Interactions of optical photons

Optical photons are produced when a charged particle traverses:

1. a dielectric material with velocity above the Čerenkov threshold;

2. a scintillating material.

Physics processes for optical photons

A photon is called optical when its wavelength is much greater than the typical atomic spacing, for instance when \(\lambda \geq 10\) nm which corresponds to an energy \(E \leq 100\) eV. Production of an optical photon in a HEP detector is primarily due to:

1. Čerenkov effect;

2. Scintillation.

Optical photons undergo three kinds of interactions:

1. Elastic (Rayleigh) scattering;

2. Absorption;

3. Medium boundary interactions.

Rayleigh scattering

For optical photons Rayleigh scattering is usually unimportant. For \(\lambda=.2 \; \mu \mathrm{m}\) we have \(\sigma_{Rayleigh} \approx .2b\) for \(N_{2}\) or \(O_{2}\) which gives a mean free path of \(\approx\) 1.7 km in air and \(\approx\) 1 m in quartz. Two important exceptions are aerogel, which is used as a Čerenkov radiator for some special applications and large water Čerenkov detectors for neutrino detection.

The differential cross section in Rayleigh scattering, \(d\sigma/d\Omega\), is proportional to \(1+\cos^{2}\theta\), where \(\theta\) is the polar angle of the new polarization with respect to the old polarization.

Absorption

Absorption 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.

Medium boundary effects

When a photon arrives at the boundary of a dielectric medium, its behaviour depends on the nature of the two materials which join at that boundary:

Case dielectric \(\rightarrow\) dielectric.

The photon can be transmitted (refracted ray) or reflected (reflected ray). In case where the photon can only be reflected, total internal reflection takes place.

Case dielectric \(\rightarrow\) metal.

The photon can be absorbed by the metal or reflected back into the dielectric. If the photon is absorbed it can be detected according to the photoelectron efficiency of the metal.

Case dielectric \(\rightarrow\) black material.

A black material is a tracking medium for which the user has not defined any optical property. In this case the photon is immediately absorbed undetected.

Photon polarization

The photon polarization is defined as a two component vector normal to the direction of the photon:

\[{a_{1}e^{i\Phi_{1}} \choose a_{2}e^{i\Phi_{2}}} = e^{\Phi_{o}} {a_{1}e^{i\Phi_{c}} \choose a_{2}e^{-i\Phi_{c}}}\]

where \(\Phi_{c}= (\Phi_{1}-\Phi_{2})/2\) is called circularity and \(\Phi_{o}=(\Phi_{1}+\Phi_{2})/2\) is called overall phase. Circularity gives the left- or right-polarization characteristic of the photon. RICH materials usually do not distinguish between the two polarizations and photons produced by the Čerenkov effect and scintillation are linearly polarized, that is \(\Phi_{c}=0\).

The overall phase is important in determining interference effects between coherent waves. These are important only in layers of thickness comparable with the wavelength, such as interference filters on mirrors. The effects of such coatings can be accounted for by the empirical reflectivity factor for the surface, and do not require a microscopic simulation. Geant4 does not keep track of the overall phase.

Vector polarization is described by the polarization angle \(\tan \Psi = a_{2}/a_{1}\). Reflection/transmission probabilities are sensitive to the state of linear polarization, so this has to be taken into account. One parameter is sufficient to describe vector polarization, but to avoid too many trigonometrical transformations, a unit vector perpendicular to the direction of the photon is used in Geant4. The polarization vector is a data member of G4DynamicParticle.

Tracking of the photons

Optical photons are subject to in flight absorption, Rayleigh scattering and boundary action. As explained above, the status of the photon is defined by two vectors, the photon momentum (\(\vec{p}=\hbar \vec{k}\)) and photon polarization (\(\vec{e}\)). By convention the direction of the polarization vector is that of the electric field. Let also \(\vec{u}\) 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 \(\rightarrow\) black material, dielectric \(\rightarrow\) metal, dielectric \(\rightarrow\) 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:

\[\begin{split}\vec{E} &= \vec{E}_{0}e^{i\vec{k} \cdot \vec{x}-i\omega t} \\ \vec{B} &= \sqrt{\mu \epsilon} \frac{\vec{k} \times \vec{E}}{k}\end{split}\]

Case dielectric \(\rightarrow\) 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:

\[\begin{split}\vec{E}' &= \vec{E}_{0}' e^{i\vec{k}'\cdot \vec{x}-i\omega t} \\ \vec{E}'' &= \vec{E}_{0}'' e^{i\vec{k}''\cdot \vec{x}-i\omega t}\end{split}\]

For the wave numbers the following relations hold:

\[|\vec{k}| = |\vec{k}''| = k = \frac{\omega}{c}\sqrt{\mu \epsilon}\]

\[|\vec{k}'| = k' = \frac{\omega}{c}\sqrt{\mu ' \epsilon '}\]

Where the speed of the wave in the medium is \(v=c/\sqrt{\mu \epsilon}\) and the quantity \(n=c/v=\sqrt{\mu \epsilon}\) 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:

\[(\vec{k} \cdot \vec{x})_{surf} = (\vec{k}' \cdot \vec{x})_{surf} = (\vec{k}'' \cdot \vec{x})_{surf}\]

\[k \sin i = k' \sin r = k'' \sin r'\]

where \(i, r, r'\) 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:

\[i = r'\]

\[\frac{\sin i}{\sin r} = \sqrt{\frac{\mu ' \epsilon '}{\mu \epsilon}} = \frac{n'}{n}\]

The dynamic properties of the wave at the boundary are derived from Maxwell’s equations which impose the continuity of the normal components of \(\vec{D}\) and \(\vec{B}\) and of the tangential components of \(\vec{E}\) and \(\vec{H}\) 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 (polarization vector) perpendicular to the plane defined by the photon direction and the normal to the boundary:

   \[\begin{split}\frac{E_{0}'}{E_{0}} &= \frac{2n\cos i}{n \cos i = \frac{\mu}{\mu '} n' \cos r} = \frac{2n \cos i}{n \cos i + n' \cos r} \\ \frac{E_{0}''}{E_{0}} &= \frac{n \cos i - \frac{\mu}{\mu '} n' \cos r}{n \cos i + \frac{\mu}{\mu '}n' \cos r} = \frac{n \cos i - n' \cos r}{n \cos i + n' \cos r}\end{split}\]

   where we suppose, as it is legitimate for visible or near-visible light, that \(\mu/\mu ' \approx 1\);

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

   \[\begin{split}\frac{E_{0}'}{E_{0}} &= \frac{2n \cos i}{\frac{\mu}{\mu '}n' \cos i + n \cos r} = \frac{2n \cos i}{n' \cos i + n \cos r} \\ \frac{E_{0}''}{E_{0}} &= \frac{\frac{\mu}{\mu '}n' \cos i - n \cos r} {\frac{\mu}{\mu '}n' \cos i + n \cos r} = \frac{n' \cos i - n \cos r}{n' \cos i + n \cos r}\end{split}\]

   with the same approximation as above.

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

\[\frac{E_{0}'}{E_{0}} = \frac{2n}{n'+n} \; , \qquad \frac{E_{0}''}{E_{0}} = \frac{n'-n}{n'+n}\]

where the sign convention for the parallel field has been adopted. This means that if \(n'>n\) there is a phase inversion for the reflected wave.

Any incoming wave can be separated into one piece polarized parallel to the plane and one polarized perpendicular, and the two components treated accordingly.

To maintain the particle description of the photon, the probability to have a refracted or reflected 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:

\[\vec{S} = \frac{1}{2}\frac{c}{4\pi}\sqrt{\mu \epsilon} \vec{E} \times \vec{H} = \frac{c}{8\pi}\sqrt{\frac{\epsilon}{\mu}} E_{0}^{2}\hat{k}\]

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

\[\vec{S} \cdot \vec{u} = \vec{S}' \cdot \vec{u} - \vec{S}'' \cdot \vec{u}\]

\[S \cos i = S' \cos r + S'' \cos i\]

\[\frac{c}{8\pi}\frac{1}{\mu}nE_{0}^{2}\cos i = \frac{c}{8\pi}\frac{1}{\mu '}n'E_{0}'^{2}\cos r + \frac{c}{8\pi}\frac{1}{\mu}nE_{0}''^{2}\cos i\]

If we set again \(\mu /\mu ' \approx 1\), then the transmission probability for the photon will be:

\[T = \left( \frac{E_{0}'}{E_{0}} \right)^{2} \frac{n' \cos r}{n \cos i}\]

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

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

\[\vec{q} = \vec{k} \times \vec{u}\]

\[\vec{e}_{\perp} = (\frac{\vec{e} \cdot \vec{q}}{|\vec{q}|}) \frac{\vec{q}}{|\vec{q}|}\]

\[\vec{e}_{\parallel} = \vec{e} - \vec{e}_{\perp}\]

\[e_{\parallel}' = e_{\parallel} \frac{2n \cos i}{n'\cos i + n \cos r}\]

\[e_{\perp|}' = e_{\perp} \frac{2n \cos i}{n \cos i + n' \cos r}\]

\[e_{\parallel}'' = \frac{n'}{n}e_{\parallel}' - e_{\parallel}\]

\[e_{\perp}'' = e_{\perp}' - e_{\perp}\]

After transmission or reflection of the photon, the polarization vector is re-normalized to 1. In the case where \(\sin r = n \sin i/n' > 1\) then there cannot be a refracted wave, and in this case we have a total internal reflection according to the following formulas:

\[\vec{k}'' = \vec{k} - 2(\vec{k} \cdot \vec{u})\vec{u}\]

\[\vec{e}'' = -\vec{e} + 2(\vec{e} \cdot \vec{u})\vec{u}\]

Case dielectric \(\rightarrow\) 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.

Mie Scattering in Henyey-Greenstein Approximation

(Author: X. Qian, 2010-07-04)

Mie Scattering (or Mie solution) is an analytical solution of Maxwell’s equations for the scattering of optical photon by spherical particles. The general introduction of Mie scattering can be found in Ref. [wik17]. The analytical express of Mie Scattering are very complicated since they are a series sum of Bessel functions [Fit14]. Therefore, the exact expression of Mie scattering is not suitable to be included in the Monte Carlo simulation.

One common approximation made is called “Henyey-Greenstein” [ZS10]. It has been used by Vlasios Vasileiou in Geant4 simulation of Milagro experiment [Col07]. In the HG approximation,

\[\frac{d\sigma}{d\Omega} \sim \frac{1-g^2}{(1+g^2-2g\cos(\theta))^{3/2}}\]

where

\[d\Omega = d\cos(\theta) d\phi\]

and \(g = \langle \cos(\theta) \rangle\) can be viewed as a free constant labeling the angular distribution.

Therefore, the normalized density function of HG approximation can be expressed as:

\[P(\cos(\theta_0)) = \frac{\int_{-1}^{\cos(\theta_0)} \frac{d\sigma}{d\Omega} d\cos(\theta) }{\int_{-1}^{1} \frac{d\sigma}{d\Omega} d\cos(\theta)} = \frac{1-g^2}{2g} \left( \frac{1}{(1+g^2-2g\cos(\theta_0))}-\frac{1}{1+g} \right)\]

Therefore,

\[\cos(\theta) = \frac{1}{2g} \cdot \left(1+g^2 - (\frac{1-g^2}{1-g+2g \cdot p})^2 \right) = 2p \frac{(1+g)^2(1-g+gp)}{(1-g+2gp)^2} -1\]

where \(p\) is a uniform random number between 0 and 1.

Similarly, the backward angle where \(\theta_b = \pi - \theta_f\) can also be simulated by replacing \(\theta_f\) to \(\theta_b\). Therefore the final differential cross section can be viewed as:

\[\frac{d\sigma}{d\Omega} = r \frac{d\sigma}{d\Omega} (\theta_f, g_f) + (1-r) \frac{d\sigma}{d\Omega} (\theta_b, g_b)\]

This is the exact approach used in Ref. [Vas]. Here \(r\) is the ratio factor between the forward angle and backward angle.

In implementing the above MC method into Geant4, the treatment of polarization and momentum are similar to that of Rayleigh scattering. We require the final polarization direction to be perpendicular to the momentum direction. We also require the final momentum, initial polarization and final polarization to be in the same plane.

Bibliography

Col07

Milagro Collaboration. Tev all-sky gamma ray observatory. http://hawc.umd.edu/, 2007. [Online; accessed 15-July-2017].

Fit14

Richard Fitzpatrick. Mie scattering lectures. http://farside.ph.utexas.edu/teaching/jk1/lectures/node103.html, 2014. [Online; accessed 26-october-2017].

Vas

Vlasios Vasileiou. Private communication.

wik17

wikipedia. Mie scattering. http://en.wikipedia.org/wiki/Mie_scattering, 2017. [Online; accessed 26-october-2017].

ZS10

Guangyuan Zhao and Xianming Sun. Error analysis of using henyey-greensterin [sic] in monte carlo radiative transfer simulations. In Progress In Electromagnetics Research Symposium Proceedings, 1449–1452. Xi'an, China, March 22nd 2010. web-site: https://www.piers.org/proceedings/home.html. URL: https://www.piers.org/pierspublications/PIERS2010XianProceedings04.pdf.