RSE allows much more freedom to optimize the detector response compared to the initial configuration. A desktop prototype RSE interferometer was constructed to demonstrate the feasibility of the proposed signal extraction scheme and the tunability of the RSE interferometer.
List of Tables
Chapter 1
Prologue
- General Relativity
- The Search for Gravitational Waves
- Resonant Bar Detectors
- Interferometers
- Interferometer Sensitivity
- Seismic Noise
- Thermal Noise
- Optical Noise
- Interferometer Configuration
- The Goal of this Work
- Chapter 2
Both thermal and seismic noise affect the position of the surface of the test mass. The mass of the test mass is m, and L is the length of the interferometer arms.
Resonant Sideband Extraction
Frequency Response
- The Three Mirror Coupled Cavity
- RSE Transfer Functions
The view from the arm cavity "sees" the reflectivity of the signal cavity change as its phase varies. The fact that there is a peak frequency is a result of the undercoupled nature of the signal cavity.
Practical Limitations
- Power Recycling
- Transfer Function Limitations
- Implications
On the other hand, as the transmission of the ITM is increased, the effect of the losses in the signal cavity decreases. However, the effect of the losses in the signal cavity decreases because its finesse decreases.
Detector Optimization for RSE
- Narrowband Response
- Optimized Broadband
For sensitivity at 1kHz, a 5kHz RSE bandwidth is likely to be required, implying an effective arm/signal cavity finesse of approx. 4. The signal cavity reflectivity, which allows this wide bandwidth, is approx. 20%. For each particular pair, the vote was determined by Eq. 2.7), the effect in the arms is calculated using Eq. 2.27), and this is multiplied by the value of the transfer function evaluated at the target frequency using Eq.
Chapter 3
Signal Extraction
Optically Heterodyned Signals
- DC Signal
- Frequency Dependent Signal
- Phasor Diagrams
- Twiddle
The ti's are the transmission functions from the input of the interferometer to the photodiode. The derivatives of the carrier and RF sidebands are assumed to be imaginary and real, respectively. Here it can be seen that the quadrature demodulation measures the differential amplitude modulation of the two RF sidebands.
The RSE Interferometer
- Asymmetry
The signal power and cavity lengths are chosen such that the round-trip phase for the RF sidebands satisfies a resonant condition. The first solution assumes that the signal power and cavity are both resonant for the RF sidebands. Power and signal cavity phases are shown as biased phases in the legend.
Additional Sidebands
First, the dark-port photodiode is not sensitive to any of the degrees of freedom except the differential modes
Proposed Signal Extraction Scheme
It is desirable that one of the RF sideband frequencies resonate in the signal cavity while the other does not. Because both the frequency of one of the RF sidebands and the detuning frequency must be resonant in the signal cavity,4. One of these pairs is not particularly useful in generating the measured signal for the signal extraction cavity, as only one of the 3f RF sidebands resonates there.
Signal Sensitivity
- Reflected and Pickoff Signals
- Dark Port Signals
In the ideal interferometer, the calculation of the demodulation phase takes a fairly simple form. To express the derivative of the Michelson mirror with respect to the degrees of freedom in the arm cavity, the following relations are needed. The derivative of the carrying field with respect to the degree of freedom cjJ_ is zero.
Matrix of Discriminants
- Optimized Broadband
- Narrowband RSE
Increasing the asymmetry does reduce the transmission of the coupled cavity for the lower PM RF sideband and lowers the effective finesse of both the power and signal cavities. The last column is the transmission from the lower PM RF sideband to the dark port. Because of the scale of the plant, the values in H represent the loop gains of each loop.
Conclusions
Asymmetry variation was used as a means to improve the diagonality of the matrix. It was found that a reasonable double gain hierarchy could be created which did not significantly degrade the performance of the Michelson degrees of freedom. In particular, some cases were found (especially with T prm < Tsem) in which changing the asymmetry did very little to improve the conditioning of the matrix.
Chapter 4
The RSE Tabletop Prototype Experiment
Prototype Design
- Optics
- Lengths and Frequencies
- Electro-Optics
- Mirror Mounting
Two input test masses (ITM) and two final test masses (ETM) form the arms of the interferometer. The shortest choice of energy recycling cavity length corresponds to a free spectral range of 54 MHz, or 2. The nominal signal cavity length for wideband RSE corresponds to a free spectral range of 81 MHz, or 1.85 m. The RF gain was usually a few hundred V/W. The other two photodiodes used were New Focus model 1811 photodiodes.
Electronics
The SR560's "-B" input is used for calibrated offset adjustments as well as some transfer function measurements. As mentioned before, the response for + and _ is done as a combination of actuators on both arms. Second, they supply local oscillators for the mixers to demodulate all signals, as well as a phase shifter for each LO to control the demodulation phase.
Input Optics and the Mach-Zender
The first Mach-Zender beam splitter is a window that transmits about 95% of the incident power. An iris, placed next to the exit Mach-Zender beam splitter, is used to block all other diffracted orders. The output beam splitter of the Mach-Zender is approximately 50/50, so that half of the light input to the Mach-Zender goes to the interferometer, while the other half is used to control the Mach-Zender.
Arm Cavities
- Visibility
- Mode Matching
The intersection of the two plots is a solution that places the waist of the same size in the same place. MM is the mode matching fraction, and Rcavity is the cavity reflection at resonance for the cavity mode, which is a function of the losses. Figure 4.6 shows a graph showing mode matching as a function of cavity loss.
Experimental Process
- Fabry-Perot Michelson
- Dual-Recycled Michelson
- Systematic Errors
Several measurements are made on the Fabry-Perot Michelson, which provide the calibration for the rest of the experiment. The output is taken from the output degree of freedom being tested. In the end, the offsets still varied, at most by a few mV outside the mixers, especially in the 81 MHz mixers.
Data and Results
- Dual-Recycled Michelson
- Gravitational Wave Transfer Function
The matrix measurement will be limited to the <1>+, ¢+, cp_ and c/Js degrees of freedom in the 81 MHz sampler and reflected and sampled 54 MHz signal ports. The measurement of the _ signal in the dark port of the 81 MHz signal port remains as a measurement of the gravitational wave transfer function. The decrease in the size of the + transfer function below 2 kHz is mainly due to saturation effects in the electronics.
Conclusions
The assumption that the response is linear is probably a bit optimistic, which would lead to an overestimation of the resulting phase shift. Several other aspects of the detuned RSE were also demonstrated: the sensitivity to the demodulation phase, as well as the compensations in the control of the signal cavity. This gives confidence in using the Twiddle model to design a control system for an RSE interferometer with unbalanced RF sidebands.
Chapter 5
Laser Noise Couplings in RSE
Conceptual Motivation
- Balanced Sidebands
- Unbalanced Sidebands
Therefore, the relative amplitude of the phase fluctuations in the carrier and RF sidebands reaching the dark gate tend to be Amplitude noise is free from this coupling mechanism in the balanced sideband case due to the orthogonality of the resulting phasors. In this case, the noise ellipse of the RF sidebands onto the carrier is misprojected.
Figure of Merit
The detuning of the signal cavity causes the upper and lower noise sidebands to also propagate unequally. It only has an equivalent displacement noise by dividing it by the detector transfer function. This is a shift spectrum times the transfer function Tsignal, which is units of watts per meters. XsignazU) is taken to be a spectral density of the source in units of m/JHZ.
Input Noise
- Laser Frequency Noise
- Laser Amplitude Noise
The consequence of this assumption is that there is coherence between the carrier and RF sideband noise terms, and that the noise amplitude scales directly with the carrier and RF sideband amplitudes. RF oscillator phase noise requires a fundamentally different approach, as it adds noise to the demodulation stage as well. The amplitude noise is modeled as a modulation of the amplitude of the input field Ei. 5.8), and low-order expansion as in the case of frequency noise, gives 5.12).
Readout
Transmission of Light
- The General Transmission Function
- The Carrier
- RF Sidebands
These terms in the relationship are the interferometer-dependent gains for the amplitude of the mismatch carrier defect. The details of the signal cavity determine the overall coupling in the bandwidth of interest. The treatment of the RF sidebands in the interferometer has two major simplifications over that of the carrier.
Calculation of the Measured Noise
- Broadband RSE
- Detuned RSE
Also, .6.g is nominally zero in the balanced RF sideband case, so it is also a measure of the imbalance of the RF sidebands in the interferometer. The in-phase demodulation noise couplings due to frequency noise are given in the following equations. Also note that the one term that would survive in the broadband RSE interferometer, R {g}, picks out the frequency and amplitude noise coupling mechanisms found in the previous section.
5. 7 Analysis
Optimized Broadband RSE
The modulation frequency is assumed to be 81 MHz, corresponding to the work in this thesis. The amplitude noise at the interferometer input must be less than 2 x 10-9/JHZ, which is a much stricter requirement than the LIGO I requirement of 10-7 /VHZ.[59] The coupling of amplitude noise to the mismatch carrier defect, as well as the increased laser power, are the main factors driving the tightening of this specification. Due to the carrier defect suppression of RSE over LIGO I, which is the dominant coupling for frequency.
Narrowband RSE
The stronger suppression of the carrier defect due to the lower transmission of the signal mirror reduces this coupling mechanism, as discussed in Section 5.5.2. The change in these figures from the optimized broadband reflects the lower finesse of the arm cavities (which, as noted, is mainly due to the improved transmission of carrier noise around the peak frequency.
5. 7.3 Conclusion
Amplitude noise has another effect on the interferometer, known as "technical radiation pressure noise". Intensity fluctuations are translated into force fluctuations on the test masses due to the recoil of the photons reflected from them. An estimate of acceptable amplitude noise, so that this is also a factor of 10 below the basic noise sensitivity, is also roughly ;S 10-9/ JHZ.
Appendix A
The Fabry-Perot Cavity
The reflectivity, as a function of the mirror parameters and the round trip phase is given by. Lowercase letters r'e and t'e will be used for amplitude reflectivity and transmittance, while uppercase letters R'e and T'e will be used for power reflectance and transmission. It should be noted that, at resonance, the cavity reflectivity is less than any of the individual cavity mirrors, while the reflectivity is greater than any of the individual cavity mirrors for most of the phase range away from resonance.
