# Emission Characteristics of Semiconductor Lasers

This is a continuation from the previous tutorial - waveguide modes in semiconductor lasers.

Previous tutorials have discussed such basic concepts as gain-loss considerations and characteristics of the longitudinal, transverse, and lateral modes supported by the semiconductor cavity.

We now consider the emission characteristics used to characterize the performance of a semiconductor laser. They can be classified into several groups: (i) light-current, (ii) spatial-mode, (iii) spectral, and (iv) transient or dynamic characteristics.

In the following discussion we consider each group separately.

## Light-Current Characteristics

The light emitted by one facet of a semiconductor laser is measured as a function of the device current $$I$$. The resulting curve is often referred to as the light-current ($$L-I$$) curve and is strongly temperature dependent.

Figure 2-6 shows the behavior schematically for a strongly index guided laser at a given temperature.

The current $$I$$ is related to the current density $$J$$ that is injected into the active layer by

$\tag{2-6-1}I=I_\text{a}+I_\text{L}=wLJ+I_\text{L}$

where $$I_\text{a}$$ is the current passing through the active region, $$L$$ is the cavity length and $$w$$ is the active-region width.

The leakage current $$I_\text{L}$$ accounts for the fact that, depending on a specific structure, a part of the total current may not pass through the active layer.

Equation (2-6-1) is applicable for a broad-area laser or a strongly index guided laser for which the injected current density is nearly uniform over the active region width.

If the current density is laterally nonuniform, such as in a gain-guided device, $$J$$ in Equation (2-6-1) should be interpreted as the average injected current density over the width $$w$$.

The form of the $$L-I$$ curve shown in Figure 2-6 is typical for any laser. The turning point, at which the light output abruptly starts increasing, corresponds to the laser threshold.

The threshold current $$I_\text{th}$$ (or equivalently the threshold current density $$J_\text{th}$$) is an important device parameter, and its minimization is often sought.

When $$I\lt{I}_\text{th}$$, the light output mainly consist of spontaneous emission and its magnitude is governed by the term $$Bn^2$$ in Equation (2-4-9) [refer to the gain and stimulated emissions in semiconductor lasers tutorial].

When $$I\gt{I}_\text{th}$$, stimulated emission, which is governed by the last term in Equation (2-4-9) [refer to the gain and stimulated emissions in semiconductor lasers tutorial], starts to dominate.

In the vicinity of the threshold, the power increases by several orders of magnitude. This phase-transition-like behavior is discussed in later tutorials after including the contribution of spontaneous emission to the lasing mode.

An expression for the threshold current density of a strongly index guided semiconductor laser can be obtained using the phenomenological model of the gain and stimulated emissions in semiconductor lasers tutorial.

As $$J$$ increases, the gain also increases owing to an increase in the carrier density $$n$$ [see Equations (2-4-3) and (2-4-8) of the gain and stimulated emission in semiconductor lasers tutorial]. The threshold is reached when $$n$$ achieves the threshold value $$n_\text{th}$$ for which the gain equals loss. Using (2-3-9) [refer to the threshold condition and longitudinal modes in semiconductor lasers tutorial] and (2-4-3) [refer to the gain and stimulated emission in semiconductor lasers tutorial], the threshold carrier density is given by

$\tag{2-6-2}n_\text{th}=n_0+(\alpha_\text{m}+\alpha_\text{int})/(a\Gamma)$

where $$n_0$$ is the transparency value at which population inversion occurs.

The threshold current density $$J_\text{th}$$ is obtained using Equations (2-4-8) and (2-4-9) [refer to the gain and stimulated emission in semiconductor laser tutorial]. Near threshold, stimulated recombination can be neglected and we obtain

$\tag{2-6-3}J_\text{th}=qdn_\text{th}/\tau_\text{e}(n_\text{th})$

where

$\tag{2-6-4}\tau_\text{e}=(A_\text{nr}+Bn+Cn^2)^{-1}$

is the carrier-recombination time that in general depends on $$n$$.

Once threshold is reached, the carrier density $$n$$ remains pinned to its threshold value $$n_\text{th}$$ and a further increase in $$J$$ leads to light emission through the process of stimulated emission.

Equations (2-4-8) and (2-4-9) [refer to the gain and stimulated emission in semiconductor lasers tutorial] can be used to obtain the intracavity photon density given by

$\tag{2-6-5}N_\text{ph}=\eta_\text{i}(\tau_\text{p}/qd)(J-J_\text{th})$

where the internal quantum efficiency $$\eta_\text{i}$$ is introduced phenomenologically, the photon lifetime $$\tau_\text{p}$$ is defined as

$\tag{2-6-6}\tau_\text{p}^{-1}=v_\text{g}(\alpha_\text{m}+\alpha_\text{int})$

and $$v_\text{g}=c/\mu_\text{g}$$ is the group velocity.

Equation (2-6-5) shows that once the threshold is reached, the intracavity photon density increases linearly with a further increase in $$J$$. The carrier-recombination time $$\tau_\text{e}$$ and the photon lifetime $$\tau_\text{p}$$ play an important role in determining the transient response of a semiconductor laser as discussed in later tutorials.

To obtain the $$L-I$$ curve, Equation (2-6-5) should be expressed in terms of the output power $$P_\text{out}$$ and the device current $$I$$. Since photons escape out of the laser cavity at a rate of $$v_\text{g}\alpha_\text{m}$$, the power emitted by each facet is related to the intracavity photon density by the relation

$\tag{2-6-7}P_\text{out}=\frac{1}{2}h\nu{v_\text{g}}\alpha_\text{m}VN_\text{ph}$

where $$V=Lwd$$ is the active volume and the factor of $$\frac{1}{2}$$ is due to the assumption of equal facet reflectivities so that the same power is emitted by the two facets. The power emitted from each facet becomes different when the reflectivities are changed by coating the laser facets.

Using Equations (2-6-1) and (2-6-7) in Equation (2-6-5), we obtain

$\tag{2-6-8}P_\text{out}=\frac{h\nu}{2q}\frac{\eta_\text{i}\alpha_\text{m}}{\alpha_\text{m}+\alpha_\text{int}}(I-I_\text{th}-\Delta{I_\text{L}})$

where Equation (2-6-6) was used to eliminate $$\tau_\text{p}$$. The quantity $$\Delta{I_\text{L}}$$ represents a possible increase in the leakage current with the current $$I$$. The threshold current $$I_\text{th}$$ is obtained using Equations (2-6-1) and (2-6-3) and is given by

$\tag{2-6-9}I_\text{th}=qVn_\text{th}/\tau_\text{e}+I_\text{L}$

Equation (2-6-8) expresses the laser output as a function of the current $$I$$, the external pumping parameter. It predicts that the output power should vary linearly with $$I$$.

The slope $$\text{d}P_\text{out}/\text{d}I$$ is a measure of the device efficiency, and a typical value for an InGaAsP laser emitting near 1.3 μm is 0.25 mW/mA per facet. In practice the slope efficiency does not remain constant, and the output power saturates for large values of $$I$$ (see Figure 2-6).

Equation (2-6-8) predicts that three factors may contribute to power saturation.

(i) The leakage current may increase with $$I$$ ($$\Delta{I}_\text{L}\ne0$$) so that a smaller fraction of the device current leads to carrier injection into the active layer.

(ii) The value of $$I_\text{th}$$ is current dependent and increases with $$I$$. A possible mechanism is junction heating that may reduce the carrier-recombination time $$\tau_\text{e}$$ as the laser power increases. The decrease in $$\tau_\text{e}$$ can be due to Auger recombination, which increases significantly with temperature.

(iii) The internal loss $$\alpha_\text{int}$$ increases with $$I$$ so that a smaller faction of generated photons is useful as the output power.

The relative importance of the three possible mechanisms is discussed in a later tutorial.

Before the power-saturation mechanisms set in, the slope $$\text{d}P_\text{out}/\text{d}I$$ is reasonably constant and may be used to obtain the differential quantum efficiency $$\eta_\text{d}$$ of the laser device. The differential (external) quantum efficiency is defined as

$\tag{2-6-10}\eta_\text{d}=\eta_\text{i}\frac{\text{photon escape rate}}{\text{photon generateion rate}}$

where $$\eta_\text{i}$$ is the internal quantum efficiency indicating what fraction of injected carriers is converted into photons.

To maintain the steady-state, the photons are generated at a rate of $$\tau_\text{p}^{-1}$$ and escape out of the laser cavity at a rate of $$v_\text{g}\alpha_\text{m}$$.

Using (2-6-6) and (2-6-10), we obtain

$\tag{2-6-11}\eta_\text{d}=\eta_\text{i}(v_\text{g}\alpha_\text{m}\tau_\text{p})=\eta_\text{i}\frac{\alpha_\text{m}}{\alpha_\text{m}+\alpha_\text{int}}$

If we use (2-6-8), we find that $$\eta_\text{d}$$ is directly proportional to the slope of the $$L-I$$ curve, i.e.,

$\tag{2-6-12}\eta_\text{d}=\frac{2q}{h\nu}\frac{\text{d}P_\text{out}}{\text{d}I}$

The internal quantum efficiency represents the fraction of injected carriers that recombine radiatively and generate photons. Using Equation (2-4-9) [refer to the gain and stimulated emission in semiconductor lasers tutorial], it is given by

$\tag{2-6-13}\eta_\text{i}=\frac{Bn^2+R_\text{st}N_\text{ph}}{A_\text{nr}n+Bn^2+cn^3+R_\text{st}N_\text{ph}}$

and is the ratio of the radiative to the total recombination rates. In the above-threshold regime the stimulated-recombination term $$R_\text{st}N_\text{ph}$$ dominates and $$\eta_\text{i}\approx1$$.

Using $$\alpha_\text{m}$$ from (2-3-10) [refer to the threshold condition and longitudinal modes in semiconductor lasers tutorial] in Equation (2-6-11), $$\eta_\text{d}$$ becomes

$\tag{2-6-14}\eta_\text{d}=\eta_\text{i}\left[1+\frac{\alpha_\text{int}L}{\ln(1/R_\text{m})}\right]^{-1}$

where $$R_\text{m}=R_1=R_2$$ and is the mirror reflectivity assumed to be equal for both facets.

Equation (2-6-14) was first obtained by Biard et al. and predicts that $$\eta_\text{d}$$ increases with a reduction in the cavity length $$L$$. The inverse dependence of $$\eta_\text{d}$$ on $$L$$ has been verified and can be used to estimate $$\eta_\text{i}$$.

For a typical strongly index guided 1.3-μm semiconductor laser with length $$L$$ of 250 μm, the slope efficiency $$\text{d}P_\text{out}/\text{d}I\approx0.25\text{mW/mA}$$; from Equation (2-6-12), $$\eta_\text{d}\approx0.5$$ if we assume that $$\eta_\text{i}\approx1$$. From Equation (2-6-14) this value corresponds to an internal loss $$\alpha_\text{int}$$ of about $$40\text{ cm}^{-1}$$.

The emitted power and the slope efficiency become different for the two facets when their reflectivities are made different by coating the laser facets. For instance, it is common to reduce the reflectivity of one facet through an antireflection coating to increase the output power from that facet. The other facet is often coated with a high-reflection coating in order to maintain the laser threshold.

The calculation of output powers emitted from each facet is slightly more involved since the photon density or the intracavity power is far from being uniform along the cavity length when the facet reflectivities are made different.

The average intracavity power is obtained by using

$\tag{2-6-15}P_\text{in}=(\alpha_\text{m}+\alpha_\text{int})\displaystyle\int\limits_0^L|A\exp(\text{i}\tilde{\beta}z)+B\exp(-i\tilde{\beta}z)|^2\text{d}z$

where $$A$$ and $$B$$ are the amplitudes of the forward and backward propagating waves normalized such that $$|A|^2$$ and $$|B|^2$$ represent the corresponding optical power at $$z=0$$. The complex propagation constant $$\tilde{\beta}$$ is obtained from Equations (2-3-2) and (2-3-4) [refer to the threshold condition and longitudinal modes in semiconductor lasers tutorial] and is given by

$\tag{2-6-16}\tilde{\beta}=\mu{k_0}-\frac{\text{i}}{2}(\Gamma{g}-\alpha_\text{int})$

The boundary conditions at the laser facets relate $$A$$ and $$B$$ such that

$\tag{2-6-17}A=\sqrt{R_1}B,\qquad{B}=\sqrt{R_2}A\exp(2\text{i}\tilde{\beta}L)$

where $$R_1$$ and $$R_2$$ are the reflectivities of the facets located at $$z=0$$ and $$z=L$$ respectively.

The threshold condition (2-3-6) [refer to the threshold condition and longitudinal modes in semiconductor lasers tutorial] is readily obtained from Equation (2-6-17). In particular, the gain $$g$$ in Equation (2-6-16) is given by Equation (2-3-9) or by $$\Gamma{g}=\alpha_\text{m}+\alpha_\text{int}$$, where the mirror loss $$\alpha_\text{m}$$ is defined as in Equation (2-3-10).

The integration in Equation (2-6-15) is easily performed. The interference term can be neglected in most cases of practical interest because of its rapidly oscillating nature. By using Equations (2-6-15) and (2-6-16) together with $$\alpha_\text{m}=\Gamma{g}-\alpha_\text{int}$$, the result is

$\tag{2-6-18}P_\text{in}=\frac{\alpha_\text{m}+\alpha_\text{int}}{\alpha_\text{m}}\frac{(\sqrt{R_1}+\sqrt{R_2})(1-\sqrt{R_1R_2})}{R_1\sqrt{R_2}}|A|^2$

The average intracavity power $$P_\text{in}$$ is related to the photon density $$N_\text{ph}$$ as

$\tag{2-6-19}P_\text{in}=h\nu{V}N_\text{ph}/\tau_\text{p}$

where $$h\nu{V}N_\text{ph}$$ represents the total energy.

Equations (2-6-18) and (2-6-19) can be used to obtain $$|A|^2$$ in terms of $$N_\text{ph}$$. The powers emitted from the facets located at $$z=0$$ and $$z=L$$ are related to $$|B|^2$$ and $$|A|^2$$ as

$\tag{2-6-20}P_1=(1-R_1)|B|^2,\qquad{P_2}=(1-R_2)|A\exp(\text{i}\tilde{\beta}L)|^2$

By using Equations (2-6-18)-(2-6-20) the output powers are related to $$N_\text{ph}$$ as

$\tag{2-6-21a}P_1=\frac{(1-R_1)\sqrt{R_2}}{(\sqrt{R_1}+\sqrt{R_2})(1-\sqrt{R_1R_2})}h\nu{v_\text{g}}\alpha_\text{m}VN_\text{ph}$

$\tag{2-6-21b}P_2=\frac{(1-R_2)\sqrt{R_1}}{(\sqrt{R_1}+\sqrt{R_2})(1-\sqrt{R_1R_2})}h\nu{v_\text{g}}\alpha_\text{m}VN_\text{ph}$

These equations should be compared with Equation (2-6-7) to which they reduce when the facet reflectivities are equal ($$R_1=R_2$$) since $$P_1=P_2=P_\text{out}$$ in that case.

Note that $$P_1+P_2=2P_\text{out}$$ for all values of $$R_1$$ and $$R_2$$, as it should since a change in facet reflectivities does not affect the total power but only its partition between the two facets. The reflectivity-dependent factor in Equations (2-6-21) shows how the total power is partitioned; it reduces to 1/2 when $$R_1=R_2$$.

The threshold current $$I_\text{th}$$ of a gain-guided laser is considerably higher than that of a strongly index guided device. Further, $$I_\text{th}$$ depends strongly on the stripe width $$w$$ over which the current is injected (see Figure 2-3).

The variation of $$I_\text{th}$$ with $$w$$ is shown in Figure 2-7 for 1.3-μm and 1.55-μm gain-guided InGaAsP lasers. The curves were obtained using a numerical model that includes carrier diffusion, current spreading, and various radiative and nonradiative recombination mechanisms. The inclusion of Auger recombination was essential to obtain a good fit between the theory and the experimental data.

The minimum value of the threshold current at 1.3-μm is more than 100 mA and occurs for $$w\approx10$$ μm. The rapid increase in $$I_\text{th}$$ for narrower stripes is due to the combined effect of index antiguiding and Auger recombination. The differential quantum efficiency $$\eta_\text{d}$$ is also lower compared to an index-guided device because of additional diffraction losses.

As discussed in the waveguide modes in semiconductor lasers tutorial, a weakly index guided device overcomes these shortcomings through a central region of a relatively higher index.

Compared to a strongly index guided device such as a buried heterostructure laser, the lateral index step $$\Delta\mu_\text{L}$$ is more than an order of magnitude smaller. Figure 2-8 shows the calculated variation of the threshold current $$I_\text{th}$$ and the differential quantum efficiency $$\eta_\text{d}$$ with the index step $$\Delta\mu_\text{L}$$ using parameters appropriate for a 1.3-μm InGaAsP laser.

In the absence of index guiding ($$\Delta\mu_\text{L}=0$$), the device is gain-guided with $$I_\text{th}\approx300$$ mA and $$\eta_\text{d}\approx0.35$$. As $$\Delta\mu_\text{L}$$ increases, $$I_\text{th}$$ rapidly decreases and $$\eta_\text{d}$$ increases. For $$\Delta\mu_\text{L}\ge5\times10^{-3}$$, the transition from gain guiding to index guiding is complete.

The slight decrease in $$\eta_\text{d}$$ with a further increase in $$\Delta\mu_\text{L}$$ is due to a mode profile that becomes narrower than the gain profile. This profile mismatch reduces the internal quantum efficiency since a part of the gain is not used for stimulated emission.

The important point to note is that an index step as small as $$\Delta\mu_\text{L}=5\times10^{-3}$$ is enough for index guiding. The limiting threshold current $$I_\text{th}$$ is higher for a weakly index guided device than that of a strongly index guided device because of a large mode volume, a weaker mode confinement, and increased carrier diffusion.

The threshold current $$I_\text{th}$$ and the differential quantum efficiency $$\eta_\text{d}$$ are two important quantities invariably used to characterize the performance of a semiconductor laser.

As Equations (2-6-2) and (2-6-11) show the mirror loss $$\alpha_\text{m}$$ associated with the cleaved-facet reflectivity $$R_\text{m}$$ plays a significant role in their determination.

Further, the selection of TE or TM modes in a heterostructure is also based on the mode reflectivity $$R_\text{m}$$, which is generally different for the two polarizations.

A considerable amount of theoretical work has been done to calculate the dependence of $$R_\text{m}$$ on the waveguide parameters such as the active-layer thickness $$d$$ and the index difference $$\mu_2-\mu_1$$. In general, a numerical approach is necessary.

The main feature is that the effect of dielectric waveguiding is to increase $$R_\text{m}$$ for the TE modes and to decrease $$R_\text{m}$$ for the TM modes relative to their plane-wave value (~0.32), which is the same for both sets of modes. Together with the confinement factor, the reflectivity difference helps to select the TE mode of operation in heterostructure semiconductor lasers.

## Spatial-Mode Characteristics

A semiconductor laser emits light in the form of a narrow spot of elliptical cross section. The spatial-intensity distribution of the emitted light near the laser facet is known as the near field. During its propagation the spot size grows in size due to beam divergence. The dimensions of the elliptical spot and its divergence angles, both parallel and perpendicular to the junction plane, are important beam parameters associated with the laser mode. The angular intensity distribution far from the laser facet is known as the far field.

In the waveguide modes in semiconductor lasers tutorial we solved the wave equation and associated the spatial distribution of the optical field with the transverse mode $$\phi(y)$$ and the lateral mode $$\psi(x)$$.

In general, several transverse and lateral modes may be excited, and the resulting near field is formed by a superposition of them. However, the active-layer dimensions are often chosen such that only the lowest-order transverse and lateral modes are supported by the waveguide. The near field just outside the laser facet can then be written as

$\tag{2-6-22}E=E_0\psi(x)\phi(y)\exp(\text{i}k_0z)$

where $$E_0$$ is a constant related to the output power ($$P_\text{out}\propto|E_0|^2$$). $$\psi(x)$$ and $$\phi(y)$$ are the lateral and transverse mode profiles discussed in the waveguide modes in semiconductor lasers tutorial.

Experimentally, $$|\psi(x)|^2$$ and $$|\phi(y)|^2$$ can be measured by scanning a photodetector across the beam dimensions. Their full widths at half maximum (FWHM), $$w_\parallel$$ and $$w_\perp$$ respectively, are taken to be a measure of the dimensions of the elliptical spot.

The far field is obtained by taking a two-dimensional Fourier transform of the near field given by Equation (2-6-22). It can be expressed as a product of the two one-dimensional Fourier transforms corresponding to the lateral and transverse directions $$x$$ and $$y$$.

The lateral far-field intensity distribution is given by

$\tag{2-6-23}|\tilde{\psi}_\text{FF}(\theta)|^2=\cos^2\theta\left|\displaystyle\int\limits_{-\infty}^{\infty}\psi(x)\exp(\text{i}k_0x\sin\theta)\text{d}x\right|^2$

with a similar expression for the transverse far field.

Again the quantity of practical interest is the width (FWHM) of the far field distribution. The angles $$\theta_\parallel$$ and $$\theta_\perp$$, corresponding to the lateral and transverse directions respectively, are used as a measure of the angular spread of the emitted light.

Near Field

The near field $$\phi(y)$$ perpendicular to the junction plane depends on the thickness and composition of the various layers used to make a heterostructure semiconductor laser.

For a three-layer waveguide mode, $$\phi(y)$$ is given by Equation (2-5-14) and is shown schematically in Figure 2-4 [refer to the waveguide modes in semiconductor lasers tutorial].

The width $$w_\perp$$ depends on the active-layer thickness $$d$$ and increases with a reduction in $$d$$ because of a loss in the strength of mode confinement. An analytic expression for $$w_\perp$$ is difficult to obtain. However, by fitting $$\phi(y)$$ with a Gaussian distribution, the width can be approximated by

$\tag{2-6-24}w_\perp\approx{d(2\ln2)^{1/2}}(0.321+2.1D^{-3/2}+4D^{-6})$

where the normalized thickness $$D$$ is given by Equation (2-5-26) [refer to the waveguide modes in semiconductor lasers tutorial]. The expression is reasonably accurate for $$1.8\lt{D}\lt6$$.

The near field $$\psi(x)$$ parallel to the junction plane depends on the lateral guiding mechanism and displays qualitatively different features for gain-guided and index-guide lasers.

For a strongly index guided laser such as a buried-heterostructure device, the near-field behavior is similar to that of $$\phi(y)$$, and Equation (2-6-24) can be applied after replacing $$D$$ with $$W$$, where $$W$$ is the normalized width given by Equation (2-5-43) [refer to the waveguide modes in semiconductor lasers tutorial].

Figure 2-9 shows typical near-field intensity profiles $$|\psi(x)|^2$$ obtained for a buried-heterostructure laser with an active layer 0.15 μm thick and 1.8 μm wide.

Curves for several drive currents are shown, and $$I_\text{th}=52$$ mA for this device. The width (FWHM) $$w_\parallel$$ is about 2 μm, indicating that the near field is largely confined within the active layer.

By contrast, the lateral near field for gain-guide lasers extends considerably beyond the strip width. For weakly index-guided lasers, the lateral confinement is remarkably improved for an index step $$\Delta\mu_\text{L}$$ as small as $$5\times10^{-3}$$, since the device then operates in the index-guided regime (see Figure 2-8).

Far Field

The far-field patterns in the directions parallel and perpendicular to the junction plane indicate the angular spread of the laser mode and are important in determining the coupling efficiency between the semiconductor laser and a fiber.

Figure 2-10 shows the far-field scans in the two directions for the same BH device of Figure 2-9. The angular widths $$\theta_\parallel$$ and $$\theta_\perp$$ at 64 mA are $$33^\circ$$ and $$57^\circ$$, respectively, and represent typical values for strongly index guided semiconductor lasers.

Mathematically the far-field pattern is obtained by taking the Fourier transform of the near field as indicated by Equation (2-6-23).

As one may expect, the lateral far field (parallel to the junction plane) depends strongly on the lateral guiding mechanism and exhibits qualitatively different behavior for gain-guide and for index-guide lasers.

In particular, for narrow-stripe gain-guided lasers the far field has two widely separated peaks with a minimum at the center. Such twin-lobe far fields are known to arise from the carrier-induced index reduction, which is governed by the antiguiding parameter $$\beta_\text{c}$$ (see Equations (2-4-4) and (2-4-6) [refer to the gain and stimulated emission of semiconductor lasers tutorial]), that leads to a curved wavefront for the lateral mode.

The introduction of weak index guiding has a dramatic effect on the far field, as shown in Figure 2-11 for a 1.3-μm InGaAsP laser. Only one of the two peaks for each curve is shown. As $$\Delta\mu_\text{L}$$ increases, the two peaks move closer and eventually merge to yield a single-lobe far-field pattern. The transition is almost complete for $$\Delta\mu_\text{L}=4\times10^{-3}$$, at which point the device is then index-guided (see also Figure 2-8).

## Spectral Characteristics

The power spectrum of a semiconductor laser is an important device characteristic since in many applications the spectral control of the laser output is required.

Below threshold the output takes the form of spontaneous emission with a large spectral width of ~ 30 nm. As the threshold is approached, the spectrum narrows considerably, and several peaks whose frequencies coincide with the longitudinal mode frequencies [see the threshold condition and longitudinal modes in semiconductor lasers tutorial] appear.

In the above-threshold regime, the longitudinal mode closest to the gain peak increases in power, while the power in remaining side peaks saturates. This behavior is shown in Figure 2-12 for a 1.3-μm buried-heterostructure InGaAsP laser.

A somewhat surprising feature of semiconductor lasers is that even though the gain profile is largely homogeneously broadened, many longitudinal modes oscillate simultaneously and a significant amount of laser power is carried out by side modes even in the above-threshold regime.

Several mechanisms such as spatial-hole burning, spectral-hole burning, and a high rate of spontaneous emission into the lasing mode may contribute to the observed spectral features. Later tutorials give a theoretical discussion based on the multimode rate equations.

For the application of a semiconductor laser in high-bit-rate, long-haul, optical fiber communication systems, the longitudinal side modes are undesirable because they cause pulse spreading in the presence of fiber dispersion, thereby degrading system performance.

An important parameter used to describe the spectral purity of a semiconductor laser is the mode suppression ratio (MSR). It is defined as the ratio of the main-mode power to the power carried by the most intense side mode.

At the 1.55-μm wavelength, MSR values exceeding 30 dB are generally desirable. However, mode discrimination in semiconductor lasers is usually poor, and such values of MSR are difficult to realize.

In recent years distributed-feedback and coupled-cavity mechanisms have been used to provide additional side-mode discrimination and to increase the MSR. Such lasers are collectively referred to as single-longitudinal-mode lasers since their power spectrum consists of a dominant single longitudinal mode. Later tutorials examine distributed-feedback and coupled-cavity semiconductor lasers in view of their increasing importance in optical communication systems.

Another quantity of practical interest is the spectral width of a single longitudinal mode. Even when a semiconductor laser operates predominantly in a single longitudinal mode, quantum fluctuations associated with the process of spontaneous emission lead to a broadening of the laser line. Spectral line width is typically in the range of 10-100 MHz and decreases inversely with the laser power.

The line width is calculated using single-mode rate equations generalized to take into account various noise mechanisms. A later tutorial gives a detailed discussion of the noise phenomena and the line width.

## Dynamic Characteristics

An understanding of the transient or dynamic response of a semiconductor laser is crucial since in some applications, particularly in optical fiber communications, the device current is modulated periodically and the laser output takes the pulse form.

When the laser is turned on by increasing the device current from 0 to its final above-threshold value, a few nanoseconds (a time interval known as the turn-on delay) elapse before the laser power starts increasing rapidly.

Further, because of the nonlinear nature of interaction between photons and charge carriers, the power does not increase monotonically. The laser output oscillates periodically for several nanoseconds before attaining its steady-state CW value. The oscillations are referred to as relaxation oscillations, and their frequency is governed by the nonlinear dynamics of photon-carrier interaction. The relaxation-oscillation frequency is typically in the gigahertz range and increases with an increase in the device current or the output power.

When the device current is modulated at frequencies approaching a few gigahertz, transient effects play an increasingly important role. The power distribution among various longitudinal modes changes with time. Under high-frequency direct modulation, the side modes are considerably enhanced compare to those in the CW case.

Further, the line width of an individual longitudinal mode increases considerably. The line broadening is related to the carrier-induced index change (governed by the parameter $$\beta_\text{c}$$ defined in Equation (2-4-6) [refer to the gain and stimulated emission in semiconductor lasers tutorial]) that leads to frequency chirping. A later tutorial considers the dynamic and modulation characteristics of semiconductor lasers in detail.

The next tutorial covers the topic of radiative recombination in semiconductors.