New fiber read-out design for the large area scintillator detectors: providing good amplitude and time resolutions
NNew fiber read-out design for the large area scintillatordetectors: providing good amplitude and timeresolutions
V. Grabski Instituto de Fisica Universidad Nacional Autonoma de Mexico, Mexico a Av Universidad 3000, Instituto de Fisica, Coyoacan, Mexico
Abstract
Most of the time-of-flight systems as well as the fast interaction trigger detectorshave large surfaces and the principal requirements for the above mentioneddetectors is a good time resolution of the order 100 − ps . The easiest solutionhere is to split the large surface into the small tiles with a read-out directlyattached to the photo-sensor. This solution is expensive because the numberof the channels grows proportional to the surface. Although if the coverage ofthe whole surface by the sensors is not sufficient, one can obtain non uniformresponse, which is not acceptable if uniformity is required. Our suggestion isbased on the usage of clear fibres mounted perpendicular on the surface of thescintillator as a matrix providing uniform surface reduction and keeping most ofthe benefices of the small tiles with the photo-sensors. In this work the designand some analytic estimations of the light collection efficiency for direct andreflected photons will be presented. Also some simple estimations are presentedfor the time-spread and for the light pulse wave form dependent on the lateralsizes and the thickness of the scintillator. Keywords:
Scintillation detector, Time-of-flight detector,Particle identification, Fast Interaction Trigger, Optical Fibre ∗ Corresponding author
Email address: [email protected] (V. Grabski)
Preprint submitted to Journal of L A TEX Templates September 4, 2019 a r X i v : . [ phy s i c s . i n s - d e t ] S e p . Introduction Introduction of WLS fibres resolve many problems of the light collection likeuniformity, compact and flexible light transport. These light collection systemswere used mainly in calorimetry and other applications where the timing char-acteristics are not important [1]. After introduction of fast WLS fibres becomespossible to use them also for the applications when good time resolution( 1ns)is required [2]. Anyway the use of WLS will include additional stochastic pro-cess, that spreads in the time of fast photons from scintillator and worsen timeresolution. Usually standard light guide designs as well as WLS systems using asingle photo-sensor includes time spread due to coordinate spread related lateralsizes of the scintillation tile. To avoid this, normally small scintillator tiles withthe direct coupling photo-sensors to the scintillator surface have been used [3].This design is good for timing, but can introduce non uniform surface responseif photo-sensor surface is considerably smaller than the scintillator surface. Onthe other hand this design will increase the number of photo-sensors as well asthe number of electronic channels. In this work the proposed design providesa uniform reduction of the scintillator surface up to the photo-sensor surfaceand conserves the fast photons timing as in the case of direct coupling. Ofcourse using long fibres will spread photons in the time, because of the angularaperture of the fibre transmission and will worsen time characteristics. Here bythe fibre density variation one can reach maximum surface reduction to achievethe required time characteristics. Sometimes it is important when the largedynamic range for the amplitude measurement is required. This design alsoprovides a volume uniformity response, which is very important for the detec-tors with large volumes and when the pinpoint light pulse should be detected.The new design is based on the usage of clear fibres having the same longitudeand are mounted perpendicularly to the large surface of the scintillator. Theusage of other types of surface reducers like concentric cone or fibre optic ta-per requires large space for the detector and do not provide uniform surfaceresponse.The principal disadvantages of this design is the large number of fibres2hat is proportional to the surface and can be of order of few tenths of thousandper square meter. Historically this scheme of the light collection was proposedfor the large area scintillator detector V0+ of FIT(Fast interaction trigger)[4] forthe ALICE experiment[6]. This scheme have been intensively studied for a fewyears and demonstrated that can be used for the V0+ detector, which is alreadyconstructed and will be installed at the end of 2020. In the chapter of ”Fibreread-out design” it will be shown basic light collection properties, which can begeneralized for the WLS design as well. In the chapter of the ”Light collectionefficiency estimation” will be presented some analytic relations for the directphotons from the light source and for some well defined reflectors like mirrorand Lambert diffuse reflection. In Appendixes some calculations for the lightcollection efficiency considering different reflection conditions are presented.
2. Fibre read-out design
Schematic view of the design is shown in Fig 1. The fibres are mountedperpendicularly to the scintillator surface like a matrix. For the simplicity, herewe demonstrate a square tiling, but one can use any type of tiling dependenton the detection task. This tiling has Cartesian coordinate symmetry and thedistance ( d f ) between neighbour fibres will define the fibre density ( D f ) on theunit surface as D f = 1 /d f . There are three principal ways to couple fibres withthe scintillator: -a direct one with an optical contact (see Fig 1(a)); -non directwith small light concentrators for each fibre shown in Fig 1(b) and betweenthe scintillator and fibre matrix there is a plastic layer Fig 1(c). The designin Fig 1(a) is applicable if the scintillator thickness is much larger than thefibre distance. For this design if the light source is close to the fibres(when oneor a few fibre are participating in the light collection) it will provide relativelarge light collection efficiency that will affect surface uniformity. Also there areshadow regions between fibres with very low light collection efficiency, becauseof limited fibre transmission angle. The design shown in Fig 1(b) providesbetter surface uniformity reducing small distance effects of the light source.3lthough this design is preferable, but is more complex for the construction.Also this design has no shadow regions like the design in Fig 1(a) which becameimportant if scintillator thickness is comparable with the fibre distance. Theshadow regions as well as the small distance effects also can be reduced usingthe design shown in Fig 1(c) that is easier for the construction than the one withlight concentrators and provides better uniformity than direct coupling of fibres.In this case the thickness of the plastic shown in Fig 1(c)can be estimated bythe fibre distance to eliminate shadow regions. All three options of mountingfibres have similar characteristics and the difference between them is due tosmall differences in uniformity and small distance effects. With these smalldifferences it can be shown(see the next chapter) that this design in generalprovides volume uniform light collection efficiency.
3. Light collection efficiency estimation
For the analytic estimation of the light collection efficiency we introduce afew definitions as: the fibre is an object that will transmit the light within someangular acceptance Θ(which depends only on the fibre type) with an efficiency ε f (which depends on the fibre type, longitude, light wave length and bendingradius); between the fibre and the scintillator there is an optical contact thatis transmitting the light with the efficiency ε sf and between the fibre and thephoto-sensor the transmission efficiency is ε fp . So for a given point of thelight source inside the scintillator volume the light collection scheme for thedirect photons (photons from the source without reflections from the scintillatorborders) DP can be drown as it is shown in Fig 2. As it can be seen from Fig2 for the DP the number of the fibres N f ,which are participating in the lightcollection can be estimated as a: N f = D f × π × h × tan Θ (1)where ( h ) is the distance of the light source from the plane of the fibre matrix.For this point light source a given fibre with the index i will have approximately4 solid angle acceptance: ∆Ω i ≈ S f × cos θ i h (2)where S f is the fibre surface and θ i is the angle between the vector connectingthe light source to the given fibre center and the vector that is perpendicularto plane of the fibre matrix(see Fig 2). So for the number of photons that canbe transmitted up to photo-sensor by the given fibre N i ( h ) dh from the lightsource(assuming uniform light emission) with the size dh can be written: N i ( h ) dh = N × F tr × S f × exp( − a ( h,l ) cosθ i ) × cos θ i × π × h × H dh (3)where N is the number of photons emitted in the whole scintillator thicknessof H with the uniform emission in 4 × πSr solid angle, a ( h, l ) = h/λ sc + l/λ f is the distance of the light source from the photo-sensor in attenuation lengthunits(where λ sc and λ f are attenuation lengths of the scintillator and the fibrecorrespondingly, l is the fibre length), F tr = ε f × ε sf × ε fp is the transmissionefficiency through the surfaces scintillator-fibre and fibre-photo-sensor. Here weassume the approximation that the transmission through the surfaces is almostindependent from the angle for the small angles θ ≤ o and from the position oflight emission. Then the number of all photons( N p ( h ) dh ) from a given point ofthe light source that will be transmitted up to the photo-sensor can be estimatedas a sum for all N f fibres assuming that the transmission properties are the samefor all fibres: N p ( h ) dh = N × F tr × S f × N f × π × h × H ×{ N f × (cid:88) N f i =0 exp( − a ( h, l )cos θ i ) × cos θ i } dh (4)Combining Eq. (1) and (4) and integrating for the scintillator thickness forthe total number of the collected direct photons N DP can be written:5 DP = (cid:90) H N p ( h ) dh = N × F tr × S f × D f × tan Θ4 × H × (cid:90) H (cid:18) N f × (cid:88) N f i =0 exp( − a ( h, l )cos θ i ) × cos θ i (cid:19) dh (5)The dependence of N DP from the distance h that is included in a ( h, l ) is as h/λ sc . If h (cid:28) λ sc then N DP is approximately independent from the distanceof the matrix plane. Of course the condition of h ≥ d f will be fulfilled havingat least few fibres involved in the light collection. Most of the fast scintillatorshave attenuation lengths larger than 100cm, so this approximation could begood enough for the scintillator thickness up to 10cm. For the designs shownin Fig 1(b and c) the condition of h ≥ d f is already fulfilled so it can beachieved better volume uniformity for entire scintillator volume. The calculationof the integral in (5) to estimate of N DP is performed in Appendix A. Obtainedanalytic expression is a large one, thats why we present here as a functionΦ dp (cos Θ , a , a ): N DP = N × F tr × S f × D f × tan Θ4 × H × Φ dp (cos Θ , a , a )where a = a (0 , l ) is the fibre longitude in attenuation longitude units and a = a ( H, l ) is the sum of the scintillator thickness and the fibre longitudein attenuation longitude units. The light collection efficiency for DP can beestimated as Υ DP (cos Θ , a , a ) = N DP / N .Υ DP = N × F tr × S f × D f × tan Θ4 × H × Φ dp (cos Θ , a , a ) (6), which converts a relation independent from H if we ignore attenuation in thescintillator:Υ DP = N × F tr × S f × D f × tan Θ4 × Φ dp (cos Θ , a , a ) (7)Ignoring also light attenuation in the fibres we obtain a simple expression.Υ DP = F tr × S f × D f × tan Θ(1 − cos Θ)16(1 − cos Θ) (8)6he simplified relation of the light collection efficiency for the DP dependson the three parameters like the fibre surface S f , the fibre aperture angle Θand the fibre density D f . One can easily increase or decrease the efficiencymanipulating with these parameters. Direct photons are only a part of the lightthat can be collected. Painting all scintillator borders with the black color onecan reject the contribution from the reflected photons. The light collection effi-ciency will increase considering the light reflection from the scintillator borders.For example in case of specular reflecting coating on the large surface of thescintillator in front of the fibre matrix will increase the light collection efficiencyapproximately twice due to the solid angle increase (see Appendix A). If matrixsurface also has specular reflective coating then the light collection efficiencycan be estimated as a sum of all reflections. The number of the reflection ispractically limited, because of the efficiency reflection from the surfaces, whichis normally smaller than 0.8 for blue scintillator light. So 10 reflections aresufficient to estimate efficiency within a few percent precision.In case of the usage of the diffuse reflector coating for the scintillator thelight collection efficiency also depends on the lateral sizes, because in this caselarge areas of the surface can have contribution in the light collection. Of courseeverything depends on the geometry and for the estimation efficiency in generalwe will perform it for the infinite lateral sizes of the scintillator, because in thatcase it is easy to obtain analytic expressions. For this estimation we shouldconsider some real construction design, for example, the one that was used forthe large scintillator area of the ALICE FIT detector[5]. The simple light col-lection drawing for the above mentioned detector for the infinite lateral sizes isshown in Fig 3. As it can be seen from the figure the whole surface against thefibre matrix as well as the fibre matrix surface participate in the light collectionprocesses if they both have diffuse reflection properties. Here we are going toconsider it a Lambert type reflection (the reflected light has cosθ low from thezenith angle) because in this case it is easier to obtain analytic estimations. As itcan be seen from Fig 3 for each small solid angle, emission from the point sourceafter reflection can contribute in the light collection process. So for the infinite7ateral sizes of the scintillator for the first reflection only the half of the lightemission can participate in the light collection. The other half will contributefor the multiple reflections, if the matrix surface also has diffuse reflective coat-ing. The number of significant reflections is limited within a few ns after morethan 10 reflections the emitted light practically will be absorbed or lost becauseof the average reflection efficiency, which is below 80%(depends on the emis-sion spectrum). So for the first reflection and for a given fibre i using similarprocedure like (3-6) for the number of transmitted photon N idf ( θ , θ i , h , H )assuming a Lambert type of reflection, can be written: N idf ( θ , θ i , h , H ) d Ω dh = N × π × H × exp( − b ( h )cos θ ) d Ω dh × (cid:18) F tr × (cid:15) dfr π × exp( − a cos θ i ) × cos θ i × d Ω i (cid:19) , (9)where the expression before the parentheses is the number of photons reach-ing the reflection surface within the solid angle d Ω = dφ × d cos θ , φ is theazimuthal angle, θ is the zenith angle (see Fig 3), expression in the parenthesesis the portion of reflected light that is transmitted up to phtosensor by the fibre i , h is the distance of the light source from the reflection plane, b ( h ) = h /λ sc , a is the same as in Eq. (6), (cid:15) dfr is the average efficiency of the reflection ofthe diffuse coating and θ i is the angle between the vector connecting the lightsource on the reflection plane to a given fibre center and the to vector that isperpendicular to the fibre matrix plane(see Fig 3), d Ω i is the solid angle of thefibre i from the light source located on the reflective surface and approximatelycan be estimated using expression (2). The total number of the transmittedphotons N df (Ψ , H ) can be estimated summing for the all participating fibres N fH , replacing value of N fH using expression (1) and performing the integra-tion of: d Ω by θ and φ ( θ from zero to Ψ and φ from zero to π ) and by h (fromzero to H ): 8 df (Ψ , H ) = (cid:88) N fH i =0 N idf = N × (cid:15) dfr × F tr × S f × D f × tan Θ4 × H × (cid:90) H (cid:90) exp( − b ( h )cos θ ) d cos θ dh × (cid:26) N fH × (cid:88) N fH i =0 exp( − a cos θ i ) × cos θ i (cid:27) (10)After integration and sum estimation in the expression (10), which is ob-tained in Appendix B, the result is shown below: N df (Ψ , H ) = N × (cid:15) dfr × F tr × S f × D f × tan Θ4 × H × I (Ψ , H ) × S dfr ( a , Θ) (11)where I (Ψ , H ) is the integral in relation (9) that is the part of the emittedlight reaching the reflection surface and S dfr ( a , Θ) is the value of the sum inthe parenthesis in the expression (10). For the infinite lateral sizes Ψ = 90 o If the plane of the fibre matrix plastic support also has a diffuse reflectivitythen we should sum all of reflections to obtain total light collection efficiency.For the matrix plane the reflection efficiency should be reduced by the factorof ε fm = 1 − S fib /S tot , which is connected with the light loose in fibres (seeAppendix A). So for the number of transmitted photons of a double reflection(at the beginning from the matrix plane and then from the plane in front of thefibres) N df (Ψ , H ), can be obtained an expression in similar way: N df (Ψ , H ) = N × (cid:15) dfr × F tr × S f × D f × tan Θ4 × H × (cid:15) dfr × ε fm × ε sfs × I (Ψ , H ) × S dfr ( a , Θ) (12)where a = a (2 H, l ) and ε sfs is the efficiency of the light transmission be-tween the scintillator and the fibre plastic support.Here we ignore the lightattenuation in the thin layer of the fibre plastic support, which is much smallerthan in the scintillator. So in this way we can estimate any time of the reflectionsjust modifying parameter a and multiplying average reflection efficiency for thesurface. Number of reflections is limited because of the value of ε dfr ≤ . ε fm ≤ . . 9he light collection efficiency with diffuse reflection for n reflections and forinfinite lateral sizes can be estimated as:Υ DF R = (cid:15) dfr × F tr × S f × D f × tan Θ4 H × I (Ψ , H ) × ( S dfr ( a , Θ) ++ (cid:15) dfr × ε sfs × ε fm × S dfr ( a , Θ) + ... + (cid:15) n − dfr × ε n − sfs × ε n − fm × S dfr ( a n , Θ)) (13)where a n = a ( nH, l ). In case of short fibre lengths the attenuation can beignored and for the light collection efficiency can be obtained a simple expressionfor the first n reflections that is shown below(see Appendix B):Υ DF R = (cid:15) dfr × F tr × S f × D f × tan Θ × (1 − cos Ψ) × (1 − cos Θ)10 × (1 − cos Θ) (1 + (cid:15) dfr × ε sfs × ε fm + ... + (cid:15) n − dfr × ε n − sfs × ε n − fm ) (14)For the average reflection efficiency about (cid:15) dfr = 0 . T OT can be estimated as a sum of the direct and the reflected lightcollection efficiencies: Υ
T OT = Υ DF + Υ DF R
All analytic expressions except when the attenuation is ignored include theExponential Integral, which can be evaluated only numerically. For this reasonwe make two plots to show the dependencies of the light collection efficiency fromthe number of reflections and from the lateral size. Here we ignore the reflections10rom the lateral sides to simplify the calculations. For these calculations thefollowing parameter values have been used( λ sc = 100 cm , λ f = 200 cm , H =4 cm , l = 50 cm ,Θ = 21, S f = 0 . π/ cm , D f = 4 /cm , F tr = 0 . (cid:15) dfr = 0 .
4. Time spread estimation
The time spread for DP photons that is important for the fast timing hasthree principal sources: -the stochastic process of light emission; -the fibre lon-gitude because of the transmission light angle aperture; -the thickness of thescintillator. The main time spread comes from the stochastic light emissionprocess of the scintillation material. To have good time resolution it is requiredthat the light collection system should have sufficient efficiency and less timespread for the photon transport. For DP the estimation of the time spread dueto photon transport is simple and is done in Appendix C using the light pathspread due to angle and photon position variation inside the scintillator. Theobtained relation for the relative value of time spread(standard deviation over11o average value) assuming that the light speed is the same in the scintillatorand in the fibre as it is shown below: σ t t = (cid:115) (1 + Hl ) × (cos Θ − ((1 + H l ) × cos Θ × ln cos Θ − t is the average value of thetime transition to the photo-sensor, l is the fibre length and H is the scintillatorthickness. As it can be seen from the formula() the main time spread comesfrom the fibre length if the thickness of the scintillator is much smaller than thefibre longitude. In this case the time spread is proportional to average transitiontime which in its turn is proportional to the fibre longitude. For the single cladfibre with NA(numerical aperture) 0.5 the coefficient is about 0.02. For themulti-clad fibres this coefficient will increase up to 0.03(using formula C6). Sothis means that for the time performance is better to use higher density of thefibres and a single clad than a multi-clad fibres. Of course this estimation shouldbe considered as a limit that can be achieved for the given H and l .For the specular reflection from the large surface of the scintillator the thick-ness of the scintillator will be doubled. In case of the diffuse reflection the timespread estimation is not as easy as in the previous case. To study the timespread for the diffuse reflection a simple simulation geometry have been used(shown in Fig 6). All important simulation parameters are mentioned in theprevious chapter. The simulation results for the pulse time wave forms of di-rect and reflected photons with the large statistics just to have smooth curveis shown in Fig. 6. As it can be seen from the figure the number of the re-flected photons is significantly larger(factor of 3) than DP and the rise timeof fast photons is less than 100 ps . The delay of the twice reflected photons isabout 400 ps , which corresponds to the thickness of scintillator. The standarddeviation of the time spread of DP is about 112 ps, which is similar to the ap-proximate value obtained from formula (C6). In the same figure is also shownthe modification of the time wave forms when the scintillator stochastic timeemission is included. As it can be seen from the figure the rise time of the light12ulse mainly depends on DP and the once reflected photons. So if the photonstatistics and the surface uniformity is not important the multiple reflectionscan be excluded painting the fibre plane surface with the black color.
5. Conclusions
In this study is presented a new light collection design using clear fibres andsome analytical expressions to estimate expected light collection efficiency. His-torically this scheme was proposed for the FIT(Fast interaction trigger) largearea scintillator detector V0+ of the ALICE experiment. The obtained approx-imate analytical expressions will be useful to understand the light collectionscheme as well as for the development of a real detectors in general. Experi-mental studies for V0+ prototypes a qualitative agreement between some of theresults of these calculations have been already observed.
6. Acknowledgements
Author acknowledge the partial support from PAPIIT-UNAM IN111117,and CONACYT 280362 grants.
7. References
Appendix A. Efficiency estimation for direct photons
Before estimating the integral in the expression (5) or Φ dp (cos Θ , a , a ) firstof all it should be estimated the sum shown below: S ( a, Θ) = 1 N f × (cid:88) N f i =0 exp( − a ( h, l )cos θ i ) × cos θ i (A.1)As it can be seen from the above mentioned expression it is an average valueof exp( − a ( h,l )cos θ ) × cos θ for the angles θ ≤ Θ. The estimation of (A.1) can beeasily performed representing it in the integral form and assuming a uniformlight emission: S ( a, Θ) = (cid:82) π (cid:82) Θ0 exp( − a ( h,l )cos θ ) × cos θd Ω (cid:82) π (cid:82) Θ0 d Ω = (cid:82) Θ0 exp( − a ( h,l )cos θ ) × cos θd cos θ (cid:82) Θ0 d cos θ (A.2)To perform the integration of the numerator of the relation (A.2) it is convenientto make a variable change like x ≡ cos θ and make the expressions shorter for a ( h, l ) we will use the notation a : f ( x, a ) = (cid:90) exp( − ax ) × x dx, (A.3)14hich after integration by parts[7] and using the relation for the exponentialintegral E i ( − x ) = − E ( x ) for f ( x, a ) can be obtained: f ( x, a ) = e − ax (cid:0) a E (cid:0) ax (cid:1) e ax + 6 x − ax + a x − a x (cid:1)
24 + const (A.4)where E (cid:0) ax (cid:1) is the well known exponential integral defined as E ( x ) = (cid:82) ∞ x exp( − u ) u du and has series representation:E ( u ) = − γ − ln( u ) − (cid:88) ∞ k =1 ( − k × u k k × k ! (A.5)where γ ≈ . u > (cid:88) ∞ k =1 ( − k × u k k × k ! = (cid:88) ∞ k =1 k (cid:89) ki =1 − ui (A.6)With this expression for the sum, 100 terms are sufficient to get stable evaluationfor u <
20 values. For u >
20 probably the double precision is not sufficient forthe stable evaluation of E using the expressions (A.5 and A.6). For this workit is not so important, because the fibre length and the scintillator thickness inreal detector designs can not be much larger than the attenuation lengths(always u < S ( a, Θ) in (A.2) we will obtain: S ( a, Θ) = f (1 , a ) − f (cos Θ , a )1 − cos Θ (A.7)So the integral in the expression (5) can be written: (cid:90) H S ( a, Θ) dh = λ sc (cid:90) a a S ( a, Θ) da = λ sc (cid:90) a a f (1 , a ) − f (cos Θ , a )1 − cos Θ da (A.8)where a = a (0 , l ) is the fibre longitude in attenuation longitude units and a = a ( H, l ) is the sum of scintillator thickness and fibre longitude also inattenuation longitude units as it has already been defined in the main text15see Eq. 6). Ignoring attenuation in the scintillator, which means S ( a, Θ) isindependent from h then for Φ dp (cos Θ , a , a ) we obtain:Φ dp (cos Θ , a , a ) = H × S ( a, Θ) (A.9)Here before evaluating the integral mentioned above, we evaluate the indef-inite integral: φ ( x, a ) = (cid:90) f ( x, a ) da (A.10)Using expression (A.4) for f ( x, a ) then for the φ ( x, a ) we obtain: φ ( x, a ) = x e − ax (cid:16) ( ax ) + 2( ax ) + 6( ax ) (cid:17) − a (cid:18) γ ax −
125 + (cid:88) ∞ k =1 k × ( k + 5) (cid:89) ki =1 − ax × i (cid:19) + const (A.11)Finally for the Φ dp (cos Θ , a , a ) using the expression (A.7) we obtain:Φ dp (cos Θ , a , a ) = λ sc ( φ (1 , a ) − φ (1 , a ) − ( φ (cos Θ , a ) − φ (cos Θ , a )1 − cos Θ (A.12)Excluding the light attenuation in the expression (A.3), (ie a=0 in A4) forΦ(cos Θ) we will obtain:Φ dp (cos Θ) = H × − cos Θ4 × (1 − cos Θ) (A.13)The usage of a specular reflection with the reflection efficiency ε sp for theplane in front of the fibre matrix is similar to the one for the direct photons.So for the function Φ sp (cos Θ , a , a ) can be used the expression (A.10) with a = a ( H, l ) and a = a (2 H, l )Φ sp (cos Θ , a , a ) = ε sp × Φ dp (cos Θ , a , a ) (A.14)If the plain of the matrix also has specular reflectivity then we should sumall of reflections to obtain total light collection efficiency. For the matrix plane16xcept reflection efficiency we have light loose connected with fibres. The effi-ciency connected with this lost ε fm = 1 − S fib /S tot where S fib is the sum of thefibres’ surface and S tot is the total matrix surface. So for the double reflection(at the beginning from the matrix plane and then from the plane in front of thefibres) for Φ spm (cos Θ , a , a ) we obtain an expression in a similar way as theprevious one: Φ spm (cos Θ , a , a ) = ε sp ε fm × Φ dp (cos Θ , a , a ) , (A.15)were a = a (3 H, l ). So in this way we can estimate any time of reflectionsand then summing them we can have total contributions from the specularreflections. The number of the reflections is limited because of the value of ε sp ≤ . ε fm ≤ . Appendix B. Efficiency estimation in case of diffuse reflection
To estimate the integral in the expression (9), which is proportional to theamount of the light that reaches the reflective surface, at the beginning it shouldbe evaluated by cos θ and then by h . L (Ψ , h ) = (cid:90) Ψ0 exp( − b ( h )cos θ ) d cos θ (B.1)This integral can be evaluated again using an indefinite integral. For thesimplification we can substitute x ≡ cos θ and again use E i ( − x ) = − E ( x ) l ( x, h ) = (cid:90) exp( − b ( h ) x ) dx = x exp( − b ( h ) x ) − b ( h ) E (cid:18) b ( h ) x (cid:19) + C (B.2)where E (cid:0) ax (cid:1) is the well known exponential integral(see Appendix A). Sothe integral (B.1) can be estimated as: L (Ψ , h ) = l (1 , h ) − l (cos Ψ , h ) (B.3)17erforming the integration for h of the expression (B.3) we obtain the esti-mation of the integral in (9): I (Ψ , H ) = (cid:90) H l (1 , h ) dh − (cid:90) H l (cos Ψ , h ) dh (B.4)Taking into account that b ( h ) = h /λ sc , the expression (A5) for E (cid:0) ax (cid:1) (seeAppendix A) and again evaluating the indefinite integral (cid:82) l ( x, h ) dh we can es-timate I (Ψ , H ).The evaluation of the indefinite integral G ( x, h ) = (cid:82) l ( x, h ) dh is shown below: G ( x, h ) = − λ sc × x exp( − h λ sc x ) − h × xλ sc (cid:32) γ h λ sc x −
14 + (cid:88) ∞ k =1 k × ( k + 2) (cid:89) ki =1 − h λ sc × x × i (cid:33) + C (B.5)So for I (Ψ , H ) we obtain the expression below: I (Ψ , H ) = G (1 , H ) − G (1 , − G (cos Ψ , H ) + G (cos Ψ ,
0) (B.6)Without consideration of the light attenuation in the expression (B.1) for I (Ψ , H ) we will obtain: I (Ψ , H ) = H × (1 − cos Ψ) (B.7)To estimate the sum in the expression (10), which is shown below: S dfr ( a , Θ) = 1 N fH × (cid:88) N fH i =0 exp( − a cos θ i ) × cos θ i (B.8)As it can be seen from the expression above it is the average value of exp( − a ( H,l )cos θ ) × cos θ for the angles θ ≤ Θ. So to estimate the sum we represent it in the inte-gral form as it is done in Appendix A, assuming Lambert type distribution forthe light reflection(cos θ low): S dfr ( a , Θ) = (cid:82) π (cid:82) Θ0 exp( − a cos θ ) × cos θd Ω (cid:82) π (cid:82) Θ0 cos θd Ω = (cid:82) Θ0 exp( − a cos θ ) × cos θd cos θ (cid:82) Θ0 cos θd cos θ (B.9)18enominator in the expression (B.9) is just for the normalization purpose. Toperform the integration of the numerator of the expression (B.9) it is convenientto make a variable change like x ≡ cos θ : f dfr ( a , x ) = (cid:90) exp( − a x ) × x dx, (B.10)which after the integration by parts[7] can be obtained for f dfr ( a , x ) : f dfr ( a , x ) = − e − a x (cid:16) a E (cid:0) a x (cid:1) e a x − x + 6 a x − a x + a x − a x (cid:17)
120 + const (B.11)where E (cid:0) a x (cid:1) is the well known exponential integral(see A.5) So for S dfr ( a , Θ)in (B.9) we will obtain: S dfr ( a , Θ) = f dfr ( a , − f dfr ( a , cos Θ)0 . × (1 − cos Θ) (B.12)Ignoring the light attenuation in the expression (B.11 ie a = 0) for S dfr (0 , Θ)we will obtain: S dfr (0 , Θ) = 2 × (1 − cos Θ)5 × (1 − cos Θ) (B.13)
Appendix C. Time spread estimation for the photon transport
The time spread estimation for DP is based on the photon path spreadestimation. At the beginning it should be estimated the average path valueand then the standard deviation using this average value for the photon pathspread estimation. The average path value is estimated the same way as in theAppendix A. For each source point inside the scintillator the photon path length L is calculated as: L ( l, h, θ ) = ( l + h )cos θ (C.1)where l is the fibre length, h is the distance of the source point from the fibreplane and θ is the angle between the vector connecting the light source to19he given fibre and the vector that is perpendicular to the plane of the fibrematrix(see Fig a1). The average value of the path length for the given fibrelength l and for the scintillator thickness H is estimated with the standardprocedure for the uniform angular photon emission: L ( l, h, θ ) = (cid:82) H (cid:82) π (cid:82) Θ0 ( l + h )cos θ d Ω dh (cid:82) H (cid:82) π (cid:82) Θ0 d Ω = (cid:82) H (cid:82) cos Θ1 ( l + h ) x dxdh (cid:82) H (cid:82) cos Θ1 dx (C.2)where Ω is the solid angle x = cos θ and integration is performed within fibreaperture angle Θ. The Integration by the h is performed for the interval fromthe zero to the scintillator thickness H . For the average value after integrationwe obtain the expression below: L ( l, H, θ ) = ( l + H/ × ln cos Θ(cos Θ −
1) (C.3)In similar way can be estimated the standard deviation σ L : σ L L = (cid:115) (1 + Hl ) × (cos Θ − ((1 + H l ) × cos Θ × ln cos Θ − H/l (cid:28) L . Appendix D. Figures igure D.1: Three different light collection schemes. igure D.2: Light collection scheme from the point like source for direct photons. Descriptionof all notations are in the text.Figure D.3: Light collection scheme from the point like source and for the reflected photons.Description of all notations are in the text. igure D.4: Light collection efficiency vs number of reflections. Lines show efficiency for DPwith and without attenuation in the scintillator and in the fibres igure D.5: Light collection efficiency vs detector lateral sizes in the scintillator thicknessunits. igure D.6: Pulse time wave forms for direct and reflected photons.igure D.6: Pulse time wave forms for direct and reflected photons.