Development of a Readout Scheme for High Frequency Gravitational Waves Jared Markowitz Mentors: Rick Savage Paul Schwinberg

Abstract The LIGO Interferometer is currently configured for optimal sensitivity at approximately 150 Hz. The sensitivity of the interferometer peaks at every FSR, leading one to consider searching for gravitational waves at higher frequencies (37.5 kHz). A readout channel for gravitational waves at 37.5 kHz will be set up, and output data will be down-converted to low frequency to match the existing data-acquisition system. The data will be examined for sources of noise in this frequency range, and also will be surveyed for gravitational waves.

The Fabry-Perot Cavity T= L/cω fsr = /T E(t) = t a E in (t) + r a r b e -2ik[L+δL(t)] E(t-2T)

Tale of Two Transfer Functions The normalized frequency to signal transfer function H  (s), pictured above, has zeros at multiples of the FSR. The normalized frequency to length transfer function H L (s), shown below, has its maxima at multiples of the FSR. This indicates that at multiples of the FSR, the sensitivity to length variations is at a maximum while the sensitivity to frequency is at a minimum. This suggests searching for gravitational waves at multiples of the FSR. However, GW response more complicated than H L (s).

Frequency to Signal Transfer Function Response at FSR

Hurdles to Clear Sources of gravitational waves at 37.5 kHz must be identified and characterized. Sources of background noise in the high frequency range of 37.5 kHz must be determined and accounted for. At multiples of the FSR, there is no response in an optimally-oriented interferometer to gravitational waves. This stems from the fact that gravitational waves affect both the light and the mirrors in the interferometer, making H  the pertinent quantity in calculating the response. However,gravitational waves may be detected with increased sensitivity at these frequencies for other orientations.

What’s Been Done Background research on Fabry-Perot Cavities, PDH locking systems. Interfaced SR830 Lock-In Amplifier with Unix terminal via RS232 interface. (Thank you Richard and Dave!) Wrote a C program to cycle lock-in reference frequencies through the serial port, allowing the generation of transfer functions remotely. Tested program on a 37.5 kHz bandpass filter, feeding output to Matlab for plotting. Obtained same results as were seen from the dynamic signal analyzer plot.

Lock-In Amplifier Used to extract a signal at a given reference frequency from background noise. Employs a PSD (phase sensitive detector) to multiply the input signal with the reference signal (a sine wave), resulting in a DC output at ω ref = ω lock. V prod = V sig V lock sin(ω ref t +  sig )sin(ω lock t +  ref ) = ½V sig V lock cos([ω ref - ω lock ]t +  sig -  ref ) - ½V sig V lock cos([ω ref + ω lock ]t +  sig +  ref ) The output signal is fed through a low-pass filter, essentially eliminating all but the DC signal. The phase dependency of the DC signal is eliminated through sending the signal through a second PSD, this time multiplying by the reference oscillator signal phase shifted by 90°. This allows the lock-in to measure both the amplitude and phase of the component of the input signal equal in frequency to the reference.

Where to go next Set up a 2kHz fast channel for down-converted higher frequency signals. Configure lock-in to down-convert high frequency signals. Calculate the band-limited RMS in low frequency bins.

For more detail, see: