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Dynamical thermal behavior and thermal self-stability of microcavities

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As stability and continuous operation are important for almost any use of a microcavity, we demonstrate here experimentally and theoretically a self-stable equilibrium solution for a pump-microcavity system. In this stable equilibrium, intensity- and wavelength-perturbations cause a small thermal resonant-drift that is enough to compensate for the perturbation (noises); consequently the cavity stays warm and loaded as perturbations are self compensated. We also compare here, our theoretical prediction for the thermal line broadening (and for the wavelength hysteretic response) to experimental results.

©2004 Optical Society of America

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Figures (5)

Fig. 1.
Fig. 1. Dynamical thermal behavior of a toroidal microcavity (a) As the pump makes a 46 Å/s wavelength scan (upper red) it approaches the cavity resonance (upper blue) and cause a thermal drift of the resonance line. In the upper plot, the right ordinate describes the temperature of the mode volume relative to the ambient temperature and the left ordinate gives the pump wavelength and cavity resonance wavelength (relative to the cold resonance). The pump-cavity transmission is presented as a function of time (middle) and as a function of the pump wavelength (bottom). Here blue dots represent experimentally measured data and lines stand for calculations. The parameters used for the fit are Ih /K=7.16°C and Ih /Cp =18000 °C/s. The cold resonance wavelength of the microcavity is λ 0 =1545nm and its quality factor is Q = 2×107, the pump power was 1.8 mW. (b) To emphasize fine details, we repeat the same calculation but using a reduced Q (Q = 5×105), For convenience, the cavity FWHM is marked on the cavity resonance wavelength (top blue).
Fig. 2.
Fig. 2. (2.4 MB) Movie of the cavity wavelength response (6.6 MB version). The scanning-pump (red) induces thermal drift of the resonant lineshape (blue). The intersection point between the pump line and the cavity Lorentzian draws the hysteretic absorption (black). Parameters here are identical to the parameters in Fig. 1(b). The movie is in slow motion as each scan cycle truly takes 80 ms
Fig. 3
Fig. 3 Equilibrium solutions: (a) Presented are the resonant thermal wavelength-drift (left ordinate) and the cavity temperature (right ordinate, relative to the ambient temperature) at equilibrium for various values of pump wavelength (horizontal axis). The 3 distinct equilibrium regimes are color coded. On the right (b, c, and d), we illustrate a solution-triplet for the pump position shown in the left panel. (Parameters are as in Fig. 1(b))
Fig. 4.
Fig. 4. Numerical calculation of dynamical noise response at equilibrium. (a) Warm stable-equilibrium: The stable warm equilibrium is reached by an upward wavelength scan, stopping at a pump wavelength of 0.56 angstrom above the cold resonance. In this equilibrium, the system overcomes Gaussian noise in the pump wavelength (with amplitude of one cavity width). The noise spectra is of random amplitude and spread Gaussianly in the Fourier space having FWHM of 100 KHz around the DC. (b) Unstable warm equilibrium: Starting in the unstable warm equilibrium (pump wavelength 0.56 angstrom and cavity thermal-drifted resonance 0.52 angstrom above the cold resonance), the smallest positive noise will take the system to the warm stable-equilibrium; while the smallest negative noise will take the system to the cold stable-equilibrium. Noise here is smaller than 1/1010 of the cavity FWHM. In this figure all parameters are as in Fig 1(b) (except for the pump wavelength), Figs. 2 and 3. All temperatures are relative to the ambient temperature and all wavelengths are relative to the cold cavity resonance.
Fig. 5.
Fig. 5. The wavelength response for a slow scan (0.002 Å/s) possesses a stable, warm equilibrium during the upward wavelength scan. A fast pass through resonance occurs during the downward wavelength scan (note that we zoom in here and show only a fraction of the wavelength scan). When in the warm stable equilibrium (in a region near 20 s in the scan), the system can recover from a perturbation as shown in the inset. We used a spherical cavity with a diameter of 0.26mm and Q = 2×106.

Equations (7)

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N = 2 πr c 1 ( 1 + ε Δ T ) λ r ( n 0 + dn dT Δ T )
λ r ( Δ T ) λ 0 [ 1 + ( ε + dn dT n 0 ) Δ T ]
λ 0 ( 1 + a Δ T ) .
q ˙ in = I η Q Q abs 1 ( λ p λ r Δ λ 2 ) 2 + 1 I h 1 ( λ p λ 0 ( 1 + a Δ T ) Δ λ 2 ) 2 + 1
Cp Δ T ˙ ( t ) = q ˙ in q ˙ out
= I h 1 ( λ p λ 0 ( 1 + a Δ T ) Δ λ 2 ) 2 + 1 K Δ T ( t ) .
0 = I h 1 ( λ p λ 0 ( 1 + a Δ T ) Δ λ 2 ) 2 + 1 K Δ T .
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