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

Abstract The LIGO Interferometer is currently configured for optimal sensitivity in the frequency band from 150 Hz to 3000 Hz. The LIGO Interferometer is currently configured for optimal sensitivity in the frequency band from 150 Hz to 3000 Hz. The sensitivity of the interferometer peaks at every FSR, leading one to consider searching for gravitational waves at higher frequencies (37.5 kHz for 4k IFO). The sensitivity of the interferometer peaks at every FSR, leading one to consider searching for gravitational waves at higher frequencies (37.5 kHz for 4k IFO). A readout channel for gravitational waves at 37.5 kHz was set up, with output data being down-converted for a 2 kHz data channel via a lock-in amplifier. A readout channel for gravitational waves at 37.5 kHz was set up, with output data being down-converted for a 2 kHz data channel via a lock-in amplifier. The channel data was calibrated and the noise levels of the setup analyzed to determine the promise of the lock-in method. The channel data was calibrated and the noise levels of the setup analyzed to determine the promise of the lock-in method.

Static Fabry-Perot Cavity T= L/c ν fsr = c/2L E E in L E out ab

Dynamic Fabry-Perot Cavity T= L/c ν fsr = c/2L E(t) = t a E in (t) + r a r b e -2ik[L+δL(t)] E(t-2T)* L E E in xaxa xbxb a b *Dynamic resonance of light in Fabry-Perot Cavities Rakhmanov, Savage, Reitze, Tanner

Maintaining Resonance A Pound-Drever-Hall negative feedback signal is generally used to lock a cavity to resonance. A Pound-Drever-Hall negative feedback signal is generally used to lock a cavity to resonance. For a static cavity, length and frequency variations are essentially equivalent. Not so for a dynamic cavity! For a static cavity, length and frequency variations are essentially equivalent. Not so for a dynamic cavity! To maintain resonance, cavity length variations must be compensated for by laser frequency variations. To maintain resonance, cavity length variations must be compensated for by laser frequency variations. The length-to-signal and frequency-to-signal transfer functions for the PDH signal reflect the differing effects of frequency and length variations in a dynamic cavity. The length-to-signal and frequency-to-signal transfer functions for the PDH signal reflect the differing effects of frequency and length variations in a dynamic cavity.

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 FSR multiples, the sensitivity to length variations is at a maximum while the sensitivity to frequency is at a minimum.

Periodic Response (H L (s))

The Best of News, the Worst of News The peaks in the length-to-signal transfer function suggest a search for gravitational waves near FSR multiples. The peaks in the length-to-signal transfer function suggest a search for gravitational waves near FSR multiples. However, the cavity response to gravitational wave strains in an optimally- oriented interferometer mimics that of H  (s) in that it goes to zero at FSR multiples. However, the cavity response to gravitational wave strains in an optimally- oriented interferometer mimics that of H  (s) in that it goes to zero at FSR multiples. The sensitivity to other gravitational wave orientations at FSR multiples still peaks at FSR multiples. The sensitivity to other gravitational wave orientations at FSR multiples still peaks at FSR multiples. The anticipated sources at this high frequency are distant and thus difficult to detect (e.g. neutron star modes). The anticipated sources at this high frequency are distant and thus difficult to detect (e.g. neutron star modes).

Readout Configuration Antisymmetric Photodetector Port Differential Driver Lock-In AmplifierDAQ AS Hz Data Channel H1:PSL-TEST2_F

The Lock-In Amplifier Traditionally used to extract the component of the input signal at a specified frequency. Traditionally used to extract the component of the input signal at a specified frequency. Employs a mixer to multiply the input with a reference frequency sine wave, resulting in a DC output at ω ref = ω lock : Employs a mixer to multiply the input with a reference frequency 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 variable low-pass filter, allowing the user to determine the output bandwidth. The output signal is fed through a variable low-pass filter, allowing the user to determine the output bandwidth. The phase of the input is determined through a second mixer, this time multiplying by the reference signal phase shifted by 90°. An amplifier follows. The phase of the input is determined through a second mixer, this time multiplying by the reference signal phase shifted by 90°. An amplifier follows. Thus the lock-in measures both the amplitude and phase of the component of the input signal at the reference frequency. Thus the lock-in measures both the amplitude and phase of the component of the input signal at the reference frequency.

Lock-In As Down Converter The lock-in reference is set to 37 kHz. The lock-in reference is set to 37 kHz. The sensitivity band of a 2 kHz data channel only encompasses difference terms of the mixer output. The sensitivity band of a 2 kHz data channel only encompasses difference terms of the mixer output. The time constant on the ensuing low pass filter is fixed to its lowest setting, preventing filtering of difference terms in the range of the channel sensitivity. The time constant on the ensuing low pass filter is fixed to its lowest setting, preventing filtering of difference terms in the range of the channel sensitivity. The output represents the superposition of amplified data from 36.1 kHz to 37 kHz and 37 kHz to 37.9 kHz in the band from Hz. The output represents the superposition of amplified data from 36.1 kHz to 37 kHz and 37 kHz to 37.9 kHz in the band from Hz. Since the sensitivity of the data channel from 36.1 kHz to 37 kHz is minimal, the net output seen is primarily the input signal from 37.1 kHz to 37.9 kHz down-converted by 37 kHz and amplified. Since the sensitivity of the data channel from 36.1 kHz to 37 kHz is minimal, the net output seen is primarily the input signal from 37.1 kHz to 37.9 kHz down-converted by 37 kHz and amplified.

ITMX Length Drive To calibrate cavity length changes, the ratio between test mass displacement and ASPD voltage signal was determined with the 4k IFO in low power common mode lock. To calibrate cavity length changes, the ratio between test mass displacement and ASPD voltage signal was determined with the 4k IFO in low power common mode lock. A Stanford SR785 provided a sinusoidal drive signal, which was split and amplified by a factor of 5 by two preamplifiers. A Stanford SR785 provided a sinusoidal drive signal, which was split and amplified by a factor of 5 by two preamplifiers. One preamplifier output was inverted, while the other remained non-inverted. One preamplifier output was inverted, while the other remained non-inverted. The inverted output was sent to the UR and LL suspension coils of the ITMX while the non-inverted output went to the LR and UL coils. The inverted output was sent to the UR and LL suspension coils of the ITMX while the non-inverted output went to the LR and UL coils. The transfer function of the ITMX amplitude response was fed out through the ASPD port and viewed on the SR785 via a swept sine measurement. The transfer function of the ITMX amplitude response was fed out through the ASPD port and viewed on the SR785 via a swept sine measurement.

Calibration Setup SR785SR560’s ITMX ASPD Port AS2 ITMX Drive Input Inv.

ITMX Drive Transfer Function Transfer function, 1600 Hz span.Transfer Function, 400 Hz span.

Length Calibration The length calibration of the setup was determined through extrapolation of the DC calibration done by Michael Landry. The length calibration of the setup was determined through extrapolation of the DC calibration done by Michael Landry. The test mass responds to the drive as a simple pendulum, with a transfer function proportional to (f o /f) 2. The ratio between ASQ voltage and ITMX displacement was found by: The test mass responds to the drive as a simple pendulum, with a transfer function proportional to (f o /f) 2. The ratio between ASQ voltage and ITMX displacement was found by: Where M = nm/count (DC Calibration Factor) f o = Hz(Pendulum Characteristic Frequency) f o = Hz(Pendulum Characteristic Frequency) T = -77 dB(Transfer Function Response) T = -77 dB(Transfer Function Response) A = 5(Signal Amplification) A = 5(Signal Amplification) The final ratio around the FSR was found to be: The final ratio around the FSR was found to be: 1 V ASQ => 5.7 x ITMX 1 V ASQ => 5.7 x m ITMX

Length Sensitivity Measurement A power spectrum of the ASPD QMON signal was taken with the IFO in low power common mode lock, generating a representation of the noise level. A power spectrum of the ASPD QMON signal was taken with the IFO in low power common mode lock, generating a representation of the noise level. The sensitivity of the channel was found by dividing the noise spectrum by the length calibration spectrum. The sensitivity of the channel was found by dividing the noise spectrum by the length calibration spectrum. The length sensitivity peaks at the cavity FSR. The length sensitivity peaks at the cavity FSR.

Noise Spectrum

Frequency to Signal Transfer Function Response at FSR

FSR Length Sensitivity, 400 Hz Bandwidth

FSR Length Sensitivity, 200 Hz Bandwidth

Conclusions The SR830 lock-in amplifier has been installed and used for down-conversion to the 2 kHz data channel H1:PSL-TEST2_F. The SR830 lock-in amplifier has been installed and used for down-conversion to the 2 kHz data channel H1:PSL-TEST2_F. The lock-in was determined to be adequate for frequency down-conversion, with its output characterized in terms of its output window and amplification. The lock-in was determined to be adequate for frequency down-conversion, with its output characterized in terms of its output window and amplification. The sensitivity of the 4k IFO to ITMX length displacement near 37.5 kHz is approximately 2 x m, with a dip down to 5 x at the FSR frequency. The sensitivity of the 4k IFO to ITMX length displacement near 37.5 kHz is approximately 2 x m, with a dip down to 5 x at the FSR frequency. The sensitivity to gravitational wave strains averaged over all orientations has been proposed to be approximately 5 times worse (D. Sigg, 2003). The sensitivity to gravitational wave strains averaged over all orientations has been proposed to be approximately 5 times worse (D. Sigg, 2003). A sample of actual channel data was plotted into a histogram to reveal a noise signal that was approximately Gaussian, as expected. A sample of actual channel data was plotted into a histogram to reveal a noise signal that was approximately Gaussian, as expected.