Cryostat

evaluation of the scattered light noise possibly ge nerated by imperfect glass baffles.


1) Introduction
Installation of small aperture glass baffles in the links between towers in order to protect the vacuum pipes from scattered light raises the question of the influence of these baffles themselves on scattered light if those are imperfect, or imperfectly installed.
Les us recall that scattered light noise (SLN) is a second order process with respect to the scattering rate. Any particular channel of SLN begins by emission of scattered light off a mirror due to local roughness, a more or less complicated path involving specular reflections, and a second scattering on a mirror (possibly the same one). The spurious noise is caused by the phase modulation undergone by the light at each reflection off an object linked to ground and moving by seismic excitation, its modulation being transmitted to the main beam after the last scattering process.
We neglect third order scattering involving a rough surface on the path because firstly that kind of surface is systematically hidden by the baffles, and secondly because even for ordinary surfaces (stainless steel, glass or other), unless a special treatment was done (grating), the scattering rate is low.
We are thus in the present case faced with three extra channels of noise caused by the presence of baffles linked to the ground : -There exists on a given baffle a zone directly reflecting the light scattered either by the nearby mirror or the far one to the emitter. This may happen if there is some splinter at the baffle surface caused by a shock during manufacture or installation.
-Either the axis of the baffle is imperfectly aligned with the optical axis, or equivalently, its inner edge is imperfectly manufactured, so that there is a zone on it able to reflect scattered light from the far mirror to the nearby one (and conversely) under grazing incidence.
-Dynamical diffraction of the beam by the finite and vibrating aperture of the baffle We first recall the theoretical tools available after [1] and [2].

2) General considerations
If we consider a mirror illuminated by a TEM00 gaussian mode, the light re-emitted has two components : a specularly reflected TEM00 wave, and a scattered wave. The incoming power is shared between the two, according to the roughness of the surface. In Virgo-like mirrors, the power carried by the scattered wave is fortunately very weak. If the rough surface is viewed as a 2D random process, the scattered wave is also a random process, and at some location x at a distance D of the mirror, we have after diffraction, a new random complex optical amplitude ( ) D s x . It has been shown [1] that the relevant quantity for scattering studies is the coherence function of the speckle at distance D : where w 0 is the waist of the TEM00 beam, ε the scattering rate (a few ppm). The scattered light is emitted under all directions ( , ) θ ϕ with respect to the optical axis, but we assume an isotropic distribution in ϕ . p(θ) is the normalized distribution, in the sense that where κ is a normalisation constant such that, for 10 ppm total losses, we have εκ ~10 -7 .

3) Reflection in general
We consider now a reflecting element (spurious mirror) at distance 1 D from a mirror γ ≡ Φ . γ is a complex random process, and its variance is (see [1]): After the last scattering process, the optical amplitude re-emitted by the mirror M 2 is The phase noise being given by: In the case of a moving reflecting element such that its surface has a motion ( ) x t δ , the reflection operator is of the form  where ϑ is the incidence angle and 0 φ an unkown phase. The coupling coefficient is therefore of the form: .For a small amplitude motion (compared to the wavelength), we have at first order: ℓ where ℓ is the length of the interferometer's arms. In terms of spectral density, owing to the unknown phase 0 φ assumed randomly and uniformly distributed over [0,2π], we get : so that the problem amounts to compute * γγ in our various situations.

3) Back reflecting surface elements
In the case of a reflecting element facing a mirror, we have 1 The element's surface will be assumed having an axis of direction ( , ) α β and a mean curvature radius C r .
It is located at 0 ( , 0) x r X Y = + (r B is in practice the inner radius of a baffle) and the reflection operator is : where the integrals are extended to the surface of the element. For obtaining an order of magnitude, it is convenient to assume a rectangular shape of the element (we expect the exact shape of marginal importance), and we shall take After some algebra, these can be expressed as: With the following notation: We have neglected here the factor cosϑ because in the cases of interest, the incidence angle is nearly zero. For mirrors having 10 ppm scattering losses, 10 Hz ( ) 4 10 Hz (7) 10 m We have now to study the influence of the form factors F',F''. We can for instance study the situation when the collimation is not perfect. Assume a perfect matching r C =D, a perfect azimuthal orientation β=0, and an approximate radial orientation α=α 0 +δα. We have two cases : -The case of a far (assume 3000 m) mirror. The waist of the main beam is located inside the long cavity and about w 0~6 mm. After crossing the substrate viewed as a diverging lens, it remains almost unchanged. Obviously the width of the beam is very different (about 6 cm), but we need only the Gaussian divergence of the beam 0 / g w θ λ π = . Then, with a=1mm and D=3000m, we get ' 169 σ ≈ which is large compared to the integration range, we can thus ignore the Gaussian factor in the integrals (5.a,b) -The case of a close mirror (say D=0.9m), then conversely, ' 0.05 σ ≈ is rather small, and the Gaussian factor becomes predominant. The result must be nevertheless numerically processed. We have a reduction factor of ' '' ' 0.09 F F σ π ∼ ∼ , with an angular width of about 8 mRd.
The conclusion is that the maximum value is significant, but unless severe collimation conditions, the noise coming from the far mirror spurious light is negligible, and with less severe collimation conditions, the noise coming from the close mirror spurious light is also negligible.
See below the dependence of the spectral density vs misalignment.

Fig2
D=3000m The next figure shows the weak dependence of the spectral density of noise on the ROC of the reflecting element as soon as the latter is larger than a meter.

Fig3
Spectral density of noise vs curvature radius of the reflecting element [m]

4) Grazing reflection off inner edges
We consider a situation in which, due to misalignment of a baffle, a piece of its inner edge is able to directly reflect the light scattered by mirror M 2 to mirror M 1 (and conversely) (see Fig.   above). The axis of the baffle is assumed making an angle µ with the optical axis. In order to retrieve the preceding situation of a mirror under quasi normal incidence, we consider the reflecting surface as a plane mirror, and using the method of images, we replace the grazing incidence by a quasi-normal incidence of a virtual mirror orthogonal to the preceding, such that the virtual incidence is now α = µ + θ 2 = θ 1 − µ = (θ 1 +θ 2 )/2. If now the baffle has not exactly this dangerous attitude, it has an inclination µ+δµ and the virtual incidence is α=(θ 1 +θ 2 )/2+δµ. The virtual mirror may be represented by where we have assumed, without loss of generality the axis of the baffle in the (xz) plane, X representing the excursion relative to the center of the virtual mirror. We are back to the preceding problem in a simplified version. The coupling coefficient has a variance: Where D 1 and D 2 are the distances of the baffle to nearby and far mirror respectively. As in the preceding section, we take ( , ) . We get thus As in the preceding section, we consider the surface integrals as being extended to the rectangular zone . But (a,b) must be interpreted as the projection of the actual reflecting surface onto the incoming/reflected beam, so that a is not the full width W of the inner edge of the baffle, but its projected value Wsinµ.
After some algebra, this is as well And the following notation: the LSD of noise is now : An expression quite similar to (6) , except that the modulation factor is cos 2 µ=1.7° The three first angles are rather large, and a careful installation should easily avoid such misalignments. It seems thus that only the last baffle could possibly present a danger if it has an obliquity angle 1.7 0.1 µ =°±°, which has however a low probability.

5) Couplers
Small splinters on the baffle edges may be oriented in such a way that they produce the same effect as the preceding, even with perfectly oriented baffles. The following figure summarizes the effect of the misalignments, and gives an idea of the angular range.

6) Dynamical diffraction : clipping noise
This question is not linked to scattered light, but has been raised in the past. The baffle has a finite inner radius a, so that there is a coupling coefficient γ between the incoming beam