Limitations on the Design of the CEDAR Light-guide Calculations of the light-collection efficiency indicate a severe limitation in the possible collection area of a light guide designed to channel light onto the active surface of a PMT

Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September Light from each of the 8 quartz windows is reflected off a 45 o mirror and illuminates the area covered by (approximately) 30 PMTs A lightguide must channel the light onto the active regions of the PMTs with high efficiency

Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September Bristol Simulation Light from quartz windows (blue) reflects off 45 o ellipsoidal mirrors to Illuminate PMTs (green). Active surface of PMTs covers about 20% of the illuminated area so lightguide (red) consisting of a set of conical reflectors (L = 25 mm) channels the light onto the PMTs. Taking account of Fresnel reflection at PMT glass surface and multiple reflection on conical surfaces about 90% of light reaches PMTs. Can we do better?

Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September First a problem to confront... MC work by Angela Romano (Birmingham) indicates non-uniform light intensity across the PMT plane. If uncorrected we would need to increase the number of PMTs to avoid the central ones saturating - and the outer PMTs would then be used inefficiently. Liverpool addressed this problem by designing a lightguide with cones of variable light- collection area to equalise the signal in each PMT. We can make it in our workshop But, will it work? How does the light-collection efficiency vary with the incident area of the cone? What is the effect of varying the cone length or changing from a flat geometry?

Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September A B C DN S Q R α β (α+β) (α+3β) A serious issue – Part I Only a small fraction of the incident light reaches the active surface of the PMT without reflection off the conical surface of the lightguide. After m reflections the angle of incidence on the PMT glass envelope is α + 2m β For large α, β, or m this angle exceeds 90 o meaning that light is reflected backwards from the conical surface and does not reach the PMT. eg AB = 8, AN = 30, CD = 24 mm No incident light reaches the PMT, even for the most central PMT with small angle α, if there are more than 2 reflections. The problem is much worse for outer PMTs where α is 0.1 – 0.2 radians

Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September n = 1.5 Averaged polarisation A serious issue – Part II Photons reaching the PMT at an incident angle > 60 o have a rapidly increasing probability of being reflected – and hence lost. This occurs for just 2 reflections off the conical surface for large diameter. Low photon statistics compound the problem because of poor quantum efficiency (20%) in converting photons to electrons. The loss of one photon may make the difference between success and failure in detection of a group of photons by a PMT.

Methodology I – Central PMT Illuminate a conical reflector with light from a point source on the axis of the cone – cylindrical symmetry: – Calculate the contributions to the effective collection area from direct light and that reflected from the conical surface, taking into account Fresnel reflection at the PMT surface; – Calculate the efficiency as the ratio of effective to total area; – Vary the ‘collection’ cone diameter while keeping constant the ‘exit’ diameter to match the active area of the PMT. – Repeat the above for different cone lengths. Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September

8 A B C DN S Q P R α β (α+β) (α+3β) Geometry: AB = 2a = 8 mm CD = 2b > 18 mm AN = L = 30 mm [typically] R 0 = 300 mm from PMT to mirror β is the angle of the cone tanβ = (b – a) / L α is angle of light incident from mirror Coordinates: D (0, 0) A ((b-a), L) B ((b+a), L) C (2b, 0) Q (y 1 tanβ, y 1 ) P (b, -(R 0 - L)) [extrapolated view]

Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September A photon will hit the PMT without reflection if: A photon impacting the cone at Q(x 1, y 1 ), where: A photon reflecting a second time at R(x 2, y 2 ) will have coordinates: Sanity Checks:If y 2 > L, then only 1 reflection has occurred; If y 2 2 reflections have occurred; If X > (a + b) then the photon is reflected backwards. will suffer just one reflection if: This photon strikes the PMT at

Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September Results for a central cone of L= 30 mm show: i) A slow decrease in efficiency for cone diameter >18 mm due to increased Fresnel reflection at the PMT; ii) A rapid decrease in efficiency for cone diameter >25 mm because an increasing fraction of light from the second reflection fails to reach the PMT; iii) The effective collection area actually decreases for CD > 25 mm.

Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September Results for central cone of reduced length L= 20 mm show: i) Greatly reduced acceptance with cone diameter mainly because of the increasing probability of photons reflecting backwards after 2 reflections; ii) The ‘safe’ cone diameter is now reduced to 21 mm, even for small α

Methodology II – Outer PMT Typical irradiance angles for outer PMTs are α = 0.2 radians. We no longer have cylindrical symmetry about the axis of the cone and an analytical treatment is too complex without approximation. One approach would be to maintain cylindrical symmetry and set R 0 to be 100 mm instead of 300 mm. However, the central part of the cone would then always be illuminated at small angles and the outer regions at large angles – a horrible systematic bias. I have chosen a different bias by using an offset angle. All parts of the cone are irradiated at angles varying around this offset angle in a symmetrical way. This is only approximately true, but does have the advantage of averaging over the whole cone. I show indicative results for offsets of 0.1 and 0.2 radians. This study must be repeated properly using MC ray-tracing. Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September

Indicative Efficiency for Outer PMTs 13 Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September 2010

Conclusions We must ensure that all photons that reflect twice on the conical surface reach the PMT with a low probability of Fresnel reflection – hence reject 20 mm cone length; We have some leeway in relaxing the spacing of central PMTs but little or none in increasing the spacing of outer ones – which we would like to do in order to increase the ratio of collection area to that of inner PMTs; To solve the above difficulties we need one or more of: – Clever mirror geometry that produces a more uniform intensity distribution to favour equal spacing of PMTs; – A reduction in the angular spread of light from the mirror; – Light-guide (and PMTs) forming a spherical rather than planar surface so that light from the mirror is parallel to the cone axis. Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September

Summary of Results 15 Limitations on the Design of Light-guides: NA62 Collaboration Meeting Brussels September 2010