Precision Interferometry in a New Shape: Higher-order Laguerre-Gauss Modes for Gravitational Wave Detection
Author: Paul Fulda
Currently at Department of Physics, University of Florida, Gainesville, USA
PhD research performed at School of Physics and Astronomy, University of Birmingham, UK.
With the approach of advanced detector science runs, the field of gravitational wave detection stands on the brink of finally making the much sought after first direct detection of gravitational waves. Never a bunch to rest on our laurels, however, we are always trying to think of ways to push the sensitivity of these kilometer-scale interferometers ever higher. A single detection, with its ramifications in the areas of astrophysics, cosmology and General Relativity, would be an incredible achievement for a scientific field long in the making.
However, since the observable astrophysical event rate scales as the cube of the observable event distance (and hence roughly as the cube of the broadband sensitivity), any sensitivity increase we can make will bring with it an ever greater number of detected gravitational wave events. The more events we have, the better statistics we can do with them, and the better we can refine our model of the universe itself. With that in mind, in this article I will focus on one of many ideas for improving the sensitivity of gravitational wave detectors beyond the 'advanced' generation; the idea of using higher-order Laguerre-Gauss modes to reduce the thermal noise of the test-masses of the interferometers.
The rest of this article won't make a lot of sense if I don't first explain what a higher-order laser mode is, but in order to understand this concept we also need to know a little bit about optical cavities. An optical cavity exists wherever there is an arrangement of mirrors that form a closed optical path. Perhaps the simplest case is a linear cavity, which consists of two partially transmissive mirrors, one flat and one curved, orientated with the reflective surfaces parallel to each other (see Fig. 1). When the length of the cavity is tuned just right relative to the frequency of the light, light entering through one partially transmissive mirror creates a standing wave inside the cavity. This condition is known as a resonance of the cavity, and it creates a higher circulating light power inside the cavity than the light power entering. The light is 'stored' inside the cavity for some time before it leaks out again.
Fig 1: A cartoon picture of an optical cavity. Light enters the cavity through the input mirror from the left, and resonates inside the cavity if the cavity length is divisible by an integer number of half wavelengths.
There is another condition on top of the length/frequency condition in order for light to resonate in an optical cavity; the amplitude cross-section of the light (i.e. the beam shape) must be unchanged after several reflections from the mirrors that make up the cavity. If this is not the case, the light will not properly add up after each reflection to make a standing wave. For a cavity made up of spherically curved mirrors, this condition is satisfied by sets of so-called "spatial modes". There are two main families of these spatial modes - Hermite-Gauss (HG) modes and Laguerre-Gauss (LG) modes (see Fig. 2). In a well designed optical cavity, each higher mode order (the order being n+m for HG modes and 2p+l for LG modes) is resonant at a different cavity tuning. These modes are mathematical solutions to the wave equation in the beam-like or 'paraxial' limit. However, far from being purely abstract mathematical constructs, they really exist and show up in interferometers all the time, as we shall soon see!
Fig 2: [click on the image to see better resolution] Intensity patterns for ideal higher-order HG (top left) and LG (top right) modes. The lower half of the image shows the higher-order LG modes generated in the lab using an SLM (in negative to show detail).
Advanced interferometric gravitational wave detectors make use of optical cavities in many places. Advanced LIGO for example has a pre- mode-cleaner cavity, an input mode cleaner cavity and an output mode cleaner cavity, in addition to the 4 coupled cavities of the central interferometer (see Fig. 3) . These optical cavities are used outside the central interferometer as 'mode cleaners' for their ability to filter the frequency and spatial properties of light, and inside the central interferometer in order to increase the circulating light power and the interaction time between the light and passing gravitational waves. LIGO would not have been able to achieve the unprecedented strain sensitivity it did without using these crucial interferometric tools.
Fig 3: [click on the image to see better resolution] The optical layout of an Advanced LIGO interferometer, showing the input and output mode cleaner cavities, the two 4km long arm cavities and the two recycling cavities.
Usually one aims to use only the lowest order 'fundamental' mode (the HG00/LG00 mode), for precision interferometric metrology. The presence of any of the higher-order modes commonly signifies an imperfection in the optical setup, for example a misalignment or a discrepancy in the characteristic size of two interfering beams. As a result, there are several activities directed towards reducing the presence of these modes in interferometers [2, 3].
Recently, however, it was shown that by using one of the higher-order Laguerre-Gauss modes instead of the funamental mode it may be possible to reduce the effects of mirror thermal noise, which is expected to be one of the limiting noise sources in gravitational wave detectors . Mirror thermal noise appears as random fluctations of the reflecting surfaces, which can mask any gravitational wave passing through the interferometer. Higher-order LG modes are able to reduce the effect of this noise by averaging better over the mirror surface, effectively smoothing out the fluctuations. Higher-order LG modes also have the interesting property of having spiral phasefronts, thereby carrying angular momentum - a property which has made them useful in the fields of cold atoms and bio-photonics .
Could it then be a case of "from villain to hero" for higher-order modes? This is the question we sought to answer with a program of research into the application of LG modes in precision interferometry. Although the thermal noise benefits of using LG modes made them seem like an attractive option, it was crucial to test their compatibility with all of the other technologies that are used in gravitational wave detectors. At the University of Birmingham we started this investigation using simulations of advanced interferometer where the main fundamental mode beam was exchanged for a LG33 beam (top right in each panel of Fig. 1) . The LG33 beam performed very favorably in the simulations, giving us the motivation to move forward to table-top lab experiments.
We used a device called a spatial light modulator (SLM) to generate the higher-order LG modes in the lab. This device works somewhat like an LCD screen, except that each pixel can change the phase of the light reflected from it instead of the intensity. By displaying a specific phase pattern on the SLM, we were able to convert a fundamental mode beam into almost any higher-order LG mode we wanted (see the lower half of Fig. 2). We confirmed the spiral phase front of the beams by interfering them with a fundamental mode beam, creating the spiral intensity patterns shown in Fig. 4. However, the mode purity we could achieve with the SLM alone was not high enough for use with gravitational wave detectors.
Fig. 4: Images of the interference pattern between various SLM generated higher-order LG beams and a fundamental mode beam. The spiral pattern indicates the presence of a spiral phasefront in the LG beams. From left: LG22, LG33, LG44, LG55.
The positive aspect of this was that it gave us the perfect opportunity to make one of the most crucial compatibility tests for the LG33 mode - if the LG33 mode was to be effective in gravitational wave detectors, it would have to be compatible with mode cleaner optical cavities and the control schemes that are commonly used with them. If it worked, we would have the added bonus of a `cleaned' LG33 mode after the mode cleaner. As predicted by the earlier simulations, the LG33 mode was compatible with a mode cleaner cavity, and we achieved mode purity in excess of 99%. Figure 5 shows images of two kinds of LG33 mode before and after the mode cleaner cavity .
Fig. 5: Images of the SLM generated LG33 mode before (top) and after (bottom) transmission through a linear optical cavity.
With these positive results we decided to move on to the next level of compatibility testing: using LG modes with a suspended 10m prototype interferometer in Glasgow. The specific aim here was to address a tricky problem with LG modes that was identified through simulations; with larger beam sizes, the imperfections in the mirrors that form an optical cavity can cause a significant reduction in the mode purity [8,9].
We set up an LG mode conversion path on the Glasgow prototype input bench (see Fig. 6), and directed the beam into the vacuum tank towards a 10m long optical cavity. We employed a sophisticated method to precisely tune the length of the cavity, while observing the transmitted light power with a high-speed infra-red camera. Since different modes are resonant in the cavity at different tunings, the high-speed camera measurements gave us a good picture of the higher-order mode content in the cavity.
Fig. 6: [click on the image to see better resolution] The LG mode conversion path for the Glasgow 10m prototype interferometer experiment. The red line shows the LG33 conversion path, the green line shows the fundamental mode 'control' path, and the purple line shows the original laser path.
We saw that even when we injected a LG33 mode, we could only observe the rectangular shaped HG modes resonating in the cavity (see Fig. 7). The work reported in  and  suggested that this behavior might be expected in a cavity where the mirrors are astigmatically curved, rather than the spherically curved ideal case. This was a setback for the use of LG modes in gravitational wave interferometers, as it was the first experimental evidence that mirror surface imperfections might have a stronger impact on mode purity for the LG33 mode than for the fundamental mode. The cavity mirrors were taken out of the prototype and independent measurements of the surface figures were made. The measured astigmatism values were used as inputs for a simulation of the 10m cavity, which reproduced the resonance structures seen in Fig. 7 very well. This was the confirmation that was needed to show that the LG33 mode really was more sensitive to mirror surface imperfections that the fundamental mode .
Fig. 7: [click on the image to see better resolution] A trace showing reflected power from the 10m cavity as a function of time as the cavity is swept across a resonance with the LG33 input beam. The resonance is split into several peaks, as predicted in  and . Images from the high-speed camera in transmission of the cavity show the resonant mode shape around several of the separate resonance peaks.
From here it was possible to derive tolerances on the mirror imperfections that would be allowable if the LG33 mode was to be used as the readout beam in a real gravitational wave detector. The requirements that the LG33 mode puts on the quality of the mirror surfaces are beyond what is achievable with the current technology, and as such they do look very tough to meet. However, coating and polishing technology is always moving forward with the demands of ever more precise experiments, so at some time in the future we may yet see the LG33 mode flashing on the projector screens in a gravitational wave detector installation.
Those of us who work in the business of laser interferometry become quite accustomed to the sight of higher-order spatial laser modes. It can be easy to take their aesthetic qualities for granted, especially when a lot of our efforts are directed towards minimizing their presence. I still remember the first time I aligned an optical cavity though, and how I was mesmerized by the small CCTV camera picture showing the strange and beautiful patterns in the cavity transmitted light. I'm very glad that my PhD research took me in the direction of studying a particular one of these higher-order laser modes in great detail. By the end of my studies, the higher-order modes weren't just obscure patterns of light to me any more; rather they seemed like cryptic hieroglyphs which when translated could provide a wealth of knowledge about the state of an optical system.
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