Abstract
For coating Brownian thermal noise reduction in future gravitational wave detectors, it is proposed to use light in the helical Laguerre-Gaussian LG3,3 mode instead of the currently used LG0,0 mode. However, the simultaneous reduction of quantum noise would then require the efficient generation of squeezed vacuum states in the LG3,3 mode. Current squeezed light generation techniques employ continuous-wave second harmonic generation (SHG). Here, we simulate the SHG for both modes numerically to derive first insights into the transferability of standard squeezed light generation techniques to the LG3,3 mode. In the first part of this paper, we therefore theoretically discuss SHG in the case of a single undepleted pump mode, which, in general, excites a superposition of harmonic modes. Based on the differential equation for the harmonic field, we derive individual phase matching conditions and hence conversion efficiencies for the excited harmonic modes. In the second part, we analyse the numerical simulations of the LG0,0 and LG3,3 SHG in a single-pass, double-pass and cavity-enhanced configuration under the influence of the focusing, the different pump intensity distributions and the individual phase matching conditions. Our results predict that the LG3,3 mode requires about 14 times the pump power of the LG0,0 mode to achieve the same SHG conversion efficiency in an ideal, realistic cavity design and mainly generates the harmonic LG6,6 mode.
Highlights
One main focus of the gravitational wave community is the steady improvement of the gravitational wave detectors
We have first theoretically investigated the continuous-wave second harmonic generation pumped by an undepleted pump mode which generally excites a superposition of harmonic modes and secondly compared the LG0,0 and LG3,3 SHG in the single, double-pass and cavity-enhanced configuration via numerical simulations
The same equation shows how the Gouy phase causes the harmonic modes which are excited by a single pump mode to exhibit different phase matching conditions
Summary
One main focus of the gravitational wave community is the steady improvement of the gravitational wave detectors. These conditions are especially important for the LG3,3 SHG and are reflected in the phase relations of the excited harmonic modes to the crystal polarisation along the crystal. Together with the subsequent double-pass simulations, our results allow for predictions regarding both the harmonic output fields and the ratio of the pump-mode dependent effective nonlinearities of the nonlinear medium We use the latter to quantify the SHG comparison. We take the double-pass results as an input for an existing simulation tool for the cavity-enhanced SHG to compute the conversion efficiencies of both the LG0,0 and LG3,3 mode under typical experimental conditions prevailing in gravitational wave detectors [16]
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