2.3 Other exotic modes

Several other types of modes have been or could be proposed in the same spirit of reducing the Brownian thermal noise and/or the thermoelastic noise.

2.3.1 Bessel beams

The search for weakly diffracting beams leads naturally to nondiffractive beams. There is an obvious solution to Helmholtz’s equation in cylindrical coordinates;

ψn (r,φ, z) = exp(iβz)Jn (αr)exp (in φ), (2.18 )

where n is an arbitrary integer, Jn(x ) a Bessel function, and α and β are numbers such that α2 + β2 = k2 (k ≡ 2π∕λ). This was first noted (in the case n = 0) by Durnin [16] and is thus called a “Durnin” beam by some. We feel that for clarity it is more appropriate to call it the “Bessel” beam. The case of n = 0 is particularly interesting in the context of hyper-resolution, for instance, due to the sharp central peak when α is large, but it is forbidden in our case for the same reason. The transverse structure of such a wave is independent on z and similar to a wave guided in the core of a cylindrical fiber (in the cladding, there is a different solution smoothly matched and of finite extension). However, the energy carried by a Bessel beam is infinite, exactly as in the case of a plane wave. In fact, the wavefront is flat in the case of n = 0. The impossibility of generating waves of infinite extension leads to truncated waves having diffractive behavior and consequent clipping losses. The result depends on the method of truncation. The wavefront of such a truncated Bessel wave after propagation is hardly compatible with a reasonable mirror shape anyway.

2.3.2 Conical-mirror or Gauss–Bessel beams

The best way of truncating a Bessel beam is to make the following construction;

∫ 1 2π ik𝜃(xcosψ+ysinψ) Ψn (x,y,0) = --- ϕ(x,y,0 )e dψ, (2.19 ) 2π 0
where ϕ(x,y,z) refers to a TEM00 mode of width parameter w0. In words, we add elementary Gaussian waves whose propagation axes generate a cone of small aperture 𝜃, having its axis along the z direction and its vertex at z = 0. We assume that this is the situation at the middle of a 3 km cavity, and are interested in the amplitude at the end (or input) mirror. A sum of elementary Gaussian beams may be propagated by propagating each component separately and summing up at the end. We get, up to some phase factors irrelevant for expressing the intensity, the following amplitude for the mode at any abscissa z;
[ Z (r2 + 𝜃2z2)] ( kr𝜃) Ψn(r,φ, z) = κexp − ------------- J0 ---- , (2.20 ) w(z)2 Z
where zR ≡ πw2 ∕λ 0 and Z ≡ 1 − iz∕zR (as above). κ is a normalization factor. We have the standard relation √ ---- w (z) = w0 Z Z. This formula makes it clear that this solution is a Bessel mode truncated by a Gaussian envelope. We therefore call these “Gauss–Bessel” modes. The intensity pattern of such modes depend obviously on the width parameter w0 and on the aperture angle 𝜃. Combinations of these exist such that the intensity pattern is spread on the mirror surface, apart from a central peak. An example is shown in Figure 5View Image. The wavefront is nearly conical (see Figure 6View Image).
View Image

Figure 5: Power distribution of a Gauss–Bessel mode of parameters 𝜃 = 54 μRd , w0 = 5.2 cm, z = 1.5 km
View Image

Figure 6: Surface of a mirror matching a Gauss–Bessel mode of parameters 𝜃 = 54 μRd , w0 = 5.2 cm, z = 1.5 km

Bondarescu et al. [4Jump To The Next Citation Point] have carried out an optimization of coating thermal noise by combining LG modes. Using the better series of coefficients, they reach a wave analogous to a Gauss–Bessel mode and with a conical wavefront of the same kind. We intend to include these kinds of modes in an update to this review. To be specific, we give, in the section related to coating thermal noise (8.3.2), the figure of merit of the mode described in Figures 5View Image and 6View Image, which is not optimal, but already exhibits a good value, regarding coating thermal noise in the infinite mirror approximation (see Section 8.3).

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