Empujes horizontales causados por sobrecargas

When we consider the Cherenkov propagation inside a fibre we have to consider the probability of survival of the created photons (Collection Efficiency, CE) in the waveg- uide. CE depends on θcrit, that in turns depends on the refractive index of core and

cladding and, therefore, on NA as given by (2.22). Finally CE is determined by NA of the fibre and by the direction of the Cherenkov photons. The Collection Efficiency is expressed as the ratio between the number of collected photons and the number of the produced photons by Cherenkov Effect in the fibre as follows:

CE = collected photons

produced photons (2.48)

CE calculation To calculate the CE, taking also into account the cleaving angle of the fibre end facet, a C++ code has been written to predict the collection efficiency of Cherenkov photons in the fiber, on the basis of the equations derived in section 2.3.2 [33].

In particular, for each pair (α,S) of incident angle and impact parameter, the code samples the distribution of all possible photons generated, scanning φ in the interval [0, 2π], see eqn. (2.33), and the x component of the position of the photon generation

Px in the interval [−

p

ρ2_−S2_, p

ρ2_−S2_{], see eqn. (2.39). This information allows}

to determine, through eqn. (2.41), if the photon is transported 1. If this is the case, the CE for this particular photon, indicated with CE(γ(φ), ~P), is set equal to the

1_{Since the total reflection angle θ}

crit depends on the fiber refractive indexes, and they in turn

depend on the wavelength of the impinging photon, a complete analysis would include a calculation of

θcrit dependent on the photon energy. However, in the transmission window of the fiber (wavelengths

longer than 200 nm), the refractive index of core and cladding, calculated as per Sellmaier equation, only changes by less than 5%, thereforeθcritand its impact can therefore be considered negligible.

transmission efficiencyηtrans, given by eqn. (2.47); otherwiseCE(γ(φ), ~P) is set to 0.

The overall CE for each (α, S) pair is then given by the average of CE(γ(φ), ~P) over all samples.

Information on the actual number of photons expected to reach the detector in a given wavelength interval per impinging particle, corresponding to each particular (α, S) pair configuration can be obtained by multiplying the CE so obtained by the number of photons produced in the chosen wavelength interval for the particular (α, S) pair configuration. To do this, eqn. (2.20), describing the Cherenkov radiation spectrum as described by the Bethe-Bloch relationship, is integrated over the wavelength interval and the distance travelled by the charged particle in the fibre.

Assuming that during its travel in the fibre the impinging particle does not lose enough energy significantly to modify its velocity (constantβ) and homogeneous medium conditions (constant n), the integration over distance reduces to a multiplication by the distance travelled in the fibre, which is given by trigonometry (with reference to Figs. 2.9 and 2.6), by:

∆x = 2

p

ρ2_−S2

cos α (2.49)

The integration of eqn. (2.20) overλis thus performed numerically (to take into account the variation ofn withλ), and followed by the multiplication by ∆x.

For the application of interest in this work, the wavelength interval is chosen to be 193nm to 1064nm, respectively the lower and upper cut-off wavelengths for the Cherenkov effect.

CE optimization To optimize the collection efficiency and to make easier the cou- pling between the fibre and the detector (whose active surface is 1 mm2), multimode step index fibres up to a total diameter of 1 mm were considered. The properties of these fibres are listed in table 2.1. Furthermore the water level (i.e. OH content) present in the silicon of the fibres was also taken in account because of the different absorption bands introduced in the attenuation spectrum. High OH content fibre (600-1000 ppm ofOH−) and low OH content fibres (<2 ppm of OH−) were considered.

Inserting the features of the fibres (i.e. core/cladding refractive indexes and di- mensions) in the C++ code, the corresponding CE relative to the incidence angle of electrons passing through the fibre and to the impact parameter was calculated as the

ratio between the photons created by the impinging electrons and the photons collected at the end of the fibre.

Furthermore, as was already mentioned, from the CE it is possible to compute the actual number of photons collected per interval of impact parameter and incidence

angle, per impinging primary charged particle. This is done automatically by the

software, assuming a wavelength interval of 193÷1064 nm, by using the Bethe-Bloch

calculations described in section 2.2.2.

The C++ computer code gives a matrix of results that can be displayed as a surface plot, where the number of collected photons (vertical axis) is expressed in terms of the incidence angle of electrons and the impact parameter (horizontal axes). The impact parameter is normalized to the fibre radius and expressed as a percentage, thus an impact parameter of 0% represents a particle passing through the fibre axis whilst an impact parameter of 100% represents a particle hitting the fibre with a distance from the axis equivalent to the radius of the core. This computation of collected photons in terms of impact parameter and incidence angle is shown in Fig. 2.11 for different numerical apertures.

Since the fibre cross-section is much smaller than the beam size, the whole cross- section is impacted by the electrons of the accelerator beam. In this assumption all the impact parameters are equally likely and thus the overall number of collected photons at a given incidence angle is obtained by averaging over all the impact parameters. The resulting curve, displaying the average number of collected photons for each incidence angle, is shown in Fig. 2.12.

All the curves have maxima between 35◦and 50◦. As a consequence, the probability of photon survival is maximized when the impinging particles cross the fibres at the Cherenkov angle. The theoretical curves exhibit a cut (i.e. a sharp drop to zero) for incidence angles approaching 0◦, due to the angle of incidence of the created photon on the wall of the fibre being less than the fibre critical angle. This happens because this angle of incidence is the vector sum of the charged particle angle of incidence and the Cherenkov cone angle; when the angle of incidence approaches 0◦, the Cherenkov cone angle dominates the summation in all directions, hence leading to the angle of impact with the wall exceeding the critical angle.

The results presented in this section can be used to identify the ideal optical fibre to be used for Cherenkov effect based beam loss monitoring in accelerator environments.

Figure 2.11: Number of photons reaching the detector per incident particle in the wave- length interval 193÷1064 nm, as a function of impact parameter and incident angle for all fibres listed in Table 2.1. Number of photons corresponding to bins of 0.9◦ incident angle and and 1% impact parameter.

Figure 2.12: Number of collected photons for each incidence angle in the wavelength interval 193÷1064 nm for all fibres listed in Table 2.1. Data is averaged over all impact parameters. Number of photons corresponding to bins of 0.9◦ _{incident angle and and 1%}

However, when making such a selection, together with collection efficiency, the effects of radiation on the fibre material need to be taken into account. Therefore a discussion including the evaluation of both these factors is postponed to the end of chapter 3.