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 LG0,0 at its waist (z = 0)
∘ ----- 2 2 2 2 2 2 ϕ(x,y,0) = πw2-exp (− r ∕w 0) (r = x + y ), (2.5 ) 0
and we take the convolution product with the characteristic function of a centered disk Δ of radius bf
∫ -κ-- Ψ (x,y,0 ) = πb2 ϕ(x − x0,y − y0,0)dx0dy0, (2.6 ) f Δ
where κ is a constant to be determined by normalization. This mode of construction allows one to compute the propagated mode. Because Ψ is a linear combination of modes, its propagated value is nothing but the same combination of propagated elementary modes
κ ∫ Ψ (x,y,z ) = --2- ϕ(x − x0,y − y0,z)dx0dy0, (2.7 ) πbf Δ
so that the mode is defined at any abscissa z by
∘ ----- κ 2 Ψ (x, y,z) = --2- ---2 exp(− itan−1(z∕zR )) πbf πw{ } ∫ Z [ 2 2] exp − -2- (x − x0) + (y − y0) dx0dy0, (2.8 ) Δ w
where zR ≡ πw20∕λ is the Rayleigh parameter, Z ≡ 1 − iz∕zR and √ ---- w ≡ w0 Z Z. After some algebra, the result being axisymmetric, this is equivalent to
----- 2κ ∘ 2w2 Ψ (r,z) = -2- ---- exp(− itan−1(z∕zR )) bf π ∫ b∕w [ ] exp − Z (r∕w − x )2 exp(− 2Zrx ∕w )I0(2Zrx ∕w )xdx. (2.9 ) 0
Normalization is easier to compute in the Fourier space. We have, after the Plancherel theorem
∫ ∞ ∫ ∞ 2 2 -1- &tidle; 2 ∥ Ψ ∥ = 2π 0 |Ψ(r,z)| rdr = 2π 0 |Ψ (ρ,z)| ρdρ. (2.10 )
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 z = 0 is
∘ ------ [ w2 ρ2] ϕ&tidle;(ρ, 0) = 2πw20 exp − --0-- , (2.11 ) 4
whereas the Fourier transform of the disk is
2J1(bfρ) &tidle;ℱΔ (ρ) = --b-ρ---, (2.12 ) f
where J1(x) is a Bessel function. Thus, we have
[ ] κ2 ∫ ∞ 4w2 κ2 ∫ ∞ w2x2 dx 2w2κ2 ∥ Ψ ∥2= --- |&tidle;ϕ(ρ,0)ℱ&tidle;Δ(ρ)|2ρdρ = ---02-- J1(x )2 exp − -0-2- ---= --02--M, (2.13 ) 2π 0 bf 0 2bf x bf
[ ] M ≡ 1 − exp (− b2f∕w20) I0(b2f∕w20) + I1(b2f∕w20) (2.14 )
and {In(x),n = 0,1,2, ...} are the modified Bessel functions. Therefore, we have
---bf--- κ = √ ----, (2.15 ) w0 2M
so that the normalized mode is simply
∫ b ∕w --2√Z---- f [ 2] Ψ (r,z) = bf πM 0 exp − Z (r∕w − x) exp(− 2Zrx ∕w )I0(2Zrx ∕w)xdx, (2.16 )
which is straightforward to numerically integrate, the function e−X I0(X ),{X ∈ ℂ, ℜ (X ) > 0 } having a simply form. One sees that the intensity profile is flat at the waist, with sharp wings (depending on the parameter w0), and that the propagated mode is also almost flat. The beam’s intensity profile (see Figure 3View Image) 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 4View Image). 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 [35Jump To The Next Citation Point]; 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 w0 and b in order to have 1 ppm clipping losses. It is possible to reduce clipping losses either by a smaller w0 or by a smaller b. However, reducing w0 too much leads to distorted wavefronts and unfeasible mirrors. We have found a possible compromise with w0 = 3.2 cm and bf = 10.7 cm, giving exactly 1 ppm clipping losses on both 35 cm diameter mirrors.

View Image

Figure 3: Solid line: Intensity profile of a normalized mesa mode of parameters b f = 10.7 cm, w0 = 3.2 cm. Dashed line: nearest flat beam profile (b = 9.1 cm).
View Image

Figure 4: Surface of a mirror matching the mesa beam of parameters b f = 10.7 cm, w0 = 3.2 cm

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

{ I (r ) = -1-- 1 (r ≤ b), (2.17 ) πb2 0 (r > b)
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 (2.16View Equation). 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 (2.16View Equation). The value of b 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 3View Image).

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 LG0,0 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 b f = 10.7 cm (1 ppm clipping loss). The third, “Ex3”, is the LG5,5 mode of w = 3.5 cm (1 ppm clipping loss). However, the analytic expressions are general.

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