ARTICLE IN PRESS

0168-9002/$ - se

doi:10.1016/j.ni

�CorrespondE-mail addr

URL: http:/

Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631

www.elsevier.com/locate/nima

Light tracking through ice and water—Scattering and absorption inheterogeneous media with PHOTONICS

J. Lundberga,�, P. Miocinovicb, K. Woschnaggc, T. Burgessd, J. Adamse,S. Hundertmarkd, P. Desiatif, P. Niesseng

aDivision of High Energy Physics, Uppsala University, Uppsala, SwedenbDepartment of Physics and Astronomy, University of Hawaii, Manoa, USA

cDepartment of Physics, University of California, Berkeley, CA, USAdDepartment of Physics, Stockholm University, Stockholm, Sweden

eDepartment of Physics and Astronomy, University of Canterbury, Christchurch, New ZealandfDepartment of Physics, University of Wisconsin, Madison, WI, USAgBartol Research Institute, University of Delaware, Newark, DE, USA

Received 5 February 2007; received in revised form 21 July 2007; accepted 23 July 2007

Available online 6 August 2007

Abstract

In the field of neutrino astronomy, large volumes of optically transparent matter like glacial ice, lake water, or deep ocean water are

used as detector media. Elementary particle interactions are studied using in situ detectors recording time distributions and fluxes of the

faint photon fields of Cherenkov radiation generated by ultra-relativistic charged particles, typically muons or electrons.

The PHOTONICS software package was developed to determine photon flux and time distributions throughout a volume containing a

light source through Monte Carlo simulation. Photons are propagated and time distributions are recorded throughout a cellular grid

constituting the simulation volume, and Mie scattering and absorption are realised using wavelength and position dependent

parameterisations. The photon tracking results are stored in binary tables for transparent access through ANSI-C and C++ interfaces. For

higher-level physics applications, like simulation or reconstruction of particle events, it is then possible to quickly acquire the light yield

and time distributions for a pre-specified set of light source and detector properties and geometries without real-time photon

propagation.

In this paper the PHOTONICS light propagation routines and methodology are presented and applied to the IceCube and ANTARES

neutrino telescopes. The way in which inhomogeneities of the Antarctic glacial ice distort the signatures of elementary particle

interactions, and how PHOTONICS can be used to account for these effects, is described.

r 2007 Elsevier B.V. All rights reserved.

PACS: 78.20.Bh; 02.70.Uu; 42.15.Dp; 91.50.Yf; 92.40.�t; 93.30.Ca; 95.85.Ry

Keywords: Numerical simulation; Optical properties; Monte Carlo method; Ray tracing; Optical; Neutrino detection

1. Introduction

In optical high energy neutrino astronomy light fromparticle physics events is observed using a large number ofdetectors placed deep in glacial ice or in ocean or lakewater. Successful simulation and reconstruction of such

e front matter r 2007 Elsevier B.V. All rights reserved.

ma.2007.07.143

ing author. Tel.: +4670 7422777; fax: +46 18 4713513.

ess: johan.lundberg@tsl.uu.se (J. Lundberg).

/photonics.tsl.uu.se (J. Lundberg).

events relies on accurate knowledge of light propagationwithin the detector medium. Light propagating througheven the clearest water or ice is affected by scattering andabsorption. For light sources and receivers separated bydistances comparable to the photon mean free path,scattering effects can neither be analytically calculatednor ignored. The typical scattering lengths in thesedetection media are tens to hundreds of metres. Since thisscale is comparable to the typical detector separation,detailed simulation of the photon propagation is required

ARTICLE IN PRESSJ. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631620

to obtain information necessary for event simulation andreconstruction. The problem is complicated further by theanisotropy of the light emitted in particle interactions andthe heterogeneity of detector media. PHOTONICS is a freelyavailable software package [1] containing routines fordetailed photon Monte Carlo simulations, which take intoaccount such complexities to provide, in tabulated form,the photon flux distribution throughout a specified mediumfor an input light source.

With PHOTONICS the photon flux and time distributionscan be tabulated for an arbitrarily large volume of apropagation medium, for a user-defined range of lightsource and detector properties. This means that once aPHOTONICS table set has been generated for a class of lightsources and detectors, it is possible to quickly andtransparently acquire the light yield and time distributionswithout any need for real-time photon propagation during,for example, particle physics event simulation or recon-struction. This is made possible by the PHOTONICS tablereader library, with which a user (program) can dynami-cally query the pre-calculated tables by specifying thelocations and geometrical relations between light sourcesand detectors. A simulation chain for a complete experi-mental setup can be achieved by using these interfaces andapplying detector specific details such as modelling ofelectronics, data acquisition, and triggers. For eventreconstruction, PHOTONICS provides probability densityfunctions for arrival times of independent photons andthe expected number of detected photons.

In this paper we first introduce the relevant physics of thephoton propagation (Section 2) and the details of thePHOTONICS implementation (Section 3). We then compareour results with observations of calibration light sources insea water and glacial ice (Section 4). In Section 5, wepresent some photon tracking results relevant to thedetection of neutrinos with the IceCube neutrino telescope.

2. Light propagation in diffuse media

Our goal was to model the transport of light throughglacial ice and water. Photon propagation depends on theoptical properties of the medium, in particular on thevelocity of light and the absorption and scattering cross-sections. Glacial ice is optically inhomogeneous because ofdepth dependent variations in temperature, pressure, andconcentrations of air bubbles and insoluble dust. Since thedust deposits track climatological changes and are there-fore assumed to be arranged in horizontal layers, theireffect is parameterised as a vertical variation of the opticalproperties. In addition to this spatial variation, thewavelength dependence of the medium parameters mustbe taken into account. Before describing the implementa-tion to achieve our goal, we review the optical quantitiesthat must be considered in the simulation, using notation inwhich the wavelength dependence is left implicit.

The time of light travel is determined by the groupvelocity of light, which is given by the group refractive

index ng, while various transmission and scatteringcoefficients depend on the phase velocity [2] and its indexof refraction np.Absorption of visible and near UV photons in pure

water and ice is due to electronic and molecular excitationprocesses and is characterised by the absorption length la.Measurements of light attenuation have been performed atrelevant wavelengths in lake water by the Baikal [3]collaboration, and in sea water by the DUMAND [4], NESTOR

[5], ANTARES [6], and NEMO [7] collaborations. The AMANDA

collaboration has developed an empirical model for opticalabsorption in deep glacial ice by combining laboratory andin situ measurements [8].Photon scattering by scattering centres of general sizes is

described by Mie scattering theory [9], which for anywavelength and scattering centre size gives the scatteringangle distribution, the phase function. Rahman scattering,where the scattering centre is affected by the scatteredphoton, and Brillouin scattering, where photons arescattered on (thermal) density fluctuations, may result ina change of photon energy. However, these processes aresubdominant to Mie scattering in both glacial ice [10] andsea water [6] where light is scattered by centres of verydifferent types and sizes: from ice crystal point defects toair bubbles and mineral grains in ice, and from biologicalmatter to sediment particles in water.In natural ice and water it is difficult to determine the

phase function from in situ measurements. Instead, theAMANDA and ANTARES collaborations have used calibrationlight sources to determine the propagation characteristicsassuming certain forms of the scattering angle distribu-tions. In the case of ice, a one parameter Henyey–Green-stein (HG) phase function is often used, approximatingMie scattering under the assumption that scattering isforward peaked [11]. For water, a two parameter phasefunction is more useful. For this paper we mostly use thesingle parameter HG phase function

pHGðcos yÞ ¼1� t2

2ð1þ t2 � 2t cos yÞ3=2(1)

which is completely characterised by the t parameter, themean of the cosine of the scattering angle y,

t � hcos yi ¼Z

pHGðcos yÞ cos y dðcos yÞ. (2)

The absorption length, la, and scattering length, ls, arethe mean free paths of exponential distributions. Theprobability density function for the path length s to thenext scatter is

f lsðsÞ �dF ðsÞ

ds¼

e�s=ls

ls(3)

where F ðsÞ is the probability distribution function.When determining ice or water optical properties, there

is a degeneracy between ls and t. One therefore considers

ARTICLE IN PRESSJ. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631 621

the effective scattering length, le, defined as

le ¼ls

1� t(4)

which in anisotropic scattering is analogous to the(geometric) scattering length ls in isotropic scattering; itis the distance which light propagates through a turbidmedium before the photon directions are completelyrandomised. Consider a collimated light pulse injected intoa non-absorbing medium. In this case, the photons are onaverage scattered at successive steps of length ls and theprojection of the net velocity vector on the originaldirection is decreased on average by t ¼ hcos yi in eachscattering step (in which all photons are scattered) [12].Hence the injected light is effectively transported a forwarddistance of

lsX1i¼0

ti !ls

1� t¼ le (5)

and le has a natural interpretation as the distance that thecentre of gravity of the photon cloud advances, in the limitof many scatters.

3. Monte Carlo simulation implementation

In this section we present the main ingredients in ourphoton Monte Carlo simulation implementation. The endproduct is a set of photon flux density tables, eachdescribing the evolution of the light field around a user-defined source at a specific location and orientation. For agiven light source, a user-specified large number of photonsis generated according to the source characteristics. Thephotons are then tracked and their contribution to theoverall light field is determined and recorded in a cellulargrid throughout a user-defined portion of space, thesimulation volume. The sensor locations are not fixed, butare dynamically specified when accessing the simulationresults. The photon intensity and time distributions arestored in a six-dimensional binary table. Four of thesedimensions are for the spatial and temporal location in thesimulation volume with respect to the emission point. Asthe acceptance of the light sensors is assumed to beazimuthally symmetric, around the vertical axis in hetero-geneous media and around any axis in homogeneousmedia, the photon impact direction is characterised by thezenith angle alone, constituting the fifth dimension. Thesixth dimension is the angle from the light source principalaxis at which a photon is emitted (this dimension can beused, for example, to reweight the flux tables for a differentemission profile). These latter two dimensions are usuallyintegrated over when the photon tables are used withdetector simulations. In this case the wavelength andangular sensitivity of the detector elements need to befolded in as the photons are recorded, using recordingweights which can be specified in functional or tabularform. For each light source position and orientation onetable is produced. A set of tables describes a range of

source locations and directions, valid for the specified classof light sources and detectors.

3.1. Media and light source properties

The parameters describing the optical mediumðng; np; t; le; laÞ are taken to be functions of wavelengthand a spatial dimension Z, typically specifying the ice orwater depth. Thus, the propagation medium is divided intohorizontal regions which differ by their optical properties.For media where the single parameter HG approximationdoes not provide an adequate description of the scatteringit can be replaced by other phase functions. An example ofthis is the treatment of sea water in Section 4.1.A single simulation run begins with the injection of a

photon with wavelength and emission direction chosenfrom user-specified probability distributions and at a user-specified location. Our procedures support point-like(infinitesimal) light sources, where all emitted photonsoriginate from the same point, and volume light sourceswhere the emission is distributed over a volume. Anexample light source, essential for neutrino astrophysics, isthat of a Cherenkov emitter. In PHOTONICS a Cherenkovemitter is a point-like light source with a Cherenkovwavelength spectrum and an angular emission in aCherenkov cone [13]. In this case the emission isazimuthally symmetric around the principal axis of theCherenkov emitter. Closely related are light sourcescomposed of many short Cherenkov emitting tracks, suchas electromagnetic cascades. Another category of point-likesources are laser or LED light sources, and in Section 4 wecompare our simulation of such sources with observations.Continuous and extended light sources are composed byintegration of infinitesimal sources. For example, the lightdistribution due to a relativistic muon is composed byintegrating a series of infinitesimal Cherenkov emitters overspace and time, as described in Section 3.4. Simulationresults for both point-like and extended Cherenkovemitters are presented in Section 5.

3.2. Coordinate systems for photon flux recording

The Cherenkov light source example possesses cylind-rical symmetry. Although this symmetry is typically brokenby the response of the propagation medium and thedetector, a cylindrical coordinate system aligned with thesource’s symmetry axis is a natural choice for the recordingof the light flux, and it is therefore used in the following. Inaddition to cylindrical ðr; l;fÞ coordinates, we have alsoincluded functionality to allow the flux to be recorded inspherical ðr;W;fÞ or Cartesian ðe1; e2; e3Þ coordinates. A gridover the spatial coordinates defines cells in which thephoton flux density and time distributions are averaged.The user-specified region of space covered by these cellsconstitutes the recording volume.In some dimensions it can be desirable to have denser

binning close to the origin. Therefore, uniform as well as

ARTICLE IN PRESS

Table 1

An example of table binning using cylindrical coordinates

Dimension Bins Low High

Radial r 30 0m 500m

Longitudinal l 51 �500m 500m

Azimuthal f 10 01 1801

Time t 50 0 ns 6000 ns

The rotational symmetry of the emitter and the horizontal symmetry of

the medium implies an azimuthal symmetry in f, so that the flux is the

same at �f and f.

J. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631622

linearly increasing bin sizes are supported for r, l, and thetime t. In addition to specifying the spatial coordinateswhere the light flux is recorded relative to the source, thelocation of the source and the direction of its axis ofsymmetry with respect to the medium are needed. Thesecan be characterised using just two coordinates, the depthZs and the zenith angle Ys. This is because of thehorizontal symmetry of the medium, as discussed inSection 3.1. Fig. 1 shows the coordinates and a recordingcell in which the average flux is recorded.

If in addition to the medium symmetry around z weassume that the light source is symmetric with respect tothe sign of f (besides any light source that is azimuthallysymmetric around its axis this is the case for some non-isotropic LED calibration light sources used in glacial ice[8,14]), the flux needs only to be tabulated for f between 0�

and 180�. For heterogeneous media we require that thesensor acceptance and the medium properties are sym-metric around the same axis z. This is not required forhomogeneous media since the coordinate system can berotated to make the sensor symmetry axis collinear with thez axis used to define source location and orientation.

Fig. 1. The recording cell geometry with variables used for binning of

photon flux data. Photons are emitted from a user-defined light source at

Zs and their flux and time distributions are recorded and averaged in all

spatial cells the photons traverse (one of which is shown as a shaded

volume). These cells are defined in a coordinate system aligned with the

source’s principle axis l, which is tilted by p�Ys with respect to the

medium symmetry axis z. The angle f is defined to be zero where the

radial vector is maximally aligned with z. The azimuthal direction Fs is

degenerate since the medium properties are assumed to be symmetric

around the z axis.

PHOTONICS can be extended to handle cases where the twosymmetry axes do not align in heterogeneous media, byadding further dimensions to the photon tables.A binning example is shown in Table 1 for a cylindrical

coordinate system. As each single precision floating pointnumber requires 4 bytes, the size of this example tablewould be 3Mbytes. To get the total table set size this mustbe multiplied with the number of light source positions Zs

and zenith angles Ys of interest. For example, with 50source depths and 20 source angles, the total table set size is3Gbytes.

3.3. Photon propagation

Starting at their point of origin, the photons are trackedand recorded along straight-line paths between successivescattering points using one of two methods. In the volume-density mode, photons are recorded at equidistant record-ing points along their paths. In the area-crossing mode,photons are recorded at every surface-crossing into a cell.In the latter mode, the propagation step is dynamicallycalculated to bring the photon to its next cell boundary.The photon’s propagation properties are always updated atmedium boundaries and scattering points, regardless ofrecording method.When a scattering point is reached, the photon’s

direction is changed by an angle randomly drawn fromthe selected phase function, typically the HG function inEq. (1), and in a uniform random azimuthal direction. Atthis point, the distance to the next scattering point is drawnfrom an exponentially distributed random variable with amean value of the local scattering length.Photon absorption is taken into account by successively

updating the photon survival weight w as the photonspropagate through regions of different absorption. For aphoton tracked n steps, each within a locally homogeneousmedium, the weight is given by

w ¼Yn

i¼0

exp �Dsi

la;i

� �(6)

where Dsi is the length of step i in a region with absorptionlength la;i. The weight is updated every time the photon isscattered, enters a new medium region, or is recorded.

ARTICLE IN PRESSJ. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631 623

When a photon enters a medium region with differentscattering and absorption parameters, these are updated,from ðls; laÞ to ðl

0s; l0aÞ, at the region boundary. At this

point, the remaining distance to the next scatter iss0 ¼ sl0s=ls, where s would have been the remainingdistance to the next scatter in the former region withscattering length ls. Refraction at the boundary issupported but reflection is ignored since it is assumed thatrefractive index variations are continuous.

During its propagation, the flux contributed by a photonis recorded either in each spatial cell it enters (in area-crossing mode) or each time it completes a propagationstep (in volume-density mode). The photon flux F(particles per area and time) at any point ðr; l;fÞ at a timeT after the emission from a light source at a depth Zs

pointing in a direction Ys is denoted FðZs;Ys;r; l;f;T � t0ðr; lÞÞ, where t0 is a reference time typically chosento be the first time causally connected to the light emittedby the source,

t0ðr; lÞ ¼nref

c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ l2

q(7)

where c is the speed of light in vacuum and nref is a user-specified reference refractive index. This reference timeconvention is appropriate for point-like stationary lightsources only. For fast moving light sources such as muonsa different expression is used which takes into account themore complicated causality condition. The residual time

t � T � t0 is the time delay caused by scattering, relative tothe propagation time for a photon travelling in a straightline. Photons are tracked until their residual times exceed auser-specified value. The tracking of a photon is terminatedif it leaves the simulation volume (which can be larger thanthe recording volume to allow the photon to scatter backinto the recording volume) or if its survival weight dropsbelow a pre-set value.

The probability density function for a photon flux attime t is given by

f pdf ðZs;Ys;r; l;f; tÞ ¼FðZs;Ys;r; l;f; tÞIðZs;Ys;r; l;fÞ

(8)

where I is the time integrated photon flux, or intensity,

IðZs;Ys; r; l;fÞ ¼Z 1�1

FðZs;Ys;r; l;f; tÞdt. (9)

Since photon fluxes are additive we can determine thetime distribution of a combination of light sources through

f pdf ðtÞ ¼

Pi I if iðtÞP

i I i

¼

Pi FiðtÞP

i I i

. (10)

The way in which the photon intensity in a spatial cell iscalculated depends on the recording method. In the area-crossing method, the contribution of each photon to thetotal flux in the cell depends on the projected surface areaof the cell as seen in the direction of the contrib-uting photon as it crosses the cell boundary. In thevolume-density method, photons contribute at equidistant

recording points along their paths, so that the contributionis proportional to the number of recording points that fallin a given cell. The respective equations for the calculationof the observed intensity, per emitted photon, are

Iarea�crossing ¼1

N

XN

g¼1

wgA�1? (11)

Ivolume�density ¼1

N

XN

g¼1

Xngk¼1

wgðkÞDs

V(12)

where the g sum runs over the N simulated photons, A? isthe recording cell area projected perpendicularly to thedirection of the photon, V is the volume of the cell, and Ds

is the recording point separation along the photon path inthe volume-density method. The quantity ng is the numberof recording points inside the cell for a given photon.Hence, in the area-crossing mode, photons are recorded atevery surface-crossing into a cell, whereas in the volume-density mode, they are recorded at equidistant recordingpoints. The photon weight wg is a product of theabsorption-induced survival weight, Eq. (6), and optionaluser-specified sensor detection efficiencies.The calculation and recording of a photon’s contribution

to the flux in a recording cell is computationally expensive,but a suitable choice of method (area-crossing or volume-density) can speed up the simulation. To optimise for speedone compares the step size Ds with the scale dimension of arecording cell Dcell. If DsoDcell, the area-crossing method iscompetitive; otherwise the volume-density method shouldbe used. Hence the area-crossing method can result infaster code execution for large detection volumes with largerecording cells, while the volume-density method is fasterfor small, dense recording cell configurations. To ensureunbiased sampling in volume-density mode even forDsbDcell, the path length to the first recording point afteremission is drawn from a uniform distribution between 0and Ds. To maximise the number of independent samplingpoints for a given execution time, Ds is balanced against thetotal number of photons N. A large Ds allows a large N,but if too large the number of recordings per executiontime will drop as more time is spent propagating photonsbetween recording points. Using Ds � ls is often a goodcompromise.Another way to improve the speed of the Monte Carlo

simulation is through importance-weighted scattering. Thismeans that the photons are propagated using scatteringparameters which are different from those of the scatteringsituation at hand in order to get higher statistics at lowprobability phase space locations. When solving a MonteCarlo problem for random numbers xi (for example,scattering angles) from a probability density distributionf 1ðxÞ, we can choose to instead sample from anotherdistribution f 2ðxÞ while applying a weight of f 1ðxiÞ=f 2ðxiÞ.As an example, a user might want to oversample straighterpaths to enhance the statistics for early photons at largesource-receiver distances. The user could then simply scale

ARTICLE IN PRESS

Fig. 2. Schematic view of flux tables for four collinear tracks (offset in x

for clarity). The light distribution of a finite track (I) is the difference

between two semi-infinite tracks (II) and (III). The dotted rectangles show

the outer limits of the semi-infinite tables. The grey lines represent example

isointensity contours. The representation of photon flux at a point p2 far

from the starting and stopping points can be done in two ways. Either, as

in (III), by considering the edge of the best matching semi-infinite table,

or, as in (IV), with an infinite track table with only a single l bin.

J. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631624

the scattering length by a factor k41, propagate photonswith scattering described by f klsðsÞ (see Eq. (3)), andreweight each photon by f ls ðsÞ=f kls ðsÞ for every scatter itexperiences. Alternatively, the mean cosine of the scatter-ing angle, t, can be modified to achieve a similar effectusing Eq. (4). The effective scattering length is made afactor k longer by t0 ¼ ðk � 1þ tÞ=k. The correspondingweight is f tðyÞ=f t0 ðyÞ.

We have described how the photon flux density iscalculated in cells throughout the simulation volume. Inneutrino astronomy, the quantity of interest is the numberof photons detected by an optical sensor such as aphotomultiplier. Since the sensor response is usuallywavelength and angle dependent but we do not want tostore the wavelengths and arrival directions of thesimulated photons, PHOTONICS includes the option to foldthe sensor response into the simulation when the tables aregenerated. When this option is selected, user-suppliedwavelength and angular efficiency files are used to weightthe photon flux density to obtain the detected photon flux.

3.4. Propagating light sources

To this point we have discussed how to obtain a set oftables describing the photon fluxes for a range of locationsand orientations of a point-like, stationary light source. Apropagating light source cannot be satisfactorily approxi-mated as a flash of light from a single spatial point. Wediscuss in the following the modelling of Cherenkov lightfrom high energy muons, which give rise to kilometre scaleCherenkov emitting tracks in water or ice. Our method,however, applies also to other line-like light sources such ashigh energy tauons.

To calculate the photon flux distribution generated by amuon we integrate over the photon flux distributions ofmany point-like Cherenkov emitters with Ys given by themuon direction. A set of point-like photon tables providethe differential light flux Fpoint at any space–time locationfrom sources at any causally connected location. Thisserves as integration kernel,

GpointðxsðtsÞ;xrec; trecÞ �qqts

FpointðxsðtsÞ;xrec; trecÞ (13)

where ðxs; tsÞ and ðxrec; trecÞ are the emission and receivercoordinates. The light distribution of a propagating muonis thus generated by convolving this kernel with the trackof the muon, xmðtmÞ, so that

Fmðxrec; trecÞ ¼

Z tmstop

tmstart

GpointðxmðtmÞ;xrec; trecÞdtm. (14)

PHOTONICS provides the capability to efficiently performthis integration for a series of fixed muon directions andlocations and to store the resulting light fluxes in tables likethose for the point-like light sources. These tables are thenused to obtain the light flux for muon tracks throughany part of the simulation volume by interpolation(Section 3.5). Although the construction of muon events

through the integration in Eq. (14) can in principle also bedone event by event in a detector simulation or eventreconstruction, such an approach would often be signifi-cantly slower since the number of considered events, andtherefore the number of required flux calculations, istypically larger than the number of fixed muon directionsand locations needed to adequately cover the relevant range.To provide the photon flux for any given track length

without having to dynamically perform the time consum-ing integration of Eq. (14), we have developed a schemebased on the subtraction of semi-infinite tracks. The flux atðxrec; trecÞ arising from a finite muon starting at xA andstopping at xB can be expressed as the difference betweentwo semi-infinite tracks. For two semi-infinite starting

tracks, one (denoted mA!Þ starting at the point xA and theother ðmB!Þ starting at xB located further along the muontrack, we write

FmABðxrec; trecÞ ¼ FmA!ðxrec; trecÞ � FmB!ðxrec; trecÞ (15)

and the probability density function can be written

f mAB¼

f mA! � f mB!

ImA! � ImB!(16)

using Eq. (8). This construction of finite tracks is depictedin Fig. 2.Consider the point p0, close to the finite track start point.

At this point, the contribution from FmB! is comparablysmall, so that FmAB

� FmA! . However, for the point p1 closeto the end of the finite track, FmA! � FmB! and thecalculation of FmAB

becomes sensitive to exact numericalcancellation. It is then numerically superior to subtract twosemi-infinite stopping tracks, m!A and m!B, and writeFmAB¼ Fm!B

� Fm!A. Our algorithm dynamically chooses

between these two descriptions depending on which of theendpoints is closer to the observation point.For applications in which tracks can be regarded as

completely infinite, the flux tables are made smaller by

ARTICLE IN PRESSJ. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631 625

removing redundant information. At a given observationpoint in the medium, the flux from an infinite muon trackwith table origin Z1, giving the point a longitudinalcoordinate l1, is identical to the flux in a table with originZ2 where the point is at l2 ¼ l1 þ ðZ2 � Z1Þ= cosYs.Therefore in each infinite muon table only one l bin isretained, typically at l ¼ 0. Fig. 2 illustrates both ways ofaccessing information about virtually infinite tracks: withinfinite tables and with semi-infinite tables.

The size of a set of infinite tables is typically about 1% ofthe size of the corresponding semi-infinite description,which can be tens of gigabytes. Hence the infinite tables caneasily be loaded into computer memory all at once, whichis particularly useful in event reconstruction where it can behard to estimate in advance the properties of an event to bereconstructed. Reconstruction and large table support isdiscussed in the following section, describing the PHOTONICS

reader library.

3.5. Using the photon flux tables for event simulation and

reconstruction

For event simulation and reconstruction, the photon fluxtables are accessed using either a set of ANSI-C procedures,or a more abstract (ROOT compliant) user interface writtenin C++, both provided with the PHOTONICS package.

The full simulation of for example an ultra-relativisticmuon crossing the detector volume is performed by firstpropagating the muon through the detector medium with acharged particle propagator such as MMC [15]. Thisresults in a list of light generating subevents such asminimum ionising muon track segments and associatedelectromagnetic showers induced by stochastic energylosses. The detector specific simulation program can thenquery the corresponding PHOTONICS table information toobtain the number and time distribution of detectedphotons in each detector module from each such subevent.PHOTONICS comes with a variety of idealised light emissionprofiles, such as those of minimum ionising muons andpoint-like electromagnetic and hadronic showers.

A set of tables is loaded into memory using the tablereader library. The user can then query the photon fluxtables by specifying the location ðxsÞ and orientationðYs;FsÞ of the source, and the location of the lightdetectors ðxrecÞ. Two additional source characteristics areoptional in the query: the source length L (applicable tofinite muons) and the energy E used to scale the lightsource intensity. In an experiment simulation, the user firstrequests the expected number of detected photonsNðxs;Ys;Fs;xrec;E;LÞ. This query also returns a tablereference which can be used to get photon arrival timesrandomly drawn from the tabulated time distribution atthe corresponding coordinates. For event reconstruction,PHOTONICS also provides the arrival time probabilitydensity function f pdf ðxs;Ys;Fs;xrec;L; tÞ, which can beused by track-fitting algorithms, for example maximum-likelihood routines. Both f pdf and the photon intensity

(giving N) are naturally continuous in L and E, and aremade continuous in xs;Ys;Fs;xrec, and t by multidimen-sional linear interpolation. The interpolation of timedistributions is flux weighted, in agreement with Eq. (10),

f pdf ¼

PioiI if iP

ioiI i

withX

i

oi ¼ 1 (17)

where oi are the interpolation weights. When the flux for arequested source direction Ys is interpolated from twosurrounding tabulated directions Y1 and Y2, these tablesare first (implicitly) rotated to Ys. The receiver coordinatesxrec are then identical for the Y1 and Y2 table. Ananalogous approach is used for the source origin xs.It is sometimes necessary to convolve the photon time

distributions with the detector time response function orthe emission time profile. This can be done with a providedroutine operating on the photon flux tables. Convolutionwith a Gaussian or with one of two light source timedistributions with longer positive tails [6] (see Section 4)has been implemented.Any number of table sets can be loaded simultaneously

(limited only by available memory), making it possible tosimulate or reconstruct the cumulative signal from differenttypes of light sources (and detectors). This is useful forexample when simulating muons with secondary electro-magnetic showers from a primary muon track or severalcoincident muons. Since detailed photon table sets areoften many times larger than the primary memory of acomputer, we provide some memory management tools.Users can dynamically load and unload tables, and selectloading of tables corresponding to limited ranges of depthsðZsÞ and light source angles ðYsÞ. In experiment simula-tions this is particularly useful since the parameters for thesimulated particles are known. In event reconstruction, theloaded tables should cover the phase space taken intoaccount in the fitting algorithms. If memory limitationsrestrict this phase space, reconstruction can be performedfor subregions, on events predetermined to lie withinsubregions small enough for the corresponding photontables to fit in memory. Such preselection can be done witha set of more coarsely binned PHOTONICS tables or withother first-guess approaches (for muons, the muchsmaller tables of the infinite description can be used, seeSection 3.4).In addition to what has already been mentioned, the

PHOTONICS package includes several other tools for proces-sing of the photon flux tables, including integration in anydimension and conversion between differential and cumu-lative time distributions.

4. Comparison with observations

In this section we use measurements with artificialcalibration light sources in sea water and glacial ice todemonstrate that PHOTONICS reproduces the general beha-viour of the observed photon distributions.

ARTICLE IN PRESSJ. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631626

4.1. Modelling of natural water and ANTARES’

Mediterranean water surveys

The ANTARES collaboration is constructing a 0.1 km3

water neutrino detector in the deep Mediterranean sea[6]. In the present design the detector has 12 verticalstrings, each of which has an instrumented height ofabout 350m and consists of 25 storeys with threeoptical sensors each. Three strings were deployed in 2006and the remaining strings are scheduled to be installedduring 2007.

Absorption and effective scattering lengths in the deepMediterranean sea water have been investigated by theANTARES collaboration [6]. The surveys were performedduring several seasons using a calibrated setup of isotropiclight sources, one in the blue at 473 nm and one in the UVat 375 nm. For blue (UV) a la of 60 (26)m and a le of 265(122)m with 15% time variability are quoted by ANTARES.The details of the experimental setup, such as the lightemission time profile of the source and detector efficiencies,play a large role for the photon flux time distributionprofile in media, like ocean water, where scattering isweaker than absorption. In addition, since the light istypically observed at distances shorter than or comparableto the effective scattering length, the scattering phasefunction plays an even larger role. ANTARES assumed aweighted sum of a Petzold and a molecular (Einstein–Smoluchowski) distribution. The Petzold distribution hast ¼ 0:924, and is here approximated by a HG distribution,while the molecular distribution is approximated byisotropic scattering (HG with t ¼ 0Þ. Fig. 3 compares oursimulation results using this model with ANTARES measure-ments from June 2000 [6]. The agreement verifies thevalidity of the model and our photon flux simulation forthis case.

–20 0 20 40 60 80 100 120 14010–3

10–2

10–1

100

Residual time [ns]

24 m

44 m

Fig. 3. Residual time distributions at two distances from monochromatic flas

while the solid curves are our simulation results using the measured water prop

measured in air, where scattering and absorption can be ignored. The distributio

and (b) Blue flash, 473 nm in water.

4.2. Modelling of ice and ICECUBE’s Antarctic glacial ice

surveys

The IceCube neutrino telescope, under construction deepin the glacial ice at the geographic South Pole, is planned tobecome a high-energy neutrino detector of 1 km3 instru-mented volume [14]. It is planned to have 80 strings, ofwhich 22 have been deployed as of January 2007, eachequipped with 60 encapsulated photomultipliers evenlydistributed over depths between 1450 and 2450m. IceCubebuilds on to the 19 strings of the AMANDA array, whichhave been in operation since 2000.A detailed study of the properties of the deep South Pole

glacial ice has been performed by the AMANDA collabora-tion [8]. The glacial ice is very clear in the optical and nearUV with absorption lengths of 20–120m depending onwavelength. At wavelengths shorter than �210 nm andlonger than �500 nm, absorption is dominated by theproperties of pure ice, while in the intermediate rangeabsorption by impurities dominates. The effective scatter-ing length is on average 25m, less for shorter wavelengths.Both scattering and absorption are strongly depth depen-dent and vary on all depth scales. The variations at depthsexceeding 1450m, where bubbles no longer exist, areexplained by varying concentrations of insoluble dustdeposits which correlate with changes in climatic condi-tions over geological time scales. By using physicsmotivated functional forms for the wavelength and depthdependences of scattering and absorption, the AMANDA

collaboration has elaborated a heterogeneous ice model [8]by investigating a large number of recorded light distribu-tions generated by in situ pulsed and steady light sources atdifferent wavelengths. The resulting effective scattering andabsorptions lengths, le and la as functions of wavelength,were averaged in 10m depth intervals.

10–3

10–2

10–1

100

Tim

e d

istr

ibu

tio

n [

s–

1]

24 m

44 m

–20 0 20 40 60 80 100 120 140

Residual time [ns]

hes in water. The circles and stars are calibration data from ANTARES [6],

erties. The emission time profiles of the light sources (dashed curves) were

ns are normalised to unity at the peak value. (a) UV flash, 375 nm in water

ARTICLE IN PRESS

0 100 200 300 400 500

Residual time [ns] Residual time [ns]

0 250 500 750 1000 1250 1500 1750 2000

10–4

10–3

10–2

Tim

e d

istr

ibution [s

–1]

Heterogeneous model

Single histogram modelHeterogeneous model

Single histogram model

10–5

10–4

10–3

Tim

e d

istr

ibution [s

–1]

Fig. 4. Residual time distributions for two monochromatic pulsed light sources in deep glacial ice. In (a), light is recorded at a horizontal distance of 75m

from an isotropic laser source located at a depth of 1825m. In (b), the detector is at a horizontal distance of 140m from an upward-pointing LED emitter

located at a depth of 1580m. The AMANDA calibration data are shown with Poissonian error bars. The intrinsic timing widths of the light sources, less than

10 ns, were not included in the simulation. The thick solid curves are PHOTONICS simulations using the scattering and absorption parameters fitted to these

particular time distributions, and the thin dashed curves represent two opposite parameter variations within the parameter uncertainties from the fits. The

thin solid curves show our simulation results with the heterogeneous ice model [8]. (a) Nd:YAG laser, 532 nm in ice and (b) Blue LED beacons, 470 nm

in ice.

J. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631 627

Using the AMANDA ice model parameters, we havegenerated simulated time distributions corresponding totwo combinations of wavelength and light source–receiverpositions, and compare them with experimental distribu-tions in Fig. 4. The thick solid curves are our results whenusing the scattering and absorption parameters fitted tothese particular delay time distributions [8], with dashedcurves representing two opposing deviations within theparameter uncertainty from these fits. The parametersfitted to the displayed distributions were le ¼ 27:6m, la ¼20:5m for the 532 nm curve, and le ¼ 22:6m, la ¼ 82:0mfor the 470 nm curve, both with t ¼ 0:94. The slightoverestimation in the tail in Fig. 4(b) is due to systematicuncertainties in the simulation of the LED emitter whichlead to an imperfect description of the data in thisparticular case. The thin solid curves are simulation resultswith the heterogeneous ice model which is based on datafrom many source–receiver combinations at differentdepths and wavelengths. The differences between the twosolid curves in each picture reflect the fact that the icemodel parameters are averages over all fitted parameters in10m depth bins and the parameters from individual fits aredistributed around these averages.

5. Application to neutrino astronomy

In neutrino astronomy, the universe is studied using highenergy neutrinos as cosmic messengers. The neutrinos canbe detected optically only after interacting with matter inthe vicinity of the neutrino telescope and producingcharged, Cherenkov light emitting particles like muons.Some of the emitted light is recorded by optical sensorsdistributed throughout the detector volume.

Ultra-relativistic muons are the primary channel throughwhich high energy neutrinos are detected by opticalneutrino telescopes. They are also the primary backgroundin the form of so-called atmospheric muons arising fromhigh energy cosmic-ray interactions in the Earth’s atmo-sphere. An extraterrestrial neutrino signal is distinguishedfrom the background of neutrinos and muons created inthe atmosphere mainly by differences in energy spectra andangular distributions. It is therefore important to establishthe particle energy and direction in every recorded event asaccurately as possible. Our software contributes to this aimby providing means for detailed photon flux simulations,using depth and wavelength dependent optical propertiesas established at a specific site. High-energy neutrinotelescopes are typically recording data for several yearswhile they are constructed by adding more optical sensors.Photon flux tables generated by PHOTONICS can coverarbitrarily large volumes and their use is therefore easilyscalable to such growing sensor arrays.To further illustrate the utility of PHOTONICS, we present

in this section photon propagation results for theinhomogeneous ice at the site of the IceCube neutrinotelescope [8]. The flux in all figures is given per emittedphoton, and is weighted with the angular and wavelengthdependent acceptance of the optical detectors used byIceCube, so that the figures display the expected number ofdetections normalised to a 1m2 detection area in thedirection of maximum optical module sensitivity. Thephoton flux tables were also convolved with a 10 ns wideGaussian to account for photomultiplier jitter.Fig. 5 shows the photon flux from a simulated in-

finitesimal electromagnetic cascade at 1730m depth ðZs ¼

0 in AMANDA detector coordinates). Cascades are initiated

ARTICLE IN PRESS

Fig. 5. Light flux generated by a simulated electromagnetic cascade near the centre of the AMANDA telescope. The upper panel shows a vertical slice of the

photon flux (left) and the probability distribution (right) at t ¼ 100ns after light emission from a cascade at the origin and oriented toward the upper left at

Ys ¼ 135�. The lower panel shows the same distributions at t ¼ 300ns. Note the different scales in the two panels. (a) F ð100nsÞ, Zs ¼ 0m;

(b) f pdf ðt ¼ 100nsÞ, Zs ¼ 0m; (c) F ð300nsÞ, Zs ¼ 0m and (d) f pdf ð300nsÞ, Zs ¼ 0m.

J. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631628

in muon energy loss processes as well as in primaryneutrino interactions. The light spectra of hadronic andelectromagnetic cascades are Cherenkov in nature, but thelight originates from many Cherenkov light emittingparticles. The Cherenkov emission cone is slightly distorted[16] since not all emitting particles travel in parallel or atthe same speed. The cascade in Fig. 5 is oriented toward thesurface, at Ys ¼ 135� pointing at the upper left corner ofthe picture. A vertical slice through the photon fluxcontaining the principal axis of the light source is dis-played. The angular distribution of emitted photons ispeaked at the Cherenkov angle, but the photon flux issmoothed out by scattering as it evolves through the ice.After 100 ns, we can still observe a peaked light distributionin the forward direction. At later times, the flux becomes

more and more isotropic, to asymptotically resemble thatof an isotropic flash.Since the IceCube sensors are pointing downward they

detect upgoing light with a higher efficiency than down-going light, which needs to be scattered to reach thephotomultiplier photocathode. As a result, a point-likelight source appears more upward. This can be seen inFig. 5 where the direction of the light source appears to beoriented at an angle larger than Ys ¼ 135�.While the ice is very clear in the centre of the AMANDA

telescope, there are other depths where stronger scatteringand absorption distort the photon flux. In the upper panelof Fig. 6 simulation results are shown for a setupanalogous to that of Fig. 5 but with the shower origin350m deeper in the ice. This location is immediately below

ARTICLE IN PRESS

Fig. 6. Light fluxes generated by two simulated electromagnetic cascades 350m below the AMANDA centre. This location is immediately below a region

with higher dust concentration that causes stronger scattering and absorption. The flux is shown for both cascades at 100 and 300 ns after light emission.

Because of the strong scattering and absorption just above the emission point, the flux from the upward pointing ðYs ¼ 135�Þ source in the upper panel,

(a) and (b), evolves from a distribution peaked at the Cherenkov angle to a distribution similar to that from a downward pointing ðYs ¼ 0�Þ source, shown

in the lower panel, (c) and (d). (a) F ð100nsÞ, Ys ¼ 135�; (b) F ð300nsÞ, Ys ¼ 135�; (c) F ð100nsÞ, Ys ¼ 0� and (d) F ð300nsÞ, Ys ¼ 0�.

J. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631 629

a region of strong scattering and absorption. The cascadedirection is again Ys ¼ 135�, but because of the particularmedium properties in this region the event shape appears tobe more isotropic and even resembles a downward pointingcascade. However, it does differ from truly downwardpointing events at the same location, shown in the lowerpanel of Fig. 6. In this particular case it is exceptionallyhard to characterise the event correctly, but by using thecorrect heterogeneous medium description we improve thepossibility to distinguish these cases. This is important forevent simulation and reconstruction of parameters such asthe zenith angle and the cascade energy.

Inhomogeneities in the detector medium also stronglyaffect the optical topology of muon events. Fig. 7 showsthe light distribution of a simulated muon moving upward

through deep South Pole ice. At the front of the track, weobserve a cross-section of the unscattered Cherenkovwavefront, followed by a diffuse light cloud as the photonsare scattered away from the geometrical Cherenkov cone.At depths with higher dust concentrations, photons areobstructed by scattering and absorbed before they cantravel very far. This deforms the conical light front, whichappears to be bent backwards. In the dusty region nearz ¼ �350m, the photon flux is depleted and the survivingphotons delayed by increased scattering. Muon trackreconstruction is often strongly dependent on the earliestrecorded photons, corresponding to the Cherenkov wave-front. Distortions in the wavefront, like in Fig. 7(b), candegrade the reconstruction accuracy unless they are takeninto account by the track fitting algorithms.

ARTICLE IN PRESS

Fig. 7. A snapshot of the light field produced by a muon which entered from below, at an angle Ys ¼ 135�. Inhomogeneities in the medium properties

distort the smoothly arched light cone, as is most easily seen in the probability density function (b) in the particularly dusty region around z ¼ �350m

which has stronger scattering and absorption. In (a), the flux is depleted in the dusty region. (a) FðtÞ and (b) f pdf ðtÞ.

Fig. 8. Light distribution from a 142m long muon track without secondary interactions. This muon was created at the origin and propagated upward at

Ys ¼ 135� until it decayed at ðx; zÞ ¼ ð�100; 100Þ. The figure shows a snapshot 147 ns after the muon disappeared. Both the photon flux (a) and the

probability density function (b) for such a comparably short track are similar to those produced by a point-like cascade. (a) FðtÞ and (b) f pdf ðtÞ.

J. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631630

Fig. 8 shows a snapshot of the light field generated by afinite muon track without secondary interactions. Themuon was created at the origin and propagated upward atYs ¼ 135� until it decayed after 142m. The probabilitydensity function for such a relatively short track ap-proaches a shape similar to that of point-like cascades,making it hard to distinguish the two cases in anexperimental situation with a limited number of pointssampled by light sensors. PHOTONICS-based simulationswith realistic medium and light source descriptions allowexperimentalists to isolate the differences in light profilesfor these (and other) distinct cases and to developappropriate reconstruction algorithms.

6. Conclusion

In this paper we have presented the concepts andmethods which combine into the PHOTONICS softwarepackage. We have explained how the program can be usedfor calculating and tabulating light distributions around astationary or moving source, as a function of time andspace in scattering and absorbing heterogeneous media.The light distributions obtained from our Monte Carlosimulation agree well with observations of calibration lightsources in deep sea water and glacial ice surveys. In the lastsection it is demonstrated how PHOTONICS can be used tomodel how optical inhomogeneities of the Antarctic ice at

ARTICLE IN PRESSJ. Lundberg et al. / Nuclear Instruments and Methods in Physics Research A 581 (2007) 619–631 631

the location of the IceCube neutrino telescope distort thefootprints of elementary particle interactions.

Acknowledgements

We are grateful to Dr. Nathalie Palanque-Delabrouillefor supplying data from the ANTARES water surveys and herhelpful advice on the implementation of water specificparameters. We also thank the members of the AMANDA

and IceCube collaborations for fruitful discussions anduseful feedback.

References

[1] PHOTONICS, online reference; hhttp://photonics.tsl.uu.sei.

[2] P.B. Price, K. Woschnagg, Astropart. Phys. 15 (2001) 97.

[3] I.A. Belolaptikov, et al., Variation of water parameters at the site of

the Baikal experiment and their effect on the detector performance,

in: Proceedings of the XXIV ICRC, Rome, vol. 1, 1995, p. 770.

[4] H. Bradner, G. Blackinton, Appl. Opt. 23 (1984) 1009.

[5] E.G. Anassontzis, et al., Nucl. Instr. and Meth. A 349 (1994)

242.

[6] J.A. Aguilar, et al., ANTARES Collaboration, Astropart. Phys. 23

(2005) 131.

[7] A. Capone, et al., NEMO Collaboration, Nucl. Instr. and Meth.

A 487 (2002) 423.

[8] M. Ackermann, et al., AMANDA Collaboration, J. Geophys. Res. 111

(2006) D13203.

[9] G. Mie, Ann. Phys. 25 (1908) 377.

[10] P.B. Price, L. Bergstrom, Appl. Opt. 36 (1997) 4181.

[11] L. Henyey, J. Greenstein, Astrophys. J. 93 (1941) 70.

[12] S. Chandrasekar, Radiative Transfer, Dover Publications, New

York, 1960.

[13] J.V. Jelly, Cerenkov Radiation and its Applications, Pergamon Press,

London, 1958.

[14] A. Achterberg, et al., IceCube Collaboration, Astropart. Phys. 26

(2006) 155.

[15] D. Chirkin, W. Rhode, Muon Monte Carlo: a new high-precision

tool for tracking of muons in medium, in: Proceedings of the XXVII

ICRC, Hamburg, 2001, p. 1017.

[16] C. Wiebusch, The detection of faint light in deep underwater neutrino

telescopes, Ph.D. Thesis, RWTH Aachen, December 1995, PITHA

95/37.