (eqn 13.5), as can be seen in Fig. 13.4. The difference between the gain at the upturn of the L–I curve and optical loss is sufficiently small that for most purposes the definition of threshold as the pumping rate when they are equal is satisfactory.
13.2.4 The influence of ${\beta}_{\text{spon}}$
The form of the L–I curve near threshold is influenced by the fraction of spontaneous emission that is coupled into the laser mode. Figure 13.5 shows that the higher the value of ${\beta}_{\text{spon}}$ the more gradual is the upturn of light output near threshold.
The influence of ${\beta}_{\text{spon}}$ is shown more clearly in Figure 13.6, which is a log–log plot of Fig. 13.3, with straight lines through the the data below and above threshold. At high current above threshold the calculated L–I curve is linear with a slope of one extra photon out for one extra electron in; a fraction ${F}_{\text{ext}}=0.91$ of these appear externally.
Below threshold the light in the mode is due to a fraction ${\beta}_{\text{spon}}$ of the total spontaneous emission rate, of which the same fraction, 0.91, appears externally. As the current increases, the spontaneous emission rate ($\approx {n}^{2}$) increases faster than the nonradiative rate ($\approx n$), so the spontaneous quantum efficiency increases with current, causing the L–I curve to be superlinear, as manifest in the slope of the log–log plot.
By extrapolating the two portions of the L–I curve to threshold it is possible to compare the spontaneous and stimulated rates in the mode at the same carrier density. All stimulated emission above threshold enters the lasing mode, which is equivalent to a β of unity, whereas spontaneous emission is “stimulated” into all freespace modes. Therefore the ratio of the intercepts of the L–I extrapolations on a vertical line at threshold is approximately equal to the fraction of spontaneous emission that enters the laser mode, ${\beta}_{\text{spon}}$. As the value of ${\beta}_{\text{spon}}$ is increased, this ratio decreases, and if ${\beta}_{\text{spon}}$ could be made equal to unity, the L–I curve would be linear, though there would still be a “threshold” for production of coherent stimulated emission.
13.2.5 Cavity roundtrip time
The time for a single round trip of the cavity is $2{L}_{\text{c}}\times (n/c)$, which for the structure modelled here is $5.8\times {10}^{12}$ s. The photon lifetime ${\mathrm{\tau}}_{\text{ph}}^{0}$, which is a measure of the optical loss of the cavity (eqn 13.4), is about $2.2\times {10}^{12}$ s, so, in the absence of gain, on average the photons are unable to complete a round trip. Above threshold the average time for net loss of photons from the cavity is increased owing to addition of photons by stimulated emission, and photons make many round trips, thereby maintaining the coherence of the emission (Exercise 13.2).
(p.221) 13.2.6 Intraband relaxation
It is implicit in the rate equations in Section 13.1 that quasiequilibrium is maintained in the upper and lower bands even above threshold. It is therefore assumed that the carrier–carrier intraband relaxation processes are much faster than the sum of recombination rates between bands.^{6} The stimulated rate per carrier (eqn 13.2) is about 10^{11} s^{−1} at 10 × threshold whereas intraband relaxation rates are of order 10^{12} s^{−1}, which is much faster. However, at very high currents the photon density builds up and the stimulated recombination rate is very large, suppressing the carrier density. Rapid intraband relaxation is responsible for pinning the global Fermi levels and carrier density above threshold, as shown in Fig. 13.4. Laser emission occurs over a narrow band, but nevertheless rapid relaxation of carriers between higher states resupplies the lasing process and maintains the whole carrier distribution in quasiequilibrium.
Most aspects of laser diode operation considered in this book fall into the regime where intraband relaxation is sufficiently rapid that quasiequilibrium is maintained. The exception is quantum dots at low temperature, which is examined in the next section.
13.3 Laser operation of quantum dots
13.3.1 Quasiequilibrium
The L–I characteristic of quantum dot lasers can be calculated using rate equations as described in Section 13.2. In quasiequilibrium the modal gain and spontaneous recombination rate as functions of the dot carrier density can be calculated for an inhomogeneous distribution of dots as described in Chapter 9 using Fermi functions. It is shown in Section 10.3.2 that the primary requirement for this to be so is that the emission rate from dot states to the wetting layer is much faster the recombination rate. This is usually the case at room temperature.
13.3.2 Rate equations for a nonthermal carrier distribution
At low temperature the emission rate to the wetting layer is so slow that quasiequilibrium is not established and Fermi statistics cannot be used; instead the occupation probability is obtained from the rate equations for the dot and wetting layer system interacting with a thermal distribution of phonons, as described in Chapter 10. The net stimulated rate is incorporated as a carrier loss in eqn 10.8:
(p.222) where ${R}^{\downarrow}$ and ${R}^{\uparrow}$ are the phononinduced rates of capture and emission of electrons in the wetting layer at the ground state. This is combined with the photon rate equation (eqn 13.5)
using the gain for the dot system as a function of carrier density to give ${R}_{\text{stim}}^{\text{net}}$ from eqn 13.2. Figure 13.7 shows Fig. 13.2 modified to show the exchange of carriers between the dots and wetting layer. These equations are solved in the steady state, first for identical dots then for an inhomogeneous distribution.
Identical dots
Consider the ground states of identical dots occupied by electron–hole pairs (Section 10.2). The average occupation probability of the conduction state at energy ${E}_{\text{g}}$ is ${f}_{\text{g}}$ and the hole occupation is also equal to ${f}_{\text{g}}$, so the electron occupation of the valence state is $1{f}_{\text{g}}$ and $f({E}_{\text{c}})f({E}_{\text{v}})=2{f}_{\text{g}}1$. The modal gain for a single ground state transition is (eqn 9.18)
where σ is the optical cross section of the dot. The spontaneous recombination rate is
(p.223) Therefore, starting with a chosen value for ${f}_{\text{g}}$, these equations give $G({f}_{\text{g}})$ and ${R}_{\text{spon}}({f}_{\text{g}})$; the photon density is given by eqn 13.6, from which the light output can be calculated (eqn 13.9). Since the recombination terms are known, the current can be calculated.
The continuous line in Fig. 13.8 shows the L–I curve for identical dots in a cavity 1.2 mm long at 300 K, where occupation is thermal. The cavity loss is not temperaturedependent and since we are considering the ground state only of a system of identical dots without homogeneous broadening, the threshold current is not temperaturedependent, and this is confirmed by the calculation at 10 K also shown in the figure. Random occupation at 10 K has no consequences for identical dots.
The dot occupation probability ${f}_{\text{g}}$ as a function of current is shown in Fig. 13.9. At both 300 and 10 K ${f}_{\text{g}}$ is clamped above threshold at the value necessary to produce the threshold gain.
The wetting layer population is given by eqn 10.20 with the occupation factor of the wetting layer, ${f}_{\text{w}}$, to maintain the required ${f}_{\text{g}}$. This is achieved through the relative capture and emission rates at the dot state, ${R}^{\downarrow}$ and ${R}^{\uparrow}$, controlled by ${f}_{\text{w}}$. Figure 13.9 shows that at 300 K the carrier density in the wetting layer increases with current, then clamps above threshold: this is because at 300 K the emission rate of electrons from dots to wetting layer is much faster than the stimulated recombination rate, so the dots and wetting layer are in quasiequilibrium with a common Fermi level (Section 10.3.2).
At 10 K the dot occupation ${f}_{\text{g}}$ is also clamped above threshold to give a gain equal to the cavity loss. However, the emission rate to the wetting layer is very slow and the dot occupation is given by the ratio of the capture rate to the recombination rate (eqn 10.19) with ${\mathrm{\tau}}_{\text{spon}}$ replaced by the total recombination rate per carrier, $1/{\mathrm{\tau}}_{\text{rec}}=1/{\mathrm{\tau}}_{\text{stim}}+1/{\mathrm{\tau}}_{\text{spon}}$:
(p.224) As the stimulated rate increases above threshold the capture rate must increase to maintain the value of ${f}_{\text{g}}$ required to produce the threshold gain and this is achieved by increasing the wetting layer carrier population. Consequently at 10 K the wetting layer population is not pinned above threshold: the system is not in quasiequilibrium.
Random population of an inhomogeneous dot distribution
This treatment can be extended to an inhomogeneous distribution of dots in the lowtemperature regime, and illustrative results are given for a distribution of ground states with a standard deviation of 20 meV in a cavity with an optical loss of 15 cm^{−1}.
As the wetting layer population ${f}_{\text{w}}{N}_{\text{w}}$ is increased, ${\mathrm{\tau}}_{\text{cap}}$ decreases for all dots (eqn 10.5) and ${f}_{\text{g}}$ rises. The dot occupation is given by eqn 10.19 and below threshold ${\mathrm{\tau}}_{\text{rec}}={\mathrm{\tau}}_{\text{spon}}$ and ${f}_{\text{g}}$ is the same for all ground states. Consequently the gain spectrum follows the inhomogeneous distribution and laser action is first initiated at the peak of this distribution, as illustrated in Fig. 13.10 The laser emission is a narrow line.
The gain is controlled by ${f}_{\text{g}}$, whereas the stimulated recombination rate is also proportional to the photon density (eqn 13.2). As the wetting layer carrier density is increased further and the photon density increases, the capture rate increases to supply the increasing stimulated rate at lasing dots, thereby maintaining their occupation probability. The gain produced by these dots matches the optical loss of the cavity.
The capture rate at dots that do not contribute to lasing also increases, because they are fed from the wetting layer, and since recombination at these dots is due only to spontaneous emission, their occupation rises with increasing wetting layer carrier density. Eventually in these dots ${f}_{\text{g}}$ is sufficient to produce gain to match the loss and the gain spectrum (and emission spectrum) broadens, clamped at the cavity loss.
(p.225) This is illustrated in Fig. 13.11, which shows gain spectra and occupation probability distributions for increasing drive current above threshold. The gain of those dots that contribute to laser action is clamped at the cavity loss and their occupation probability increases with energy away from the peak to compensate for the decreasing dot density. Their occupation does not change once lasing is initiated. The occupation of dots that are not lasing increases with increasing wetting layer carrier density until their gain matches the optical loss, when their occupation is then fixed. The carriers required to invert more dots are provided by the external current and not by depleting the occupation of other dots.
In contrast to dots in thermal equilibrium, the laser emission spectrum broadens with increasing current above threshold, as is observed experimentally, for example by Sugawara et al. 2000.
True random population occurs only at low temperatures; however, once the stimulated emission rate of a group of dots exceeds the emission rate to the wetting layer, the system departs from quasiequilibrium. In such situations clamping of the wetting layer population is not complete, with implications for the external differential efficiency.
13.4 Small signal modulation
Many practical applications of laser diodes, particularly for transmission and storage of digital information, require the light output to be modulated, usually as quickly as possible! The most direct way to do this is by modulating the drive current; however, the modulation is not applied directly to the active region where the light is produced, but to a pair of contacts from which the electrons and holes are transported though the structure to the gain layer. In quantum well and quantum (p.226) dot lasers carrier transport through the barrier alongside the well influences the modulation response. The modulation in current produces a modulation in the carrier density, which modulates the effective index. In order to satisfy the phase condition for laser action (eqn 5.16) this brings about small changes in the laser wavelength over each cycle of the modulation, known as chirp, increasing the linewidth. This presents a serious difficulty for direct modulation of diode lasers in optical fibre communication systems, which has led to the development of integrated modulators.
Rate equations are the means by which these matters are analysed and the topic is covered in many textbooks. Here we give a short summary of the key results and then consider the influence of carrier transport in quantum well lasers. This gives further insight into relaxation oscillations and resonance frequency introduced in Section 5.2.4.
These concepts also apply to quantum dot lasers, though an explicit account of the modulation response of these devices is not given here.
13.4.1 Small signal equations
Through the rate equations, we analyse the light output ${P}_{L\text{out}}$ in response to a small modulation of the drive current, $\mathrm{\Delta}J(t)$, about an average value J_{0}: $J(t)={J}_{0}+\mathrm{\Delta}J(t)$, as illustrated in Fig. 13.12. This produces modulations^{7} in the carrier density $n(t)={n}_{0}+\mathrm{\Delta}n(t)$ and in the gain $G(t)={G}_{0}+{G}^{\prime}({n}_{0})\mathrm{\Delta}n(t)$, where G_{0} is the gain at n_{0} and ${G}^{\prime}({n}_{0})$ is the change in gain for a change in carrier density at the carrier density n_{0}, known as the differential gain. This produces a modulation in the photon density ${N}_{\text{ph}}(t)={N}_{\text{ph}0}+\mathrm{\Delta}{N}_{\text{ph}}(t)$ and light output, $\mathrm{\Delta}{P}_{L\text{out}}$. We seek the modulation response: the ratio of the amplitude of the modulated output to the amplitude of the current modulation as a function of the modulation frequency. Small signal modulation implies all the changes are linear with respect to the change in carrier density.
To ease the notation we combine the spontaneous and nonradiative carrier recombination into a single carrier lifetime ${\mathrm{\tau}}_{n}(n)$:
Although the lifetime is dependent upon carrier density, for calculation of the small signal response n is taken to be constant over the amplitude of the modulation and ${\mathrm{\tau}}_{n}(n)$ in this equation is the differential lifetime introduced in Section E.2.2. The value of ${\mathrm{\tau}}_{n}(n)$ may be an input parameter or calculated for assumed radiative and nonradiative processes.
The values n_{0} and ${N}_{\text{ph}0}$ correspond to the steadystate solutions at the current density J_{0} and are related by eqns 13.6 and 13.7.
When the timedependent quantities are substituted into the rate equations the steadystate terms cancel, leaving the timedependent (p.227) terms. Writing ${R}_{\text{stim}}^{\text{net}}$ in terms of the gain (eqn 13.2), eqn 13.3 becomes
and eqn 13.5 becomes
where it is understood that ${G}^{\prime}$ is evaluated at n_{0}. There are two contributions to the modulation in the stimulated emission rate: modulations in carrier density and photon density.
13.4.2 Modulation response
The modulations are represented by a sinusoidal time dependence of the form $\mathrm{\Delta}n(t)=\mathrm{\Delta}{n}_{0}(\omega ){\text{e}}^{\text{i}\omega t}$, where $\mathrm{\Delta}{n}_{0}(\omega )$ is the frequencydependent amplitude of the modulation. The light output is proportional to the photon density (eqn 13.9), so the response is expressed as the ratio of the amplitude of modulation in photon density to the amplitude of modulation in current:
The time constants controlling the carrier and photon densities may not permit them to respond promptly at the modulation frequency, which results in a phase shift of the light output relative to the current, and this is indicated by the presence of an imaginary term in the equation for $\mathrm{\Delta}{N}_{\text{ph}}$: $M(\omega )$ is therefore complex.
The amplitude of the modulation response is given by $M{}^{2}=M{M}^{\ast}$ (Exercise 13.7),
and the phase shift is
The new terms in these equations are the resonance frequency, given by
and the damping rate, given by
(p.228) To a very close approximation $G({n}_{0})$ can be represented by the threshold gain, which at threshold is equal to the cavity loss, so ${v}_{\text{g}}G({n}_{0})={v}_{\text{g}}({\mathrm{\alpha}}_{\text{i}}+{\mathrm{\alpha}}_{\text{m}})$, which from eqn 13.4 is equal to $1/{\mathrm{\tau}}_{\text{ph}}^{0}$. The second term in eqn 13.21 is also a rate, characteristic of the coupling between carriers and photons, which we write as $1/{\mathrm{\tau}}_{nN}$; then the resonance frequency can be seen to be controlled by these two rates:
The modulation characteristics are illustrated in Fig. 13.13, for the relative modulation amplitude $M(\omega )/M(0)$, using input values obtained from gain calculations for four drive currents above threshold.
The response is flat at low frequencies, has a sharp peak at the resonance frequency, and then falls off at higher frequencies. Generally the damping rate ${\mathrm{\gamma}}_{\text{m}}$ is less than ${\omega}_{\text{r}}$, so when $\omega \ll {\omega}_{\text{r}}$ the frequencydependent term in eqn 13.19 is constant and equal to $1/{\omega}_{\text{r}}^{2}$. Above the resonance frequency ${\omega}^{2}\gg {\mathrm{\gamma}}_{\text{m}}^{2}$ and the response falls off as $1/{\omega}^{2}$.
The damping rate is proportional to ${\omega}_{\text{r}}^{2}$, so, as the photon density and modulation frequency increase, the damping rate increases such that at high drive current the resonance peak is completely damped and the response rolls off monotonically with increasing frequency.
To achieve a flat modulation response over a wide frequency range the resonance frequency should be as high as possible. Usually ${\omega}_{\text{r}}$ is limited by the ${v}_{\text{g}}{N}_{\text{ph}0}{G}^{\prime}$ term, which is about two orders of magnitude smaller than ${v}_{\text{g}}G$; therefore the critical parameter is the differential gain ${G}^{\prime}$, which should be as large as possible.
(p.229) 13.4.3 The resonance or relaxation frequency
When the drive current is increased there is an initial increase in carrier density and from eqn 13.17 this causes an increase in the rate at which photons are stimulated into the laser mode. The increase in ${N}_{\text{ph}}$ increases the rate at which carriers recombine by stimulated emission and this causes a decrease in the rate of change of carrier density (eqn 13.16). The resonance arises from the coupling of photons to carrier density through τ_{nN} and carriers to photons through ${\mathrm{\tau}}_{\text{ph}}^{0}$, as in eqn 13.23.
Equation 13.21 shows that ${\omega}_{\text{r}}^{2}$ is proportional to ${N}_{\text{ph}0}$, so ${\omega}_{\text{r}}$ increases with increasing steady current. If the L–I curve is linear above threshold, ${N}_{\text{ph}0}\propto (I{I}_{\text{th}})$, so ${\omega}_{\text{r}}\propto \sqrt{I{I}_{\text{th}}}$.
13.4.4 Gain compression
We have assumed that the gain coefficient itself is not modified by the high photon density that builds up above threshold. Intraband relaxation is fast, but at high photon density not sufficiently fast to supply stimulated emission, so the gain coefficient is suppressed, or compressed. Although the effect of gain compression on threshold itself is negligible, above threshold, where there is only a small difference between the gain and the cavity loss (eqn 13.6), its effect becomes significant.
The dependence of G on photon density is incorporated into the small signal rate equations using a compression factor ε_{G} as^{8}
After neglecting small terms the relaxation frequency becomes^{9}
At high current, once ${N}_{\text{ph}}\gg {\u03f5}_{G}^{1}$, gain compression reduces the resonant frequency and increases the damping rate, as in Fig. 13.14.
(p.230) 13.5 Carrier transport
The rate equation for carriers (eqn 13.3) assumes the carrier population in the gain material responds instantly to a change in current at the contacts. Figure 3.15 shows the band diagram of a forwardbiased quantum well laser, reproduced in simplified form in Fig. 13.15. This is a separate confinement heterostructure (SCH), because the light and carriers are confined by two different double heterostructures, forming the waveguide and the well. The waveguide core is the barrier of the well.
Electrons are injected from the Ncladding layer into the SCH region and are transported to the well, where they are captured and thermalise to the lowest subband. To calculate the response of the light output to modulation of current at the terminals it is necessary to take account of the time for carriers to be transported through the SCH region. This was set out by Nagarajan et al. (1992) for quantum wells and his approach is summarised here.
13.5.1 Transport time
At high carrier densities the distribution of electrons and holes in the SCH region is controlled by the ambipolar diffusion length ${L}_{\text{a}}$ (see Wolfe et al. 1989, Section 8.3.1). The current flowing into the well, ${J}_{\text{w}}$ (Fig. 13.15), is given by the concentration gradient of carriers in the SCH at the position of the well, and the small signal response of the well current to the SCH current ${J}_{\text{s}}$ is
where it is assumed that the halfwidth of the SCH region, ${L}_{\text{s}}/2$, is less than the ambipolar diffusion length; ${D}_{\text{a}}$ is the ambipolar diffusion constant. The magnitude of this transport response function is
which shows that the modulation of the current entering the well is controlled by an ambipolar transport time given by (eqn 13.26)
The ambipolar diffusion constant for AlGaAs is about 10 cm^{2} s^{−1}, so if the fullwidth of the waveguide core is about 300 nm, ${\mathrm{\tau}}_{\text{s}}$ is about 10 ps.
(p.231) 13.5.2 Rate equations
The response of the output to modulation in the external current ${J}_{\text{s}}$ is obtained by solving three rate equations, for the SCH, for the quantum well, and for the photons, using the same principles as in Section 13.1 and quantities per unit area as in Fig. 13.1.
SCH
It is assumed the carrier capture time into the well is much shorter than the transport time through the SCH and can be neglected. The loss rate of carriers from the SCH is therefore ${n}_{\text{s}}/{\mathrm{\tau}}_{\text{s}}$, where ${n}_{\text{s}}$ is the population per unit area. Carriers are reemitted from the well to the SCH at a rate ${n}_{\text{w}}/{\mathrm{\tau}}_{\text{e}}$ and the rate equation for the SCH carrier population is^{10}
Carrier recombination in the SCH is ignored: it could be included as a further loss term in the rate equation.
Quantum well
Electrons are added to the well from the SCH and lost by reemission, they are also lost by recombination represented by a lifetime τ_{n} (radiative and nonradiative, eqn 13.15), and by stimulated emission. Including gain compression, the rate equation in the well is
Photons
Photons are added to the mode by stimulated emission and lost by scattering and through the mirrors, represented by the photon lifetime ${\mathrm{\tau}}_{\text{ph}}^{0}$. As these equations are only used above threshold, spontaneous emission into the mode (${\beta}_{\text{spon}}{R}_{\text{spon}}(n)$) is neglected (eqn 13.5). The photon rate equation with gain compression is therefore
13.5.3 Solution
These three equations contain the essential physics of the problem. They can be can put into the small signal form and the modulations represented by terms such as, for the current, $\mathrm{\Delta}J(t)=\mathrm{\Delta}J(\omega ){\text{e}}^{\text{i}\omega t}$ and, for the photons, ${N}_{\text{ph}}(t)=\mathrm{\Delta}{N}_{\text{ph}}(\omega ){\text{e}}^{\text{i}\omega t}$, as in Section 13.4. A full solution involves a thirdorder polynomial in ω; however, by making a number (p.232) of reasonable approximations it is possible to obtain an expression for the modulation response in a similar form to eqn 13.19, with modified expressions for the resonance frequency and damping rate. The response relative to that at low frequency is
with
and
The transport effects appear through the factor
which is related to the lifetimes for carrier loss and gain between the well and SCH region. This becomes large when ${\mathrm{\tau}}_{\text{s}}\gg {\mathrm{\tau}}_{\text{e}}$.
Carrier transport through the SCH modifies the modulation response in several ways, producing the net effects illustrated in Fig. 13.16
:• The response is modified by a rolloff given by the first square bracket of eqn 13.32 due to the transport time alone, shown as the dashed line in the figure. This term is indistinguishable from the effects of capacitance in the electrical circuit of the diode.

(p.233) • The second term of the response function, eqn 13.32, is modified through the effect of the transport factor on the resonance frequency. Comparison with eqn 13.25 shows that the effective differential gain, represented by the square bracket in eqn 13.33, is reduced by the factor $1/\mathrm{\chi}$.
13.6 What is a laser?
Imagine the scenario: after weeks toiling making a device, you rush to a colleague, proclaiming with excitement: “I’ve made a laser!”. Without looking up from the desk, the reply comes back “How can you tell?”^{11}
Indeed, how can you tell?
The coupling between the photon and carrier populations through stimulated emission and optical feedback to maintain coherence are key distinguishing features of the lasing process. These are incorporated in the rate equations, so it is appropriate here to revisit briefly the evidence for laser action, following the remarks in Section 1.2.
Light–current threshold
The evidence usually cited is an abrupt upturn of the light output as a function of drive current (Fig. 13.3), or the incident power when optical excitation is used. However, the output due simply to amplification along a pumped stripe, with no optical feedback, also increases superlinearly with the pumping rate owing to amplified spontaneous emission (ASE) (Section 5.1.2) from the facet of the laser, as illustrated in Fig. 13.17. When presented with real experimental data with scatter the optimistic investigator may be tempted to draw a line through the points at high current and conclude that this is a laser with a soft turnon due to a high value of ${\beta}_{\text{spon}}$ (Fig. 13.5). Several approaches are available to check whether or not an L–I characteristic is due to laser action:
• When presented as a log–log plot the L–I curve of a laser has a characteristic S shape shown in Fig. 13.6. Figure 13.18 shows such plots for a normal laser with optical feedback, $R=0.3$, and a structure with effectively no optical feedback, having $R=0.3\times {10}^{5}$. Nonradiative recombination is not included in the calculation, so the L–I curve due to spontaneous emission alone should be linear (Section 15.5). Both structures have a superlinear characteristic due to stimulated emission, but this only develops into an S shape in the device with optical feedback. There is stimulated emission into the guided mode of the structure without feedback: this is amplified spontaneous emission; but in the absence of feedback the photon density does not build up to develop the S shape characteristic of a laser.

• The threshold is a determined by the cavity loss, so changing the loss, for example by taking a device of different cavity length, (p.234) changes the threshold current density if it is indeed a laser. The ASE intensity also changes (increases) with pumped length (Section 18.2.1) but the shape of the L–I characteristic is not changed.

• Laser action is accompanied by Fermi level pinning, which causes a change of slope in the current–voltage characteristic at threshold, which is usually detected as the second derivative of the voltage with respect to current (Section 15.1.2). This arises from coupling of carrier and photon densities above threshold and is good evidence for laser action.
Linewidth and fringes
Stimulated emission produces an increasing number of photons at the same energy, causing a narrowing of the emission spectrum. This was the evidence presented in one of the first reports of laser action in a semiconductor junction by Nathan et al. 1962, reproduced in Fig. 13.19; see also Nathan (2012). This is accompanied by closely spaced fringes within the spectral envelope of the laser emission. Such features are characteristic of the coherent feedback necessary for laser action, but fringes also appear below threshold as the stimulated emission overtakes spontaneous emission in the mode (Fig. 13.4) and are seen in roundtrip amplified spontaneous emission spectra (Section 17.2) below threshold. How sharp must the line be and what fringe contrast defines laser action?
Coherence
While interference fringes are an indicator of coherence, the quantitative evidence for laser action is provided by the secondorder correlation function of the intensity, which has a value of unity for perfectly coherent light (Fox (2006), Section 6.3). This has been discussed by Chow (2013), and Chow et al. 2014 have examined both the form of the L–I curve and the coherence of nanolasers.^{12}
(p.235) In conclusion . . .
. . . an abrupt L–I characteristic is satisfactory evidence for laser action in most cases, but is not definitive. It is not satisfactory for lasers with high ${\beta}_{\text{spon}}$, or for nanolasers, and must be replaced by methods based on the statistical properties of the light, such as the correlation function. Claims of laser action in new materials such as organic semiconductors has aroused debate how best to recognise laser action, as has been described by Samuel et al. 2009.
Chapter summary
• The light output as a function of current is given by the solutions of rate equations in the steady state, which show that the carrier density is pinned above threshold for a system in quasiequilibrium.
• Solutions for carriers on inhomogeneous dot states interacting with a bath of phonons show that when quasiequilibrium is not established pinning does not occur over the whole distribution. This causes a broadening of the laser emission spectrum in the random population regime.
• The rate equations give the response of the light output to a small modulation in drive current. The response rolls off above the resonance frequency.
• A high resonance frequency requires a high differential gain (rate of change of gain with carrier density).
• The resonance frequency, and modulation response, are modified by the transport time between the contact and the gain region.
Further reading
Bibliography references:
A full derivation of the modulation response is given by Chuang (2009), Chapter 12, and Coldren and Corzine 1995, Chapter 5. A fuller account of some of the issues that surround the question “What is a laser?” is to be found in the group of papers by Blood (2013), Chow (2013), Coldren (2013), and Ning (2013)
Exercises
The first six exercises are a linked sequence. Use the following relations for ${R}_{\text{spon}}(n)$ and G(n), typical of a quantum well with a transition energy of 1.54 eV:
with n in units of cm^{−2}. Unless stated otherwise, use a refractive index of 3.5, ${\beta}_{\text{spon}}={10}^{4}$, and a Fabry–Perot cavity with ${\mathrm{\alpha}}_{\text{i}}=5$ cm^{−1}, $R=0.3$, and ${L}_{\text{c}}=300\text{\mu}$m.
The photon density is given by the reciprocal of the difference between $1/{\mathrm{\tau}}_{\text{ph}}^{0}$ and ${v}_{\text{g}}G(n)$ (eqn 13.6), so it is necessary to work to six figures to obtain accurate (p.236) values above threshold. Construct a table of the results while working through the exercises.
(13.1) What are the cavity loss and coldcavity photon lifetime?
What are the threshold carrier density and radiative recombination current obtained by equating the gain to the optical loss?
What is the fraction of light extracted from this cavity through both end mirrors? If the Fermi levels pin above threshold, what is the external differential quantum efficiency measured from one mirror above threshold?

(13.2) What is the expression for the average lifetime of a photon in the presence of optical gain?
Calculate the photon lifetime when the carrier density is $n=1.98000\times {10}^{12}$ cm^{−2}, which is slightly less than ${n}_{\text{th}}$ obtained by equating gain and loss. On average how many round trips of the cavity are made by photons at this injection and is this device operating as a laser at this carrier density? Compare this with the number of round trips in a cold cavity.
(Take ${v}_{\text{g}}=c/n=8.57143\times {10}^{9}$ cm s^{−1} and the cold cavity photon lifetime from Exercise 13.1.)
(13.3) The “threshold” carrier density obtained in Exercise 13.1 by equating the gain to the cavity loss is $1.98886\times {10}^{12}$ cm^{−2}; however, when used in eqn 13.6 these values give an infinite photon density because $1/{\mathrm{\tau}}_{\text{ph}}^{0}={v}_{\text{g}}G({n}_{\text{th}})$. The rate equations show that the gain never becomes exactly equal to the loss, as illustrated by this exercise.
(1) Calculate the gain and photon density for the cavity loss in Exercise 13.1 for a carrier density of $n=1.98884\times {10}^{12}$ cm^{−2}. Work to six figures.
(2) Calculate the average number of round trips of the cavity and compare the spontaneous and stimulated emission rates into the laser mode. Is this device operating as a laser, even though the gain is slightly less than the optical loss?
(3) Show that the value of the rate of loss of photons is equal to the sum of the stimulated rate and spontaneous rate into the mode.
(4) What is the total radiative recombination current density at this carrier density and at “threshold” (gain = loss)?
(13.4) What is the laser output from one facet under the pumping conditions in Exercise 13.3 (carrier density of $n=1.98884\times {10}^{12}$ cm^{−2}) for a cavity length of 300 μm, $R=0.3$ (as in previous exercises), and a stripe width of 10 μm?
Calculate the overall external quantum efficiency of emission in the lasing mode (both facets) (photons out divided by electron–hole pairs in) at this carrier density. There is no nonradiative recombination or leakage in this example, so why is this external efficiency not equal to unity?
(13.5) To determine the modulation response it is necessary to know the differential gain. What is the differential gain for $n=1.988\times {10}^{12}$ cm^{−2}.
(13.6) The resonance frequency increases with photon density. At a carrier density of $1.98884\times {10}^{12}$ cm^{−2} as used in Exercise 13.3, and using data collected in the table from other relevant exercises, calculate the rates $1/{\mathrm{\tau}}_{\text{ph}}^{0}$ and $1/{\mathrm{\tau}}_{nN}$ (eqn 13.23) and hence calculate the resonance frequency and damping rate.
(13.7) § Derive eqn 13.19 for the small signal modulation response from eqns 13.16 and 13.17.
Begin with the time dependence of the photon density and show that by assuming certain terms are small enough to be neglected,
(13.38)$$\mathrm{\Delta}n(\omega )\approx \frac{\text{i}\omega}{{v}_{\text{g}}{N}_{\text{ph}0}{G}^{\prime}}\text{}\mathrm{\Delta}{N}_{\text{ph}}(\omega )$$Then obtain the modulation response (eqn 13.18) using the carrier rate equation and quantities defined in Section 13.4.2.

(13.8) At what frequency do you expect carrier transport to cause a rolloff in the modulation response of a GaAs/AlGaAs quantum well laser? (If necessary, do a search for any numbers required.) Will a resonance peak still appear in the response function? If the emission time of electrons from the well is about 30 ps, what is the effect of transport on the resonance frequency.
Notes:
(^{1}) We assume the energy velocity is equal to the group velocity, which is reasonable for weak guiding (Sections 4.1.3 and 4.4.3).
(^{3}) In some publications for quantum wells this is done by writing ${R}_{\text{spon}}=B{n}^{2}$, which reduces the calculation to simply choosing a value for the radiative recombination coefficient B (Section 11.6.2). However, the value for B must be consistent with the band structure used to calculate the gain.
(^{4}) ${\mathrm{\alpha}}_{\text{m}}/{\mathrm{\alpha}}_{\text{cav}}=48.2/53.2=0.906$. Since the current density is that entering the well, the relation between the slope efficiency and internal differential efficiency is not modified by current spreading and carrier injection (Section 5.5.2) in this example.
(^{5}) In this calculation the L–I curve is linear above threshold. This does not occur if the gain is nonuniform along the length of the cavity. Owing to amplification, the photon density is not uniform, and if the stimulated rate is sufficiently high to suppress the local carrier density then the gain in that region is also reduced (see Section 13.2.6). This gain nonuniformity is manifest as a reduction in the slope of the L–I curve at high current.
(^{6}) Carrier recombination that attempts to restore equilibrium between conduction and valence bands is interband relaxation.
(^{7}) The modulations are represented as a frequencydependent amplitude and an oscillating, timedependent term; e.g. for the carrier density $\mathrm{\Delta}n(t)=\mathrm{\Delta}n(\omega ){\text{e}}^{\text{i}\omega t}$.
(^{8}) In this book the gain compression factor is defined in terms of the number of photons per unit area. It therefore has dimensions $[\text{L}{]}^{2}$.
(^{9}) See Chuang (2009), Section 12.2.2. This is the same as the “approximate" result obtained by Coldren and Corzine 1995, eqn 5.51.
(^{10}) Since we are dealing with a subband in the well and a band of states in the SCH, it is assumed that there are ample empty states in both regions to accept electrons captured into the well or emitted to the SCH.
(^{11}) Calvin Coolidge, the 30th president of the United States, was a man of few words and a reputation for laissezfaire government, qualities encapsulated by a campaign poster: “Stay cool with Coolidge”. On hearing of his death, Dorothy Parker (1893–1967), critic and satirist, proclaimed: “How can they tell?”
(^{12}) At the 2014 International Conference on Semiconductor Lasers this work was presented under the title “Searching for lasing threshold in the thresholdless laser”!