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detectors comprising scintillating fibres , packed together to form arrays , and multi - channel photo - detectors have been used since the 1990s to track charged particles , see e.g.  @xcite . a fibre array is made from layers of fibres , often with alternating layers off - set relative to each other .
the design of such detectors is governed by the relatively small light yield of thin ( @xmath0 1 mm ) fibres .
multi - channel photomultiplier tubes are especially suitable for fibre read - out because of the good matching between the pixel size of the photomultiplier and common fibre diameters , offering a significant reduction in size and cabling with respect to conventional tubes . since the pioneering work of kuroda  @xcite such tubes have been developed in order to meet the demands on precise and reliable tracking devices under high - rate  @xcite . in recent years they have been continuously improved .
the drawbacks of early devices have been greatly reduced ; modern multi - anode tubes exhibit little cross - talk and a high gain uniformity between pixels . the hamamatsu photonics
r5900 series has been chosen by many experimental groups for fibre read - out .    in modern experiments , fibre bundles involving a rather large number of channels
are easily read out via multi - channel photomultiplier tubes , while the use of single - channel photomultiplier tubes is no longer economical in terms of cost and space requirements .
accordingly , multi - layered structures of packed scintillating fibres coupled to multi - anode photomultiplier tubes became the preferred choice for some fast detectors , the fibre trackers in the compass experiment  @xcite at cern being one recent example .
fibres of a `` column '' along the geometrical trajectory of incident particles are typically grouped  with one fibre from each layer  to one common pixel of the multi - channel read - out device .
if the light yield per crossing particle is too small to be detected with the required efficiency , layers need to be added to the array . in case
charged particles are crossing the fibre array at right angle to the detector base the total thickness of the array can be increased arbitrarily until a physical limit or restrictions in terms of small angle scattering are reached .
this condition applies to many applications , as the angular acceptance for fibre hodoscopes is generally small , typically up to @xmath1 3@xmath2 .
however , in some applications the incident particles will cross the detection plane with large angles , @xmath3 , w.r.t .
the normal to the layers , and would traverse several neighbouring columns .
the resultant large hit multiplicities would compromise the tracking capabilities of such a detector . for the fibre hodoscopes of the dirac experiment
the detector response to particles with incident angles up to @xmath4 45@xmath2 was studied for background issues  @xcite .
the read - out electronics of that detector lead to a strong suppression of detection efficiency for larger angles . for restoring the original spatial resolution the packing of the fibres
needs to be adapted to the incident particles direction and the columns need to be inclined
. only the experiment is known in which the fibre column angle , @xmath5 , w.r.t .  the normal to the layers is adapted to large ( @xmath6 45@xmath2 ) incident angles of the particles for deliberately matching the geometry  @xcite . in the spectrometer s electron arm hodoscope the average incident angle is @xmath1 65@xmath2 .
it has been shown that a high detection efficiency and a good spatial resolution can be achieved using detectors with inclined columns .
fibre array geometries with different column angles , @xmath5 , ranging from 10@xmath2 to 80@xmath2 and four fibres per column are shown schematically in fig .  [ fig : geometries ] . in the scheme
the base of each detector is turned by an angle of 50@xmath2 to the horizontal . the most efficient way to pack fibres together to form inclined columns
is not obvious .
`` rows '' of fibres are defined along the layers in the direction of the detector base . for column angles between 30@xmath2 and 60@xmath2
the arrays can be formed in a way that all fibres of a column touch each other ( closed columns ) , but leaving gaps to the corresponding fibres in neighbouring columns .
alternatively , the arrays can be formed with corresponding fibres of neighbouring columns touching ( closed rows ) , leaving gaps between fibres within a column . finally , for column angles @xmath5 below 30@xmath2 both columns and rows can be closed . for column angles of 30@xmath2 and 60@xmath2
the fibre centres are forming a hexagonal lattice . for a column angle of 45@xmath2 the fibre centres are forming a square lattice .
the hexagonal packing , in which each fibre is surrounded by 6 touching fibres , has the highest packing density of @xmath7 0.90 .
square and hexagonal packing geometry are used most often for fibre arrays .
each geometry corresponds to a different overlap fraction , a different column pitch , a different detector width and length for a given number of layers and read - out pixels , and a different average thickness .
the relevance of these design criteria for choosing the detector geometry vary with the specific application .
primarily , the fibre array needs to provide a high enough light yield per read - out pixel to discriminate the signals against noise .
light yield and detection efficiency depend on the diameter and overlap fraction as well as on the number of layers .
secondly , the spatial resolution depends on fibre diameter and column pitch . a detector geometry with closed columns
as shown in the figure will have a relatively large column pitch and a relatively small spatial overlap .
this leads to low spatial resolution and low detection efficiency .
both can be avoided by placing the layers not along the base of the detector , but with a smaller column pitch and larger spatial overlap .
that will create a shift between the first fibres of each column and the base - line it was supposed to track . by occasionally ( every 3 or more columns ) moving the column position one full fibre diameter in the direction of the base - line
the shift will be corrected .
these geometries have the advantage of being better adapted to practical applications , but have less symmetry . for column angles above 60@xmath2 geometries with neither columns nor rows
closed can be created in a similar way . in practice , only those fibre arrays can be built which allow precise mounting and alignment during all stages of the processing : gluing , bending , and installation .
[ fig : design](left ) shows the ratio of the column pitch to the fibre radius , @xmath8 , i.e.  the distance between two fibre centres along the base direction in units of fibre radius , as a function of the column angle , @xmath5 .
the main branch starting at a ratio @xmath9 2 corresponds to a geometry with both , closed rows and closed columns , for which @xmath10 .
the branch splitting off at @xmath11 corresponds to a geometry with closed columns , where @xmath12 .
the continuous curve is calculated for the geometry in which the rows are closed .
the branch splitting off at @xmath13 follows a @xmath14 dependence and corresponds to closed columns .
spatial overlap is important to avoid relying on the detection of events with only a grazing contact of the charged particle with the fibres .
a particle which crosses the gap between two fibres in one of the layers needs to traverse a significant portion of the full fibre diameter in the other layer . fig .
[ fig : design](right ) shows the overlap fraction , @xmath15 , which is the overlap of two neighbouring fibre columns in units of the fibre radius .
it is directly related to the pitch to radius ratio by @xmath16 .
fibre diameter and overlap fraction relate to the theoretical spatial resolution of fibre arrays expected from the geometry . for events
in which exclusively one column was hit , the the spatial resolution is @xmath17 . for events in which the particle trajectory covered purely the inter - column region ,
the spatial resolution is given by @xmath18 .
averaging both event types leads to a combined resolution of @xmath19 . for fibre detectors in hexagonal packed arrays
the total overlap is @xmath20 with a pitch of @xmath21 .
the theoretical spatial resolution of such an array is @xmath22 .
the spread of energy deposition in a fibre bundle is proportional to the detector thickness variation .
[ fig : thickness ] shows how the double layer thickness varies as a function of the base coordinate . in the hexagonal packing with a column angle of @xmath23 60@xmath2
the variation is @xmath24 ( minimum to maximum ) per double layer . for
all other column angles the variation can become smaller , depending on the fibre array geometry .
the design criteria discussed so far only apply when all particles cross the detector with the same incident angle .
the expected response of a detector to diverging particles can be evaluated in a simulation .
different fibre arrays , in which particles were crossing the detector with finite incident angles with respect to the column angle , were included in a detector simulation within the geant4 framework  @xcite .
incident electron trajectories were averaged over the base .
the simulation gave information on the energy deposition in individual fibres and on interactions of the particles with the material , e.g.  small angle scattering , ionisation and bremsstrahlung .
the total energy deposited in the active cores of the four corresponding fibres of each column was calculated .
signals above a given threshold were assigned to the corresponding read - out pixel .
the increase in channel multiplicity for detectors with column angles of @xmath25 and @xmath26 is presented in fig .
[ fig : multiplicities ] as a function of the incident angle @xmath3 .
it is shown that the multiplicity increases gradually to smaller incident angles but steeper to higher angles .
owing to the symmetry of the fibre arrays , a local maximum of the multiplicity appears for @xmath13 at @xmath1 30@xmath2 , and for @xmath27 at @xmath1 0@xmath2 .
it is obvious that the columns must be aligned with the incident particle direction and the particle s divergence must be small to achieve a minimal multiplicity and an optimal spatial resolution .
the simulation can further help to define the optimum threshold so that a detector becomes insensitive to incoming particles with large angles w.r.t  the nominal angle .
that is important when a detector is exposed to a large number of background particles .
the angular dependence of the detector efficiency is plotted in fig .
[ fig : efficiency ] for 4 different relative thresholds .
the relative threshold of 100% corresponds to the mean signal of a detector pixel for events with particles hitting the detector with nominal angle .
the plateau width for 100% efficiency depends crucially on the threshold . for incident angles much smaller than the nominal angle
only one fibre of each column is hit , but many neighbouring channels .
thus , the probability for having a signal in at least one pixel above threshold is strongly dependent on the distribution of energy deposition and the value of the threshold , but almost independent on geometry and incident angle .
when a set of fibres from a multi - layered fibre detector is coupled to a multi - channel read - out device a multitude of different fibre array geometries is possible .
it is the experimenter s choice to select the geometry , which is matching the application best , according to the relevant design criteria . in this paper the dependence of column pitch and overlap ratio on the fibre array column angle
is discussed .
both criteria are also reflected by the thickness variation of a double layer , which is shown for a selection of column angles .
the effect of diverging particles incident on fibre arrays with inclined columns was studied with the help of a monte carlo simulation .
the resulting curves of channel multiplicity versus incident angle can be folded with the incident angle distribution of an experiment to get the average channel multiplicity .
large multiplicities compromise the position resolution of the fibre arrays , which is the key issue for tracking detectors . with the choice of the detection threshold , the sensitivity of the detector to particles with large deviations to the nominal angle can be varied .
this work was supported by the federal state of rhineland - palatinate and by the deutsche forschungsgemeinschaft with the collaborative research center 443 .
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p.  achenbach , probing hypernuclei at @xmath28anda and at mami - c , in : j.  pochodzalla , th .
walcher ( eds . ) , proc . of the ix .
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7984 .         range from 10@xmath2 to 80@xmath2 with a detector base angle of 50@xmath2 .
geometries with closed rows and closed columns are separately drawn . for column angles of 30@xmath2 and 60@xmath2
the fibre centres are forming a hexagonal lattice . for a column angle of 45@xmath2 the fibre centres are forming a square lattice . ]    .
the right plot shows the overlap of two neighbouring fibre columns in units of the fibre radius .
the continuous curve corresponds to a geometry with closed rows , the two branches that are splitting off to geometries with closed columns.,title="fig:",scaledwidth=48.0% ] .
the right plot shows the overlap of two neighbouring fibre columns in units of the fibre radius .
the continuous curve corresponds to a geometry with closed rows , the two branches that are splitting off to geometries with closed columns.,title="fig:",scaledwidth=48.0% ]     ( top left ) , @xmath29 ( top right ) , @xmath30 ( bottom left ) , and @xmath31 ( bottom right ) .
the two curves correspond to geometries with closed columns ( full curve ) and closed rows ( dotted curve).,title="fig:",scaledwidth=48.0% ]   ( top left ) , @xmath29 ( top right ) , @xmath30 ( bottom left ) , and @xmath31 ( bottom right ) .
the two curves correspond to geometries with closed columns ( full curve ) and closed rows ( dotted curve).,title="fig:",scaledwidth=48.0% ] +   ( top left ) , @xmath29 ( top right ) , @xmath30 ( bottom left ) , and @xmath31 ( bottom right ) .
the two curves correspond to geometries with closed columns ( full curve ) and closed rows ( dotted curve).,title="fig:",scaledwidth=48.0% ]   ( top left ) , @xmath29 ( top right ) , @xmath30 ( bottom left ) , and @xmath31 ( bottom right ) .
the two curves correspond to geometries with closed columns ( full curve ) and closed rows ( dotted curve).,title="fig:",scaledwidth=48.0% ]       ( left ) and 60@xmath2 ( right ) .
each symbol represents a different threshold relative to the mean signal of particles with nominal incident angle.,title="fig:",scaledwidth=49.0% ]   ( left ) and 60@xmath2 ( right ) .
each symbol represents a different threshold relative to the mean signal of particles with nominal incident angle.,title="fig:",scaledwidth=49.0% ]
multi - layered scintillating fibre arrays read - out are commonly used as high resolution charged particle hodoscopes . 
 fibres of a `` column '' along the geometrical trajectory of incident particles are typically grouped to one pixel of a multi - channel read - out device . in some applications the incident particles will cross the detection plane with large angles w.r.t .  
 the normal to the layers . 
 then , the packing of the fibres needs to be adapted to the incident particles and the columns need to be inclined . in this 
 paper possible fibre array geometries are shown , relevant design criteria for detectors are discussed , and the effect of diverging particles incident on fibre arrays was studied using a monte carlo simulation .    ,    , and    tracking and position - sensitive detectors , scintillating fibres , particle detector design 29.40.gx , 29.40.mc , 85.60.ha