# Coupled-Cavity Semiconductor Lasers

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

## Introduction

Distributed-feedback mechanism can provide single-frequency semiconductor lasers with a high degree of side-mode suppression. The operating wavelength is relatively unaffected by external perturbations since it is determined by the spatial period of a permanently etched grating.

Although wavelength stability is an attractive feature of distributed-feedback lasers, it is achieved at the expense of tunability. For some applications it is desirable to have a semiconductor laser whose wavelength can be tuned over a wide range, at least discretely.

Coupled-cavity semiconductor lasers have the potential of offering mode selectivity together with wavelength tunability.

The basic concept of using the coupled-cavity scheme for longitudinal mode selection is known from the early work on gas lasers and has been widely used for mode selection. In the case of semiconductor lasers, considerable work was carried out by coupling the semiconductor laser to an external cavity.

Monolithically integrated coupled-cavity devices were considered occasionally in relation to their specific properties such as optical bistability, optical amplification, and longitudinal-mode selectivity. Such devices attracted wide attention during the 1980s in an attempt to obtain single-frequency semiconductor lasers that are useful for optical communications in the 1.55-μm wavelength region.

At the same time, interest in semiconductor lasers coupled to a small-length external cavity has also revived. To distinguish them from monolithic coupled-cavity devices, such lasers are often referred to as external-cavity semiconductor lasers.

The mechanism of mode selectivity in coupled-cavity lasers can be understood by referring to Figure 8-1. The effect of feedback from the external cavity can be modeled through an effective wavelength-dependent reflectivity of the facet facing the external cavity. As a result, the cavity loss is different for different Fabry-Perot (FP) modes of the laser cavity. In general, the loss profile is periodic as shown schematically in the bottom part of Figure 8-1.

The mode selected by the coupled-cavity device is the FP mode that has the lowest cavity loss and is closest to the peak of the gain profile. Because of the periodic nature of the loss profile, other FP modes with relatively low cavity losses may exist. Such modes are discriminated by the gain roll-off because of their large separation from each other. The power carried by these side modes is typically 20-30 dB below that of the main mode occurring in the vicinity of the gain peak.

## Coupled-Cavity Schemes

As mentioned above, coupled-cavity semiconductor lasers can be classified into two broad categories. We shall refer to them as ** active-passive** and

**schemes, depending on whether the second cavity (sometimes referred to as the "controller') remains unpumped or can be pumped to provide gain.**

*active-active*The devices in the latter category are also called ** three-terminal** devices since three electrical contacts are used to pump the two optically coupled but electrically isolated cavity sections. Figures 8-2 and 8-3 show a specific example of each kind of device.

In the active-passive scheme, the semiconductor laser is coupled to an external cavity that is unpumped and plays a passive role. In the simplest design, a plane or spherical mirror is placed at a short distance from one of the laser facets; the facet may have an antireflection coating to increase the coupling between the two cavity sections.

The spherical-mirror geometry for InGaAsP lasers has attracted considerable attention and has been used in a transmission experiment at 1.52 μm. Several variations of the plane-mirror geometry have been adopted for InGaAsP lasers.

In the short coupled-cavity scheme, a short laser cavity (50-80 μm) is coupled to a short external cavity (30-80 μm). The gold-coated facet of a semiconductor chip acts as a plane mirror.

In another scheme the external cavity consists of a graded-index fiber lens to provide improved coupling and to avoid diffraction loss. A short rod (~ 100-200 μm) of graded-index multimode fiber with one end gold-coated is used for this purpose (see Figure 8-2).

Etching techniques have also been used to make monolithic coupled-cavity devices with integrated and passive sections.

In the active-active scheme shown in Figure 8-3 both sections can be independently pumped, giving an additional degree of freedom that can be used to control the behavior of the device.

A natural choice is to use identical active materials for both cavities. Furthermore, alignment between the active regions is automatically achieved if the two cavity sections are created by forming a gap in a conventional semiconductor laser. Cleaving and etching techniques have been used for this purpose.

In another scheme the interferometric property of a coupled-cavity device is obtained by bending the active region; although the cavities are not physically separated, the \(p\) contact can be separated into two parts by chemical etching to accomplish separate pumping.

Whatever the technique employed, the qualitative behavior of such three-terminal coupled-cavity lasers is similar with respect to mode selectivity and wavelength tunability.

The advantage of the cleaving technique lies in providing a device whose four facets are mirror-like and parallel to each other. The cleaved-facet reflectivity (about 32%) allows reasonable coupling between the two cavity sections as long as the gap is not too wide.

Coupled-cavity lasers made by the cleaving technique are sometimes referred to as ** cleaved-coupled-cavity**, or \(\text{C}^3\) lasers (see Figure 8-3). Their operating characteristics have been extensively studied. We shall refer to \(\text{C}^3\) lasers while considering three-terminal devices after keeping in mind that similar behavior would occur for other kinds of active-active devices.

The mechanism of mode selectivity is the same for both active-active and active-passive devices and can be understood from Figure 8-1. The main difference between the two kinds of coupled-cavity lasers lies in the external means used to shift the FP modes of the two cavities.

In the active-passive case, the modes of the external cavity can be shifted by changing its length or its temperature.

By contrast, the active-active case offers the possibility of electronic shifting since the current in the two cavities can be independently controlled. If one of the cavities is operated below threshold, a change in its drive current significantly changes the carrier density inside the active region. Since the refractive index of a semiconductor laser changes along with the carrier density, the longitudinal modes shift with a change in the drive current, and different FP modes of the laser cavity can be selected.

In the design of a coupled-cavity laser, the cavity lengths \(L_1\) and \(L_2\) are adjustable to some extent. The performance of such lasers depends on the relative lengths \(\mu_1L_1\) and \(\mu_2L_2\) of the two cavities, where \(\mu_1\) and \(\mu_2\) are the mode indices.

Two cases should be distinguished depending on whether the optical lengths are similar (\(\mu_1L_1\approx\mu_2L_2\)) or differ significantly (\(\mu_1L_1\gg\mu_2L_2\)); they are referred to as the long-long and long-short geometries, respectively.

Although both geometries are capable of mode selection and wavelength tuning, they differ in one important aspect of wavelength stability. For the long-long geometry, even a relatively small shift of the FP mode can lead to "mode hopping." By contrast, a shift of about one mode spacing is required to achieve mode hopping in the long-short geometry.

Clearly, unintentional mode hopping because of temperature and current fluctuations is less likely to occur in the long-short case. By the same token, wavelength tuning would require larger current or temperature changes for a long-short than for a long-long device.

## Theory

A theoretical analysis of the coupled-cavity semiconductor lasers requires simultaneous consideration of the gain and loss in the two cavities after taking into account their mutual optical feedback.

In contrast to a single-cavity laser for which the facet loss is wavelength-independent (and hence the same for all FP modes), in a coupled-cavity laser the effective facet loss becomes different for different FP modes.

One objective of the theory is to find the longitudinal modes of the coupled system and their respective gains required to reach threshold. The threshold gains can then be used in the rate equations to study the transient behavior and the modulation response.

For three-terminal devices the gain and index variations in both cavities should be considered with respect to individual current changes in the two coupled cavities.

A general analysis of coupled-cavity semiconductor lasers is extremely complicated, so it is necessary to make several simplifying assumptions.

We assume that the field distribution associated with the fundamental lateral and transverse mode of the waveguide is unaffected by the intercavity coupling and that only axial propagation in each cavity needs to be considered. This reduces the problem to one dimension.

Note, however, that the coupling depends upon the mode width and other related parameters, and mode-conversion losses occur as the optical mode leaves cavity 1, diffracts in the air inside the gap, and then reenters cavity 2. Such losses would be incorporated through an effective gap loss \(\alpha_\text{g}\).

In this tutorial we introduce the concept of the coupling constant and then obtain the longitudinal modes and their respective threshold gains for the coupled system. The gain margin between the lowest-threshold mode and the mode with the second-lowest threshold gain determines the extent of side-mode suppression. Particular attention is paid to the modulation characteristics, and under certain conditions the intercavity coupling is found to reduce the frequency chirp.

### 1. Coupling Constant

The first step in the analysis is to determine the extent of coupling between the two cavities. We consider the general case applicable for both active-active and active-passive devices.

Figure 8-4 shows the geometry and notation.

The coupling between the cavities is governed by an air gap of width \(L_\text{g}\). The air gap itself forms a third FP cavity, and the intercavity coupling is affected by the loss and phase shift experienced by the optical field while traversing the gap.

In the scattering-matrix approach the fields in the two cavities are related by

\[\tag{8-3-1}\begin{pmatrix}E_1'\\E_2'\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}E_1\\E_2\end{pmatrix}\]

The scattering matrix elements can be obtained by considering multiple reflections inside the gap.

More explicitly, the gap is treated as an FP cavity on which the field \(E_1\) is incident from the left (Figure 8-4) and the reflected field \(E_1'\) and the transmitted field \(E_2'\) are obtained in terms of \(E_1\) after considering multiple round trips inside the cavity. The procedure yields \(S_{11}=E_1'/E_1\) and \(S_{21}=E_2'/E_1\).

The same procedure is used to obtain \(S_{22}\) and \(S_{12}\) after assuming that only the field \(E_2\) is incident on the FP cavity from the right.

The result is

\[\tag{8-3-2}S_{11}=r_1-\frac{r_2(1-r_1^2)t_\text{g}}{1-r_1r_2t_\text{g}}\]

\[\tag{8-3-3}S_{22}=r_2-\frac{r_1(1-r_2^2)t_\text{g}}{1-r_1r_2t_\text{g}}\]

\[\tag{8-3-4}S_{12}=S_{21}=\frac{[t_\text{g}(1-r_1^2)(1-r_2^2)]^{1/2}}{1-r_1r_2t_\text{g}}\]

where

\[\tag{8-3-5}t_\text{g}=\exp(2\text{i}\beta_\text{g}L_\text{g})=\exp(2\text{i}k_0L_\text{g})\exp(-\alpha_\text{g}L_\text{g})\]

Here \(\beta_\text{g}=k_0+\text{i}\alpha_\text{g}/2\) and accounts for the phase shift and the loss inside the gap.

Further, \(r_1\) and \(r_2\) are the amplitude-reflection coefficients at the two facets forming the gap; i.e., \(r_n=(\mu_n-1)/(\mu_n+1)\), where \(\mu_1\) and \(\mu_2\) are the mode indices in the two cavities.

Physically the gap has been replaced by an interface whose effective reflection coefficients are \(S_{11}\) and \(S_{22}\), while \(S_{12}\) and \(S_{21}\) are the effective-transmission coefficients from cavity 1 to cavity 2 and vice versa.

It is useful to define a complex coupling parameter

\[\tag{8-3-6}\tilde{C}=C\exp(\text{i}\theta)=\left(\frac{S_{12}S_{21}}{S_{11}S_{22}}\right)^{1/2}\]

where \(C\) governs the strength of mutual coupling and \(\theta\) is the coupling phase, which will be seen later to play an important role.

The magnitudes of \(C\) and \(\theta\) depend on a large number of device parameters.

The simplest case occurs for a semiconductor laser coupled to an external mirror. In this case the coupling element is just the laser-air interface. Since \(L_\text{g}=0\), \(t_\text{g}=1\). Furthermore \(\mu_2=1\), and therefore \(r_2=0\). Using these values in Equations (8-3-2) to (8-3-6), we find that \(C=(1-r_1^2)^{1/2}/r_1\) and \(\theta=\pi/2\).

For a \(\text{C}^3\) laser, the two cavities have nearly equal indices of refraction, and therefore \(r_1=r_2=r\). Using Equations (8-3-2) to (8-3-6), we now obtain

\[\tag{8-3-7}C\exp(\text{i}\theta)=\left(\frac{1-r^2}{r}\right)\left(\frac{t_\text{g}^{1/2}}{1-t_\text{g}}\right)\]

The coupling depends on \(t_\text{g}\), and both \(C\) and \(\theta\) vary with the gap width \(L_\text{g}\).

The evaluation of \(C\) and \(\theta\) requires a knowledge of the gap loss \(\alpha_\text{g}\). These losses arise mainly from diffraction spreading of the beam inside the gap and have been estimated using a simple diffraction-spreading model perpendicular to the junction plane.

Figure 8-5 shows the variation of \(C\) and \(\theta\) with the gap width using this model after taking \(|r|^2=0.31\) as the cleaved-facet reflectivity.

Both \(C\) and \(\theta\) vary considerably with small changes in \(L_\text{g}\). The in-phase coupling (\(\theta=0\)) occurs whenever \(L_\text{g}=m\lambda/2\) (\(m\) is an integer), and \(C\) also goes through a maximum for that value of \(L_\text{g}\).

In practice, however, the gap width may vary from device to device. The corresponding large variations occurring in \(C\) and \(\theta\) imply that the performance of \(\text{C}^3\) lasers would also be device-dependent.

### 2. Longitudinal Modes and Threshold Gain

In this section we obtain an eigenvalue equation whose solutions yield the wavelength and threshold gain associated with the longitudinal modes of the coupled system.

A simple way to do this is to consider the relationship between \(E_1\) and \(E_1'\) using Figure 8-4. The field \(E_1'\) results from reflection of \(E_1\) and transmission of \(E_2\) and is given by [see Equation (8-3-1)]

\[\tag{8-3-8}E_1'=S_{11}E_1+S_{12}E_2\]

On the other hand, the round trip through cavity 1 results in the relation

\[\tag{8-3-9}E_1=r_1'\exp(2\text{i}\beta_1L_1)E_1'=r_1't_1E_1'\]

where the complex propagation constant

\[\tag{8-3-10}\beta_n=\mu_nk_0-\text{i}\bar{\alpha}_n/2\qquad(n=1,2)\]

governs wave propagation in \(n\)th cavity, \(k_0=2\pi/\lambda\), \(\lambda\) is the device wavelength, and \(\alpha_n\) is the mode gain.

The definition of \(\beta_n\) is similar to Equation (7-3-5) [refer to the DFB semiconductor lasers tutorial] with the difference that both cavities should be considered and the mode-propagation constant is in general different in each cavity.

As in Equation (7-3-6) in the DFB semiconductor lasers tutorial, the mode gain is related to the material gain \(g_n\) by

\[\tag{8-3-11}\bar{\alpha}_n=\Gamma{g_n}-\alpha_n^\text{int}\]

where \(\Gamma\) is the confinement factor and \(\alpha_n^\text{int}\) is the internal loss.

For an active-active device, \(g_1\) and \(g_2\) can be individually varied by changing the current passing through each cavity section. For an active-passive device, \(g_2=0\) and \(\bar{\alpha}_2\) accounts for the absorption loss in the passive cavity.

Equations (8-3-8) and (8-3-9) can be combined to obtain the relation

\[\tag{8-3-12}(1-r_1't_1S_{11})E_1=r_1't_1S_{12}E_2\]

Similar considerations for cavity 2 lead to

\[\tag{8-3-13}(1-r_2't_2S_{22})E_2=r_2't_2S_{21}E_1\]

These two homogeneous equations have nontrivial solutions only if the secular condition

\[\tag{8-3-14}(1-r_1't_1S_{11})(1-r_2't_2S_{22})=r_1'r_2't_1t_2S_{12}S_{21}\]

is satisfied.

Equation (8-3-14) is the desired eigenvalue equations for the coupled system and has been extensively studied.

In the absence of coupling, \(S_{12}=0\), \(S_{nn}=r_n\), and we recover the threshold condition for uncoupled cavities.

Note that

\[\tag{8-3-15}t_n=\exp(2\text{i}\beta_nL_n)=\exp(2\text{i}\mu_nk_0L_n)\exp(-\bar{\alpha}_nL_n)\]

incorporates the phase shift and the gain (or loss) experienced by the optical field during a round trip in each cavity.

The loss and the phase shift inside the gap are included through \(t_\text{g}\) and \(S_{ij}\), as given by Equations (8-3-2) to (8-3-5).

The eigenvalue equation (8-3-14) is applicable for all kinds of coupled-cavity devices with arbitrary reflectivities at four interfaces (see Figure 8-4).

Before proceeding, it is useful to introduce the concept of an effective mirror reflectivity.

In many cases the role of one cavity (say cavity 2) is to provide a control through which a single FP mode of cavity 1 is selected. The effect of cavity 2 on mode selectivity can be treated by an effective reflectivity for the laser facet facing cavity 2.

Equation (8-3-14) can be written in the equivalent form of

\[\tag{8-3-16}(1-r_1'R_\text{eff}t_1)=0\]

where the effective reflectivity

\[\tag{8-3-17}R_\text{eff}=S_{11}+\frac{r_2't_2S_{12}S_{21}}{1-r_2't_2S_{22}}=S_{11}\left(1+\frac{\tilde{C}^2f_2}{1-f_2}\right)\]

and \(f_2=r_2't_2S_{22}\) and is the fraction of the amplitude coupled back into the laser cavity after a round trip in cavity 2.

Equation (8-3-16) suggest that as far as mode selectivity is concerned, the coupled-cavity laser is equivalent to a single-cavity laser with facet reflection coefficients \(r_1'\) and \(R_\text{eff}\).

Although Equation (8-3-16) is formally exact, its practical utility is limited to the case of weak coupling so that a change in \(t_1\) (through operating conditions of cavity 1) does not affect \(R_\text{eff}\) through a change in the feedback fraction \(f_2\). This is often the case for active-passive devices. In the case of active-active devices, the effective reflectivity concept is valid when cavity 2 is biased below threshold.

To illustrate the extent of mode selectivity offered by the coupled-cavity mechanism, we consider solutions of the eigenvalue equation (8-3-14) for a specific \(\text{C}^3\)-type active-active device for which \(r_1=r_1'=r_2=r_2'\approx0.56\).

We assume that cavity 2 is biased below threshold such that \(\bar{\alpha}_2=0\) (in the absence of coupling) and that it does not change significantly with coupling.

Equation (8-3-14) is used to obtain \(\bar{\alpha}_1\), and the wavelength \(\lambda=2\pi/k_0\) corresponding to various longitudinal modes.

Figure 8-6 shows the longitudinal modes and their respective gains for a specific gap width (\(L_\text{g}=1.55\) μm) and for two sets of cavity lengths \(L_1\) and \(L_2\) corresponding to long-long and long-short geometries.

Since the concept of effective reflectivity is approximately valid, \(|R_\text{eff}|\) versus \(\lambda\) is also shown.

The effect of the second cavity is to modulate the effective reflectivity, and the minimum threshold gain is required for modes for which \(R_\text{eff}\) is maximum.

In the absence of coupling, the same threshold gain (\(\bar{\alpha}_1=48\text{ cm}^{-1}\)) is required for all modes. With coupling, the gain difference \(\Delta\alpha\) between the lowest-gain mode and the neighboring mode provides mode discrimination and leads to side-mode suppression.

Figure 8-6 is drawn for the optimum case (\(L_\text{g}=\lambda\)) using \(S_{11}=S_{22}=0.409\) and \(S_{12}=S_{21}=0.371\); from Equation (8-3-6) these two relations give \(C=0.907\) and \(\theta=0\).

As mentioned before, \(C\) and \(\theta\) vary considerably with the gap width \(L_\text{g}\). The effect of gap width on the threshold gains has been extensively studied.

For a lossy gap the best mode selectivity occurs for the case of in-phase coupling (\(\theta=0\)), which requires that \(L_\text{g}\) be an integer multiple of \(\lambda/2\). The worst case (\(\theta=\pi/2\)) occurs wherever \(L_\text{g}\) is an odd multiple of \(\lambda/4\) (see Figure 8-5).

For a lossless gap the situation is reversed; i.e., the best and worst selectivity occur for \(\theta=\pi/2\) and \(\theta=0\), respectively. This is the case for external-cavity devices where a single semiconductor-air interface acts as the coupling element. As discussed in the **Coupling Constant** section above, \(\theta=\pi/2\) for such devices. Reasonable discrimination can thus be expected and has been observed experimentally.

A comparison of long-long and long-short devices in Figure 8-6 reveals an interesting feature. Even though the wavelength variation of \(|R_\text{eff}|\) is much faster in the long-long case, the two devices behave similarly as far as the threshold gains are concerned.

In Figure 8-6 in particular, the mode pattern repeats after eight modes (\(M=8\)). For the long-short geometry (\(L_1\gg{L_2}\)), the repeat mode is determined by the ratio of the optical lengths in each cavity (\(M\approx\mu_1L_1/\mu_2L_2\)). For the long-long geometry, by contrast, \(M\approx\mu_1L_1/(\mu_1L_1-\mu_2L_2)\). For a given coupled-cavity device the repeat mode is thus determined by \(\mu_2L_2\) or \(\mu_1L_1-\mu_2L_2\) depending on which quantity is smaller.

However, the long-long and long-short devices, differ with respect to tunability and stability of the mode wavelength.

We briefly consider the general case in which both cavities are close to or above threshold and the gain and index variations in one cavity influence the behavior of the other.

The eigenvalue equation (8-3-14) can be used to obtain the threshold gains \(\bar{\alpha}_1\) and \(\bar{\alpha}_2\) for a given mode wavelength \(\lambda\). Because of the intercavity coupling, the mode gains \(\bar{\alpha}_1\) and \(\bar{\alpha}_2\) are not pinned in the above-threshold regime, but are related for each mode in the form of a curve in the \(\bar{\alpha}_1-\bar{\alpha}_2\) plane.

The numerical results show that as the currents in the two sections are varied, the laser wavelength can be tuned discretely by wavelength hopping from one mode to another.

The current range over which the laser maintains the same longitudinal mode, however, depends on the gap width as well as the cavity lengths. This range is considerably larger for long-short devices (\(L_1\gg{L_2}\)) compared to those for which \(L_1\approx{L_2}\). The same behavior holds when mode hopping occurs because of temperature changes. The long-short geometry is generally preferable if the objective is to obtain a laser with a stable wavelength.

A detailed theoretical understanding for three-terminal \(\text{C}^3\) devices requires that the mode thresholds for various longitudinal modes be known as a function of the currents \(I_1\) and \(I_2\) passing through the two cavity sections. This behavior has been studied using a simple model for the gain and index variations with the current.

Single-mode operation above threshold is described by zones in the \(I_1-I_2\) plane, and mode hopping occurs at the zone boundaries. However, these zones are extremely sensitive to the coupling phase \(\theta\); optimum performance for \(\text{C}^3\) lasers is expected to occur for in-phase coupling (\(\theta=0\)). For other values of \(\theta\), the steady-state operation can become unstable or display bistable behavior.

### 3. Side-Mode Suppression

The mode suppression ratio (MSR), defined as the ratio of the power in the main mode to that of the most intense side mode, is often used to express the extent of mode selectivity for coupled-cavity devices.

It can be related to the threshold gains in a similar way as was done for DFB lasers in the performance of DFB lasers tutorial. Using Equations (7-4-1) to (7-4-4) [refer to the performance of DFB lasers tutorial], the MSR can be written in the form

\[\tag{8-3-18}\text{MSR}=1+\frac{\Delta\alpha+\Gamma\Delta{g}}{(\bar{\alpha}_1+\alpha_1^\text{int})\delta}\]

where \(\Delta\alpha\) is the threshold-gain margin (see Figure 8-6), \(\Gamma\) is the confinement factor, \(\Delta{g}\) is the gain roll-off from its peak value, and \(\bar{\alpha}_1+\alpha_1^\text{int}\) is the total mode gain for the lowest-loss mode.

The parameter \(\delta\) is a small dimensionless number and decreases with an increase in the main-mode power [see Equation (6-3-11) in the steady-state characteristics of semiconductor lasers tutorial]. Typically, \(\delta\approx5\times10^{-5}\) at a power level of few milliwatts.

Mode discrimination in coupled-cavity lasers can be understood using Equation (8-3-18) and Figure 8-6.

Lasing occurs for the lowest-loss mode closest to the gain peak. The next mode sees practically no gain roll-off (\(\Delta{g}\approx0\)) since the longitudinal-mode spacing is very small relative to the gain bandwidth. However, the cavity-loss discrimination (\(\Delta\alpha\approx4-5\text{ cm}^{-1}\)) can provide MSR values of \(\sim30\text{ dB}\).

As we move away from the lasing mode, \(\Delta\alpha\) decreases and \(\Delta{g}\) increases. Nonetheless, their sum can be large enough to provide sufficient mode discrimination.

For the next pair of coincident modes, \(\Delta\alpha=0\) and mode discrimination is completely determined by the gain roll-off. However, one can expect \(\Gamma\Delta{g}\) to be \(\sim5\text{ cm}^{-1}\) if the repeat mode is more than 4 or 5 mode spacings away from the gain peak.

This discussion applies for CW operation of a coupled-cavity device. For single-cavity lasers the MSR degrades severely under transient conditions, as discussed in the transient response of semiconductor lasers tutorial.

The multimode rate equations used there can be applied for coupled-cavity lasers after incorporating different photon lifetimes for different modes. Similar to the case of DFB lasers, side modes remain suppressed by \(\sim30\text{ dB}\) even under transient conditions if \(\Delta\alpha\approx8-10\text{ cm}^{-1}\).

Because of their capability for dynamic side-mode suppression, coupled-cavity lasers are useful as single-frequency sources for optical communication systems employing dispersive fibers at the 1.55-μm wavelength.

### 4. Modulation Response

For their application in optical fiber communications, semiconductor lasers are directly modulated to provide pulsed output. Important features such as modulation bandwidth and frequency chirp were discussed in the modulation response of semiconductor lasers tutorial for single-cavity lasers.

In this section we consider the effect of intercavity coupling on the modulation response of coupled cavity lasers. Three-terminal devices such as the \(\text{C}^3\) laser offer an additional degree of control since the bias and modulation currents in each cavity can be individually adjusted.

It has been observed experimentally that the chirp for \(\text{C}^3\) lasers can be reduced, typically by a factor of 2, through proper adjustment of the device currents. Chirp reduction is also expected to occur for external-cavity devices, even though the passive cavity remains unbiased.

Although one generally should consider multimode rate equations, modulation analysis is considerably simplified by following the dynamics of the dominant single-longitudinal mode.

This is justified since practically useful devices must maintain the same longitudinal mode with sufficient side-mode suppression (greater than ~ 20 dB) throughout modulation. Such weak side modes do not significantly affect the modulation response of the main mode.

However, one should generalize the single-cavity, single-mode rate equation of the rate equations for semiconductor lasers tutorial to incorporate the feedback owing to intercavity coupling.

This generalization can be carried out using Equation (8-3-8), which shows that on every reflection a relative fraction \(S_{12}E_2/S_{11}E_1\) of the field in cavity 2 enters cavity 1. Since this fraction is complex, both the power and the phase in cavity 1 are affected.

Adding the additional term representing the field entering from the neighboring cavity to Equation (6-2-9) and following the analysis of the rate equations for semiconductor lasers tutorial with the above-mentioned modification, we obtain the generalized rate equations

\[\tag{8-3-19}\dot{P}_j=(G_j-\gamma_j)P_j+R_\text{sp}(N_j)+\kappa_j\cos(\theta\pm\phi)\]

\[\tag{8-3-20}\dot{N}_j=I_j/q-\gamma_\text{e}(N_j)N_j-G_jP_j\]

\[\tag{8-3-21}\dot{\phi}_j=-\frac{\mu_j}{\mu_{\text{g}j}}(\omega-\Omega_j)+\frac{1}{2}\beta_\text{c}(G_j-\gamma_j)+\frac{\kappa_j}{2P_j}\sin(\theta\pm\phi)\]

where \(\phi=\phi_1-\phi_2\) and is the relative phase between the two cavities.

The feedback rate

\[\tag{8-3-22}\kappa_j=\frac{c}{\mu_{\text{g}j}L_j}(P_1P_2)^{1/2}C\]

where the coupling constant \(C\) and the coupling phase \(\theta\) are defined by Equation (8-3-6).

The other symbols have the same meanings as in the rate equations for semiconductor lasers tutorial except for the subscript \(j\) that has been added to distinguish the two cavities.

Consider the case of a \(\text{C}^3\) laser under direct sinusoidal modulation at the frequency \(\nu_\text{m}\). In the general case, a fraction \(m_\text{s}\) of the modulation current is applied to cavity 2. The device currents in that case are

\[\tag{8-3-23}I_1(t)=I_1^\text{b}+(1-m_\text{s})I_\text{m}\sin(2\pi\nu_\text{m}t)\]

\[\tag{8-3-24}I_2(t)=I_2^\text{b}+m_\text{s}I_\text{m}\sin(2\pi\nu_\text{m}t+\theta_\text{m})\]

where \(I_1^\text{b}\) and \(I_2^\text{b}\) are the bias levels, \(I_\text{m}\) is the peak value of the modulation current, and \(\theta_\text{m}\) is the phase shift between the modulation current applied to the two cavities.

Equations (8-3-19) to (8-3-24) can be used to model the modulation response of a \(\text{C}^3\) laser of arbitrary cavity lengths.

The large-signal regime requires a numerical solution of the rate equations. However, a qualitative understanding can be developed by considering the small-signal solution of the generalized rate equations.

In this approach, \(P_j\), \(N_j\), and \(\phi_j\) are expanded around their steady-state values corresponding to the bias levels, and Equations (8-3-19) to (8-3-21) are linearized in terms of the small deviations \(\delta{P}_j\), \(\delta{N}_j\), and \(\delta\phi_j\), which vary sinusoidally in response to direct modulation.

Two quantities of practical interest are the normalized power response

\[\tag{8-3-25}H(\nu_\text{m})=\left|\frac{\delta{P_1}(\nu_\text{m})}{\delta{P_1}(0)}\right|\]

and the frequency chirp per milliampere of the modulation current

\[\tag{8-3-26}\delta\nu=\left|\frac{\delta\dot{\phi}_1}{2\pi{I_\text{m}}}\right|\]

for laser cavity 1.

The dependence of \(H(\nu_\text{m})\) and \(\delta\nu\) on various device parameters has been considered for both active-passive and active-active schemes.

To demonstrate the effect of intercavity coupling, we consider a specific \(\text{C}^3\) laser such that \(L_1\) = 200 μm and \(L_2\) = 50 μm, and we assume in-phase coupling (\(\theta=0\)).

Figure 8-7 shows the power response and the chirp when both sections are biased at 1.5 times above threshold and the modulation current is split equally with no phase shift (\(m_\text{s}=0.5\), \(\theta_\text{m}=0\)).

The power response is similar to the single-cavity case and peaks at the relaxation-oscillation frequency \(\nu_\text{R}\). However, the frequency \(\nu_\text{R}\) increases with an increase in the coupling constant \(C\). This can be understood by noting that the effect of coupling is to decrease the laser threshold and that at the fixed bias current the output power and hence \(\nu_\text{R}\) increase.

The most notable feature of the coupled-cavity laser is the chirp reduction occurring at high modulation frequencies. Figure 8-7 shows that when the coupling is strong, the chirp initially decreases before peaking at the relaxation-oscillation frequency \(\nu_\text{R}\).

Thus, depending on the modulation frequency, the chirp may be higher or lower relative to its single-cavity value (dashed curve). A reduction by a factor of 2 is predicted when \(\nu_\text{m}=1\) GHz and \(C=1\). Figure 8-5 shows that such values of \(C\) can be achieved for narrow-gap \(\text{C}^3\) devices.

The chirp reduction has also been observed experimentally. The origin of chirp reduction can be understood by noting that the second and third terms in Equation (8-3-21) contribute to chirp, whereas the third term is absent for single-cavity lasers. Depending on operating conditions, the two contributions may interfere to increase or decrease the total chirp.

The chirp increase for \(\text{C}^3\) lasers at low modulation frequencies is another attractive feature that can be used for analog frequency modulation with negligible spurious intensity modulation.

Frequency-excursion rates in the range of 1-10 GHz/mA have been experimentally observed and are in agreement with the theoretical results shown in Figure 8-7. The numerical value for a given device depends, among other things, on the coupling constant \(C\) and the coupling phase \(\theta\).

Figure 8-7 is drawn for a specific set of device parameters. Both \(H(\nu_\text{m})\) and \(\delta\nu\) change with the bias currents, with the modulation splitting (fraction \(m_\text{s}\), phase shift \(\theta_\text{m}\)), and with the coupling parameters \(C\) and \(\theta\).

Qualitatively speaking, the in-phase coupling is desirable to minimize chirp for \(\text{C}^3\) devices. Since the in-phase condition is also optimum from the viewpoint of mode stability and dynamic side-mode suppression, the \(\text{C}^3\) devices for which the gap is nearly an integer multiple of \(\lambda/2\) is expected to provide the best performance.

The next tutorial discusses the ** operating characteristics of coupled-cavity semiconductor lasers**.