Abstract

To emulate a spiking neuron, a photonic component needs to be excitable. In this paper, we theoretically simulate and experimentally demonstrate cascadable excitability near a self-pulsation regime in high-Q-factor silicon-on-insulator microrings. For the theoretical study we use Coupled Mode Theory. While neglecting the fast energy and phase dynamics of the cavity light, we can still preserve the most important microring dynamics, by only keeping the temperature difference with the surroundings and the amount of free carriers as dynamical variables of the system. Therefore we can analyse the microring dynamics in a 2D phase portrait. For some wavelengths, when changing the input power, the microring undergoes a subcritical Andronov-Hopf bifurcation at the self-pulsation onset. As a consequence the system shows class II excitability. Experimental single ring excitability and self-pulsation behaviour follows the theoretic predictions. Moreover, simulations and experiments show that this excitation mechanism is cascadable.

© 2012 OSA

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References

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  1. K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw.22, 1469–1481 (2011).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [PubMed]
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    [CrossRef]
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    [CrossRef]
  6. G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express13, 9623–9628 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  15. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Time-domain and frequency-domain modeling of nonlinear optical components on circuit-level using a node-based approach,” Comp. Phys. Commun.181, 687–702 (2010).
    [CrossRef]

2012

M. Brunstein, A. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A85, 1–5 (2012).
[CrossRef]

2011

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw.22, 1469–1481 (2011).
[CrossRef] [PubMed]

M. Fiers, T. Van Vaerenbergh, K. Caluwaerts, D. Vande Ginste, B. Schrauwen, J. Dambre, and P. Bienstman, “Time-domain and frequency-domain modeling of nonlinear optical components on circuit-level using a node-based approach,” J. Opt. Soc. Am. B29, 896–900 (2011).
[CrossRef]

2010

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Time-domain and frequency-domain modeling of nonlinear optical components on circuit-level using a node-based approach,” Comp. Phys. Commun.181, 687–702 (2010).
[CrossRef]

W. H. P. Pernice, M. Li, and H. X. Tang, “Time-domain measurement of optical transport in silicon micro-ring resonators,” Opt. Express18, 18438–18452 (2010).
[CrossRef] [PubMed]

2008

2006

A. Yacomotti, P. Monnier, F. Raineri, B. Bakir, C. Seassal, R. Raj, and J. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett.97, 6–9 (2006).
[CrossRef]

T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express14, 817–831 (2006).
[CrossRef] [PubMed]

2005

2002

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: A new framework for neural computation based on perturbations,” Neural Comput.14, 2531–2560 (2002).
[CrossRef] [PubMed]

Baets, R.

Bakir, B.

A. Yacomotti, P. Monnier, F. Raineri, B. Bakir, C. Seassal, R. Raj, and J. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett.97, 6–9 (2006).
[CrossRef]

Barclay, P.

Beri, S.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Time-domain and frequency-domain modeling of nonlinear optical components on circuit-level using a node-based approach,” Comp. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Bienstman, P.

Bigot, L.

M. Brunstein, A. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A85, 1–5 (2012).
[CrossRef]

Bogaerts, W.

Borselli, M.

Brunstein, M.

M. Brunstein, A. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A85, 1–5 (2012).
[CrossRef]

Caluwaerts, K.

Coomans, W.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

Dambre, J.

Danckaert, J.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

Dumon, P.

Fiers, M.

Gelens, L.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

Guckenheimer, J.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, 2nd ed. (Springer-Verlag, 1983).

Holmes, P.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, 2nd ed. (Springer-Verlag, 1983).

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Time-domain and frequency-domain modeling of nonlinear optical components on circuit-level using a node-based approach,” Comp. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Izhikevich, E. M.

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Computational Neuroscience), 1st ed. (The MIT Press, 2006).
[PubMed]

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Time-domain and frequency-domain modeling of nonlinear optical components on circuit-level using a node-based approach,” Comp. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Time-domain and frequency-domain modeling of nonlinear optical components on circuit-level using a node-based approach,” Comp. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Johnson, T. J.

Levenson, A.

M. Brunstein, A. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A85, 1–5 (2012).
[CrossRef]

Levenson, J.

A. Yacomotti, P. Monnier, F. Raineri, B. Bakir, C. Seassal, R. Raj, and J. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett.97, 6–9 (2006).
[CrossRef]

Li, M.

Lin, Q.

Maass, W.

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: A new framework for neural computation based on perturbations,” Neural Comput.14, 2531–2560 (2002).
[CrossRef] [PubMed]

Markram, H.

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: A new framework for neural computation based on perturbations,” Neural Comput.14, 2531–2560 (2002).
[CrossRef] [PubMed]

Michael, C. P.

Monnier, P.

A. Yacomotti, P. Monnier, F. Raineri, B. Bakir, C. Seassal, R. Raj, and J. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett.97, 6–9 (2006).
[CrossRef]

Morthier, G.

Natschläger, T.

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: A new framework for neural computation based on perturbations,” Neural Comput.14, 2531–2560 (2002).
[CrossRef] [PubMed]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Time-domain and frequency-domain modeling of nonlinear optical components on circuit-level using a node-based approach,” Comp. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Painter, O.

Pernice, W. H. P.

Priem, G.

Raineri, F.

M. Brunstein, A. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A85, 1–5 (2012).
[CrossRef]

A. Yacomotti, P. Monnier, F. Raineri, B. Bakir, C. Seassal, R. Raj, and J. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett.97, 6–9 (2006).
[CrossRef]

Raj, R.

A. Yacomotti, P. Monnier, F. Raineri, B. Bakir, C. Seassal, R. Raj, and J. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett.97, 6–9 (2006).
[CrossRef]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Time-domain and frequency-domain modeling of nonlinear optical components on circuit-level using a node-based approach,” Comp. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Sagnes, I.

M. Brunstein, A. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A85, 1–5 (2012).
[CrossRef]

Schrauwen, B.

Seassal, C.

A. Yacomotti, P. Monnier, F. Raineri, B. Bakir, C. Seassal, R. Raj, and J. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett.97, 6–9 (2006).
[CrossRef]

Srinivasan, K.

Tang, H. X.

Van der Sande, G.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

Van Thourhout, D.

Van Vaerenbergh, T.

Vande Ginste, D.

Vandoorne, K.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw.22, 1469–1481 (2011).
[CrossRef] [PubMed]

Verstraeten, D.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw.22, 1469–1481 (2011).
[CrossRef] [PubMed]

Yacomotti, A.

M. Brunstein, A. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A85, 1–5 (2012).
[CrossRef]

A. Yacomotti, P. Monnier, F. Raineri, B. Bakir, C. Seassal, R. Raj, and J. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett.97, 6–9 (2006).
[CrossRef]

Comp. Phys. Commun.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Time-domain and frequency-domain modeling of nonlinear optical components on circuit-level using a node-based approach,” Comp. Phys. Commun.181, 687–702 (2010).
[CrossRef]

IEEE Trans. Neural Netw.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw.22, 1469–1481 (2011).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

Neural Comput.

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: A new framework for neural computation based on perturbations,” Neural Comput.14, 2531–2560 (2002).
[CrossRef] [PubMed]

Opt. Express

Phys. Rev. A

M. Brunstein, A. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A85, 1–5 (2012).
[CrossRef]

Phys. Rev. E

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

Phys. Rev. Lett.

A. Yacomotti, P. Monnier, F. Raineri, B. Bakir, C. Seassal, R. Raj, and J. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett.97, 6–9 (2006).
[CrossRef]

Other

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Computational Neuroscience), 1st ed. (The MIT Press, 2006).
[PubMed]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, 2nd ed. (Springer-Verlag, 1983).

R. H. Clewley, W. E. Sherwood, M. D. LaMar, and J. M. Guckenheimer“PyDSTool, a software environment for dynamical systems modeling,” http://pydstool.sourceforge.net , (2007).

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Figures (14)

Fig. 1
Fig. 1

For detuning λλr = 62 pm, Pout (Pin) is bistable (left figure), for Pin > 191μW the lower Pout-branch becomes unstable, which is an indication of self-pulsation. For Pin = 0.6 mW and (a, ΔT, N)(t=0)=(0, 0.7, 0) this gives the self-pulsation time-traces on the right.

Fig. 2
Fig. 2

On the phase portrait for Pin = 0.6 mW and a 62 pm detuning, the d(N, a)/dt = 0, dT, a)/dt = 0 nullclines only intersect at the three FPs (orange circles). In correspondence with Fig. 1 two of those FPs are unstable (open circle), while one is stable (filled circle). The example time-trace from Fig. 1 (black line) clearly follows both the dT, N)/dt directions on the da/dt = 0-surface (grey arrows) and the corresponding direction changes indicated by the nullclines. Moreover, (grey) contour lines of da/dt = 0 for |a|2 = 1 fJ – 31 fJ are elliptic and do not overlap (Appendix C).

Fig. 3
Fig. 3

The phase portrait obtained by neglecting the TPA-contribution in γloss in Eq. (6), looks similar to Fig. 2 and still explains (approximately) the dynamic behaviour of the time-trace of the 4D-system from Fig. 1 (black line). Furthermore, the time-trace with a corresponding initial condition in the 2D-approximation (dashed magenta lines) follows qualitative the 4D-behaviour, both in phase-plane and in time-domain, although the shape of the limit cycle (LC) is slightly different. The yellow line is the separatrix of the simplified system.

Fig. 4
Fig. 4

At the red side of the resonance (e.g., left: δλ = 62 pm) the Andronov-Hopf (AH) bifurcation (blue dot) tends to be supercritical, while it can be subcritical at the blue side of the resonance (e.g., right: δλ = −16 pm). FPs (black) and the extreme values of the LCs (magenta) in a ΔT (Pin)-bifurcation diagram, calculated using PyDSTool [14], illustrate this. Moreover, at δλ = 62 pm the ring is bistable in-between two Saddle-Node (SN) bifurcations (red dots), while at δλ = −16 pm a stable and unstable LC annihilate in a LC Fold bifurcation at Pin = 2.836 mW (black dots). Relevant Pin-values used in the other figures are indicated.

Fig. 5
Fig. 5

For some input powers and wavelength settings the LC encloses a stable FP (filled circle) in the (ΔT, N) phase-plane. This indicates a subcritical AH bifurcation. We illustrate this here for Pin = 2.85 mW and δλ = −16 pm. Depending on the initial conditions, the trajectory will converge to the LC (black curve (ΔT, N)(t=0) = (1.5 K, 8e16 cm−3)) or to the FP (magenta curve (ΔT, N)(t=0) = (1.2 K, 8e16 cm−3)).

Fig. 6
Fig. 6

A temporary increase from Pin = 1.8 mW to 2.9 mW at δλ = −16 pm, during 2 ns, triggers an excitation. Although for this input power no LC is present, the excitation can be seen as a reminiscent of the nearby LC from Fig. 5.

Fig. 7
Fig. 7

The excitability-threshold power Pthreshold is more λ than Pin dependent. Trigger pulses are Ttr = 10 ns long and have been send in the opposite direction as the pump light.

Fig. 8
Fig. 8

The refractory time Trf is the time after a pulse during which the ring is insensitive to a second perturbation (a). It is on the order of magnitude of τth, and is not much power dependent for δλ = −35 pm (b), while there is a clear wavelength dependency for Pin = 1.8 mW (c). The refractory time can be predicted by looking at the time needed for ΔT (t) to relax to the rest state (Trf,predict.). Moreover, the width of the pulse Twidth is proportional to the rise time of the temperature, i.e., the time needed to reach the maximum temperature after a pulse. In the phase portrait we indicate the trajectory the ring makes during the external perturbations with cyan, while we use black for the rest of the response.

Fig. 9
Fig. 9

If a ring is excited by a trigger signal, this excitation can excite another ring. To demonstrate this we send a CW pump signal with Pin = 1.8 mW and δλ = −16 pm through the common bus of a series of two AD filters. By exciting the first ring via the drop port (with a 10 ns trigger with Ptr = 250μW, λtr = λ) we guarantee that the external trigger pulse never reaches the second ring. The second pulse in the circuit’s output, which corresponds to the second ring’s excitation, is thus triggered by the first pulse, originating from the first ring. In contrast to the perturbation of the first ring (caused by the trigger), the second ring is initially perturbed (by the first ring) towards lower ΔT and N (right phase portrait). This causes the delay between the two excitations to be bigger than the delay between the trigger and the first pulse (time-trace bottom left).

Fig. 10
Fig. 10

Schematic of the setup for a single ring measurement. Light of a tunable laser (TL), polarized with polarization controllers (PC) is coupled in and out the chip via grating couplers (GC). The ring output is measured with a 10 GHz photodiode and visualized with a 1 GHz real-time scope. In the excitability experiment a second TL is used, mostly coupled in the opposite direction via a circulator. The pulses are created using a signal generator (SG) and a pulse pattern generator (PPG) and an electro-optical modulator (EOM). At the bottom, spectral details of both the single ring (left figure) and double ring resonances (right figure), used in this paper, are included.

Fig. 11
Fig. 11

Both the input power and wavelength clearly change the pulse shape and period of the self-pulsation in an AP ring with a 550 nm × 220 nm cross section, a 4.5μm radius, near the resonance wavelength at 1530.708 nm. (a) Input power sweep with pump wavelength detuning δλ = λλr = 40 pm. Power values are those at the output of the laser. Due to the grating coupler the on-chip input power of the ring is expected to be ∼ 6 dB lower. (b) Detuning sweep of the same ring with 5.0 dB output power at the TL laser. The self-pulsation period is in the order of ∼ 50 ns.

Fig. 12
Fig. 12

(a) If the trigger power is sufficiently high (≥ 7 dBm@TL) the ring excites with a fixed pulse shape, while for lower trigger powers subthreshold oscillations are visible. The 4 dBm pump light is detuned at δλ = −4 pm from the λr = 1530.708 nm resonance. The trigger light is tuned δλtr = 9 pm near another ring resonance at λr = 1550.671 nm. (b) The refractory time is on the order of magnitude of the self-pulsation period. The pump settings are similar to (a), while the trigger pulse settings are δλtr = 9 pm and Ptr = 5 dBm. Mentioned power values are those at the output of the lasers, due to GCs and EOM the on-chip input power of the ring is expected to be ∼ 6 dB lower for the pump light and ∼ 14 dB lower for the trigger signal.

Fig. 13
Fig. 13

If the resonances of two identical AP rings with common bus waveguide are close enough to each other they will show self-pulsation (a) and excitability (b) for the same pump wavelength and power. Both rings have a 5.0μm radius. The self-pulsation is measured at 10.5 dBm@TL (this starts at ∼ 1529.120 nm and ends around 1529.260 nm in hysteresis with single ring self-pulsation), the excitability with the pump at λ = 1529.007 nm and Pin = 13.60 dBm, while Ptr = 12.00 dBm. Trigger pulse and pump light are now co-directional. On-chip powers are therefore expected to be resp. 10.00 dB and 18.00 dB lower, as ∼ 4 dB is lost in a splitter used to combine pump and trigger signals.

Fig. 14
Fig. 14

Triggering two cascaded AP rings through the common bus with a 5 ns power increase from 1.8 mW to 2.59 mW at δ λ = −16 pm results in a similar time-lapse between two pulses in the time-trace (left) as in Fig 9. The phase-plane (right) clearly illustrates how the excitation of the first ring delays the excitation of the second ring, by kicking its trajectory towards lower N.

Tables (1)

Tables Icon

Table 1 Values of the parameters used in the simulations.

Equations (9)

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d a d t = [ j ( ω r + δ ω n l ω ) γ loss 2 ] a + κ s in ,
d Δ T d t = Δ T τ t h + Γ th γ a b s | a | 2 ρ S i c p , S i V t h ,
d N d t = N τ f c + Γ F C A β S i c 2 | a | 4 2 h ¯ ω V F C A 2 n g 2 ,
s out = e j ϕ c s in + κ a ,
γ loss = γ coup + γ rad + γ abs ,
γ abs = γ abs , lin + Γ T P A β S i c 2 | a | 2 n g 2 V T P A + Γ F C A σ S i c n g N ,
δ ω n l ω r = 1 n g ( d n S i d T Δ T + d n S i d N N ) .
[ j ( ω r + δ ω n l ( Δ T , N ) ) ω ) γ loss ( | a | , N ) 2 ] a = κ s in .
[ ( ω r + δ ω n l ( Δ T , N ) ω 2 ) + ( γ loss ( | a | , N ) 2 ) 2 ] | a | 2 = | κ s in | 2 .

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