Dynamical Characteristics of External Cavity Quantum Cascade Lasers

In this paper, we calculate the dynamical characteristics such as delay time, rise time, and time to steady state establishment, of a mid infrared quantum cascade laser coupled to external cavity. The approach is based on the three-level rate equations including the dependence of the loss on external cavity parameters. We find in particular that the threshold current of external cavity is strongly influenced by the external reflectivity and external cavity length. In addition, the equations that allow for the determination of the dynamical characteristics are derived within the premises of our model in the general case. The effects of the external cavity parameters on dynamical characteristics are also explored.

It is well known that EC strongly affects the losses, photon lifetime and threshold current of QC laser, and thus influence the turn on delay previous work reported in Ref [15]. Figure 1 shows the schematic illustration of a cavity model for the QC laser with external cavity that we treat here. The output power from a Fabry-Perot QC laser cavity (length L and reflectivities R 1 and R 2 ) is reflected by an external mirror of reflectance R ext , which is located at a distance L ext from the front facet of the QC laser. The dynamics of EC-QC laser can be described by a three-level model. The upper and lower states will be taken as levels 3 and 2, respectively, while the ground state used to empty the lower state through longitudinal optical phonon (LO) emission will be called level 1.

The Model
The system of rate equations for electron numbers N 1 , N 2 and N 3 in levels 1, 2 and 3, and the photon numbers FP S and EC S in the FP and in the EC is used to model the dynamic behavior of EC-QC laser in this work [15].
Where inj I is the injected current, e is the electron charge, 32 τ , 31 τ , and 21 τ are the nonradiative scattering times between the corresponding levels due to LO-phonon emission, sp τ is the radiative spontaneous relaxation time between levels 3 and 2, 3 τ is the lifetime of the upper level and defined as 3 32 31 τ is the electron escape time between two adjacent stages [16], β defines the fraction of the spontaneous emission light emitted in the lasing mode [17], p N is the number is the ratio of optical path lengths of the FP and the external cavity [18] where n eff is the effective refractive index of FP active region, and EC G and EC G are the gain coefficients per period in the FP and in the external cavity respectively. The latter is defined through [14].
( ) Where eff R is the effective reflectivity of the equivalent EC-QC laser and can be written as [19].
In the some way, the threshold current for the FP can be defined as Next using the theory developed above we estimate numerically FP th I , FP p τ , EC G , and EC sat S using the following experimental QC laser parameters as reported in Refs. [14][15][16]: L = 1.5 mm, R 1 = 0.8, R 2 = 0.01, α w = 14 cm -1 , N p = 48, β = 1 × 10 -3 , n eff = 3.27, τ 32 = 2.4 ps, τ 31 = 3 ps, τ 21 = 0.4 ps, τ out = 1 ps, τ sp =140 ns, λ = 8 µm, Our results are as follows: = 3.6  cm that produced minimum threshold current of 198.7 mA. In Figure 3, the threshold current EC th I is plotted versus external cavity reflectivity ext R for an external cavity length = 8 ext L cm. We can easily see that the threshold current is a decreasing function of external cavity reflectivity. This is attributed to the increases of photon lifetime in the EC with ext R . We show also that by increasing Where ω is the laser angular frequency, = 2 / ext L c τ is the round-trip time of light in the external cavity and where c is the speed of light in vacuum.
In Eq. (3) we have assumed for simplicity that the waveguide loss of the EC mode is the same as that for the FP mode i.e., Under steady state conditions, the electron numbers in the upper and lower laser levels obey the following relations ( Where we introduced the photon saturation number for both modes i sat S given by Where the superscript i refer to the FP or EC mode. The population inversion N ∆ between the upper and lower levels as a function of the FP and EC photon numbers can be then written by using Eqs. (5a-b) as Combining Eqs.(5a), (6), and (7) with Eq.(1e), the following expression is then obtained for EC :

Numerical analysis
The temporal evolution of the photon numbers FP S and EC S is analyzed by numerically solving the system of nonlinear differential equations the external cavity reflectivity from 10% to 100%, the threshold current decreased from 0.275 A to 0.195 A. laser reaches its stationary regime. In the conditions of Figure 4, this goes at about 3ns after the start of current injection. This period is called the TSSE ( EC ss t ). The results of Figure 4 show that the FP number of photons reaches its maximum and stabilizes within less then 0.5 ns, followed by a delay time for the EC number of photons to buildup and to reach its maximum, accompanied by the simultaneous decay of the FP number of photons. This delay time is of around of 1.8 ns.

Derivation of the dynamical characteristics of EC-QC laser
In this section, we first derive approximate analytical time-dependent solutions of the rate equations above external cavity QC laser threshold and we present our results to derive the expressions that allow for the determination of the dynamical characteristics of external cavity such as EC      [20].
According to the results obtained in [20], we can make the time-dependent of photon numbers ( der the effect of external cavity as ( )  I  I  I  I  S  t  S  I  I  I  I  I