## Quantum light generation on a silicon chip using waveguides and resonators |

Optics Express, Vol. 21, Issue 4, pp. 5171-5181 (2013)

http://dx.doi.org/10.1364/OE.21.005171

Acrobat PDF (2692 KB)

### 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

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]

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]

*μ*m wavelength from a silicon chip at room temperature [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]

*γ*= 2

*πn*

_{2}/

*λA*

_{eff}), in terms of the Kerr nonlinear index

*n*

_{2}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

*μ*m

^{2}has a nonlinearity coefficient

*γ*

_{Si}= 100–200 W

^{−1}m

^{−1}(five orders of magnitude greater than optical fiber) around a wavelength of 1.5

*μ*m [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]

*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.

## 2. Photon pair generation

### 2.1. Single micro-resonators

### 2.2. Coupled micro-resonator waveguide

*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]

*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*) which describes the phase-matching and pump spectral envelope, is the joint spectral amplitude [13

_{i}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]

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]

*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

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]

*α*does not vary significantly with wavelength over the bandwidth of interest. To account for nonlinear absorption losses in silicon [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]

*PL*

_{eff}→

*P̄L*where

*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*∝

*S*

^{2}

*β*

_{0}. For an apodized structure, which we define as the case where the boundary coupling coefficients are matched to the input/output waveguides [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]

*S*= 1/|

*κ*|, where |

*κ*| is the inter-resonator coupling coefficient in the transfer-matrix formalism. The bandwidth of the photon generation process, Δ

*ν*, 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*,

*R*= 5

*μ*m, waveguide loss = 1 dB/cm,

*γ*

_{0}= 200 W

^{−1}m

^{−1},

*β*

_{0}= 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 Δ

*ν*. 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).

*γ*

_{eff}

*P̄L*for each value of

*S*and

*N*. For a

*γ*

_{eff}

*P̄L*≪ 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]

*S*and small

*N*).

## 3. Discussion

### 3.1. Scaling difference between single rings and coupled ring waveguides

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]

*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*(from an injected pump power

_{i,SP}*P*at optical carrier frequency

_{p}*ω*) and a key assumption, that the ring quality factor

_{p}*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], where

*a*= exp(−

_{rt}*αL*/2) and

*L*= 2

*πR*, we examine two limiting cases as examples. In the first case, we examine a weakly coupled resonator (

*a*≈ 1 −

_{rt}*αL*/2) in which case the quality factor can be expressed as, which is the intrinsic

*Q*limit. In this case,

*Q*is indeed independent of

*R*, and

*P*scales as

_{i,SP}*R*

^{−2}. In the second case, however, we assume that the loaded

*Q*is dominated by the coupling coefficient (|

*κ*| ≠ 0) and then In this case, the ratio

*Q/R*in Eq. (10) is length-invariant, and

*P*increases linearly with

_{i,SP}*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]

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]

### 3.2. Joint spectral intensity (JSI)

*K*= 1/∑Λ

^{2}, 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]

*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]

*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]

*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

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]

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]

*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.*

_{p}## 4. Conclusion

## Appendix: Slowing factor in CROWs

*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]

*κ*| (see below). Here,

*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

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]

*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

## References and links

1. | J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics |

2. | Q. Lin and G. P. Agrawal, “Silicon waveguides for creating quantum-correlated photon pairs,” Opt. Lett. |

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 |

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 |

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 |

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. |

7. | M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. |

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. |

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. |

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. |

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 |

12. | Y. J. Lu and Z. Y. Ou, “Optical parametric oscillator far below threshold: Experiment versus theory,” Phys. Rev. A |

13. | Y. Jeronimo-Moreno, S. Rodriguez-Benavides, and A. B. U’Ren, “Theory of cavity-enhanced spontaneous parametric downconversion,” Laser Phys. |

14. | M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-q coupled nanocavities,” Nat. Photonics |

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. |

16. | L. G. Helt, Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous four-wave mixing in microring resonators,” Opt. Lett. |

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 |

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 |

19. | C.-S. Chuu and S. E. Harris, “Ultrabright backward-wave biphoton source,” Phys. Rev. A |

20. | J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. i. gain enhancement and noise,” J. Opt. Soc. Am. B |

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. |

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. |

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 |

24. | M. L. Cooper and S. Mookherjea, “Modeling of multiband transmission in long silicon coupled-resonator optical waveguides,” IEEE Photon. Technol. Lett. |

25. | H.-C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (crows),” Opt. Express |

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 |

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. |

28. | Y.-C. Hung, S. Kim, B. Bortnik, B.-J. Seo, H. Tazawa, H. R. Fetterman, and W. H. Steier, |

29. | S. Mookherjea and M. A. Schneider, “Avoiding bandwidth collapse in long chains of coupled optical microresonators,” Opt. Lett. |

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. |

31. | C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. |

32. | R. Aguinaldo, Y. Shen, and S. Mookherjea, “Large dispersion of silicon directional couplers obtained via wideband microring parametric characterization,” IEEE Photon. Technol. Lett. |

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. |

**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

- J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics3, 687–695 (2009). [CrossRef]
- Q. Lin and G. P. Agrawal, “Silicon waveguides for creating quantum-correlated photon pairs,” Opt. Lett.31, 3140–3142 (2006). [CrossRef] [PubMed]
- 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. Express14, 12388–12393 (2006). [CrossRef] [PubMed]
- 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. Express16, 20368–20373 (2008). [CrossRef] [PubMed]
- 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. Express17, 16558–16570 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
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