Edge Emitting Lasers
Thus far, we have discussed semiconductors and their properties. Now we turn to aspects of semiconductor lasers and their operation. The first semiconductor lasers consisted of heavily doped p-n junctions with cleaved facets. This construction is inefficient, as there is no defined region in which recombination can take place. Carriers can be lost to diffusion before recombination occurs. As a result, these early devices required a lot of current to reach threshold. Threshold currents were reduced with the developements of the double-heterostructure laser, which has a thin region of semiconductor with a smaller energy gap sandwiched between two oppositely doped semiconductors with a wide bandgap energy. When forward biased, carriers flow into the active region and recombine more efficiently because of the potential barriers of the heterostructure confine the carriers to the active region. There is an added advantage of guided the laser light, because the refractive index of the cladding layers is less than the active region. A typical structure of a double heterostructure laser is shown below:
.
The addition of a stripe contact to the top surface of the device crudely allows for the current flow to be confined to a small region of the device. The size of the device is actually quite small. The cavity is formed by cleaving the semiconductor along a plane of atoms. Nature has been kind here and the semiconductor redily forms very flat edges. Typical dimensions for such a device are 200um x 50um x 100um with a threshold current of 10mA. The highly elongated area of emission causes diffraction of the light to result in a high astigmatic beam of light. Unlike, most lasers, semiconductor lasers have widely divergent beams. This is a problem when trying to couple the light into an optical fibre requiring expensive packaging techniques such as micro-lenses or beam guides.
Device Operation
When the device is forward biased, the carriers flow into the active layer and recombine. The key to laser operation is the interaction between the excited population of electrons and photons. If the energy of the light is the same as the difference in energy of the electrons in the conduction band and valence band, then stimulated emission will occur. The cleaved facets of the device acts as mirrors, reflecting photons back into the active layer of the semiconductor where they interact with the excited electrons. The cavity is designed to resonate at a frequency corresponding to the energy of the stimulated emission. The cleaved facets of edge emitting devices have a reflectivity of 30% allowing some of the light to escape.
These type of devices described above make use of the Fabry-Perot resonant cavity, this is essentially a rectangular cavity with highly polished surfaces at the ends of the cavity. For a wave to exist within the cavity, it must be of an integer number of half wavelengths in length or in mathematical terms:
(4.1)
Where L is the length of the cavity and q=1,2,3,4... etc. and l is the wavelength of the light in the material. The spacing between the modes is given by:
(4.2)
Given that L >>l there will be a large number of modes within the cavity and they will be closely spaced. Below threshold the emission come largely from spontaneous emission. A frequency spectrum of a semiconductor laser is shown below. The Fabry-Perot modes are clearly shown below threshold. Above threshold, a single mode become dominant and becomes the lasing frequency.
Let us now move on to examine the rate equations governing the gain below and above threshold, by considering the injection of carriers and the mechanisms for loss. The rate equation below defined how the number of carriers changes in the active region with respect to the generation and recombination processes.
(4.3)
Where Ggen is the rate of injection of electrons and Rrec is the rate at which electrons recombine per unit volume. The rate at which electrons are being injected is given by
(4.4)
where h i is the injection efficiency i.e. the percentage of electrons injected into the device that reach the active region, I is the injection current, e the electron charge and V is the volume of the active layer. Recombination consists of several elements to take account of the various radiative and non-radiative mechanisms by which carriers may lose their excess energy.
(4.5)
In the above equation, Rsp is the rate of spontaneous emission, Rl is the current leakage from the active layer, Rnr is the non-radiative recombination rate and finally, Rst is the rate of stimulated emission. The first three terms of (4.5) all allude to natural decay processes and as such can be defined in terms of a time constant or carrier lifetime t . Stimulated emission requires the presence of photons to take place. Alternatively, the recombination terms can be written in terms of the carrier density N. In this case, the recombination rate becomes:
(4.6)
With A, B and C constants depending on the semiconductor material. Using the notion of a time constant to describe the natural carrier decay mechanisms, (4.3) becomes using (4.2),
(4.7)
Below threshold the rate of stimulated emission is negligible and can be neglected. For a steady state conditions, 
and (4.7) becomes
(4.8)
The spontaneously generated optical power is obtained by multiplying the number of photons generated per unit volume, Rsp by the energy per photon, hu and the volume of the active region. The dependence of Rnr and Rl with current is removed by grouping these losses by defining a radiative efficiency h r.
(4.9)
(4.10)
Above threshold the main photon generation term is the rate of stimulated emission. This rate must be adjusted because the photons can occupy a volume larger than that occupied by the electrons which are confined to the active medium. The ratio of the volume of the cavity to the volume occupied by the photons in the cavity is called the confinement factor and is given the symbol G .
Photon loss occurs within the cavity due to optical absorption and scattering out of the mode, and also at the output coupling mirror where a portion of the resonant mode is usefully coupled to some output medium. These losses can be characterised by a photon lifetime. The photon rate equation takes the form
(4.11)
Where b sp is the spontaneous emission factor. For uniform coupling to all modes, b sp is just the reciprocal of the number of optical modes within the bandwidth of the spontaneous emission. Before equations (4.3) and (4.11) can be solved for the steady and dynamic responce of a semiconductor laser, several of the terms need to be expressed in terms of the number of photons Np and the number of electrons N. Consider Rst, this represents the photon-stimulated net electron-hole recombination which generates more photons. For a small section of gain medium, the number of photon increase from Np to Np+D Np. This growth can be express as the gain per unit length, g, by
(4.12)
For D
z sufficiently small, the Exponent term can be approximated by 
and D
z=vgD
, where vg is the group velocity, we find
Threshold or Steady-state gain in Lasers
The optical energy of a mode propagates in a dielectric waveguide mode which is confined transversely and laterally as defined by the normalised transverse electric field profile, U(x,y). In the axial direction, this mode propagates as exp(-jb z) where b is the complex propagation constant which includes any loss or gain. Thus, the time and space varying electric field can be written as
(4.x)
where 
is the unit vector indicating TE polarisation and E0 is the magnitude of the field. The complex propagation constant 
includes the incremental transverse modal gain 
and internal modal loss 
. That is,
(4.x)
where the real part of 
and 
is an effective index of refraction for the mode. The transverse modal gain and loss are found from weighted averages of the gain and loss across the modal shape U(x,y). Both are related to power; thus, the factor of 1/2 in this equation for the amplitude propagation coefficient factor, if g(x,y) is constant across the active region and zero elsewhere. This is generally valid for in-plane lasers, but not for VCSELs. Also, for notational convenience, we shall let =
.
Most lasers can be divided into two sections, an active section of length La and a pasive section Lp. The propagating mode is reflected by the two end mirrors, which have amplitude reflection coefficients of r1 and r2, to provide a resonant cavity. The amount transmitted is potentially useful output. In order for the mode of the laser to reach threshold, the gain in the active region must be increased to the point that all the propagation losses and mirror losses are compensated, so that the electric field exactly replicates itself after one round trip in the cavity. Equivalently, we can unravel the round-trip to lie along the z-axis and require that E(z=2L)=E(z=0), provided we insert the mode reflection coefficients at z=0 and z=L. As a consequence of inserting these boundaries into (4.x) we obtain
(4.x)
Using (4.x) we can break this equation into two separate equations for magnitude and phase. For the magnitude
(4.x)
where we have chosen reference planes to make the mirror reflectivities real. Solving for 
we obtain
(4.x)
where the mean mirror reflection coefficient 
. Dividing the last equation by the total cavity length, L, realising that 
(exact for La>>l
), and defining the average internal loss 
as
we have
(4.x)
It is important to realise that the last two equations give only the cavity loss parameters necessary to calculate the threshold gain. They have nothing to do with the stimulated emission physics which determine what the gain is for a given injection current.
For the phase part of (4.x) 
requires that 
which gives a condition on the modal wavelength,
(4.x)
where m is the longitudinal mode number. It should be noted that 
varies with wavelength and it generally is also dependent upon the carrier density. Thus, when making computations these dependences must be included. That is, to
determine at a wavelength 
and a carrier density, 
, we use
(4.x)
Typically, 
and 
when 
is the index in the active region. The wavelength separation between two modes, m and m+1, to be
(4.x)
where the group effective index for the jth section 
. The group index is typically ~20-30% larger than the index of refraction, depending on the specific wavelength relative to the band edge. From experiments, the values of 
for the active sections of GaAs and InGaAsP DH plane lasers are near 4.5 and 4, respectively. Finally it is important to note that the steady-state gain in a laser operating above threshold must also equal its threshold value as given by (4.x). That is, in a laser cavity,
(4.x)
If the gain were higher than gth then the field amplitude would continue to increase without bound, and this clearly cannot exist in the steady state. Furthermore, since the gain is monotonically related to the carrier density, this implies that the carrier density must also clamp at its threshold value. That is
(4.x)
Actually, the current is increased to a value above threshold is the carrier density and gain initially (for on the order of a nanosecond) increase to values above their threshold levels, and the photon density grows. But then stimulated recombination term Rst also increases, reducing the carrier density and gain until a new steady-state dynamic balance is struck where (4.x) and (4.x) are again satisfied. Put another way, the stimulated recombination term in (4.x) uses up all additional carrier injection above threshold. The figure below summarises this carrier clamping effect in a laser cavity. The physics of the g vs. N curve never changes. The feedback effect causes the carrier density to clamp, in order to keep the gain at its threshold value.