# Waveguide Modes in Semiconductor Lasers

This is a continuation from the previous tutorial - ** gain and stimulated emission in semiconductor lasers**.

The discussions in the gain and stimulated emission in semiconductor lasers tutorial is based on the plane-wave solutions of the wave equation (2-2-19) [refer to the Maxwell's equations for semiconductor lasers tutorial].

However, the light emitted by a laser has finite transverse dimensions, since it should be confined in the vicinity of the thin active region, which provides gain for stimulated emission.

In semiconductor lasers the output is in the form of a narrow beam with an elliptic cross section. Depending on the laser structure, the field distribution across the beam can take certain well-defined forms, often referred to as the ** laser modes**.

Mathematically, a laser mode is the specific solution of the wave equation (2-2-19) [refer to the Maxwell's equations for semiconductor lasers tutorial] that satisfies all the boundary conditions imposed by the laser structure.

In the general multimode case the optical field is denoted by \(E_{pqm}\), where the subscript \(m\) denotes the longitudinal or axial modes encountered in the threshold condition and longitudinal modes of semiconductor lasers tutorial. The subscripts \(p\) and \(q\) take integer values; they stand for the ** transverse** and

**modes specifying the field distribution in the direction perpendicular and parallel to the junction plane, respectively.**

*lateral*An understanding of the number of allowed modes and the resulting field distributions is essential for their control, since it is often desired to design semiconductor lasers that emit light predominantly in a single mode.

Further, important laser characteristics such as the near- and far-field widths depend on details of laser modes. For semiconductor lasers the far-field distribution plays an important role as it controls the amount of power coupled into a fiber.

In heterostructure semiconductor lasers the field confinement in the transverse direction, perpendicular to the junction plane, occurs through dielectric wave guiding. This mechanism is often referred to as ** index guiding** since the refractive-index discontinuity between the active and cladding layers is responsible for the mode confinement through the total internal reflection occurring at the interface.

However, field confinement in the lateral direction, which is parallel to the junction plane, is not always due to index guiding. Semiconductor lasers can be classified as gain-guided or index-guided depending on whether it is the lateral variation of the optical gain or the refractive index that confines the mode. Index-guided lasers can further be subclassified as weakly or strongly index guided depending on the magnitude of the lateral index step.

Figure 2-3 below shows schematically the three kinds of devices.

Historically, gain-guided devices based on the stripe geometry were developed first in view of their ease of fabrication. However, such devices have a number of undesirable characteristics that become worse as the laser wavelength increases. Index guiding is therefore almost invariably used in most practical semiconductor lasers.

## Effective Index Approximation

The mathematical description of the laser modes is based on the time-independent wave equation [see Equation (2-2-19) in the Maxwell's equations for semiconductor lasers tutorial]

\[\tag{2-5-1}\boldsymbol{\nabla}^2\mathbf{E}+\boldsymbol{\epsilon}(x,y)k_0^2\mathbf{E}=0\]

where the \(x\) axis is parallel and the \(y\) axis is perpendicular to the heterojunction in Figure 2-3.

The dielectric constant \(\boldsymbol{\epsilon}\) may vary with \(x\) and \(y\) but is assumed to be independent of \(z\), the direction of field propagation.

For some semiconductor lasers (e.g., distributed feedback lasers), \(\boldsymbol{\epsilon}\) varies with \(z\). As discussed in later tutorials, such variations are small enough that they can be ignored in the discussion of waveguide modes.

The spatially varying dielectric constant is generally of the form

\[\tag{2-5-2}\boldsymbol{\epsilon}(x,y)=\boldsymbol{\epsilon}_j(x)\]

where the subscript \(j\) numbers various layers in a heterostructure laser.

To account for the absorption, the dielectric constant \(\boldsymbol{\epsilon}_j\) is complex in each layer. Further, as shown in Equation (2-2-20) [refer to the Maxwell's equations for semiconductor lasers tutorial], within the active layer it also varies with external pumping.

To obtain an exact solution of Equation (2-5-1) is a difficult task. It is essential to make certain simplifying assumptions whose nature and validity vary from device to device.

For a strongly index guided laser such as the buried heterostructure laser shown schematically in Figure 2-3, the effect of gain or loss on the passive-cavity modes can often be ignored. The resulting rectangular waveguide problem can then be solved to obtain the transverse and lateral modes supported by the device.

This approach, however, is not suitable for gain-guided lasers where the lateral modes arise precisely because of the active-region gain.

An alternative approach is based on the effective index approximation. Instead of solving the two-dimensional wave equation, the problem is split into two one-dimensional parts whose solutions are relatively easy to obtain.

Such an approach is partially successful for both gain-guided and index-guided lasers. At the same time, it is helpful for a physical understanding of the guiding mechanism.

The physical motivation behind the effective index approximation is that often the dielectric constant \(\boldsymbol{\epsilon}(x,y)\) varies slowly in the lateral \(x\) direction compared to its variation in the transverse \(y\) direction.

To a good approximation, the slab-waveguide problem in the \(y\) direction can be solved for each \(x\) and the resulting solution can then be used to account for the lateral variation.

The electric field in Equation (2-5-1) is thus approximated by

\[\tag{2-5-3}\mathbf{E}\approx\hat{\mathbf{e}}\phi(y;x)\psi(x)\exp(\text{i}\beta{z})\]

where \(\beta\) is the propagation constant of the mode and \(\hat{\mathbf{e}}\) is the unit vector in the direction along which the mode is polarized.

On substituting Equation (2-5-3) in Equation (2-5-1) we obtain

\[\tag{2-5-4}\frac{1}{\psi}\frac{\partial^2\psi}{\partial{x^2}}+\frac{1}{\phi}\frac{\partial^2\phi}{\partial{y^2}}+[\boldsymbol{\epsilon}(x,y)k_0^2-\beta^2]=0\]

In the effective index approximation, the transverse field distribution \(\phi(y;x)\) is obtained first by solving

\[\tag{2-5-5}\frac{\partial^2\phi}{\partial{y^2}}+[\boldsymbol{\epsilon}(x,y)k_0^2-\beta_\text{eff}^2(x)]\phi=0\]

where \(\beta_\text{eff}(x)\) is the effective propagation constant for a fixed value of \(x\).

The lateral field distribution \(\psi(x)\) is then obtained by solving

\[\tag{2-5-6}\frac{\partial^2\psi}{\partial{x^2}}+[\beta_\text{eff}^2(x)-\beta^2]\psi=0\]

For a given laser structure Equation (2-5-5) and (2-5-6) can be used to obtain the transverse and lateral modes, respectively.

Since \(\boldsymbol{\epsilon}(x,y)\) is generally complex, \(\beta_\text{eff}(x)\) is also complex.

The effective index of refraction is defined as

\[\tag{2-5-7}\mu_\text{eff}(x)=\beta_\text{eff}(x)/k_0\]

and is itself complex.

Equation (2-5-5) is a one-dimensional eigenvalue equation and can be solved using the methods developed for dielectric slab waveguides.

Although it is in general possible to include the gain or loss occurring in each layer, the resulting analysis is cumbersome. A simpler approach is to treat the effect of gain or loss as a small perturbation to the eigenvalue problem. This is justified for heterostructure semiconductor lasers since the mode confinement in the \(y\) direction occurs mainly because of the index step at the heterostructure interfaces.

The dielectric constant \(\boldsymbol{\epsilon}(x,y)\) is of the form

\[\tag{2-5-8}\boldsymbol{\epsilon}(x,y)=\mu_\text{b}^2(y)+\Delta\boldsymbol{\epsilon}(x,y)\]

where \(\mu_\text{b}\) is the background (real) refractive index, constant for each layer. The small perturbation \(|\Delta\boldsymbol{\epsilon}|\ll\mu_\text{b}^2\) includes the loss and the contribution of external pumping.

If we use the first-order perturbation theory, the eigenvalue given by \(\mu_\text{eff}\) becomes

\[\tag{2-5-9}\mu_\text{eff}(x)=\mu_\text{e}(x)+\Delta\mu_\text{e}(x)\]

where \(\mu_\text{e}(x)\) is obtained by solving the unperturbed eigenvalue equation

\[\tag{2-5-10}\frac{\text{d}^2\phi}{\text{d}y^2}+k_0^2[\mu_\text{b}^2(y)-\mu_\text{e}^2(x)]\phi=0\]

and the perturbation \(\Delta\mu_\text{e}\) is obtained using

\[\tag{2-5-11}2\mu_\text{e}\Delta\mu_\text{e}=\frac{\displaystyle\int\limits_{-\infty}^\infty\Delta\boldsymbol{\epsilon}(x,y)\phi^2(y;z)\text{d}y}{\displaystyle\int\limits_{-\infty}^\infty\phi^2(y;x)\text{d}y}\]

Since \(\Delta\boldsymbol{\epsilon}(x,y)\) is constant within each layer, Equation (2-5-11) can be simplified to become

\[\tag{2-5-12}\Delta\mu_\text{e}(x)=\frac{1}{2\mu_\text{e}}\sum_j\Gamma_j(x)\Delta\boldsymbol{\epsilon}_j(x)\]

where the sum is over the number of layers, \(\Delta\boldsymbol{\epsilon}_j\) is the dielectric perturbation of \(j\)th layer, and

\[\tag{2-5-13}\Gamma_j(x)=\frac{\displaystyle\int\limits_{d_j}\phi^2(y;x)\text{d}y}{\displaystyle\int\limits_{-\infty}^\infty\phi^2(y;x)\text{d}y}\]

is the fraction of the mode intensity contained in that layer.

For the active layer, \(\Gamma_j\) is referred to as the confinement or filling factor since it indicates the extent to which the mode is confined to the active region.

Both \(\Gamma_j\) and the effective index \(\mu_\text{e}\) vary with \(x\) if the active layer is not laterally uniform in thickness. This is the case, for example, for a channeled-substrate device discussed in later tutorials.

## Transverse Modes

The transverse modes are obtained by solving Equation (2-5-10) and depend on the thickness and refractive indices of the various layers used to fabricate a semiconductor laser.

The number of layers to be considered depends on the specific laser structure, and its is often necessary to consider four or five layers for a reasonably accurate description of the transverse modes.

However, the basic concepts involved in dielectric waveguiding may be understood using a symmetric three-layer slab waveguide shown schematically in Figure 2-4.

The active layer of thickness \(d\) is surrounded on both sides by cladding layers. If the cladding layers are sufficiently thick such that the mode is largely confined within the three layers, the remaining layers can be ignored.

The slab-waveguide problem has been extensively studied. Starting from Maxwell's equations (2-2-1) - (2-2-4) [refer to the Maxwell's equations for semiconductor lasers tutorial], a slab waveguide is found to support two sets of modes, the TE and TM modes, which are distinguished on the basis of their polarization.

For TE modes the electric field \(E\) is polarized along the heterojunction plane, i.e., the polarization vector in Equation (2-5-3) is along the \(x\) axis. For TM modes, it is the magnetic field \(H\) that is polarized along the \(x\) axis.

In both cases the same equation (2-5-10) can be used to obtain the field distribution.

The boundary condition that the tangential components of the electric and magnetic fields be continuous at the dielectric interface requires that for TE modes \(\phi\) and \(\text{d}\phi/\text{d}y\) should match at \(|y|=d/2\) (see Figure 2-4).

The other boundary condition for the guided modes is that the field \(\phi(y)\) should vanish as \(y\) tends to infinity.

It should be pointed out that a slab waveguide also supports unguided modes, the so-called radiation modes, for which the latter boundary condition does not apply.

For a semiconductor laser, these radiation modes are not of significant interest and would not be considered. However, their inclusion is crucial if an arbitrary optical field is expanded in terms of a complete set of waveguide modes.

The mode analysis is considerably simplified if the even and odd solutions of the eigenvalue equation (2-5-10) are considered separately.

Let us first consider the even TE modes. A general solution of Equation (2-5-10) is of the form

\[\tag{2-5-14}\phi(y)=\begin{cases}A_\text{e}\cos(\kappa{y})\quad\qquad\qquad\qquad\text{for }|y|\le{d/2}\\B_\text{e}\exp[-\gamma(|y|-d/2)]\qquad\text{for }|y|\ge{d/2}\end{cases}\]

where

\[\tag{2-5-15}\kappa=k_0(\mu_2^2-\mu_\text{e}^2)^{1/2}\]

\[\tag{2-5-16}\gamma=k_0(\mu_\text{e}^2-\mu_1^2)^{1/2}\]

and \(\mu_2\) and \(\mu_1\) are the material refractive indices for the active and cladding layers, respectively, with \(\mu_2\gt\mu_1\).

The continuity of \(\phi\) and \(\text{d}\phi/\text{d}y\) at \(|y|=d/2\) requires that

\[\tag{2-5-17}B_\text{e}=A_\text{e}\cos(\kappa{d/2})\]

\[\tag{2-5-18}\gamma{B_\text{e}}=\kappa{A_\text{e}}\sin(\kappa{d/2})\]

If we divide Equation (2-5-18) by Equation (2-5-17), we obtain the eigenvalue equation

\[\tag{2-5-19}\gamma=\kappa\tan(\kappa{d/2})\]

whose solutions yield the effective mode index \(\mu_\text{e}\). In general, multiple solutions are possible corresponding to different even TE modes.

A similar analysis can be carried out for the odd TE modes with the only difference that the \(\cos(\kappa{y})\) in Equation (2-5-14) is replaced by \(\sin(\kappa{y})\). The application of the boundary conditions yields the eigenvalue equation

\[\tag{2-5-20}\gamma=-\kappa\cot(\kappa{d/2})\]

whose solutions yield \(\mu_\text{e}\) for the odd TE modes.

For all guided modes, the inequality \(\mu_2\gt\mu_\text{e}\gt\mu_1\) is satisfied.

The TM modes are obtained using the same procedure. The only difference lies in the application of the boundary conditions.

The continuity of the tangential components of the electric field, \(E_z\), requires that \(\mu_j^{-2}(\text{d}\phi/\text{d}y)\) be continuous across the heterostructure interfaces at \(|y|=d/2\). Here \(\mu_j\) is either \(\mu_1\) or \(\mu_2\) depending on the side from which the interface is approached.

The origin of this difference between the TE and TM modes can be traced back to Maxwell's equations (2-2-1) and (2-2-2) [refer to the Maxwell's equations for semiconductor lasers tutorial].

Similar to TE-mode analysis, the even and odd TM modes are considered separately. The resulting eigenvalue equations are

\[\tag{2-5-21}\mu_2^2\gamma=\mu_1^2\kappa\tan(\kappa{d/2})\]

\[\tag{2-5-22}\mu_2^2\gamma=-\mu_1^2\kappa\cot(\kappa{d/2})\]

for the even and odd TM modes respectively.

In heterostructure semiconductor lasers, the TE modes are generally favored over the TM modes since the facet reflectivity is higher for TE modes, as discussed in later tutorials.

In the following discussion we therefore consider only TE modes.

The TE-mode eigenvalues are obtained using Equations (2-5-19) and (2-5-20) together with the relation

\[\tag{2-5-23}\kappa^2+\gamma^2=k_0^2(\mu_2^2-\mu_1^2)\]

obtained by squaring and adding Equations (2-5-15) and (2-5-16).

Equation (2-5-23) describes a circle in the \(\kappa-\gamma\) plane, and its intersection with the curves obtained using Equations (2-5-19) and (2-5-20) yields \(\kappa_p\) and \(\gamma_p\) values of the \(p\)th TE mode.

Multiple solutions occur because of the periodic nature of trigonometric functions. The number of allowed waveguide modes can be determined by noting that a solution is no longer bounded if \(\gamma\le0\) since it leads to an exponential growth of the field distribution \(\phi\) in the cladding layers.

The cut-off condition is thus determined by \(\gamma=0\) and occurs when

\[\tag{2-5-24}\kappa{d}=p\pi\]

where \(p\) is an integer whose even and odd values correspond to even and odd TE modes governed respectively by Equations (2-5-19) and (2-5-20).

If we substitute \(\kappa\) from (2-5-23) after using \(\gamma=0\), we obtain the simple relation

\[\tag{2-5-25}D=p\pi\]

where

\[\tag{2-5-26}D=k_0(\mu_2^2-\mu_1^2)^{1/2}d\]

is the normalized waveguide thickness.

This parameter plays a central role in determining the mode characteristics of a three-layer slab waveguide. In particular, if \(D\lt\pi\), the waveguide can support only the lowest-order (\(p=0\)), fundamental TE mode.

A heterostructure semiconductor laser can therefore be made to emit light in a single transverse mode if the active layer thickness \(d\) is chosen to satisfy the condition \(D\lt\pi\) or from Equation (2-5-26) after using \(k_0=2\pi/\lambda\),

\[\tag{2-5-27}d\lt\frac{\lambda}{2}(\mu_2^2-\mu_1^2)^{-1/2}\]

Since typically \(d\lesssim0.2\) μm, the single-transverse-mode condition is almost always satisfied in practical devices.

For an InGaAsP laser with InP cladding layers, it has been noted that to a good approximation

\[\tag{2-5-28}\lambda(\mu_2^2-\mu_1^2)^{-1/2}\approx0.95\text{ μm}\]

for the wavelength range of 1.1-1.65 μm. Condition (2-5-27) then simply becomes \(d\lt0.48\) μm.

Similarly, using Equation (2-5-26) and (2-5-28), one finds that

\[\tag{2-5-29}D\approx6.6d\]

where \(d\) is the active-layer thickness in micrometers.

This relation is independent of \(\lambda\), \(\mu_1\), and \(\mu_2\) in the entire wavelength region of 1.1-1.65 μm for InGaAsP lasers with InP cladding layers.

As mentioned before, a quantity that plays an important role for heterostructure semiconductor lasers is the transverse confinement factor \(\Gamma_\text{T}\) because it represents the fraction of the mode energy within the active layer that is available for interaction with the injected charge carriers.

Using \(\phi(y)\) from Equation (2-5-14) in

\[\tag{2-5-30}\Gamma_\text{T}=\frac{\displaystyle\int\limits_{-d/2}^{d/2}\phi^2(y)\text{d}y}{\displaystyle\int\limits_{-\infty}^{\infty}\phi^2(y)\text{d}y}\]

and carrying out the integrations, we obtain

\[\tag{2-5-31}\Gamma_\text{T}=\frac{1+2\gamma{d}/D^2}{1+2/\gamma{d}}\]

The evaluation of \(\Gamma_\text{T}\) requires the knowledge of \(\kappa\) and \(\gamma\), and in general it is necessary to solve the eigenvalue equation (2-5-19) numerically.

For the fundamental transverse mode, however, a remarkably simple expression

\[\tag{2-5-32}\Gamma_\text{T}\approx{D}^2/(2+D^2)\]

was found to be accurate to within 1.5%, and can be used to obtain \(\Gamma_\text{T}\) for any slab waveguide with the help of Equation (2-5-26).

For an InGaAsP laser with a 0.15-μm thick active layer, \(D\approx1\) from Equation (2-5-29) and \(\Gamma_\text{T}\approx\frac{1}{3}\).

In Figure 2-5 we have shown the variation of \(\Gamma_\text{T}\) with the active-layer thickness for a 1.3-μm InGaAsP laser. The transverse confinement factor for the fundamental TM mode is also shown for comparison. It is found to be lower, indicating that the TE mode is confined more than the TM mode.

Finally, the effective index \(\mu_\text{e}\) of the fundamental TE mode can be approximated by

\[\tag{2-5-33}\mu_\text{e}^2\approx\mu_1^2+\Gamma_\text{T}(\mu_2^2-\mu_1^2)\]

Both \(\mu_\text{e}\) and \(\Gamma_\text{T}\) are required for our discussion of the lateral modes.

## Lateral Modes

The lateral modes are obtained by solving Equation (2-5-6) which after using Equation (2-5-7) and (2-5-9) becomes

\[\tag{2-5-34}\frac{\partial^2\psi}{\partial{x^2}}+\{k_0^2[\mu_\text{e}(x)+\Delta\mu_\text{e}(x)]^2-\beta^2\}\psi=0\]

The lateral-mode behavior in semiconductor lasers is different depending on whether gain guiding or index guiding is used to confine the lateral modes.

In a gain-guided device, \(\mu_\text{e}(x)\) is a constant given by Equation (2-5-33). By contrast, in an index-guided device, structural lateral variations are used to make \(\mu_\text{e}\) larger in a central region of width \(w\).

For the latter case the slab-waveguide problem discussed in the transverse modes section above is solved separately in the two regions, and

\[\tag{2-5-35}\mu_\text{e}(x)=\begin{cases}\mu_\text{e}^\text{in}\qquad\text{if }|x|\le{w/2}\\\mu_\text{e}^\text{out}\qquad\text{otherwise}\end{cases}\]

where \(\mu_\text{e}^\text{in}\) and \(\mu_\text{e}^\text{out}\) are the effective indices corresponding to the two regions. Their magnitude depends on structural details, and the lateral index step

\[\tag{2-5-36}\Delta\mu_\text{L}=\mu_\text{e}^\text{in}-\mu_\text{e}^\text{out}\]

determines the extent of index guiding.

Whether the lateral mode is index-guided or gain-guided depends on the relative magnitudes of \(\Delta\mu_\text{L}\) and \(\Delta\mu_\text{e}(x)\), and in general both should be considered.

**Strongly Index Guided Semiconductor Lasers**

Index-guided semiconductor lasers can often be classified as strongly index guided or weakly index guided depending on the structural modifications that gie rise to \(\Delta\mu_\text{L}\).

A buried heterostructure device (see Figure 2-3) falls in the category of strongly index guided devices since the index step \(\Delta\mu_\text{L}\gg|\Delta\mu_\text{e}(x)|\).

In that case the effect of gain can be treated as a small perturbation to the index-guided lateral mode and one can follow a perturbation procedure similar to that outlined in Equations (2-5-9)-(2-5-13).

The mode-propagation constant is given by

\[\tag{2-5-37}\beta=k_0\bar{\mu}+\text{i}\bar{\alpha}/2\]

where \(\bar{\mu}\) and \(\bar{\alpha}\) are the refractive index and the absorption coefficient of the mode supported by the rectangular waveguide of width \(w\) and thickness \(d\).

The lateral modes are obtained by solving the three-layer slab-waveguide problem

\[\tag{2-5-38}\frac{\partial^2\psi}{\partial{x^2}}+k_0^2[\mu_\text{e}^2(x)-\bar{\mu}^2]\psi=0\]

The mode-absorption coefficient \(\bar{\alpha}\) is obtained using the first-order perturbation theory and is given by

\[\tag{2-5-39}\bar{\alpha}=\frac{k_0}{\bar{\mu}}\text{Im}\left(\frac{\displaystyle\int2\mu_\text{e}(x)\Delta\mu_\text{e}(x)\psi^2(x)\text{d}x}{\displaystyle\int\psi^2(x)\text{d}x}\right)\]

where \(\Delta\mu_\text{e}(x)\) is given by Equation (2-5-12).

For the symmetric three-layer waveguide considered in the Transverse Modes section above, \(\Gamma_2=\Gamma_\text{T}\), and \(\Gamma_1=\Gamma_3=(1-\Gamma_\text{T})/2\).

If we use Equations (2-2-20)-(2-2-25) [refer to the Maxwell's equations for semiconductor lasers tutorial] to obtain \(\Delta\boldsymbol{\epsilon}_j\) for the active and cladding layers, \(\Delta\mu_\text{e}(x)\) becomes

\[\tag{2-5-40}\Delta\mu_\text{e}=\frac{1}{2\mu_\text{e}}[\Gamma_\text{T}\mu_2(2\Delta\mu_\text{p}-\text{i}g/k_0)+\text{i}(1-\Gamma_\text{T})\mu_1\alpha_\text{c}/k_0]\]

where \(\alpha_\text{c}\) is the cladding-layer absorption coefficient, \(\Delta\mu_\text{p}\) is the carrier-induced index change given by Equation (2-3-3) [refer to the threshold condition and longitudinal modes of semiconductor lasers tutorial], and \(g\) is the active-layer gain given by Equation (2-3-4) [refer to the threshold condition and longitudinal modes of semiconductor lasers tutorial].

In general, both \(\Delta\mu_\text{p}\) and \(g\) vary with \(x\) due to carrier diffusion. For buried heterostructure lasers (see Figure 2-3), only the central region of width \(w\) is externally pumped. Furthermore, if \(w\) is small compared to the diffusion length, the carrier density is approximately uniform across the active width. In that case, \(g\) and \(\Delta\mu_\text{p}\) can be assumed to be independent of \(x\) for \(|x|\le{w/2}\). Outside the active region, \(g=\alpha_\text{c}\).

Substituting Equation (2-5-40) in Equation (2-5-39) and carrying out the integrations, we obtain the simple expression

\[\tag{2-5-41}\bar{\alpha}\approx-\Gamma_\text{L}\Gamma_\text{T}g+(1-\Gamma_\text{L}\Gamma_\text{T})\alpha_\text{c}\]

where we used the approximation \(\bar{\mu}\approx\mu_1\approx\mu_2\) and

\[\tag{2-5-42}\Gamma_\text{L}=\frac{\displaystyle\int\limits_{-w/2}^{w/2}|\psi(x)|^2\text{d}x}{\displaystyle\int\limits_{-\infty}^{\infty}|\psi(x)|^2\text{d}x}\]

is the lateral confinement factor.

The lateral modes are obtained by solving the waveguide problem governed by Equations (2-5-35) and (2-5-38).

Note that the injected carriers reduce \(\mu_\text{e}^\text{in}\) by a small amount approximately given by \(\Gamma_\text{T}\Delta\mu_\text{p}\) [see Equation (2-5-40)]. For strongly index guided lasers, this is a minor effect and can be neglected.

It is clear from the form of Equation (2-5-35) that a three-layer waveguide problem is to be solved. The properties of the lateral modes for a strongly index guided buried heterostructure laser can therefore by described using the results of the Transverse Modes section above.

In analogy to Equations (2-5-25) and (2-5-26), one can define the normalized waveguide width \(W\) as

\[\tag{2-5-43}W=k_0w[(\mu_\text{e}^\text{in})^2-(\mu_\text{e}^\text{out})^2]^{1/2}\]

and the cut-off condition becomes

\[\tag{2-5-44}W=q\pi\]

where \(q\) is an integer whose even and odd values correspond to even and odd lateral modes, respectively.

In particular, only the lowest-order (\(q=0\)) lateral mode is supported by the waveguide if \(W\le\pi\) or equivalently, the active-layer width

\[\tag{2-5-45}w\le\lambda/(8\mu_\text{e}\Delta\mu_\text{L})^{1/2}\]

where \(\Delta\mu_\text{L}\) is the lateral index step given by Equation (2-5-36) and \(\mu_\text{e}\) is the average effective index.

In analogy to Equations (2-5-32) and (2-5-33), the lateral confinement factor \(\Gamma_\text{L}\) and the mode refractive index \(\bar{\mu}\) for the fundamental lateral mode are given by

\[\tag{2-5-46}\Gamma_\text{L}\approx{W^2}/(2+W^2)\]

and

\[\tag{2-5-47}\bar{\mu}^2\approx(\mu_\text{e}^\text{out})^2+\Gamma_\text{L}[(\mu_\text{e}^\text{in})^2-(\mu_\text{e}^\text{out})^2]\]

The description of waveguiding in strongly index-guided lasers is now complete.

If the active-layer dimensions are chosen to satisfy Equations (2-5-27) and (2-5-45), only a single transverse and lateral mode is supported by the passive waveguide. Its complex propagation constant is given by Equation (2-5-37), where the mode index \(\bar{\mu}\) is obtained using Equation (2-5-47). The mode loss is governed by \(\bar{\alpha}\) given by Equation (2-5-41), which can be rewritten as

\[\tag{2-5-48}\bar{\alpha}=-\Gamma{g}+\alpha_\text{int}\]

where \(\Gamma=\Gamma_\text{L}\Gamma_\text{T}\) and is the mode confinement factor and \(\alpha_\text{int}\) is the internal loss due to absorption in the cladding layers.

Equation (2-5-48) should be compared with Equation (2-3-5) [refer to the threshold condition and longitudinal modes of semiconductor lasers tutorial], its counterpart obtained in the plane-wave approximation.

Whereas the confinement factor \(\Gamma\) in Equation (2-3-5) [refer to the threshold condition and longitudinal modes of semiconductor lasers tutorial] was introduced phenomenologically, in the waveguide description

\[\tag{2-5-49}\Gamma=\Gamma_\text{L}\Gamma_\text{T}\]

where \(\Gamma_\text{T}\) and \(\Gamma_\text{L}\) are the transverse and lateral confinement factors defined by Equations (2-5-30) and (2-5-42), respectively.

Clearly \(\Gamma\) represents the fraction of the mode energy contained within the active region. For typically used values \(w\approx2\) μm, \(\Gamma_\text{L}\approx1\) and \(\Gamma_\text{T}\) can be used for \(\Gamma\).

The internal loss \(\alpha_\text{int}\) in Equation (2-5-48) should be generalized to include all sources of loss. Its suitable form is

\[\tag{2-5-50}\alpha_\text{int}=\Gamma\alpha_\text{a}+(1-\Gamma)\alpha_\text{c}+\alpha_\text{scat}\]

where \(\alpha_\text{a}\) is the active-layer loss (mainly due to free-carrier absorption) and \(\alpha_\text{scat}\) is the scattering loss at heterostructure interfaces.

**Gain-guided Semiconductor Lasers**

For gain-guided devices the effective index \(\mu_\text{e}\) in Equation (2-5-34) is constant along the lateral direction \(x\), and the mode confinement occurs through \(\Delta\mu_\text{e}(x)\) given by Equation (2-5-12).

In contrast to index-guided lasers where the index changes discontinuously [see equation (2-5-35)], \(\Delta\mu_\text{e}(x)\) varies continuously.

For the three-layer waveguide model, \(\Delta\mu_\text{e}(x)\) is obtained using Equation (2-5-40). Equation (2-5-34) then becomes

\[\tag{2-5-51}\frac{\partial^2\psi}{\partial{x^2}}+k_0^2[\mu_\text{e}^2+\Gamma\mu_2(2\Delta\mu_\text{p}-\text{i}g/k_0)+\text{i}\mu_1\alpha_\text{int}/k_0]\psi=\beta^2\psi\]

where \(\alpha_\text{int}\) is given by Equation (2-5-50) and \(|\Delta\mu_\text{e}|\ll\mu_\text{e}\) is assumed. Both \(g\) and \(\Delta\mu_\text{p}\) vary with the carrier density \(n\) as given by Equations (2-4-3) and (2-4-4) [refer to the gain and stimulated emission of semiconductor lasers tutorial].

The most distinctive feature of gain-guided lasers is that carrier diffusion plays an important role and \(n\), obtained by solving Equation (2-4-7) [refer to the gain and stimulated emission of semiconductor lasers tutorial], is laterally nonuniform.

It is precisely the variation in the optical gain \(g(x)\) that leads to the confinement of the lateral modes.

For a known carrier-density profile \(n(x)\), the eigenvalue equation (2-5-51) can be solved to obtain the complex propagation constant \(\beta\) and the mode profile \(\psi(x)\) corresponding to various lateral modes.

The problem of lateral-mode determination for a gain-guided semiconductor laser is, however, exceedingly complex, and in general a numerical approach is necessary.

This is so because stimulated emission couples the carrier-diffusion equation (2-4-7) [refer to the gain and stimulated emission in semiconductor lasers tutorial] and the wave equation (2-5-51), and the two should be solved self-consistently for each value of the device current.

A further complication is that the injected current density \(J(x)\) is itself laterally nonuniform because of current spreading in the cladding layer below the contact stripe (see Figure 2-3).

Several numerical models have been developed for gain-guided lasers and are reasonably successful in predicting the operating characteristics of a gaind-guided semiconductor laser.

Most of these models have been applied to AlGaAs lasers. They can be generalized for InGaAsP and other long-wavelength semiconductor lasers with some modifications such as the inclusion of Auger recombination. A combination of increased Auger recombination and increased index antiguiding [governed by the parameter \(\beta_\text{c}\) defined in Equation (2-4-6)] makes gain-guided semiconductor lasers relatively unattractive at wavelengths greater than 1 μm.

To obtain some physical insight in the process of gain guiding, it is useful to consider a specific carrier-density profile. One excellent example is provided by the parabolic profile

\[\tag{2-5-52}n(x)=n(0)-n_2x^2\]

which is often a good approximation near the stripe center at \(x=0\).

If we substitute Equation (2-5-52) in Equations (2-4-3) and (2-4-4) [refer to the gain and stimulated emission in semiconductor lasers tutorial] and then use \(g\) and \(\Delta\mu_\text{p}\) in Equation (2-5-51), we obtain

\[\tag{2-5-53}\frac{\partial^2\psi}{\partial{x^2}}+k_0^2[\boldsymbol{\epsilon}(0)-\bar{a}^2x^2]\psi=\beta^2\psi\]

where

\[\tag{2-5-54}\boldsymbol{\epsilon}(0)=\epsilon_\text{0r}+\text{i}\epsilon_\text{0i}\]

is the complex dielectric constant at the stripe center and

\[\tag{2-5-55}\bar{a}=a_\text{r}+\text{i}a_\text{i}\]

governs its quadratic decrease with \(x\).

The parameters \(\epsilon_\text{0r}\), \(\epsilon_\text{0i}\), \(a_\text{r}\), and \(a_\text{i}\) are determined by the material constants and phenomenological parameters introduced in the gain and stimulated emission in semiconductor lasers tutorial.

The eigenvalue equation (2-5-53) can be readily solved to obtain the eigenfunctions \(\psi(x)\) and the eigenvalues \(\beta\) corresponding to various lateral modes labeled by the subscript \(q\).

The lateral modes are the well-known Hermite-Gaussian functions given by

\[\tag{2-5-56}\psi_q(x)=H_q[(k_0\bar{a})^{1/2}x]\exp(-\frac{1}{2}k_0\bar{a}x^2)\]

where \(H_q(z)\) is the \(q\)th-order Hermite polynomial for its complex argument \(z\).

The corresponding eigenvalues are

\[\tag{2-5-57}\beta_q^2\approx{k}_0^2\boldsymbol{\epsilon}(0)-(2q+1)\bar{a}k_0\]

The real and imaginary parts of \(\beta_q\) can be used to obtain the mode index \(\bar{\mu}\) and the absorption coefficient \(\bar{\alpha}\) using Equation (2-5-37).

The Hermite-Gaussian modes in a homostructure semiconductor laser were observed as early as 1967. However, their origin was unclear at that time and a quadratic variation of the refractive index was used to explain the experimental results.

It was only in 1973 that the mechanism of gain guiding was identified as being responsible for the lateral confinement of the optical mode in stripe-geometry heterostructure semiconductor lasers.

A characteristic feature of gain guiding is that the phase front is curved, in contrast to index guiding that leads to a planar phase front [see Equation (2-5-14)]. This can be seen in Equation (2-5-56) by noting that \(\bar{a}\) is complex.

For the lowest-order lateral mode (\(q=0\)), the phase varies quadratically. Its curvature can be accounted by defining a radius of curvature \(R_\text{c}\) tha tis given by

\[\tag{2-5-58}R_\text{c}=\bar{\mu}/a_\text{i}\]

where \(\bar{\mu}\) is the mode index of the fundamental lateral mode.

Physically, the origin of a curved wavefront in gain-guided devices can be traced back to the carrier-induced index change \(\Delta\mu_\text{p}\) given by Equation (2-4-4) [refer to the gain and stimulated emission in semiconductor lasers tutorial].

The antiguiding parameter \(\beta_\text{c}\) defined by Equation (2-4-6) [refer to the gain and stimulated emission in semiconductor lasers tutorial] therefore plays an important role in gain-guided devices, and it is easy to verify that \(a_\text{i}=0\) if \(\beta_\text{c}=0\).

The number of lateral modes excited in a gain-guided device depends on the stripe width \(w\), which is a rough equivalent of the active-layer width \(w\) used in the discussion of index-guided devices.

Although the quadratic model used above does not predict a mode cut-off condition such as given by Equation (2-5-45), experimentally it is observed that only the fundamental lateral mode is excited when \(w\lesssim10\) μm.

For narrow-stripe lasers, the parabolic model is generally not suitable and a numerical solution of Equation (2-5-51) is often necessary. Some analytical insight can be gained using an alternative model that assumes that the complex dielectric constant varies as \(\text{sech}^2(x)\).

**Weakly Index Guided Semiconductor Lasers**

Gain-guided semiconductor lasers have an inherent drawback. The lateral-gain profile responsible for the mode confinement in general changes with external pumping, and the lateral-mode control is difficult to achieve in the above-threshold regime.

Strongly index guided lasers such as the buried heterostructure do not suffer from this problem but are relatively difficult to fabricate.

During the development of GaAs lasers it was realized that some index guiding can be induced in gain-guided lasers if the cladding-layer thickness is designed to be laterally nonuniform (see Figure 2-3).

A number of structures, including the rib waveguide, ridge waveguide, and channeled-substrate planar waveguide devices, have been proposed. We shall collectively call them weakly index guided devices.

From the viewpoint of the lateral-mode control, the important point is that the effective index \(\mu_\text{e}\) in Equation (2-5-34) is slightly larger over a narrow central region of width \(w\).

It should be stressed that \(w\) is not the active-layer width (see Figure 2-3). Further, carrier diffusion inside the active layer is responsible for the continuous lateral variation of the gain and the refractive index.

The mathematical description of lateral modes is based on Equation (2-5-34). The lateral variation of \(\mu_\text{e}\) is represented by Equation (2-5-35). The index step \(\Delta\mu_\text{L}\) given by (2-5-36) governs the extent of index guiding.

However, in contrast to strongly index guided lasers where \(\Delta\mu_\text{L}\gg|\Delta\mu_\text{e}|\), now the two terms are comparable in magnitude and should be considered simultaneously. The resulting lateral-mode behavior is complex since gain guiding, carrier-induced antiguiding, and built-in index guiding all participate in the formation of lateral modes.

A numerical solution of Equation (2-5-34) is generally necessary to analyze a weakly index guided device. Depending on the magnitude of \(\Delta\mu_\text{L}\), such a device exhibits lateral-mode features reminiscent of pure gain guiding or pure index guiding.

As \(\Delta\mu_\text{L}\) progressively increases, a transition from gain guiding to index guiding occurs. Clearly the device is of practical use only when it is designed to operate in the index-guided regime.

Numerical calculations for a 1.3-μm InGaAsP laser shows that an index step \(\Delta\mu_\text{L}\) in the range of 0.005-0.01 is enough to achieve a stable lateral mode.

The next tutorial covers the topic of ** emission characteristics of semiconductor lasers**.