## Entangled photon generation using four-wave mixing in azimuthally symmetric microresonators |

Optics Express, Vol. 20, Issue 20, pp. 21977-21991 (2012)

http://dx.doi.org/10.1364/OE.20.021977

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### Abstract

A novel quantum mechanical formulation of the bi-photon wavefunction and spectra resulting from four-wave mixing is developed for azimuthally symmetric systems. Numerical calculations are performed verifying the use of the angular group velocity and angular group velocity dispersion in such systems, as opposed their commonly used linear counterparts. The dispersion profile and bi-photon spectra of two illustrative examples are given, emphasizing the physical origin of the effects leading to the conditions for angular momentum and energy conservation. A scheme is proposed in which widely spaced narrowband entangled photons may be produced through a four-wave mixing process in a chip-scale ring resonator. The entangled photon pairs are found to conserve energy and momentum in the four-wave mixing interaction, even though both photon modes lie in spectral regions of steep angular group velocity dispersion.

© 2012 OSA

## 1. Introduction

1. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. **78**, 3221–3224 (1997). [CrossRef]

2. T. C. Ralph and G. J. Pryde, “Optical quantum computation,” in *Prog. Optics*,vol. 54, E. Wolf, ed. (Elsevier Science2010), pp. 209–269. [CrossRef]

3. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature **423**, 731–734 (2003). [CrossRef] [PubMed]

4. A. Haase, N. Piro, J. Eschner, and M. W. Mitchell, “Tunable narrowband entangled photon pair source for resonant single-photon single-atom interaction,” Opt. Lett. **34**, 55–57 (2009). [CrossRef]

5. K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys. **11**, 013049 (2009). [CrossRef]

6. Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric Down-Conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. **83**, 2556–2559 (1999). [CrossRef]

7. 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]

9. 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]

10. 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 (2002). [CrossRef]

12. H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. I. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. **91**, 201108 (2007). [CrossRef]

16. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature **450**, 1214–1217 (2007). [CrossRef]

22. A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100GHz repetition rates,” Opt. Lett. **37**, 875–877 (2012). [CrossRef] [PubMed]

23. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. **25**, 554–556 (2000). [CrossRef]

24. A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express **16**, 4881–4887 (2008). [CrossRef] [PubMed]

*ω*=

_{p}*ω*+

_{s}*ω*, where

_{i}*ω*(

_{j}*j*=

*p*,

*s*,

*i*) are the frequencies of the pump, signal and idler waves, respectively.

25. Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A **82**, 033801 (2010). [CrossRef]

26. Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. **104**, 103902 (2010). [CrossRef] [PubMed]

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

9. 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]

9. 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]

*L*to enforce phase matching. In a cylindrical system, however, linear momentum is zero and the linear wavevector and group velocities are not well defined, especially as the radius of the resonator is reduced. This is because no translational symmetry exists in the system, meaning that attempting to model the propagating waves using a linear wavevector (a vector that describes

*linear*translation of the optical waves) is at best an approximation. Thus, a more accurate approach would involve the angular wavevector and angular group velocities rather than their linear counterparts.

*m*:

**E**

*(*

_{m}**r**,

*t*) =

**E**

*(*

_{m}*ρ*,

*z*)

*e*

^{−i(ωmt+mϕ)}. The basic geometry considered is similar to the ring resonator shown in Fig. 1, along with examples of the radial field profiles

*E*(

_{ρ}*ρ*,

*z*) for three TE-like resonant modes with differing angular wavenumbers and radial orders. In the numerical calculations performed in this paper, we only consider the fundamental radial TE-like modes. If the ring is constructed of a material with a third order nonlinear suceptiblity

*χ*

^{(3)}(

*ρ*,

*z*), then when a strong pump beam with frequency

*ω*is coupled into mode

_{p}*m*of the ring, two pump photons can be annihilated to produce entangled signal and idler photons at

*ω*and

_{s}*ω*via spontaneous FWM, so long as the process conserves both energy and momentum. The conservation of energy requires that

_{i}*ω*+

_{s}*ω*= 2

_{i}*ω*, while the conservation of momentum requires that

_{p}*m*+

_{s}*m*= 2

_{i}*m*.

_{p}*m*is an integer; for every integer Δ

*m*there will be a pair of phase matched modes at

*m*

_{+}=

*m*+ Δ

_{p}*m*and

*m*

_{−}=

*m*− Δ

_{p}*m*. The conservation of energy, however, is not guaranteed, since

*ω*, while discrete, can take on a continuum of values which are not necessarily equally spaced. Thus, the dispersion of the optical modes (i.e their spectral spacing) is the most important determinant as to whether spontaneous FWM will occur and produce entangled photon pairs.

_{m}## 2. Calculation of entangled photon spectra

*k*in may be written where

**u**

*(*

_{k}**r**) is spatial function describing the mode profile, normalized such that Note the explicit inclusion of the group and phase velocities in the normalization of the electric field operator, essential to include if the total energy of of the system is to be accounted for in a Hamiltonian formulation (see Appendix 1).

*m*

_{±}≡

*m*± Δ

_{p}*m*ensures momentum conservation and Δ

_{0}≡ 2

*ω*− (

_{p}*ω*

_{ms}+

*ω*

_{mi}) is the spectral walk-off characterizing energy conservation. In deriving Eq. (8), we have assumed a monochromatic pump with a classical amplitude, and made use of a two-port cavity input/output formalism [28] in which

*κ*

_{1}represents an intrinsic loss rate and

*κ*is the total loss rate (intrinsic and extrinsic). We have also made the approximation that the cavity linewidth is much less than the FSR (

_{tot}*κ*≪

_{tot}*f*), such that only pair-wise products of modes

_{m}*m*

_{±}occur in the summation (see Appendix 2). Notice the absence of an effective area, as well as the absence of a sinc function to account for linear phase matching, terms commonly found in many waveguide formulations of nonlinear optical interactions [9

**19**, 1470–1483 (2011). [CrossRef] [PubMed]

*v*. Phase matching is automatically accounted for by only summing over those signal and idler modes with angular wavenumbers equidistant to that of the pump. With the two-photon state in hand, we can calculate quantities such as the bi-photon production rate 〈

_{p}/v_{g}*ψ*|

*ψ*〉, the single photon spectrum 〈

*ψ*|

*â*

^{†}(

*ω*)

*â*(

*ω*)

*ψ*|〉, and the coincidence spectrum.

*ω*with

_{s}*ω*. We now proceed to calculate the spectra of entangled bi-photons for two illustrative examples.

_{i}## 3. Numerical examples

*μ*m and a thickness of 250 nm. Silicon nitride is an important optical material whose spectral bandgap lies above the energy required for two-photon absorption at 1550 nm. The field profiles

**u**

*(*

_{m}**r**) and eigenfrequencies

*ω*are calculated using a fully vectorial 2-D axially symmetric weighted residual formulation of Maxwell’s equations implemented in Comsol Multiphysics software. An iterative approach is used to incorporate material dispersion, which is expressed using a Sellmeier equation for the wavelength dependent dielctric constant [29

_{m}29. T. Bååk, “Silicon oxynitride; a material for GRIN optics,” Appl. Optics **21**, 1069–1072 (1982). [CrossRef]

*C*

_{1}= 2.8939 and

*C*

_{2}= 139.67×10

^{−3}. From the field profiles and eigenfrequencies, we calculate the local FSR

*f*for each eigenmode, the spectral walk-off Δ

_{m}_{0}, and the overlap integrals necessary to calculate the spectra of the signal and idler photons. The result is shown in Fig. 2.

*πω*of the fundamental TE-like mode (with odd vector symmetry about the z-axis) of the disk as a function of the angular wavevector

_{m}*m*. Physically,

*m*represents the number of wavelengths that fit around the disk for a given mode. Hence, at small values of

*m*, the optical wavelength is large (low frequency), and thus spills out into the surrounding air. At larger values of

*m*, the wavelength is much smaller and the optical mode is pulled into the dielectric. As a reference, light lines

*m*tends to zero or infinity. Mode dispersion occurs as a result of the transition of the mode frequency from the air line to the dielectric line.

*πω*, which physically correspond to FSR

_{m}*f*(red) and the change in the FSR per mode Δ

_{m}*f*(blue). The FSR is calculated two ways: using a direct difference of two eigenfrequenies [Eq. (3b)] and using the field profile from a single solution [Eq. (4)], and the two methods are found to have excellent agreement. As one might expect, the FSR decreases as the mode transitions from the air line to the dielectric line, eventually leveling off to a value near 1.2 THz. An important observation is that the FSR decreases

_{m}*monotonically*for this range in

*m*, as can most easily be seen by looking at its derivative, which is always negative. As a result, there is no point of zero angular group velocity dispersion, and hence no set of modes available that can perfectly conserve energy in a FWM interaction. Figure 2(c) shows an example of the entangled photon spectra calculated using Eq. (9) that results when the disk is pumped near 1.55

*μ*m (m = 109). An optical quality factor of 10

^{5}(i.e.

*κ*

_{1}=

*κ*=

_{tot}*ω*/10

_{m}^{5}) has been assumed for the disk, a value easily obtainable experimentally for silicon nitride disks with this geometry. The spectra has been normalized such that peaks have a value of unity when the mode pair

*ω*

_{+}and

*ω*

_{−}corresponding to that peak exactly conserve energy (Δ

_{0}= 0), have identical angular group velocities (

*f*

_{+}=

*f*

_{−}), have a perfect spatial overlap [

**u**

_{+}(

**r**) =

**u**

_{−}(

**r**) =

**u**

*(*

_{p}**r**)], and the approximation

*m*

_{+}≈

*m*

_{−}≈

*m*is valid. Any reduction in the amplitude of a peak in the spectra results from one of these criteria not being fulfilled, and is dominated by nonzero values of Δ

_{p}_{0}. The dotted reference line in Figure 2(b) shows the location of the pump beam on the dispersion walk-off curve Δ

*f*. As can be seen, the change in the FSR for each successive mode is approximately 2 GHz, which corresponds to approximately a full linewidth at these frequencies (i.e. 200 THz/10

_{m}^{5}= 2 GHz). As a result, the first order side-bands in the spectra are reduced by almost half from the ideal case, and the second order sidebands and beyond are strongly suppressed owing to the walk-off.

*r*= 20

*μ*m), clad with silicon dioxide, and with a width of

*w*= 1.1

*μ*m and a thickness of

*h*= 750 nm. The wavelength dependent dielectric constant of SiO

_{2}is given by [30

30. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. **55**, 1205–1208 (1965). [CrossRef]

*C*

_{1}= 0.6961663,

*C*

_{2}= 4.67914826×10

^{3},

*C*

_{3}= 0.4079426,

*C*

_{4}= 1.35120631×10

^{2},

*C*

_{5}= 0.8974794, and

*C*

_{6}= 97.9340025×10

^{2}. A similar set of plots to those produced for the silicon nitride disk are shown in Fig. 3, highlighting some key qualitative differences in the resulting spectra. As shown in Fig. 3(a), the mode frequency is again bounded by two light lines (in this case SiO

_{2}and Si

_{3}N

_{4}), and the mode frequency transitions from one light line to the other as

*m*increases. However, unlike the case of the thin disk, the FSR does not decrease monotonically, but rather undergoes a steep decrease, followed by a shallow increase (i.e. the mode frequency changes from negative to positive curvature as it switches from one asymptote to the other). In fact, such a change in curvature also occurs in the thin disk, but at much higher frequencies when the light is much more strongly confined the high index dielectric and approaches the dielectric light line more closely. For the ring, the result is a zero-crossing in Δ

*f*(i.e. a point of zero angular group velocity dispersion), which can be utilized to produce more desirable entangled photon spectra. The geometry of the ring has been chosen such that this point occurs near a wavelength of 1550 nm. Figure 3(c) shows the resulting signal spectra when the resonator is pumped at the point of zero Δ

_{m}*f*, with no angular group velocity dispersion (m = 137). The spectrum consists of many comb-like pairs which are pair-wise entangled and cover a broad spectrum with near unity relative amplitude. It was noted by Chen et al. [9

_{m}**19**, 1470–1483 (2011). [CrossRef] [PubMed]

*f*point. The number of lines in the comb-like spectra is reduced owing to the faster dispersion walk-off (nonzero Δ

_{m}_{0}), but two large peaks in the spectra appear at large values of Δ

*m*. The peaks originate from an accidental degeneracy in the energy separation of the signal and idler modes from the pump, and may be physically understood as follows: The total frequency separation between the pump mode

*ω*and any other mode

_{p}*ω*is given by the discrete sum of the FSRs of each mode between

_{m}*ω*and

_{p}*ω*. This sum is shown graphically as the area under the

_{m}*f*curve in Fig. 3(b). When the pump beam is slightly detuned from the minimum dispersion point to the side with a shallow slope (higher wavenumber), there exists a unique Δ

_{m}*m*at which the area under the curve to the left and to the right of the pump are equal. No such point exists when the pump is detuned to smaller wavenumbers, as the area on the left will always be larger than that on the right.

^{6}(which is experimentally feasible [17

17. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics **4**, 37–40 (2010). [CrossRef]

## 4. Conclusion

## 5. Appendix 1: quantization of the field

31. D. A. B. Miller, *Quantum Mechanics for Scientists and Engineers* (Cambridge University Press, 2008). [CrossRef]

*ε*, and calculate the energy as We immediately see that the effect of the dielectric is to increase the total energy of each mode by a factor of

*ε*. Note that we have used the relation expressing the equal sharing of energy among the electric and magnetic fields for lossless, harmonic modes.

*μ*(

*ω*) = 1, then we have where

*v*

_{gm}=

*c*/[

*n*(

*ω*)+d

*n*(

*ω*)/d

*ω*] evaluated at

*ω*. We see from this that the effect of the material dispersion is to increase the energy by an additional factor of

_{m}*v*

_{pm}/

*v*

_{gm}. An expression similar to Eq. (16) was used by Milonni to quantize the electromagnetic field in uniform dispersive media [32

32. P. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Optic. **42**, 1991–2004 (1995). [CrossRef]

*v*

_{pm}/

*v*

_{gm}, while maintaing the same power flow. The energy density must therefore increase by a factor of

*v*

_{pm}/

*v*

_{gm}. Even though the material dispersion is weak in our system, the intuition gained from calculating the dispersive energy can be used to quantify the effect of geometric dispersion on the total energy of the system.

*ε*(

**r**) can provide a potential well for bound modes, so that we may integrate over all space and obtain a finite result. The non-uniform dielectric constant has another important consequence: geometric dispersion, which, in addition to material dispersion, modifies the group velocity. We make the physical argument that, owing to conservation of energy in the volume, the relation

*U*

_{ε(r, ω)}= [

*v*

_{pm}/

*v*

_{gm}]

*U*still holds.

_{ε}*c*/[

*n*(

*ω*) + d

*n*(

*ω*)/d

*ω*] as in the case of uniform dielectrics, but is more generally

**v**

_{gm}= ∇

*, where ∇*

_{m}ω_{m}*is the gradient with respect to*

_{m}*m*. To find the group and phase velocities, one must solve Maxwell’s equations for the eigen-mode frequencies or field profiles and use either Eq. (3b) or Eq. (4). Also note that because

*ε*(

**r**) is no longer a constant in space, we cannot factor it from the energy density. This specifically precludes the possibility of normalizing the spatial function

**u**

*(*

_{m}**r**) such that ∫

*|*

_{V}**u**

*(*

_{m}**r**)|

^{2}= 1.

*p*(

_{m}*t*) and

*q*(

_{m}*t*) are functions of time,

*D*is a constant, and

_{m}**u**

*(*

_{m}**r**) and

**v**

*(*

_{m}**r**) are spatial functions normalized such that.

*material*dispersion is present (though geometric dispersion does not effect orthogonality). The reason for this can be most easily seen by writing down the mode solutions as an eigenvalue problem: The solutions to this generalized eigenvalue problem will only be orthogonal if both of the operators ∇ × ∇ × and

*ε*(

**r**,

*ω*) are Hermitian. While ∇ × ∇ × is always Hermitian,

*ε*(

**r**,

*ω*) is not Hermitian if it is frequency dependent. Owing to causality, the imaginary part of the dielectric constant must be nonzero: A complex valued dielectric function usually precludes its Hermiticity, and typically arises from frequency dependent loss. It is important to note, however, that geometric dispersion (e.g. boundaries) does not affect the Hermiticity of the operators, and hence the orthogonality of the modes, even though the effective index and phase velocity for each mode can vary greatly. For dielectric materials used to construct high quality optical resonators, the imaginary part of the dielectric constant is typically more than six (6) orders of magnitude smaller than the real part, making the approximation of orthogonality very good. In metallic resonators, however, the situation is quite different and great care should be taken when quantizing the field.

*v*multiplier (this can also be accomplished by considering the total power flux rather than the total energy [33

_{p}/v_{g}33. B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. **16**, 177–182 (1991). [CrossRef]

*D*to unity and instead normalized the spatial functions

_{m}**u**

*(*

_{m}**r**), and

**v**

*(*

_{m}**r**) such that and such a formulation would still satisfy Hamilton’s equations, though it would bury the explicit dependence of the field operators on the phase and group velocities. Postulating a momentum operator

*p̂*= −

_{m}*ih̄∂*/

*∂q*(

_{m}*t*), the quantum Hamiltonian can then be written where the creation and annihilation operators are with

*D*and

_{m}*p*̂(

*t*) into Eq. (17a), we find the quantized electric field operator for each mode to be

## 6. Appendix 2: two-photon wavefunction

*â*(

*t*) in Eq. (1) are functions of time and the resonator is an open system in which photons enter and leave through ports. We may incorporate this fact phenomenologically by using an input/output formalism [28]. It should be noted, however, that while incorporating the resonator loss via phenomenological decay rates allows a simple way to correctly calculate the signal and idler spectrum, it is not as rigorous as including loss via a direct coupling of the electric field operators to a reservoir of excitations. Physically, this is because the signal and idler photons from a generated pair may experience loss differently, (i.e. one may be absorbed or outcoupled while the other is not) affecting quantities such as coherence functions (e.g.

*g*

^{2}).

*a*|

^{2}is the intracavity energy and |

*s*

_{l}_{+}|

^{2}is the input power at port

*l*,

*κ*

_{0}is the intrinsic linewidth of the resonator and

*κ*is the coupling rate of the resonator to port

_{l}*l*. Here, |

*s*

_{l}_{−}|

^{2}is the power flowing away from port

*l*. Specializing to the case of launching power only into port (1) we can rewrite Eqs. (32a)–(32b) as The steady state fields in the cavity and moving away from the cavity (in the absence of nonlinear frequency conversion) are then The Fourier components of

*â*(

_{m}*t*) can thus be represented as The electric field creation operator is then Using first order perturbation theory, we may write the two-photon wavefunction as

*ϕ*to enforce the conservation of angular momentum, and making the approximation that the cavity linewidth is much less than the FSR (

*κ*≪

_{tot}*f*), we find that only pair-wise products of modes

_{m}*m*

_{±}occur, reducing the wavefunction to a single summation: where

*m*

_{±}=

*m*± Δ

_{p}*m*and we have used the identity Finally, we explicitly integrate over time to enforce energy conservation: where we have used the identity

## Acknowledgments

## References and links

1. | J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. |

2. | T. C. Ralph and G. J. Pryde, “Optical quantum computation,” in |

3. | A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature |

4. | A. Haase, N. Piro, J. Eschner, and M. W. Mitchell, “Tunable narrowband entangled photon pair source for resonant single-photon single-atom interaction,” Opt. Lett. |

5. | K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys. |

6. | Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric Down-Conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. |

7. | 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 |

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

9. | 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 |

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

11. | 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 |

12. | H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. I. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. |

13. | X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber source of Polarization-Entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. |

14. | H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-i. Itabashi, “Generation of polarization entangledphoton pairs using silicon wirewaveguide,” Opt. Express |

15. | 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 wire waveguide,” Opt. Express |

16. | P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature |

17. | J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics |

18. | L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics |

19. | M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun |

20. | T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based optical frequency combs,” Science |

21. | F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics |

22. | A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100GHz repetition rates,” Opt. Lett. |

23. | P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. |

24. | A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express |

25. | Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A |

26. | Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. |

27. | M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal/idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. |

28. | D. Walls and G. J. Milburn, |

29. | T. Bååk, “Silicon oxynitride; a material for GRIN optics,” Appl. Optics |

30. | I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. |

31. | D. A. B. Miller, |

32. | P. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Optic. |

33. | B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. |

**OCIS Codes**

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

(230.3990) Optical devices : Micro-optical devices

(230.5750) Optical devices : Resonators

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 24, 2012

Revised Manuscript: August 17, 2012

Manuscript Accepted: August 17, 2012

Published: September 11, 2012

**Citation**

Ryan M. Camacho, "Entangled photon generation using four-wave mixing in azimuthally symmetric microresonators," Opt. Express **20**, 21977-21991 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-21977

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### References

- J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett.78, 3221–3224 (1997). [CrossRef]
- T. C. Ralph and G. J. Pryde, “Optical quantum computation,” in Prog. Optics,vol. 54, E. Wolf, ed. (Elsevier Science2010), pp. 209–269. [CrossRef]
- A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature423, 731–734 (2003). [CrossRef] [PubMed]
- A. Haase, N. Piro, J. Eschner, and M. W. Mitchell, “Tunable narrowband entangled photon pair source for resonant single-photon single-atom interaction,” Opt. Lett.34, 55–57 (2009). [CrossRef]
- K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys.11, 013049 (2009). [CrossRef]
- Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric Down-Conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett.83, 2556–2559 (1999). [CrossRef]
- 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]
- 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]
- 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]
- 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 (2002). [CrossRef]
- 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]
- H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. I. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett.91, 201108 (2007). [CrossRef]
- X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber source of Polarization-Entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett.94, 053601 (2005). [CrossRef] [PubMed]
- H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-i. Itabashi, “Generation of polarization entangledphoton pairs using silicon wirewaveguide,” Opt. Express16, 5721–5727 (2008). [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 wire waveguide,” Opt. Express16, 20368–20373 (2008). [CrossRef] [PubMed]
- P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007). [CrossRef]
- J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4, 37–40 (2010). [CrossRef]
- L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010). [CrossRef]
- M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun1, 29 (2010). [CrossRef] [PubMed]
- T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based optical frequency combs,” Science332, 555 –559 (2011). [CrossRef] [PubMed]
- F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics5, 770–776 (2011). [CrossRef]
- A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100GHz repetition rates,” Opt. Lett.37, 875–877 (2012). [CrossRef] [PubMed]
- P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett.25, 554–556 (2000). [CrossRef]
- A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express16, 4881–4887 (2008). [CrossRef] [PubMed]
- Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A82, 033801 (2010). [CrossRef]
- Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett.104, 103902 (2010). [CrossRef] [PubMed]
- M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal/idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun.282, 3518–3523 (2009). [CrossRef]
- D. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).
- T. Bååk, “Silicon oxynitride; a material for GRIN optics,” Appl. Optics21, 1069–1072 (1982). [CrossRef]
- I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am.55, 1205–1208 (1965). [CrossRef]
- D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge University Press, 2008). [CrossRef]
- P. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Optic.42, 1991–2004 (1995). [CrossRef]
- B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett.16, 177–182 (1991). [CrossRef]

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