Analytical investigation

The matrix in Equation 7.5 can be solved analytically, and simplied using Equation 7.17. The transfer function H(s)can be dened using the Laplace transform :

∆np(s) =H(s)∆I(s) (7.18)

which relates modulation of intra-cavity photon density to input current modulation . To relate it to the output power, one can use Equation 5.12. The resulting expression has the form:

H(s) = ΓQWG 0 mnp,0 1 τef f +s ηi qVQW (s+ 2ξωn) h (s+Bnp,0) 1 τef f +s + 2CDn2 p,0 i +_τ1 ef f +s ω2 n (7.19) Where the natural frequency ωn and the damping factor ξ are dened as:

ω_n2 = ΓQW Γgeom G0_mnp,0αT (7.20) ξ = 1 τr + ΓQW ΓgeomG 0 mnp,0 2ωn (7.21) and the total loss rate, linear and nonlinear, is dened as:

αT =α+Bnp,0+CnSi,0 (7.22)

The transfer function of Equation 7.19 has in general one zero and three poles. Unfortunately, the roots of the third-order polynomial in the denominator cannot be

expressed analytically. I will study this transfer function by looking at two dierent regimes: low and high nonlinear losses.

7.2.1.1 Low nonlinear loss regime:

Written explicitly, the second term in the denominator is: 2CDn2_p,0 = 2 τef f vgσanSi,0ΓSi = 2 τef f αF CA (7.23)

Where we have used the steady-state silicon carrier-density from Equation 5.1, and dened the loss rate due to FCA αF CA in units of [sec−1]. I can now compare a few

terms in the square bracket of Equation 7.19. If the FCA loss rate is slower than some frequencyω of interest:

2αF CA ω (7.24)

then the term 2CDn2

p,0 can be neglected. In this case, the

1

τef f +s

term cancels out everywhere, and the transfer function is reduced to the familiar form:

H(s) = ΓQWG 0 mnp,0 s2_{+ 2ξω} ns+ω2n (7.25) which is the typical second-order low-pass lter.

The exact location of the poles ωp of the transfer-function is given by:

ωp =ωn

−ξ±pξ2₋₁ (7.26)

When the damping factor is smaller than unity, the knee frequency can be well- approximated by the value of ωn and the resonance is under-damped. When the

damping factor is larger than unity, there will be two real poles and the system will be highly damped.

It is interesting to study how the relaxation resonance behavior changes with dierent spacer designs. Most textbooks have a similar expression as Equation 7.20, except that for a traditional laser ΓQW

obscured. I will use the relationship between the photon density and the total loss rate: np,0 =η (I −Ith) eVpαT (7.27) such that: ω_n2 = ΓQW Γgeom G0_mη(I−Ith) eVp (7.28) It is now clear that in our spacer design, where unlike the case of a conventional laser, the quantum wells reside deeply in the exponential tail of the mode, such thatΓQW

Γgeom

1, and we expect to see very small relaxation oscillation resonance frequencies with the trend:

ωn∼

p

ΓQW (7.29)

The damping factor in our system can also be evaluated. Assuming we are op- erating in a regime where 1

τr <

ΓQW

ΓgeomG

0

mnp,0, which is a reasonable assumption for

practical photon densities and connements, we can express the damping factor as:

ξ≈ 1 2 s ΓQW Γgeom np,0 αT (7.30) Since the total loss is also a function of ΓIII−V in the regime where the total-Q is not

saturated by the silicon-Q, we can write: np,0 ∼ _α1

T ∼

1

ΓQW and get the form: ξ ∼ _p1

ΓQW

(7.31) and we expect an increasingly damped response with reduced III-V connement. 7.2.1.2 High nonlinear loss regime:

When the FCA loss is high and the second term in the square bracket of Equation 7.19 cannot be neglected, we are back to the regime in which an analytical expression cannot be obtained. However, there are some conclusions we can draw from the general form of the transfer function:

1. The existence of a zero of the transfer function atωz = _τ1

ef f. In the low nonlinear

regime, this zero was eectively masked by a pole at the same frequency. In the high nonlinear regime this is no longer the case, and we expect to see a zero of the transfer function.

2. The existence of three poles of the transfer function. In general, these can be three real poles, or a pair of complex-conjugate poles and a real pole. Several types of behavior will be possible, depending on strength of FCA, and on the location of the zero/poles. For example, if the system is highly damped we can expect to see a pole of the transfer function at low frequency, then a zero at an intermediate frequency, and the pole pair at high frequencies. If the system is less damped, we expect to see the zero at low frequency and three poles at higher frequencies. Due to the complex nature of the transfer function, and the fact thatnp,0,ΓQW,QSi, andτef f all depend on each other, a numerical analysis

is needed.