9.3 Frequency noise
9.3.3 Noise due to the plasma eect in silicon
In this section I will consider only noise due to uctuations of free-carriers in Si that aect the frequency noise spectrum through the plasma eect (free-carrier-dispersion):
∆ν=−vg λ0
ΓSi∂∆nef f
∂nSi ∆nSi
(9.50) I will compare the resulting spectrum from two dierent models:
1. Shot-noise only model - In this model I treat the eective lifetime of carriers in Si as a recombination lifetime. I will neglect carrier diusion altogether, and assign recombination properties to the carriers that diuse away from the mode.
2. Shot-noise and diusion noise - I consider diusion of carriers together with re- combination (bulk and surface). However, I neglect noise due to cross-correlation of photons and Si carriers, and only treat intrinsic carrier uctuations.
In the second model, we need to take into account spatial variation. The correlation of Langevin noise force (shot-noise) now becomes [50]:
< fnSi(t, r)fn∗Si(t0, r0)>=
nSi(r)
τr,Si +βThνΓSiVg2 2 n2p(r)
δ3(r−r0)δ(t−t0) (9.51)
while the correlation for the Langevin diusion source is:
< fD(t, r)fD∗(t0, r0)>=−2Da∇r· ∇r0h
nSi(r)δ3(r−r0) i
δ(t−t0) (9.52) To estimate the frequency uctuation, I need to know the optical mode prole i =
|e(r)|2. For purposes of simplicity, I will assume a Gaussian prole with the same FWHM as the simulated mode. I will then compute the spatial Fourier transform of the energy distributionI(kx, ky, kz).Notice that the Langevin force correlations in the Fourier domain have the form:
< fnSi(ω,k)fn∗
Si(ω0,k0)>=
nSi(k+k0)
τr,Si + βThνΓSiVg2
2 G(k+k0)
2πδ(ω−ω0) (9.53)
< fD(ω,k)fD∗(ω0,k0)>=−2Dak·k0nSi(k+k0)2πδ(ω−ω0) (9.54) where nSi(k), G(k) are the Fourier transforms of nSi(r), n2P(r), respectively. The frequency uctuations can be calculated by taking the Fourier transform of Equation 9.50, and by using the Fourier-domain Si carrier uctuations from Equation 9.31.
Finally, I will calculate the PSD of the frequency uctuations using Equation 9.12, and plug in the correlations of Equations 9.51 and 9.52. Notice that this can be done numerically using no other assumptions. However, this brute-force approach requires calculating six nested integrals numerically. Performing this with sucient resolution to obtain a reliable result is almost prohibitive, especially if one wishes to sweep other parameters, e.g., the pump power. I will therefore simplify this six-dimensional integral by using the following assumptions:
1. Instead of assuming a Gaussian prole I assume a three-dimensional sinc func- tion for the optical mode with the same width. The Fourier transform of a sinc function is a window function. This will change the limits of integration to a nite window (an ellipsoid), and make the integrand managable.
2. I assume that the Si carrier density prole nSi(r) is constant over the mode
Figure 9.6: The eect of Si-carrier uctuation on the frequency noise spectrum:
Eective recombination model vs recombination-diusion model. 150nm Spacer, QSi= 106, I = 2Ith
prole. We have previously seen in Chapter 3 that this is a good approximation due to the large diusion length of carriers in Si. The Fourier transform will be a delta function (compared to the Fourier transform of the optical mode), which will reduce the dimensionality of the resulting integral.
3. I will work in spherical coordinates since the term |k|2 appears everywhere in the integration. The only non-spherical symmetric part of the integration is its boundary; the integration is over an ellipsoid and not a sphere.
Figure 9.6 compares frequency noise from the recombination-only model and the recombination-diusion model. Some interesting conclusions can be drawn from this gure:
1. At the very low frequencies they both have same order of magnitude at re- sponse. In the steady-state, the rate of carrier generation due to TPA equals the rate of carrier (eective) recombination. In both models, the generation term is identical so the shot-noise generated by this process is likewise identical.
2. The recombination-diusion model has a pole at a much lower frequency. This
Figure 9.7: Frequency noise spectrum due to Si carrier uctuations for dierent pump currents. 150nm Spacer, QSi= 106
is due to the much lower recombination lifetime of carriers in Si, compared to the eective lifetime that includes diusion.
3. The high-frequency noise due to diusion drops o much more slowly than that of the recombination-only model. This is a characteristic feature of diusion noise. It demonstrates why it is so important to take the diusion noise into consideration; at the high frequencies (∼1 GHz) it might be dominant.
In the example of Figure 9.6 the diusion noise at 1GHz for the 150nm spacer corre- sponds to a linewidth of 200Hz. In the high-Q platform, this is comparable to the S-T linewidth we would expect from this laser. This suggests that diusion noise might be a dominant noise-source in our platform.
Figure 9.7 shows the eect of pump current on frequency noise due to Si carrier uctuations. As we increase the pump current, the intra-cavity optical intensity increases together with the two-photon-absorption rate. This increases the free-carrier density in silicon, and their uctuations increasingly degrade the frequency spectrum.
The increase in noise level with pump current highlights an important feature of the high-Q hybrid Si/III-V platform: Noise that stems from uctuations of carriers generated at the silicon slab due to TPA, does not behave as the noise of a conventional
Figure 9.8: Frequency noise spectrum due to Si carrier uctuations for dierent spac- ers. QSi = 106, I = 2·Ith
laser. In a conventional laser, the linewidth scales inversely with power. Fluctuation of the QW carriers yield the Henry linewidth enhancement, but the overall noise spectrum goes down at high frequencies with increasing power. However, in the high- Q hybrid platform the noise due to silicon carriers increases with power. This work therefore predicts that as we increase the laser's power, the linewidth will eventually start to broaden. This model predicts a sub-KHz equivalent linewidth as a rough limit. A better estimate will follow in the next sections. Figure 9.8 shows that this conclusion also holds for spacer thickness. For the thicker spacer, as we attempt to reduce loss and store more photons in the laser cavity, uctuations of Si carriers limit the achievable linewidth.