Determination and optimization of mode matching into optical cavities by heterodyne detection

Single and coupled cavity mode sensing schemes using a diagnostic field

Precise optical mode matching is of critical importance in experiments using squeezed-vacuum states. Automatic spatial-mode matching schemes have the potential to reduce losses and improve loss stability. However, in quantum-enhanced coupled-cavity experiments, such as gravitational-wave detectors, one must also ensure that the sub-cavities are also mode matched. We propose what we believe to be a new mode sensing scheme, which works for simple and coupled cavities. The scheme requires no moving parts, nor tuning of Gouy phases. Instead a diagnostic field tuned to the HG20/LG10 mode frequency is used. The error signals are derived to be proportional to the difference in waist position, and difference in Rayleigh ranges, between the sub-cavity eigenmodes. The two error signals are separable by 90 degrees of demodulation phase. We demonstrate reasonable error signals for a simplified Einstein Telescope optical design. This work will facilitate routine use of extremely high levels of squeezing in current and future gravitational-wave detectors.

Differential wavefront sensing and control using radio-frequency optical demodulation

Differential wavefront sensing is an essential technique for optimising the performance of many precision interferometric experiments. Perhaps the most extensive application of this is for alignment sensing using radio-frequency beats measured with quadrant photodiodes. Here we present a new technique that uses optical demodulation to measure such optical beats at high resolutions using commercial laboratory equipment. We experimentally demonstrate that the images captured can be digitally processed to generate wavefront error signals and use these in a closed loop control system for correct wavefront errors for alignment and mode-matching a beam into an optical cavity to 99.9%. This experiment paves the way for the correction of even higher order errors when paired with higher order wavefront actuators. Such a sensing scheme could find use in optimizing complex interferometers consisting of coupled cavities, such as those found in gravitational wave detectors, or simply just for sensing higher order wavefront errors in heterodyne interferometric table-top experiments.

Mode matching error signals using radio-frequency beam shape modulation

Precise mode matching is needed to maximize performance in coupled cavity interferometers such as Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO). In this paper, we present a new mode matching sensing scheme, to the best of our knowledge, that uses a single radio-frequency higher-order-mode sideband and single-element photodiodes. It is first-order insensitive to misalignment and can serve as an error signal in a closed loop control system for a set of mode matching actuators. We also discuss how it may be implemented in Advanced LIGO. The proposed mode matching error signal has been successfully demonstrated on a tabletop experiment, where the error signal increased the mode matching of a beam to a cavity to 99.9%.

Optical lock-in camera for gravitational wave detectors

Knowledge of the intensity and phase profiles of spectral components in a coherent optical field is critical for a wide range of high-precision optical applications. One of these is interferometric gravitational wave detectors, which rely on the optical beats between these fields for precise control of the experiment. Here we describe an optical lock-in camera and show that it can be used to record optical beats at MHz or greater frequencies with higher spatial and temporal resolution than previously possible. This improvement is achieved using a Pockels cell as a fast optical switch to transform each pixel on a sCMOS array into an optical lock-in amplifier. We demonstrate that the optical lock-in camera can record fields with 2 Mpx resolution at 10 Hz with a sensitivity of -62 dBc when averaged over 2s.

High dynamic range spatial mode decomposition

An accurate readout of low-power optical higher-order spatial modes is of increasing importance to the precision metrology community. Mode sensors are used to prevent mode mismatches from degrading quantum and thermal noise mitigation strategies. Direct mode analysis sensors (MODAN) are a promising technology for real-time monitoring of arbitrary higher-order modes. We demonstrate MODAN with photo-diode readout to mitigate the typically low dynamic range of CCDs. We look for asymmetries in the response of our sensor to break degeneracies in the relative alignment of the MODAN and photo-diode and consequently improve the dynamic range of the mode sensor. We provide a tolerance analysis and show methodology that can be applied for sensors beyond first order spatial modes.

Interferometer techniques for gravitational-wave detection

Several km-scale gravitational-wave detectors have been constructed worldwide. These instruments combine a number of advanced technologies to push the limits of precision length measurement. The core devices are laser interferometers of a new kind; developed from the classical Michelson topology these interferometers integrate additional optical elements, which significantly change the properties of the optical system. Much of the design and analysis of these laser interferometers can be performed using well-known classical optical techniques; however, the complex optical layouts provide a new challenge. In this review, we give a textbook-style introduction to the optical science required for the understanding of modern gravitational wave detectors, as well as other high-precision laser interferometers. In addition, we provide a number of examples for a freely available interferometer simulation software and encourage the reader to use these examples to gain hands-on experience with the discussed optical methods.

Audio-Band Frequency-Dependent Squeezing for Gravitational-Wave Detectors

Quantum vacuum fluctuations impose strict limits on precision displacement measurements, those of interferometric gravitational-wave detectors among them. Introducing squeezed states into an interferometer's readout port can improve the sensitivity of the instrument, leading to richer astrophysical observations. However, optomechanical interactions dictate that the vacuum's squeezed quadrature must rotate by 90° around 50 Hz. Here we use a 2-m-long, high-finesse optical resonator to produce frequency-dependent rotation around 1.2 kHz. This demonstration of audio-band frequency-dependent squeezing uses technology and methods that are scalable to the required rotation frequency and validates previously developed theoretical models, heralding application of the technique in future gravitational-wave detectors.

Length control of an optical resonator using second-order transverse modes

We present the analysis of an unorthodox technique for locking a laser to a resonant optical cavity. Error signals are derived from the interference between the fundamental cavity mode and higher-order spatial modes of order two excited by mode mismatch. This scheme is simple, inexpensive, and, in contrast to similar techniques, first-order insensitive to beam jitter. After mitigating sources of technical noise, performance is fundamentally limited by quantum shot noise.

Feedback control of optical beam spatial profiles using thermal lensing

A method for active control of the spatial profile of a laser beam using adaptive thermal lensing is described. A segmented electrical heater was used to generate thermal gradients across a transmissive optical element, resulting in a controllable thermal lens. The segmented heater also allows the generation of cylindrical lenses, and provides the capability to steer the beam in both horizontal and vertical planes. Using this device as an actuator, a feedback control loop was developed to stabilize the beam size and position.

Performance of a thermally deformable mirror for correction of low-order aberrations in laser beams

The thermally deformable mirror is a device aiming at correcting beam-wavefront distortions for applications where classical mechanical methods are precluded by noise considerations, as in advanced gravitational wave interferometric detectors. This moderately low-cost technology can be easily implemented and controlled thanks to the good reproducibility of the actuation. By using a flexible printed circuit board technology, we demonstrate experimentally that a device of 61 actuators in thermal contact with the back surface of a high-reflective mirror is able to correct the low-order aberrations of a laser beam at 1064 nm and could be used to optimize the mode matching into Fabry-Perot cavities.

Measurement of gouy phase evolution by use of spatial mode interference

An experimental technique to observe and accurately measure the Gouy phase evolution of Hermite-Gaussian modes is presented. Because of the unique features of spatial mode interference frequency-locking error signals, we are able to readily perform explicit measurement of the Gouy phase in a simple and highly accurate manner. We present these data and discuss the technique and its implications.

Frequency-resolving spatiotemporal wave-front sensor

We report on a high-resolution wave-front sensor that measures the complete spatial profile of any frequency component of a laser field containing multiple frequencies. This probe technique was developed to address the necessity of measuring the spatial overlap of the carrier field with each sideband component of the field exiting the output port of a gravitational-wave interferometer. We present the results of an experimental test of the probe, where we were able to construct the spatial profile of a single radio-frequency sideband at the level of -50 dBc.