Optical self cooling of a deformable Fabry-Perot cavity in the classical limit

We investigated the optomechanical properties of a Fabry-Perot cavity with a mirror mounted on a spring. Such a structure allows the cavity length to change elastically under the effect of light-induced forces. This optomechanical coupling is exploited to control the amplitude of the mechanical fluctuation of the mirror, a situation referred to as optical self cooling or passive optical cooling. We present a model developed in the classical limit and discuss data obtained in the particular case in which photothermal forces are dominant.

Figure 1

(a) Schematic model of a deformable Fabry-Perot cavity. (b) After discreet step-shaped changes in mirror distance $z$, the light-induced force $F$ grows after a characteristic delay time $\tau$.

Figure 2

A FP cavity with a mirror attached on a spring is illuminated with a laser beam. The input laser intensity is assumed to be noiseless. The transmitted light shows amplitude fluctuations, which are impressed on the original amplitude by the thermal fluctuation of the mirror. More importantly, the transmitted laser light has an averaged intensity enhanced by the fluctuations added by the mechanical resonator. The mirror vibrational motion has been cooled and the excess energy turns into photons.

Figure 3

(a) Schematic of the experimental setup. Inset (b) shows a photograph of a similar cavity as used in the experiment but with a lever to mirror separation greatly increased for the picture. During measurements, the cavity length was $34\mu \text{m}$.

Figure 4

(a) Dependence of the cantilever’s resonance linewidth at full width at half maximum on pressure and laser power. The triangles show data taken with a HeNe laser at moderate pressure of about ${10}^{-3}\text{mbar}$, while the circles show data taken at the minimum pressure of $5\times {10}^{-6}\text{mbar}$. The squares represent data taken also at the minimum pressure but using a red (670 nm) diode laser instead. Clearly, the linewidth of the cantilever is much smaller at $5\times {10}^{-6}\text{mbar}$. (b) Mechanical quality factors of different cantilevers with various thicknesses of evaporated gold. The gray line is a guide to the eye.

Figure 5

(a) Normalized reflectivity of plan-plan cavity setup shown in Fig. 3. The cavity detuning is calibrated in units of the wavelength $\lambda$. The finesse is 4, with the parameter $\text{g}=2.5$. (b) Amplitude fluctuation spectral density near vibrational resonance of the cantilever with $f_0=7.3\text{kHz}$ with different laser powers taken at cavity detuning of $+\lambda /25$ from a cavity resonance. The largest amplitude corresponds to thermal fluctuation at 300 K with mechanical damping $\Gamma =28\text{Hz}$, measured with reflected laser power of $3.1\mu \text{W}$. The other measurements correspond to reflected laser powers of 0.87, 1.3, and 3.6 mW with damping of $\Gamma =98$, 131, and 263 Hz. The effective temperatures of the spectra from top to bottom are 300, 86, 64, and 32 K.

Figure 6

Response measurement of driven cantilever amplitude. Frequency sweep is from 0 to 100 kHz. The upper inset shows a zoom of the region around the cantilever resonance frequency at 7.3 kHz. The inset at the bottom shows a zoom of the enhanced response at $f=284\text{Hz}$ arising at the frequency where $\omega \tau =1$. The response frequency corresponds to ${\tau }_{\text{pth}}=560\mu \text{s}$.

Figure 7

(a) Cooling behavior of fundamental vibrational mode at 8.7 kHz. Laser powers coupled in the fiber before the cavity are $0.16\mu \text{W}$ for Brownian peak, then 2.25, 5.8, and $7.6\mu \text{W}$, corresponding to 300, 174, 102, and 94 K, respectively. The fits were made according to Eq. (28). The effective damping for the spectra is shown in (c). (b) Cooling of the first harmonic at 60.6 kHz, with laser powers of 0.31, 0.49, 3.14, 4.19, and $4.53\mu \text{W}$ corresponding to 300, 290, 251, 240, and 239 K, respectively. The offset of the spectra shows $1/\sqrt{P}$ dependence and is caused by the shot noise of the laser. (c) Effective damping ${\Gamma }_{\text{eff}}$ with laser power before cavity for the ground mode at 8.7 kHz. ${\Gamma }_{\text{eff}}$ shows linear power dependence according to Eq. (14). (d) Effective damping ${\Gamma }_{\text{eff}}$ with laser power before cavity for the first harmonics at 60.6 kHz.