## Abstract

In this paper, 1.5μm correlated photon pairs are generated under continuous wave (CW) pumping in a silicon micro-ring cavity with a Q factor of 8.1 × 10^{4}. The ratio of coincidences to accidental coincidences (CAR) is up to 200 under a coincidence time bin width of 5ns. The experiment result of single side photon count shows that the generation rate does not increase as the square of the pump level due to the nonlinear losses in the cavity which reduce the Q factor and impact the field enhancement effect in the cavity under high pump level. Theoretical analysis shows that the photon pair generation rate in the cavity is proportional to the seventh power of the Q factor, which agrees well with the experiment result. It provides a way to analyze the performance of CW pumping correlated photon pair generation in silicon micro-ring cavities under high pump levels.

© 2014 Optical Society of America

## 1. Introduction

Correlated photon pairs have important applications in quantum information and quantum communication. In recent years, spontaneous four wave mixing (SFWM) in silicon wire waveguides attracts much attention for its potential to realize correlated photon pair generation in 1.5μm band [1]. The nonlinear coefficient of silicon wire waveguides is about five magnitudes higher than silica fibers thanks to their ultra-small modal field areas and the high Kerr nonlinearity *n*_{2} of silicon material [2]. On the other hand, since the spectrum of spontaneous Raman scattering in a silicon wire waveguide is quite narrow, the noise photons induced by spontaneous Raman scattering are easy to be filtered out, which is effective for realizing low noise correlated photon pair generation at room temperature [3]. Furthermore, silicon wire waveguides can be fabricated by standard CMOS process, indicating the potential on developing silicon integrated quantum light sources [4]. Hence, silicon waveguide has been widely investigated as an important scheme for 1.5 μm correlated photon pair generation [5,6].

Micro-ring cavities [7–9] based on silicon wire waveguide are important silicon device structures. If the input light is at the resonance wavelength of a cavity mode, the light field in the cavity would be enhanced significantly, leading to a significant improvement of the efficiency of correlated photon pair generation [10,11]. A micro-ring cavity with a circumference of several tens of micrometers could generate correlated photon pairs effectively [12]. Comparatively, for simple silicon waveguides without micro-ring structures, samples of several millimeters or 1~2 centimeters in length are required. The light field enhancement in the cavities can reduce the requirement of input pump power [13–15] because it not only reduces the difficulty in the residual pump suppression but also significantly reduces the noise photons generated by the spontaneous Raman scattering in optical fibers, such as pigtails of optical components and the lensed fibers in the experiment setup. As a result, correlated photon pair generation under CW pumping can be effectively realized in silicon micro-ring cavities [16]. However, the light field enhancement would also increases the nonlinear loss in the cavity such as the two-photon absorption (TPA) and free carrier absorption (FCA) in the silicon materials. The nonlinear loss reduces the Q factor of the cavity, which would impact the correlated photon pair generation in the cavity. In this paper, 1.5 μm correlated photon pair generation is realized in a silicon micro-ring cavity under CW pumping at room temperature. The setup description and results of the experiment are shown in section 2. The impact of the nonlinear loss in the cavity is analyzed in Section 3. Section 4 is the conclusion.

## 2. Experiment of correlated photon pair generation in a silicon micro-ring cavity under CW pumping

The silicon micro-ring cavity is fabricated by standard CMOS processes (by Institute of Microelectronics, Singapore). Its sketch is shown in Fig. 1(a). The ring has a shape of racetrack, with a circumference of 132.5um. It is coupled to a bus waveguide, which is 3 mm in length. The cross section of the bus waveguides and ring is depicted in Fig. 1(b).

The transmission spectrum of the cavity is measured under low probe light power by an optical device measurement system with a tunable laser (TSL510, Santec Inc.) and a power meter (Agilent 8164B). Inverted taper structures are fabricated at both ends of the bus waveguide to reduce the coupling loss between the waveguide and the lensed fibers. Figure 1(c) shows the measured spectrum of the quasi-transverse electrical mode (TE) near the resonance wavelength. The black dots are experiment results and the red line is the result of Lorentz fitting. It can be seen that the insertion loss of the bus waveguide is 5.6dB, shown by the off-resonance transmission value. A narrow dip appears at the resonance wavelength with an extinction ratio of 19 dB. The full width of half maximum (FWHM) of the resonance dip is 0.019nm, showing a Q factor of 8.1 × 10^{4}, estimated by the ratio between the resonance wavelength and the FWHM.

The experiment setup for the correlated photon pair generation in the micro-ring cavity is shown in Fig. 2. The tunable laser (TSL510, Santec Inc.) is used as the CW pump source. Its wavelength is tuned to the resonance wavelength shown in Fig. 1(c). Two cascaded tunable filters are used to improve the side band rejection of the pump light up to 130dB. The pump light is injected into the bus waveguide as quasi-TE mode by a polarization controller (PC). It generates signal and idler photons in two cavity modes near the mode for the pump light. The generated photon pairs are separated by a filter system including a coarse wavelength division multiplexing (CWDM) device and several tunable filters. The center wavelengths of the filter systems for the signal and idler photons are tuned to 1537.05 nm and 1554.66 nm, respectively, corresponding to the resonance wavelength of two cavity modes for the signal and idler photons. The transmission losses of the filter system are 4.1dB and 4.3dB for the signal and idler photons, respectively. Then the signal and idler photons are detected by two free running single photon detectors (SPDs, Id220, IDQ, Inc.). Their detection efficiencies and dead times are set as 10% and 10 μs, respectively.

Firstly, the signal side photon count rate is measured under different pump levels and plotted in Fig. 3 as the black squares. In the measurement, the pump light is always tuned to the resonance wavelength of the cavity mode under each pump level. It can be seen that the count rate rises with increasing pump level. However, it is saturated by the dead time of the single photon detector. Actually, the generation rate of signal photons at the output end of the bus waveguide should be calculated by

Where*η*is the collection and detection efficiency at signal side, including the detection efficiency of the SPD and the transmission efficiency between the output end of the bus waveguide and the SPD. C’

_{s}_{s}has the same unit as the measured signal side count rate (denoted by C

_{s}) and is calculated byWhere, d

_{s}and τ

_{s}are the dark count rate and the dead time of the signal side SPD, respectively. It can be seen that C’

_{s}is a modified signal side count rate proportional to R

_{s}, in which the impacts of dead time and dark count of the SPD are taken out. The blue dots in Fig. 3 are the calculated C’

_{s}according to the measured signal side count rate. It can be seen that C’

_{s}is close to C

_{s}at low pump level, while far higher than C

_{s}at high pump level.

Previous theoretical works have shown that the correlated photon pair generation rate in a micro-ring cavity is proportional to P_{p}^{2}Q^{3}, where P_{p} is the input pump power and Q is the quality factor of the micro-cavity. Under low pump level, it can be expected that R is proportional to P_{p}^{2} (so is C’_{s}) since Q factor can be looked as a constant. The C’_{s} measured under an input pump powers lower than 0.1mW are fitted by the function of AP_{p}^{2}Q^{3} and the curve is then extended to high pump power which is shown as the red line in Fig. 3. In the curve fitting Q is set as a constant 8.1 × 10^{4} and A is the fitting parameter. The inset figure in Fig. 3 is the detailed fitting result at low pump level. It can be seen that although the curve fitting has high precision at low pump level, it overestimates the C’_{s} under a high pump level. The reason for this overestimation is that the Q factor of the cavity would reduce if the pump level is high enough to introduce obvious nonlinear loss in the cavity [17]. Its impact on the correlated photon pair generation will be analyzed comprehensively and shown in section 3.

The quantum correlation between the signal and the idler photons is demonstrated by the measurement of coincidence count and accidental coincidence count, which are shown in Fig. 4. The output of the signal side SPD is connected to the multi-channel scalar (SR-430, Stanford Scientific Research, Inc) as the trigger signal. The output signal of the idler side SPD is connected to the multi-channel scalar with a time delay induced by a digital delay generator (DG-645, Stanford Scientific Research, Inc). The time delay, time bin width and the total trigger number are set as 180ns, 5ns, 30000, respectively. Figure 4(a) shows a typical result of a coincidence count measurement. The correlation between the generated photon pairs leads to an obvious high counts in a specific time bin, which is recorded as the coincidence count. The average count of tens of time bins nearby is calculated as the accidental coincidence count. The measurement results of the coincidence and accidental coincidence counts under different signal side count rate are shown in Fig. 4(b). Theoretically, the coincidence count should be a constant under different signal side count rate C_{s}, which is determined by the collection and detection efficiencies of the generated idler side photons. However, the coincidence count reduces rapidly with decreasing pump level when pump power is low due to the impact of dark counts of the SPD1. It also reduces with increasing pump level when pump power is high, which may be due to the impact of the dead time of the SPD2. On the other hand, the accidental coincidence count is proportional to the signal side count rate.

The CAR under different signal side count rate is shown in Fig. 5. It can be seen that the CAR increases with decreasing signal side count rate-under high pump level. However, it reduces rapidly at low pump level due to the dark counts of SPD1. Usually, the performance of the experiment setup for correlated photon pair generation can be evaluated by the maximum CAR it achieves. It can be seen that the maximum CAR is as high as 200 in the experiment. Since relatively large time bin (5ns) is used in the experiment, such a high CAR is owe to the excellent isolation of the residual pump light and spontaneous Raman noise photons, low insertion loss of the micro-ring cavity sample and high collection efficiencies of generated signal and idler photons.

## 3. The impact of nonlinear losses in silicon micro-ring cavities on the CW pumping correlated photon pair generation

Figure 3 has indicated that the generation rate does not increase as the square of the pump level due to the impact of nonlinear losses in the silicon micro-ring cavity. In previous theoretical analysis, Q factor of the cavity is always considered as a constant to simplify the relation between the photon pair generation rate and the pump level [18,19]. However, it is not suitable to analyze the photon pair generation rate under high pump level, since the intrinsic Q factor of the cavity changes due to the nonlinear loss in the cavity. In this section, the impact of the nonlinear loss on the relation between the photon pair generation rate and the pump level is investigated.

The light propagation in the silicon waveguide can be described by

Where, P is the light power propagates in the silicon waveguide. The total waveguide loss coefficient, which is denoted by an effective loss coefficient*α*, has three contributions: the first one is the linear loss, which is expressed by a linear loss coefficient

_{eff}*α.*The second one is the two-photon absorption (TPA) effect of silicon, which is expressed by the TPA coefficient

*β.*The last one is the free carrier absorption (FCA) effect of silicon, which can be expressed through the FCA cross section

*σ*and free carrier recombination life

*τ*.

*A*are the effective area of the silicon waveguide, the light frequency and the Planck constant, respectively.

_{eff,}ν, hFor a micro-ring cavity coupled to a bus waveguide, the Q factor of a specific cavity mode can be expressed as [20]

Where*Q*and

_{0}*Q*are the intrinsic and external Q factor of the cavity mode, respectively.

_{e}*Q*can be calculated by Eq. (6), in which

_{0}*α*is the effective loss coefficient of the waveguide in the cavity, v

_{eff}*is the group velocity of the light propagating in the cavity and*

_{g}*ω*is the angular frequency of the light.

*Q _{e}* can be calculated by Eq. (7), in which

*κ*is the coupling coefficient between the cavity and the bus waveguide,

*L*is the circumference of the cavity, while

*L*/v

_{g}is the round trip time of the light in the cavity

In experiments, the Q factor of a specific cavity mode is usually estimated through the transmission spectrum near the resonance wavelength of the mode, which can be measured by a tunable laser with low power. If the Q factor is high (*Q>>1*), it can be estimated by the ratio between the resonance wavelength *λ* and line width *Δλ* of the dip shown in the spectrum

If both *Q _{0}* and

*Q*are far higher than 1, it can be deduced that the extinction ratio (denoted by Γ) can be expressed as (the derivation is shown in the appendix)

_{e}*Q*and

_{0}*Q*of the mode also can be calculated according to the transmission spectrum through Eqs. (5) and (9). For the micro-ring cavity sample used in our experiment, the

_{e}*Q*,

*Q*and

_{0}*Q*of the specific mode can be calculated according to the measured transmission spectrum shown in Fig. 1(c), which are 8.1 × 10

_{e}^{4}, 14.6 × 10

^{4}and 18.3 × 10

^{4}, respectively. However, this method can’t be used to analyze the cavity under high light power due to strong light field enhancement effect in the cavity. On one hand, it will lead to obvious nonlinear losses in the cavity, which will change the intrinsic Q factor of the cavity. On the other hand, the index of the silicon waveguide in the cavity also changes under high light power, leading to a resonance wavelength shift of the cavity mode.

To investigate the impact of nonlinear loss in the cavity on the photon pair generation rate, we calculate the value of *Q _{0}* under different pump level according to the parameters of micro-ring cavity sample and parameters of TPA and FCA in silicon material. Equation (6) shows the relation between the

*Q*and the effective loss coefficient

_{0}*α*, which includes the contribution of nonlinear losses. The definition of

_{eff}*α*is shown in Eq. (4). In the following calculation,

_{eff}*β*

**,**

*σ*and

*τ*are set as 6.7 × 10

^{−12}m/W 、1.97 × 10

^{−21}m

^{2}and 4ns respectively according to the parameters of bulk silicon material [21]. The

*A*is 0.1μm

_{eff}^{2}, calculated according to the parameters of the waveguide cross section. ν is frequency of the pump light.

In Eq. (4), *α* is the linear loss coefficient. Under low light power, nonlinear losses can be neglected and the *α _{eff}* is simplified to

*α*. Hence, according to the measured

*Q*under low light power,

_{0}*α*of the waveguide in the cavity can be calculated as 1.14dB/cm by Eq. (4). In the calculation, v

_{g}is c/4.12 calculated by the expression of v

_{g}= c/n

_{g}= c × L × Δλ

_{FSR}/λ

^{2}[22], where c is the light speed in vacuum, the cavity circumference L is 132.5μm, Δλ

_{FSR}is the free spectrum region (FSR) of the cavity, which is measured as 4.4nm.

If the light power in the cavity is high, the contributions of FCA and TPA can’t be neglected. The light power circling in the cavity can be expressed by

Where*P*is the light power in the bus waveguide before coupled to the ring.

_{in}*F(ω)*is the field enhancement factor of the cavity [23] under the light frequency

*ω*. If the input light is close to resonance wavelength, it can be expressed as

*F*, can be expressed as

_{p}*ω*is the angular frequency of the pump light.

_{p}Equation (7) shows that *Q _{e}* is determined by the coupling coefficient κ, which does not change with the pump level. Substituting the expression of

*P*(Eqs. (10) and (12)) into Eq. (3) and Eq. (4), it can be seen that the nonlinear loss coefficient

_{c}*α*in the cavity is a function of

_{eff}*P*and

_{in}*Q*. Substituting this expression of

_{0}*α*into Eq. (6), the relation between

_{eff}*P*and

_{in}*Q*can be derived and

_{0}*Q*under different pump level can be calculated. Figure 6 shows the calculated

_{0}*Q*,

_{0}*Q*and

*Q*under different input pump power, which are plotted as the lines with squares, circles and triangles, respectively. In this figure, the power of the pump source is used as the horizontal axis, which is 2.8dB higher than

_{e}*P*estimated by the coupling efficiency between the input lensed fiber and the bus waveguide. It can be seen that due to the nonlinear losses in the micro-ring cavity,

_{in}*Q*decreases with increasing pump level, while,

_{0}*Q*is unchanged. As a result, the total

_{e}*Q*also decreases with increasing pump level due to the nonlinear losses in the cavity.

*Q*is lower than

_{0}*Q*under all the pump levels, showing that the cavity used in the experiment is under-coupling.

_{e}In the experiment of correlated photon pair generation in micro-ring cavities, the pump light is always tuned on resonance for a specific cavity mode, the signal and idler photons generated in two cavity modes symmetrical to the pump wavelength are detected. The field enhancement effects of the pump light, signal photons and idler photons all affect the photon pair generation rate [24,25]. Since the modes for pump light, signal photons and idler photons are close in frequency, it can be assumed that they have the same *Q _{e}* and

*Q*. Previous theoretical analysis [19] has shown that the

_{0}*P*which is the average power associated with half of the generated photons due to the SFWM, can be expressed by

_{I},Where *F _{p}* is calculated by Eq. (12).

*γ*is the nonlinear coefficient of the silicon waveguide,

*T*is the bandwidth of the generated photons by SFWM. In the case of micro-ring cavities,

^{−1}*T*is calculated by

*ω*. Considering that the signal frequency

_{s}≈ω_{i}≈ω_{p}= ω_{0}*ω*and idler frequency

_{s}*ω*satisfy that

_{i}*ω*,

_{s}+ ω_{i}= 2ω_{p}*Δω*is defined as

*Δω*=

*ω*- ω

_{s}_{0,s}=

*ω*,

_{0,i}- ω_{i}*ω*and

_{0,s}*ω*are resonance frequency of the cavity modes for the signal and idler photons, respectively. Substituting Eq. (11) into Eq. (13) and Eq. (14), it can be calculated that,

_{0,i}*R ∝P*.

_{I}∝P_{p}^{2}Q^{7}It is worthwhile to note that if the cavity satisfies the critical coupling condition, i.e. *Q _{0}*

**=**

*Q*, this relation would be simplified to

_{e}*R ∝P*. Another simplification usually used in some condition is to ignore the intrinsic loss of the cavity, i.e.

_{p}^{2}Q^{3}*Q*

**=**

*Q*, it also leads to

_{e}*R ∝P*. In both cases, it is supposed that Q factor of the cavity is constant and the nonlinear losses in the cavity are not considered.

_{p}^{2}Q^{3}The analysis in section 2 has shown that the modified signal side count rate C’ in the experiment is proportional to the photon pair generation rate *R*. Figure 3 has shown that *R ∝P _{p}^{2}Q^{3}* does not make a decent description of the relation between the photon pair generation rate and pump level when the pump power is high. The theoretical analysis in this section shows that under high pump level, the reduction of the Q factor due to the nonlinear losses in the cavity should be considered, leading to the relation of

*R ∝P*. To demonstrate it, the C’ measured under an input pump power lower than 0.1mW are fitted by the function of

_{p}^{2}Q^{7}*C’ = BP*. B is the fitting parameter. The value of

_{p}^{2}Q^{7}*Q*utilizes the calculated results shown in Fig. 6, in which the nonlinear losses due to the FCA and TPA in the cavity are considered. The fitting result is shown in Fig. 7, while the fitting result of C’ = A

*P*in Fig. 3 is also plotted for comparison. It can be seen that compared with the results of C’ = A

_{p}^{2}Q^{3}*P*the fitting result of

_{p}^{2}Q^{3},*C’ = BP*not only has a high precision at low pump level, but also agrees well with the C’ at high pump level. The comparison shows that under high pump level, the relation between the photon pair generation rate and pump level should be described by Eq. (15) or

_{p}^{2}Q^{7}*R ∝P*, in which the Q factor reduces with increasing pump level due to the nonlinear losses in the cavity.

_{p}^{2}Q^{7}## 4. Conclusion

In this paper, 1.5μm correlated photon pairs are generated under CW pumping in a silicon micro ring cavity with a Q factor of 8.1 × 10^{4}. The measured CAR is up to 200 under a coincidence time bin width of 5ns. The Experiment results show that the photon pair generation rate does not increase as the square of the pump level due to the nonlinear losses in the silicon micro-ring cavity. Theoretical analysis shows that the relation between the photon pair generation rate and the pump level should be described by R∝P_{p}^{2}Q^{7}, meanwhile, the Q factor reduces with increasing pump level. This relation agrees well with the experiment results, providing a way to analyze the performance of the correlated photon pair generation in silicon micro-ring cavities at high pump level.

## 5. Appendix

The transmission of a bus waveguide with a micro-ring cavity can be expressed as

*β*is the light propagation constant in the micro-ring cavity.

*t*is the field straight-through coupling coefficient and Λ is the round trip loss coefficient of resonator. In a high Q cavity, they can be simplified as $t=\sqrt{1-{\left|\kappa \right|}^{2}}\approx 1-\frac{{\left|\kappa \right|}^{2}}{2}$ and $\Lambda =\mathrm{exp}\left(-\frac{{\alpha}_{eff}L}{2}\right)\approx 1-\frac{{\alpha}_{eff}L}{2}$ according to |κ|

^{2}<<1 and

*α*L/2 << 1, where κ is the field cross-coupling coefficient.

_{eff}If the input light is on resonance, according to Eqs. (6) and (7), equation (16) can be simplified as:

## Acknowledgment

This work is supported in part by 973 Programs of China under Contract No. 2011CBA00303 and 2010CB327606, Tsinghua University Initiative Scientific Research Program, Basic Research Foundation of Tsinghua National Laboratory for Information Science and Technology (TNList), and China Postdoctoral Science Foundation.

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