# Semiconductor Lasers

This is a continuation from the previous tutorial - **semiconductor optical amplifiers (SOAs)**.

Semiconductor lasers, also called ** laser diodes** or

**, are compact and efficient laser that have found many important application in optical communications, optical data storage, optical signal processing, compact-disk players, laser printers, and medical instruments.**

*diode lasers*High-power semiconductor lasers are also used as highly efficient pump sources for other optically pumped lasers and amplifiers, such as solid-state lasers, optical fiber lasers, and fiber amplifiers.

Unlike the light-emitting active region of an LED, which can be made of either a direct-gap or an indirect-gap semiconductor, the active region of a semiconductor laser has to be made of a direct-gap semiconductor of high radiative efficiency though indirect-gap semiconductors can be used for its cladding layers.

Practical semiconductor lasers are based on III-V semiconductors though lasers based on IV-VI compounds have also been developed. In principle, any direct-gap semiconductor can be used as a laser material. In reality, however, there are many other issues to be considered.

For most lasers, the active region has to be lattice matched to the substrate in order to avoid defects in the active layer, which can act as nonradiative recombination centers. In addition, lattice-matched cladding layers that have larger bandgaps and lower indices than the active layer are required for heterostructures.

These requirements limit the spectral ranges of certain materials for laser applications. As a consequence, not all ternary and quaternary compounds that have direct bandgaps are successfully used to make lasers.

One notable exception is the InGaN lasers fabricated on sapphire substrates. Another exception is strained QW (Quantum Well) lasers. The wavelength ranges of major lasers based on III-V semiconductors are listed in Table 13-2.

There are many significant differences, in fundamental principles, structures, and characteristics, between a semiconductor laser and an LED though they use similar semiconductor materials and share the basic structures discussed in the semiconductor junction structures tutorial.

The fundamental principles of a semiconductor laser differ from those of an LED in that a laser requires an optical gain for stimulated amplification of the emitted photons and a resonant cavity for optical feedback, both of which are not needed for an LED.

Because of the need for an optical gain, the active region of a semiconductor laser has to be made of a direct-gap semiconductor of high radiative efficiency. An indirect-gap semiconductor simply cannot be pumped efficiently to reach the condition of population inversion for an optical gain.

To reach population inversion, a semiconductor laser requires a higher current density than that required by a typical LED. Therefore, confinement of the current flow for efficiently injecting carriers into the active region and confinement of injected carriers for reducing carrier leakage are important factors to be considered in designing the structure of a semiconductor laser.

As discussed in the semiconductor junction structures tutorial, the structures that serve these purposes well are DH (Double Heterostructures) junctions and quantum wells in the vertical direction and, for edge-emitting lasers, index-guiding structures in the lateral direction [refer to the lateral structures tutorial].

For this reason, efficient semiconductor lasers are commonly index-guided devices with DH junctions or quantum wells.

The need of a resonant cavity to provide optical feedback for the laser action leads to many different designs of laser structures.

The combination of stimulated emission and optical feedback results in many characteristics of a laser, including the presence of a laser threshold, the existence of laser modes, the coherence and narrow linewidth of the laser emission, the high quantum efficiency of a laser, and the large modulation bandwidth of a laser, that are absent from the characteristics of an LED.

In terms of the mechanism for optical feedback, there are two basic types of resonant cavities for semiconductor lasers: the Fabry-Perot cavity and the grating-feedback cavity.

Though both types of cavities serve the same purpose of providing optical feedback for laser oscillation, they are based on very different principles and have very different characteristics.

Each type of cavity can have a number of different variations. Hybrids of the two types are also used in some lasers. Both types of cavities and their variations and hybrids can be used to make either edge-emitting or surface-emitting lasers.

In terms of structural geometry, the cavity of a semiconductor laser can be a horizontal cavity, formed in a direction parallel to the junction plane, a vertical cavity, formed in the direction perpendicular to the junction plane, or a folded cavity.

An edge-emitting laser normally has a horizontal cavity. In contrast, a surface-emitting laser can have a horizontal cavity, a vertical cavity, or a folded cavity.

The general discussions on laser oscillation in the laser oscillation tutorial, including the concepts of laser threshold, mode pulling, and longitudinal modes, apply to semiconductor lasers as well.

For a laser to oscillate at a particular frequency, the general concept is that the round-trip gain has to exactly balance the round-trip loss while the round-trip phase shift is a multiple of \(2\pi\).

This concept is also applied to determine the threshold gain and the oscillating modes of a semiconductor laser. However, some special considerations are often needed in the application of this concept because of the structural variation among different kinds of semiconductor lasers.

Two structural factors are most significant for semiconductor lasers.

First, the overlap between the laser field distribution and the active gain medium in a semiconductor can be small, it has to be considered for the gain of the laser.

Second, many semiconductor lasers use gratings for their optical feedback; the effects of a grating on the phase and amplitude of a laser field have to be considered in such cases.

In a semiconductor laser, the volume of a laser mode is generally larger than that of the active region where the gain exists. The ** gain overlap factor**, or the

**, of a laser mode is thus defined as**

*gain filling factor*\[\tag{13-79}\Gamma=\frac{\displaystyle\iiint\limits_\text{active}|\mathbf{E}|^2\text{d}x\text{d}y\text{d}z}{\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty|\mathbf{E}|^2\text{d}x\text{d}y\text{d}z}\approx\frac{\mathcal{V}_\text{active}}{\mathcal{V}_\text{mode}}\]

where \(\mathbf{E}\) is the intracavity laser field, \(\mathcal{V}_\text{active}\) is the volume of the active region, and \(\mathcal{V}_\text{mode}\) is the effective volume of the laser mode under consideration.

With the notable exception of the vertical-cavity surface-emitting laser (VCSEL), most semiconductor lasers are basically waveguide lasers of stripe geometry.

For a typical stripe-geometry laser that has a thin active layer, the laser waveguide has a thickness much less than its width, \(d\ll{w}\). It can be considered as a single-mode slab waveguide that has a small \(V\) number.

Then, according to (93) in the symmetric slab waveguides tutorial, the confinement factor, \(\Gamma_\text{mode}\), of the laser mode can be approximated by

\[\tag{13-80}\Gamma_\text{mode}=\frac{V^2}{2+V^2}\approx\frac{4\pi^2n\Delta{n}d^2/\lambda^2}{1+4\pi^2n\Delta{n}d^2/\lambda^2}\]

where \(n\) and \(\Delta{n}\) are, respectively, the refractive index and the index step of the laser waveguide.

Fro a DH (double heterostructures) semiconductor laser that does not contain quantum wells, the carriers distribute almost uniformly in the thickness \(d\) of the active layer, which also serves as the optical waveguide. Then the overlap factor \(\Gamma\) is the same as the mode confinement factor \(\Gamma_\text{mode}\).

For a QW (quantum well) laser, the carriers are confined in the width \(d_\text{QW}\) of a quantum well but the laser mode is confined by the waveguide width \(d\). Because \(d_\text{QW}\ll{d}\) in general for a QW laser, the overlap factor \(\Gamma\) is not the same as, but is smaller than, the mode confinement factor \(\Gamma_\text{mode}\). We thus have

\[\tag{13-81}\Gamma=\begin{cases}\Gamma_\text{mode},\qquad\qquad\qquad\text{for DH lasers}\\\frac{M_\text{QW}d_\text{QW}}{d}\Gamma_\text{mode},\qquad\quad\text{for QW lasers}\end{cases}\]

where \(M_\text{QW}\) is the number of quantum wells in the active layer of a QW laser.

For a VCSEL, which is generally a QW laser, the overlap factor has the form of the filling factor defined in the laser oscillation tutorial. It takes the following simple form:

\[\tag{13-82}\Gamma=a\frac{M_\text{QW}d_\text{QW}}{l}\]

where \(l\) is the length of the laser cavity and \(a\) is a factor between 1 and 2.

The confinement factor of a stripe-geometry laser is independent of the cavity length, but that of a VCSEL can be increased by reducing the length of the VCSEL cavity.

The threshold gain coefficient for each mode of a semiconductor laser can be found by applying the concept of balancing the gain with the loss of the laser mode. Because some semiconductor lasers, such as DFB lasers, do not use localized cavity mirrors, the threshold gain coefficient of a given laser mode can be generally expressed as

\[\tag{13-83}\Gamma{g}_\text{th}=\bar{\alpha}+\alpha_\text{out}\]

where \(\bar{\alpha}\) is the internal distributed loss as defined in (11-56) [refer to the laser oscillation tutorial] and \(\alpha_\text{out}\) is the output coupling loss of the laser oscillator.

**Example 13-14**

A GaAs/AlGaAs laser emits at 850 nm wavelength. The refractive index of GaAs at 850 nm is \(n=3.65\).

(a) Find the gain overlap factor \(\Gamma\) if the laser is a stripe-geometry DH laser that has an active waveguide thickness of \(d=0.2\text{ μm}\) defined by an index step of \(\Delta{n}=0.2\).

(b) Find the gain overlap factor \(\Gamma\) if the laser is a tripe-geometry MQW laser that contains three quantum wells each of a thickness of \(d_\text{QW}=10\text{ nm}\) in a waveguide of \(d=0.2\text{ μm}\) defined by an index step of \(\Delta{n}=0.2\).

(c) Find the gain overlap factor \(\Gamma\) if the laser is an MQW VCSEL that contains three quantum wells each of a thickness of \(d_\text{QW}=10\text{ nm}\) in a cavity of \(l=1\text{ μm}\). Take the factor \(a=2\).

**(a)**

For \(\lambda=850\text{ nm}\), \(d=0.2\text{ μm}=200\text{ nm}\), \(n=3.65\), and \(\Delta{n}=0.2\), we find that

\[\Gamma_\text{mode}=\frac{4\times\pi^2\times3.65\times0.2\times(200/850)^2}{1+4\times\pi^2\times3.65\times0.2\times(200/850)^2}=0.61\]

Thus, \(\Gamma=\Gamma_\text{mode}=61\%\) for the DH laser without quantum wells.

**(b)**

For the MQW laser, we have \(M_\text{QW}=3\) and \(d_\text{QW}=10\text{ nm}\). We also have \(\Gamma_\text{mode}=61\%\) from (a). From (13-81), we find the following overlap factor:

\[\Gamma=\frac{3\times10}{200}\times61\%=9.2\%\]

**(c)**

For the MQW VCSEL, we have \(M_\text{QW}=3\), \(d_\text{QW}=10\text{ nm}\), \(l=1\text{ μm}\), and \(a=2\). From (13-82), we find the following gain overlap factor:

\[\Gamma=2\times\frac{3\times10\times10^{-9}}{1\times10^{-6}}=6\%\]

We find that the gain overlap factor of the MQW VCSEL and that of the stripe-geometry MQW laser are both less than \(10\%\) and much smaller than that of the DH laser. This small gain overlap factor is normally compensated by the much higher gain of quantum wells in comparison to that of an ordinary DH structure.

**Edge-Emitting Lasers**

Most edge-emitting lasers are stripe-geometry lasers, though some broad-area edge-emitting laser are still useful. The cavity of an edge-emitting laser is normally a horizontal cavity with a longitudinal axis defined by a gain-guiding or index-guiding stripe.

There are three different kinds of edge-emitting semiconductor lasers: the ** Fabry-Perot laser**, the

**(DBR laser), and the**

*distributed Bragg reflector laser***(DFB laser).**

*distributed feedback laser*

**Fabry-Perot Lasers**

A Fabry-Perot resonant cavity for an edge-emitting semiconductor laser, shown in Figure 13-30, can be realized by simply cleaving end facets.

Because the entire structure of a semiconductor laser forms a single crystal, the cleaved facets are guaranteed to be perfectly parallel and vertical if they are cleaved along one of the crystalline planes. Typical III-V semiconductor lasers have end facets cleaved along the (110) plane of the crystal.

Because a III-V semiconductor has a high refractive index, a cleaved facet in the air has a reflectivity typically in the range of 25-35%, depending on the specific composition of the semiconductor and the polarization and the mode of the laser field.

Because of the high optical gain of a typical semiconductor laser, the optical feedback provided by such natural reflectivity from the cleaved facets is normally sufficient for laser oscillation.

With such a Fabry-Perot cavity formed by two cleaved facets, the laser emits equally from both ends. To increase the laser output in one direction, the back facet can be coated with total-reflection coating so that all of the laser power is emitted from the uncoated front facet.

Without an additional spectrum-filtering or frequency-selecting mechanism incorporated into the device, a Fabry-Perot semiconductor laser tends to oscillate in multiple longitudinal modes with a mode spacing of \(\Delta\nu_\text{L}\) given in (11-43) [refer to the resonant optical cavities tutorial] but modified by the mode-pulling effect.

The threshold gain coefficient for each mode of a Fabry-Perot semiconductor laser is found by taking \(\alpha_\text{out}=-(\ln\sqrt{R_1R_2})/l\) in (13-83) for the output-coupling loss to be the mirror loss, thus reducing (13-83) to the form of (11-56) [refer to the laser oscillation tutorial].

**Distributed Bragg Reflector Lasers**

Both DBR and DFB lasers use built-in gratings for optical feedback, but they have some basic structural differences and thus different characteristics.

A DBR laser simply utilizes one or two Bragg reflectors as end mirrors in a manner similar to the mirrors of a Fabry-Perot laser. In contrast, a DFB laser uses a grating not as an end mirror but as a distributed feedback mechanism.

The principle and detailed characteristics of grating waveguide couplers are discussed in the grating waveguide couplers tutorial. By properly choosing the grating period \(\Lambda\), a DBR can be designed to have a peak reflectivity at a desired Bragg frequency of

\[\tag{13-84}\nu_\text{B}=\frac{c}{\lambda_\text{B}}\]

where \(\lambda_\text{B}\) is the Bragg wavelength defined in (23) [refer to the grating waveguide couplers tutorial].

At the Bragg frequency, the Bragg reflector has a peak reflectivity of

\[\tag{13-85}R_\text{DBR}=\tanh^2|\kappa|l_\text{DBR}\]

where \(l_\text{DBR}\) is the ** actual physical length** of the DBR.

From (24) [refer to the grating waveguide couplers tutorial], we know that the phase shift on reflection from a DBR for a wave that has a propagation constant \(\beta(\omega)\) at a frequency \(\omega\) is

\[\tag{13-86}\varphi_\text{DBR}=\varphi_\text{B}+2[\beta(\omega)-\beta_\text{B}]l_\text{DBR}^\text{eff}\]

where \(\beta_\text{B}=\beta(\omega_\text{B})\) and

\[\tag{13-87}l_\text{DBR}^\text{eff}=\frac{\tanh|\kappa|l_\text{DBR}}{2|\kappa|}=\frac{R_\text{DBR}^{1/2}}{2|\kappa|}\]

is the ** effective phase length** of the DBR for its reflection phase shift.

We see from (13-86) that \(\varphi_\text{DBR}=\varphi_\text{B}\) at the Bragg frequency.

According to (29) [refer to the grating waveguide couplers tutorial], the frequency bandwidth for high reflectivity of a DBR is approximately

\[\tag{13-88}\Delta\nu_\text{DBR}\approx\frac{|\kappa|c}{\pi{N}_\beta}\]

where \(N_\beta\) is the effective group refractive index of the mode field at the Bragg frequency.

The peak reflectivity and the bandwidth of a DBR can be chosen by properly choosing the coupling coefficient \(\kappa\) and the physical length \(l_\text{DBR}\) of the DBR.

The effective phase length \(l_\text{DBR}^\text{eff}\) is then determined by both \(\kappa\) and \(l_\text{DBR}\). Thus the phase shift \(\varphi_\text{DBR}\) is only a function of optical frequency once the physical parameters \(\kappa\) and \(l_\text{DBR}\) of the DBR are given.

To make a DBR laser, one or both of the reflective facets of a Fabry-Perot laser are replaced with DBR mirrors. Figure 13-31 shows a DBR laser with two Bragg reflectors. Note that in a DBR laser, the grating is placed outside the active region of the laser.

A DBR laser cavity of a length \(l\) between the two DBR mirrors such as the one shown in Figure 13-31 has many longitudinal modes at the frequencies defined by the following resonance condition:

\[\tag{13-89}2\beta{l}+\varphi_\text{DBR1}+\varphi_\text{DBR2}=2q\pi\]

where \(\beta\) is the propagation constant of the laser field in the laser waveguide; \(\varphi_\text{DBR1}\) and \(\varphi_\text{DBR2}\) are the phase shifts of the field upon reflection from the left and the right DBR mirrors, respectively, and \(q\) is the integral longitudinal mode number of the resonant cavity (not to be confused with the grating order of the DBR).

Using (13-86) for the DBR phase shift, this resonance condition for the longitudinal mode frequencies of a DBR laser with DBR mirrors on both ends can be expressed as

\[\tag{13-90}2\beta(\omega_\text{q})l_\text{eff}=2q\pi-\varphi_\text{B1}-\varphi_\text{B2}+2\beta_\text{B}(l_\text{DBR1}^\text{eff}+l_\text{DBR2}^\text{eff})\]

where

\[\tag{13-91}l_\text{eff}=l+l_\text{DBR1}^\text{eff}+l_\text{DBR2}^\text{eff}\]

is the effective phase length of the DBR laser cavity including the effect of the DBR mirrors. Note that all the parameters on the right-hand side of (13-90) are frequency-independent constants.

We see form (13-90) that the Bragg frequency \(\nu_\text{B}=\omega_\text{B}/2\pi\) is not necessarily a longitudinal mode frequency of the DBR laser. Therefore, ** a DBR laser does not generally oscillate exactly at the Bragg frequency of its DBR mirrors**.

However, in normal situations when the gain spectrum of the semiconductor material peaks near the Bragg frequency with a bandwidth much broader than the spectral bandwidth of the DBR mirrors, the oscillating longitudinal mode frequency must be the one that is closest to the Bragg frequency because the cavity has the lowest loss at the Bragg frequency where the DBR reflectivity has a maximum value.

Near the Bragg frequency, \(\beta(\omega)=n_\beta\omega/c\) is approximately linearly proportional to the optical frequency. Therefore, the longitudinal mode spacing of a DBR laser cavity can be given approximately by

\[\tag{13-92}\Delta\nu_\text{L}=\frac{c}{2n_\beta{l}_\text{eff}}\]

where \(n_\beta\) is the effective phase index of the mode field.

Though a DBR laser cavity has multiple longitudinal modes similarly to a Fabry-Perot cavity, the DBR mirrors are much more frequency selective than the Fabry-Perot cavity mirrors. Therefore, if the DBR bandwidth \(\Delta\nu_\text{DBR}\) is made sufficiently narrow, a DBR laser will oscillate in a single longitudinal mode at a frequency that is closest to the Bragg frequency \(\nu_\text{B}\).

To find the threshold gain coefficient of a DBR laser, we consider the balance of the gain with the loss of the laser.

The optical gain for a pass through the laser cavity is \(\Gamma{g}l\) because the length of the gain medium is \(l\) and the gain overlap factor is \(\Gamma\).

The loss for a pass is \(\bar{\alpha}l-\ln\sqrt{R_1R_2}\), where \(\bar{\alpha}l\) is the total distributed loss including that contributed by the scattering and absorption, but not the transmission, of the laser field in the DBRs and \(-\ln\sqrt{R_1R_2}\) is the transmission loss of the DBRs.

Therefore, for a DBR laser, \(\bar{\alpha}\) is a weighted average of the distributed loss of the entire structure divided only by the length \(l\).

By equating the gain with the loss for the laser threshold, we then find that the threshold gain coefficient of a DBR semiconductor laser can be expressed in the form of (13-83) by taking \(\alpha_\text{out}=-(\ln\sqrt{R_1R_2})/l\) like that of a Fabry-Perot laser but by using the reflectivities of the Bragg mirrors at the oscillating laser frequency for \(R_1\) and \(R_2\).

Note that \(l_\text{eff}\) defined in (13-91) is used to find \(\Delta\nu_\text{L}\) but is not used to evaluate \(\alpha_\text{out}\) for a DBR laser because \(l_\text{eff}\) is an effective ** phase** length, which only determines the phase shift of the laser field but does not determine the amplification or attenuation of the laser intensity.

**Example 13-15**

An InGaAsP DBR laser consists of a gain section of a length \(l=300\text{ μm}\) and two identical DBRs as end mirrors, each of a length \(l_\text{DBR}=150\text{ μm}\). The Bragg wavelength of the DBRs is \(\lambda_\text{B}=1.53000\text{ μm}\). The effective indices for the laser modes are taken to be \(n_\beta=N_\beta=3.45\). The gain overlap factor is \(\Gamma=0.4\). The DBR coupling coefficient is \(|\kappa|=50\text{ cm}^{-1}\). The laser has a distributed loss of \(\bar{\alpha}=40\text{ cm}^{-1}\), which includes the contributions from the DBRs and scattering at the junctions between the gain section and the DBR sections.

(a) Find the peak reflectivity and the bandwidth of the two identical DBRs.

(b) Find the effective phase length of the DBRs and that of the DBR laser to determine the longitudinal mode spacing of the laser.

(c) How many longitudinal modes fall within the DBR bandwidth? If the laser is pumped in such a way that only one longitudinal mode oscillates, what is its wavelength?

(d) What is the threshold gain coefficient of the oscillating mode?

(e) If the gain medium has a gain cross section of \(\sigma=3\times10^{-20}\text{ m}^2\), what is the required carrier density above transparency for the laser to reach its threshold?

**(a)**

For \(|\kappa|=50\text{ cm}^{-1}=5000\text{ m}^{-1}\) and \(l_\text{DBR}=150\text{ μm}\), we have \(|\kappa|l_\text{DBR}=0.75\). Therefore, the DBR peak reflectivity at the Bragg wavelength is

\[R_\text{DBR}=\tanh^2|\kappa|l_\text{DBR}=\tanh^2{0.75}=0.403\]

The bandwidth of the Bragg reflectors is

\[\Delta\nu_\text{DBR}\approx\frac{|\kappa|c}{\pi{N}_\beta}=\frac{5000\times3\times10^8}{\pi\times3.45}\text{ Hz}=138.4\text{ GHz}\]

**(b)**

The effective phase length, \(l_\text{DBR}^\text{eff}=l_\text{DBR1}^\text{eff}=l_\text{DBR2}^\text{eff}\), of both DBRs is

\[l_\text{DBR}^\text{eff}=\frac{R_\text{DBR}^{1/2}}{2|\kappa|}=\frac{0.403^{1/2}}{2\times5000}\text{ m}=63.5\text{ μm}\]

Thus, the effective phase length of the laser is

\[l_\text{eff}=l+2l_\text{DBR}^\text{eff}=427\text{ μm}\]

We then find the following longitudinal mode spacing:

\[\Delta\nu_\text{L}=\frac{c}{2n_\beta{l}_\text{eff}}=\frac{3\times10^8}{2\times3.45\times427\times10^{-6}}\text{ Hz}=101.8\text{ GHz}\]

**(c)**

Because \(2\Delta\nu_\text{L}\gt\Delta\nu_\text{DBR}\gt\Delta\nu_\text{L}\), there is at least one longitudinal mode, but at most two modes, within the reflector bandwidth. Whether one or two modes fall within the DBR bandwidth depends on where the mode frequencies are located with respect to the Bragg frequency.

To answer this question, we need to find the mode number \(q\) for the longitudinal mode frequency \(\nu_q\) that is closest to \(\nu_\text{B}\). Using (13-90), we find that the phase mismatch for a mode at \(\nu_q\) can be expressed as

\[\delta_q=-\beta(\omega_q)+\beta_\text{B}=\frac{1}{2l_\text{eff}}[2\beta_\text{B}l-(2q-1)\pi]\]

The mode frequency that is closest to \(\nu_\text{B}\) is found by finding the number \(q\) that minimizes the value of \(|\delta_q|\). Using \(n_\beta=3.45\) and \(\lambda_\text{B}=1.53000\text{ μm}\) for \(\beta_\text{B}=2\pi{n}_\beta/\lambda_\text{B}\), we find that the value of \(|\delta_q|\) is minimized with \(q=1353\) for \(\delta=32.46\text{ cm}^{-1}=3246\text{ m}^{-1}\). Thus

\[\nu-\nu_\text{B}=-\frac{c\delta}{2\pi{n}_\beta}=-\frac{3\times10^8\times3246}{2\times\pi\times3.45}\text{ Hz}=-44.9\text{ GHz}\]

For the two neighboring modes, corresponding to \(q+1=1354\) and \(q-1=1352\), the \(q+1=1354\) mode falls within the DBR bandwidth because \(|\nu_{q+1}-\nu_\text{B}|=|-44.9+101.8|\text{ GHz}\lt\Delta\nu_\text{DBR}/2=69.2\text{ GHz}\), but the \(q-1=1352\) mode falls outside the DBR bandwidth because \(|\nu_{q-1}-\nu_\text{B}|=|-44.9-101.8|\text{ GHz}\gt\Delta\nu_\text{DBR}/2=69.2\text{ GHz}\).

Therefore, there are two modes that fall within the DBR bandwidth. If the laser is pumped right at the threshold so that only one mode oscillates, the \(q=1353\) mode will oscillate because it has the smallest phase mismatch, thus the lowest threshold. Its wavelength

\[\begin{align}\lambda&=\frac{c}{\nu}=\frac{c\lambda_\text{B}}{c+(\nu-\nu_\text{B})\lambda_\text{B}}\\&=\frac{3\times10^8\times1.53\times10^{-6}}{3\times10^8-44.9\times10^9\times1.53\times10^{-6}}\text{ m}\\&=1.53035\text{ μm}\end{align}\]

**(d)**

With \(|\kappa|=50\text{ cm}^{-1}\), \(\delta=32.46\text{ cm}^{-1}\), and \(l_\text{DBR}=150\text{ μm}\), we have \(|\kappa|l=0.75\) and \(|\delta/\kappa|^2=(32.46/50)^2\). By using (91) [refer to the two-mode coupling theory tutorial] for the contradirectional coupling efficiency in the presence of phase mismatch, the DBR reflectivity at the oscillating mode frequency can be found as

\[R=\frac{\sinh^2\left(|\kappa|l\sqrt{1-|\delta/\kappa|^2}\right)}{\cosh^2\left(|\kappa|l\sqrt{1-|\delta/\kappa|^2}\right)-|\delta/\kappa|^2}=0.385\]

which is somewhat smaller than \(R_\text{DBR}\) because the mode frequency does not fall right at \(\nu_\text{B}\). Taking \(R_1=R_2=R=0.385\), the output coupling loss for the DBR laser is

\[\alpha_\text{out}=-\frac{\ln\sqrt{R_1R_2}}{l}=-\frac{\ln0.385}{300}\text{ μm}^{-1}=3.18\times10^{-3}\text{ μm}^{-1}=31.8\text{ cm}^{-1}\]

Thus the threshold gain coefficient

\[g_\text{th}=\frac{\bar{\alpha}+\alpha_\text{out}}{\Gamma}=\frac{40+31.8}{0.4}\text{ cm}^{-1}=179.5\text{ cm}^{-1}\]

**(e)**

For \(g_\text{th}=179.5\text{ cm}^{-1}=1.795\times10^4\text{ m}^{-1}\) with \(\sigma=3\times10^{-20}\text{ m}^2\), the threshold carrier density above transparency is

\[N_\text{th}-N_\text{tr}=\frac{g_\text{th}}{\sigma}=\frac{1.795\times10^4}{3\times10^{-20}}\text{ m}^{-3}=5.98\times10^{23}\text{ m}^{-3}\]

For this laser, there are two modes within the DBR bandwidth. In this particular case, the threshold of the second mode is not much higher than the first mode considered above. Therefore, this DBR laser can possibly oscillate in two modes. This problem can be avoided by proper design of a DBR laser.

**Distributed Feedback Lasers**

In a DFB laser, the grating is placed right next to the waveguiding layer along the length of the active region, as shown in Figure 13-32(a).

We learn from the discussions in the grating waveguide couplers tutorial that it can be placed either above or below the active layer for the same effect.

The grating provides all of the optical feedback for laser oscillation. The end facets of a DFB laser are coated with antireflection coating to eliminate any reflection from the facets.

Because the grating in a DFB laser runs along the length of the active region where optical gain exists, it does not function as a simple passive reflector like that in a DBR laser but rather as a frequency-selective contradirectionally coupled amplifier for the intracavity laser field. Consequently, the characteristics of a DFB laser are more complicated than those of a DBR laser.

A DFB laser also has multiple longitudinal modes whose frequencies are still determined by the basic requirement that the round-trip phase shift be an integral multiple of \(2\pi\).

Meanwhile, its threshold gain coefficient is still subject to the relation in (13-83) under the requirement that the gain and loss of an oscillating laser mode exactly balance each other.

Because of the distributed nature of the optical feedback in a DFB laser, however, it is not feasible to apply these concepts by simply following the round-trip propagation of a laser field as is done for Fabry-Perot lasers and ring lasers. Instead, we have to consider coupling of the contrapropagating fields in a DFB laser cavity by using the concepts discussed in the two-mode coupling tutorial and the grating waveguide couplers tutorial.

A DFB laser that haws a continuous grating across its entire length \(l\) and perfect antireflection coating on its end facets, as shown in Figure 13-32(a), is considered.

This structure is basically a DBR. From the discussions in the two-mode coupling tutorial and the grating waveguide couplers tutorial, we know that its complex reflection coefficient can be found by replacing \(\kappa_{ab}\) with \(\kappa\) and \(\kappa_{ba}\) with \(\kappa^*\) in (72) for \(\alpha_\text{c}\) and in (78) for \(r\) [refer to the two-mode coupling tutorial], where \(\kappa=\kappa_{ab}(q)\) as defined in (8) and (10) [refer to the grating waveguide couplers tutorial].

Thus, we have

\[\tag{13-93}r=\frac{\text{i}\kappa^*\sinh\alpha_\text{c}l}{\alpha_\text{c}\cosh\alpha_\text{c}l+\text{i}\delta\sinh\alpha_\text{c}l}\]

where

\[\tag{13-94}\alpha_\text{c}=(|\kappa|^2-\delta^2)^{1/2}\]

A DFB laser differs from a passive DBR in that an optical field propagating in a DFB laser sees an optical gain coefficient of \(\Gamma{g}\) and a distributed loss of \(\bar{\alpha}\) just like a field in any laser.

To account for the effects of such gain to and loss from the laser medium, the laser field at a frequency \(\omega\) has a complex propagation constant \(\beta\) of the following form:

\[\tag{13-95}\beta=\frac{n_\beta\omega}{c}-\text{i}\frac{\Gamma{g}-\bar{\alpha}}{2}\]

Then, from (21) and (22) [refer to the grating waveguide couplers tutorial], we have the following frequency-dependent phase mismatch:

\[\tag{13-96}\delta=-\beta(\omega)+\beta_\text{B}=-\frac{n_\beta\Delta\omega}{c}+\text{i}\frac{\Gamma{g}-\bar{\alpha}}{2}\]

where \(\Delta\omega=\omega-\omega_\text{B}\) and \(\omega_\text{B}=2\pi\nu_\text{B}\).

Note that \(\alpha_\text{c}\) given in (13-94) is frequency dependent because of the frequency dependence of \(\delta\), and \(\alpha_\text{c}\) is complex when \(\Delta\omega\ne0\). Clearly, the reflection coefficient \(r\) given in (13-93) is highly frequency dependent.

The oscillation condition of a DFB laser can be found by considering the fact that when a laser mode is oscillating, there is a laser output without an optical input at that particular mode frequency. This condition is met when \(r=\infty\) at the oscillating mode frequency.

Because the numerator of \(r\) in (13-93) is always finite for finite values of \(|\kappa|\), \(\delta\), and \(l\), the oscillation condition of a DFB laser is found by setting its denominator to zero:

\[\tag{13-97}\alpha_\text{c}\cosh\alpha_\text{c}l+\text{i}\delta\sinh\alpha_\text{c}l=0\]

This oscillation condition for a DFB laser can be transformed to the following simple form:

\[\tag{13-98}|\kappa|\sinh\alpha_\text{c}l=\text{i}\alpha_\text{c}\]

The longitudinal mode frequencies and the threshold gain coefficient of each mode for a DFB laser of a given \(|\kappa|l\) value can be found by solving the oscillation condition in (13-97) or, equivalently, that in (13-98).

The complex transcendental equations in (13-97) and (13-98) have no simple analytical solutions. Some of their characteristics can be obtained from approximate solutions, but accurate solutions must be obtained numerically.

It can be shown analytically, however, that the Bragg frequency \(\nu_\text{B}\) is not a longitudinal mode of the DFB laser and that the longitudinal modes are symmetrically distributed on both sides of \(\nu_\text{B}\).

By solving (13-97), or (13-98), for a longitudinal mode frequency of a DFB laser that has a given \(|\kappa|l\) value, the value of \((\Gamma{g_\text{th}}-\bar{\alpha})l=\alpha_\text{out}l\) at the oscillation threshold of that particular mode is obtained simultaneously.

Numerical solutions show that the longitudinal modes have the following longitudinal mode frequencies:

\[\tag{13-99}\nu_q\approx\nu_\text{B}\pm(q+\mu)\frac{c}{2n_\beta{l}}\]

where \(q=0,1,2,\ldots\) is an integral mode number for the DFB laser modes (not to be confused with the grating order) and \(\mu\) is a constant that is a function of \(|\kappa|l\).

The longitudinal mode spacing of a DFB laser is approximately, but not exactly,

\[\tag{13-100}\Delta\nu_\text{L}\approx\frac{c}{2n_\beta{l}}\]

which is similar to that of a Fabry-Perot laser of an effective index of \(n_\beta\) and a cavity length of \(l\).

*The optical gain required for a DFB laser to oscillate is lowest at the Bragg frequency, but a DFB laser that does not have a structural phase shift in its grating does not oscillate at its Bragg frequency because \(\nu_\text{B}\) is not one of its mode frequencies.*

In an ideal situation, the two lowest-order frequencies on the two sides of \(\nu_\text{B}\), corresponding to \(q=0\), have the same lowest oscillation threshold. Therefore, in a normal operating condition, the spectral feature of a DFB laser often consists of the two longitudinal modes at

\[\tag{13-101}\nu=\nu_\text{B}\pm\frac{\mu{c}}{2n_\beta{l}}=\nu_\text{B}\pm\mu\Delta\nu_\text{L}\]

which have the following two wavelengths:

\[\tag{13-102}\lambda\approx\lambda_\text{B}\pm\frac{\mu\lambda_\text{B}^2}{2n_\beta{l}}\]

Clearly, there is a ** stop band** that is centered at the Bragg frequency between these two fundamental mode frequencies:

\[\tag{13-103}\Delta\nu_\text{SB}=\frac{\mu{c}}{n_\beta{l}}=2\mu\Delta\nu_\text{L}\]

Figure 13-33 shows the numerically solved value of \(\mu=\Delta\nu_\text{SB}/2\Delta\nu_\text{L}\) and the output coupling loss \(\alpha_\text{out}l=(\Gamma{g}_\text{th}-\bar{\alpha})l\) of a DFB laser at the threshold of its fundamental mode frequencies, both as a function of the value of \(|\kappa|l\). We see that \(\mu\rightarrow1/2\) for \(|\kappa|l\rightarrow0\), but \(\mu\gt1/2\) for \(|\kappa|l\ne0\).

Figure 13-34 shows two longitudinal mode spectra of a DFB laser for \(|\kappa|l=1.5\) and \(|\kappa|l=1\), respectively, when the laser oscillates at its fundamental mode frequencies. This spectrum is obtained by plotting \(R=|r|^2\) as a function of the frequency difference \(\Delta\nu=\nu-\nu_\text{B}\) normalized to the mode spacing \(\Delta\nu_\text{L}\).

The symmetry of the two lowest-order modes for a DFB laser can be upset by introducing a fixed phase shift in the DFB structure. A phase shift at either end facet, whether introduced intentionally by selective coating or unintentionally by the cleaving process, can remove the degeneracy to result in single-mode, or quasi-single-mode, oscillation.

The most effective is to incorporate a \(\pi/2\) phase shift in the grating. In such a so-called \(\lambda/4\)-shifted DFB laser as shown in Figure 13-32(b), a longitudinal mode appears exactly at the Bragg frequency \(\nu_\text{B}\) where the cavity has the lowest loss and the laser has the corresponding lowest threshold.

Consequently, ** a \(\lambda/4\) phase-shifted DFB laser oscillates in a single longitudinal mode at \(\nu_\text{B}\)**.

**Example 13-16**

An InGaAsP DFB laser has its grating fabricated along its gain section, which has a length of \(l=300\,\mu\text{m}\). There is no intentional or unintentional structural phase shift in the grating. It has the same \(l=300\,\mu\text{m}\) length of the gain section as that of the DBR laser considered in Example 13-15. Its grating length of \(300\,\mu\text{m}\) is also the same as the total grating length of the two DBRs of the DBR laser. Most of the parameters of this DFB laser are the same as those of the DBR laser described in Example 13-15 with \(\lambda_\text{B}=1.53000\,\mu\text{m}\), \(n_\beta=N_\beta=3.45\), \(\Gamma=0.4\), and \(|\kappa|=50\text{ cm}^{-1}\). It has a distributed loss of \(\bar{\alpha}=10\text{ cm}^{-1}\), which is smaller than that of the DBR laser because the DBR laser has additional losses contributed by the external DBRs.

(a) Find the longitudinal mode spacing and the stop band of the DFB laser.

(b) How many longitudinal modes will oscillate if the laser is pumped to its lowest threshold? What are their wavelengths?

(c) What is the threshold gain coefficient of the oscillating modes?

(d) If the gain medium has a gain cross section of \(\sigma=3\times10^{-20}\text{ m}^2\), what is the required carrier density above transparency for this laser to reach its threshold?

Compare the characteristics of this DFB laser to those of the DBR laser in Example 13-15 while answering these questions.

**(a)**

The longitudinal mode spacing

\[\Delta\nu_\text{L}\approx\frac{c}{2n_\beta{l}}=\frac{3\times10^8}{2\times3.45\times300\times10^{-6}}\text{ Hz}=144.9\text{ GHz}\]

For this laser, we have \(|\kappa|l=1.5\). From Figure 13-33, we find that \(\mu=0.96\) for \(|\kappa|l=1.5\). Thus, the stop band

\[\Delta\nu_\text{SB}=2\mu\Delta\nu_\text{L}=278.2\text{ GHz}\]

The longitudinal mode spacing \(\Delta\nu_\text{L}\) of this DFB laser is much larger than that of the DBR laser considered in Example 13-15 because of the larger effective phase length of the DBR laser. The spacing between the two fundamental longitudinal modes of the DFB laser is \(\Delta\nu_\text{SB}\), which is even larger than \(\Delta\nu_\text{L}\).

**(b)**

Because there is no structural phase shift in the grating, both fundamental modes have the same threshold. Therefore, both of them should oscillate when the laser is pumped to reach its lowest threshold. The wavelengths of these two modes are

\[\lambda\approx\lambda_\text{B}\pm\frac{\mu\lambda_\text{B}^2}{2n_\beta{l}}=\left(1.53\pm\frac{0.96\times1.53^2}{2\times3.45\times300}\right)\text{ μm}=(1.53\pm1.09\times10^{-3})\text{ μm}\]

Thus we find two wavelengths at 1.52891 and 1.53109 μm. Both of these two modes will oscillate once the DFB laser reaches its threshold because both of them have the same threshold.

In comparison, the DBR laser oscillates in only one wavelength at its threshold, though two modes fall in the DBR bandwidth and the second mode may have a threshold only slightly higher than the first.

**(c)**

From Figure 13-33, we find that \(\alpha_\text{out}l=2.574\) at the DFB laser threshold for \(|\kappa|l=1.5\). For \(l=300\text{ μm}\), we have

\[\alpha_\text{out}=\frac{2.574}{300}\text{ μm}^{-1}=8.58\times10^{-3}\text{ μm}^{-1}=85.8\text{ cm}^{-1}\]

Therefore,

\[g_\text{th}=\frac{\bar{\alpha}+\alpha_\text{out}}{\Gamma}=\frac{10+85.8}{0.4}\text{ cm}^{-1}=240\text{ cm}^{-1}\]

This threshold gain coefficient is higher than that of the DBR laser despite the fact that the DBR laser has a much larger distributed loss than this DFB laser.

**(d)**

For \(g_\text{th}=240\text{ cm}^{-1}=2.4\times10^4\text{ m}^{-1}\) with \(\sigma=3\times10^{-20}\text{ m}^2\), the threshold carrier density above transparency is

\[N_\text{th}-N_\text{tr}=\frac{g_\text{th}}{\sigma}=\frac{2.4\times10^{4}}{3\times10^{-20}}\text{ m}^{-3}=8\times10^{23}\text{ m}^{-3}\]

The threshold carrier density above transparency for this DFB laser is higher than that for the DBR laser because of the higher threshold gain coefficient for the DFB laser.

Clearly, a DFB laser without a structural phase shift in its grating has a high threshold and oscillates in two modes. These characteristics are inferior to those of a similar DBR laser.

A DFB laser with a proper structural phase shift, such as the \(\lambda/4\)-shifted DFB laser, has a lower threshold with only one oscillating mode. It is then competitive to the DBR laser in performance and indeed is favored over the DBR laser because of its simpler and shorter structure than the DBR laser structure.

**Surface-Emitting Lasers**

The common feature for all surface-emitting lasers irrespective of their structures is that the laser output is emitted in a direction perpendicular to the semiconductor substrate.

In comparison to edge-emitting lasers, a unique advantage of surface-emitting lasers is that they can be made in a two-dimensional array on a common substrate, which is very useful for applications in parallel optical interconnects and parallel optical signal processing.

Nevertheless, a surface-emitting laser can be packaged separately and used as a discrete laser as well.

The cavity of a surface-emitting laser can be a horizontal cavity, a vertical cavity, or a folded cavity. Each type of cavity can have different variations. Different cavity structures lead to different characteristics and different applications for the surface-emitting lasers.

Considering only the basic structures, there are three kinds of surface-emitting semiconductor lasers: the ** folded-cavity surface-emitting laser** (FCSEL), the

**(GCSEL), and the**

*grating-coupled surface-emitting laser***(VCSEL).**

*vertical-cavity surface-emitting laser*Both FCSELs and GCSELs are typically stripe-geometry lasers similar to the edge-emitting lasers, but VCSELs are quite different from all of them.

**Folded-Cavity Surface-Emitting Lasers**

A FCSEL can be constructed by modifying the output-coupling geometry of an edge-emitting laser with the addition of 45° internal total-reflection mirrors to direct the laser output to the surface-emitting direction, as illustrated in Figure 13-35.

A 45° semiconductor facet serves as an internal total-reflection mirror for the intracavity laser field because the critical angle for internal reflection at a semiconductor-air interface is much smaller than 45°, due to the large refractive index of a semiconductor.

In principle, each of the three concepts for edge-emitting lasers can be used for a FCSEL.

Except for the advantages associated with its surface-emitting geometry, the general characteristics of a FCSEL are similar to those of a corresponding edge-emitting laser based on the same concept.

Uncoated semiconductor surfaces are sufficient as surface-emitting output mirrors for a Fabry-Perot FCSEL.

For a DBR FCSEL, high-reflection DBR surface-facing mirrors for output coupling can be constructed with alternating thin layers of semiconductors that have different compositions, thus different refractive indices. Such multilayer DBR reflectors are also used in the VCSELs discussed below. Because these vertically alternating layers are parallel to the substrate surface, they can be fabricated using crystal growth technology with more ease and control than a horizontal grating.

**Grating-Coupled Surface-Emitting Lasers**

The concept of grating surface output coupling discussed in the surface input and output couplers tutorial can be applied to a horizontal-cavity semiconductor laser for vertical emission through grating coupling.

Because of the use of grating coupling, a GCSEL is normally based on a DBR laser. It is possible to use the same set of gratings for both optical feedback and surface output coupling. For better control of the output beam characteristics, however, separate gratings are often used to serve the functions of optical feedback and output coupling, respectively.

Figure 13-36 shows a GCSEL that is a DBR laser with two first-order DBR mirrors and a separate section of second-order grating for surface output coupling. The surface output-coupling grating does not provide optical feedback and thus does not participate in laser oscillation.

GCSELs have a few unique features because of their structures. As a DBR laser, a GCSEL oscillates in a single longitudinal mode close to the Bragg frequency of its grating.

Sophisticated two-dimensional geometry for the output-coupling grating can be used for output beam shaping and control.

A curved or circular output-coupling grating can be used to emit a collimated output laser beam with a very small divergence. A large grating can be used to increase the output-coupling efficiency for a high-power laser.

To concentrate most of the output power into one surface-emitting beam, a blazed grating can be used to reduce the emission in the substrate direction. On the other hand, as we have learned in the surface input and output couplers tutorial, it is also possible to choose a grating for multiple output beams emitting in different directions if desired.

**Vertical-Cavity Surface-Emitting Lasers**

Uniquely among all edge-emitting and surface-emitting lasers, the resonant cavity of a VCSEL is formed in the direction perpendicular to the junction plane and the substrate.

The uniqueness of a VCSEL is that it has a very short cavity made possible by its vertical orientation. This feature has several important implications for the structure and the performance characteristics of a VCSEL.

Because of the short cavity of a VCSEL, it is required that its gain section be thin but highly efficient and its mirror reflectivities be high in order for the VCSEL to function.

A thin and efficient gain section is achieved by using quantum wells. High mirror reflectivities are achieved by using semiconductor DBRs, such as AlAS/GaAs DBRs, or dielectric DBRs, such as SiO_{2}/TiO_{2} or Si/Al_{2}O_{3} DBRs. A metal layer can also be deposited on top or below to increase the reflectivity of a DBR.

Consequently, a VCSEL is normally a QW (quantum well) DBR laser. Figure 13-37 shows the basic structure of a VCSEL. A typical VCSEL contains a very thin active region of one to four quantum wells, each of which has a typical thickness of 5-10 nm. This active region is sandwiched between two spacer layers. Optical feedback is provided by monolithically integrated DBRs.

The DBR mirrors of a VCSEL are index-modulation gratings containing thin layers of alternating compositions and refractive indices. Because of their vertical stacking geometry, the thickness and composition of each layer, as well as the sharp transition between neighboring layers, can be precisely controlled using advanced fabrication technology.

Such a square index grating with alternating layers of indices of \(\bar{n}\pm\Delta{n}/2\) has an index step of \(\Delta{n}\) and an average index of \(\bar{n}\) between two neighboring layers.

It can be shown that a first-order square grating of a 50% duty factor has the largest coupling coefficient with a magnitude given by

\[\tag{13-104}|\kappa|=\frac{2\Delta{n}}{\lambda}\]

where \(\lambda\) is the optical wavelength in free space.

The period of a first-order DBR is \(\Lambda=\lambda_\text{B}/2\bar{n}\) for \(q=1\) in (23) [refer to the grating waveguide couplers tutorial]. Therefore, the thickness of each alternating layer for the first-order square grating of a 50% duty factor is \(\lambda_\text{B}/4\bar{n}\). The physical length of such a DBR is

\[\tag{13-105}l_\text{DBR}=N_\text{DBR}\Lambda=\frac{N_\text{DBR}\lambda_\text{B}}{2\bar{n}}\]

where \(N_\text{DBR}\) is the number of pairs of alternating layers that define the grating periods. The value of \(N_\text{DBR}\) can only be an integer or half-integer because there is an integral number of layers.

The number of pairs for a semiconductor DBR ranges between 10 and 40, but that for a dielectric DBR is typically less than 10, sometimes only a few.

The peak reflectivity of the DBR at the Bragg wavelength can be expressed as

\[\tag{13-106}R_\text{DBR}=\tanh^2|\kappa|l_\text{DBR}=\tanh^2\frac{N_\text{DBR}\Delta{n}}{\bar{n}}\]

The DBR reflectivities for VCSELs are required to be very high, normally higher than 98% but often higher than 99%. The two DBRs for a VCSEL are not symmetric. The top DBR has a lower reflectivity to accommodate the output coupling from the top of the laser, whereas the bottom DBR normally has a very high reflectivity as close to 100% as possible to compensate for the loss of the top DBR and to direct all laser energy to the output window on top.

These DBRs have very large bandwidths because of their high reflectivities and small lengths that lead to correspondingly large values for \(|\kappa|\). Because current is injected vertically through these DBR mirrors, the top DBR is heavily p doped and the bottom DBR is heavily n doped for high conductivity.

In a VCSEL, the laser field propagates perpendicularly to the active layer. This feature has profound implications for the structure and characteristics of a VCSEL. As seen in Figure 13-37, the cavity length \(l\) of a VCSEL is defined by the active layer and two spacer layers around the active layer. From (13-82), the gain overlap factor of a VCSEL is

\[\tag{13-107}\Gamma=a\frac{M_\text{QW}d_\text{QW}}{l}\]

The factor \(a\) in (13-107) has a value between 1 and 2 that depends on the overlap of the active layer with the field pattern in the cavity.

Because the intracavity laser field forms a standing wave in the direction perpendicular to the active layer, it is important to locate the active layer at a crest of the standing wave so that the optical gain in the active layer is most efficiently used for stimulated amplification of the laser field.

Furthermore, to ensure that the entire active layer stays within the high-intensity crest region of the standing wave, the thickness \(d\) of the active layer, which consists of the quantum wells and the barrier layers between wells, is normally limited to one-quarter of the wavelength in the medium, \(\lambda/4n\), where \(n\) is the refractive index in the laser cavity and is normally different from \(\bar{n}\) of either of the two DBRs.

In the case when \(d\ll\lambda/4n\), the factor \(a\) has a value of 2 if the quantum wells are located properly at a crest of the standing wave pattern. For a GaAs VCSEL emitting at \(\lambda=850\) nm, the thickness of its active layer is thus restricted to 60 nm or less because \(n\approx3.65\).

The overlap between the intracavity field and the gain region, which defines the gain overlap factor \(\Gamma\) of a laser, is independent of the cross section of the cavity but increases as the length of the cavity is reduced.

It is advantageous from the standpoint of reducing the laser threshold and increasing the laser efficiency that a VCSEL have a very short cavity. The spacer layers are required, however, for two reasons. First, they are required for the standing-wave pattern to form in the cavity with a peak located at the active layer. Second, they can be tailored to guide current injection into the active layer, thus improving the efficiency of carrier injection and reducing the laser threshold. These characteristics are illustrated in Figure 13-37.

For a VCSEL, the total length \(l\) for the active layer and the spacer layers between the two DBR mirrors is typically on the order of one or a few optical wavelengths in the medium, depending on the thickness of the spacer layers. For example, a \(1\lambda\) cavity has \(l=\lambda/n\), and a \(3\lambda\) cavity has \(l=3\lambda/n\).

In order to maintain a high \(Q\) for such a short cavity, the reflectivity of the DBR mirrors has to be very high. The required reflectivity is higher than 99% for a VCSEL with a single quantum but can be slightly lower for a VCSEL with multiple quantum wells. Thus, a VCSEL is a microcavity QW laser with relatively thick DBR mirrors.

The general characteristics of a DBR laser described earlier are still valid for a VCSEL. The longitudinal mode spacing is given by (13-92), and the threshold gain coefficient is given by (13-83) with \(\alpha_\text{out}=-(\ln\sqrt{R_1R_2})/l\). In a VCSEL, both \(l_\text{DBR1}^\text{eff}\) and \(l_\text{DBR2}^\text{eff}\) can be larger than \(l\), but the total effective phase length of \(l_\text{eff}=l+l_\text{DBR1}^\text{eff}+l_\text{DBR2}^\text{eff}\) is still on the order of 1-2 μm.

Therefore, the longitudinal mode spacing is very large. It can be either lager or smaller than the bandwidths of the highly reflective DBRs, but it can be easily made comparable to, or even larger than, the entire gain bandwidth of a semiconductor quantum well. Consequently, ** a microcavity VCSEL inherently oscillates in a single longitudinal mode**.

The requirement for the confinement of electric current and optical field in a VCSEL is fulfilled by controlling the transverse dimension and geometry of the vertical cavity. Symmetric transverse geometry of circular or square shape is normally used for a VCSEL so that its emission has a nice, round pattern with a symmetric beam divergence, making its coupling to an optical fiber easy and efficient. The transverse dimension of the cavity is typically in the range of 3-10 μm in diameter.

One significant problem associated with the symmetric optical guiding structure is polarization instability in the VCSEL emission because it does not provide any polarization discrimination. For polarization control, anisotropy in the structure or in the gain medium has to be introduced.

**Example 13-17**

A GaAs/AlAs QW VCSEL is designed to emit at 850 nm wavelength. It consists of two first-order AlAs/Al_{0.2}Ga_{0.8}As DBR mirrors of a 50% duty factor. The p-side top DBR has 21 pairs of alternating quarter-wavelength AlAs and Al_{0.2}Ga_{0.8}As layers, and the n-side bottom DBR has 24 pairs. The active layer consists of three 8-nm wide GaAs quantum wells separated by 4-nm wide Al_{0.2}Ga_{0.8}As barriers, embedded in a \(1\lambda\) cavity with Al_{0.2}Ga_{0.8}As spacer layers. The refractive indices at 850 nm are \(n=3.65\) for GaAs, \(n=3.00\) for AlAs, and \(n=3.52\) for Al_{0.2}Ga_{0.8}As.

(a) Find the coupling coefficient and the reflectivities of the DBRs.

(b) Find the length of the cavity and those of the DBRs. What is the total length of the device?

(c) If the device has a distributed loss of \(\bar{\alpha}=18\text{ cm}^{-1}\), what is its threshold gain coefficient?

(d) Find the longitudinal mode spacing. Compare it to the DBR bandwidth and the gain bandwidth.

**(a)**

For the first-order AlAs/Al_{0.2}Ga_{0.8}As DBRs, we have \(\Delta{n}=0.52\) and \(\bar{n}=3.26\) because \(n=3.00\) for AlAs and \(n=3.52\) for Al_{0.2}Ga_{0.8}As. Therefore, for \(\lambda=\lambda_\text{B}=850\text{ nm}=0.85\text{ μm}\), we have

\[|\kappa|=\frac{2\Delta{n}}{\lambda}=\frac{2\times0.52}{0.85}\text{ μm}^{-1}=1.224\text{ μm}^{-1}\]

We have \(N_\text{DBR1}=21\) and \(N_\text{DBR2}=24\). Thus

\[R_\text{DBR1}=\tanh^2\frac{N_\text{DBR1}\Delta{n}}{\bar{n}}=\tanh^2\frac{21\times0.52}{3.26}=99.5\%\]

and

\[R_\text{DBR2}=\tanh^2\frac{N_\text{DBR2}\Delta{n}}{\bar{n}}=\tanh^2\frac{24\times0.52}{3.26}=99.8\%\]

The bottom DBR has a higher reflectivity because it has three more pairs than the top DBR.

**(b)**

The cavity consists mostly of Al_{0.2}Ga_{0.8}As, which has \(n=3.52\). Because it is a \(1\lambda\) cavity, its length

\[l=\frac{\lambda}{n}=\frac{850}{3.52}\text{ nm}=241.5\text{ nm}\]

For the DBRs, the average index is \(\bar{n}=3.26\). Therefore,

\[l_\text{DBR1}=N_\text{DBR1}\Lambda=21\times\frac{850}{2\times3.26}\text{ nm}=2737.7\text{ nm}\]

and

\[l_\text{DBR2}=N_\text{DBR2}\Lambda=24\times\frac{850}{2\times3.26}\text{ nm}=3128.8\text{ nm}\]

The total length of the structure is \(l_\text{device}=l+l_\text{DBR1}+l_\text{DBR2}=6108\text{ nm}=6.108\text{ μm}\).

We see that the DBRs occupy 96% of the device structure.

**(c)**

The active layer consists of three quantum wells and two barriers with a total thickness of \(d=3\times8\text{ nm}+2\times4\text{ nm}=32\text{ nm}\). In this case, \(d\ll\lambda/4n\approx60\text{ nm}\). Thus, we have \(a\approx2\) for the following gain overlap factor:

\[\Gamma=a\frac{M_\text{QW}d_\text{QW}}{l}=2\times\frac{3\times8}{241.5}=20\%\]

With \(R_1=R_\text{DBR1}=99.5\%\) and \(R_2=R_\text{DBR2}=99.8\%\), the output coupling loss of this device is

\[\alpha_\text{out}=-\frac{\ln\sqrt{R_1R_2}}{l}=-\frac{\ln\sqrt{0.995\times0.998}}{241.5\times10^{-9}}\text{ m}^{-1}=1.452\times10^4\text{ m}^{-1}=145.2\text{ cm}^{-1}\]

With \(\bar{\alpha}=18\text{ cm}^{-1}\), the threshold gain coefficient

\[g_\text{th}=\frac{\bar{\alpha}+\alpha_\text{out}}{\Gamma}=\frac{18+145.2}{0.2}\text{ cm}^{-1}=816\text{ cm}^{-1}\]

**(d)**

First, we calculate the bandwidth of the DBRs by taking \(\bar{n}\) for \(N_\beta\) in (13-88). Because both DBRs have the same \(|\kappa|=1.224\text{ μm}^{-1}=1.224\times10^6\text{ m}^{-1}\) and \(\bar{n}=3.26\), they have the same bandwidth of

\[\Delta\nu_\text{DBR}=\frac{|\kappa|c}{\pi\bar{n}}=\frac{1.224\times10^6\times3\times10^8}{\pi\times3.26}\text{ Hz}=35.9\text{ THz}\]

To find \(\Delta\nu_\text{L}\), we need to find \(l_\text{eff}\) first. Because \(R_\text{DBR1}\approx{R}_\text{DBR2}\approx1\), we have

\[l_\text{DBR1}^\text{eff}\approx{l}_\text{DBR2}^\text{eff}\approx\frac{1}{2|\kappa|}=\frac{1}{2\times1.224}\text{ μm}=408.5\text{ nm}\]

Therefore, \(l_\text{eff}=l+l_\text{DBR1}^\text{eff}+l_\text{DBR2}^\text{eff}=241.5\text{ nm}+2\times408.5\text{ nm}=1058.5\text{ nm}=1.0585\text{ μm}\). We find that \(l_\text{eff}\) is on the order of 1 μm, which is much smaller than the 6.108 μm length of the device but is much larger than the 241.8 nm length of the cavity.

Then,

\[\Delta\nu_\text{L}=\frac{c}{2nl_\text{eff}}=\frac{3\times10^8}{2\times3.52\times1.0585\times10^{-6}}\text{ Hz}=40.3\text{ THz}\]

As discussed in the semiconductor junction structures tutorial, the gain bandwidth of a quantum well is typically in the range of 20 - 40 THz. We thus find that the longitudinal mode spacing of this QW VCSEL is larger than both its DBR bandwidth and its gain bandwidth.

Clearly, this VCSEL will oscillate in a single longitudinal mode.

The next tutorial covers the topic of **semiconductor laser characteristics**.