lasers. The lasers presented in this thesis aim to minimize these components of loss.
A side effect of increasing Qabs and Qe is that the other components of loss begin to limit the totalQ. Thus, all of these components should be considered during the de- sign of a high-Q hybrid laser. This section will show that the components other than absorption are independent of ΓIII-V, so the assumption that Qpassive is independent of ΓIII-V in equation 3.4 is a good approximation.
A.3.1 Absorption
Optical absorption is weighted by the confinement factor of the mode in each material, as given by equation 2.6. III-V is much more absorbing than silicon; therefore, Qabs is maximized by shifting as much modal energy away from lossy III-V into low-loss silicon as is possible, either through replacing the III-V with silicon or through modal engineering. The total absorptionQ is
Q−1abs = ΓIII-V
QIII-V +1−ΓIII-V
QSi (A.8)
≈ ΓIII-V QIII-V.
Here, QIII-V is the material absorption of III-V, approximately 10k for highly-doped active material [36–42].
A.3.2 Scattering
Scattering is most significant from the rough etched waveguide sidewalls. Scattering is reduced in the lasers presented in this thesis by guiding the mode with a very shallow etch in silicon, as seen in the cross-section of the waveguide (figures 4.1 and 5.1).
The shallow etch results in very little sidewall and, therefore, little scattering [51;52].
Similarly, the waveguide is designed to be 2µm wide to make the mode weak at the sidewall, also reducing scattering.
Silicon dioxide is easily grown from silicon, providing one further way to reduce scattering. Thin layers of thermal oxide (∼25 nm) are grown on the etched silicon
chips used to make the lasers from this thesis. Scattering is reduced because the ther- mal oxide physically smooths rough surfaces and reduces the index contrast between the rough silicon waveguide and its cladding [55].
Scattering depends on the intensity of the mode at the sidewall, so it may depend on ΓIII-V. Under the replacement transformation, the shape of the mode doesn’t change. Under the modal engineering transformation, the mode concentrates in the silicon waveguide, but the etched sidewalls are far from the most intense region of the mode. For this reason,Qsc is assumed to be independent of ΓIII-V.
A.3.3 Radiation
The biggest potential problem with using a shallow-etch rib waveguide is that the guided mode may easily couple to radiation modes in the adjacent silicon slab, losing stored energy [103]. The strength of the coupling can be found by finding the longi- tudinal mode profile in the resonator and transforming it into k-space. Waves with small values ofk along the propagation direction are not guided by the waveguide, so an integral of the k values which lie across the light line yields the coupling to leaky modes [98].
It is clear that coupling to radiation modes will be minimized by having a very narrow distribution of k values. This is the primary advantage of the 1D modegap- type resonator discussed at the beginning of the section: it is easy to tailor the envelope of the mode along the propagation direction by deterministically modulating the valence bandedge. By designing a quadratic photonic well, the field envelope is Gaussian in both real space and k space. A broad, shallow photonic well thus has a very tight distribution ofk values, minimizing the potential coupling to leaky modes.
The quadratic well can be formed in either the valence band or the conduction band. Continuing the earlier analogy to quantum mechanics, an upward-opening well in the conduction band is a “donor”-type well and a downward-opening well in the valence band is a “acceptor”-type well. Acceptor-type wells are slightly farther from the light line, so they are chosen for these resonators.
10−4 10−3 10−2 10−1 100
Field en velop e (| a ( z )|
2, a.u.)
Quarter-wave shift DFB Quadratic well
modegap
Distance along resonator (z)
Figure A.4: Mode envelope profiles for a 1D modegap grating resonator and a stan- dard QWS DFB. The QWS DFB has κthe same as in the uniform modegap mirrors.
Envelopes have been normalized to have the same area.
Conveniently, the broad field envelope along the resonator also reduces spatial hole burning compared to the field profile of quarter-wave shift (QWS) DFB gratings, which are sharply-peaked at the center of the cavity, as seen in figure A.4. Modegap cavities therefore improve upon QWS DFBs in two ways: (1) the field envelope is smoother, causing a tighter distribution in k-space, reducing coupling to radiation modes, and (2) the field is broader in real space, reducing spatial hole burning.
Radiation loss depends on the longitudinal parameters of the grating, so it is also independent of ΓIII-V.
A.3.4 External coupling
External coupling is determined by the uniform mirror sections on each end of the quadratic well. The field envelope decays exponentially in the mirrors, such that the number of mirror holes determines the amount of stored energy that couples to the output. External coupling is often considered separately from the other sources of loss because it provides useful output; however, external coupling reduces the number
of photons stored in the cavity just the same as the other sources of loss, thereby increasing phase noise, and should be considered as just another source of loss.
For minimum phase noise, lasers should operate in a regime in which external coupling is minor compared to the dominant source of loss in order to store as many photons as possible. In such a cavity, more photons will leave the cavity and be lost than the number of photons leaving as useful output. The external efficiency describing the fraction of photons leaving the cavity which are useful output is given by [2, p.696]
ηex = Q−1e
Q−1e +Q−1i , (A.9)
where Qe is the Q for external coupling and Qi is the “intrinsic” Q, accounting for all other sources of loss. The optical output power in steady state can be found by multiplying the external efficiency by the stimulated transition rate into the mode and the energy of a single photon. Therefore, there is a tradeoff between large stored energies in the cavity and high external efficiency.
The field envelope is exponentially decaying in the mirror section, so external coupling decreases exponentially with an increasing number of mirror holes on both sides of the photonic well. External coupling is therefore highly sensitive to deviations in mirror coupling strength, so the devices made for this thesis are fabricated as several different designs, each with a different number of mirror holes.
External coupling can be controlled independently of the other cavity parameters by tuning the number of mirror holes, making Qe a useful parameter to vary in an experiment. It can be made independent of ΓIII-V by adjusting the mirror length for designs with different ΓIII-V.