Since we have a significantly higher mechanical mode density for ET than we did for Advanced LIGO, the calculations will be slower. We therefore make use of clipping effects to reduce the maximum number of higher order optical modes which must be included in the calculation to get representative results, as described in [31]. The method is summarised below.
4.3.1
Apertures as a Source of Loss
By default, all Finesse simulations assume that optics have infinite extent in directions transverse to the beam path. This means that a mirror is treated as a reflective plane, perhaps with a defined radius of curvature. Any light incident on the mirror will be reflected or transmitted, and continue to propagate through the system. However in reality mirrors are of finite size, and therefore act as apertures. Light incident on such a mirror can therefore escape from the system, depending on the transverse distribution of the beam and the dimensions of the aperture. We describe this loss
as clipping.
For a fundamental (HG00) gaussian beam centred on a circular mirror, the fractional loss due to
clipping is given by:
θclip = P0− Pmirror P0 =P0− RD/2 0 I0e −2r2/w2 2πrdr P0 = e−D2/2w2, (4.2)
where w is the radius of the beam as defined in equation2.48, D is the diameter of the mirror, and I0 is the intensity at the centre of the incident beam. P0 is the total power in the incident beam,
while Pmirroris the power inside the locus of the mirror. Typically, we design optical systems such
that clipping losses are less than 1 ppm per optic. This corresponds to D & 5 × w.
Figure4.7 shows how the clipping loss and power reflected from a perfectly reflecting flat mirror changes with the mirror diameter, expressed as a multiple of the beam spot size. The effects of circular and square apertures on curved mirrors in optical resonators are explored in detail in [17].
Figure 4.7: Fractional power reflected and lost when a gaussian beam of radius w is reflected off a flat, circular mirror of diameter D, calculated analytically from equation4.2. A ratio of D/w ∼ 5.3 corresponds to clipping losses of less than 1 ppm.
Finesse models apertures by using higher order optical modes to compute the coupling. This modal method is not ideal, since in principal an infinite number of higher order modes is required to recreate a sharp edge. Figure4.8illustrates how the power reflected from the mirror is approxi- mated by Finesse when higher order modes up to different orders are included in the calculation. We see that even including 15th order modes is insufficient to recreate the analytical result when
there is significant clipping. Including more higher order modes also increases the number of cal- culations Finesse must do; the total number of calculations scales as 12(m + 1)(m + 2) where m is
the maximum mode order included (the ‘maxtem’). Therefore a compromise may need to be made between computation time and simulation accuracy: the maxtem included in the model should be sufficient for the numerical result to match the analytical solution to a chosen level. However for cases with minimal clipping loss (D/w > 5), a low maxtem is sufficient to accurately model clipping effects.
Figure 4.8: Numerical computation of power reflected from a mirror of diameter D when illumi- nated by a 1 W gaussian beam of diameter w, using Finesse. For low D/w ratios with significant clipping losses, the numerical calculation is unable to completely match the analytical result. In- creasing the maximum higher order mode included in the calculation (‘maxtem’) improves the accuracy of the model, but also increases the computation time.
4.3.2
Clipping of Higher Order Modes
As described in section2.5, higher order optical modes describe a range of different intensity dis- tributions transverse to the beam propagation direction. Figure4.9shows three example Hermite- Gauss modes, using the same colour map scaling in all three cases. The dashed line marks a locus of D = 6 × w, which we know (from figure 4.7) has clipping losses of ∼ 10−8 for HG00. Since
the higher order modes have a broader intensity distribution, clipping losses from the same mirror when illuminated by these will increase with mode order, as illustrated in figure4.10.
This allows a limit to be placed on the number of higher order optical modes that should reasonably be included in optical simulations. This limit differs from the compromise discussed in section4.3.1: here, the motivation to use a higher maxtem comes from effects in the simulation other than the aperture map, and we assume that we have designed our interferometer such that clipping losses for
(a) HG00 (b) HG21 (c) HG33
Figure 4.9: The transverse intensity distribution of higher order modes spreads out with mode order. The three figures show different Hermite-Gauss modes with the same fundamental beam parameter and input power. The dashed circle marks a locus of diameter D = 6 × w. Since the higher order modes have more intensity further from the centre of the mirror, they will experience more significant clipping.
Figure 4.10: Loss due to clipping increases with mode order. Here, a pure Hermite-Gauss mode is shone on a flat mirror of diameter D = 6 × w. The solid line shows the loss due to clipping for modes of type HGx0. These are the most asymmetric modes of their order, with a narrow intensity
distribution in one axis and broad in the other. This means that these modes have the greatest clipping loss of their order. For comparison, all of the HG modes at 3rd, 5th and 9th order are also shown–the difference between modes of the same order is far smaller than the difference in loss between mode orders. This plot was produced numerically using Finesse with maxtem 15.
the fundamental mode are small. Instead, some additional effect, such as a parametric instability, results in some higher order mode content also scattering into the optical system. Sufficient maxtem must be used to represent these new scattered fields. However, if no apertures are included in the model, no light will be lost from the system, and so the power in the new fields is artificially inflated. Including apertures introduces clipping losses, which will have a negligible effect on the fundamental mode but an increasingly significant effect for higher order modes, as shown in figure4.10. Above a certain threshold order, the optical mode will experience clipping losses that are sufficient to reduce its power to below the level at which it makes a significant contribution to the physical effect being studied. Therefore this threshold becomes the limiting order for the
simulation.
4.3.3
Using Clipping Effects to Improve and Speed Up PI Modelling
The study of PIs at Advanced LIGO described in chapter 3 does not include aperture effects. Mirrors without maps are treated as perfect reflective planes, and the maps generated for the mechanical modes are defined with surface distortions out to the diameter of the mirror, and contain zeros beyond this, i.e. beyond the mirror diameter, they are again treated as a perfect reflective plane.
This means that the numbers of unstable modes given by plots such as figure3.7 will experience the artificial inflation described above: mechanical modes that interact primarily with high optical mode orders will appear to have a higher gain than they would in reality, since parametric gain depends on the amount of power in the optical field and how much this field is amplified as it cycles through the interferometer.
Figure 4.11 demonstrates this effect for the case of a single LIGO arm. First, the file is run as in figure 3.7, using no apertures and including higher order modes up to 10th order. This allows a list of unstable modes to be collected and used as a subset for faster modelling. Then the same procedure is followed for four cases: no apertures, a single aperture on the ETM (i.e. the mirror which experiences the PI), a single aperture on the ITM, and apertures on both arm cavity mirrors. The radius of the aperture, 17 cm, is 5 times the spotsize on the ETM, and so clipping of the fundamental mode is of order 1 ppm and can be considered negligible. The code is run with different maximum optical mode orders, and the total number of unstable modes, as well as the time take, recorded.
The upper plot shows the total number of unstable modes in each case. While the original version, with no apertures, shows no sign of converging, we see that applying an aperture to the ETM significantly reduces the number of modes we expect to be unstable, and that this number does not substantially change when optical modes above 6th order are included. We also see that including an aperture on the ITM does not have a significant effect on this number. The lower plot tracks the total computation time of each run. Increasing the maxtem has the biggest effect on the time taken, but we also see that applying more apertures increases the integration time. Therefore since the number of unstable mechanical modes does not appear to be affected much by applying a map to the ITM, we can optimise the running time of this type of simulation, while
Figure 4.11: Number of unstable mechanical modes in a LIGO-like Fabry-Perot cavity, and corre- sponding computation time, resulting from changing the maximum order of HOM coupling (‘max- tem’) included in the model. Including aperture effects in the model limits the number of unstable modes found, resulting in a convergence that does not occur when the mirrors are treated as infinite reflective surfaces. Four cases are compared: no apertures (blue), apertures on one mirror (red: ETM, black: ITM), and apertures on both mirrors forming the cavity. In this example specifying an aperture for the ETM has the most significant effect. Since the number of unstable modes has converged once 6th order optical modes are included, the required computation time can be significantly reduced.
producing more realistic results, by choosing to model PIs with apertures in the arm cavities, and, in the case of LIGO mirrors, running numerical calculations including higher order optical modes up to 6th order.