3.3 Modal engineering
3.3.1 Supermode model
Modal engineering could greatly benefit from a “knob” to tune the strength of the electric field in the III-V region of the hybrid Si/III-V laser cavity. Typical III-V active material forms a waveguide in order to confine photons to the same region as the excited electrons. Silicon that has replaced some of the lossy III-V forms a second waveguide that is coupled to the III-V waveguide.
The eigenmodes of the coupled waveguide system can be viewed as linear combi- nations of the individual waveguide modes called “supermodes” [65, p.526]. Some of the early research on hybrid Si/III-V lasers in this group concentrated on using super- modes to increase the modal confinement in III-V to increase modal gain, reducing threshold [66–70]. Now, the goal has changed to reducing the modal confinement in
III-V
Si even odd
supermodes
isolated III-V waveguide
isolated silicon waveguide
Figure 3.8: Supermodes are formed by linear combinations of the individual waveguide modes, producing odd and even modes. These two supermodes have equal weight.
III-V to reduce the spontaneous emission rate into the mode.
Figure 3.8 shows a cartoon version of a pair of slab waveguides arrayed in the ˆ
y direction, with the mode propagating in the ˆz direction. The supermode can be represented as a linear combination of the individual waveguide modes [65, p.527],
E(y, z) =AIII-VEIII-V(y) exp(−iβIII-Vz) +ASiESi(y) exp(−iβSiz), (3.18) where Ei(y) is the transverse electric field of the individual (non-coupled) waveguide mode in materiali,Ai is a weighting factor applied to the individual modes in material ito form the supermode, and the complex exponentials represent a phase term. There are two supermodes in a system of two coupled waveguides: one with even symmetry and one with odd symmetry (see figure3.8). The amplitude weighting factors for the even supermode are given by
AIII-V
ASi
=
iκ∗ δ−S
1
, (3.19)
where the various parameters are defined as κ≡ −iω0
4 Z ∞
−∞
EIII-V∗ (y) ∆n2 ESi(y) dy, (3.20)
2δ≡βSi−βIII-V, (3.21)
S ≡p
δ2 +|κ|2. (3.22)
Physically, κ is the coupling strength between the two individual waveguides, ∆n is the perturbation of the index resulting from the presence of the other waveguide [2, p.613], and δ is the phase detuning between the two individual waveguide modes.
An interesting case arises for δ < 0 and |δ| |κ|. In this case, the amplitude weighting factors for the supermode are given by [66]
AIII-V
ASi
=
iκ∗ 2δ
1
. (3.23)
This is the desired result for modal engineering: the amplitude weighting factor for the supermode in the III-V waveguide is very small (by assumption of |δ| |κ|) and is tunable with the parameters κ and δ.
The field in the III-V waveguide is the sum of the exponentially decaying field from the silicon waveguide and the weighted field from the III-V waveguide (equation3.18).
The steep exponential decay of the silicon portion of the sum is expected to make the total field in the III-V waveguide dominated by the III-V portion of the sum.
Equation3.23indicates two parameters that could be tuned in order to reduceAIII-V, thereby reducing the total field in the III-V portion of the waveguide:
Waveguide coupling (κ): Decreasing coupling between the silicon and III-V waveg- uides would reduce the supermode field in the III-V waveguide. Coupling could be decreased by moving the waveguides farther from each other, separating them with a low-index material.
Phase detuning (δ): Changing the transverse geometry of the silicon or III-V waveg- uide will change its individual propagation constant (βi) thereby changing the
Normalized mode intensity (|E0l|2, a.u.) Low-loss silicon
Lossy III-V
Gain
Reducing AIII-V
Figure 3.9: Simulated mode in a hybrid Si/III-V waveguide. As βIII-V is decreased, AIII-V decreases, and modal energy is removed from the III-V part of the hybrid Si/III-V waveguide.
phase detuning between the two waveguides. This was the technique previously used in our group toincrease the electric field strength in the III-V region [66–
70].
Figure 3.9 shows a simulated mode in a hybrid Si/III-V waveguide. In this fig- ure, the propagation constant of the individual III-V waveguide (βIII-V) is decreased, causing AIII-V to decrease, shifting modal energy away from the III-V and towards the low-loss silicon.
Reducing the laser mode intensity at the active region via modal engineering will decrease the single-carrier spontaneous transition rate into the laser mode according to Fermi’s Golden Rule (Wsp, equation 2.26). The total spontaneous emission rate into the mode also depends on the total number of excited carriers (equation 2.55), so care must be taken that modal engineering does not cause the number of excited carriers at threshold to increase. Modal gain is proportional to the confinement factor of the mode in the active region (Γact) and optical absorption in III-V is proportional to the confinement factor of the mode in III-V (ΓIII-V). The supermode model shows
that these two confinement factors are expected to be proportional to A2III-V, so the ratio between the two should be constant under the modal engineering transformation.
In the end, supermode theory only provides a simple model for calculating how the confinement factors vary based on waveguide parameters. Numerical simulation of the actual transverse Si/III-V structure should be performed to verify the assumption that ΓIII-V/Γact is constant for modal engineering.