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 μm can output an entangled photon comb of 21 pairwise-correlated peaks (42 comb lines) spanning from 1.3 μm to 1.8 μm. 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

Quantum-entangled photon pairs play a key role in quantum information processing applications such as quantum key distribution, quantum teleportation, and linear optical quantum computing [1]. Traditional sources produce such photon pairs using spontaneous parametric down conversion (SPDC) in bulk χ(2) nonlinear crystals [2, 3], and more recently, spontaneous four-wave mixing (FWM) in χ(3) standard optical fibers [46] and photonic crystal fibers [7,8]. However, the common drawback for all these sources is that they do not offer an easy path to scalable integration. As quantum optical circuits become increasingly complex to handle more and more qubits, it is highly desirable to develop a compact and efficient way to produce and propagate photonic qubits while preserving quantum coherence among them. Silica-on-Silicon waveguide quantum circuits [9] have emerged as an efficient passive guide for single photons. On the other hand, compact active devices for generating photon pairs demonstrated so far include quantum dots [10], SPDC in quasi-phase-matched microphotonic waveguides [1114], and FWM in Silicon-on-Insulator (SOI) nanophotonic straight waveguides [15, 16] and microring resonators [17]. Among these, SOI devices appear most integration-compatible by leveraging the mature Si fabrication technology from the integrated circuit industry. As the bending loss decreases exponentially with increasing core-cladding refractive index difference, the high index contrast between crystalline Si and its surroundings (air and/or SiO2) enable ultrasmall bending radii (several μm) without incurring high loss, thus enabling large-density integration of SOI devices in a single platform.

Both SOI straight waveguides and microring resonators are of great interest: the former are capable of generating broadband FWM photon pairs, whereas the latter output narrowband photons. Classically, the field enhancement effect by a microring resonator has been studied in Ref. [18] with GaAs microring resonators, and in Ref. [19] with SOI microring resonators. Quantum mechanically, there are several experimental demonstrations of photon pair generation in straight SOI waveguides [15, 16], and a quantum theoretical description has also appeared [20]. For mircoring resonators, Ref. [17] demonstrates photon pair generation experimentally, but uses classical arguments from Ref. [18] to explain their experimental results which leads to certain discrepancies. This highlights the need for a quantum theory for describing resonator-enhanced photon-pair generation in SOI micro-cavities, which we present in Section 2. Our theory is also markedly different from the straight-waveguide quantum theory in Ref. [20] in that our theory deals with FWM in a resonant cavity while Ref. [20] treats FWM in a non-resonant structure, with the emission bandwidth from the former being at least 3 orders of magnitude narrower than the latter. This helps address the issue of mode-mismatch between photonic and material qubit systems by narrowing the difference in inherent bandwidths, which is now an area of much research interest [21]. In addition, instead of using Heisenburg picture with operator evolution as done in Ref. [20], our theory uses Schrödinger picture with state vector evolution, which facilitates explicit calculation of the final two-photon wave function and spectral characteristics. Section 3 is devoted to discussing important design principles for SOI microring resonators developed through extensive numerical modelling. Straight-cavity resonators are studied in Section 4, and are found to be generally inferior to their ring-shaped counterparts in terms of enhancing FWM photon-pair production. We conclude with an outlook for future work in Section 5.

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

FWM is a third-order [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 2ωp = ωs + ωi and 2k⃗p = k⃗s + k⃗i, where ωp,s,i and k⃗p,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]. Emitted simultaneously as a pair, signal and idler photons are strongly time-energy entangled, forming a two-photon state.

The integrated SOI devices under consideration consist of a microring side-coupled to a straight bus waveguide [Fig. 1(a)]. The bus waveguide can be fabricated on the same chip, but it can also be a tapered optical fiber. The coupling strength between the two can be tuned by changing their separation, which also tunes the cavity damping rate γ. Strong, CW pump light at λp is injected into the microring from the bus waveguide [Fig. 1(b)]. The wavelength λp is chosen to be near the zero-dispersion wavelength λZDW 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.

 

Fig. 1 (a) Schematic of a microring resonator side-coupled to a bus waveguide, both integrated on a SOI chip. (b) Top-down view of photon pair production in the SOI device shown in (a). Pump is injected into the microring via the bus waveguide; copolarized photon pairs are generated and resonantly enhanced and evanescently coupled out of the microring. Waves propagate in the z direction in the bus waveguide. Inside the microring, there exist two polarization eigenmodes: TM (Electric field perpendicular to the plane of propagation) and TE (Electric field in the plane of propagation but perpendicular to the propagation direction). R1, inner ring radius; R2, outer ring radius. (c) Cross section of both the microring and the bus waveguide. The crystallographic axes are designated for the bus waveguide only. (d) An entangled comb of photon pairs is generated when pump frequency is tuned to mode number Mp. A signal photon occupying mode ms can always find its partner idler photon symmetrically placed around pump occupying mode mi = ms. Also shown is the simulated signal output with a relative mode number ms = 1, which has a full width at half-maximum of 20 GHz for a cavity damping rate of 31.25 GHz.

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For simplicity we focus on the case where both the microring and the bus waveguide share the same cross-sectional dimensions. This monolithic design will minimize spatial mode mismatch between the microring and the bus waveguide if the bending radius is not too small. The cross section is shown in Fig. 1(c), where a 750 nm by 750 nm square-shaped Si block sits on top of a SiO2 buried oxide (BOX) layer (with a thickness ≥ 0.5 μm to prevent leakage into the Si substrate) with an air top cladding. The bus waveguide is fabricated along the [0 1̄ 1] direction on a (1 0 0) silicon wafer for cleaving convenience. This particular geometry is chosen because numerical simulations show that the microring’s zero-dispersion wavelengths for both quasi-TE [transverse electric, Eρ in Fig. 1(b)] and quasi-TM [transverse magnetic, Ex in Fig. 1(b)] fundamental modes are within the 1550 nm telecommunications band (see Section 3). Although shown as a ring, the microcavity could also be shaped as a racetrack to improve coupling between the resonator and the bus waveguide. Both the microring and the bus waveguide are multimode over a broad spectral range due to the deep etch down to the BOX layer; however, higher-order modes have quite different spatial mode profiles, polarization, and dispersion characteristics compared to the fundamental modes. As a result, FWM involving higher-order modes have negligible contributions to the production of correlated photon pairs inside the wavelength range of interest. Thus we can limit our study of the generation of photon pairs to fundamental modes only.

There are a myriad of nonlinear optical processes that may occur when intense pump light circulates around the SOI microring. These include FWM, Raman scattering, Brillouin scattering, two-photon absorption (TPA), and free carrier absorption (FCA). A general description of these processes can be found in Ref. [23]. For the purpose of generating photon pairs, we are concerned only with spontaneous FWM, as the pump power is kept low to avoid stimulated scattering. Processes such as TPA and FCA are only efficient when the pump power is high, and therefore can be neglected in the low pump power regime. Brillouin scattering is two orders of magnitude weaker in strength compared with first-order Raman scattering in Si [23], and therefore is negligible here. Due to the intrinsic crystal symmetry of Si and the ring structure under consideration, Raman scattering is absent for the process wherein a quasi-TM pump transfers its energy to another quasi-TM mode (TM-TM), but is allowed for all other processes such as TM-TE, TE-TM, and TE-TE to transfer part of the pump energy to a mode that is 15.6 THz down-shifted from the pump (the so-called Stokes wave) with a full width at half-maximum (FWHM) of 105 GHz [23]. Thanks to the narrow-band nature of Raman scattering in Si, it is possible to neglect Raman scattering altogether by simply designing the microring cavity to have its resonances detuned from the pump-derived Raman Stokes wave (see Section 3 for details).

The following theory is developed with general applicability to all cavities, including straight cavities and microring cavities as special cases. The notation, however, uses that of a straight cavity as default. When applied to a microring cavity, care should be taken as there are subtleties of whispering gallery modes (WGM) [24] that are not entirely captured in the following formalism. For example, the physical length 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.

The physical cavity length 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: ngjL = Mjλj (j=p, s, i), where Mj are integer mode numbers (Ms < Mp < Mi) and ngj 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:

Ep(+)(z,t)=Epei[kp(ωp)zωpt]eiΓPz,
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 eiΓPz is explicitly included. The resonant signal field is quantized, and can be written as [25]:
Es()(z,t)=h¯ωs2ɛ0nscAeff,sγsΔωs2πmsdΩsas(ωs,ms+Ωs)γs/2iΩsei[ksz(ωs,ms+Ωs)t].

The idler equation can be obtained by exchanging the subscripts s and i. Here Aeff,s = |∫∫|Fs|2dxdy|2 / ∫ ∫ |Fs|4 dxdy is the effective mode area, with Fs being the transverse mode profile of the signal field. γs 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). Δωs is the free spectral range (FSR), ωs,ms is the msth central frequency, Ωs is the deviation from ωs,ms, and as is the creation operator for signal. Lastly, we use positive integers ms = MpMs and mi = MiMp to represent the relative mode number for signal and idler, respectively.

Using first-order perturbation theory, the two-photon state |Ψ=(ih¯)1dtHI|0 can be calculated through the interaction Hamiltonian HI=ζχ(3)L0dzEp(+)Ep(+)Es()Ei()+h.c. where 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:

|Ψ=ηLmsmidΩsdΩiγsγiδ(msΔωs+miΔωi+Ωs+Ωi)(γs/2iΩs)(γi/2iΩi)eiL[k4(msΔωsΩs)2+k4(miΔωi+Ωi)2+ΓP]as(ωs,ms+Ωs)ai(ωi,mi+Ωi)|0sinc{L[k4(msΔωsΩs)2+k4(miΔωi+Ωi)2+ΓP]},
where η is another proportionality constant, and k=d2kdω2|ω=ωp is the group velocity dispersion at the pump frequency ωp. In deriving Eq. (3), we have used the following mathematical identities:
eiutdt=2πδ(u),
L0eiβxdx=LeiβL/2sinc(βL/2).

We can further simplify Eq. (3) by taking into account the following reasonable assumptions: (i) the cavity bandwidth is much smaller than the FSR (|Ωs(i)| ≪ Δωs(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 (ms = mi = m, Ωs = −Ωi = Ω, Δωs = Δωi = Δω). The following simplified two-photon state is thus obtained:

|Ψ=ηLmdΩγsγieiL[k(mΔωΩ)2/2+ΓP](γs/2iΩ)(γi/2+iΩ)sinc{L[k(mΔωΩ)2/2+ΓP]}as(ωpmΔω+Ω)ai(ωp+mΔωΩ)|0.

Equation 6 demonstrates the energy-entangled nature of the two-photon state through the arguments of its two creation operators (which sum up to twice the pump frequency). The two-photon state also exhibits a “frequency-bin entangled comb” structure with discrete spectral peaks that are correlated pairwise in energy, all in a superposition state. The frequency-bin entanglement notion is exactly analogous to time-bin entanglement [26] as a discrete form of the more familiar continuous frequency entanglement [27]. Photon pairs in Eq. (6) can be thought of as existing in discrete “Schmidt modes” [27]: If one photon is found to be in the mode |ωpmΔω + Ω〉, then its partner must be found in the mode |ωp + mΔω − Ω〉. The “comb” term signifies the nearly equidistant nature of these mode pairs when the dispersion in the cavity is low (or compensated). We emphasize that the frequency-bin entangled state introduced here is different from the ones recently demonstrated in Ref. [28] (two-level frequency-entangled qubits) and Ref. [29] (discrete, one-to-many frequency correlation). In comparison, our state features multi-level, one-to-one frequency correlation with explicitly-generated (instead of mathematically decomposed) Schmidt mode pairs.

Although simplified, Eq. (6) is still useful in most practical cases, and can be used to calculate the biphoton production rate (∼ 〈Ψ|Ψ〉), the single-photon spectrum, and the coincidence spectrum. We calculate the single-photon spectrum below as an example. The spectrum of signal photons is obtained via S(ωs)=Ψ|as(ωs)as(ωs)|Ψ. Using Eq. (6), we obtain:

S(ωs)=(ηL)2mγsγisinc2{L[k(ωpωs)2/2+ΓP]}|γs/2i(ωsωp+mΔω)|2|γi/2+i(ωsωp+mΔω)|2.

The idler photon spectrum can be derived in a similar fashion:

S(ωi)=Ψ|ai(ωi)ai(ωi)|Ψ=(ηL)2mγsγisinc2{L[k(ωiωp)2/2+ΓP]}|γs/2i(ωpωi+mΔω)|2|γi/2+i(ωpωi+mΔω)|2.

We can see that the single-photon spectrum is defined by two frequency characteristics that are wildly different. First of all, the sinc function defines the frequency range in which the FWM phase matching condition is satisfied. It can extend to almost 100 THz for some cases (see Fig. 5 for examples). On the other hand, the product of the two Lorentzians (a characteristic of the cavity) defines the narrow-band output of each resonant peak under the sinc-function envelope. Comparison between Eq. (7) and Eq. (8) leads to the observation that signal and idler spectra consist of identically-shaped spectral peaks that are pairwise energy-entangled, forming an entangled photon comb. Idler damping (γi) plays a role in shaping the signal spectrum, and vice versa. Each comb peak has a FWHM that is related to the cavity damping rate through: FWHM=21γ0.64γ, where we assume γs = γi = γ. A numerically simulated signal peak for ms = 1 is shown in Fig. 1(d), where γ = 31.25 GHz is assumed (which leads to a resonant peak FWHM of 20 GHz).

 

Fig. 5 (a) Frequency mismatch for TM modes for a microring resonator of R1 = 8μm. Blue, λp = 1.498 μm with Mp = 114; red, λp = 1.555 μm with Mp = 109; black, λp = 1.616 μm with Mp = 104. One can see that Mp = 109 corresponds to the optimal pump mode. Frequency mismatch when pump is chosen optimally for TE (black) and TM (red) for several different bending radii: (b) R1 = 8μm, Mp = 111 for TE, Mp = 109 for TM; (c) R1 = 7μm, Mp = 97 for TE, Mp = 96 for TM; and (d) R1 = 5μm, Mp = 69 for TE, Mp = 68 for TM. Optimal pump wavelengths are labelled on the figures.

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3. Design principles of SOI microring resonators

A microring resonator has several key design parameters: the cross section dimensions and the bending curvature (quantified by the ring inner radius R1). 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 [3032]. However, the effect of the bending curvature on dispersion has not been fully explored, to the best of our knowledge. Here we take the ring’s cross section to be fixed at 750 nm ×750 nm, and gradually change R1 (from ∞ for a straight waveguide to 3 μm) to find its effect on the microring dispersion.

The microring is modelled as a bent waveguide, and a conformal transformation approach is used to transform the bent waveguide to its equivalent straight waveguide [33]. The procedure is illustrated in Fig. 2, where the conformal transformation is described by

u=R2ln(ρ/R2),
v=R2θ.
Here ρ and θ are polar coordinates depicted in Fig. 1(b), with ρ=y2+z2 and 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 R1 (R2), width WR2R1, and height H is transformed to an equivalent straight waveguide of width −R2 ln(R1/R2) and height H. The refractive index of each material is also transformed according to na = naeu/R2, a = Air, Si, SiO2. For the refractive indices, we use
nAir=1,
nSiO2=1+0.6961663λ2λ20.06840432+0.4079426λ2λ20.11624142+0.8974794λ2λ29.8961612,
nSi=3.41906+0.123172λ20.028+0.0265456(λ20.028)22.66511×108λ2+5.45852×1014λ2,
where we have used the Sellmeier equations for air and SiO2, 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.

 

Fig. 2 Conformal transformation from a bent waveguide to its equivalent straight waveguide, along with the fundamental TE mode shape (R 1 = 8μm, λ = 1.528 μm) in its corresponding coordinate.

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We use a commercial finite-element solver (COMSOL [35, 36]) to numerically calculate the eigenmodes and propagation constants for each waveguide, from which we obtain the group index (ng=cdkdω) and the group velocity dispersion (k=d2kdω2) through numerical differentiation, where 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 R1 = 8μm and 3μm, 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 R1 = 8μm changes the relative magnitude of the group index for TE and TM modes, and a further bending to R1 = 3μm reverses the relative positions of the group velocity dispersion curves for TE and TM modes. The bending effect on dispersion is more pronounced when R1 becomes comparable to the waveguide cross dimension (0.75 μm in this case).

 

Fig. 3 Numerically simulated group index ng and group velocity dispersion k″ for (a) straight waveguide, (b) bent waveguide with R1 = 8μm, and (c) bent waveguide with R1 = 3μm. TE mode: black; TM mode: red. The curves for bulk Si are plotted in blue for reference.

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The above bending effect can be intuitively understood if we notice from Fig. 2 that the equivalent straight waveguide has a smaller index contrast (Δn′ = Δneu/R2) than a straight waveguide (Δn = nSinsurroundings) for u ≤ 0, where the mode is predominantly located. Furthermore, Δn′ reduces with decreasing R1 (or R2). This means that mode confinement is strongest for a straight waveguide; as the bending increases (R1 decreases), the mode confinement becomes weaker. Therefore, both group index and group velocity dispersion approach that of bulk Si with decreasing R1, as can be seen in Fig. 3.

Since the zero-dispersion wavelength (λZDW) is of great interest for phase-matching considerations, we plot its dependence on the inverse of the bending curvature (1/R1) 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 (R1 = 8μm), the TE and TM zero-dispersion wavelengths overlap (λZDWTE=λZDWTM=1530nm). This can also be seen in Fig. 3(b), as the TE and TM k″ curves almost completely overlap each other throughout the wavelength range from 1300 nm to 1900 nm. At a small bending radius of R1=3μm,λZDWTE far exceeds λZDWTM by over 50 nm. We also see that bending affects TE modes more than TM modes, as the slope for λZDWTEvs.1/R1 is larger than that for λZDWTMvs.1/R1. This is because TE polarization is within the bending plane, while TM polarization is perpendicular to it, making the two polarizations see the bending effect differently. The above information can help us determine where the optimal pump wavelength should be for a microring resonator of a given size, as can be seen in the next part of our simulation.

 

Fig. 4 Zero-dispersion wavelength vs. the inverse of the bending radius of an SOI waveguide for both TE (black dots) and TM (red squares) modes.

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For an azimuthally symmetric resonator such as a microring, each resonant WGM mode exhibits a unique angular momentum with a field distribution of EM(ρ,θ,x) ∼ eiMθ where 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 2Mp = Ms + Mi. This is clearly satisfied for signal/idler pairs that are symmetrically placed around the pump mode, i.e., Ms/i = Mp ± 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]:

Δ=12π(2ω0pω0sω0i),
where ω0j (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]). For each R1, we numerically simulate the resonant modes using the weak-form formulation in COMSOL developed by Oxborrow [39]. An iterative routine is needed to arrive at self-consistent resonances when the wavelength-dependent refractive indices of SiO2 and Si (cf. Eqs. 12 and 13) are taken into account [40]. Frequency mismatch is then calculated via Eq. (14) for various pump wavelengths.

Typical results are shown in Fig. 5 for several values of the bending radii. For R1 = 8μm, Fig. 5(a) shows that there is an optimal pump for which frequency mismatch is minimized over a large wavelength range (from 1.3 μm to 1.8 μm), which we denote as λopt. 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 λopt, even though for R1 = 8μm they have the same zero-dispersion wavelength: for TM mode λopt = 1.555μm, and for TE mode λopt = 1.528μm. Nevertheless, both are very close to λZDW = 1530nm, with negligible k″ values ( k ″ = −0.097 ps2/m for TM and k″ = 0.005 ps2/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 R1 = 7μm, Fig. 5(c) shows that λopt = 1.556μm with k″ = −0.08 ps2/m for TM and λopt = 1.539μm with k″ = −0.007 ps2/m for TE. Figures 5(d) depicts the case for R1 = 5μm, where we have λopt = 1.597μm with k″ = −0.17 ps2/m for TM and λopt = 1.572μm with k″ = −0.026 ps2/m for TE. In all these examples, the optimal pump resides close to λZDW, and often (but not always) in the anomalous dispersion regime (i.e., k″ < 0).

Take an SOI microring with R1 = 8μm for example. From Fig. 5(b) we can see that the effective FWM range is roughly estimated to be [1.3 μm, 1.8 μm], 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.

In addition to the linear frequency mismatch in Eq. (14), there is a nonlinear contribution to frequency mismatch due to the intensity-dependent refractive index [38], which is given by

Δnl=2n2In(ωp)ωp2π,
where n2 = 2.5 × 10−18 m2/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 Aeff = 0.33μm2, the intensity I is estimated to be about 3 × 1011 W/m2. Δnl 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 104). Therefore, the nonlinear contribution to frequency mismatch can be ignored for most practical cases.

Let us also estimate the product ΓPLeff, which is important in gauging the importance of nonlinear loss terms such as TPA and FCA. According to the studies done in Ref. [20], as long as ΓPLeff ≤ 0.2 the effect of TPA and FCA remains small. In the above example, the nonlinear parameter Γ = 2πn2/(λAeff) ≃ 30.7 W−1 m−1. The effective propagation length, Leff, is determined through the cavity linewidth (FWHM = 20 GHz) and photon life-time [τ = 1/(2πFWHM) ≃ 8 ps] in such a cavity, and is given by Leff = /ng ≃ 0.65 mm. With P = 0.1 W, we estimate ΓPLeff ≃ 0.002. At this ΓPLeff level, we can safely ignore the nonlinear loss terms such as TPA and FCA.

Raman scattering in an SOI microring deserves some special attention, partly because of the ring geometry which mandates transformation of χ(3) tensor components. Here we adopt the formalism in Ref. [23] to systematically derive the transformed 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]):

Rijkl(3)(τ)=γeδ(τ)[σ3(δijδkl+δikδjl+δilδjk)+(1σ)δijδjkδkl]+γRhR(τ)(δikδjl+δilδjk2δijδjkδkl).
Here γe (γR) is the electronic (Raman) part of the third-order nonlinearity, δij is the Kronecker delta, δ(τ) is the Dirac delta function, hR(τ) is the Raman response function, and σ ≈ 1.27 is the nonlinear anisotropy at 1550 nm [23].

The ring geometry requires that the nonlinear response function be transformed into the polar coordinate system, where x = x, y = ρ cosθ, z = ρ sinθ. The transformation matrix is given by

[axaρaθ]=[MxxMxyMxzMρxMρyMρzMθxMθyMθz][axayaz],
where aj is the unit vector along axis j, and the (un)primed notations represent the (Cartesian) polar coordinate. The matrix M is calculated to be:
[MxxMxyMxzMρxMρyMρzMθxMθyMθz]=[1000cosθsinθ0sinθρcosθρ].
Tensor transformation obeys Rqrst(3)=Rijkl(3)MqiMrjMskMtl, where Einstein summation convention is assumed. Using Eq. (18), we obtain the four nonlinear response components pertaining to the TE and TM modes in the polar coordinate system:
Rρρρρ(3)=γeδ(τ)(cos4θ+sin4θ+2σsin2θcos2θ)+4γRhR(τ)sin2θcos2θ,
Rxxxx(3)=γeδ(τ),
Rxρρx(3)=Rρxxρ(3)=σ3γeδ(τ)+γRhR(τ).

One can see that Raman component is absent only for the TM-TM process (corresponding to Rxxxx(3) and is present for all other processes (TM-TE, TE-TM, TE-TE). The design rule for avoiding Raman scattering can therefore be summarized as: (i) for TM pump, there is no TE resonance 15.6 THz away; or (ii) for TE pump, there is neither a TE nor TM resonance 15.6 THz away. Both rules are satisfied, for example, with the design of R1 = 8μm. 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 R1 = 10μm, 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

For a straight cavity, its cavity length is fixed (i.e., independent of the mode number) in contrast to the varying cavity lengths for WGM modes. Therefore, we can calculate the resonances of a straight cavity by numerically solving the following nonlinear equation:

ngL=Mλ,
where M is the mode number, λ is the resonant wavelength, L is the cavity length, and ng is the group index for the resonant wavelength in a straight waveguide [such as the one shown in Fig. 3(a)]. Taking into account ng=cdkdω and λ = 2πc/ω, Eq. (22) is reduced to
ωdkdω=2πML.
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 = 100μm with air top cladding) for both its quasi-TE and quasi-TM modes, as shown in Fig. 6.

 

Fig. 6 Frequency mismatch in a straight cavity for quasi-TE modes with λp = 1.407 μm (solid blue), 1.516 μm (solid red), and 1.649 μm (solid black), and quasi-TM modes with λp = 1.405 μm (hollow blue), 1.514 μm (hollow red), and 1.646 μm (hollow black).

Download Full Size | PPT Slide | PDF

It can be seen that the frequency mismatch 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., λopt). 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

We have developed a quantum mechanical theory for the two-photon state generated via spontaneous FWM inside an SOI micro-cavity. The two-photon state is shown to be a frequency-bin entangled comb of photon pairs. We have also provided practical design principles for SOI microring-based photon-pair sources through extensive numerical simulations. More specifically, an SOI microring with an inner radius of 8 μm can output an entangled photon comb of 21 pairwise-correlated peaks (42 comb lines) spanning from 1.3 μm to 1.8 μm. Microring resonators are found to be superior than straight cavities at enhancing FWM photon-pair generation, as the latter lack a wide dispersion-compensated wavelength region. Such integrated-optic devices are a building block of future quantum integrated circuits, which consist of passive waveguides, interferometers, modulators, and quantum logic gates etc., all integrated on a single chip. We believe Silicon-on-Insulator is the platform of choice. Future research is aimed at fabricating the proposed devices, and experimentally demonstrating cavity-enhanced frequency-bin entangled photon-pair comb generation in SOI microring resonators.

Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

References and links

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

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]  

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]  

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 70, 031802(R) (2004). [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]  

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]  

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 15, 7479 (2007). [CrossRef]   [PubMed]  

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 17, 6727 (2009). [CrossRef]   [PubMed]  

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 KTiOPO4 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]  

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]  

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]  

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]  

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.

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]  

26. 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]  

27. 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]  

28. S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009). [CrossRef]  

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 82, 013804 (2010). [CrossRef]  

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

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 14, 4357 (2006). [CrossRef]   [PubMed]  

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 18, 1904 (2010).

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, Handbook of Optical Constants of Solids, (Academic Press1985), p. 548.

35. http://www.comsol.com

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 16, 10596 (2008). [CrossRef]   [PubMed]  

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]  

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]  

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

References

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  • |
  • |

  1. D. Bouwmeester, A. K. Ekert, and A. Zeilinger, The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, 1st Ed., (Springer 2000).
    [PubMed]
  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]
  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]
  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 70, 031802(R) (2004).
    [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]
  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,” Nat. Photonics 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]
  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 15, 7479 (2007).
    [CrossRef] [PubMed]
  13. 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]
  14. 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]
  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]
  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]
  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]
  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]
  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.
  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]
  26. 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]
  27. 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]
  28. S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
    [CrossRef]
  29. 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]
  30. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31, 1295 (2006).
    [CrossRef] [PubMed]
  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 14, 4357 (2006).
    [CrossRef] [PubMed]
  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 18, 1904 (2010).
  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, Handbook of Optical Constants of Solids (Academic Press 1985), p. 548.
  35. http://www.comsol.com
  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 16, 10596 (2008).
    [CrossRef] [PubMed]
  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]
  39. 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]
  40. Private communications with M. Oxborrow and P. Del’Haye,

2010

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]

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]

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).

2009

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]

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[CrossRef]

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]

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]

2008

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]

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]

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

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]

2007

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]

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]

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]

2006

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]

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

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]

2005

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]

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]

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]

2004

. 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 70, 031802(R) (2004).
[CrossRef]

2002

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]

2001

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]

2000

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]

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]

1999

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]

1995

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]

1993

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]

1975

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

Absil, P. P.

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]

Agha, I. H.

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]

Agrawal, G. P.

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]

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

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

Alley, C. O.

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]

Baets, R. G.

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]

Banaszek, K.

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]

Battle, P.

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]

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]

Beausoleil, R. G.

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]

Benson, O.

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]

Bogaerts, W.

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]

Brendel, J.

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]

Chen, J.

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]

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]

Cho, P. S.

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]

Clemmen, S.

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]

Cryan, M. J.

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]

Cussey, J.

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]

Duligall, J.

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]

Dwir, B.

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]

Eberly, J. H.

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]

Emplit, P.

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]

Emplit, Ph.

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]

Faist, J.

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]

Fan, J.

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]

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

Fedrizzi, A.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[CrossRef]

Felici, M.

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]

Fiorentino, M.

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]

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]

Foster, M. A.

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).

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]

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]

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]

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]

Fukuda, H.

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]

Fulconis, J.

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]

Gaeta, A. L.

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).

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]

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]

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]

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]

Gallo, P.

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]

Gisin, N.

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]

Harada, K.

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]

Harris, J. H.

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

Heiblum, M.

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

Ho, P.-T.

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]

Hryniewicz, J. V.

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]

Huy, K. P.

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]

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]

Inoue, K.

. 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 70, 031802(R) (2004).
[CrossRef]

Itabashi, S.

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]

Johnson, T. J.

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]

Joneckis, L. G.

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]

Kapon, E.

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]

Kiess, T. E.

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]

Kimble, H. J.

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

Koch, L.

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]

Kumar, P.

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]

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]

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]

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]

Kwiat, P. G.

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]

Langford, N. K.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[CrossRef]

Law, C. K.

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]

Lee, K. F.

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]

Li, X.

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]

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]

Lin, Q.

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]

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]

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

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

Ling, A.

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]

Lipson, M.

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).

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]

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]

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]

Little, B. E.

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]

Manolatou, C.

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]

Massar, S.

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]

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]

Mattle, K.

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]

Merolla, J.-M.

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]

Michael, C. P.

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]

Migdall, A.

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]

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

Mohan, A.

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]

Munro, M. W.

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]

Nguyen, A. T.

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]

O’Brien, J. L.

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]

Okawachi, Y.

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]

Olislager, L.

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]

Oxborrow, M.

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]

Painter, O. J.

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]

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]

Pearlman, A.

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]

Perahia, R.

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]

Politi, A.

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]

Ramelow, S.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[CrossRef]

Rarity, J. G.

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]

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]

Ratschbacher, L.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[CrossRef]

Roberts, T. D.

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]

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]

Rudra, A.

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]

Russell, P. S.

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]

Salem, R.

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).

Schmidt, B. S.

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]

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]

Scholtz, M.

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]

Sergienko, A. V.

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]

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]

Sharping, J. E.

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]

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]

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]

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]

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]

Shih, Y.

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]

Shih, Y. H.

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]

Spillane, S. M.

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]

Takesue, H.

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]

. 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 70, 031802(R) (2004).
[CrossRef]

Tittel, W.

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]

Tokura, Y.

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]

Tsuchizawa, T.

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]

Turner, A. C.

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]

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]

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]

Turner-Foster, A. C.

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).

Voss, P. L.

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]

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]

Wadsworth, W. J.

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]

Walmsley, I. A.

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]

Wang, L. J.

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

Watanabe, T.

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]

Weinfurter, H.

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]

Wilson, R. A.

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]

Wong, F. N. C.

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]

Yamada, K.

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]

Yin, L.

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

Yu, S.

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]

Zbinden, H.

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]

Zeilinger, A.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[CrossRef]

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]

Zhong, T.

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]

IEEE J. Quantum Electron.

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

IEEE Photon. Technol. Lett.

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]

IEEE Trans. Microwave Theory Tech.

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]

Nat. Photonics

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]

Nature

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

Opt. Commun.

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]

Opt. Express

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]

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).

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]

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]

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]

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]

Opt. Lett.

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

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]

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

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]

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

Phys. Rev. A

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]

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]

. 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 70, 031802(R) (2004).
[CrossRef]

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]

Phys. Rev. Lett.

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]

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]

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]

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]

Science

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]

Other

D. Bouwmeester, A. K. Ekert, and A. Zeilinger, The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, 1st Ed., (Springer 2000).
[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.

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.

Private communications with M. Oxborrow and P. Del’Haye,

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic of a microring resonator side-coupled to a bus waveguide, both integrated on a SOI chip. (b) Top-down view of photon pair production in the SOI device shown in (a). Pump is injected into the microring via the bus waveguide; copolarized photon pairs are generated and resonantly enhanced and evanescently coupled out of the microring. Waves propagate in the z direction in the bus waveguide. Inside the microring, there exist two polarization eigenmodes: TM (Electric field perpendicular to the plane of propagation) and TE (Electric field in the plane of propagation but perpendicular to the propagation direction). R1, inner ring radius; R2, outer ring radius. (c) Cross section of both the microring and the bus waveguide. The crystallographic axes are designated for the bus waveguide only. (d) An entangled comb of photon pairs is generated when pump frequency is tuned to mode number Mp. A signal photon occupying mode ms can always find its partner idler photon symmetrically placed around pump occupying mode mi = ms. Also shown is the simulated signal output with a relative mode number ms = 1, which has a full width at half-maximum of 20 GHz for a cavity damping rate of 31.25 GHz.

Fig. 5
Fig. 5

(a) Frequency mismatch for TM modes for a microring resonator of R1 = 8μm. Blue, λp = 1.498 μm with Mp = 114; red, λp = 1.555 μm with Mp = 109; black, λp = 1.616 μm with Mp = 104. One can see that Mp = 109 corresponds to the optimal pump mode. Frequency mismatch when pump is chosen optimally for TE (black) and TM (red) for several different bending radii: (b) R1 = 8μm, Mp = 111 for TE, Mp = 109 for TM; (c) R1 = 7μm, Mp = 97 for TE, Mp = 96 for TM; and (d) R1 = 5μm, Mp = 69 for TE, Mp = 68 for TM. Optimal pump wavelengths are labelled on the figures.

Fig. 2
Fig. 2

Conformal transformation from a bent waveguide to its equivalent straight waveguide, along with the fundamental TE mode shape (R 1 = 8μm, λ = 1.528 μm) in its corresponding coordinate.

Fig. 3
Fig. 3

Numerically simulated group index ng and group velocity dispersion k″ for (a) straight waveguide, (b) bent waveguide with R1 = 8μm, and (c) bent waveguide with R1 = 3μm. TE mode: black; TM mode: red. The curves for bulk Si are plotted in blue for reference.

Fig. 4
Fig. 4

Zero-dispersion wavelength vs. the inverse of the bending radius of an SOI waveguide for both TE (black dots) and TM (red squares) modes.

Fig. 6
Fig. 6

Frequency mismatch in a straight cavity for quasi-TE modes with λp = 1.407 μm (solid blue), 1.516 μm (solid red), and 1.649 μm (solid black), and quasi-TM modes with λp = 1.405 μm (hollow blue), 1.514 μm (hollow red), and 1.646 μm (hollow black).

Equations (23)

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E p ( + ) ( z , t ) = E p e i [ k p ( ω p ) z ω p t ] e i Γ P z ,
E s ( ) ( z , t ) = h ¯ ω s 2 ɛ 0 n s c A eff , s γ s Δ ω s 2 π m s d Ω s a s ( ω s , m s + Ω s ) γ s / 2 i Ω s e i [ k s z ( ω s , m s + Ω s ) t ] .
| Ψ = η L m s m i d Ω s d Ω i γ s γ i δ ( m s Δ ω s + m i Δ ω i + Ω s + Ω i ) ( γ s / 2 i Ω s ) ( γ i / 2 i Ω i ) e i L [ k 4 ( m s Δ ω s Ω s ) 2 + k 4 ( m i Δ ω i + Ω i ) 2 + Γ P ] a s ( ω s , m s + Ω s ) a i ( ω i , m i + Ω i ) | 0 sinc { L [ k 4 ( m s Δ ω s Ω s ) 2 + k 4 ( m i Δ ω i + Ω i ) 2 + Γ P ] } ,
e i u t d t = 2 π δ ( u ) ,
L 0 e i β x d x = L e i β L / 2 sinc ( β L / 2 ) .
| Ψ = η L m d Ω γ s γ i e i L [ k ( m Δ ω Ω ) 2 / 2 + Γ P ] ( γ s / 2 i Ω ) ( γ i / 2 + i Ω ) sinc { L [ k ( m Δ ω Ω ) 2 / 2 + Γ P ] } a s ( ω p m Δ ω + Ω ) a i ( ω p + m Δ ω Ω ) | 0 .
S ( ω s ) = ( η L ) 2 m γ s γ i sinc 2 { L [ k ( ω p ω s ) 2 / 2 + Γ P ] } | γ s / 2 i ( ω s ω p + m Δ ω ) | 2 | γ i / 2 + i ( ω s ω p + m Δ ω ) | 2 .
S ( ω i ) = Ψ | a i ( ω i ) a i ( ω i ) | Ψ = ( η L ) 2 m γ s γ i sinc 2 { L [ k ( ω i ω p ) 2 / 2 + Γ P ] } | γ s / 2 i ( ω p ω i + m Δ ω ) | 2 | γ i / 2 + i ( ω p ω i + m Δ ω ) | 2 .
u = R 2 ln ( ρ / R 2 ) ,
v = R 2 θ .
n Air = 1 ,
n SiO 2 = 1 + 0.6961663 λ 2 λ 2 0.0684043 2 + 0.4079426 λ 2 λ 2 0.1162414 2 + 0.8974794 λ 2 λ 2 9.896161 2 ,
n Si = 3.41906 + 0.123172 λ 2 0.028 + 0.0265456 ( λ 2 0.028 ) 2 2.66511 × 10 8 λ 2 + 5.45852 × 10 14 λ 2 ,
Δ = 1 2 π ( 2 ω 0 p ω 0 s ω 0 i ) ,
Δ nl = 2 n 2 I n ( ω p ) ω p 2 π ,
R ijkl ( 3 ) ( τ ) = γ e δ ( τ ) [ σ 3 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) + ( 1 σ ) δ ij δ jk δ kl ] + γ R h R ( τ ) ( δ ik δ jl + δ il δ jk 2 δ ij δ jk δ kl ) .
[ a x a ρ a θ ] = [ M xx M xy M xz M ρ x M ρ y M ρ z M θ x M θ y M θ z ] [ a x a y a z ] ,
[ M xx M xy M xz M ρ x M ρ y M ρ z M θ x M θ y M θ z ] = [ 1 0 0 0 cos θ sin θ 0 sin θ ρ cos θ ρ ] .
R ρ ρ ρ ρ ( 3 ) = γ e δ ( τ ) ( cos 4 θ + sin 4 θ + 2 σ sin 2 θ cos 2 θ ) + 4 γ R h R ( τ ) sin 2 θ cos 2 θ ,
R xxxx ( 3 ) = γ e δ ( τ ) ,
R x ρ ρ x ( 3 ) = R ρ xx ρ ( 3 ) = σ 3 γ e δ ( τ ) + γ R h R ( τ ) .
n g L = M λ ,
ω d k d ω = 2 π M L .

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