### 2.2 Mesa and flat beams

We shall see in a following Section 8 that thermal noise is reduced by widening the beam on a mirror in such a way that the fluctuations of the surface are significantly cancelled by averaging on the readout beam cross section. A way of obtaining almost “flat” beam profiles was proposed by a Caltech team led by Thorne and O’Shaughnessy [123311]. To understand the proposal, we start from a fundamental mode at its waist ()
and we take the convolution product with the characteristic function of a centered disk of radius
where is a constant to be determined by normalization. This mode of construction allows one to compute the propagated mode. Because is a linear combination of modes, its propagated value is nothing but the same combination of propagated elementary modes
so that the mode is defined at any abscissa by
where is the Rayleigh parameter, and . After some algebra, the result being axisymmetric, this is equivalent to
Normalization is easier to compute in the Fourier space. We have, after the Plancherel theorem
Now, the Fourier transform of the mode is nothing but the product of the Fourier transform of the elementary mode by the Fourier transform of the characteristic function of the disk. The Fourier transform of the mode at is
whereas the Fourier transform of the disk is
where is a Bessel function. Thus, we have
where
and are the modified Bessel functions. Therefore, we have
so that the normalized mode is simply
which is straightforward to numerically integrate, the function having a simply form. One sees that the intensity profile is flat at the waist, with sharp wings (depending on the parameter ), and that the propagated mode is also almost flat. The beam’s intensity profile (see Figure 3) is similar to a flat bump with rather sharp edges, so that the beam was called “mesa” by the previously mentioned Caltech team. The same mode propagated over kilometer-long distances exhibits a very weak distortion of its intensity profile despite diffraction. In foregoing numerical examples, we shall assume a symmetric cavity having a pair of identical mirrors matched to that kind of mode. The wavefront at 1.5 km from the waist in a 3 km long cavity determines the mirror’s shape (see Figure 4). This particular construction scheme gives a nearly flat mirror, apart from a small departure. This kind of mirror has been tested for the issues of angular alignment requirements and not found satisfactory [35]; this is why a new version has been proposed starting from spherical wave fronts in a nearly concentric cavity geometry. There is a duality relation, found by Savov et al. [35], which allows one to map the properties of this kind of beam to that of a “mesa” beam. In particular the intensity profile is identical on the mirror coating, so that the analysis we propose here is valid for the mesa beam model presented above and for the “nearly concentric cavity mode” as well (see Equation (16) in [3]).

We choose the parameters and in order to have 1 ppm clipping losses. It is possible to reduce clipping losses either by a smaller or by a smaller . However, reducing too much leads to distorted wavefronts and unfeasible mirrors. We have found a possible compromise with  = 3.2 cm and  = 10.7 cm, giving exactly 1 ppm clipping losses on both 35 cm diameter mirrors.

For most of our purposes regarding thermal problems, a crude model (flat), reduced to the characteristic function of the disk, namely an intensity function of the form

is sufficient. However, if need be, we will check that the conclusions drawn from the crude model (flat) are still valid for the realistic model (mesa), defined by Equation (2.16). However, in some specific cases (e.g., thermodynamical noise), the crude model leads to mathematical problems and cannot be used at all; thus, we must work with Equation (2.16). The value of is chosen such that the flat beam gives the darkest fringe when interfering with the mesa beam (minimizing the Hermitian distance). This gives, for the mesa beam described above, an effective value b = 9.1 cm (see Figure 3).

Throughout the following discussions, we shall numerically treat three examples. The first, “Ex1”, is the current situation for the Virgo input mirrors, i.e., an mode of w = 2 cm. The second, “Ex2”, is the flat mode described above of b = 9.1 cm, or, when needed, the mesa mode with  = 10.7 cm (1 ppm clipping loss). The third, “Ex3”, is the mode of w = 3.5 cm (1 ppm clipping loss). However, the analytic expressions are general.