High-power laser beam in higher-order Hermite–Gaussian modes

The only gravitational-wave detectors (GWDs), which could already measure gravitational waves, are highly sophisticated Michelson interferometers, like Advanced LIGO¹ and Advanced Virgo,² which measure differential changes in the arm lengths at the subatomic scale. This small effect can easily be masked by a variety of noise sources, and their mitigation is of fundamental importance. Coating Brownian thermal noise limits the detector sensitivities at signal frequencies around 100 Hz. To mitigate the coating Brownian noise, it has been proposed to operate the detectors in a higher-order Hermite–Gaussian (HG) laser mode instead of the currently used fundamental Gaussian $TEM 0 , 0$ mode.³ The more uniform intensity distributions of these higher-order modes, compared to the $TEM 0 , 0$ mode, would yield a better averaging over the thermally induced mirror fluctuations.

Whether higher-order HG modes satisfy all of the requirements set by laser interferometric gravitational-wave detectors (GWDs) is an active field of research, e.g., regarding their higher sensitivity to spatial mismatches,^4,5 their robustness in a Michelson interferometer against mirror distortions, arm losses, and contrast defects,⁶ and their efficiency in reducing quantum noise via squeezed states of light.⁷ One fundamental part of this research is the generation of these higher-order modes, especially at the demanded high optical power in a continuous-wave (cw) laser beam (up to 500 W for future detectors like the Einstein Telescope⁸). However, so far, only the generation of the Laguerre–Gaussian $LG 3 , 3$ mode has been examined under these conditions.⁹ A conversion efficiency of about 59% was achieved using a transmissive diffractive optical element at up to 138 W of optical input power.

Here, we extend this research to higher-order HG modes and computer-controlled spatial light modulators (SLMs). SLMs offer the advantage over diffractive optical elements in that their output can easily be adjusted during operation if needed. SLMs have already been used with pulsed high-power lasers,¹⁰ but only for low-power continuous-wave experiments (⁠ $≤ 1 W$⁠). In the low-power regime, conversion efficiencies of 61% could be achieved with two subsequent SLMs in the $HG 9 , 0$ mode,¹¹ while a single SLM only reached a conversion efficiency of less than 7% for the $HG 5 , 5$ mode.¹² 

Here, we discuss the investigation of an SLM operated with a high-power cw laser beam of up to 85 W. We especially focus on the dependence of the conversion efficiencies on the incident optical power for the modes $HG 2 , 2 , HG 3 , 3$⁠, and $HG 4 , 4$ generated from the $TEM 0 , 0$ mode. These modes are commonly investigated in GWD-related research as a trade-off between a high thermal noise mitigation and challenging aspects like mode-matching, mode purity, contrast defects, and quantum noise reduction, which show up when using higher-order laser modes.^5–7,13

In the experiment, a computer-controlled, reflective, phase-only SLM¹⁴ (type LCOS-Hamamatsu, model “X10468–03” with water cooling) manipulates the transverse phase distribution of a $TEM 0 , 0$ laser beam to generate the higher-order modes. According to the manufacturer, the SLM has a reflectivity of 95% at the used wavelength. Its resolution is 792 × 600 pixels on an active area of $15.8 × 12.0 cm 2$⁠. This should not influence the conversion as the displayed phase images are simple checkerboard-like patterns of alternating areas, which are large compared to the size of the pixels, with phase values of zero and π.¹⁵ The conversion efficiency depends on the intensity overlap between the incident $TEM 0 , 0$ mode and the generated mode.¹⁶ Since the intensities of higher-order modes are more widely distributed over the transverse plane than the intensity of the $TEM 0 , 0$ mode, we set the waist size of the generated mode via the phase pattern on the SLM to be smaller than the waist size of the incident $TEM 0 , 0$ mode to optimize this overlap. The optimum ratio depends on the generated mode and, as a compromise between the used modes $HG 2 , 2 , HG 3 , 3$⁠, and $HG 4 , 4$⁠, we chose a ratio of about 2.3.

Figure 1 shows the experimental setup for the conversion efficiency measurement. The laser source is based on a non-planar ring oscillator that is amplified, stabilized, and mode-filtered similar to the pre-stabilized laser system of the Advanced LIGO detector.^17–19 It reaches an optical power of up to 85 W at a wavelength of 1064 nm in a fundamental $TEM 0 , 0$ mode with a purity above 98.8%.

The photodetector $PD in$⁠, calibrated using the setup's powermeter, measures the input power $P in$⁠. The SLM then changes the phase distribution of the input laser beam, which is collimated to a waist size of roughly 3 mm on the SLM, to the phase distribution of the respective higher-order mode. In reflection of the SLM, the light field is mode-cleaned using an impedance-matched triangular cavity with an estimated power transmission above 98% for the $TEM 0 , 0$ mode. The cavity is stabilized in length to the resonance condition of the generated mode by a feedback control loop using the dither lock technique.²⁰ In transmission of the cavity, a water-cooled powermeter measures the output power $P out$⁠, and a diagnostic breadboard (DBB)²¹ analyses the mode purity of the output beam. By scanning the length of a diagnostic cavity, the DBB can automatically analyze the mode content of the injected laser beam via the power transmission spectrum.

The conversion efficiencies were first measured by comparing the input power $P in$⁠, incident on the SLM, to the output power $P out$ in transmission of the triangular mode-cleaner cavity. For clarity, this will be called the approximated conversion efficiency, $η appr = P out / P in$⁠, as it disregards the mode purity of the output beam. The results are shown in Fig. 2, where the significant uncertainty at $P in = 1 W$ is due to the inherent uncertainty of the water-cooled powermeter, which is designed to measure higher optical powers. Up to an optical input power of about 85 W, the approximated conversion efficiency is 45.7 ± 1.6% for the $HG 2 , 2$ mode, 47.3 ± 1.0% for the $HG 3 , 3$ mode, and 44.9 ± 0.8% for the $HG 4 , 4$ mode on average. Neither measurement shows a dependence on the optical input power. The fluctuations around the average are most likely caused by thermal effects on the mode matching to the triangular cavity, which we had to adjust several times during the input power increase. The approximated conversion efficiencies at the highest input power are 44.4 ± 0.6% for the $HG 2 , 2$ mode at 84.6 ± 0.5 W input power, 46.7 ± 0.6% for the $HG 3 , 3$ mode at 77.8 ± 1.0 W input power, and 44.5 ± 0.3% for the $HG 4 , 4$ mode at 83.9 ± 0.5 W input power.

Lower limits for the mode purities μ of the output beams in transmission of the triangular mode-cleaner cavity were measured at a high input power of about 84 W by injecting about 1% of the output power $P out$ into the diagnostic breadboard (DBB). The corresponding mode spectra can be seen in Fig. 3. The most significant impurities in the mode spectra appear close to the middle of the free spectral range with a normalized power of about 0.02–0.07. This mismatch most likely comes from a small rotation angle between the symmetry axes of the mode cleaner cavity and of the DBB that is a small rotation around the propagation axis. Given that the HG modes are not symmetric under such arbitrary rotations, the injected HG mode partially couples to additional eigenmodes of the DBB cavity in the middle of the free spectral range.

A residual misalignment and waist mismatch of the laser beam to the DBB cavity cause peaks at around 0.15 and 0.66 of the free spectral range. The alignment was optimized and actively controlled; still, slight misalignments occurred over the total DBB measurement time of 180 s.

The measured mode purities are 93.5 ± 0.3% for the $HG 2 , 2$ mode, 89.1 ± 1.0% for the $HG 3 , 3$ mode, and 90.5 ± 0.5% for the $HG 4 , 4$ mode. These results can be seen as lower limits to the mode purities as the other modes could as well be caused by mismatches in alignment, mode-matching and rotation. Neglecting the mismatch due to rotation, which was unexpected and could not be acted upon in our setup, these results are around 94%, which is a typical value for mode purities in currently operating gravitational-wave detector's laser systems.¹⁹ 

Combining the approximated conversion efficiency and mode purity yields the effective conversion efficiency $η eff = η appr · μ$⁠. Table I summarizes the results of the measurements. These results are fundamentally limited by the imperfect overlap of the intensity distributions of the incident $TEM 0 , 0$ mode and the generated mode.¹⁶ The corresponding theoretical limit to the effective conversion efficiencies is between 50% and 60% for the used modes and, thus, the dominating effect.

In summary, we presented the generation of Hermite–Gaussian modes at high optical continuous-wave power, using one computer-controlled and water-cooled spatial light modulator (SLM). We achieved the highest reported conversion efficiencies for higher-order Hermite–Gaussian modes utilizing a single SLM with 42.7 ± 1.6% for the $HG 2 , 2$ mode, 42.1 ± 1.0% for the $HG 3 , 3$ mode, and 40.6 ± 0.7% for the $HG 4 , 4$ mode. These results are a factor of at least 6.2 above previous publications. The SLM operates robustly against the thermal impact of the incident power up to a maximum of 85 W, and the mode purities are comparable to the mode purities of the input beams of current gravitational-wave detectors. These are promising results for the usage of SLMs in future gravitational-wave detectors, where higher-order spatial laser modes could mitigate mirror thermal noise and where high-power input beams are required. Furthermore, the conversion efficiency and, therefore, the power in the generated mode can further be increased by using two subsequent SLMs. This yields the possibility of changing the phase distribution as well as the intensity distribution and has already been demonstrated at low optical power.¹¹ 

We acknowledge the funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2123 QuantumFrontiers (390837967).