## Frequency-bin entangled comb of photon pairs from a Silicon-on-Insulator micro-resonator |

Optics Express, Vol. 19, Issue 2, pp. 1470-1483 (2011)

http://dx.doi.org/10.1364/OE.19.001470

Acrobat PDF (1233 KB)

### Abstract

We present a quantum-mechanical theory to describe narrow-band photon-pair generation via four-wave mixing in a Silicon-on-Insulator (SOI) micro-resonator. We also provide design principles for efficient photon-pair generation in an SOI micro-resonator through extensive numerical simulations. Microring cavities are shown to have a much wider dispersion-compensated frequency range than straight cavities. A microring with an inner radius of 8 *μ*m can output an entangled photon comb of 21 pairwise-correlated peaks (42 comb lines) spanning from 1.3 *μ*m to 1.8 *μ*m. Such on-chip quantum photonic devices offer a path toward future integrated quantum photonics and quantum integrated circuits.

© 2011 Optical Society of America

## 1. Introduction

1. D. Bouwmeester, A. K. Ekert, and A. Zeilinger, *The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation*, 1st Ed., (Springer2000). [PubMed]

*χ*

^{(2)}nonlinear crystals [2

2. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. **75**, 4337 (1995). [CrossRef] [PubMed]

3. T. E. Kiess, Y. H. Shih, A. V. Sergienko, and C. O. Alley, “Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta Produced by Type-II Parametric Down-conversion,” Phys. Rev. Lett. **71**, 3893 (1993). [CrossRef] [PubMed]

*χ*

^{(3)}standard optical fibers [4

4. 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]

6. 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]

7. J. Fan, A. Migdall, and L. J. Wang, “Efficient generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. **30**, 3368 (2005). [CrossRef]

8. J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express **13**, 534 (2005). [CrossRef] [PubMed]

9. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science **320**, 646 (2008). [CrossRef] [PubMed]

*passive*guide for single photons. On the other hand, compact

*active*devices for generating photon pairs demonstrated so far include quantum dots [10

10. A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. **4**, 302 (2010). [CrossRef]

11. K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides,” Opt. Lett. **26**, 1367 (2001). [CrossRef]

14. T. Zhong, F. N. C. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled KTiOPO_{4} waveguide,” Opt. Express **17**, 12019 (2009). [CrossRef] [PubMed]

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

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

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

_{2}) enable ultrasmall bending radii (several

*μ*m) without incurring high loss, thus enabling large-density integration of SOI devices in a single platform.

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

19. 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 (2008). [CrossRef] [PubMed]

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

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

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

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

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

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

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

21. H. J. Kimble, “The quantum internet,” Nature **453**, 1023 (2008). [CrossRef] [PubMed]

**31**, 3140 (2006). [CrossRef] [PubMed]

## 2. Quantum theory of two-photon state generation via cavity-enhanced four-wave mixing

*χ*

^{(3)}] nonlinear process, in which 2 pump photons are absorbed and a pair of energy- and momentum-conserving daughter photons (referred to as signal and idler) are generated, satisfying 2

*ω*

_{p}=

*ω*

_{s}+

*ω*

_{i}and 2

*k⃗*

_{p}=

*k⃗*

_{s}+

*k⃗*

_{i}, where

*ω*

_{p,s,i}and

*k⃗*

_{p,s,i}are the photon frequencies and wavevectors, and the subscripts p, s, and i stand for pump, signal, and idler, respectively. Here we focus on a continuous-wave (CW) pump scenario, which is also the most relevant pumping scheme for a micro-resonator cavity. A treatment of the pulsed-pump case can be done by following Ref. [22

22. J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A **72**, 033801 (2005). [CrossRef]

*γ*. Strong, CW pump light at

*λ*

_{p}is injected into the microring from the bus waveguide [Fig. 1(b)]. The wavelength

*λ*

_{p}is chosen to be near the zero-dispersion wavelength

*λ*

_{ZDW}of the microring (or equivalently, a bent waveguide) so that FWM phase matching produces a broad gain spectrum without taking into account the cavity resonances. The resonator effectively acts like an active filter, enhancing its resonant frequencies while suppressing all other non-resonant ones.

23. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express **15**, 16604 (2007). [CrossRef] [PubMed]

23. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express **15**, 16604 (2007). [CrossRef] [PubMed]

23. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express **15**, 16604 (2007). [CrossRef] [PubMed]

24. We note that the optical modes in our proposed microring resonators are not ideal whispering gallery modes, in the sense that the modes have non-zero interatction with the inner wall of the ring resonators. However, the electric field strength at the inner wall is typically no more than 1% of its peak value near the outer wall (see Fig. 2). Even though this interaction is negligibly small, we used the numerically determined modes in our calculations, rather than analytical expressions of ideal whispering gallery modes. We still refer to our modes as whispering gallery modes throughout the text, but we recognize this is an approximation.

*L*of a straight cavity mode remains fixed, while for the microring WGMs it shortens slightly for shorter resonant wavelengths. Another example is the transverse mode overlap between optical fields, which for straight cavity modes can be assumed to be close to unity, but for WGMs it is generally less than 1 because the mode profiles shift slightly with wavelength. Nevertheless, for signal and idler wavelengths close to the pump the above differences are minute, and can be safely disregarded.

*L*is typically several tens of micrometers long. We further require that all optical fields participating in the FWM process are fundamental resonant modes of the cavity, satisfying:

*n*

_{gj}

*L*=

*M*

_{j}

*λ*

_{j}(j=p, s, i), where

*M*

_{j}are integer mode numbers (

*M*

_{s}<

*M*

_{p}<

*M*

_{i}) and

*n*

_{gj}are group indices for each field. The propagation direction is denoted as

*z*(which should be replaced by

*ρθ*for the microring case). Without loss of generality, the transverse spatial mode of each field is taken to be the fundamental TM mode of the cavity, and we assume that all fields are copolarized so that scalar notation can be used. The pump field is a classical wave: where Γ is the nonlinear parameter for the Si waveguide and

*P*is the circulating power inside the cavity. Note that the pump self-phase modulation term

*e*

^{−iΓPz}is explicitly included. The resonant signal field is quantized, and can be written as [25

25. M. Scholtz, 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 (2009). [CrossRef]

*A*

_{eff,s}= |∫∫|

*F*

_{s}|

^{2}

*dxdy*|

^{2}/ ∫ ∫ |

*F*

_{s}|

^{4}

*dxdy*is the effective mode area, with

*F*

_{s}being the transverse mode profile of the signal field.

*γ*

_{s}is the cavity damping rate for the signal, which represents all possible loss mechanisms including both linear loss (propagation loss, out-coupling loss) and nonlinear loss (TPA, FCA). In practice, the nonlinear loss terms play a minor role at the pump power levels we use (see Section 3 for loss estimation). Δ

*ω*

_{s}is the free spectral range (FSR),

*ω*

_{s,ms}is the

*m*

_{s}th central frequency, Ω

_{s}is the deviation from

*ω*

_{s,ms}, and

*positive*integers

*m*

_{s}=

*M*

_{p}−

*M*

_{s}and

*m*

_{i}=

*M*

_{i}−

*M*

_{p}to represent the

*relative*mode number for signal and idler, respectively.

*ζ*is a proportionality constant, and

*h.c*. stands for Hermitian conjugate. For simplicity, we only consider signal and idler frequencies close to the pump, so that a Taylor expansion of the propagation constant

*k*around the pump frequency can be employed, and we keep the expansion series up to second order (

*k*″). A straightforward calculation yields the following expression for the two-photon state:

*η*is another proportionality constant, and

*ω*

_{p}. In deriving Eq. (3), we have used the following mathematical identities:

_{s(i)}| ≪ Δ

*ω*

_{s(i)}) and (ii) all participating frequencies are close so that there is negligible difference in their group indices and the FSR can be considered constant (

*m*

_{s}=

*m*

_{i}=

*m*, Ω

_{s}= −Ω

_{i}= Ω, Δ

*ω*

_{s}= Δ

*ω*

_{i}= Δ

*ω*). The following simplified two-photon state is thus obtained:

## 3. Design principles of SOI microring resonators

*R*

_{1}). All of them dramatically affect the microring’s dispersion properties, and in particular its zero-dispersion wavelength, which has significant implications on where the pump wavelength should be. The effect of the cross section (both size and aspect ratio) on the dispersion of a straight Si waveguide has been studied extensively [30

30. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. **31**, 1295 (2006). [CrossRef] [PubMed]

*R*

_{1}(from ∞ for a straight waveguide to 3

*μ*m) to find its effect on the microring dispersion.

33. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron.11, 75 (1975). [CrossRef]

*ρ*and

*θ*are polar coordinates depicted in Fig. 1(b), with

*θ*= arctan(

*z*/

*y*) (

*y*,

*z*are Cartesian coordinates).

*v*is the direction of propagation (perpendicular to the cross section in Fig. 2). As shown in Fig. 2, a bent waveguide with inner (outer) radius

*R*

_{1}(

*R*

_{2}), width

*W*≡

*R*

_{2}−

*R*

_{1}, and height

*H*is transformed to an equivalent straight waveguide of width −

*R*

_{2}ln(

*R*

_{1}/

*R*

_{2}) and height

*H*. The refractive index of each material is also transformed according to

*n*′

_{a}=

*n*

_{a}e^{u/R2}, a = Air, Si, SiO

_{2}. For the refractive indices, we use where we have used the Sellmeier equations for air and SiO

_{2}, and the Herzberger equation for Si [34].

*λ*is the free-space wavelength in units of micrometers. Note that the equivalent straight waveguide is situated in the negative-

*u*plane.

36. Certain trade names and company products are mentioned in the text or identified in an illustration in order to specify adequately the experimental procedure and equipment used. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it necessarily imply that the products are the best available for the purpose.

*c*is the vacuum speed of light. The results are shown in Fig. 3, for a straight waveguide and two bent waveguides with its inner radius

*R*

_{1}= 8

*μ*m and 3

*μ*m, respectively. Bending the waveguide has a significant influence on both the group index and group velocity dispersion of its fundamental TE and TM modes. Bending a straight waveguide to

*R*

_{1}= 8

*μ*m changes the relative magnitude of the group index for TE and TM modes, and a further bending to

*R*

_{1}= 3

*μ*m reverses the relative positions of the group velocity dispersion curves for TE and TM modes. The bending effect on dispersion is more pronounced when

*R*

_{1}becomes comparable to the waveguide cross dimension (0.75

*μ*m in this case).

*n*′ = Δ

*ne*

^{u/R2}) than a straight waveguide (Δ

*n*=

*n*

_{Si}−

*n*

_{surroundings}) for

*u*≤ 0, where the mode is predominantly located. Furthermore, Δ

*n*′ reduces with decreasing

*R*

_{1}(or

*R*

_{2}). This means that mode confinement is strongest for a straight waveguide; as the bending increases (

*R*

_{1}decreases), the mode confinement becomes weaker. Therefore, both group index and group velocity dispersion approach that of bulk Si with decreasing

*R*

_{1}, as can be seen in Fig. 3.

*λ*

_{ZDW}) is of great interest for phase-matching considerations, we plot its dependence on the inverse of the bending curvature (1/

*R*

_{1}) in Fig. 4. We can see that the bending curvature has a dramatic effect on the location of zero-dispersion wavelengths for both TE and TM modes. At a suitable bending radius (

*R*

_{1}= 8

*μ*m), the TE and TM zero-dispersion wavelengths overlap

*k*″ curves almost completely overlap each other throughout the wavelength range from 1300 nm to 1900 nm. At a small bending radius of

*E*

_{M}(

*ρ*,

*θ*,

*x*) ∼

*e*where

^{iMθ}*M*is an integer [see Fig. 1(b)]. The FWM phase-matching requirement of linear momentum conservation in a straight geometry is adapted to angular momentum conservation in a curved geometry, which stipulates 2

*M*

_{p}=

*M*

_{s}+

*M*

_{i}. This is clearly satisfied for signal/idler pairs that are symmetrically placed around the pump mode, i.e.,

*M*

_{s/i}=

*M*

_{p}±

*m*(

*m*is an integer). Due to the dispersion inside the microring, the equally-important energy conservation requirement is not always satisfied for those adjacent modes, which is quantified by the

*frequency mismatch*[37

37. Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express **16**, 10596 (2008). [CrossRef] [PubMed]

*ω*

_{0j}(j = p, s, i) are resonant frequencies of the microring. It is clear that if all optical fields participating in FWM are resonant modes of the cavity, the FWM efficiency will be greatly enhanced. Therefore, it is important to reduce the frequency mismatch to below the cavity linewidth, in which case the FWM efficiency will suffer the least (referred to as the dispersion-compensated regime in Ref. [38

38. I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q silica microspheres,” Phys. Rev. A **76**, 043837 (2007). [CrossRef]

*R*

_{1}, we numerically simulate the resonant modes using the weak-form formulation in COMSOL developed by Oxborrow [39

39. M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Tran. Micro. Theory **55**, 1209 (2007). [CrossRef]

_{2}and Si (cf. Eqs. 12 and 13) are taken into account [40]. Frequency mismatch is then calculated via Eq. (14) for various pump wavelengths.

*R*

_{1}= 8

*μ*m, Fig. 5(a) shows that there is an optimal pump for which frequency mismatch is minimized over a large wavelength range (from 1.3

*μ*m to 1.8

*μ*m), which we denote as

*λ*

_{opt}. Any deviation from the optimal pump results in reduction of the effective FWM range. As shown in Fig. 5(b), TM and TE modes generally have different

*λ*

_{opt}, even though for

*R*

_{1}= 8

*μ*m they have the same zero-dispersion wavelength: for TM mode

*λ*

_{opt}= 1.555

*μ*m, and for TE mode

*λ*

_{opt}= 1.528

*μ*m. Nevertheless, both are very close to

*λ*

_{ZDW}= 1530nm, with negligible

*k*″ values (

*k*″ = −0.097 ps

^{2}/m for TM and

*k*″ = 0.005 ps

^{2}/m for TE). Extensive numerical study shows that this statement holds true for other bending radii as well; Fig. 5(c) and (d) depict two such examples. For

*R*

_{1}= 7

*μ*m, Fig. 5(c) shows that

*λ*

_{opt}= 1.556

*μ*m with

*k*″ = −0.08 ps

^{2}/m for TM and

*λ*

_{opt}= 1.539

*μ*m with

*k*″ = −0.007 ps

^{2}/m for TE. Figures 5(d) depicts the case for

*R*

_{1}= 5

*μ*m, where we have

*λ*

_{opt}= 1.597

*μ*m with

*k*″ = −0.17 ps

^{2}/m for TM and

*λ*

_{opt}= 1.572

*μ*m with

*k*″ = −0.026 ps

^{2}/m for TE. In all these examples, the optimal pump resides close to

*λ*

_{ZDW}, and often (but not always) in the anomalous dispersion regime (i.e.,

*k*″ < 0).

*R*

_{1}= 8

*μ*m for example. From Fig. 5(b) we can see that the effective FWM range is roughly estimated to be [1.3

*μ*m, 1.8

*μ*m], where the frequency mismatch |Δ| ≤ 4 GHz for TE modes and |Δ| ≤ 10 GHz for TM modes. Inside this 500-nm broad wavelength range, there exist 21 TE-comb and 19 TM-comb peak pairs. Each peak has a FWHM of 20 GHz, and peaks are separated by a free spectral range of about 12 nm. These are important design parameters that are relevant for experimentally testing a fabricated device.

38. I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q silica microspheres,” Phys. Rev. A **76**, 043837 (2007). [CrossRef]

*n*

_{2}= 2.5 × 10

^{−18}m

^{2}/W is the nonlinear index coefficient of Si and

*I*is the intensity of the pump circulating the cavity. Assuming a pump circulating power of 100 mW inside the microring (very high for the purpose of photon-pair production) and an effective area

*A*

_{eff}= 0.33

*μ*m

^{2}, the intensity

*I*is estimated to be about 3 × 10

^{11}W/m

^{2}. Δ

_{nl}is then estimated to be 86 MHz at a wavelength of 1550 nm, which is much less than the cavity linewidth (20 GHz, corresponding to a moderate cavity

*Q*of 10

^{4}). Therefore, the nonlinear contribution to frequency mismatch can be ignored for most practical cases.

*PL*

_{eff}, which is important in gauging the importance of nonlinear loss terms such as TPA and FCA. According to the studies done in Ref. [20

**31**, 3140 (2006). [CrossRef] [PubMed]

*PL*

_{eff}≤ 0.2 the effect of TPA and FCA remains small. In the above example, the nonlinear parameter Γ = 2

*πn*

_{2}/(

*λA*

_{eff}) ≃ 30.7 W

^{−1}m

^{−1}. The effective propagation length,

*L*

_{eff}, is determined through the cavity linewidth (FWHM = 20 GHz) and photon life-time [

*τ*= 1/(2

*π*FWHM) ≃ 8 ps] in such a cavity, and is given by

*L*

_{eff}=

*cτ*/

*n*

_{g}≃ 0.65 mm. With

*P*= 0.1 W, we estimate Γ

*PL*

_{eff}≃ 0.002. At this Γ

*PL*

_{eff}level, we can safely ignore the nonlinear loss terms such as TPA and FCA.

*χ*

^{(3)}tensor components. Here we adopt the formalism in Ref. [23

**15**, 16604 (2007). [CrossRef] [PubMed]

*χ*

^{(3)}tensor components for a microring resonator fabricated on Si (1 0 0) wafer whose coordinates are shown in Fig. 1(b). In the Cartesian coordinate system where

*x*= [1 0 0],

*y*= [0 1 0],

*z*= [0 0 1] are the crystallographic axes, the third-order nonlinear response function is given by (Eq. (39) in Ref. [23

**15**, 16604 (2007). [CrossRef] [PubMed]

*γ*

_{e}(

*γ*

_{R}) is the electronic (Raman) part of the third-order nonlinearity,

*δ*

_{ij}is the Kronecker delta,

*δ*(

*τ*) is the Dirac delta function,

*h*

_{R}(

*τ*) is the Raman response function, and

*σ ≈*1.27 is the nonlinear anisotropy at 1550 nm [23

**15**, 16604 (2007). [CrossRef] [PubMed]

*x*=

*x*,

*y*=

*ρ*cos

*θ*,

*z*=

*ρ*sin

*θ*. The transformation matrix is given by where

*a*

_{j}is the unit vector along axis j, and the (un)primed notations represent the (Cartesian) polar coordinate. The matrix

*M*is calculated to be: Tensor transformation obeys

*R*

_{1}= 8

*μ*m. Calculations show that neither TM-pump nor TE-pump-derived Raman Stokes wave occurs near any TE or TM resonances for this particular design. It is not the case, however, for the design with

*R*

_{1}= 10

*μ*m, where the TE-pump-derived Raman Stokes wave occurs too close to a TE resonance (less than 120 GHz away), violating the above rule (ii). It can still be used to enhance FWM photon pairs with a TM-pump, since its Raman Stokes wave occurs far enough from any TE resonances.

## 4. Straight cavities

*M*is the mode number,

*λ*is the resonant wavelength,

*L*is the cavity length, and

*n*

_{g}is the group index for the resonant wavelength in a straight waveguide [such as the one shown in Fig. 3(a)]. Taking into account

*λ*= 2

*πc*/

*ω*, Eq. (22) is reduced to Resonant frequency of mode number

*M*is obtained by numerically solving Eq. (23). Following the discussions in Section 3, we calculate the frequency mismatch of a straight cavity (cross dimension 750 nm by 750 nm and length

*L*= 100

*μ*m with air top cladding) for both its quasi-TE and quasi-TM modes, as shown in Fig. 6.

*always*grows with mode separation, independent of the pump wavelength. This is in stark contrast with the microring case, where there is a flat dispersion-compensated frequency range for a properly-chosen pump wavelength (i.e.,

*λ*

_{opt}). For the straight-cavity case, there are only a few resonant modes for which the frequency mismatch is below the cavity linewidth (20 GHz in our case). For those modes cavity-enhanced FWM still play a role. One can, of course, broaden the cavity linewidth to incorporate more dispersion-compensated modes, at the expense of a lower cavity

*Q*. We have also explored several other cross-sectional dimensions of a straight cavity, and find that the same conclusion holds: In general, for a given cavity linewidth, the straight-cavity design is inferior to its ring-cavity alternative for enhancing FWM.

## 5. Conclusion

## References and links

1. | D. Bouwmeester, A. K. Ekert, and A. Zeilinger, |

2. | P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. |

3. | T. E. Kiess, Y. H. Shih, A. V. Sergienko, and C. O. Alley, “Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta Produced by Type-II Parametric Down-conversion,” Phys. Rev. Lett. |

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

5. | H. Takesue and K. Inoue, “Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop,” Phys. Rev. A |

6. | 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. |

7. | J. Fan, A. Migdall, and L. J. Wang, “Efficient generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. |

8. | J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express |

9. | A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science |

10. | A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. |

11. | K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides,” Opt. Lett. |

12. | M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled KTP waveguides and bulk crystals,” Opt. Express |

13. | J. Chen, A. Pearlman, A. Ling, J. Fan, and A. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express |

14. | T. Zhong, F. N. C. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled KTiOPO |

15. | 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 |

16. | K. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S. Itabashi, “Generation of high-purity entangled photon pairs using silicon wire waveguide,” Opt. Express |

17. | S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, Ph. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express |

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

19. | 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 |

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

21. | H. J. Kimble, “The quantum internet,” Nature |

22. | J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A |

23. | Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express |

24. | We note that the optical modes in our proposed microring resonators are not ideal whispering gallery modes, in the sense that the modes have non-zero interatction with the inner wall of the ring resonators. However, the electric field strength at the inner wall is typically no more than 1% of its peak value near the outer wall (see Fig. 2). Even though this interaction is negligibly small, we used the numerically determined modes in our calculations, rather than analytical expressions of ideal whispering gallery modes. We still refer to our modes as whispering gallery modes throughout the text, but we recognize this is an approximation. |

25. | M. Scholtz, 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. |

26. | J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. |

27. | C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. |

28. | S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. |

29. | L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy, “Frequency-bin entangled photons,” Phys. Rev. A |

30. | L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. |

31. | A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguide,” Opt. Express |

32. | A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express |

33. | M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron.11, 75 (1975). [CrossRef] |

34. | E. D. Palik, |

35. | |

36. | Certain trade names and company products are mentioned in the text or identified in an illustration in order to specify adequately the experimental procedure and equipment used. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it necessarily imply that the products are the best available for the purpose. |

37. | Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express |

38. | I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q silica microspheres,” Phys. Rev. A |

39. | M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Tran. Micro. Theory |

40. | Private communications with M. Oxborrow and P. Del’Haye. |

**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: August 24, 2010

Revised Manuscript: November 2, 2010

Manuscript Accepted: November 2, 2010

Published: January 13, 2011

**Citation**

Jun Chen, Zachary H. Levine, Jingyun Fan, and Alan L. Migdall, "Frequency-bin entangled comb of photon pairs from a Silicon-on-Insulator
micro-resonator," Opt. Express **19**, 1470-1483 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-1470

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### References

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- J. Fan, A. Migdall, and L. J. Wang, “Efficient generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. 30, 3368 (2005). [CrossRef]
- J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534 (2005). [CrossRef] [PubMed]
- A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646 (2008). [CrossRef] [PubMed]
- A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nat. Photonics 4, 302 (2010). [CrossRef]
- K. Banaszek, “A. B. U’Ren, and I. A.Walmsley, “Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides,” Opt. Lett. 26, 1367 (2001). [CrossRef]
- M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled KTP waveguides and bulk crystals,” Opt. Express 15, 7479 (2007). [CrossRef] [PubMed]
- J. Chen, A. Pearlman, A. Ling, J. Fan, and A. Migdall, “A versatile waveguide source of photon pairs for chipscale quantum information processing,” Opt. Express 17, 6727 (2009). [CrossRef] [PubMed]
- T. Zhong, F. N. C. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fibercoupled periodically poled KTiOPO4 waveguide,” Opt. Express 17, 12019 (2009). [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. Express 14, 12388 (2006). [CrossRef] [PubMed]
- K. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S. Itabashi, “Generation of high-purity entangled photon pairs using silicon wire waveguide,” Opt. Express 16, 20368 (2008). [CrossRef] [PubMed]
- S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, Ph. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17, 16558 (2009). [CrossRef] [PubMed]
- P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554 (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. Express 16, 4881 (2008). [CrossRef] [PubMed]
- Q. Lin, and G. P. Agrawal, “Silicon waveguides for creating quantum-correlated photon pairs,” Opt. Lett. 31, 3140 (2006). [CrossRef] [PubMed]
- H. J. Kimble, “The quantum internet,” Nature 453, 1023 (2008). [CrossRef] [PubMed]
- J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72, 033801 (2005). [CrossRef]
- Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604 (2007). [CrossRef] [PubMed]
- We note that the optical modes in our proposed microring resonators are not ideal whispering gallery modes, in the sense that the modes have non-zero interatction with the inner wall of the ring resonators. However, the electric field strength at the inner wall is typically no more than 1% of its peak value near the outer wall (see Fig. 2). Even though this interaction is negligibly small, we used the numerically determined modes in our calculations, rather than analytical expressions of ideal whispering gallery modes. We still refer to our modes as whispering gallery modes throughout the text, but we recognize this is an approximation.
- M. Scholtz, 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 (2009). [CrossRef]
- J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999). [CrossRef]
- C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304 (2000). [CrossRef] [PubMed]
- S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009). [CrossRef]
- L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. P. Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010). [CrossRef]
- L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31, 1295 (2006). [CrossRef] [PubMed]
- A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguide,” Opt. Express 14, 4357 (2006). [CrossRef] [PubMed]
- A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express 18, 1904 (2010).
- M. Heiblum, and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75 (1975). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press 1985), p. 548.
- http://www.comsol.com
- Certain trade names and company products are mentioned in the text or identified in an illustration in order to specify adequately the experimental procedure and equipment used. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it necessarily imply that the products are the best available for the purpose.
- Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express 16, 10596 (2008). [CrossRef] [PubMed]
- I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q silica microspheres,” Phys. Rev. A 76, 043837 (2007). [CrossRef]
- M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theory Tech. 55, 1209 (2007). [CrossRef]
- Private communications with M. Oxborrow and P. Del’Haye,

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