Tracking of the photons

Cerenkov photons are tracked in the routine GTCKOV. These particles are subject to in flight absorption (process LABS, number 101) and boundary action (process LREF, number 102, see above). As explained above, the status of the photon is defined by 2 vectors, the photon momentum ($p=k$ ) 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 which the photon is leaving and toward the one which the photon is entering. The behaviour of a photon at the surface boundary is determined by three quantities:

1. refraction or reflection angle, this represents the kinematics of the effect;
2. amplitude of the reflected and refracted waves, this is the dynamics of the effect;
3. probability of the photon to be refracted or reflected, this is the quantum mechanical effect which we have to take into account if we want to describe the photon as a particle and not as a wave;

As said above, we distinguish three kinds of boundary action, dielectric $\to$ black material, dielectric $\to$ metal, dielectric $\to$ 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 homogeneous monochromatic plane wave:

$E$	$=$	$E$0ei kx- i ωt
$B$	$=$	$\mu \epsilon \{kxEk\}$

Case dielectric → dielectric

In the classical description the incoming wave splits into a reflected wave (quantities with a double prime) and a refracted wave (quantities with a single prime). Our problem is solved if we find the following quantities:

$E\text{'}$	$=$	$E\text{'}$0ei k'x- i ωt
$E\text{'}\text{'}$	$=$	$E\text{'}\text{'}$0ei k''x- i ωt

For the wave numbers the following relations hold:

$|k|$	$=$	$|k\text{'}\text{'}|\; =\; k\; =\; \{\omega c\}\mu \epsilon$
$|k\text{'}|$	$=$	$k\text{'}\; =\; \{\omega c\}\mu \text{'}\; \epsilon \text{'}$

Where the speed of the wave in the medium is $v=c/\mu \epsilon$

and the quantity $n=c/v=\mu \epsilon$ is called refractive 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:

$(kx)$surf	$=$	$(k\text{'}x)$surf=(k'' x)surf
$k\; sini$	$=$	$k\text{'}\; sinr=\; k\text{'}\text{'}\; sinr\text{'}$

where $i,\; r,\; r\text{'}$ are, respectively, the angle of the incident, refracted and reflected ray with the normal to the surface. From this formula the well known condition emerges:

$i$	$=$	$r\text{'}$
$\{sinisinr\}$	$=$	$\{\mu \text{'}\; \epsilon \text{'}\mu \epsilon \}=\{n\text{'}n\}$

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 expressed in the two following cases:

1. a plane wave with the electric field (polarisation vector) perpendicular to the plane defined by the photon direction and the normal to the boundary:

   $\{E$0'E0}	$=$	$\{2\; n\; cosin\; cosi+\; \{\mu \mu \text{'}\}n\text{'}\; cosr\}=\; \{2\; n\; cosin\; cosi+\; n\text{'}\; cosr\}$
   $\{E$0''E0}	$=$	$\{n\; cosi-\; \{\mu \mu \text{'}\}n\text{'}\; cosrn\; cosi+\; \{\mu \mu \text{'}\}n\text{'}\; cosr\}=\; \{n\; cosi-\; n\text{'}\; cosrn\; cosi+\; n\text{'}\; cosr\}$

   where we suppose, as it is legitimate for visible or near-visible light, that $\mu /\mu \text{'}\; \approx 1$ ;

2. a plane wave with the electric field parallel to the above surface:

   $\{E$0'E0}	$=$	$\{2\; n\; cosi\{\mu \mu \text{'}\}n\text{'}\; cosi+\; n\; cosr\}=\{2\; n\; cosin\text{'}\; cosi+\; n\; cosr\}$
   $\{E$0''E0}	$=$	$\{\{\mu \mu \text{'}\}n\text{'}\; cosi-\; n\; cosr\{\mu \mu \text{'}\}n\text{'}\; cosi+\; n\; cosr\}=\; \{n\text{'}\; cosi-\; n\; cosrn\text{'}\; cosi+\; n\; cosr\}$

   whith the same approximation as above.

We note that in case of photon perpendicular to the surface, the following relations hold:

$\{E$0'E0}

$=$

$\{2nn\text{'}+n\}$

$\{E$0''E0}

$=$

$\{n\text{'}-nn\text{'}+n\}$

where the sign convention for the parallel field has been adopted. This means that if $n\text{'}\; >\; n$ there is a phase inversion for the reflected wave.

Any incoming wave can be separated into one piece 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:

$S$

$=$

$\{12\}\{c4\; \pi \}\mu \epsilon ExH*=\; \{c8\; \pi \}\{\epsilon \mu \}E$02k

and the energy balance on a unit area of the boundary requires that:

$Su$	$=$	$S\text{'}u-S\text{'}\text{'}u$
$S\; cosi$	$=$	$S\text{'}\; cosr+\; S\text{'}\text{'}\; cosi$
$\{c8\; \pi \}\{1\mu \}n\; E$02cosi	$=$	$\{c8\; \pi \}\{1\mu \text{'}\}n\text{'}\; E$0'2cosr+{c8 π}{1μ}n E0''2cosi

If we set again $\mu /\mu \text{'}\; \approx 1$ , then the transmission probability for the photon will be:

$T\; =\; (\; \{E$₀'E₀}) ²{ n' cosrn cosi}

and the corresponding probability to be reflected will be R=1-T.

In case of reflection the relation between the incoming photon ($k,e$ ), the refracted one ( ) and the reflected one ( ) is given by the following relations:

$q$	$=$	$kxu$
$e$	$=$	$\{eu|q|\}$
$e$⊥	$=$	$\{eq|q|\}$
$e$'	$=$	$e${2 n cosin' cosi+ n cosr}
$e$⊥'	$=$	$e$⊥{2 n cosin cosi+ n' cosr}
$e$''	$=$	$\{n\text{'}n\}e$' - e
$e$⊥''	$=$	$e$⊥' - e⊥

After transmission or reflection of the photon, the polarisation vector is renormalised to 1. In the case where then there cannot be a refracted wave, and in this case we have a total internal reflection according to the following formulas:

$k\text{'}\text{'}$	$=$	$k-\; 2\; (ku)u$
	$=$	$-e+\; 2\; (eu)u$

Case dielectric → metal

In this case the photon cannot be transmitted. 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 reflected.