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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 4 — Feb. 25, 2013
  • pp: 5171–5181
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Quantum light generation on a silicon chip using waveguides and resonators

Jun Rong Ong and Shayan Mookherjea  »View Author Affiliations


Optics Express, Vol. 21, Issue 4, pp. 5171-5181 (2013)
http://dx.doi.org/10.1364/OE.21.005171


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Abstract

Integrated optical devices may replace bulk crystal or fiber based assemblies with a more compact and controllable photon pair and heralded single photon source and generate quantum light at telecommunications wavelengths. Here, we propose that a periodic waveguide consisting of a sequence of optical resonators can outperform conventional waveguides or single resonators and generate more than 1 Giga-pairs per second from a sub-millimeter-long room-temperature silicon device, pumped with only about 10 milliwatts of optical power. Furthermore, the spectral properties of such devices provide novel opportunities for chip-scale quantum light sources.

© 2013 OSA

1. Introduction

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/λAeff), in terms of the Kerr nonlinear index n2 and the effective area of the waveguide mode (Aeff), 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 γSi= 100–200 W−1m−1 (five orders of magnitude greater than optical fiber) around a wavelength of 1.5 μm [9

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]

]. 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 Leff = [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

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]

]. 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

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]

] 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

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

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]

].

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 1τe1 and 1τe2 (the latter is not shown, at the output side of the chip). The resonator loss is indicated by the damping rate 1τl.

2. Photon pair generation

2.1. Single micro-resonators

2.2. Coupled micro-resonator waveguide

Extending to the case of N coupled cavities [20

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]

], we have the following matrix equation,
[as,1as,2ai,1ai,2]2N×1=iμT[as,in0ai,in0]2N×1
(5a)
where
T=[MsCCMi]2N×2N1
(5b)
Ms=[i(ωsΩs,1)+1τl+1τe1iκs,200iκs,2i(ωsΩs,2)+1τl.00....0.i(ωsΩs,N)+1τl+1τe2]N×N
(5c)
C=[iχ1000iχ2000..iχN]N×N
(5d)
and we have assumed a single sided input/output.

Similar to single ring case, we have for the coupled-resonator waveguide,
aout,s(ωs)=μ1μ2[TN,1(ωs,ωi)ain,s(ωs)+TN,N+1(ωs,ωi)ain,i(ωi)]
(6a)
aout,i(ωi)=μ1μ2[T2N,1(ωs,ωi)ain,s(ωs)+T2N,N+1(ωs,ωi)ain,i(ωi)]
(6b)
and the joint spectral intensity σ(ωs,ωi)=μ12μ22|TN,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=|0s|0i+gdωsdωiSs(ωs)Si(ωi)Sp(ωs,ωi)×f(ωs,ωi)a(ωs)a(ωi)|0s|0i
(7)
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(ωs, ωi) which describes the phase-matching and pump spectral envelope, is the joint spectral amplitude [13

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]

]. S are the field enhancement factors [21

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]

] and are equivalent to the slowing factors used in [22

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]

].

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 γeffP̄L 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 Nopt = 25 for which F = 4 MHz/mW2.

For a heralded single photon source we require low multi-photon probability. Figure 2(b) shows the value of the quantity γeffP̄L for each value of S and N. For a γeffP̄L ≪ 1, the level of stimulated scattering events is kept relatively low [2

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

] 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

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

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]

], 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 Pi,SP (from an injected pump power Pp at optical carrier frequency ωp)
Pi,SP=(γ2πR)2(QvgωpπR)3h¯ωpvg4πRPp2,
(10)
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

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).

],
Q=πartτ1artτngLλ
(11a)
where art = exp(−αL/2) and L = 2πR, we examine two limiting cases as examples. In the first case, we examine a weakly coupled resonator ( τ=1|κ|2=1) with low loss (art ≈ 1 − αL/2) in which case the quality factor can be expressed as,
Q=2πngλα
(11b)
which is the intrinsic Q limit. In this case, Q is indeed independent of R, and Pi,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=2πngλ|κ|2/L.
(11c)
In this case, the ratio Q/R in Eq. (10) is length-invariant, and Pi,SP increases linearly with R. As previously shown [29

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]

], 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

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]

]. 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

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]

]. 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

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]

]. 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.

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

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]

] 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

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]

].

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

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]

], 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

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]

]. The spectrum of the two photon state for a cw pump placed at the resonance Ωp 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 Ωp. (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.

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

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]

]. The increased nonlinear interaction can be described as an effective nonlinear parameter,
γeff1LdΦCROWdPin=dΦCROWdϕdϕdPringdPringdPin=SS+12γSS2γ
(A1)
the approximation being valid for small |κ| (see below). Here, dΦCROWdϕ is the group delay enhancement of the CROW, dϕdPring is the nonlinear phase shift in the waveguide and dPringdPin 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=1+t2|κ|22/|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

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]

],
Δν=2FSRπsin1|κ|.
(A2)
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.

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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]

OCIS Codes
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(270.0270) Quantum optics : Quantum optics
(230.4555) Optical devices : Coupled resonators

ToC Category:
Quantum Optics

History
Original Manuscript: December 14, 2012
Revised Manuscript: February 4, 2013
Manuscript Accepted: February 13, 2013
Published: February 22, 2013

Citation
Jun Rong Ong and Shayan Mookherjea, "Quantum light generation on a silicon chip using waveguides and resonators," Opt. Express 21, 5171-5181 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-5171


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References

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  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. Express19, 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. A31, 3761–3774 (1985). [CrossRef] [PubMed]
  19. C.-S. Chuu and S. E. Harris, “Ultrabright backward-wave biphoton source,” Phys. Rev. A83, 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. B24, 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. Express13, 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. Express19, 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. B29, 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]

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