## Quantum Cascade Laser

Figure of Merit Derivation

### 1 Figure of merit the easy way

The figure of merit (FoM) for a quantum cascade (QC) laser (QCL) is a tool used to maximize performance from a QC design. The FoM represents the parameters from the QC laser gain coefficient that are affected by the quantum design of a QC structure. The QCL gain coefficient is generally given as:

| (1) |

where τ_{u} is the lifetime of the upper laser state, τ_{ℓ} is the lifetime of the lower laser state, q is the electron charge, τ_{uℓ} is the transition time between the upper and
lower laser state, z_{uℓ} is the optical dipole matrix element (having
units of length), λ_{0} is the free space wavelength of the lasing
transition, n_{eff} is the effective refractive index of the optical
mode, L_{p} is the length of a single active-injector region
period, and δɛ_{uℓ} is the full-width-at-half-maximum (FWHM) of
transition’s spontaneous emission (usually taken as the measured FWHM of a QC
structure’s electroluminescence, and in units of energy). The gain coefficient
as defined this way has units of , which is generally
.

Since the optical dipole matrix element z_{uℓ} and the energy state lifetimes τ_{i} are QC design-dependent parameters, the most
direct approach to deriving a FoM is simply to pull these factors from the
gain coefficient.

| (2) |

Thus, the FoM has units of time x length^{2}; the standard units are ps Å^{2}. While this is the simplest method for deriving a
FoM, it has severe limitations when practically implemented. QC lasers after
all, comprise a system of coupled quantum wells with significant intermixing
(anticrossing) of quantum states; in Eq. (2), we have assumed only a
single upper laser state and a single lower laser state.

### 2 Intermixed lower energy states

For a proper calculation of the laser gain coefficient—and thus FoM—we must include all optical transitions that might contribute photons into the lasing mode. One might address this problem by perturbing the field at which the FoM parameters are calculated, so as to prevent the upper and lower laser states from intermixing with nearby states. However, the FoM itself is a field-dependent parameter, so this approach is not ideal for calculating laser gain at current turn-on (i.e., the design field).

To more broadly address the problem of multiple charge carrier transitions contributing photons to the lasing mode, we need to consider each potential transition’s contribution to the optical gain. As an example, let’s examine an optical transition with one upper energy state and two closely spaced, intermixed lower energy states.

We need to remember that each transition has
its own spontaneous emission spectrum with a finite FWHM; in intersubband
heterostructure emitters, a good estimate for δɛ_{uℓ} is 10% of the transition energy. If the two
transitions are close enough together, the spontaneous emission will overlap
in energy. Let’s assume the spontaneous emission lineshape for a transition with
energy ɛ_{uℓ} has a Lorentzian form.

| (3) |

Here, ɛ_{0} is the lasing photon energy. Now, each transition
will be able to contribute to lasing. The lasing wavelength would not be exactly
that of a single optical transition, but somewhere between the two transitions.
To find the lasing wavelength, we need to keep in mind that the transitions
might have different spontaneous emission rates. That is, one transition might
be able to emit photons faster than the other. The spontaneous emission rate W_{sp} [] is the inverse of the
spontaneous emission lifetime τ_{sp} [time].

| (4) |

Notice that the rate, or the strength of the transition, is proportional to z_{uℓ}^{2}. If we add up the spontaneous emission spectrum of
each transition multiplied by z_{uℓ}^{2}, the peak gain and thus the lasing energy ɛ_{0} can be found.

| (5) |

Now that we know what the lasing energy is
going to be, let’s look at stimulated emission for each of the transitions.
Gain γ [] is simply the charge carrier population
difference N = N_{u} - N_{ℓ} [] times the transition cross-section σ
[area].

| (6) |

Let’s take each of the two contributions to γ individually. In a simple two level optical transition
system with an upper state pumped at rate R_{u} [], the carrier population
difference is

| (7) |

where J is pumping current density [] and N_{p} is the number of active region-injector periods in
the QCL active core. The transition cross-section can be found by recognizing
that, at threshold where optical gain clamps,

| (8) |

where α_{total} is the total optical loss (waveguide and
mirror loss), Γ is the gain region confinement factor for the optical mode in
the waveguide, and J_{th} is the threshold current density []. By applying
Eqs. (1), (6) and (7) to Eq. (8), we get

| (9) |

While this is close for the gain cross-section, we’ve got one correction to
make. The gain coefficient from Eq. (1) was derived assuming two
discreet states. To correct for this, we need to multiply the original g_{c} by our lineshape function
(ɛ_{uℓ}) from Eq. (3). Thus, we get the
quantum cascade laser transition cross-section.

| (10) |

Now, getting back to our example with one upper energy state and two closely spaced lower energy states, we can pull from the components of Eq. (6) those elements that are influenced by quantum design.

FoM | = τ_{u}1 -z_{uℓ,a}^{2}(ɛ_{
uℓ,a}) + τ_{u}1 -z_{uℓ,b}^{2}(ɛ_{
uℓ,b}) | ||

= ∑
_{ℓ}τ_{u}1 -z_{uℓ}^{2}(ɛ_{
uℓ}) | (11) |

### 3 Intermixed upper energy states

Calculating a proper FoM for energy transitions with upper state intermixing takes consideration similar to the lower state intermixing case. However, with upper state intermixing, we have an additional complication. We cannot assume that each upper energy state is equally populated with electrons; the relative populations of each upper state will influence the total gain contributed by that transition.

Recall that our basic relation for gain,

| (12) |

has a term R_{u} that describes the rate at which the upper energy
level u is populated with electrons. For a FoM calculation,
where we are focused on obtaining a number that allows us to compare different
QC designs, we are not concerned about the absolute pumping rate. Rather, what
we need is a weighting factor that reflects the relative population of each of
the upper energy states. Now, we can write our FoM as

| (13) |

with an upper state weighting factor C_{u}. As we’ve previously shown, the pumping rate R_{u} of an energy state is proportional to the current
density passing through the state. In a QCL system with strong coupling between
energy states,

| (14) |

where n_{u} is the sheet density [] of electrons populating the state. Thus,
our weighting factor C_{u} ∝ n_{u}∕τ_{u}. The energy state population n_{u} follows the Fermi-Dirac distribution

| (15) |

where Δɛ = ɛ_{u} -ɛ_{F} is approximately the energy difference between the
state u and the injector ground state, k_{B} is the Boltzmann constant, and T is temperature. For most quantum cascade laser systems,
with relatively low doping (carrier densities) and high temperatures, the Fermi
distribution is well-approximated by the Boltzmann distribution, so

| (16) |

and

| (17) |

Since C_{u} is a weighting factor, the restriction ∑
_{u}C_{u} = 1 must hold.

| (18) |

Finally, we arrive at the general equation for FoM by combining Eqs. (11) and (13), the generalized cases for transitions between intermixed upper and lower energy states.

| (19) |