Large laser-interferometers are being constructed to create a new type of astronomy based on gravitational waves. Their sensitivities, as for many other high-precision experiments, are approaching fundamental noise limits such as the atomic vibration of their components. We are pioneering technologies to overcome these limits using novel laser beam shapes.
Thermal noise in high-reflectivity mirrors is a major impediment for several types of high-precision interferometric experiments that aim to reach the standard quantum limit or to cool mechanical systems to their quantum ground state. This is for example the case of future gravitational wave observatories, whose sensitivity to gravitational wave signals is expected to be limited in the most sensitive frequency band, by atomic vibration of their mirror masses. One promising approach being pursued to overcome this limitation is to employ higher-order Laguerre-Gauss (LG) optical beams in place of the conventionally used fundamental mode. Owing to their more homogeneous light intensity distribution these beams average more effectively over the thermally driven fluctuations of the mirror surface, which in turn reduces the uncertainty in the mirror position sensed by the laser light.
We demonstrate a promising method to generate higher-order LG beams by shaping a fundamental Gaussian beam with the help of diffractive optical elements. We show that with conventional sensing and control techniques that are known for stabilizing fundamental laser beams, higher-order LG modes can be purified and stabilized just as well at a comparably high level. A set of diagnostic tools allows us to control and tailor the properties of generated LG beams. This enabled us to produce an LG beam with the highest purity reported to date. The demonstrated compatibility of higher-order LG modes with standard interferometry techniques and with the use of standard spherical optics makes them an ideal candidate for application in a future generation of high-precision interferometry.
During the past decades high-precision interferometric experiments were pushed towards an ultimate sensitivity regime where quantum effects are starting to play a decisive role. In these ongoing and future experiments, such as laser cooling of mechanical oscillators 1, optical traps for mirrors 2, generation of entangled test masses 3, quantum non-demolition interferometry 4, frequency stabilization of lasers with rigid cavities 5, and gravitational wave detection 6, 7, 8, researchers are facing a multitude of limiting fundamental and technical noise sources. One of the most severe problems is the thermal noise of the cavity mirrors of the interferometric setups, which is caused by the thermal excitation of the atoms that make up the mirror substrates and the mirror reflective coatings 7, 8, 9. This effect, also called Brownian motion, will cause an uncertainty in the phase of the light reflected from any test masses and will therefore manifest as a fundamental noise limitation in the interferometer output. For instance, the projected design sensitivity of advanced gravitational wave antennae, such as Advanced LIGO, Advanced VIRGO, and the Einstein Telescope, is limited by this type of noise in the most sensitive region of the observation frequency band 10, 11, 12.
Experimental physicists in the community work hard in a continuous effort to minimize these noise contributions and to improve the sensitivity of their instruments. In the particular case of mirror Brownian noise, one method for mitigation is to employ a larger beam spot size of the currently used standard fundamental HG00 beam on the test mass surfaces, since a larger beam averages more effectively over the random motions of the surface 13, 14. The power spectral density of the mirror thermal noise has been shown to scale with the inverse of the Gaussian beam size for the mirror substrate and with the inverse square for the mirror surface 9. However, as the beam spots are made larger, a larger fraction of the light power is lost over the edge of the reflective surface. If one uses a beam with a more homogeneous radial intensity distribution than the commonly used HG00 beam (see for example Figure 1), the Brownian thermal noise level can be reduced without increasing this type of loss. Amongst all the more homogeneous beam types that have been suggested for new versions of high-precision interferometry, for example Mesa beams or conical modes 13, 14, the most promising are higher-order LG beams due to their potential compatibility with the currently used spherical mirror surfaces 15. For instance, the detection rate of binary neutron star in spiral systems - which are considered the most promising astrophysical sources for a first GW detection - could be enhanced by about a factor of 2 or more 16 at the cost of a minimal amount of modifications in the design of second-generation interferometers currently under construction 10, 11. In addition to the thermal noise benefits, the wider intensity distributions of higher-order LG beams (see as an example Figure 2) have been shown to mitigate the magnitude of thermal aberrations of optics within the interferometers. This would reduce the extent to which thermal compensation systems are relied upon in future experiments to reach design sensitivities 19.
We have investigated and successfully demonstrated the feasibility of generating LG beams at the levels of purity and stability required to successfully operate GW interferometers at the best of their sensitivity 16, 18, 19, 20, 21, 22. The proposed method combines techniques and expertise developed in diverse areas of physics and optics such as the generation of high stability, low noise single mode laser beams 23, the use of spatial light modulators and diffractive optical elements for the manipulation of the spatial profiles of light beams 18, 22, 24, 25, 26, and the use of advanced techniques for the sensing, control and stabilization of resonant optical cavities 27 aiming at a further purification and stabilization of the laser light. This method has been successfully demonstrated in the laboratory experiments, exported for tests in large-scale prototype interferometers 20, and for generating LG modes at high laser powers up to 80 W 21. In this article we present the details of the method of generating higher order LG beams and discuss a methodology for the characterization and validation of the resulting beam. Further, in Step 4 a method for numerical investigations of cavities with non-perfect mirrors 19 is outlined.
Preamble: In this protocol section we assume that a pure, low noise, power-stabilized fundamental mode Gaussian beam is provided, for instance by means of the standard setup as shown in Figure 3 containing: a commercial Nd:YAG laser to generate continuous-wave infrared light at 1064 nm wavelength; a Faraday Isolator (FI) to avoid back-reflection of the light towards the laser source; and an Electro-Optic Modulator (EOM) to modulate the phase of the light. The resulting beam is injected into a triangular optical cavity, where the laser frequency and the light power are stabilized by means of active control loops 27, while the resonant cavity provides spatial filtering for unwanted beam shapes.
The setup described above and shown in Figure 3 is a conventional experimental arrangement that is used in scientific apparatuses demanding low noise laser stabilization for precision measurements 1-8. The protocol section below describes how this fundamental mode Gaussian beam can be efficiently converted into a higher order Laguerre-Gauss type optical beam with comparable performances, if not identical, in terms of purity, noise, and stability. This is implemented by means of the apparatus shown in Figure 4, whose design, construction, and operation is described in the sections below. In this example presented in this work the generated mode will be a LG33. However it is worth stressing that the technique has general validity and that the described protocol applies to any desired higher order LG mode.
1. Designing and Prototyping the Optical Mode Converter for Optimal Conversion of Fundamental Mode Laser Beam into Higher Order LG Beams
The requirement for a phase modulation profile to convert a fundamental mode beam into a higher-order LG beam is to replicate the phase cross-section of the desired LG mode, which will be imprinted via a proportional phase-shift onto the wavefront of the incident beam 26. Two types of mode-converters work in this way: Spatial Light Modulators (SLM) - computer-controlled liquid-crystal displays whose pixels can be controlled to imprint phase shifts on the incident light - and diffractive phase plates - etched glass substrates where the desired phase shifts are produced in transmission by the purposely varying thickness of the glass element. SLMs are flexible but lack stability and efficiency, while phase plates are stable and efficient, but lack flexibility. Therefore we advise the use of the SLM for initial studies and prototyping and the use of a phase plate for long-term operations.
Optimal conversion relies on the precise choice of the parameters (waist size and position) of the beam to be shaped. Therefore before injecting it onto a mode converter, the initial fundamental mode beam must be characterized, and its parameters re-shaped to match the ones offering optimal conversion - this operation is called 'mode-matching'.
During interaction with the phase modulating device, some of the injected light remains unmodulated due to the quantization of the phase modulation levels. This unconverted light propagates along the same axis of the converted beam, spoiling the desired phase modulation effects. To circumvent this problem one can overlay a blazed grating profile on the LG mode conversion phase image. The modulated light carrying the LG mode phase profile will be deflected by the blazed grating, whereas the unmodulated light, which does not interact with the substrate, will proceed undisturbed. This causes a spatial separation between the two types of beams.
2. Operation of the Phase Plate, Mode Conversion and Purity Enhancement
The inability of the discussed phase plate designs to modulate amplitude as well as phase means that they will not convert all of the incoming fundamental beam into the desired mode. The result is a composite beam with a dominant desired LG beam over a background of other higher-order modes of minor intensity, as shown in Figure 7. In order to spatially filter out unwanted LG modes and enhance the mode purity, the converted beam can be injected into an optical resonant cavity. Such a cavity can operate as a 'mode selector' allowing only specific optical modes to be transmitted, depending on the cavity length relative to the light wavelength.
Once the optimal alignment of the beam into the mode cleaner cavity has been achieved, and the mode content of the injected beam has been analyzed, 'mode-cleaning' and enhancement of the purity of the composite LG beam can be finally implemented. A Pound-Drever-Hall locking scheme 27 can be used to stabilize the cavity length to the desired resonant mode. The light transmitted by the mode cleaner cavity can be read by a photodiode, which can provide the error signal necessary for the control loop that controls the cavity length.
3. Diagnostics and Characterization of the Generated LG Beam
In this experiment, two main properties define the quality of a 'good' beam for the successful implementation in high-precision interferometric measurements: the beam power and the beam purity. Other relevant properties such as the frequency or the power stability can be preserved making use of the same control techniques implemented on the fundamental mode beam, as described above.
Two important figures of merit are useful to evaluate the quality of the entire mode conversion process: the conversion efficiencies of the phase plate and of the overall setup.
4. Injection into Large Interferometers: Simulation Investigation
One application of this protocol is to investigate LG beams for their use in gravitational wave detectors. These are long baseline high precision interferometers. The baseline requires relatively large mirrors and beam sizes. This, however, enhances the effects of imperfect optics, especially when using higher order modes. This section describes a simulation based approach to investigate the behavior of higher order LG modes in realistic detectors.
The use of higher order beams introduces a 'degeneracy' to the optical cavities as there are several different beam shapes fighting for dominance. An optical cavity resonant for a Gaussian mode is resonant for all modes of that order. An HG00 mode is the only mode of order 0, so all other modes are suppressed. For example, the LG33 mode is one of ten modes of order 9, all of which will be enhanced in the interferometer. Mirror surface distortions that are always present in real interferometers could couple the incident mode into other ones. If these new modes are of the same order as the incident beam they are enhanced in the arm cavities, resulting in highly distorted circulating beams. This can eventually deteriorate the instrument sensitivity.
All the experimental results so far described in the text and shown in the figures constitute a representative example of a successful execution of the beam conversion protocol. The most representative result is the purity of the generated beam: a successful beam conversion should lead to a beam purity on the order of 95% or above. A good example of successful beam conversion is the measurement of the intensity profile of an 82.8 Watts, 96% pure LG33 beam obtained in 21 and here shown in Figure 12.
Similarly, as discussed in protocol sections, the mode conversion efficiencies of the phase plate and of the overall experimental setup are a good indicator of the successful design of the experimental apparatus, including the phase plate and the mode cleaner cavity. Values of order 50% to 60% and above are generally considered a good value for the mode conversion efficiency. The highest conversion efficiency reported so far with this type of setup is about 70% 21.
The simulation investigation described in Protocol Sec 4 should result in numbers for beam purity with realistic mirrors, suggested mirror specifications and the resulting beam purity when these specifications are adopted. An example of the results you can expect with realistic mirror maps are shown in 19 where an original LG33 purity of 89% is obtained, compared to a purity of >99% for HG00. A purity of >99% for the LG33 mode is achieved using specific mirror requirements with a major reduction of astigmatism in the mirror surface.
Figure 1. Intensity patterns for Hermite-Gauss (HG) modes up to order 6. The intensity patterns are normalized to have the same peak intensity, for visibility.
Figure 2. Intensity patterns for helical LG modes up to order 9. The intensity patterns are normalized to have the same peak intensity, for visibility.
Figure 3. Sketch of a conventional setup for production and stabilization of HG00 beams.
Figure 4. A sketch of the experimental setup discussed in this paper. The HG00 beam is first mode-matched to a desired waist size via a telescope then injected on the phase plate. The main diffracted beam is separated from the higher diffraction orders with an aperture and then sent to the Mode Cleaner cavity. A photodiode is used to extract the error signal for controlling the cavity length. The beam intensity is analyzed by a CCD camera.
Figure 5. Phase modulation profile to convert a HG00 mode to LG33 mode.
Figure 6. Example of blazed phase modulation profiles for generating LG33 modes.
Figure 7. Comparison between the intensity distribution of the composite beam generated by the phase plate (left) and the theoretical intensity distribution for a pure LG33 beam with same parameters.
Figure 8. Example of beam intensity profile fitting applied to a real LG33 beam transmitted from a phase plate (left) compared to fit results (center) and residuals of fit (right). Click here to view larger figure.
Figure 9. Profile of an LG33 beam with Gaussian fit shown for comparison.
Figure 10. Light power transmitted by a linear cavity as a function of the cavity length, when injecting a beam generated by the phase plate. The resonant peaks at 0 and 1 FSR correspond to the desired LG33 mode. A fit to this dominant mode is shown for comparison (blue line). The red curve shows the result of the numerical model, based on the modal content described in Table 2. Pictures of the unwanted beams to be filtered by the cavity are shown in the insets.
Figure 11. An example of a mirror surface map for one of the Advanced LIGO optical mirrors 19.
Figure 12. Intensity profile of a 82.8 W LG33 beam transmitted by a linear cavity (left) compared with fit residuals (right).
p | l | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
0 | 1.0 | 2.8 | 1.7 | 2.0 | 2.2 | 2.5 | 2.6 | 3.1 | 3.0 | 3.2 | |
1 | 1.7 | 2.7 | 2.2 | 2.4 | 2.6 | 2.8 | 2.9 | 2.8 | Â | Â | |
2 | 2.2 | 2.4 | 2.5 | 2.7 | 2.9 | 3.0 | Â | Â | Â | Â | |
3 | 2.5 | 2.7 | 2.8 | 3.0 | Â | Â | Â | Â | Â | Â | |
4 | 2.9 | 3.0 | Â | Â | Â | Â | Â | Â | Â | Â |
Table 1. Optimum ratio between input HG00 beam size and LGpl phase image beam size for LG modes up to the order 9.
Mode | LG33 | LG63 | LG43 | LG53 | LG32 | LG62 |
Power | 75% | 8% | 4% | 4% | 4% | 1% |
Table 2. Mode content analysis described by the cavity scan shown in Figure 10.
The output beams of most lasers used in high-precision measurements are designed to have a shape well described as a fundamental Gaussian mode. This particular beam geometry combines low diffraction with a spherical wave front. While the low diffraction is one of the key advantages of laser light, the spherical wave front is equally important, as it allows the low-loss transformation of the laser beam by standard optical components with spherical surfaces. Different beam shapes can be created as well, and recently Laguerre-Gauss beams have become of interest for their potential application in high-precision interferometry.
In this paper we demonstrated the experimental procedure to create higher-order Laguerre-Gauss modes with 95% purity for high-power, ultra stable laser beams. To achieve this, we have combined standard techniques from different aspects of optical research, namely diffractive phase plates and laser pre-stabilization to mode cleaner cavities. Our experiment provides a simple, modular and very reliable method to create high power beams in user defined higher-order modes. A commercial ultra-stable laser is used as the light source. Its output is injected to a diffractive phase plate, which can convert up to 75% of the light into the desired Laguerre-Gauss mode. This light is then injected to a small optical cavity and an electronic feedback loop is used to stabilize the laser frequency of the laser to the cavity length. The beam transmitted by the cavity is to 95% in the desired mode and, like the fundamental mode beam at the origin of the setup, has very good frequency stability at audio frequencies. All the parts represent standard components in modern optical experiments. We have successfully demonstrated this technique for laser powers up to 80 W pure Laguerre-Gauss 33 mode.
It could be possible to achieve similar results by replacing the phase plate with another mode-converting element (for example, other diffractive elements or astigmatic mode converters). Alternatively a laser could be setup with an optical resonator tuned for the desired Laguerre-Gauss modes, using for example, an amplitude mask. Finally the laser frequency stabilization to the reference optical cavity could be exchanged with a similar scheme that uses an atomic reference. The need for an electronic feedback system is probably the main disadvantage, but this is inevitable for any light source used for precision interferometer.
However, we believe that the method demonstrated in this paper provides a simple and modular scheme which can be scaled to all ranges of required laser frequency, power, or shape and thus presents a powerful and versatile method. Each part, the laser source, the diffractive element, as well as the optical cavity can be changed or optimized individually, which means that also existing laser injection systems can be upgraded to use Laguerre-Gauss modes.
This work was funded by the Science and Technology Facilities Council (STFC).
Name | Company | Catalog Number | Comments |
The experimental apparatus discussed in this paper requires the following types of instruments: | |||
Instrument | |||
Solid state Laser source, Nd:YAG 1064 nm CW laser | Quantity: 1 | ||
Faraday Isolator | Quantity: 1 | ||
Electro-Optic Modulator (EOM) | Quantity: 1 | ||
CCDcamera beam profiler | Quantity: 1 | ||
Lenses | Quantity: depending on apparatus design | ||
Steering Mirrors | Quantity: depending on apparatus design | ||
Aperture | Quantity: 1 | ||
High reflectivity mirrors (for normal incidence) | Quantity: 2 | ||
Piezoelectric ring | Quantity: 1 | ||
Cavity spacer | Quantity: 1 | ||
Photodiodes and related control electronics | Quantity: 1 or more, depending on apparatus design | ||
Spatial light modulator | Quantity: 1 Holoeye LCR-2500 | ||
All the above instruments are commercially available and no particular specification is required. We leave the choice of the most suitable instruments to the experimenter's discretion. | |||
For the interest of the experimenter interested in reproducing the protocol, we recommend the following tools used in our experiment: | |||
Tools | |||
Innolight OEM 300NE, 1064 nm, 300 mW | Laser Source: | ||
SIMTOOLs | Software for data analysis, available at www.gwoptics.org/simtools/ | ||
FINESSE | Software for optical simulations, www.gwoptics.org/finesse/ | ||
Finally, the phase plate employed in the present experiment was manufactured by Jenoptik GmbH, based on a custom design provided by the Authors. |
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