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| Geant 3.16 | GEANT User's Guide | PHYS260 ##
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Author(s) : F.Carminati, R.Jones Submitted: 03.02.93
Origin : R.Jones Revised: 22.04.93
Cerenkov photons are produced when a charged particle traverses a dielectric material.
A photon is called optical when its wavelength is >> the typical atomic spacing, for instance when lambda>=10nm )E< =100eV. Production of an optical photon is due to:
Fluorescence is taken into account in GEANT in the context of the photoelectric effect (PHYS???), but only above the energy cut CUTGAM. Scintillation is not yet simulated by GEANT.
Optical photons undergo three kinds of interaction:
For optical photons elastic scattering is relatively unimportant. For lambda=.200mum we have sigma #.2b for N or O which gives a Rayleigh 2 2 mean free path of #1.7km in air and # 1m in quartz. Not only the cross section is small, but elastic scattering is also forward peaked. This interaction is ignored in GEANT.
Absorbtion is important for optical photons because it determines the lower lambda limit in the window of transparency of the radiator. Absorption competes with photo-ionisation in producing the signal in the detector, so it must be treated properly in the tracking of optical photons.
When a photon arrives at the boundary of the dielectric medium in which it travels, its behaviour, here called boundary effect, depends on the nature of the two materials which join at that boundary:
The photon polarisation is defined as a two component vector normal to the direction of the photon:
iphi iphi
a e 1 phi a e c
( 1 iphi )= e o( 1 -iphi )
a e 2 a e c
2 2
where phi = ((phi -phi )/ (2)) is called circularity and
c 1 2
phi = ((phi + phi )/(2)) is called overall phase. Circularity gives the
o 1 2
left- or right-polarisation characteristic of the photon. RICH materials usually do not distinguish between the two polarisations and photons produced by the Cerenkov effect are linearly polarised, that is phi = 0. c The circularity of the photon is ignored by GEANT.
The overall phase is important in determining interference effects between coherent waves. These are important only in layers of thickness comparable with the wavelength. GEANT ignores the overall phase.
Vector polarisation is described by the polarisation angle tan psi= ((a )/(a )). Reflection/transmission probabilities are sensitive 2 1 to the state of linear polarisation, so this has to be takent into account. One parameter is sufficient to describe vector polarisation, but to avoid too many trigonometrical transformations, a unit vector perpendicular to the direction of the photon is used in GEANT.
The polarisation vectors are stored in a special track structure which is lifted at the link LQ(JSTACK-1) when the first Cerenkov photon is stored in the stack.
For the formulas contained in this chapter, please see [bib-JACK]. Let n be the refraction index of the dielectric material acting as a radiator (n= c/c' where c' is the group velocity of light in the material: it follows that 1<=n). In a dispersive material n is an increasing function of the photon momentum p ,((dn)/(dp))> =0. A particle travelling with speed beta= v/c will gamma emit photons with an angle given by the relation:
cos theta= ((1)/(betan))
from which follows the limitation for the momentum of the emitted photons:
min
n(p ) =((1)/ (beta))
gamma
Additionally, the photons must be within the window of transparency of the radiator. All the photons will be contained in a cone of opening max cos(theta )= ((1)/(betan(p ))). max gamma
The average number of photons produced is given by the relations:
2 2 2 2 2 2 2
dN =((2piz )/ (hc))sin theta((dnu)/(c))dx= ((2piz )/(hc))(1- cos theta)((dnu)/ (c))dx= ((2piz )/(hc))(1-((1)/ (n beta )))((dnu)/(c))dx =
2 2 2 2 2 2 2 2
= ((z )/(h c ))(1- ((1)/(n beta )))dp dx# 370z ((photons)/(cm eV))(1-((1)/ (n beta )))dp dx
gamma gamma
and
max max
p p
2 gamma 2 2 2 max min 2 gamma 2
((dN)/ (dx))# 370z int dp (1-((1)/(n beta )))= 370z (p -p - ((1)/(beta )) int dp ((1)/(n(p ) )))
min gamma min gamma gamma
p gamma gamma p
gamma gamma
The number of photons produced is calculated from a poissonian distribution with average value ((dN)/(dx))STEP. The momentum distribution of the photon is then sampled from the density function:
2 2
f(p ) =(1- ((1)/(n (p )beta )))
gamma gamma
Cerenkov photons are tracked via the routine GTCKOV. These particles are subject to in flight absorbtion (process 101) and boundary action (process 102, see above). As explained above, the status of the photon is defined by 2 vectors, the photon momentum (p= hvk) and photon polarisation (e). By convention the direction of the polarisation vector is that of the electric field. Let also u be the normal to the material boundary at the point of intersection, pointing out of the material in which the photon is and toward the one which the photon is entering. The behaviour of a photon at the surface boundary is determined by three class of phenomena:
As said above, we distinguish three kinds of boundary action, dielectric -> black material, dielectric -> metal, dielectric -> dielectric. The first case is trivial, in the sense that the photon is immediately absorbed and it goes undetected.
To determine the behaviour of the photon at the boundary, we will at first treat it as an homogeneus monocromatic plane wave:
ik#x-iomegat
E = E e
0
B = B sqrt(muepsilon)((k xE)/ (k))
0
In the classical description the incoming wave splits into a reflected wave (quantities with a double apex) and a reflected wave (quantities with a single apex). Our problem is solved if we find the following quantities:
ik'#x-iomegat
E' = E' e
0
ik''#x-iomegat
E'' = E'' e
0
For the wave numbers the following relations hold:
|k| = |k''|= k= ((omega)/(c))sqrt(muepsilon)
|k'| = k' =((omega)/ (c))sqrt(omega'epsilon')
Where the speed of the wave in the medium is v= ((c)/(sqrt(muepsilon))) and the quantity n= ((c)/(v))= sqrt(muepsilon) is called refraction index of the medium. The condition that the three waves, refracted, reflected and incident have the same phase at the surface of the medium, gives us the well known Fresnel law:
(k #x) = (k' #x) =(k'' #x)
surf surf surf
ksini = k' sinr= k''sin r'
where i,r,r' are, respectively, the angle of the incident, refracted and reflected ray with the normal to the surface. The formula results in the well known condition:
i = r'
((sin i)/(sinr)) = sqrt(((mu'epsilon')/ (muepsilon)))= ((n')/(n))
The dynamic properties of the wave at the boundary are derived from Maxwell's equations which impose the continuity of the normal components of D and B and of the tangential components of E and H at the surface boundary. The resulting ratios between the amplitudes of the the generated waves with respect to the incoming one are simply expressed in the two following cases:
((E ')/ (E )) = ((2ncosi)/ (ncosi +((mu)/ (mu'))n'cosr))
0 0
((E '')/ (E )) = ((ncosi- ((mu)/(mu'))n'cos r)/(ncos i+ ((mu)/(mu'))n'cos r))
0 0
((E ')/ (E )) = ((2ncosi)/ (((mu)/(mu'))n'cos i+ ncosr))
0 0
((E '')/ (E )) = ((((mu)/(mu'))n'cos i-ncos r)/(((mu)/(mu'))n' cosi +ncos r))
0 0
Any incoming wave can be separated in one polarised parallel to the plane and one polarised perpendicular, and the two components treated accordingly.
To mantain the particle description of the photon, the probability to have a refracted (mechanism 107) or reflected (mechanism 106) photon must be calculated. The constraint is that the number of photons be conserved, and this can be imposed via the conservation of the energy flux at the boundary, as the number of photons is proportional to the energy. The energy current is given by the expression:
* 2
S = ((1)/ (2))((c)/(4pi))sqrt(muepsilon)Ex H = ((c)/(8pi))sqrt(((epsilon)/ (mu)))E k
0
and the energy balance on a unit area of the boundary requires that:
S #n = S' #n- S''#n
Scosi = S' cosr+ S''cos i
2 2 2
((c)/ (8pi))munE cosi = ((c)/ (8pi))mu'n'E ' cos r+((c)/ (8pi))munE '' cos i
0 0 0
If we suppose, as it is legitimate for visible or near-visible light, that ((mu)/(mu'))# 1, then the probability for the photon to be refracted will be:
2
P =(((E ')/ (E ))) ((n'cos r)/(ncos i))
0 0
and the corresponding probability to be reflected will be 1-P.
In case of reflection the relation between the incoming photon (k,e) and the reflected one (k'',e'') is given by the following relations:
q = e xu
e = ((e #u)/ (|q|))
||
e = ((e #q)/ (|q|))
?
e ' = e ((2n cosi)/(n' cosi +ncos r))
|| ||
e ' = e ((2n cosi)/(n cosi +n'cos r))
? ?
e '' = ((n')/ (n))e '-e
|| || ||
e '' = e '- e
? ? ?
In the case where sinr= ((n)/(n')) sini>1 then there cannot be a
refracted wave, and in this case we have a total internal reflection
according to the following formulas:
k'' = k- 2(k#u)u
e'' = - e+2(e #u)u
In this case the photon cannot be refracted. So the probability for the photon to be absorbed by the metal is estimated according to the table provided by the user. If the photon is not absorbed, it is totally reflected internally.
In the case where the surface between two bodies is perfectly polished, then the normal provided by the program is the normal to the surface defined by the body boundary. This is indicated by the the value POLISH=1 as returned by the GUPLSH function. When the value returned is <1, then a random point is generated in a sphere of radius 1-POLISH, and the corresponding vector is added to the normal. This new normal is accepted if the reflected wave is still inside the volume.
+-----------------------------------------------------------------------------+ |CALL GSCKOV (ITMED, NPCKOV, PPCKOV, ABSCO, EFFIC, RINDEX) | +-----------------------------------------------------------------------------+
This routine declares a tracking medium either as a radiator or as a metal and stores the tables provided by the user. In the case of a metal the RINDEX array does not need to be of length NPCKOV, as long as it is set to 0. The user should call this routine if he wants to use Cerenkov photons. Please note that for the moment only BOXes, TUBEs and CONEs can be assigned optical properties due to the current limitations of the GGPERP routine described below.
+----------------------------------------------------------------------+ |CALL GLISUR (X0, X1, MEDI0, MEDI1, U, PDOTU, IERR) | +----------------------------------------------------------------------+
This routine simulates the surface profile between two media as seen by an approaching particle with coordinate and direction given by X0. The surface is identified by the arguments MEDI0 and MEDI1 which are the tracking medium indices of the region in which the track is presently and the one which it approaches, respectively. The input vector X1 contains the coordinates of a point on the other side of the boundary from X0 and lying in within medium MEDI1. The result is a unit vector U normal to the surface of reflection at X0 and pointing into the medium from which the track is approaching.
The quality of the surface finish is given by the parameter returned by the user function GUPLSH (see below).
VALUE = GUPLSH(MEDI0, MEDI1)
This function must be supplied by the user. It returns a value between 0 and 1 which decsribes the quality of the surface finish between MEDI0 and MEDI1. The value 0 means maximum roughness with effective plane of reflection distributed as cosalpha where alpha is the angle between the unit normal to the effective plane of reflection and the normal to the nominal medium boundary at X0. The value 1 means perfect smoothness. In between the surface is modelled as a bell-shaped distribution in alpha with limits given by:
sin alpha= #(1-GUPLSH)
At the interface between two media the routine is called to evaluate the surface. The default version in GEANT returns 1, i.e. a perfectly polished surface is assumed. When GUPLSH = 0 the distribution of the normal to the surface is #costheta.
+--------------------------------+
|CALL GGCKOV |
+--------------------------------+
This routine handles the generation of Cerenkov photons and is called from GTHADR, GTMUON and GTELEC in radiative materials for which the optical characteristics have been defined via the routine GSCKOV.
+--------------------------------------+
|CALL GSCKOV (IPHO) |
+--------------------------------------+
This routines takes the Cerenkov photon IPHO generated during the current step and stores it in the stack for subsequent tracking. This routine performs for Cerenkov photons the same function that the GSKING routine performs for all the other particles. The generated photons are stored in the common /GCKIN2/ ([BASE030]).
+--------------------------------+
|CALL GTCKOV |
+--------------------------------+
This routine is called to track Cerenkov photons. The user routine GUSTEP is called at every step of tracking. When ISTOP = 2 the photon has been absorbed. If DESTEP !=0 then the photon has been detected.
+--------------------------------------------+
|CALL GGPERP (X, U, IERR) |
+--------------------------------------------+
This routine solves the general problem of calculating the unit vector normal to the surface of the current volume at the point X. The result is returned in the array U. X is assumed to be outside the current volume and near a boundary (within EPSIL). The current volume is indicated by the common /GCVOLU/. U points from inside to outside in that neighbourhood. If X is within EPSIL of more than one boundary (i.e. in a corner) an arbitrary choice is made. If X is inside the current volume or if the current volume is not handled by the routine, the routine returns with IERR=1, otherwise IERR=0. At the moment the routine only handles BOXes, TUBEs and CONEs.
The process of generating a Cerenkov photon is called CKOV and corresponds to the process value 105 (variable LMEC in common /GCTRAK/). This process is activated only in a radiator defined via the routine GSCKOV.
The process of photon absorbtion (name LABS, code 101) is controlled by the LABS FFREAD data record. By default the process is activated for all the materials for which optical properties have been defined.
The action taken at the boundary is identified by the process name LREF, code 102.
At a boundary a photon can be either reflected (name REFL, code 106) or refracted (name REFR, code 107).