# DFB (Distributed Feedback) Semiconductor Lasers

This is a continuation from the previous tutorial - ** effects of external optical feedback on semiconductor lasers**.

## Introduction to distributed-feedback semiconductor lasers

As we have seen in the previous tutorials a conventional semiconductor laser does not emit light in a single longitudinal mode. In general, the mode closest to the gain peak is most intense, and a few percent of the output power is carried by other longitudinal modes lying close to the gain peak.

Furthermore, even when these side modes are reasonably suppressed under CW operation, their power content increases significantly when the laser is pulsed rapidly. Direct modulation of a semiconductor laser at frequencies in the gigahertz range is commonly employed in optical communication systems. In the presence of chromatic dispersion, the unwanted side modes limit the information transmitting rate by reducing the fiber bandwidth.

It is therefore desirable to devise means for a semiconductor laser to emit light predominantly in a single longitudinal mode even under high-speed modulation. Such lasers are referred to as single-frequency or single-longitudinal-mode lasers and in view of their potential application in optical communication systems were studied extensively during the 1980s. They are used in commercial light wave systems by 1990.

In conventional FP-type semiconductor lasers, the feedback is provided by facet reflections whose magnitude remains the same for all longitudinal modes. The only longitudinal-mode discrimination in such a laser is provided by the gain spectrum itself. However, since the gain spectrum is usually much wider than the longitudinal-mode spacing, the resulting mode discrimination is poor.

One way of improving the mode selectivity is to make the feedback frequency-dependent so that the cavity loss is different for different longitudinal modes. Two mechanisms have been found useful in this respect and are known as the distributed feedback and the coupled-cavity mechanisms.

As the name implies, the feedback necessary for the lasing action in a DFB laser is not localized at the cavity facets but is distributed throughout the cavity length. This is achieved through the use of a grating etched so that the thickness of one layer (participating in the heterostructure) varies periodically along the cavity length.

The resulting periodic perturbation of the refractive index provides feedback by means of backward Bragg scattering, which couples the forward- and backward-propagating waves.

Mode selectivity of the DFB mechanism results from the Bragg condition. According to the Bragg condition, coherent coupling between counterpropagating waves occurs only for wavelengths such that the grating period \(\Lambda=m\lambda_\text{m}/2\) where \(\lambda_\text{m}\) is the wavelength inside the laser medium and the integer \(m\) is the order of Bragg diffraction induced by the grating. By choosing \(\Lambda\) appropriately, such a device can be made to provide distributed feedback only at selected wavelengths.

Kogelnik and Shank were the first to observe the lasing action in a periodic structure that utilized the DFB mechanism. Since then, DFB semiconductor lasers have attracted considerable attention both experimentally and theoretically. Although most of the early work related to GaAs lasers, the need for a single-frequency semiconductor laser operating at the minimum-fiber-loss wavelength of 1.55 μm has resulted in the development of InGaAsP DFB lasers.

From the viewpoint of device operation, semiconductor lasers employing distributed feedback can be classified into two broad categories: DFB lasers and ** distributed Bragg reflector** (DBR) lasers. These are shown schematically in Figure 7-1.

In DBR lasers, the grating is etched near the cavity ends and distributed feedback does not take place in the central active region. The unpumped corrugated end regions act as effective mirrors whose reflectivity is of DFB origin and is therefore wavelength-dependent.

## DFB Laser Structures

In the previous tutorials we discussed various semiconductor laser structures that are commonly used [refer to the gain-guided lasers, weakly index-guided lasers, and strongly index-guided lasers tutorial]. Any of these structures can be employed to make a DFB semiconductor laser by etching a grating onto one of the layers.

The direct etching of the active layer is generally not preferred since it can increase the rate of nonradiative recombination by introducing defects in the active region. This would affect the device performance by producing a higher threshold current.

The grating is therefore etched onto one of the cladding layers. Since only the evanescence field associated with the fundamental transverse mode interacts with the grating, it is clear that the exact location of the grating with respect to the active layer and the corrugation depth are critical in determining the effectiveness of the grating.

The grating period \(\Lambda\) is determined by the device wavelength in the medium and the order of Bragg diffraction used for the distributed feedback. The Bragg condition for the \(m\)th-order coupling between the forward- and backward-propagating waves is

\[\tag{7-2-1}\Lambda=m\lambda/2\bar{\mu}\]

where \(\bar{\mu}\) is the effective-mode index and \(\lambda/\bar{\mu}\) is the wavelength inside the medium.

For a 1.55-μm InGaAsP laser, \(\Lambda\approx0.23\text{ μm}\) if we use \(m=1\) (first order grating), and a typical value \(\bar{\mu}\approx3.4\). This value doubles if a second-order grating is used. Both first-order and second-order gratings are employed to fabricate InGaAsP DFB lasers.

Two techniques have been used for the formation of a grating with sub-micrometer periodicity.

In the holographic technique, optical interference is used to for a fringe pattern on the photoresist deposited on the wafer surface. Two optical beams obtained from an ultraviolet laser (e.g., an HeCd laser) are made to interfere on the wafer; the grating period can be adjusted by changing the angle between the two beams.

In the electron-beam lithographic technique, an electron beam scans the wafer surface and “writes” the desired pattern on the electron-beam resist.

Both methods generally use chemical etching to produce the grating corrugations, with the patterned resist acting as a mask. Once the grating has been etched on the substrate or on an epitaxial layer, the wafer can be processed in the usual way to obtain a specific layer structure.

The choice of whether to use the upper or the lower cladding layer for forming the grating requires careful consideration. In practice the device wavelength \(\lambda\) may vary from wafer to wafer because of variations in the location of the gain peak that are due to variations in alloy composition. The placement of the grating on the upper cladding layer has the advantage that the grating period \(\Lambda\) can be adjusted after identifying the effective-mode index \(\bar{\mu}\) and the wavelength \(\lambda\) corresponding to the gain peak. At the same time, however, this choice has the disadvantage that it requires an additional epitaxial step.

As discussed in the waveguide modes in semiconductor lasers tutorial, semiconductor lasers can be classified into three broad categories based on the lateral-mode guiding mechanism: gain-guided, weakly index-guided, and strongly index-guided. Although the initial work on GaAs DFB lasers related to gain-guided structures, gain-guided InGaAsP DFB lasers are unattractive; many undesirable characteristics of gain-guided lasers worsen in the long-wavelength regime [refer to the gain-guided lasers tutorial]. Strongly index-guided devices such as the buried-heterostructure types are most suitable for distributed feedback and are commonly used to make 1.55-μm InGaAsP DFB lasers.

Figure 7-2 shows schematics and a photomicrograph of a 1.5-μm InGaAsP double-channel planar buried-heterostructure (DCPBH) laser.

These DFB lasers have attracted considerable attention at both 1.3-μm and 1.55-μm wavelengths. Their performance is nearly comparable to the conventional FP lasers as far as characteristics such as threshold current, output power, and high-temperature performance are concerned. At the same time, the use of the DFB mechanism leads to a high degree of longitudinal-mode selectivity, and side modes of such laser are suppressed as much as 30-35 dB relative to the main lasing mode.

In Figure 7-3 we have shown schematically a 1.55-μm InGaAsP ridge waveguide DFB laser. These lasers also exhibit a side-mode suppression ratio exceeding 30-35 dB. Their threshold current is generally higher than the DCPBH-DFB lasers; this is mainly due to a larger mode volume and to some extend because of current spreading and carrier diffusion occurring in any weakly index-guided structure.

## DFB Laser Theory

To understand the operating characteristics of a DFB laser, it is first necessary to consider wave propagation in periodic structures.

Grating-induced dielectric perturbation leads to a coupling between the forward- and backward-propagating waves associated with a particular laser mode. Two equivalent approaches have been used to analyze the device behavior.

In the coupled-wave approach, a set of two equations corresponding to the counterpropagating forward and backward waves is solved subject to the specific boundary conditions applicable to a given device.

In an alternative but equivalent approach, one first obtains the Bloch-type eigensolutions by assuming an infinitely long structure; one then applies these solutions to a specific device.

In both approaches the objective is to obtain the threshold gain and the longitudinal-mode spectrum of a DFB laser.

We follow an approach that combines the essential features of both. Starting from the wave equation, we derive the coupled-wave equations whose general solution, without reference to any particular boundary conditions, is then used to construct Bloch-type eigensolutions. These are then used to obtain the DFB longitudinal modes and their threshold gains for the general case where both the grating and the cleaved facets contribute to the feedback.

### 1. Coupled-Wave Equations

The starting point of the analysis is the time-independent wave equation [refer to Equation (2-2-19) in the Maxwell's equations for semiconductor lasers tutorial]

\[\tag{7-3-1}\boldsymbol{\nabla}^2\mathbf{E}+\boldsymbol{\epsilon}(x,y,z)k_0^2\mathbf{E}=0\]

where \(k_0=\omega/c\) and \(\omega\) is the mode frequency.

The main difference from the waveguide-mode theory presented in the waveguide modes in semiconductor lasers tutorial is that in the grating region, \(\boldsymbol{\epsilon}\) is a periodic function of \(z\). It is useful to write

\[\tag{7-3-2}\boldsymbol{\epsilon}(x,y,z)=\bar{\boldsymbol{\epsilon}}(x,y)+\Delta\boldsymbol{\epsilon}(x,y,z)\]

where \(\bar{\boldsymbol{\epsilon}}(x,y)\) is the average value of \(\boldsymbol{\epsilon}\) and the dielectric perturbation \(\Delta\boldsymbol{\epsilon}\) is nonzero only over the grating region whose thickness is equal to the corrugation depth.

In the absence of a grating (\(\Delta\boldsymbol{\epsilon}=0\)), the general solution of Equation (7-3-1) takes the form

\[\tag{7-3-3}\mathbf{E}(x,y,z)=\hat{\mathbf{x}}U(x,y)[E_\text{f}\exp(\text{i}\beta{z})+E_\text{b}\exp(-\text{i}\beta{z})]\]

where \(\hat{\mathbf{x}}\) is the unit vector (along the junction plane for the TE mode).

The field distribution corresponding to a specific waveguide mode is obtained by solving

\[\tag{7-3-4}\frac{\partial^2U}{\partial{x}^2}+\frac{\partial^2U}{\partial{y}^2}+[\bar{\boldsymbol{\epsilon}}(x,y)k_0^2-\beta^2]U=0\]

for a given device structure, where \(\beta\) is the mode-propagation constant.

We assume that the device supports only the fundamental waveguide mode and the corresponding \(U(x,y)\) and \(\beta\) have been obtained following the analysis of the waveguide modes in semiconductor lasers tutorial.

The mode-propagation constant \(\beta\) is complex:

\[\tag{7-3-5}\beta=\bar{\mu}k_0-\text{i}\bar{\alpha}/2\]

where \(\bar{\mu}\) is the effective-mode index and \(\bar{\alpha}\) is the mode-gain coefficient given by

\[\tag{7-3-6}\bar{\alpha}=\Gamma{g}-\alpha_\text{int}\]

\(\Gamma\) is the confinement factor, \(g\) is the active-region gain, and \(\alpha_\text{int}\) accounts for the internal loss.

These relations are identical to Equations (2-5-37) and (2-5-48) [refer to the waveguide modes in semiconductor lasers tutorial] except that in those equations \(\bar{\alpha}\) represents the mode-absorption coefficient rather than the mode-gain coefficient.

In the presence of dielectric perturbation \(\Delta\boldsymbol{\epsilon}\), the amplitudes \(E_\text{f}\) and \(E_\text{b}\) of the forward and backward waves becomes \(z\)-dependent because a periodic structure produces Bragg diffraction.

A major simplification occurs if we assume that the spatial distribution \(U(x,y)\) is unaffected by the weak perturbation \(\Delta\boldsymbol{\epsilon}\).

We substitute Equation (7-3-3) in (7-3-1) and allow for slow axial variations of \(E_\text{f}\) and \(E_\text{b}\). Multiplying the resulting equation with \(U(x,y)\) and integrating over \(x\) and \(y\), we obtain

\[\tag{7-3-7}\begin{align}\frac{\text{d}E_\text{f}}{\text{d}z}\exp(\text{i}\beta{z})-\frac{\text{d}E_\text{b}}{\text{d}z}\exp(-\text{i}\beta{z})=&\frac{\text{i}k_0^2}{2\beta{V}}\displaystyle\iint\Delta\boldsymbol{\epsilon}(x,y,z)U^2(x,y)\\&\times[E_\text{f}\exp(\text{i}\beta{z})+E_\text{b}\exp(-\text{i}\beta{z})]\text{d}x\text{d}y\end{align}\]

where \(V=\displaystyle\iint{U^2(x,y)}\text{d}x\text{d}y\) and is a normalization constant.

Since \(\Delta\boldsymbol{\epsilon}\) is periodic in \(z\) with the grating period \(\Lambda\), it can be expanded in a Fourier series to

\[\tag{7-3-8}\Delta\boldsymbol{\epsilon}(x,y,z)=\sum_{l\ne0}\Delta\boldsymbol{\epsilon}_l(x,y)\exp[\text{i}(2\pi/\Lambda)lz]\]

We substitute Equation (7-3-8) into (7-3-7) and equate the coefficients of \(\exp(\pm\text{i}\beta{z})\) on both sides. We assume for simplicity that \(\Delta\boldsymbol{\epsilon}\) is real. Collecting only the terms that are approximately phase-matched, we obtain

\[\tag{7-3-9}\frac{\text{d}E_\text{f}}{\text{d}z}=\text{i}\kappa{E_\text{b}}\exp(-2\text{i}\Delta\beta{z})\]

\[\tag{7-3-10}\frac{\text{d}E_\text{b}}{\text{d}z}=-\text{i}\kappa^*{E_\text{f}}\exp(2\text{i}\Delta\beta{z})\]

where

\[\tag{7-3-11}\Delta\beta=\beta-m\pi/\Lambda=\beta-\beta_0\]

is the phase mismatch and was assumed to be smallest for the term \(l=m\) in Equation (7-3-8).

The contribution of other terms has been neglected because of their large phase mismatch.

The coupling coefficient \(\kappa\) is given by

\[\tag{7-3-12}\kappa=\frac{k_0^2}{2\beta}\frac{\displaystyle\iint\Delta\boldsymbol{\epsilon}_m(x,y)U^2(x,y)\text{d}x\text{d}y}{\displaystyle\iint{U^2}(x,y)\text{d}x\text{d}y}\]

It shows that the Fourier component \(\Delta\boldsymbol{\epsilon}_m\), for which the Bragg condition (\(\beta\approx{m}\pi/\Lambda\)) is approximately satisfied, couples the forward- and backward-propagating waves.

The magnitude of \(\kappa\) is governed by the weighted average of \(\Delta\boldsymbol{\epsilon}_m\) with the mode intensity over the grating region.

In general, \(\kappa\) may be complex even when \(\Delta\boldsymbol{\epsilon}\) is real. This is the case, for example, for gain-guided lasers where \(U(x,y)\) is complex because of the wave-front curvature.

The small imaginary part of \(\beta\) in Equation (7-3-5) also makes \(\kappa\) complex. However, this is a minor effect, and \(\kappa\) can be assumed to be real for index-guided lasers by replacing \(\beta\) with \(\bar{\mu}k_0\) in Equation (7-3-12).

Instead of using complex \(\beta\) as the propagation constant in Equation (7-3-3), it is more convenient to use the Bragg wave number \(\beta_0=m\pi/\Lambda\) and write the axially varying field as

\[\tag{7-3-13}E(z)=A(z)\exp(\text{i}\beta_0z)+B(z)\exp(-\text{i}\beta_0z)\]

where

\[\tag{7-3-14}A=E_\text{f}\exp(\text{i}\Delta\beta{z})\qquad\text{and}\qquad{B}=E_\text{b}\exp(-\text{i}\Delta\beta{z})\]

The coupled-wave equations for \(A\) and \(B\) then become

\[\tag{7-3-15}\frac{\text{d}A}{\text{d}z}=\text{i}\Delta\beta{A}+\text{i}\kappa{B}\]

\[\tag{7-3-16}-\frac{\text{d}B}{\text{d}z}=\text{i}\Delta\beta{B}+\text{i}\kappa{A}\]

where we have used Equations (7-3-9) to (7-3-11).

A general solution of these equations takes the form

\[\tag{7-3-17}A(z)=A_1\exp(\text{i}qz)+A_2\exp(-\text{i}qz)\]

\[\tag{7-3-18}B(z)=B_1\exp(\text{i}qz)+B_2\exp(-\text{i}qz)\]

where \(q\) is the complex wave number to be determined from the boundary conditions.

The constants \(A_1\), \(A_2\), \(B_1\), and \(B_2\) are interdependent. If we substitute the general solution into Equation (7-3-15) and (7-3-16) and equate the coefficients of \(\exp(\pm\text{i}qz)\), we obtain

\[\tag{7-3-19}(q-\Delta\beta)A_1=\kappa{B_1}\qquad(q+\Delta\beta)B_1=-\kappa{A_1}\]

\[\tag{7-3-20}(q-\Delta\beta)B_2=\kappa{A_2}\qquad(q+\Delta\beta)A_2=-\kappa{B_2}\]

These relations are satisfied with nonzero values of \(A_1\), \(A_2\), \(B_1\), and \(B_2\) if the possible values of \(q\) obey the following dispersion relation (obtained by setting the determinant of the coefficient matrix to be zero):

\[\tag{7-3-21}q=\pm[(\Delta\beta)^2-\kappa^2]^{1/2}\]

The plus and minus signs correspond to forward- and backward-propagating waves, respectively.

Furthermore, we can define the DFB reflection coefficient as

\[\tag{7-3-22}r(q)=\frac{q-\Delta\beta}{\kappa}=-\frac{\kappa}{q+\Delta\beta}\]

and eliminate \(A_2\) and \(B_1\) in Equations (7-3-17) and (7-3-18) in favor of \(r(q)\).

The general solution of the coupled-wave equations then becomes

\[\tag{7-3-23}A(z)=A_1\exp(\text{i}qz)+r(q)B_2\exp(-\text{i}qz)\]

\[\tag{7-3-24}B(z)=B_2\exp(-\text{i}qz)+r(q)A_1\exp(\text{i}qz)\]

Since \(r=0\) if \(\kappa=0\), it is evident that \(r(q)\) represents the fraction of the forward-wave amplitude that is reflected back toward the backward wave and vice versa.

The sign ambiguity in the expression (7-3-21) can be resolved by choosing the sign of \(q\) such that \(|r(q)|\le1\).

Mode selectivity of DFB lasers stems, as will become clear, from the dependence of \(r\) on \(q\). The eigenvalue \(q\) and the ratio \(B_2/A_1\) are determined by applying the appropriate boundary conditions at the laser facets.

Before considering a finite-length DFB laser, it is instructive to discuss wave propagation in an infinite-length periodic structure.

The dispersion relation (7-3-21) governs the possible \(q\) values in such a medium. In general, \(q\) is complex since \(\Delta\beta\) is complex in the presence of gain or loss associated with the medium. If we neglect \(\bar{\alpha}\) in Equation (7-3-5), \(\Delta\beta=\beta-\beta_0\) and is real.

According to Equation (7-3-21), \(q\) is pure imaginary when \(|\Delta\beta|\le\kappa\), and the medium cannot support a propagating wave. Consequently, an infinite periodic structure has a strop band; i.e., only waves of certain frequencies for which the propagation constant \(\beta\) satisfies the inequality \(|\beta-\beta_0|\gt\kappa\) can propagate inside the medium.

It will be seen later that the stop band exists even for a finite DFB laser, although under specific conditions the device may support a "gap mode" that lies inside the stop band.

An infinite periodic structure is conceptually similar to a one-dimensional crystal, and the concept of stop band is analogous to the band gap occurring in a crystal.

The analogy can be extended further by noting that the field \(E(z)\) in Equation (7-3-13) is in the form of a Bloch wave. This is seen more clearly if we substitute \(A\) and \(B\) from Equation (7-3-23) and (7-3-24) into Equation (7-3-13) and rearrange the terms to obtain

\[\tag{7-3-25}\begin{align}E(z)=&A_1[\exp(\text{i}\beta_0z)+r(q)\exp(-\text{i}\beta_0z)]\exp(\text{i}qz)\\&+B_2[\exp(-\text{i}\beta_0z)+r(q)\exp(\text{i}\beta_0z)]\exp(-\text{i}qz)\end{align}\]

The expression inside each pair of square brackets is a periodic function of \(z\) with periodicity \(\Lambda\).

The Bloch-type eigenmodes are expected in view of the periodic variation of the dielectric constant and could have been obtained directly by applying Floquet's theorem to Equation (7-3-1).

### 2. Longitudinal Modes and Threshold Gain

**Threshold Condition**

For a finite-length DFB laser, the boundary conditions at the facets are satisfied only for discrete values of \(q\). The real and imaginary parts of \(q\) give, respectively, the DFB longitudinal-mode frequency and the corresponding threshold gain.

Consider a laser of length \(L\). Since \(L\) is not necessarily an exact multiple of the grating period \(\Lambda\), the last period of the grating close to the facet is generally not complete.

Even though no significant distributed feedback occurs over these incomplete grating periods, the phase shift in this region plays an important role in determining DFB laser characteristics and should be properly accounted for.

A simple way to do this is to assume that the effective-facet-reflection coefficient is complex, i.e.,

\[\tag{7-3-26}r_j=R_j^{1/2}\exp(\text{i}\phi_j)\qquad{j=1,2}\]

where \(R_j\) is the facet reflectivity and \(\phi_j\) is the round-trip phase shift encountered by the field in traveling the extra distance (a fraction of \(\Lambda\)) corresponding to the last incomplete grating period.

In applying the boundary conditions, one can assume that the mirrors are located at the last complete grating period. The constant phase shift of about \(180^\circ\) occurring during facet reflections can also be incorporated into \(\phi_j\).

A characteristic feature of a DFB semiconductor laser is that the phase shifts \(\phi_1\) and \(\phi_2\) are expected to vary from device to device since the relative distance between the cleaved facet and the last complete grating period (a small fraction of 1 μm) is uncontrollable at present except through the use of controlled etching or coating of individual facets.

The boundary conditions at the two facets are

\[\tag{7-3-27}A(0)=r_1B(0)\qquad\text{and}\qquad{B}(L)=r_2A(L)\]

If we use them in the general solution given by Equations (7-3-23) and (7-3-24), we obtain the following two homogeneous equations for the unknown constants \(A_1\) and \(B_2\):

\[\tag{7-3-28}(r_1-r)B_2-(1-rr_1)A_1=0\]

\[\tag{7-3-29}(r_2-r)\exp(2\text{i}qL)A_1-(1-rr_2)B_2=0\]

These equations have a consistent nontrivial solution only for values of \(q\) satisfying the eigenvalue equation

\[\tag{7-3-30}\left(\frac{r_1-r}{1-rr_1}\right)\left(\frac{r_2-r}{1-rr_2}\right)\exp(2\text{i}qL)=1\]

This is the threshold condition for DFB lasers.

It is similar to the threshold condition obtained for an FP laser [refer to Equation (2-3-6) in the threshold condition and longitudinal modes of semiconductor lasers tutorial] and reduces to the latter if the DFB contribution is neglected by setting \(r\) to zero. On the other hand, if facet reflectivities are neglected by setting \(r_1\) and \(r_2\) to zero, we obtain

\[\tag{7-3-31}r^2\exp(2\text{i}qL)=1\]

implying that \(r\) is the effective-reflection coefficient for a DFB laser.

The DFB eigenvalues \(q\) are obtained from Equation (7-3-30) after noting that \(r\) as given by Equation (7-3-22) also depends on \(q\).

Each eigenvalue \(q\) is complex and can be used to calculate

\[\tag{7-3-32}\Delta\beta=\pm(q^2+\kappa^2)^{1/2}\]

after using Equation (7-3-21).

From Equations (7-3-5) and (7-3-11) the real and imaginary parts of the relation

\[\tag{7-3-33}\Delta\beta=\delta-\text{i}\bar{\alpha}/2\]

yield the mode detuning

\[\tag{7-3-34}\delta=\bar{\mu}k_0-\beta_0=\frac{-2\pi\mu_\text{g}}{\lambda^2}\Delta\lambda\]

and the threshold gain \(\bar{\alpha}\).

In Equation (7-3-34), \(\Delta\lambda\) is the deviation of the mode wavelength from the Bragg wavelength (\(\lambda_0=2\bar{\mu}\Lambda/m\)), and care is taken to account for the chromatic dispersion (wavelength variation of \(\bar{\mu}\)).

Note that \(\bar{\alpha}\) is the power gain. In previous work the amplitude gain (\(\alpha=\bar{\alpha}/2\)) has often been used.

The threshold material gain is determined by using Equation (7-3-6) and is given by

\[\tag{7-3-35}g_\text{th}=\frac{\bar{\alpha}+\alpha_\text{int}}{\Gamma}\]

The values of \(\delta\) corresponding to different eigenvalues indicate how far that longitudinal mode is displaced from the Bragg wavelength \(\lambda_0\). According to Equations (7-3-32) to (7-3-34), the DFB longitudinal modes are situated symmetrically with respect to \(\lambda_0\).

In contrast to FP lasers, however, different modes have different threshold gains as determined by the eigenvalue equation (7-3-30) and by Equation (7-3-35).

The physical interpretation of the threshold condition (7-3-35) is clear. The mode gain \(\bar{\alpha}\) can be thought of as the gain required to overcome both the distributed and the facet losses; and it plays the same role as the mirror loss

\[\tag{7-3-36}\alpha_\text{m}=\frac{1}{2L}\ln\left(\frac{1}{R_1R_2}\right)\]

for an FP laser [refer to Equation (2-3-10) in the threshold condition and longitudinal modes of semiconductor lasers tutorial].

The thresholds for DFB and FP lasers can thus be compared through \(\bar{\alpha}\) and \(\alpha_\text{m}\) provided the internal loss \(\alpha_\text{int}\) is assumed to be identical for both kinds of lasers.

Using cleaved-facet reflectivities \(R_1=R_2\approx0.32\), we obtain \(\alpha_\text{m}L\approx1.1\), and a DFB laser would have a lower threshold than an FP laser if \(\bar{\alpha}L\lt1.1\) for the lowest-threshold mode.

**Nonreflecting Facets**

Numerical solution of the eigenvalue equation (7-3-30) is generally required to obtain the longitudinal-mode frequencies and their respective threshold gains.

For some physical insight, it is instructive to consider the ideal case in which the facet reflectivities \(R_1\) and \(R_2\) are set to zero. Experimentally this can be realized by etching, coating, or burying the laser facets.

Equations (7-3-22) and (7-3-31) can be combined to yield the transcendental equation

\[\tag{7-3-37}\kappa\sin(qL)=\pm\text{i}q\]

The DFB modes are obtained by solving this equation for a given value of the coupling coefficient \(\kappa\). To make the results independent of the cavity length \(L\), it is useful to consider the dimensionless parameter \(\kappa{L}\) and plot \(\bar{\alpha}L\) versus \(\delta{L}\).

Figure 7-4 shows the mode gain \(\bar{\alpha}L\) versus the mode detuning \(\delta{L}\) for the three modes closest to the Bragg wavelength \(\lambda_0\). Since the spectrum is symmetric with respect to \(\lambda_0\), only half of the spectrum is shown.

Each solid curve is obtained by varying \(\kappa{L}\). For a given value of \(\kappa{L}\), the intersections of the corresponding dashed line with the solid lines provide the values for mode gain and mode frequency.

In particular, the mode closest to the Bragg wavelength (\(m=1\)) has the lowest threshold gain and would reach the threshold first.

The gain difference between the modes \(m=2\) and \(m=1\) is a measure of the mode selectivity and determines the extent to which other side modes are suppressed.

The horizontal dashed line in Figure 7-4 corresponds to an FP laser (which has identical gain for all modes) and shows that the DFB mechanism would lead to a lower threshold only if \(\kappa{L}\gt3.2\).

A noteworthy feature of a DFB laser with nonreflecting facets is that, because of the symmetric nature of the DFB mode spectrum, the two modes closest to the Bragg frequency (one on each side) have the lowest threshold gain and would lase simultaneously.

The frequency or wavelength separation between these two degenerate modes corresponds to the stop band discussed earlier. Simultaneous oscillation of the two DFB modes separated by the stop band has been observed.

The width of the stop band is related to the coupling coefficient \(\kappa\) and can be used for its estimation. This can be seen clearly in the low-gain limit. When the trivial solution \(q=0\) of Equation (7-3-37) is used in Equations (7-3-32) and (7-3-33), we find that \(\delta\approx\pm\kappa\) for the mode with lowest threshold gain. Thus in the low-gain limit, the width of the stop band is approximately \(2\kappa\).

The eigenvalue equation (7-3-31) can be solved approximately in an analytic form for the high-gain case that is likely to occur under weak coupling, i.e., when \(\kappa{L}\ll1\).

Using Equations (7-3-21) and (7-3-22), \(q\approx\Delta\beta\) and the DFB reflection coefficient \(r\approx-\kappa/(2\Delta\beta)\). With the help of Equations (7-3-31) and (7-3-33), the eigenvalue equation becomes

\[\tag{7-3-38}\kappa^2\exp[(2\text{i}\delta+\bar{\alpha})L]=-(2\text{i}\delta+\bar{\alpha})^2\]

Equating the modulus and the phase on the two sides, we obtain

\[\tag{7-3-39}\bar{\alpha}^2+4\delta^2=\kappa^2\exp(\bar{\alpha}L)\]

\[\tag{7-3-40}\delta{L}=(m-\frac{1}{2})\pi+\tan^{-1}\left(\frac{2\delta}{\bar{\alpha}}\right)\]

where the integer \(m\) labels various longitudinal modes (see Figure 7-4).

If we neglect the last term in Equation (7-3-40) and use Equation (7-3-34), the longitudinal-mode spacing is given by

\[\tag{7-3-41}\Delta\lambda=\lambda^2/(2\mu_\text{g}L)\]

This value is identical to that obtained for FP lasers in the threshold conditions and longitudinal modes of semiconductor lasers tutorial [refer to Equation (2-3-14)].

Mode selectivity of DFB lasers is evident from Equation (7-3-39). As \(\delta\) increases, \(\bar{\alpha}\) increases for a fixed value of \(\kappa\), and higher-order modes require higher threshold gains.

**Cleaved Facets**

From a practical viewpoint it is important to consider the effect of facet reflections on the DFB mode spectrum. This is accomplished by using Equation (7-3-26) in (7-3-30) and solving for the eigenvalue \(q\).

As mentioned before, the phases \(\phi_1\) and \(\phi_2\) are arbitrary and would vary from device to device. The threshold characteristics of DFB lasers with cleaved facets are therefore susceptible to considerable variations due to uncontrollable phase shifts. Such variations have been observed experimentally.

In the following discussion we examine a few representative cases to show the new qualitative features arising from facet reflections.

We first consider the case where \(R_1=R_2=0.32\) in order to include reflections from both cleaved facets. This is the case commonly encountered.

Figure 7-5 shows the mode gain \(\bar{\alpha}L\) and the detuning \(\delta{L}\) for six DFB modes as \(\phi_1\) is varied over its \(2\pi\) range for a fixed value of \(\phi_2\).

Two curves for different values of \(\kappa{L}\) are shown. The horizontal dashed line shows the FP laser threshold in the absence of distributed feedback (\(\kappa=0\)). The open circles denote the location of DFB modes for \(\phi_1=0\). As \(\phi_1\) increases, the modes shift to the left on the curve. Their position for the specific case where \(\phi_1=\pi\) is marked by closed circles. Arrows denote the modes for the intermediate case where \(\phi_1=\pi/2\).

Several conclusions can be drawn from Figure 7-5. It is evident that facet reflections break the gain degeneracy that occurs in the absence of reflections for the two modes separated by the stop band.

Which of the two modes has the lower threshold gain depends on the relative values of phases \(\phi_1\) and \(\phi_2\). For the case where \(\phi_2=\pi/2\) shown in Figure 7-5, the mode on the low-frequency or long-wavelength side with respect to the Bragg wavelength (negative value of \(\delta\)) has a lower gain for \(-\pi/2\le\phi_1\le\pi/2\), while the reverse occurs (positive values of \(\delta\)) for \(\pi/2\le\phi_1\le3\pi/2\).

The mode spectrum is degenerate for \(\phi_1=\pi/2\) and \(\phi_1=3\pi/2\). Arrows in Figure 7-5 mark the location of degenerate modes for \(\phi_1=\pi/2\).

Another feature arising from facet reflections is the existence of a high-gain mode inside the stop band.

It is of interest to compare the threshold of the DFB mode that has the lowest threshold gain with the threshold of an FP laser. Figure 7-5 shows that the DFB threshold is always lower than the FP threshold and that the gain margin increases with the coupling coefficient.

It should be stressed, however, that the answer to the question of whether the DFB or FP mode would lase in a given device also depends on the material-gain spectral profile. The DFB mode would lase if the gain peak occurred in the vicinity of the Bragg wavelength. When the gain peak and Bragg wavelength are significantly far apart, an FP mode close to the gain peak might have an overall lower threshold gain and reach threshold first.

The case of one reflecting facet and one nonreflecting facet has also drawn considerable attention. It is shown in Figure 7-6 for the case where \(R_2=0\) and \(R_1=0.32\).

Similar to Figure 7-5, the phase \(\phi_1\) is varied over its entire \(2\pi\) range. The qualitative features are similar to the case of two cleaved facets.

The most significant difference occurs for the higher-order DFB modes, which have much higher thresholds compared with those shown in Figure 7-5. For a given \(\kappa{L}\), the threshold of the lowest-gain mode is also higher when compared with the case of two cleaved facets.

A comparison of Figures 7-5 and 7-6 shows that the effect of reducing the reflectivity of one facet is to improve the mode selectivity at the expense of a threshold increase.

It is clear from Figures 7-5 and 7-6 that facet reflections give rise to a mode in the stop band. This mode generally has a high threshold and is sometimes referred to as the ** gap mode**.

The reason for the high threshold is that the gap mode has a considerable amplitude only in the vicinity of the reflecting laser facet. However, it is possible to design lasers in which the gap mode has the lowest threshold. For example, this can be accomplished by making one facet high-reflecting and the other facet low-reflecting. If \(R_1\) in Figure 7-6 is increased to \(0.9\), one finds that the peak at \(\delta=0\) is replaced with a broad minimum.

From a conceptual point of view, such a laser is essentially a folded version of a DFB laser with two low-reflecting ends and in which the grating is phase-shifted in the center.

Such phase-shifted gratings were originally proposed by Haus and Shank and drew considerable attention during the 1980s in relation to 1.55-μm InGaAsP DFB lasers. The optimum phase shift is \(\pi/2\), for which the gap mode occurs exactly at the Bragg wavelength (\(\delta=0\)).

Several different approaches have been followed to introduce the required phase shift into the middle of the laser cavity. In one approach, electron-beam lithography was used to change the phase at the central groove. Simultaneous holographic exposure of positive and negative photoresists has also been used for this purpose.

Alternatively, a small uncorrugated section (typical length of \(\sim10\) μm) is introduced into the center such that the optical phase changes by \(\pi/2\) during wave propagation over this region.

In another approach, the phase shift is introduced indirectly by using a nonuniform stripe width. A different active-region width over a central region (typical length of \(\sim60\) μm) changes the effective mode index, which in turn changes the optical phase.

All of the schemes have been used to demonstrate lasing action near the Bragg wavelength.

### 3. Coupling Coefficient

We have seen in the previous section that the threshold gain of a DFB laser depends on the magnitude of the coupling coefficient \(\kappa\) given by Equation (7-3-12).

The numerical value of \(\kappa\) depends on such grating parameters as shape, depth, and period of the corrugation. Furthermore, since the coupling occurs as a result of the interaction between the grating and the evanescent part of the transverse mode, \(\kappa\) is also affected by the thickness and composition of the active and cladding layers.

The evaluation of \(\kappa\) for GaAs lasers has been discussed extensively, and the techniques used there have been applied for InGaAsP lasers as well.

In this section, we discuss briefly how \(\kappa\) varies with various device parameters.

We assume that waveguide dimensions are chosen such that the DFB semiconductor laser supports only the fundamental lateral and transverse modes. Then \(U(x,y)=\psi(x)\phi(y)\) in Equation (7-3-12), where \(\psi(x)\) and \(\phi(y)\) are the corresponding field distributions introduced in the waveguide modes in semiconductor lasers tutorial.

If we assume that the grating is laterally uniform so that the dielectric perturbation \(\Delta\boldsymbol{\epsilon}_m\) is independent of \(x\), the integration over \(x\) in Equation (7-3-12) is readily performed and we obtain

\[\tag{7-3-42}\kappa=\frac{k_0}{2\bar{\mu}}\frac{\displaystyle\int\limits_{-\infty}^{\infty}\Delta\boldsymbol{\epsilon}_m(y)\phi^2(y)\text{d}y}{\displaystyle\int\limits_{-\infty}^{\infty}\phi^2(y)\text{d}y}\]

where we used \(\beta=\bar{\mu}k_0\) and \(\bar{\mu}\) is the effective-mode index.

In some devices the corrugation depth may be laterally nonuniform. For example, this may happen in a weakly index-guided ridge-waveguide-type device if the grating corrugation is partially filled during the ridge overgrowth.

Equation (7-3-42) is then used separately in the two regions to obtain \(\kappa_1\) and \(\kappa_2\), and \(\kappa\) is obtained by a weighted average:

\[\tag{7-3-43}\kappa=\Gamma_\text{L}\kappa_1+(1-\Gamma_\text{L})\kappa_2\]

where \(\Gamma_\text{L}\) is the lateral confinement factor. This equation can be easily obtained using the general expression (7-3-12) for \(\kappa\).

In Equation (7-3-42) the Fourier coefficient \(\Delta\boldsymbol{\epsilon}_m\) for the \(m\)th-order Bragg diffraction is obtained using Equation (7-3-8) and is given by

\[\tag{7-3-44}\Delta\boldsymbol{\epsilon}_m(y)=\frac{1}{\Lambda}\displaystyle\int\limits_0^\Lambda\Delta\boldsymbol{\epsilon}(y,z)\exp\left(\frac{-2\pi\text{i}mz}{\Lambda}\right)\text{d}z\]

where \(\Lambda\) is the corrugation period.

In general, \(\kappa\) is evaluated numerically using Equations (7-3-42) and (7-3-44) for a specific corrugation profile \(\Delta\boldsymbol{\epsilon}(y,z)\).

A simple expression for \(\kappa\) can be obtained for rectangular-shaped grooves. Consider the geometry shown in Figure 7-7 for a typical DFB semiconductor laser where \(d\) is the active-layer thickness, \(h\) is the thickness of the p-type cladding layer, \(d_\text{g}\) is the corrugation depth, and \(d_\text{c}\) is the average thickness of the n-type cladding layer.

We note that \(\Delta\boldsymbol{\epsilon}(y,z)=\mu_2^2-\mu_1^2\) when \(d_\text{c}-d_\text{g}/2\le{y}\le{d}_\text{c}+d_\text{g}/2\) and \(-\Lambda_1/2\le{z}\le\Lambda_1/2\). The origin of \(z\) is chosen at the center of the groove of width \(\Lambda_1\).

Integration in Equation (7-3-44) leads to

\[\tag{7-3-45}\Delta\boldsymbol{\epsilon}_m(y)=(\mu_2^2-\mu_1^2)\frac{\sin(\pi{m}\Lambda_1/\Lambda)}{\pi{m}}\]

inside the grating region and \(\Delta\boldsymbol{\epsilon}_m(y)=0\) outisde.

Using this value in Equation (7-3-42), we obtain

\[\tag{7-3-46}\kappa\approx{k_0}\Delta\mu\Gamma_\text{g}\frac{\sin(\pi{m}\Lambda_1/\Lambda)}{\pi{m}}\]

where we assumed that \(\mu_1+\mu_2\approx2\bar{\mu}\).

Here

\[\tag{7-3-47}\Gamma_\text{g}=\frac{\displaystyle\int\limits_{d_\text{c}-d_\text{g}/2}^{d_\text{c}+d_\text{g}/2}\phi^2(y)\text{d}y}{\displaystyle\int\limits_{-\infty}^{\infty}\phi^2(y)\text{d}y}\]

and is the fraction of the mode energy inside the grating region. Further, \(\Delta\mu=\mu_2-\mu_1\) and is the index difference on the two sides of the grating.

Several features of Equation (7-3-46) are noteworthy. For the first-order grating (\(m=1\)), \(\kappa\) is maximum for a symmetric grating such that \(\Lambda_1=\Lambda/2\). However, a symmetric second-order grating (\(m=2\)) leads to \(\kappa=0\). The maximum coupling for \(m=2\) occurs when \(\Lambda_1=\Lambda/4\) or \(3\Lambda/4\).

To evaluate \(\Gamma_\text{g}\), it is necessary to solve the five-layer slab-waveguide problem (see Figure 7-7). This is done by using the method discussed in the waveguide modes in semiconductor lasers tutorial that requires the matching of \(\phi\) and \(\text{d}\phi/\text{d}y\) at various interfaces.

Figure 7-8 shows the calculated values of \(\kappa\) for the TE mode of a 1.55-μm InGaAsP laser with 1.3-μm cladding layers using \(d=0.1\) μm, \(d_\text{c}=0.1\) μm, and \(h=0.1\) μm. The corrugation depth \(d_\text{g}\) is varied.

For comparison, the results of a sinusoidal grating are shown by dashed lines. The case of a symmetric grating is considered (\(\Lambda_1=\Lambda/2\)) and \(\kappa=0\) for a second-order rectangular grating. However, \(\kappa\) can become appreciable even when \(m=2\) if other corrugation shapes are employed.

Figure 7-8 shows that a coupling coefficient of about \(100\text{ cm}^{-1}\) can be achieved for a first-order grating with corrugation depths as small as 0.05 μm. However, \(\kappa\) is expected to be smaller for a second-order grating and values in the range of \(40-50\text{ cm}^{-1}\) can be obtained using \(d_\text{g}\approx0.15\) μm.

For a 250-μm-long laser, one then obtains \(\kappa{L}\approx1\) and, as Figures 7-5 and 7-6 show, such values of \(\kappa\) are sufficient to operate a DFB laser with a reasonable mode discrimination.

### 4. Threshold Behavior

The required threshold gain of a DFB laser given by Equation (7-3-35) can be used to obtain the threshold current density \(J_\text{th}\). If we assume that the material gain varies linearly with the carrier density, i.e.,

\[\tag{7-3-48}g(n)=a(n-n_0)\]

then the threshold value of the carrier density obtained using Equations (7-3-35) and (7-3-48) is given by

\[\tag{7-3-49}n_\text{th}=n_0+\frac{\bar{\alpha}+\alpha_\text{int}}{a\Gamma}\]

Here \(a\) is the gain coefficient and \(n_0\) is the transparency value of the carrier density corresponding to the onset of population inversion.

If carrier diffusion is ignored, the threshold current density can be obtained using the analysis of the emission characteristics of semiconductor lasers tutorial [see Equation (2-6-3)] and is given by

\[\tag{7-3-50}J_\text{th}=qdn_\text{th}(A_\text{nr}+Bn_\text{th}+Cn_\text{th}^2)\]

where \(A_\text{nr}\), \(B\), and \(C\) take into account various radiative and nonradiative recombination mechanisms.

To evaluate \(J_\text{th}\), consider the device structure shown schematically in Figure 7-7 for a 1.55-μm InGaAsP laser.

- The coupling coefficient \(\kappa\) for this structure is first calculated using the procedure described in the coupling coefficient section above.
- For the given value of \(\kappa{L}\), one can then obtain the DFB mode spectrum as discussed in the longitudinal modes and threshold gain section above. This provides us with the value of \(\bar{\alpha}\) for the lowest-gain mode that would reach threshold first.
- The confinement factor \(\Gamma\) can be calculated for the specific structure shown in Figure 7-7 by solving the five-layer waveguide problem.
- Equations (7-3-49) and (7-3-50) are then used to evaluate \(J_\text{th}\).

The same method can be followed for either the TE or the TM mode after making use of the appropriate boundary conditions [refer to the waveguide modes in semiconductor lasers tutorial].

For a DFB laser, it should be noted that both \(\Gamma\) and \(\kappa\) are different for TE and TM modes.

In particular, while \(\Gamma\) is generally smaller for the TM mode, the coupling coefficient can be larger since a larger fraction of the mode (due to weaker mode confinement) interacts with the grating; i.e., \(\Gamma_\text{g}\) in Equation (7-3-46) is larger.

This suggest that the threshold margin between the TE and TM modes is likely to be smaller when compared with that of FP lasers.

The threshold current density for an FP laser can also be obtained using Equations (7-3-49) and (7-3-50) after replacing \(\bar{\alpha}\) by \(\alpha_\text{m}\) given by Equation (7-3-36).

To illustrate the threshold behavior, we consider the device shown in Figure 7-7 and assume that a rectangular second-order grating of 0.1-μm thickness such that \(\Lambda_1=0.3\Lambda\) is etched on the InP substrate.

The cladding-layer thickness \(h\) and \(d_\text{c}\) are each 0.15 μm. The other device parameters are as follows: \(L=250\) μm, \(\alpha_\text{int}=40\text{ cm}^{-1}\), \(a=2.5\times10^{-16}\text{ cm}^2\), \(n_0=1\times10^{18}\text{ cm}^{-3}\), \(A_\text{nr}=6.6\times10^7\text{ s}^{-1}\), \(B=0.9\times10^{-10}\text{ cm}^3/\text{s}\), and \(C=4\times10^{-29}\text{ cm}^6/\text{s}\). We also assume that the DFB laser has negligible facet reflectivities.

Figure 7-9 shows the variation of \(J_\text{th}\) with the active-layer thickness for the TE and TM modes of the DFB semiconductor laser. For comparison, the corresponding threshold curves for a similar FP semiconductor laser are also shown.

For the FP laser, the facet loss \(\alpha_\text{m}\) was calculated using \(R_\text{m}=0.36\) and \(R_\text{m}=0.28\) for TE and TM modes, respectively, while all other parameters were taken to be the same.

The threshold current density \(J_\text{th}\) in Figure 7-9 is minimum at an optimum value of \(d\); as \(d\) decreases \(J_\text{th}\) increases sharply, since \(\Gamma\) reduces substantially due to poor mode confinement. This behavior is identical for DFB and FP lasers, although the minima occur at different values of \(d\).

From a practical point of view, the important quantity is the threshold margin \(\Delta{J}_\text{th}\) between the TE and TM modes. Devices with a high value of \(\Delta{J}_\text{th}\) are preferred because the TE mode power at which the TM mode starts to lase (leading to a kink in the \(L-I\) curve) is pushed upward.

Figure 7-9 shows that \(\Delta{J}_\text{th}\) is considerably smaller for a DFB laser in comparison to that of an FP laser. Furthermore, \(\Delta{J}_\text{th}\) varies with \(d\). For a 0.2-μm-thick active layer, \(\Delta{J}_\text{th}\) is so small that the TE and TM modes would lase almost simultaneously. By contrast, a choice of \(d\approx0.1\) μm provides a significant margin and lowers the TE-mode threshold.

The main point to note is that the design of a good DFB laser requires consideration and optimization of a large number of device parameters.

In the preceding discussion we did not include the effect of internal stress that is invariably present in actual heterostructure devices. It can be incorporated through the stress dependence of the material gain \(g\).

In the presence of a compressive stress \(S\) normal to the junction plane, the gain \(g(n)\) given by Equation (7-3-48) decreases for the TE mode and increases for the TM mode. It can be approximated by

\[\tag{7-3-51a}g_\text{TE}(n)=a(n-n_0)-\gamma_1S\]

\[\tag{7-3-51b}g_\text{TM}(n)=a(n-n_0)+2\gamma_2S\]

where the constants \(\gamma_1\) and \(\gamma_2\) are related to the elastic tensor components.

The physical reason for this gain difference is related to the splitting of the light-hole and heavy-hole valence bands as the uniaxial stress lowers the crystal symmetry from cubic to tetragonal.

Figure 7-10 shows the effect of stress on the threshold margin \(\Delta{J}_\text{th}\) with the same parameter values used for Figure 7-9. We assumed that \(\gamma_1=\gamma_2=\gamma/3\) and that \(\gamma=2\times10^{-7}\text{ cm}/\text{dyne}\).

As expected, the threshold margin decreases in the presence of stress. Furthermore, above a critical value of the active-layer thickness, the TM mode has a lower threshold and thus would lase first, before the TE mode. This behavior has been observed in some DFB lasers.

### 5. Light-Current Characteristics

Similar to an FP semiconductor laser, the light-current (\(L-I\)) characteristics of a DFB laser yield information about the threshold current and the differential quantum efficiency \(\eta_\text{d}\) that is related to the slope of the \(L-I\) curve.

In the previous section we have seen that the threshold current density \(J_\text{th}\) has similar magnitudes for FP and DFB lasers. The difference between the two kinds of devices is governed by the relative magnitudes of feedback losses \(\bar{\alpha}\) and \(\alpha_\text{m}\).

For a 250-μm-long FP laser with 32% facet reflectivities, \(\alpha_\text{m}\approx45\text{ cm}^{-1}\) [see Equation (7-3-36)].

For a DFB laser, the magnitude of \(\bar{\alpha}\) depends on the coupling coefficient \(\kappa\) as well as on the facet reflectivities. For a laser with nonreflecting facets, values of \(\kappa{L}\) greater than 3.2 are necessary to make \(\bar{\alpha}\) less than or equal to \(\alpha_\text{m}\) (see Figure 7-4). However, for cleaved-facet DFB lasers, small values of \(\kappa\) can lead to the situation where \(\bar{\alpha}\lt\alpha_\text{m}\) (see Figure 7-5).

In contrast to FP lasers, however, the DFB threshold (or \(\bar{\alpha}\)) is sensitive to phase shifts at the facets and can vary significantly from device to device.

The differential quantum efficiency \(\eta_\text{d}\) of a DFB laser can also be obtained using an expression similar to that derived in the emission characteristics of semiconductor lasers tutorial [refer to Equation (2-6-11)] for an FP laser and is given by

\[\tag{7-3-52}\eta_\text{d}=\eta_\text{i}\frac{\bar{\alpha}}{\bar{\alpha}+\alpha_\text{int}}\]

where \(\eta_\text{i}\) is the internal quantum efficiency and is nearly 100% when stimulated emission dominates in the above-threshold regime.

The internal loss \(\alpha_\text{int}\) incorporates losses from all possible mechanisms such as free-carrier absorption and scattering at the heterostructure interfaces.

For a DFB laser, two additional mechanisms may contribute to increase \(\alpha_\text{int}\) and decrease \(\eta_\text{d}\) relative to their values for an FP laser. These are

- Scattering losses due to grating imperfections
- Radiation losses from periodic structures

The latter mechanism does not occur for a first-order grating but may contribute a loss of \(\sim5\text{ cm}^{-1}\) in the case of a second-order grating since first-order Bragg diffraction then radiates perpendicular to the junction plane.

Although numerical values of \(\bar{\alpha}\) and \(\alpha_\text{int}\) are generally device-dependent, \(\eta_\text{d}\) for a DFB laser is expected to be lower than that of an FP laser when both lasers have similar threshold densities.

An interesting aspect of DFB lasers is that the differential quantum efficiency (or the slope of the \(L-I\) curves) can be different for the light emitted by the two facets when the device is apparently axially symmetric. This feature is related to the phase shifts \(\phi_1\) and \(\phi_2\) introduced in Equation (7-3-26) to account for the relative positions of the facet and the last complete grating corrugation.

Equation (7-3-52) is not helpful for obtaining individual facet efficiencies \(\eta_1\) and \(\eta_2\). Furthermore, its derivation implicitly assumes that the internal power is axially uniform. To overcome these deficiencies, we use the general definition that the differential quantum efficiency for each facet is the power transmitted through that facet divided by the total power generated by stimulated emission throughout the cavity length.

Using Equation (7-3-13) for the intracavity field, the differential quantum efficiency for the facets becomes

\[\tag{7-3-53}\begin{align}\eta_1=\frac{(1-R_1)|B(0)|^2}{(\bar{\alpha}+\alpha_\text{int})\displaystyle\int\limits_0^L|E(z)|^2\text{d}z}\\\\\eta_2=\frac{(1-R_2)|A(L)|^2}{(\bar{\alpha}+\alpha_\text{int})\displaystyle\int\limits_0^L|E(z)|^2\text{d}z}\end{align}\]

Equations (7-3-27) - (7-3-30) can be used to obtain \(\eta_1\) and \(\eta_2\) for a DFB laser with arbitrary facet reflectivities \(R_1\) and \(R_2\) and phase shifts \(\phi_1\) and \(\phi_2\).

The effect of facet phases on the differential quantum efficiencies has been discussed. The individual efficiencies \(\eta_1\) and \(\eta_2\) as well as the total efficiency (\(\eta_\text{d}=\eta_1+\eta_2\)) vary significantly with the phases \(\phi_1\) and \(\phi_2\).

The range of variation depends on the coupling coefficient \(\kappa\). For typical values of \(\kappa{L}\) of \(\sim1\), one can expect a 20% variation in the differential quantum efficiency from device to device.

For a laser with one cleaved facet and one nonreflecting facet, powers emitted by the two facets differ considerably; their ratio can be in the range of \(2-5\) depending on the coupling coefficient and the corrugation phase at the cleaved facet.

The next tutorial discusses about the ** performance of DFB semiconductor lasers**.