Parris E. Trahanas, Chunnong Zhao, Li Ju, and David G. Blair, "Double end-mirror sloshing cavity for optical dilution of thermal noise in mechanical resonators," J. Opt. Soc. Am. B 37, 1643-1652 (2020)
The sensitivity of gravitational wave detectors at high frequencies is currently limited by quantum shot noise. It has been shown theoretically that white light signal recycling using optomechanical negative dispersion filter cavities can increase the gain-bandwidth product of gravitational wave detectors that is usually limited by conventional signal recycling. This sensitivity enhancement is most pronounced at high frequencies above 500 Hz. This technology could be implemented in current facilities or future detectors, but requires thermal noise dilution of the mechanical resonator. Here we provide a theoretical analysis of a double end-mirror sloshing (DEMS) cavity to achieve strong thermal noise dilution with low radiation pressure noise and suppression of optical anti-damping. We demonstrate experimentally that the DEMS cavity can be tuned to a regime that is expected to support these favorable conditions.
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Examples of DEMS Cavity Parameters That Minimize Radiation Pressure Noise and Achieve the Required Optical Spring Stiffnessa
Parameter
Symbol
Value
HR coating loss
10 ppm
AR coating loss
50/35/20 ppm
Input mirror transmissivity
97/82/67 ppm
Effective mass
40 ng
Beam spot radius
6.35 µm
Max power density
Circulating Power
31.7/47.5/63.3 W
Sloshing mirror transmissivity
Bare mechanical frequency
Optical spring shifted frequency
Laser wavelength
1064 nm
Half-cavity length
L
1 cm
Effective temperature
4.42–5.41 K
Diffraction loss
5.00 ppm
Bare mechanical quality factor
Optical damping
The HR coating loss parameter estimates the combined loss due to transmission, scattering, and absorption at high-reflectivity AlGaAs/GaAs coatings. The AR coating loss estimates combined loss due to reflection, scattering, and absorption effects at the sloshing mirror AR coating. The maximum power density (and beam spot radius) dictates the maximum circulating power and, for each value of AR coating loss, the input transmissivity is adjusted to maximize the cavity gain. Therefore, there are only nine different parameter combinations, six of which are plotted in Fig. 5. We include the range of ${T_{{\rm eff}}}$ [see Eq. (10)] over all parameter combinations.
Table 2.
Thermal Noise Calculation Parameters and Amplitude Spectra at the 170 KHz Operating Frequencya
Parameter
Symbol
Value
Poisson ratio
0.29/0.32
Young’s modulus
400/100 GPa
Loss angle
Effective TCC
Effective substrate TEC
Effective coating TEC
Effective TRC
Specific heat capacity
Coating thickness
5.8 µm
Coating Brownian noise
Substrate Brownian noise
Substrate thermoelastic noise
Coating thermo-optic noise
Total mirror thermal noise
Resonator thermal noise
TCC (TEC) stands for thermal conductivity (expansion) coefficient. TRC is the thermorefractive coefficient. Subscripts s and c denote the substrate and coating parameter values, respectively. Each displacement noise value is the square root of the sum of the power spectral density contributions from HR coatings on all mirrors (see Fig. 1). $\sqrt {{S_{{\rm tot}}}}$ is the amplitude spectrum of all four noise sources from six HR faces. Specific heat ${C_p}$ capacities given are at constant pressure, the second subscript establishes the substrate versus coating. The bar above coefficients indicates that the “effective” quantity is an average over all coating layers or all substrate materials [54].
The radiation pressure noise force amplitude spectrum and optical damping are given at the operating frequency. We also include the effective temperature ${T_{{\rm eff}}}$ (cf. Table 1). Note also the magnitude of the negative damping relative to the mechanical damping of the resonator and the optical damping of the DEMS cavity.
Tables (3)
Table 1.
Examples of DEMS Cavity Parameters That Minimize Radiation Pressure Noise and Achieve the Required Optical Spring Stiffnessa
Parameter
Symbol
Value
HR coating loss
10 ppm
AR coating loss
50/35/20 ppm
Input mirror transmissivity
97/82/67 ppm
Effective mass
40 ng
Beam spot radius
6.35 µm
Max power density
Circulating Power
31.7/47.5/63.3 W
Sloshing mirror transmissivity
Bare mechanical frequency
Optical spring shifted frequency
Laser wavelength
1064 nm
Half-cavity length
L
1 cm
Effective temperature
4.42–5.41 K
Diffraction loss
5.00 ppm
Bare mechanical quality factor
Optical damping
The HR coating loss parameter estimates the combined loss due to transmission, scattering, and absorption at high-reflectivity AlGaAs/GaAs coatings. The AR coating loss estimates combined loss due to reflection, scattering, and absorption effects at the sloshing mirror AR coating. The maximum power density (and beam spot radius) dictates the maximum circulating power and, for each value of AR coating loss, the input transmissivity is adjusted to maximize the cavity gain. Therefore, there are only nine different parameter combinations, six of which are plotted in Fig. 5. We include the range of ${T_{{\rm eff}}}$ [see Eq. (10)] over all parameter combinations.
Table 2.
Thermal Noise Calculation Parameters and Amplitude Spectra at the 170 KHz Operating Frequencya
Parameter
Symbol
Value
Poisson ratio
0.29/0.32
Young’s modulus
400/100 GPa
Loss angle
Effective TCC
Effective substrate TEC
Effective coating TEC
Effective TRC
Specific heat capacity
Coating thickness
5.8 µm
Coating Brownian noise
Substrate Brownian noise
Substrate thermoelastic noise
Coating thermo-optic noise
Total mirror thermal noise
Resonator thermal noise
TCC (TEC) stands for thermal conductivity (expansion) coefficient. TRC is the thermorefractive coefficient. Subscripts s and c denote the substrate and coating parameter values, respectively. Each displacement noise value is the square root of the sum of the power spectral density contributions from HR coatings on all mirrors (see Fig. 1). $\sqrt {{S_{{\rm tot}}}}$ is the amplitude spectrum of all four noise sources from six HR faces. Specific heat ${C_p}$ capacities given are at constant pressure, the second subscript establishes the substrate versus coating. The bar above coefficients indicates that the “effective” quantity is an average over all coating layers or all substrate materials [54].
The radiation pressure noise force amplitude spectrum and optical damping are given at the operating frequency. We also include the effective temperature ${T_{{\rm eff}}}$ (cf. Table 1). Note also the magnitude of the negative damping relative to the mechanical damping of the resonator and the optical damping of the DEMS cavity.
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