Eect of Quantum Well carriers

The modulation of input current yields a change of the carrier density in the quantum wells. This in turn causes refractive index modulation through the plasma eect [44], which results in frequency chirping. In an ideal laser, the carrier density is clamped to its threshold value. This fact would mean that a DC modulation should result in zero frequency chirp. In practical laser systems this is not the case. Even at DC, the plasma eect causes frequency tuning, typically few hundred MHz per mA of input modulation [123, 78, 13, 126]. To explain this discrepancy I will have to consider gain compression in our model. This eect was of no signicance in previous analysis. However, since the laser is extremely sensitive to refractive index changes, it is important to consider it in the analysis of the frequency modulation response. 7.3.1.1 Gain compression

It is an approximation to view the gain as clamped to its threshold value. In practice, the high photon density will compress the (unsaturated) gain. This non-linearity of the gain is often attributed to spectral hole burning and carrier heating (intra-band re-absorption of photons) [129]. A good model for this eect can be given by the

expression:

G0_m(np) =

G0_m

1 + ΓQWcnp

(7.34) where c is the gain compression coecient, which is derived empirically. When

the laser pump current is modulated, the photon density responds, as described in section 7.2. The resulting compression of the gain would force the QW carriers to follow in order to compensate for the change in dierential gain, and frequency chirp will be observed. In light of this modication to the model, the dierential rate equations should be altered. Gain compression should be included in the dierential rate Equations 7.6-7.13 by making the dierential gain dependent on the photon density, thus making the substitutionG0_m →G0_m(np). Moreover, the derivative of the

gain with respect to the photon density has to be included. Specically, this results in changes to two terms in the small-signal matrix 7.5:

A12= ΓQW Γgeom G0_m(np)·(ne,0−ntr) 1− ΓQWcnp,0 1 + ΓQWcnp,0 (7.35) A22=iω+α+ 2Bnp,0+CnSi,0−ΓQWG 0 m(np)·(ne,0−ntr) 1− ΓQWcnp,0 1 + ΓQWcnp,0 (7.36) 7.3.1.2 Henry's alpha parameter

In laser analysis, it is useful to express changes to the refractive indexnr using changes

to the imaginary part of the refractive indexni, which is related to the material gain

g. Henry's alpha parameter αH can be dened as:

αH =

dnr dne/dni

dne (7.37)

and is used to connect the two using the expression [115]:

dnr dne =−αH λ0 4π · G0_m 2vg (7.38)

Figure 7.4: Frequency modulation response due to quantum well electrons for dierent values of spacer thickness. αH = 7, I = 2·Ith, QSi = 106

where we have used the explicit linear form of the gain (Equation 2.15) to calculate

dg

dne. The minus was added to forceαH to be positive for the expected blue shift with

carrier-density.

7.3.1.3 Frequency modulation response curve

I can now calculate the eect of quantum well carriers on the frequency modulation response. By combining Equations 7.33 and 7.38 I can express the frequency response as: ∆νQW ∆I =− vg λ0 dnr dne ∆ne= αH 8πΓQWG 0 m ∆ne ∆I (7.39) where ∆ne

∆I is calculated directly from the small signal matrix 7.5. The exact expression

can be presented analytically using the same assumption we have used in section 7.1: ∆νQW ∆I (s) = αH 8πΓQWG 0 m ηi eVQW s+_τ1 ef f h s+Bnp,0+αT ΓQWcnp,0 1+ΓQWcnp,0 i + 2CDn2 p,0 (s+ 2ξωn) h (s+Bnp,0) 1 τef f +s + 2CDn2 p,0 i +_τ1 ef f +s ω2 n (7.40) The resulting response due to the plasma eect in the quantum wells can be seen in Figure 7.4. Several prominent new features are evident:

1. For the thin 30nm spacer, the response resembles that of a conventional semi- conductor laser [43]. A at response up to a few hundred MHz, of magnitude of roughly few hundred MHz/mA are very common for III-V lasers.

2. As the spacer thickness increases, say in the case of 100nm, the entire curve maintains its general shape, but decreases in magnitude. This is due to the decreased overlap between the the mode and the QW. Changes in the QW's refractive index have a smaller eect on the mode due to the low connement factor.

3. The resonance frequency decreases with increasing spacer thickness, for the same reasons that were discussed in section 7.2.

4. For very thick spacers, i.e., 150nm, the response curve changes: a shallow dip due to FCA is revealed at ω= _τ1

ef f.

5. In the case of the 150nm spacer, the magnitude of the response at low frequen- cies is comparable to the 100nm spacer, despite the reduced overlap with the quantum wells. This is due to nonlinear loss. TPA and FCA act as eective gain compression mechanisms; increased photon density increases nonlinear loss, and QW carrier density has to grow to increase the gain, such that gain=loss.