Backscatter and Spontaneous Four-Wave Mixing in Micro-Ring Resonators

We model backscatter for electric fields propagating through optical micro-ring resonators, as occurring both in-ring and in-coupler. These provide useful tools for modelling transmission and in-ring fields in these optical devices. We then discuss spontaneous four-wave mixing and use the models to obtain heralding efficiencies and rates. We observe a trade-off between these, which gets more extreme as the rings become more strongly backscattered.


I. INTRODUCTION
We need sources of discrete numbers of photons to create photonic circuits for quantum computing. Two established ways we can generate photons are using neardeterministic single-photon emitters (e.g. colour centres [1] and quantum dots [2]) and spontaneous generation using parametric nonlinearities (e.g. four-wave mixing in optical fibre [3] and silicon photonics [4]). Although parametric generation of photon pairs is probabilistic, we can mitigate this by using one half of a photon pair to herald its partner's presence [5]. Four-wave mixing occurs in a variety of devices, but is most conveniently produced in integrated circuits by micro-ring resonators (MRRs). These allow higher generation rates, due to resonant field enhancement. [6][7][8][9][10][11].
Typically, their transmission displays Lorentzianshaped resonant peaks, reaching a minimum when the ring circumference is an integer multiple of the wavelength [12]. However, these devices are vulnerable to backscatter [13].
This occurs when light couples between the forwards and backwards modes within the ring, either due to reflections in the coupling between bus and ring, or from the surface roughness of the waveguide. This causes a splitting of the resonance peak, reducing resonant enhancement and changing the shape of the spectral response. Furthermore, one member of a generated twophoton pair could be backscattered and lost, reducing the heralding efficiency of the photon-pair source [14].
However, maintaining high heralding efficiency is essential to overcoming the randomness inherent to parametric photon-pair generation. We therefore investigate how this loss mechanism will limit performance in ringresonator sources.
While some, such as Li et al, have considered the effects of backscatter [15], there isn't yet a full analytic model for its effects on field propagation through a ring.
We construct this unified analytic model for backscatter in both the ring and the coupler, and we apply it to * jonte.hance@bristol.ac.uk It shows the characteristic asymmetric split peaks, which any analytic model of backscatter must be able to explain in order to fully model the effect.
spontaneous four-wave mixing in an MRR. This allows us to analyse the trade-offs between heralding rate and heralding efficiency. While previous studies have looked at how the heralding efficiency is limited by design parameters [14] and material properties such as cross two-photon absorption in silicon [16,17], this is the first study that investigate the role of backscatter, which can result from fabrication imperfections or non-optimal design. Therefore, this research will be useful to anyone designing MRRs for generation of photon pairs.

A. Matrix Formalism
We model the system as a scattering matrix over six modes, representing the forwards and backwards fields in the ring, bus and loss channel, as per Fig. 2.
Our matrix model of an MRR, with three two-mode vectors (R, B and L), each with a forwards and a backwards component, all linked by the 6 × 6 unitary matrix U. As all interaction occurs in this matrix, R1 = R2.
where R, B and L correspond to ring, bus and loss modes, → and ← to forwards-and backwards-travelling fields, and 1 and 2 to entering and leaving the scattering matrix. By modelling loss via coupling to a fictional mode, we conserve unitarity, and so the commutation relations, making the model suitable for later adaption for quantum analysis.
Note L 1→ = L 1← = 0 by definition; and, for consistency (as all interaction takes place within U) Our interaction can be thought of as consisting of a number of compiled smaller interactions between two modes. Therefore, the 6 × 6 matrices representing each process Cpl BR , Loss RL , Back R and Back C (bus-ring coupling, ring-loss coupling, and backscatter, modelled in-ring and in-coupler respectively) are where sub-matrices (on two modes, here a and b), are where α is the loss coefficient (one with no loss in the ring, nil with complete loss), ξ the backscatter coefficient (one when no backscatter, zero when entirely backscattered), and t the coupling coefficient (one with no ringbus coupling, zero with total coupling), and ζ and φ the respective phases on backscatter and ring-bus coupling.
Note backscatter modelled in-ring and in-coupler (in C R ab and C C ab ) differ by backscatter in-ring having all phase occurring on the backwards component, and backscatter in-coupler having opposite phase for the forwards and backwards components. Both have the same phase difference, ζ, between forwards and backwards components.
θ is the phase the field accrues over one trip around the ring (in a lossless environment), where n e is the effective refractive index, r is the ring radius, τ is any phase offset caused by the loss α, and λ is the field wavelength.
For instance, Back R is From this, the two unitary matrices, representing the entire interaction (for in-ring and in-coupler backscatter respectively), are Note, the above ordering doesn't matter, as these pro-cess matrices are commutative (up to arbitrary powers of −1).

B. Transmission
Now we can obtain the transmission, to compare with observations (e.g. Fig. 1) by, (where row-column subscript notation picks out 2 × 2 sub-matrices), This gives the transmission, B 2→ , for in-ring backscatter and in-coupler backscatter, when B 1→ is normalised to 1, and B 1← is zero.

C. Field In-Ring
Due to how the scattering matrices were composed, the ring field values they give, R, are those after a full ring round-trip. However, we need the average power in the ring to work out the photon pairs generated by spontaneous four-wave mixing -and so the field halfway through the ring. Therefore, we need to reorder the full interaction matrices, giving These give the same transmission as before, but now also the average in-ring fields. Focusing on the submatrices, considering the previous method, the relations required are For both in-ring and in-coupler backscatter, this gives equations for the forwards and backwards fields. Unfortunately, these expressions are far more complicated than that for transmission -their derivation, and graphs of produced responses, are far more informative than their actual formulae.
From this, we get forwards and backwards pump power in-ring as (14) D. Comparison of In-Ring and In-Coupler Scattering Fig. 3 shows the above models are the same when the backscatter phase is nil. However, when a phase is applied, the in-coupler model peaks show asymmetry, while the in-ring model shows none.
The in-ring model's lack of asymmetric split peaks is un-physical, given we see asymmetric peak in transmission spectra of MRRs (e.g. [15], and Fig. 1). This symmetry could be as we don't allow the backscatter coefficient, ξ, to have a phase for the in-ring model, given, unlike for in-coupler, it makes no sense for backscatter to apply a phase to the forwards-coupled component when scattering occurs in-ring.
To replicate experimental data, it makes sense to combine the models, to have backscatter within both ring and coupler. This makes sense, as backscatter has been associated with different things in each case: for inring, waveguide-roughness; and for in-coupler, modemismatch between the straight and curved coupling regions, and increased roughness-induced backscatter due to higher field intensity in the coupling region.
However, both models give the same amount of splitting for a given backscatter coefficient, and are equivalent when phase is nil. This means it makes sense to just use the backscatter in-coupler model, and proportionally reduce the phase on the backscatter coefficient. This removes unnecessary degrees of freedom, and allows us to more easily equate coupling constants with observables.

A. Paired Photon Generation Rate
We now want to take this model for backscatter in ring resonators, and obtain the heralding rate and efficiency. To do this, we first need the photon-pair generation rate.
Wang et al's work on spontaneous four-wave mixing [18], adapted to an MRR (setting the length L to one round-trip around the ring, 2πr), gives the average photon-pair generation rate as The backscatter coefficient ξ has magnitude 0.99 and phase ζ of 0 (red), magnitude 0.99 and phase π/20 (blue), and magnitude 1 (no backscatter, green). The x-axis is the ring-bus coupling coefficient t, and the y-axis is the logarithm in base-10 of the rate, with loss coefficient α of 0.98, for normalised input field B1→. where n p is the effective index for the pump wavelength, A ef f is the cross-sectional area of the waveguide, and χ (3) is the third-order nonlinear response of silicon. In the backscatter-free limit, this is identical to the result Vernon et al obtain [14]. As shown by Wang et al [18], these photons are always generated in pairs, so we don't need to consider its effects on purity or efficiency.

B. Ring Effects on Generated Photons
We now want the proportion of photons emitted from the device, out of those created into a given mode. As photon number proportions behave similarly to light field intensity, we can, similarly to when deriving the field strength in the ring, set Here, R, B and L correspond to ring, bus and loss modes, → and ← to forwards-and backwards-travelling fields, and the unitary U is determined for the in-ring and in-coupler models by Eq.12. For this, we assume the various coupling and scattering parameters remain the same for all modes -that they don't vary with frequency. This means the parameters in U are identical to those earlier, with the exception of C C ab , which changes to as photons coming from the backwards into the forwards mode won't be part of a coherent pair.
Here, q is the ratio of the power backwards (from the backwards-travelling pump field) to that forwards, By rearranging, we get This gives the proportion of photons emitted, in a given mode, to those created. This is maximised when on resonance, which occurs when θ is an integer multiple of π, minus the phases on any elements (e.g. minus ζ/2 if asymmetrically split). Assuming the wavelengths for signal and idler obey both this resonance matching, and the four-wave mixing conditions from pump frequency, we can assume that this maximum is constant (as Vernon et al do), if we neglect effects of spectral correlation [14].

C. Heralding Rate
By taking the maximal output proportion, P r B→ , and squaring it, we get the proportion of signal-idler pairs where both photons are emitted. Multiplying this by the average photon-pair number generated, J 4W M , gives the average photon-pair rate -the heralding rate, J Herald as shown in Fig.s 4 and 5. This shows, even at its peak, a relatively minor amount of backscatter reduces FIG. 5. The heralding rate J Herald /β 2 , given for backscatter coefficient ξ of magnitude 0.99 and phase ζ of 0 (red), magnitude 0.99 and phase π/20 (blue), and magnitude 1 (no backscatter, green). The x-axis is the pump wavelength in µm, and the y-axis is the logarithm in base-10 of the rate, with ring-bus coupling coefficient t and loss coefficient α of magnitude 0.98 and ring-bus coupling phase φ of 0, for for normalised input field B1→.
the heralding rate to nearly a tenth of its backscatterfree value. Alongside this, we define the rate of photons being in the Heralding Mode, J HM , by for which the maximal rate is again shown in Fig. 4, again showing a large drop (here a reduction to one-third of the original value) just to backscatter.

D. Heralding Efficiency
From Vernon et al, we define the heralding efficiency, η, as the ratio between there being a heralded output photon, and a heralding photon being emitted: The maximal output proportion and efficiency are the same. This efficiency, and so the maximal output proportion, is shown for different ring-bus coupling rates in Fig. 6. This shows that this efficiency drops heavily as the coupling between ring and bus is reduced to nil (as t goes to unity), with it going below 0.4 as we reach critical coupling (when in-ring loss and bus-ring coupling are equal, and so typical resonant peaks for bus transmission are deepest). This shows, despite critical coupling being when the most power goes into the ring when onresonance, it most probably isn't the optimal coupling for paired photon generation. To investigate this further, we need to obtain a relationship directly between heralding rate and efficiency.

E. Relationship between Rate and Efficiency
We want to define the relationship between heralding rate, J Herald , and heralding efficiency, η. From the relationship between maximal output proportion and efficiency above, As the sub-matrices commute with one another, Therefore, we can still write the heralding rate as This is the relationship shown by Vernon et al [14]. However, as opposed to their conclusion, Fig. 7 shows that this M does not remain constant across all all possible ring-bus coupling strengths.
This leads us to ask if we could write the heralding rate J Herald as some function of the heralding efficiency η, multiplied by some constant of the ring-bus coupling t.
Attempting this numerically, in Fig. 8 we get a plot of reduced heralding rate J Herald /β 2 (y-axis) against heralding efficiency (x-axis). While similar to that in [14], it shows the differences both between our model and theirs, and cases with and without backscatter. However, it does still support their conclusion -that there is a trade-off between heralding efficiency η and heralding rate J Herald , while also further suggesting this tradeoff becomes more pronounced the greater the splitting 1 − ξ 2 becomes. It also shows that, given the position of this optimal heralding rate with respect to efficiency, that critical coupling isn't optimal for either -for both, increasing bus-ring coupling above loss (reducing t to below α) is beneficial.

IV. CONCLUSION
We presented two models for backscatter in MRRs -it occurring in-ring, and in-coupler. While this was to allow analytic analysis of backscatter effects on four-wave mixing in these structures, they can be applied wherever MRRs are used (e.g. frequency combing, wavelength-filtering, and modulation of non-linear optical effects) [19,20]. Also, this analysis could be abstracted to model backscatter in any resonant cavity -which, despite being one of the key sources of difficulty in controlling their use, hasn't been heavily investigated.
Alongside this, we suggested a spectrum could act as though entirely un-split given a large enough phase, due to the split peak asymmetry this gives. This could potentially mitigate the negative effects of backscatter.
Further, we calculated the effects of backscatter on spontaneous four-wave mixing rates, heralding rates and efficiencies. A step would be to link these parameters to ring design, so these values could be optimised for given material parameters, to mitigate the effects of backscatter. Given how essential such sources will be for any form of optical quantum computing or quantum communication, this research will revolutionise the efficiency of these processes.