Abstract

In a passive cavity geometry, there exists a trade-off between resonant enhancement and response time, which is inherently limited by the cavity photon lifetime. We demonstrate frequency-selective, dynamic control of the photon lifetime using a silicon-nitride coupled-ring resonator. The photon lifetime is tuned by controlling an avoided mode crossing using thermo-optic tuning of the cavity resonance with integrated heaters. Using this effect, we achieve fast turn-on/off of a ${\chi ^{(3)}}$ degenerate optical parametric oscillator (DOPO) and on-chip true random number generation. Our approach allows us to overcome the $Q$-limited generation rate of a single-ring-based DOPO and offers a path toward the development of a scalable integrated high-quality entropy source for modern cryptographic systems.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Many modern cryptographic protocols require a pseudo-random-number generation, which is initially seeded by a high-quality entropy source, or a true random-number generator (RNG) [1]. While a wide range of entropy sources have been explored [214], an ideal true RNG remains elusive. For example, noise-based sources require threshold adjustments to allow for uniformity in the random numbers, which are susceptible to environmental variations and thermal fluctuations and can require post-processing to extract the randomness [15]. Recently, a novel approach for RNG has been demonstrated using an optical parametric oscillator [1618]. Since the system operates above threshold, RNGs based on degenerate optical parametric oscillators (DOPOs) allow for direct detection of a strong classical signal initiated from quantum noise [16,19], reducing the overall complexity of the system with minimal post-processing. The randomness is derived from the generation of an unbiased bi-phase state that results from a nonequilibrium phase transition that occurs at the oscillation threshold [2022]. Studies have shown that the probability distribution of the above-threshold DOPO is central symmetric and corresponds to two lobes in the quadrature-phase space with essentially no overlap above threshold [23]. As our detection scheme is binary, the separation of the lobes remains large, and the predictable fluctuations do not translate into the output bits. In order to ensure that each bit is generated independently, the cavity field of the previous bit must be suppressed to the quantum-noise level, allowing the DOPO to build up from perfectly random noise. For example, in a bulk cavity, a chopper is implemented to turn the DOPO on and off [16,18].

DOPOs based on the ${\chi ^{(3)}}$ nonlinearity rely on inverse four-wave mixing (IFWM), where two frequency non-degenerate pumps are injected into a nonlinear resonator for generation of a degenerate signal/idler pair [Fig. 1(a)] [17,2427]. Previous work achieved random number generation using an integrated silicon nitride (SiN) microresonator driven with a pump that is modulated using an external acousto-optic modulator with an amplified RF driver [17]. Low-power DOPO generation can be realized with higher-$Q$ devices with pump power scaling as $1/{Q^2}$ [22]. However, the generation rate using a single resonator is ultimately limited by the DOPO suppression rate, which is dependent on the cavity photon lifetime [16].

 figure: Fig. 1.

Fig. 1. (a) Energy level diagram for DOPO via four-wave mixing. Device geometry and corresponding resonance dynamics when a control signal is injected into the ring for (b) a single ring and (c) a coupled ring. Red (dashed) corresponds to control off (DOPO on) and blue (solid) corresponds to control on (DOPO off). Each pump is located 3 free-spectral ranges (FSRs) away from the DOPO resonance.

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 figure: Fig. 2.

Fig. 2. (a) Microscope image of SiN coupled-ring device. The main and auxiliary rings have FSRs of 201.86 GHz and 207.8 GHz, respectively. We implement a drop port on the auxiliary ring. The ring-to-ring gap and the gap between the auxiliary ring and drop port are 250 nm, while the through port to main ring gap is 550 nm. (b) Coupled-ring resonance interaction at the oscillation wavelength as a function of the auxiliary ring heater power. With increased power, the auxiliary ring resonance (red circles) redshifts toward the main ring resonance (blue circles) resulting in mode coupling and splitting. (c) Transmission measurements at specific auxiliary ring-heater powers. The frequency degeneracy point is at 166.4 mW with a mode splitting of 15.8 GHz.

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In a passive cavity geometry, there exists a trade-off between resonant enhancement and response time, which is fundamentally limited by the cavity decay rate or photon lifetime. There has been much work investigating techniques for overcoming the intrinsic linewidth limitations for applications including modulators [28,29], slow, and stopped light [3032]. Moreover, devices enabling frequency-selective control of the decay rate is paramount for large-scale integrated photonic circuits with applications in logic gates and switches for information processing and optical neural networks [3335].

Here, we demonstrate frequency-selective, dynamic control of the photon lifetime using a SiN coupled-ring resonator to induce an avoided mode crossing, where the coupling between the frequency-degenerate modes in the main and auxiliary rings results in mode splitting at the degeneracy point [36]. We control the mode interaction position by tuning the auxiliary ring resonance using integrated heaters. With this technique, we show fast DOPO turn-on/off and on-chip random number generation at 505 kbit/s using continuous-wave (cw) pumps. We experimentally show that our scheme overcomes the $Q$-limited generation rate of a thermally controlled single-ring DOPO by a factor ${\gt}2.7 \times$. In addition, our numerical modeling shows that the dynamic reduction in the photon lifetime can be ${\gt}100 \times$ and is largely dictated by the coupling strength between the auxiliary ring and the drop port, which allows for adding loss to the auxiliary ring without sacrificing the $Q$ of the main ring.

Figures 1(b) and 1(c) show the single-ring and coupled-ring geometries and the corresponding behavior of the resonances. When a single-ring device is driven with a modulation signal, all of the resonances shift together and the cavity decay rate is inversely proportional to the $Q$ factor. In a coupled-ring device, with main and auxiliary rings, the modulation signal is sent to the auxiliary ring. When an auxiliary resonance is tuned to spectrally overlap with the main resonance, an avoided mode crossing is generated, resulting in splitting of the central resonance. The mode interaction between the two fields in the main and auxiliary rings results in localized changes in the cavity losses and the dispersion [36,37], altering the IFWM phase-matching conditions (see Supplement 1), and enables dynamic control of the photon lifetime for the DOPO mode.

Figure 2(a) shows a microscope image of the SiN device. The resonator cross section is $730 \times 1030 \; {\rm nm}$ such that the device has normal group-velocity dispersion at the pump wavelengths. In order to have a single mode-crossing interaction near the operating wavelength, we apply the Vernier effect [36] and design the two resonators to have slightly different free-spectral ranges (FSRs) such that the periodicity of the mode-crossing interaction is ${\gt}50$ nm. We also implement a drop port on the auxiliary ring, which is terminated on both ends with a taper design to minimize backreflections. The resonances of the rings and consequently the mode interaction position is controlled using integrated platinum heaters directly above the rings. In order to tune the mode interaction wavelength, we control the heater power for the auxiliary ring. Figures 2(b) and 2(c) show the coupled-ring resonance interaction at the DOPO mode as the auxiliary ring is tuned. In the absence of applied heater power, the DOPO mode resonance is unaffected by the auxiliary ring and is critically coupled to the bus waveguide with a loaded $Q$ of 717,000. As the heater power is increased, the auxiliary ring resonance is redshifted toward the main ring resonance, resulting in an avoided mode crossing. At the frequency degeneracy point (heater ${\rm power} = 166.4\; {\rm mW}$), we observe a mode splistting of 15.8 GHz as shown in Fig. 2(c). This is accompanied by a reduction in the loaded $Q$ of the device by a factor of 4.24 at the degeneracy point, which is attributed to the coupling loss due to the mode coupling to the auxiliary ring along with the drop port. Furthermore, this mode splitting alters the phase-matching conditions, suppressing IFWM. Thus, by tuning the auxiliary cavity into resonance with the DOPO mode, we can eject all of the light from the cavity. In our system, due to the strong coupling between the rings, the mode interaction spans ${\gt}7$ modes as shown in Fig. 3. While the two pump modes border the mode interaction region, the reduction of the intracavity pump intensity due to coupling between the rings is negligible.

 figure: Fig. 3.

Fig. 3. Wide bandwidth transmission measurements for three different auxiliary ring-heater powers. The strong coupling between the rings results in the mode interaction spanning ${\gt}7$ modes (highlighted). With increasing heater power, the mode interaction region blueshifts toward the DOPO oscillation wavelength (1549.4 nm).

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The details of the experimental setup are shown in Supplement 1. Two tunable single-frequency cw lasers centered at 1544.6 nm and 1554.2 nm are amplified and sent into the SiN coupled ring device for DOPO generation. Each pump is located 3 FSRs away from the frequency degeneracy point. For DOPO generation and suppression, we drive the integrated heater with 505 kHz repetition rate, 156.3 mW peak power, 820 ns RF pulses derived from an arbitrary waveform generator, such that the DOPO is generated during the “off” state of the RF pulse. Figure 4(a) shows the generated DOPO spectrum, where the DOPO signal is generated at 1549.4 nm. For temporal characterization of the DOPO, the pumps are removed using a 1 nm tunable bandpass filter centered at the DOPO signal wavelength. To measure the interference between adjacent bits, the DOPO output is sent to an asymmetric Mach–Zehnder interferometer and measured with a real-time oscilloscope.

 figure: Fig. 4.

Fig. 4. (a) Generated DOPO spectrum. (b) Time-domain measurement of DOPO-RNG. (c) Statistical test results on our 101,000 bit sequence, divided into 20 equal samples, using the NIST Statistical Test Suite.

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Figure 4(b) shows the temporal measurement of the interferometer output that reveals the distinct binary sequence. The generation rate is 505 kbit/s, corresponding to the modulation frequency driving the integrated heaters. We verify the randomness of our DOPO-RNG by performing a series of statistical tests on a 101,000 bit sequence that is divided into 20 equal samples using the NIST Statistical Test Suite (STS-2.1.2) to determine the uniformity and independence [38]. The results of the Test Suite are shown in Fig. 4(c). P-value$_{\rm T}$ is computed using a ${\chi ^2}$ statistic and corresponds to the uniformity of P-values, which is the probability that a perfect RNG produces a sequence that is less random than the test sequence. To pass the uniformity test, P-value$_{\rm T}$ $\gt 0.0001$ must be satisfied. Proportion corresponds to the percentage of sequences that pass with a P-value $\ge 0.01$. For our sample size, the minimum pass rate for each test is 90%. Our bit sequence passed all of the statistical tests indicating that the NIST Suite accepts our sequence as random. As a comparison, we have experimentally characterized generation rates in a single ring resonator with a photon lifetime of 477 ps. Our measurements for the single-ring system (see Supplement 1) show that beyond 184 kHz modulations rates, the generated DOPO signal is seeded by the previous pulse, resulting in a correlated output, indicating that the coupled-ring geometry allows us to overcome the $Q$-limited cavity decay rate.

Lastly, we theoretically compare the cavity decay rate, or equivalently the photon lifetime, in our coupled-ring resonator to that of a single ring and investigate the scalability to higher-$Q$ systems. To model the coupled ring, we use coupled time-domain equations for the electric fields in the main (${a_1}$) and auxiliary (${a_2}$) rings normalized to energy, similar to derivations in [39]

$$\begin{array}{l}\frac{{d{a_1}}}{{dt}} = \left({i{\omega _1} - \frac{1}{{{\tau _e}}} - \frac{1}{{{\tau _1}}}} \right){a_1} - i{\kappa _{12}}{a_2} - i{\kappa _1}{s_i},\\\frac{{d{a_2}}}{{dt}} = \left({i{\omega _2} - \frac{1}{{{\tau _d}}} - \frac{1}{{{\tau _2}}}} \right){a_2} - i{\kappa _{12}}{a_1}.\end{array}$$
Here, $1/{\tau _e}$ and $1/{\tau _d}$ are the amplitude decay rates corresponding to the coupling to the bus waveguide and drop ports, respectively. We define ${{\rm{FSR}}_{1,2}}$ as the FSRs of the main and auxiliary rings; ${\omega _{1,2}}$ as the resonance frequencies of the main and auxiliary rings; and ${\theta _1}$, ${\theta _2}$, ${\theta _3}$ as the power coupling coefficients between the bus and the main ring, the main ring and the auxiliary ring, and the auxiliary ring and the drop port, respectively. The coupling decay rates obey the relations $2/{\tau _e} = {\theta _1}{{\rm{FSR}}_1}$ and $2/{\tau _d} = {\theta _3}{{\rm{FSR}}_2}$. The amplitude decay rates in the main and auxiliary rings due to absorption and scattering losses in the resonator are expressed as $1/{\tau _{1,2}} = {\omega _0}/2{Q_{1,2}} = {\alpha _{1,2}}{{\rm{FSR}}_{1,2}}/2$, where ${\omega _0}$ is the pump frequency, ${Q_{1,2}}$ are the intrinsic quality factors of the main and auxiliary rings, and ${\alpha _{1,2}}$ are the linear attenuation coefficients. Additionally, ${s_i}$ is the input field amplitude normalized to power, ${\kappa _1}$ corresponds to the pump coupling coefficient from the bus waveguide, and ${\kappa _{12}}$ corresponds to the mutual coupling coefficient between the main and auxiliary rings. The coupling coefficients can be expressed in terms of the power coupling coefficients as ${\kappa _1} = \sqrt {{\theta _1}{{\rm{FSR}}_1}}$ and ${\kappa _{12}} = \sqrt {{\theta _2}{{\rm{FSR}}_1}{{\rm{FSR}}_2}}$.
 figure: Fig. 5.

Fig. 5. (a) Characterization of the photon lifetime for single- and coupled-ring resonators as a function of the intrinsic $Q$ factor ${Q_i}$ and (b) the corresponding photon lifetime reduction factor with respect to the single ring. For all curves, ${\theta _1}$ is chosen such that the resonance extinction is 95%, which is consistent with experimental conditions. From a fit to our experiment, ${\theta _1}/{\theta _2} = 0.1035$ at ${Q_i} = 1.5 \times {10^6}$. For a photon, lifetimes are shown for a single ring (black) and a coupled ring with three different conditions: ${\theta _2} = {\theta _3} = 0.0613$ (red, same as the experimental parameters); ${\theta _2} = 0.0613$ and ${\theta _3} = 2{\theta _2}$ (orange); ${\theta _2} = 0.0613$ and ${\theta _3} = 5{\theta _2}$ (blue).

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To model the single ring, we set ${{\rm{FSR}}_1} = 200\; {\rm GHz}$, ${\theta _2} = 0$, and choose ${\theta _1}$ such that we are in the slightly overcoupled regime, corresponding to a resonance extinction of 95%. To extract the photon lifetime, we first build up the intracavity power to a steady state value using ${\omega _1} = {\omega _0}$. We then offset ${\omega _1}$ by $2\pi {{\rm{FSR}}_1}/4$. For optimal DOPO generation, ${\theta _1}$ must follow the intrinsic $Q$ factor ${Q_i}$ to maintain the same coupling condition. For the coupled-ring simulations, we use ${{\rm{FSR}}_1} = 200$ GHz, ${{\rm{FSR}}_2} = 206$ GHz, and ${\theta _1}$ is kept the same as the single ring. We choose ${Q_1} = {Q_2} = {Q_i}$. To model the photon lifetime using our coupled-ring system, the power is built up to the steady state using ${\omega _1} = {\omega _0}$ and ${\omega _2} = {\omega _0} + 2\pi {{\rm{FSR}}_2}/2$. The input field is then turned off (${s_i} = 0$), and we set the auxiliary ring resonance such that it overlaps with the main ring resonance ${\omega _2} = {\omega _0}$. We then measure the decay time until the intracavity power falls off to $1/e$ of the steady-state value. Figure 5(a) shows the simulated photon lifetime with respect to ${Q_i}$, and 5(b) shows the reduction factor of the photon lifetime when compared to a single ring. We model conditions similar to our experiment (Fig. 5, red), where we choose ${\theta _2} = {\theta _3} = 0.0613$, based on a fit to experimental values for ${Q_i} = 1.5 \times {10^6}$ [Fig. 2(c)]. For comparison, our modeling shows a $3.7 \times$ decrease in the lifetime as compared to a single ring for this $Q$, indicating that the generation rate can reach 680 kHz. Larger lifetime reductions can be achieved by increasing ${\theta _3}$, and we plot two additional curves in Fig. 5 for ${\theta _3} = 2{\theta _2}$ (orange) and ${\theta _3} = 5{\theta _2}$ (blue) and see reductions in the lifetime as high as $16 \times$ compared to a single ring for ${Q_i} = 1.5 \times {10^6}$, with a corresponding generation rate reaching 3 MHz. The power decay rate $1/{T_{\rm{p}}}$ of the coupled-ring system can be expressed as $1/{T_{\rm{p}}} = 1/{\tau _e} + 1/{\tau _1} + 1/{\tau _2} + 1/{\tau _d}$ and approaches $1/{\tau _d}$ for large ${Q_i}$. Compared to a single ring for which the lifetime is linearly dependent on ${Q_i}$, we see as much as a $100 \times$ reduction in lifetime for ${Q_i} = 10 \times {10^6}$. Since the pump power scales as $1/{Q^2}$, the coupled-ring geometry allows for low power operation while allowing for fast DOPO suppression. Further photon lifetime reduction can be achieved by using racetrack coupling or Mach–Zehnder interferometer-based coupling [28]. While our current system is limited by the tuning speed of the integrated heaters, faster on-chip tuning mechanisms such as the electro-optic effect in ${\chi ^{(2)}}$ materials (e.g., lithium niobate) allows for a tuning response of 45 GHz [40], such that the generation rate would be limited by the photon lifetime.

In conclusion, we demonstrate frequency-selective, dynamic control of the photon lifetime using coupled-ring-induced mode interactions. Using this approach we achieve fast turn-on/off of a SiN-based coupled-ring DOPO, overcoming the $Q$-limited random number generation rate of a single-ring device. Our system generates random numbers that passes the NIST statistical tests, offering potential toward the realization of a chip-scale entropy source for the development of a modern cryptographic system.

Funding

Air Force Office of Scientific Research (FA9550-15-1-0303); Semiconductor Research Corporation (SRS 2016-EP-2693-A); National Science Foundation (CCF-1640108); Army Research Office (W911NF-17-1-0016).

Acknowledgment

This work was performed in part at the Cornell Nano-Scale Facility, which is a member of the National Nanotechnology Infrastructure Network, supported by the NSF, and at the CUNY Advanced Science Research Center NanoFabrication Facility. We also thank J. K. Jang and M. Yu for useful discussions.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this Letter are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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References

  • View by:

  1. M. S. Turan, E. Barker, J. Kelsey, K. A. McKay, M. L. Baish, and M. Boyle, “Recommendation for the entropy sources used for random bit generation,” NIST Special Publ. 800-90B, Second draft (2018).
  2. T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, Rev. Sci. Instrum. 71, 1675 (2000).
    [Crossref]
  3. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, Nat. Photonics 2, 728 (2008).
    [Crossref]
  4. S. Pironio, A. Acín, S. Massar, A. B. de La Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Nature 464, 1021 (2010).
    [Crossref]
  5. C. R. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, Opt. Express 18, 23584 (2010).
    [Crossref]
  6. P. J. Bustard, D. Moffatt, R. Lausten, G. Wu, I. A. Walmsley, and B. J. Sussman, Opt. Express 19, 25173 (2011).
    [Crossref]
  7. T. Yamazaki and A. Uchida, IEEE J. Sel. Top. Quantum Electron. 19, 0600309 (2013).
    [Crossref]
  8. M. Stipčević and Ç. K. Koç, True Random Number Generators (Springer, 2014), pp. 275–315.
  9. M. Collins, A. Clark, C. Xiong, E. Mägi, M. Steel, and B. Eggleton, Appl. Phys. Lett. 107, 141112 (2015).
    [Crossref]
  10. X. Ma, X. Yuan, Z. Cao, B. Qi, and Z. Zhang, npj Quantum Inf. 2, 1 (2016).
    [Crossref]
  11. C. Abellan, W. Amaya, D. Domenech, P. Muñoz, J. Capmany, S. Longhi, M. W. Mitchell, and V. Pruneri, Optica 3, 989 (2016).
    [Crossref]
  12. A. Acín and L. Masanes, Nature 540, 213 (2016).
    [Crossref]
  13. H. Jiang, D. Belkin, S. E. Savel’ev, S. Lin, Z. Wang, Y. Li, S. Joshi, R. Midya, C. Li, M. Rao, M. Barnell, Q. Wu, J. J. Yang, and Q. Xia, Nat. Commun. 8, 1 (2017).
    [Crossref]
  14. M. Herrero-Collantes and J. C. Garcia-Escartin, Rev. Mod. Phys. 89, 015004 (2017).
    [Crossref]
  15. L. Gong, J. Zhang, H. Liu, L. Sang, and Y. Wang, IEEE Access 7, 125796 (2019).
    [Crossref]
  16. A. Marandi, N. C. Leindecker, K. L. Vodopyanov, and R. L. Byer, Opt. Express 20, 19322 (2012).
    [Crossref]
  17. Y. Okawachi, M. Yu, K. Luke, D. O. Carvalho, M. Lipson, and A. L. Gaeta, Opt. Lett. 41, 4194 (2016).
    [Crossref]
  18. T. Steinle, J. N. Greiner, J. Wrachtrup, H. Giessen, and I. Gerhardt, Phys. Rev. X 7, 041050 (2017).
    [Crossref]
  19. W. Louisell, A. Yariv, and A. Siegman, Phys. Rev. 124, 1646 (1961).
    [Crossref]
  20. T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, Nat. Photonics 10, 415 (2016).
    [Crossref]
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IEEE Access (1)

L. Gong, J. Zhang, H. Liu, L. Sang, and Y. Wang, IEEE Access 7, 125796 (2019).
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T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, Nat. Photonics 10, 415 (2016).
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Supplementary Material (1)

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Supplement 1       Supplemental document

Data Availability

Data underlying the results presented in this Letter are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)

Fig. 1.
Fig. 1. (a) Energy level diagram for DOPO via four-wave mixing. Device geometry and corresponding resonance dynamics when a control signal is injected into the ring for (b) a single ring and (c) a coupled ring. Red (dashed) corresponds to control off (DOPO on) and blue (solid) corresponds to control on (DOPO off). Each pump is located 3 free-spectral ranges (FSRs) away from the DOPO resonance.
Fig. 2.
Fig. 2. (a) Microscope image of SiN coupled-ring device. The main and auxiliary rings have FSRs of 201.86 GHz and 207.8 GHz, respectively. We implement a drop port on the auxiliary ring. The ring-to-ring gap and the gap between the auxiliary ring and drop port are 250 nm, while the through port to main ring gap is 550 nm. (b) Coupled-ring resonance interaction at the oscillation wavelength as a function of the auxiliary ring heater power. With increased power, the auxiliary ring resonance (red circles) redshifts toward the main ring resonance (blue circles) resulting in mode coupling and splitting. (c) Transmission measurements at specific auxiliary ring-heater powers. The frequency degeneracy point is at 166.4 mW with a mode splitting of 15.8 GHz.
Fig. 3.
Fig. 3. Wide bandwidth transmission measurements for three different auxiliary ring-heater powers. The strong coupling between the rings results in the mode interaction spanning ${\gt}7$ modes (highlighted). With increasing heater power, the mode interaction region blueshifts toward the DOPO oscillation wavelength (1549.4 nm).
Fig. 4.
Fig. 4. (a) Generated DOPO spectrum. (b) Time-domain measurement of DOPO-RNG. (c) Statistical test results on our 101,000 bit sequence, divided into 20 equal samples, using the NIST Statistical Test Suite.
Fig. 5.
Fig. 5. (a) Characterization of the photon lifetime for single- and coupled-ring resonators as a function of the intrinsic $Q$ factor ${Q_i}$ and (b) the corresponding photon lifetime reduction factor with respect to the single ring. For all curves, ${\theta _1}$ is chosen such that the resonance extinction is 95%, which is consistent with experimental conditions. From a fit to our experiment, ${\theta _1}/{\theta _2} = 0.1035$ at ${Q_i} = 1.5 \times {10^6}$. For a photon, lifetimes are shown for a single ring (black) and a coupled ring with three different conditions: ${\theta _2} = {\theta _3} = 0.0613$ (red, same as the experimental parameters); ${\theta _2} = 0.0613$ and ${\theta _3} = 2{\theta _2}$ (orange); ${\theta _2} = 0.0613$ and ${\theta _3} = 5{\theta _2}$ (blue).

Equations (1)

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d a 1 d t = ( i ω 1 1 τ e 1 τ 1 ) a 1 i κ 12 a 2 i κ 1 s i , d a 2 d t = ( i ω 2 1 τ d 1 τ 2 ) a 2 i κ 12 a 1 .