Quantum light generation on a silicon chip using waveguides and resonators

Jun Rong Ong; Shayan Mookherjea

doi:10.1364/OE.21.005171

1. Introduction

Trends in quantum optics are evolving towards chip-scale photonics [1], with one of the eventual goals being the full-fledged combination of sources, circuits, and detectors on a single chip. Regarding chip-scale sources, researchers have predicted and shown that an optically-pumped spontaneous four-wave mixing (SFWM) process in silicon can be used to generate entangled photon pairs in waveguides and resonators [2–5]. This third-order nonlinear process is similar to the second-order spontaneous nonlinear optical processes induced in bulk optical crystals (except scaling with the square of the pump power instead of linearly), and before being investigated in lithographically-fabricated waveguides, has been demonstrated in optical fiber [6, 7]. As a further step, we have explicitly shown heralded single photons at 1.55 μm wavelength from a silicon chip at room temperature [8]. Given the maturity of integrated optics technology, it is realistic to envision on-chip high-brightness single-photon sources at wavelengths compatible with the worldwide fiber optic internet infrastructure.

However, an important open question is: What is the optimal device for generating quantum light using an integrated photonic structure? To be specific, we focus on devices made using silicon. The photon pair generation rate depends on the intrinsic four-wave mixing nonlinear coefficient (γ = 2πn₂/λA_eff), in terms of the Kerr nonlinear index n₂ and the effective area of the waveguide mode (A_eff), the waveguide length (L), the pump power (P), and the loss coefficient of lithographically-fabricated waveguides (α). Silicon nanophotonic waveguides are already quite promising, compared to optical fiber or bulk crystals, since a single mode “ wire” waveguide with cross-sectional dimensions of about 0.5 × 0.25 μm² has a nonlinearity coefficient γ_Si= 100–200 W⁻¹m⁻¹ (five orders of magnitude greater than optical fiber) around a wavelength of 1.5 μm [9]. But chip-scale devices present special challenges as L is limited to only a few centimeters within a typical die site, and on-chip footprint is a highly valuable resource in CMOS fabrication. Moreover, for a waveguide that is fabricated with loss coefficent α, the effective interaction length of nonlinear interactions L_eff = [1 − exp(−αL)]/α can be significantly smaller than L when αL ≥ 1. Also, pump powers P in silicon are limited to a few milliwatts to minimize the probability of multi-photon generation and avoid two-photon absorption and free-carrier generation losses.

The indistinguishability of output single photons is also an important consideration [10]. In silicon waveguides, the phase-matching bandwidth of the SFWM process is generally quite broad, on the order of tens of nanometers. As such, the generated photon pair emerges as an entangled state, and detection of the heralding photon projects the signal photon into a mixed state. Purity may be enhanced by spectrally filtering the output, the disadvantage being a reduction in photon count rate since unused pairs are discarded. Recent work [11] has also shown that through the careful control of waveguide dispersion, photon pairs may be generated in factorable states which are spectrally de-correlated. Alternatively, one may limit the modes available for SFWM process by placing the nonlinear material in a cavity, thereby providing both spectral filtering of output states as well as local intensity enhancement of the pump [12, 13].

Based on these considerations, we study whether a micro-resonator is preferable to a conventional waveguide as a heralded single photon source with specific reference to silicon devices. We then show that a particular type of hybrid device [Fig. 1], which consists of a linear array of nearest-neighbor coupled microresonators, can possibly generate in excess of 1 Giga-pairs per second for 10–20 mW of optical pump power, from a waveguide that is only 0.1 mm long, thus outperforming existing photon pair sources by 1–2 orders of magnitude in generation rates and by 2–3 orders of magnitude in device size.

Fig. 1 A coupled resonator waveguide consisting of N directly-coupled microring resonators. The waveguide eigenmode is a Bloch excitation, i.e., a collective oscillation of all N resonators, with a fixed relationship between adjacent resonators [14]. The direction of light circulation in each resonator is as indicated for the specified input. In the notation used in this paper, the field operator of successive resonators are a₁, a₂, a₃, ..., the resonance radial frequencies are Ω₁, Ω₂, Ω₃, ..., the inter-ring coupling coefficients are labeled κ₂, κ₃, κ₄, ..., and the input/output external coupling coefficients are labeled by their rates $\frac{1}{τ_{e 1}}$ and $\frac{1}{τ_{e 2}}$ (the latter is not shown, at the output side of the chip). The resonator loss is indicated by the damping rate $\frac{1}{τ_{l}}$ .

Download Full Size | PDF

2. Photon pair generation

2.1. Single micro-resonators

In single ring resonators, the theory of both parametric downconversion (second order nonlinearity) as well as SFWM (third order nonlinearity) has been studied [15–17]. Here we extend the previously described methods to develop the output state of the photon pair from a series of directly coupled rings, so that waveguides, rings and coupled-ring waveguides can be compared. We begin with the phenomenological Hamiltonian,

\begin{array}{l} H = \sum_{m} \sum_{l = s, i} \bar{h} Ω_{m} a_{l, m}^{†} a_{l, m} + \bar{h} κ_{l, m} a_{l, m}^{†} a_{l, m - 1} \\ + \bar{h} κ_{l, m + 1} a_{l, m}^{†} a_{l, m + 1} + \bar{h} χ_{m} a_{s, m}^{†} a_{i, m}^{†} \end{array}

where

a_{l, m}^{†}

are the field operators of the resonator modes l = s, i at the resonator site m, Ω_m are the resonance frequencies, κ_l,m₊₁ are the inter-resonator coupling coefficients and χ_m is the coefficient proportional to the Kerr nonlinearity. In general χ_m may be time-dependent,

χ_{m} (t) = \frac{γ_{0} v_{g}}{T_{c}} {[A_{p, m} (t)]}^{2}

, where A_p,m(t) = a_p,m(t)e^−iΩ_pt is the classical pump with a slowly-varying amplitude at carrier-frequency Ω_p, γ₀ is the usual waveguide nonlinear parameter, v_g is the waveguide group velocity, T_c = 1/FSR is the round-trip time (inverse of the free-spectral range). We adopt the approach of Collett and Gardiner [18] (i.e., time-domain coupled mode theory) to obtain the equations of motion in the Heisenberg picture. In the single resonator case, these may be written explicitly as,

[\frac{1}{τ_{s}} - i (ω_{s} - Ω_{s})] a_{s} (ω_{s}) = - i \int χ (ω_{s} + ω_{i}) a_{i}^{†} (ω_{i}) d ω_{i} - i μ a_{s, in}

[\frac{1}{τ_{i}} + i (ω_{i} - Ω_{i})] a_{i}^{†} (ω_{i}) = + i \int χ^{†} (ω_{s} + ω_{i}) a_{s} (ω_{s}) d ω_{s} + i μ a_{i, i n}^{†}

where a_s(ω_s) are the frequency components of the time-dependent field operator a_s(t) and

\frac{1}{τ_{s}} = \frac{1}{τ_{l}} + \frac{1}{τ_{e}}

is the damping coefficient which includes effects of loss and external coupling. These equations contain the same information as the joint-spectral amplitude, modified by the cavity enhancement effects. In the quasi-cw limit, one may forgo the integral and solve the coupled equations as was done in [19].

a_{out, s} (ω_{s}) = - μ^{2} [A (ω_{s}, ω_{i}) a_{in, s} (ω_{s}) + B (ω_{s}, ω_{i}) a_{in, i}^{†} (ω_{i})]

a_{out, i}^{†} (ω_{i}) = - μ^{2} [C (ω_{s}, ω_{i}) a_{in, s} (ω_{s}) + D (ω_{s}, ω_{i}) a_{in, i}^{†} (ω_{i})]

We have used the boundary condition |a_out|² = μ²|a|², where

μ^{2} = \frac{2}{τ_{e}}

is the input mode coupling coefficient. In the case of vacuum input and low gain the power spectral density of the output photons,

σ (ω_{s}, ω_{i}) = 〈 a_{out, s}^{†} a_{out, s} 〉 = \frac{μ^{4} {| χ (ω_{s} + ω_{i}) |}^{2}}{{| \frac{1}{τ_{s}} - i (ω_{s} - Ω_{s}) |}^{2} {| \frac{1}{τ_{i}} + i (ω_{i} - Ω_{i}) |}^{2}}

and the total signal flux is

F = \frac{1}{2 π} \int σ (ω) d ω

, where the idler frequency is implicitly related by the energy conservation 2ω_p = ω_s + ω_i. Alternatively, by taking χ(ω_s + ω_i) as the pump distribution in the pulsed pump regime, σ(ω_s, ω_i) is interpreted as the joint spectral intensity.

2.2. Coupled micro-resonator waveguide

Extending to the case of N coupled cavities [20], we have the following matrix equation,

{[\begin{matrix} a_{s, 1} \\ a_{s, 2} \\ ⋮ \\ a_{i, 1}^{†} \\ a_{i, 2}^{†} \\ ⋮ \end{matrix}]}_{2 N \times 1} = - i μ \vec{T} {[\begin{matrix} a_{s, in} \\ 0 \\ ⋮ \\ a_{i, in}^{†} \\ 0 \\ ⋮ \end{matrix}]}_{2 N \times 1}

where

\vec{T} = {[\begin{matrix} M_{s} & C \\ C^{†} & M_{i} \end{matrix}]}_{2 N \times 2 N}^{- 1}

M_{s} = {[\begin{matrix} - i (ω_{s} - Ω_{s, 1}) + \frac{1}{τ_{l}} + \frac{1}{τ_{e 1}} & - i κ_{s, 2} & 0 & \dots & 0 \\ - i κ_{s, 2} & - i (ω_{s} - Ω_{s, 2}) + \frac{1}{τ_{l}} & . & \dots & 0 \\ 0 & . & . & . & . \\ ⋮ & ⋮ & 0 & . & - i (ω_{s} - Ω_{s, N}) + \frac{1}{τ_{l}} + \frac{1}{τ_{e 2}} \end{matrix}]}_{N \times N}

C = {[\begin{matrix} - i χ_{1} & 0 & \dots & 0 \\ 0 & - i χ_{2} & \dots & 0 \\ 0 & 0 & ⋱ & . \\ ⋮ & ⋮ & . & - i χ_{N} \end{matrix}]}_{N \times N}

and we have assumed a single sided input/output.

Similar to single ring case, we have for the coupled-resonator waveguide,

a_{out, s} (ω_{s}) = - μ_{1} μ_{2} [T_{N, 1} (ω_{s}, ω_{i}) a_{in, s} (ω_{s}) + T_{N, N + 1} (ω_{s}, ω_{i}) a_{in, i}^{†} (ω_{i})]

a_{out, i}^{†} (ω_{i}) = - μ_{1} μ_{2} [T_{2 N, 1} (ω_{s}, ω_{i}) a_{in, s} (ω_{s}) + T_{2 N, N + 1} (ω_{s}, ω_{i}) a_{in, i}^{†} (ω_{i})]

and the joint spectral intensity

σ (ω_{s}, ω_{i}) = μ_{1}^{2} μ_{2}^{2} {| T_{N, N + 1} |}^{2}

. We note here that the coupled mode theory result is equivalent to the first-order perturbation theory with a cavity modified joint spectral amplitude,

| ψ 〉 = {| 0 〉}_{s} {| 0 〉}_{i} + g \iint d ω_{s} d ω_{i} S_{s} (ω_{s}) S_{i} (ω_{i}) S_{p} (ω_{s}, ω_{i}) \times f (ω_{s}, ω_{i}) a^{†} (ω_{s}) a^{†} (ω_{i}) {| 0 〉}_{s} {| 0 〉}_{i}

where the subscripts s and i refer to the signal and idler frequencies, g is proportional to the photon-pair production rate, and the function f(ω_s, ω_i) which describes the phase-matching and pump spectral envelope, is the joint spectral amplitude [13]. S are the field enhancement factors [21] and are equivalent to the slowing factors used in [22].

We verify the agreement between the time-domain coupled mode equations and the slowing factor enhanced pair generation equations by comparing the calculated pair flux. In the discussion below, we will assume a simplified picture with flat spectral filtering about the desired signal and idler modes, as was done in previous experiments [23]. The number of photon pairs generated per second is given in the low pump power regime by

F = Δ ν {(γ_{eff} P L_{eff})}^{2} exp (- α L)

where

γ_{eff}^{2} = S_{s} S_{i} {(\frac{S_{p} + 1}{2})}^{- 2} γ_{0}^{2}

, S_{_p,s,i_} are the slowing factors at the pump (p), signal (s) and idler (i) wavelengths, and L_eff = [1 − exp(−αL)]/α represents an effective propagation length, defined as the geometric length L = NπR normalized by the loss coefficient, α. R is the radius of the micro-resonator. An experimentally-validated transfer-matrix method can be used to calculate the α coefficient which scales linearly with the slowing factor [24]. We assume that the linear loss coefficienct α does not vary significantly with wavelength over the bandwidth of interest. To account for nonlinear absorption losses in silicon [9] we substitute

α \to α + 2 \frac{{\bar{P}}_{p}}{A_{eff}} β L

and PL_eff → P̄L where

\bar{P} = [log (1 + \frac{β}{A eff} P L_{eff})] / \frac{β}{A eff} L

and β is the effective TPA coefficient of the coupled-resonator waveguide which scales in the same way as γ_eff with S, i.e. β ∝ S²β₀. For an apodized structure, which we define as the case where the boundary coupling coefficients are matched to the input/output waveguides [25], we have at resonance S = 1/|κ|, where |κ| is the inter-resonator coupling coefficient in the transfer-matrix formalism. The bandwidth of the photon generation process, Δν, is assumed to be the linewidth of a Bloch eigenmode of the coupled-resonator waveguide, which scales inversely with the number of resonators in the chain, N,

Δ ν \approx \frac{1}{N} \frac{2 FSR}{π} {sin}^{- 1} | κ | .

Calculations were performed using the following parameters, R = 5 μm, waveguide loss = 1 dB/cm, γ₀ = 200 W⁻¹m⁻¹, β₀ = 0.75 cm/GW, P = 1 mW to obtain F over a range of values of S and N, showing good agreement between the pair generation equations and coupled mode equations [Figs. 2(a) and 2(c) respectively]. We assume that slowing factors at the pump, signal and idler wavelengths are approximately equal, S_{_p,s,i_} = S. Resonator chains that are in excess of the optimum length, or with too high a value of S incur penalties because of the exponential loss factor in Eq. (8), and the collapse of the bandwidth Δν. Too small values of S do not fully utilize the slow-light enhancement of the nonlinear FWM coefficient, which scales as a higher power of S than the corresponding decrease of bandwidth, unlike in a (linear) slow-light delay line. The optimum parameters are large S and small N, i.e. towards the single resonator configuration, for which the maximum pair flux rate exceeds 10 MHz at 1 mW pump power (and scaling quadratically with the pump power, i.e. 1 GHz at 10 mW).

Fig. 2 (a) Calculated photon pair flux F using pair generation equations, Eq. (8). The white trend-line follows the optimum number of resonators for a given slowing factor. (b) Corresponding values of γ_effP̄L for each S and N, showing the low multiphoton generation probability along the white line. (c) Calculated photon pair flux F using coupled mode equations, Eq. (5a). The top region of the contour plot represents a single resonator, while the far left approaches that of a conventional silicon nanowire waveguide. For S = 50, the optimum number of resonators is N_opt = 25 for which F = 4 MHz/mW².

Download Full Size | PDF

For a heralded single photon source we require low multi-photon probability. Figure 2(b) shows the value of the quantity γ_effP̄L for each value of S and N. For a γ_effP̄L ≪ 1, the level of stimulated scattering events is kept relatively low [2] which is true for the regions of highest pair flux (large S and small N).

3. Discussion

3.1. Scaling difference between single rings and coupled ring waveguides

One of the questions regarding the optimum device geometry for generating photon pairs is the appropriate size of resonators. Recently, the efficiency of classical and spontaneous four-wave mixing in single microring resonators has been compared [26, 27], with the conclusion being that in both cases, the conversion efficiency scales with the ring radius as R⁻², i.e., smaller rings are better than larger rings in generating photon pairs. This results from the analytically derived expression for the spontaneously-generated idler power P_i,SP (from an injected pump power P_p at optical carrier frequency ω_p)

P_{i, S P} = {(γ 2 π R)}^{2} {(\frac{Q v_{g}}{ω_{p} π R})}^{3} \frac{\bar{h} ω_{p} v_{g}}{4 π R} P_{p}^{2},

and a key assumption, that the ring quality factor Q is independent of the ring radius R. Starting with the equation for the (loaded) quality factor of a ring resonator side-coupled to a waveguide [28],

Q = \frac{π \sqrt{a_{r t}} τ}{1 - a_{r t} τ} \frac{n_{g} L}{λ}

where a_rt = exp(−αL/2) and L = 2πR, we examine two limiting cases as examples. In the first case, we examine a weakly coupled resonator (

τ = \sqrt{1 - {| κ |}^{2}} = 1

) with low loss (a_rt ≈ 1 − αL/2) in which case the quality factor can be expressed as,

Q = \frac{2 π n_{g}}{λ α}

which is the intrinsic Q limit. In this case, Q is indeed independent of R, and P_i,SP scales as R⁻². In the second case, however, we assume that the loaded Q is dominated by the coupling coefficient (|κ| ≠ 0) and then

Q = \frac{2 π n_{g}}{λ {| κ |}^{2} / L} .

In this case, the ratio Q/R in Eq. (10) is length-invariant, and P_i,SP increases linearly with R. As previously shown [29], coupled-resonator waveguides are more disorder tolerant in the large-coupling regime, and therefore, Eq. (11c) is more appropriate in describing performance rather than Eq. (11b). In fact, the agreement between Fig. 2(a), calculated using the conventional waveguide theory with nonlinearities scaled by the slowing factor, and Fig. 2(c), calculated using the first-principles time-domain coupled-mode theory model, shows that coupled-resonator waveguides are, in fact, more similar to waveguides than single resonators in many ways, with the attendant benefits of a slowing factor in enhancing the nonlinearity per unit length. Here, it is useful to recall, as shown in the classical domain, that coupled-resonator waveguides break the traditional trade-offs between parametric conversion efficiency and bandwidth, and are more robust against chromatic dispersion and propagation loss, compared to conventional waveguides [30]. Similarly, in the quantum domain, coupled-resonator waveguides may outperform conventional waveguides as pair and heralded single photon sources.

3.2. Joint spectral intensity (JSI)

To evaluate the spectral characteristics of the signal-idler photon pair, we calculate the Joint Spectral Intensity, and also the Schimdt number K = 1/∑Λ², which is the sum of the squares of the Schmidt eigenvalues (for a pure state K = 1) [31]. In Figs. 3(a) and 3(b), we plot the joint spectral intensities of an unapodized and apodized coupled-resonator waveguide of similar inter-resonator coupling coefficients. The shape of the spectrum reflects the number of resonators chosen N = 5, with the peaks corresponding to the locations of maximum transmission, which are also the Bloch eigenmodes. The pump pulse width is taken as 10 ps in both cases and we obtain K = 4.47 for the unapodized device and K = 3.31 for the apodized device. However, we note that choosing shorter pulses does not significantly change the Schmidt number in contrast with the single ring case [16]. In order to herald pure state single photons, filtering will be necessary. Choosing a filter bandwidth equal to the Bloch eigenmode width given by Eq. (9), we are able to obtain approximately a single Schimdt mode output.

Fig. 3 Joint spectral intensity (JSI) plots for various coupling coefficient configurations, assuming that the coupling coefficients between adjacent resonators, shown in Fig. 1, can be individually altered. (a) Unapodized (b) Apodized (c),(d) Chosen from a sample of Monte Carlo simulations with random coupling coefficients. (e) JSI for coupling coefficients chosen so as to realize a Butterworth filter response and (f) Bessel filter response in the linear transmission regime.

Download Full Size | PDF

On the other hand, if we have control over each individual inter-resonator coupling coefficients we are able to synthesize a large variety of different joint spectral amplitudes with different Schimdt numbers. In Figs. 3(c) and 3(d) we plot two interesting contours taken from a sample of different inter-resonator coupling configurations, each coefficient being a pseudo-random number ranging from 0 to 1. Clearly, with the added control over individual couplers we can obtain a large variety of corresponding K values. Of special interest are the configurations giving maximally flat transmission (Butterworth) and maximally flat group delay (Bessel) [25] since these quantities define the overall shape of the output joint spectrum (see Figs. 3(e) and 3(f)). Without additional filtering, we are able to obtain close to a pure heralded state for both the Butterworth filter configuration (K = 1.18) and the Bessel filter configuration (K = 1.09). Of course, filtering will still be required before the detectors, to separate the signal and idler photons and reject any unused pumps from reaching the SPADs [8].

3.3. Dispersion effects

While we have focused on the details of a single resonance in the prior discussion, as was predicted for for the case of a single resonator [17], the full two-photon state generated by the coupled resonator device is expected to form a “comb” structure with peaks centered around the resonance frequencies. In Fig. 4(a) we plot the transmission spectra around five particular resonances of a 5-ring unapodized coupled resonator waveguide, taking into account both the dispersion of the intrinsic constituent waveguides as well as the dispersion of the directional couplers [32]. The spectrum of the two photon state for a cw pump placed at the resonance Ω_p is given in Fig. 4(b), showing a fine structure characteristic of the number of resonators. While the general structure remains consistent, the peaks near the edges are reduced more quickly than those near the middle. This can be attributed to the large directional coupler dispersion which give rise to non-uniform transmission bandwidths. Careful inspection of Fig. 4(a) shows that the bandwidths increase gradually with frequency. The further apart the bands are, the more misaligned the transmission peaks become which in turn reduces the effective nonlinearity (see Eq. 8), since transmission peaks correspond also to peaks in slowing factor. The band edge peaks are most adversely affected since they are also the narrowest. In Figs. 4(c) and 4(d), we plot the JSI with signal and idler in the adjacent resonances as well as being two resonances apart from the pump. As compared to Fig. 3(a), we can see that the band edge peaks have become more distorted. Clearly, the uniformity of the two photon state generated over the “comb” for the coupled resonator configuration is limited by the dispersion of the directional couplers, the suppression of which is a problem of interest not only for chip-scale quantum optics but in ”classical” photonics as well.

Fig. 4 (a) Spectra of the transmission bands of a coupled resonator waveguide consisting of five microrings. (b) Spectrum of the two photon state when a cw pump is placed at the resonance Ω_p. (vertical axes are in logarithmic scale for both (a) and (b)) (c),(d) JSI of the transmission bands adjacent to the pump as well as two bands away.

Download Full Size | PDF

4. Conclusion

In summary, we have calculated the expected photon pair flux rates from a silicon coupled-microring waveguide device based on spontaneous four-wave mixing, a nonlinear process which scales quadratically with the optical pump power. This hybrid structure may significantly outperform conventional waveguides of much longer length at realistic waveguide losses and inter-resonator coupling strengths, and also outperform single ring resonators. We also developed a quantum mechanical coupled-mode theory which may be applicable to a generic class of waveguide or resonator based integrated photonic quantum light source, and evaluated the expected joint spectral intensities for the apodized and unapodized cases. Spectral filtering to isolate individual Bloch eigenmodes will help for heralding to a pure state. We also introduced a concept of tunability of the output Schimdt number, given control over individual inter-resonator coupling coefficients. The special cases of flat transmission and flat group delay may give nearly pure heralded states without need for additional filtering.

Appendix: Slowing factor in CROWs

A coupled-resonator optical waveguide (CROW), consists of a periodic sequence of nearest-neighbor coupled micro-resonators. The micro-resonators increase the transit time of light at certain wavelengths, where the slowing factor S > 1 is defined as the ratio of the group delay of the CROW to that of the unfolded waveguide of the same geometric length as the rings. The slowing factors near resonance are equal to the intensity buildup and is equivalent to the field enhancement squared, i.e., (FE)² in the terminology of Ferrara et al. [21]. The increased nonlinear interaction can be described as an effective nonlinear parameter,

γ_{eff} \equiv \frac{1}{L} \frac{d Φ_{CROW}}{d P_{i n}} = \frac{d Φ_{CROW}}{d ϕ} \frac{d ϕ}{d P_{ring}} \frac{d P_{ring}}{d P_{in}} = S \frac{S + 1}{2} γ \approx S \frac{S}{2} γ

the approximation being valid for small |κ| (see below). Here,

\frac{d Φ_{CROW}}{d ϕ}

is the group delay enhancement of the CROW,

\frac{d ϕ}{d P_{ring}}

is the nonlinear phase shift in the waveguide and

\frac{d P_{ring}}{d P_{in}}

is the intensity enhancement in the micro-resonators.

Some care must be taken to differentiate the slowing factors of apodized CROW with well tapered boundary coupling coefficients and an unapodized CROW which is perfectly periodic. For a well apodized CROW, with minimum transmission spectra ripple, the band-center S is inversely proportional to the magnitude of the coupling coefficient |κ| between adjacent resonators, S = 1/|κ|. Clearly, N ≥ 2 rings are necessary in order to taper coupling coefficients at the boundaries. For an unapodized (perfectly periodic) CROW on the other hand, band-center S = 1/|κ|². Some elaboration is needed for the special cases of N ≤ 2, for which there is no so-called “band-center” Bloch mode. For a single resonator N = 1, i.e. the single ring in the add/drop filter configuration, no apodization is possible. For N = 2, apodization is possible, however for the unapodized case there is no mid-band Bloch mode and instead there is resonance “splitting”. In these two cases, the same Eq. (A1) applies with $S = \frac{1 + t^{2}}{{| κ |}^{2}} \approx 2 / {| κ |}^{2}$ .

The coupling coefficient can be determined by lithographic design, or controlled in real-time e.g., electro-optically. We can estimate the coupling coefficient experimentally from the passband width [33],

Δ ν = \frac{2 FSR}{π} {sin}^{- 1} | κ | .

The effect of increasing S is to decrease the bandwidth over which resonant enhancement can be attained. The trade-off is between a conventional straight waveguide, which has the widest bandwidth but no resonance enhancement, and resonator-enhanced nonlinearities, which achieve higher photon pair generation rates but over a narrow band centered around the resonant wavelength of the microresonator (or the mid-band wavelength of the CROW).

Acknowledgments

This work was supported by the National Science Foundation under grants ECCS-0642603, ECCS-0925399, ECCS-1201308, NSF-GOALI collaboration with IBM, NSF-NIST supplement, and UCSD-Calit2. J. R. Ong acknowledges support from Agency for Science, Technology and Research (A*STAR) Singapore.

References and links

1. J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009). [CrossRef]

2. Q. Lin and G. P. Agrawal, “Silicon waveguides for creating quantum-correlated photon pairs,” Opt. Lett. 31, 3140–3142 (2006). [CrossRef] [PubMed]

3. J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388–12393 (2006). [CrossRef] [PubMed]

4. K.-i. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-i. Itabashi, “Generation of high-purity entangled photon pairs using silicon wirewaveguide,” Opt. Express 16, 20368–20373 (2008). [CrossRef] [PubMed]

5. S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, P. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17, 16558–16570 (2009). [CrossRef] [PubMed]

6. P. Grangier, G. Roger, and A. Aspect, “Experimental evidence for a photon anticorrelation effect on a beam splitter - a new light on single-photon interferences,” Europhys. Lett. 1, 173–179 (1986). [CrossRef]

7. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002). [CrossRef]

8. M. Davanco, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett. 100, 261104 (2012). [CrossRef]

9. J. Osgood, R. M., N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. W. Hsieh, E. Dulkeith, W. M. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photon. 1, 162–235 (2009). [CrossRef]

10. J. Fulconis, O. Alibart, J. L. O’Brien, W. J. Wadsworth, and J. G. Rarity, “Nonclassical interference and entanglement generation using a photonic crystal fiber pair photon source,” Phys. Rev. Lett. 99, 120501 (2007). [CrossRef] [PubMed]

11. K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14870–14886 (2007). [CrossRef] [PubMed]

12. Y. J. Lu and Z. Y. Ou, “Optical parametric oscillator far below threshold: Experiment versus theory,” Phys. Rev. A 62, 033804 (2000). [CrossRef]

13. Y. Jeronimo-Moreno, S. Rodriguez-Benavides, and A. B. U’Ren, “Theory of cavity-enhanced spontaneous parametric downconversion,” Laser Phys. 20, 1221–1233 (2010). [CrossRef]

14. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-q coupled nanocavities,” Nat. Photonics 2, 741–747 (2008). [CrossRef]

15. M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signalidler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518–3523 (2009). [CrossRef]

16. L. G. Helt, Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous four-wave mixing in microring resonators,” Opt. Lett. 35, 3006–3008 (2010). [CrossRef] [PubMed]

17. J. Chen, Z. H. Levine, J. Fan, and A. L. Migdall, “Frequency-bin entangled comb of photon pairs from a silicon-on-insulator micro-resonator,” Opt. Express 19, 1470–1483 (2011). [CrossRef] [PubMed]

18. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985). [CrossRef] [PubMed]

19. C.-S. Chuu and S. E. Harris, “Ultrabright backward-wave biphoton source,” Phys. Rev. A 83, 061803 (2011). [CrossRef]

20. J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. i. gain enhancement and noise,” J. Opt. Soc. Am. B 24, 2378–2388 (2007). [CrossRef]

21. M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nature Photon. 2, 737–740 (2008). [CrossRef]

22. J. R. Ong, M. L. Cooper, G. Gupta, W. M. J. Green, S. Assefa, F. Xia, and S. Mookherjea, “Low-power continuous-wave four-wave mixing in silicon coupled-resonator optical waveguides,” Opt. Lett. 36, 2964–2966 (2011). [CrossRef] [PubMed]

23. J. Fulconis, O. Alibart, W. Wadsworth, P. Russell, and J. Rarity, “High brightness single mode source of correlated photon pairs using a photonic crystal fiber,” Opt. Express 13, 7572–7582 (2005). [CrossRef] [PubMed]

24. M. L. Cooper and S. Mookherjea, “Modeling of multiband transmission in long silicon coupled-resonator optical waveguides,” IEEE Photon. Technol. Lett. 23, 872–874 (2011). [CrossRef]

25. H.-C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (crows),” Opt. Express 19, 17653–17668 (2011). [CrossRef] [PubMed]

26. L. G. Helt, M. Liscidini, and J. E. Sipe, “How does it scale? comparing quantum and classical nonlinear optical processes in integrated devices,” J. Opt. Soc. Am. B 29, 2199–2212 (2012). [CrossRef]

27. S. Azzini, D. Grassani, M. Galli, L. C. Andreani, M. Sorel, M. J. Strain, L. G. Helt, J. E. Sipe, M. Liscidini, and D. Bajoni, “From classical four-wave mixing to parametric fluorescence in silicon microring resonators,” Opt. Lett. 37, 3807–3809 (2012). [PubMed]

28. Y.-C. Hung, S. Kim, B. Bortnik, B.-J. Seo, H. Tazawa, H. R. Fetterman, and W. H. Steier, Practical Applications of Microresonators in Optics and Photonics (CRC Press, 2009).

29. S. Mookherjea and M. A. Schneider, “Avoiding bandwidth collapse in long chains of coupled optical microresonators,” Opt. Lett. 36, 4557–4559 (2011). [CrossRef] [PubMed]

30. F. Morichetti, A. Canciamilla, C. Ferrari, A. Samarelli, M. Sorel, and A. Melloni, “Travelling-wave resonant four-wave mixing breaks the limits of cavity-enhanced all-optical wavelength conversion,” Nat. Commun. 2, 296 (2011). [CrossRef]

31. C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004). [CrossRef] [PubMed]

32. R. Aguinaldo, Y. Shen, and S. Mookherjea, “Large dispersion of silicon directional couplers obtained via wideband microring parametric characterization,” IEEE Photon. Technol. Lett. 24, 1242–1244 (2012). [CrossRef]

33. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35, 3030–3032 (2010). [CrossRef] [PubMed]

Quantum light generation on a silicon chip using waveguides and resonators

Abstract

1. Introduction

2. Photon pair generation

2.1. Single micro-resonators

2.2. Coupled micro-resonator waveguide

3. Discussion

3.1. Scaling difference between single rings and coupled ring waveguides

3.2. Joint spectral intensity (JSI)

3.3. Dispersion effects

4. Conclusion

Appendix: Slowing factor in CROWs

Acknowledgments

References and links

Cited By

Figures (4)

Equations (21)

Optics Express