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Home All issues Volume 519 (September 2010) A&A, 519 (2010) A39 Online Material
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Issue
A&A
Volume 519, September 2010
Article Number A39
Number of page(s) 24
Section Interstellar and circumstellar matter
DOI https://doi.org/10.1051/0004-6361/200913906
Published online 09 September 2010

Online Material

Appendix A: Emission by an electric dipole in an infinite medium

We first study the electric field $\vec{E}$ created by a single electric dipole, with dipole moment $\vec{p}$, in an infinite homogenous medium, and its decomposition into s- and p-polarized components. The configuration is described by Fig. A.1 and many notations used are listed in Table A.1.
\begin{figure} \par\includegraphics[width=12cm,clip]{13906fgA1} \end{figure} Figure A.1:

Coordinate system and angles used to calculate the decomposition of the electric field $\vec{E}$, created by an electrical dipole $\vec{p}$, into s- and p-polarized components.

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Table A.1:   Definition of axis, angles, and other notations.

The refractive index $\tilde{n}_{j}$ is complex if the medium j is absorbing. In the case of considering a non-absorbing medium j, the imaginary part of the index, $\kappa_{j}$, is zero. The refractive index $\tilde{n}_{j}$ is given by the relation A.1, and the wave vector $\tilde{k}_{j}$ (in the direction of the observator) is linked to the complex refractive index by Eq. (A.2):

                                                 $\displaystyle \tilde{n}_{j} = n_{j}\ +\ {\rm i} \ \kappa_{j}$ (A.1)
    $\displaystyle \tilde{k}_{j} = \tilde{n}_{j} \cdot \frac{2\pi}{\lambda} \qquad \textrm{where $\lambda$\space is the wave length in vacuum.}$ (A.2)

A.1 Polarization of the electric field

The electric field $\vec{E}$ is perpendicular to the direction of observation $\vec{k}$ and can be decomposed into two orthogonal components, i.e., two polarizations: the s-polarized one is perpendicular to the observation plane and the (p)-polarized one is in this plane:

                                $\displaystyle \vec{E}^{\rm (s)} = E^{\rm (s)}\ \vec{u_{y}}$ (A.3)
    $\displaystyle \vec{E}^{\rm (p)} = E^{\rm (p)} \left(-\cos \alpha\ \vec{u_{x}} + \sin \alpha\ \vec{u_{z}}\right)$ (A.4)

                                           $\displaystyle E^{\rm (s)} = E\ \sin \psi \qquad \textrm{and} \qquad E^{\rm (p)} = E\ \cos \psi$ (A.5)
    $\displaystyle \vec{E} = E\ \left\vert \begin{array}{ll}-\cos \psi \cdot \cos \a... ...i & \vec{u_{y}}\\ \cos \psi \cdot \sin \alpha & \vec{u_{z}} \end{array}\right.$ (A.6)
    $\displaystyle E^{\rm (s)} = E_{y}\ \qquad\ \textrm{and} \qquad E^{\rm (p)} = E_{z}\ \sin \alpha - E_{x}\ \cos \alpha$ (A.7)

                                            $\displaystyle \vec{u_{p}} = \sin \beta\ \vec{u_{E}} + \cos \beta\ \vec{u_{k}}$  
    $\displaystyle \vec{E} = E\ \left[\frac{1}{\sin \beta}\ \vec{u_{p}} - \frac{\cos \beta}{\sin \beta}\ \vec{u_{k}}\right]$  
    $\displaystyle \vec{E} = \frac{E}{\sin \beta}\ \left( \left\vert \begin{array}{l... ...{x}}\\ 0 & \vec{u_{y}}\\ \cos \alpha & \vec{u_{z}} \end{array}\right. \right)$  
    $\displaystyle \vec{E} = \frac{E}{\sin \beta}\ \left\vert \begin{array}{ll}\sin ... ...\\ \cos \theta - \cos \alpha \cdot \cos \beta& \vec{u_{z}}. \end{array}\right.$ (A.8)

With Eqs. (A.7) and (A.8), we find
                                 $\displaystyle E^{\rm (s)}$ = $\displaystyle \frac{E}{\sin \beta}\ \sin \theta\ \sin \phi$ (A.9)
$\displaystyle E^{\rm (p)}$ = $\displaystyle \frac{E}{\sin \beta}\ (\cos \theta\ \sin \alpha - \sin \theta \cos \phi \cos \alpha ).$ (A.10)

A.2 Radiant intensity in an infinite medium

The next equations show how to obtain the intensity from the electric field amplitude. We can linked the electric field to the irradiance that is the incident power of electromagnetic radiation at a surface, per unit area.

$\displaystyle \textrm{Specific\footnotemark\ irradiance (W $\cdot$\space m$^{-2}$\space $\cdot$ ~nm$^{-1}$ ): }$   $\displaystyle {\rm Irr} = \langle \vec{\Pi}\rangle = \frac{1}{2} Re(\vec{E} \times \vec{H^{\ast}})$  
    $\displaystyle {\rm Irr} = \frac{n \epsilon_{0} c}{2}\ \Vert E\Vert^2,$  

where $\epsilon_{0}$ is the vacuum permittivity, c the speed of light in vacuum, and n the refractive index of the medium.

Radiant intensity (incident power per unit solid angle) I and irradiance created by the electric dipole $\vec{p}$ are normalized as

$\displaystyle \int\!\!\!\int_{4 \pi} \left(I_{\vec p}^{\rm (s)} + I_{\vec p}^{\rm (p)}\right) \cdot {\rm d}\Omega$ = $\displaystyle \int\!\!\!\int_{4 \pi r^2} \left({\rm Irr}_{\vec p}^{\rm (s)} + {\rm Irr}_{\vec p}^{\rm (p)}\right) \cdot {\rm d}S = P(\lambda)$ (A.11)
$\displaystyle I_{\vec p}^{\rm (s,p)} \cdot {\rm d}\Omega$ = $\displaystyle {\rm Irr}_{\vec p}^{\rm (s,p)} \cdot {\rm d}S\ =\ {\rm Irr}_{\vec p}^{\rm (s,p)} \cdot r^2 \ {\rm d}\Omega,$  

where r is the radius of the sphere where the irradiance is calculated, and $P(\lambda)$ is the total[*] power emitted at the wavelength $\lambda $ by a single dipole $\vec{p}$, in all directions of the space.
$\displaystyle \textrm{Specific radiant intensity (W $\cdot$\space sr$^{-1}$\space $\cdot$ ~nm$^{-1}$ ): } I_{\vec p}^{\rm (s,p)}$ = $\displaystyle {\rm Irr}_{\vec p}^{\rm (s,p)} \cdot r^2\ =\ r^2 \cdot \frac{n \epsilon_{0} c}{2}\ \Vert E^{\rm (s,p)}\Vert^2$  
$\displaystyle I_{\vec p}^{\rm (s,p)}$ = $\displaystyle F \cdot \frac{\sin^2 \beta}{E^2} \cdot \Vert E^{\rm (s,p)}\Vert^2$ (A.12)

               $\displaystyle I_{\vec p}^{\rm (s)} = F \sin^{2} \theta\ \sin^{2} \phi \qquad$ and $\displaystyle I_{\vec p}^{\rm (p)} = F (\cos \theta\ \sin \alpha - \sin \theta \cos \phi \cos \alpha)^{2}.$  

The normalization Eq. (A.11) let us obtain factor F:

\begin{displaymath}F \int_{0}^{\pi} \!\!\! \int_{0}^{2\pi}\left( \sin^{2} \theta... ...ht) \cdot \sin \alpha\ {\rm d}\alpha\ {\rm d}\phi = P(\lambda) \end{displaymath}

                                             $\displaystyle \frac{P(\lambda)}{F} = \sin^{2}\theta \int_{0}^{\pi}\!\!\! \sin \... ...0}^{\pi}\!\!\! \sin^{3} \alpha\ {\rm d}\alpha \int_{0}^{2\pi}\!\!\! {\rm d}\phi$  
    $\displaystyle \qquad\quad +\ \sin^{2}\theta \int_{0}^{\pi}\!\!\!\!\!\! \cos^{2}... ...\ \cos \alpha\ {\rm d}\alpha \int_{0}^{2\pi}\!\!\!\!\!\! \cos \phi\ {\rm d}\phi$  
    $\displaystyle \frac{P(\lambda)}{F} = \sin^{2}\theta \bigg[-\cos \alpha \bigg]_{...  
    $\displaystyle \qquad\quad +\ \sin^{2}\theta \left[-\frac{\cos^{3} \alpha}{3}\ri...  
    $\displaystyle \frac{P(\lambda)}{F} = 2\pi \sin^{2}\theta \quad +\quad \frac{8\pi}{3} \cos^{2}\theta\quad +\quad \frac{2\pi}{3} \sin^{2}\theta \quad +\quad 0$  
    $\displaystyle F = \frac{3}{8\pi}\ P(\lambda)\ =\ r^2 \cdot \frac{n \epsilon_{0} c}{2}\ \frac{E^2}{\sin^2 \beta} \cdot$  

The radiant intensity emitted by one electric dipole $\vec{p}$, in the direction $\vec{k}$ given by the angles ($\alpha$,$\phi$), is given by the next equation (whatever the distance from source dipole since the medium is infinite, homogenous and not absorbing):
          $\displaystyle I_{\vec p}^{\rm (s)}(\lambda) = \frac{3}{8\pi}\ P(\lambda)\ \sin^{2} \theta\ \sin^{2} \phi \quad$   $\displaystyle I_{\vec p}^{\rm (p)}(\lambda) = \frac{3}{8\pi}\ P(\lambda)\ (\cos \theta\ \sin \alpha - \sin \theta \cos \phi \cos \alpha)^{2}.$ (A.13)

Consider now a luminescent layer of thickness H, and thus an ensemble of incoherently radiating electric dipoles having equal dipole moment $\vec{p}$ but randomly oriented. To take this random orientation of the dipoles into account, we have to average $I_{\vec p}^{\rm (s,p)}$ over the direction of $\vec{p}$, i.e., $\theta$ and $\phi$. The intensity obtained is named $I_{\langle \vec p \rangle}^{\rm (s,p)}$:

    $\displaystyle I_{\langle \vec p \rangle}^{\rm (s,p)}(\alpha, \lambda) = \frac{1... ...^{2\pi} I_{\vec p}^{\rm (s,p)}(\lambda) \sin \theta\ {\rm d}\theta\ {\rm d}\phi$  
    $\displaystyle I_{\langle \vec p \rangle}^{\rm (s)}(\alpha, \lambda) = I_{\langle \vec p \rangle}^{\rm (p)}(\alpha, \lambda) = \frac{1}{8\pi}\ P(\lambda).$  

The dipoles are located in the layer of thickness H; the total intensity emitted from this luminescent layer is thus obtained by integrating the product of $I_{\langle \vec p \rangle}^{\rm (s,p)}$ with Lz(h), the linear density of the radiating dipoles (m-1), over the layer depth h. As the layer is formed by a homogenous medium, Lz(h) is constant with h:

                                        $\displaystyle I^{\rm (s,p)}(\alpha, \lambda) = \int_{0}^{H} I_{\langle \vec p \rangle}^{\rm (s,p)}(\alpha, \lambda) \cdot L_z(h)\ {\rm d}h$  
    $\displaystyle I^{\rm (s)}(\alpha, \lambda) = I^{\rm (p)}(\alpha, \lambda) = \frac{L_z}{8\pi}\int_{0}^{H} P(\lambda, h)\ {\rm d}h$  
    $\displaystyle I(\alpha, \lambda) = I^{\rm (s)}(\alpha, \lambda) + I^{\rm (p)}(\alpha, \lambda) = \frac{L_z}{4\pi}\int_{0}^{H} P(\lambda, h)\ {\rm d}h.$  

Let us consider now an absorbing medium with an index $\tilde n$ that is thus complex. The following equations show the effect of propagation of an electromagnetic wave (wave vector $\tilde{k}$) on a length r in this medium of complex refractive index $\tilde n$
  • on the wave amplitude E:
                               E' = $\displaystyle E \cdot \exp \left({\rm i} \vec{\tilde{k}}\cdot \vec{r}\right) = E \cdot \exp \left({\rm i} \frac{2\pi}{\lambda}\ \tilde{n}\ r \right)$  
    E' = $\displaystyle E \cdot \underbrace{\exp \left({\rm i} \frac{2\pi}{\lambda}\ n\ r... ...rac{2\pi}{\lambda}\ \kappa\ r \right)}_{{\rm absorption}\ =\ {\rm e}^{-\tau/2}}$  

  • on the wave intensity I:
                                         I' $\textstyle \propto$ $\displaystyle \Vert E'\Vert^2$  
    I' = $\displaystyle I \cdot \exp \left(- \frac{4\pi}{\lambda}\ \kappa \cdot r \right)... ...p \left(- \alpha \cdot r \right)\quad = \quad I \cdot \exp \left(- \tau \right)$  

where $\alpha$ is the absorption coefficient of the medium (in cm-1) and $\tau$ is the optical thickness.

Appendix B: Effects of the interfaces of the film

In this appendix, we introduce the effects of the interfaces of the film and calculate the emission amplitude in a medium 1 by summing all the transmitted components (of each polarization), taking the phase shift and absorption due to propagation in each medium into account. As a result, the radiation pattern are obtained: it is the angular distribution of the light emitted by the film into the half-space 1.

Until now, we have considered an infinite medium. We now consider three different homogenous media (medium 0 is a film between semi-infinite media 1 and 2) and their interfaces. The emitting dipole $\vec{p}$ is now located in the film 0. Our aim is to determine the radiation transmitted in medium 1 under the angle of observation $\alpha _{1}$.

We separate the light transmitted in medium 1 into two beams: the direct beam D is the one emitted directly in the direction of medium 1 (with an emission angle $\alpha_{0}$ given by the angle of observation $\alpha _{1}$ and the Snell-Descartes relation of refraction). The reflected beam R is the beam that is first directed toward medium 2 and reflected off the interface 0/2 before being transmitted in medium 1. Each of these beams is reflected multiple times between each interface of the film 0. In this paragraph, we see that interference effects appear because of these multiple reflections and also because of the sum of the two different beams, D and R. This latter case of interference is known as the wide-angle effect.

\begin{figure} \par\includegraphics[width=12cm,clip]{13906fgB1} \end{figure} Figure B.1:

Schematic diagram of the emission geometry for a dipole embedded in a thin film (medium 0), at a distance h from the surface, for an emission angle $\alpha _{1}$ in medium 1. This emission is the combinaison of light from the direct beam D (red solid line) and from the reflected beam R (green dashed line).

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We now consider that the medium 1 is not absorbing, this means that

\begin{displaymath}\kappa_{1} = 0 \qquad \qquad \tilde{n}_{1} = n_{1} \end{displaymath}

B.1 The direct beam D

We first consider the the direct beam D, with its multiple reflection back and forth in layer 0, transmitted in medium 1 as shown in Fig. B.2:

\begin{displaymath}E_{D}^{\rm (s,p)} = E^{\rm (s,p)}(\alpha = \alpha_{0}) \nonumber \end{displaymath}

                                     $\displaystyle E_{1,D}^{\rm (s,p)}$ = $\displaystyle E_{D}^{\rm (s,p)} \cdot \exp \left({\rm i} \ \tilde{k}_{0} \frac{h}{\cos \alpha_{0}}\right) \cdot t_{01}^{\rm (s,p)}$  
    $\displaystyle \times \left[1 + \left(r_{01}^{\rm (s,p)} \ r_{02}^{\rm (s,p)} \ ... ...n \alpha_{1} \big)}_{\rm external\ phase\ shift} \right) + (...)^2 + ...\right]$  

                                     $\displaystyle E_{1,D}^{\rm (s,p)}$ = $\displaystyle E_{D}^{\rm (s,p)} \cdot t_{01}^{\rm (s,p)} \cdot \exp \big({\rm i} \delta_{0}(h)\big) \cdot \exp \big(- \tau_{0}(h)/2 \big)$ (B.1)
    $\displaystyle \times \left[ 1 + \sum_{l=1}^\infty \left( r_{01}^{\rm (s,p)}\ r_... ...\tau_{0}(H) \big)\ \exp \big({\rm i}\ \delta_{1}(H) \big)\ \right)^{l} \right].$  

The following notations are used to lighten expressions.

Remark: $\tau_{0}$ is defined with a factor $\frac{1}{2}$ to obtain a power attenuation of ${\rm e}^{-\tau_{0}}$.

                                              absorption:  
    $\displaystyle \frac{\tau_{0}(h)}{2} = \frac{2\pi}{\lambda} \kappa_{0} \frac{h}{\cos \alpha_{0}}$  
    phase shift:  
    $\displaystyle \delta_{0}(h) = \frac{2\pi}{\lambda} n_{0} \frac{h}{\cos \alpha_{0}}$  
    $\displaystyle \delta_{1}(h) = - \frac{2\pi}{\lambda} n_{1} 2 h \tan \alpha_{0} ... ...quad - \frac{4\pi}{\lambda} n_{0} h \frac{\sin^{2} \alpha_{0}}{\cos \alpha_{0}}$  
    $\displaystyle \delta(h) = 2 \delta_{0}(h) + \delta_{1}(h) \quad=\quad \frac{4\pi}{\lambda} n_{0} h \cos \alpha_{0}$  
    $\displaystyle r_{ij}^{\rm (s,p)} = \rho_{ij}^{\rm (s,p)}\ \exp \left( {\rm i}\ \delta_{ij}^{\rm (s,p)}\right) \qquad \textrm{: Fresnel reflection coefficients}$  
    $\displaystyle \Delta^{\rm (s,p)}(h) = \delta(h) + \delta_{01}^{\rm (s,p)}+\delt... ...delta_{0}(h) + \delta_{1}(h) + \delta_{01}^{\rm (s,p)}+\delta_{02}^{\rm (s,p)},$  

where $\Delta(H)$ and $\tau(H)$ are the phase shift and the absorption due to only one wave back and forth in layer 0 before being transmitted in medium 1.

Hence, one derives

    $\displaystyle E_{1,D}^{\rm (s,p)} = E_{D}^{\rm (s,p)} \cdot t_{01}^{\rm (s,p)} ... (B.2)
    $\displaystyle \textrm{with: }\quad m^{\rm (s,p)} = \frac{1}{\left[ 1\ -\ \rho_{...  

\begin{figure} \par\includegraphics[width=12cm,clip]{13906fgB2} \end{figure} Figure B.2:

Schematic diagram of the emission geometry for the direct beam D.

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B.2 The reflected beam R

\begin{figure} \par\includegraphics[width=12cm,clip]{13906fgB3} \vspace*{7mm} \end{figure} Figure B.3:

Schematic diagram of the emission geometry of the reflected beam R.

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For the reflected beam (see Fig. B.3), the factor due to the multiple reflections in the film stay the same as for the direct beam. We must keep in mind that the phase reference in medium 1 is defined by the direct beam:

\begin{displaymath}E_{R}^{\rm (s,p)} = E^{\rm (s,p)}(\alpha = \pi - \alpha_{0}) \nonumber \end{displaymath}

                                        $\displaystyle E_{1,R}^{\rm (s,p)}$ = $\displaystyle E_{R}^{\rm (s,p)}\! \exp \left(\! {\rm i} \tilde{k}_{0} \frac{H-h... ...\big(\! -\! {\rm i} \tilde{k}_{1} 2 (H-h) \tan \alpha_{0} \sin \alpha_{1} \big)$  
    $\displaystyle \times \left[1 + \left(r_{01}^{\rm (s,p)} \ r_{02}^{\rm (s,p)} \ ... ...{k}_{1} 2H \tan \alpha_{0} \sin \alpha_{1} \big) \right) + (...)^2 + ...\right]$  
$\displaystyle E_{1,R}^{\rm (s,p)}$ = $\displaystyle E_{R}^{\rm (s,p)} \cdot t_{01}^{\rm (s,p)} r_{02}^{\rm (s,p)} \cd... ... \big[{\rm i}\ \big(\delta_{0}(H-h) + \delta_{0}(H) + \delta_{1}(H-h)\big)\big]$  
    $\displaystyle \times \exp \big[- \frac{1}{2} \big(\tau_{0}(H-h) + \tau_{0}(H) \big)\big] \cdot m^{\rm (s,p)}$  
$\displaystyle E_{1,R}^{\rm (s,p)}$ = $\displaystyle E_{R}^{\rm (s,p)} \cdot t_{01}^{\rm (s,p)} r_{02}^{\rm (s,p)} \cd... ...ig)} \cdot {\rm e}^{- \frac{1}{2} \big(\tau_{0}(2H-h)\big)}\cdot m^{\rm (s,p)}.$ (B.3)

B.3 Combination of the D and R beams

We calculated the expression of the electric field amplitude transmitted in medium 1 of the direct beam $E_{1,D}^{\rm (s,p)}$ and of the reflected beam $E_{1,R}^{\rm (s,p)}$, for each polarization, (s) and (p). These amplitudes are exprimed as a function of $E_{D}^{\rm (s,p)}$ and $E_{R}^{\rm (s,p)}$, which are the electric field amplitude emitted by an electric dipole in the direction $\alpha_{D}= \alpha_{0} $ and $\alpha_{R} = \pi - \alpha_{0}$, respectively:

\begin{displaymath}E_{D}^{\rm (s,p)} = E^{\rm (s,p)}(\alpha = \alpha_{0}) \qquad... ... (s,p)} = E^{\rm (s,p)}(\alpha = \pi - \alpha_{0}). \nonumber \end{displaymath}

The Eqs. (A.9) and (A.10) give the influence of the angle $\alpha$ on $E^{\rm (s,p)}$. One obtain

                                                  $\displaystyle E^{\rm (s)}_{D} = E^{\rm (s)}_{R} \ =\ E^{\rm (s)}\ =\ \frac{E}{\sin \beta}\ \sin \theta\ \sin \phi$ (B.4)
    $\displaystyle E^{\rm (p)}_{D} = \frac{E}{\sin \beta}\ (\cos \theta\ \sin \alpha_{0} - \sin \theta \cos \phi \cos \alpha_{0} )$ (B.5)
    $\displaystyle E^{\rm (p)}_{R} = \frac{E}{\sin \beta}\ (\cos \theta\ \sin (\pi -\alpha_{0}) - \sin \theta \cos \phi \cos (\pi - \alpha_{0}) )$  
    $\displaystyle E^{\rm (p)}_{R} = \frac{E}{\sin \beta}\ (\cos \theta\ \sin \alpha_{0} + \sin \theta \cos \phi \cos \alpha_{0} )$ (B.6)
    $\displaystyle \Rightarrow E^{\rm (p)}_{R} \neq E^{\rm (p)}_{D}.$  

To combine the electric fields in medium 1 due to the direct and reflected beams $\vec{E}_{1,D}^{\rm (s,p)}$ and $\vec{E}_{1,R}^{\rm (s,p)}$, we have to pay attention to the direction of these vectors and not only to their amplitude,
$\displaystyle \vec{E}_{1}^{\rm (s,p)} = \vec{E}_{1,D}^{\rm (s,p)} + \vec{E}_{1,R}^{\rm (s,p)}.$    

Let us consider the direction of the vectors $\vec{E}^{\rm (s)}_{D}$, $\vec{E}^{\rm (p)}_{D}$, $\vec{E}^{\rm (s)}_{R}$ and $\vec{E}^{\rm (p)}_{R}$ in medium 0, before any reflection or transmission at the film interfaces. The Eqs. (A.3) and (A.4) show that

\begin{displaymath}\vec{E}^{\rm (s)}_{D} = \vec{E}^{\rm (s)}_{R} \qquad {\rm and... ...^{\rm (p)}_{{D},z} &=& E^{\rm (p)}_{{R},z}. \end{array}\right. \end{displaymath}

The reflection at the interface 0/2 of the reflected beam changes the sign of $E^{\rm (p)}_{{R},z}$, so the direction of $\vec{E}^{\rm (p)}_{D}$ and $\vec{E}^{\rm (p)}_{R}$ become opposite. As one can see in Figs. B.1-B.3, the direct and reflected beam transmitted in medium 1, $\vec{E}_{1,D}$ and $\vec{E}_{1,R}$, respectively, are
  • in the same direction for the s-polarization
  • in opposite direction for the p-polarization.
As a result, the combination of the direct and reflected beams amplitude is done as
$\displaystyle E_{1}^{\rm (s)} = E_{1,D}^{\rm (s)} + E_{1,R}^{\rm (s)} \qquad$ and: $\displaystyle \qquad E_{1}^{\rm (p)} = E_{1,D}^{\rm (p)} - E_{1,R}^{\rm (p)}$  

                                        $\displaystyle E_{1}^{\rm (s,p)}$ = $\displaystyle t_{01}^{\rm (s,p)} \cdot m^{\rm (s,p)} \left[ E_{D}^{\rm (s,p)} \... ...xp \big({\rm i} \delta_{0}(h)\big) \cdot \exp \big(- \tau_{0}(h)/2\big) \right.$  
    $\displaystyle \left. \pm_{(p)}^{\rm (s)} \ E_{R}^{\rm (s,p)} \cdot r_{02}^{\rm ... ...ta_{1}(H-h)\big)\big] \cdot \exp \big[- \frac{1}{2} \tau_{0}(2H-h)\big] \right]$  
$\displaystyle E_{1}^{\rm (s,p)}$ = $\displaystyle t_{01}^{\rm (s,p)} m^{\rm (s,p)} \exp \big({\rm i} \delta_{0}(h)\... ...m i} \delta_{w}^{\rm (s,p)}\big) \exp \big(\!-\!\frac{\tau_{w}}{2}\big) \right]$ (B.7)

where $\qquad \delta_{w}^{\rm (s,p)} = 2\ \delta_{0}(H-h) + \delta_{1}(H-h) + \delta_{02}^{\rm (s,p)} \qquad$ and $ \qquad \tau_{w} = 2\ \tau_{0}(H-h)$.

B.3.1 s-polarized

                                                $\displaystyle E_{1}^{\rm (s)} = E^{\rm (s)} \cdot \exp \big({\rm i} \delta_{0}(... ..._{0}(h)}{2}\big) \cdot t_{01}^{\rm (s)} \cdot m^{\rm (s)} \cdot w_{+}^{\rm (s)}$  
    $\displaystyle E_{1}^{\rm (s)} = \frac{E}{\sin \beta}\ \sin \theta\ \sin \phi \c... ..._{0}(h)}{2}\big) \cdot t_{01}^{\rm (s)} \cdot m^{\rm (s)} \cdot w_{+}^{\rm (s)}$ (B.8)
    $\displaystyle \textrm{where: }\quad w_{+}^{\rm (s)} = \left[ 1 + \rho_{02}^{\rm... ...i}\ \delta_{w}^{\rm (s)}\big) \cdot \exp \big(- \frac{\tau_{w}}{2}\big) \right]$  

B.3.2 p-polarized

    $\displaystyle E_{1}^{\rm (p)} = \exp \big({\rm i} \delta_{0}(h)\big) \exp \big(... ...}\ \delta_{w}^{\rm (p)}\big) \cdot \exp \big(- \frac{\tau_{w}}{2} \big) \right]$  
    $\displaystyle E_{1}^{\rm (p)} = \frac{E}{\sin \beta}\ {\rm e}^{{\rm i} \delta_{... ...w_{-}^{\rm (p)} - \sin \theta \cos \phi \cos \alpha_{0}\ w_{+}^{\rm (p)}\right]$ (B.9)

$w_{+}^{\rm (p)}= \left[ 1 + \rho_{02}^{\rm (p)} \cdot {\rm e}^{{\rm i}\ \delta_{w}^{\rm (p)}} \cdot {\rm e}^{- \frac{\tau_{w}}{2}} \right] \qquad $ and $\qquad w_{-}^{\rm (p)}= \left[ 1 - \rho_{02}^{\rm (p)} \cdot {\rm e}^{{\rm i}\ \delta_{w}^{\rm (p)}} \cdot {\rm e}^{- \frac{\tau_{w}}{2}} \right]$.

B.4 Radiation pattern

The radiation pattern is the angular distribution of the light emitted by the source $\vec{p}$ located in layer 0 into the half-space 1, expressed by $I_{1, h, \vec p(\theta, \phi)}^{\rm (s,p)}(\alpha_0, \phi, \lambda)$ the intensity ( $W\cdot sr^{-1} \cdot~nm^{-1}$) emitted in the direction given the angles $(\alpha_0, \phi)$ into the half-space 1.

                               $\displaystyle I_{1,h, \vec p(\theta, \phi)}^{\rm (s)}(\alpha_{0}, \phi, \lambda)$ = $\displaystyle \frac{3}{8\pi}\ P(\lambda, h)\ \sin^{2} \theta\ \sin^{2} \phi \cd... ...{+}^{\rm (s)} \right\Vert^{2} \cdot \frac{{\rm d}\Omega_{0}}{{\rm d}\Omega_{1}}$  
$\displaystyle I_{1,h, \vec p(\theta, \phi)}^{\rm (s)}(\alpha_{0}, \phi, \lambda)$ = $\displaystyle \frac{3}{8\pi}\ P(\lambda, h)\ \sin^{2} \theta\ \sin^{2} \phi \cd... ...Vert w_{+}^{\rm (s)} \Vert^{2}\cdot \frac{{\rm d}\Omega_{0}}{{\rm d}\Omega_{1}}$ (B.10)

                                  $\displaystyle I_{1,h, \vec p(\theta, \phi)}^{\rm (p)}(\alpha_{0}, \phi, \lambda)$ = $\displaystyle \frac{3}{8\pi} P(\lambda, h) {\rm e}^{\!-\! \tau_{0}(h)}T_{01}^{\... ...rm (p)} - \sin \theta \cos \phi \cos \alpha_{0}\ w_{+}^{\rm (p)}\right\Vert^{2}$  
$\displaystyle I_{1,h, \vec p(\theta, \phi)}^{\rm (p)}(\alpha_{0}, \phi, \lambda)$ = $\displaystyle \frac{3}{8\pi}\ P(\lambda, h)\ {\rm e}^{\!-\! \tau_{0}(h)}\ T_{01... ... (p)} \ \Vert m^{\rm (p)}\Vert^{2}\ \frac{{\rm d}\Omega_{0}}{{\rm d}\Omega_{1}}$ (B.11)
    $\displaystyle \times \left[\cos^{2} \theta\ \sin^{2} \alpha_{0}\ \left\Vert w_{... ...frac{1}{2} \sin{ 2 \theta} \cos \phi \sin{2 \alpha_{0}}\ w_{0}^{\rm (p)}\right]$  

where: $\quad w_{0}^{\rm (p)} \quad = \quad \Re {\rm e}(w_{-}^{\rm (p)})\ \Re {\rm e}(w... ...\rm (p)}) \quad = \quad 1 - \rho_{02}^{(p)\ 2} \cdot \exp \big(- \tau_{w} \big)$.

B.4.1 Solid angle modification at the interface

The ratio $\frac{{\rm d}\Omega_{0}}{{\rm d}\Omega_{1}}$ represents the modification of solid angle when the medium changes, $I_{i}\ {\rm d}\Omega_{i}$ is the radiated power in medium i into the solid angle ${\rm d}\Omega_{i}$. When the wave goes through the interface i/j, the solid angle is modified and the power conservation is given by Eq. (B.12) (when the transmission coefficient is 1):

$\displaystyle I_{i}\ {\rm d}\Omega_{i} = I_{j}\ {\rm d}\Omega_{j}.$    

This is the origin of the factor $\frac{{\rm d}\Omega_{0}}{{\rm d}\Omega_{1}}$ in the expression of the intensity $I_{1}^{\rm (s,p)}$,
$\displaystyle \frac{{\rm d}\Omega_{0}}{{\rm d}\Omega_{1}} = \frac{\sin \alpha_{... ...phi}\quad = \quad \frac{n_{1}^{2}\ \cos \alpha_{1}}{n_{0}^{2}\ \cos \alpha_{0}}$   (B.12)

using the derivative of the Snell-Descartes law.

B.4.2 Random orientation of the dipoles

To take a random orientation for the electric dipoles embedded in the film account, the intensity is averaged over $\theta$ and $\phi$. The geometry is thus no more dependent on the angle $\phi$; I1 is now only depending on the angle of observation $\alpha_{0}$. $P(\lambda, h)$, which is the total power emitted in all directions of the space, at the wavelength $\lambda $, by a single dipole $\vec{p}$ at the depth h, does not vary with the direction of the dipole.

The intensity ( $W\ sr^{-1}\~nm^{-1}\ m^{-1}$) emitted by a film layer of thickness dh, at the film depth h, is

                                             $\displaystyle I_{1,h}^{\rm (s,p)}(\alpha_0, \lambda) = \frac{1}{4 \pi} \int_{0}... ...)}^{\rm (s,p)}(\alpha_0, \phi, \lambda) \sin \theta\ {\rm d}\theta\ {\rm d}\phi$  
    $\displaystyle I_{1,h}^{\rm (s)}(\alpha_0, \lambda) = \frac{3}{32\pi^2}\ L_z(h)\... ...{\pi} \sin^{3} \theta\ {\rm d}\theta \int_{0}^{2\pi} \sin^{2} \phi\ {\rm d}\phi$  
    $\displaystyle I_{1,h}^{\rm (s)}(\alpha_0, \lambda) = \frac{3}{32\pi^2}\ L_z(h)\... ...ht]_{0}^{\pi} \ \left[\frac{\phi}{2} - \frac{\sin (2\phi)}{4}\right]_{0}^{2\pi}$  
    $\displaystyle I_{1,h}^{\rm (s)}(\alpha_0, \lambda) = \frac{1}{8 \pi}\ L_z(h)\ P... ...{2} \Vert w_{+}^{\rm (s)} \Vert^{2} \frac{{\rm d}\Omega_{0}}{{\rm d}\Omega_{1}}$ (B.13)

                                  $\displaystyle I_{1,h}^{\rm (p)}(\alpha_0, \lambda)$ = $\displaystyle \frac{3}{32\pi^2}\ L_z(h)\ P(\lambda, h)\ {\rm e}^{\!-\! \tau_{0}... ...} \theta \sin \theta\ {\rm d}\theta \int_{0}^{2\pi}\!\!\!\! {\rm d}\phi \right.$  
    $\displaystyle \left. + \cos^{2} \alpha_{0}\ \Vert w_{+}^{\rm (p)} \Vert^{2}\!\!... ...\theta\ {\rm d}\theta \int_{0}^{2\pi}\!\!\!\!\! \cos \phi \ {\rm d}\phi \right]$  
$\displaystyle I_{1,h}^{\rm (p)}(\alpha_0, \lambda)$ = $\displaystyle \frac{3}{32\pi^2}\ L_z(h)\ P(\lambda, h)\ {\rm e}^{\!-\! \tau_{0}... ...+ \cos^{2} \alpha_{0}\ \Vert w_{+}^{\rm (p)} \Vert^{2}\ \frac{4}{3}\ \pi\right]$  
$\displaystyle I_{1,h}^{\rm (p)}(\alpha_0, \lambda)$ = $\displaystyle \frac{1}{8 \pi}\ L_z(h)\ P(\lambda, h)\ {\rm e}^{\!-\! \tau_{0}(h... ...m (p)} \Vert^{2} + \cos^{2} \alpha_{0}\ \Vert w_{+}^{\rm (p)} \Vert^{2}\right].$ (B.14)

Finally, the specific radiant intensity in medium 1 and created by electric dipoles of random orientation at the film depth h is

                                                  $\displaystyle I_{1,h}^{\rm (s,p)}(\alpha_0, \lambda) = \frac{1}{8 \pi}\ {\rm e}... ...{n_{1}^{2}\ \cos \alpha_{1}}{n_{0}^{2}\ \cos \alpha_{0}}\ L_z(h)\ P(\lambda, h)$ (B.15)
    $\displaystyle \textrm{where }\quad M^{\rm (s,p)} = \Vert m^{\rm (s,p)}\Vert^{2}... ... \Delta^{\rm (s,p)}(H) \big)\cdot \exp \big( -\tau_{0}(H)\big) \right\Vert^{-2}$  
    $\displaystyle \qquad\qquad \qquad\; = 1\ +\ \rho_{01}^{\rm (s,p)\ 2} \rho_{02}^... ...\rm (s,p)} \cdot {\rm e}^{-\tau_{0}(H)} \cos \left(\Delta^{\rm (s,p)}(H)\right)$  
    $\displaystyle \textrm{and } \quad W^{\rm (s)} = \Vert w_{+}^{\rm (s)}\Vert^{2} ... ...ta_{w}^{\rm (s)}\big) \cdot \exp \big(- \frac{\tau_{w}}{2}\big) \right\Vert^{2}$  
    $\displaystyle \qquad\qquad\;\; = 1\ +\ \rho_{02}^{\rm (s,p)\ 2} \cdot {\rm e}^{... ...^{\rm (s,p)} \cdot {\rm e}^{-\tau_{w}/2} \cos \left(\delta_{w}^{\rm (s)}\right)$  
    $\displaystyle W^{\rm (p)} = \sin^{2} \alpha_{0}\ \Vert w_{-}^{\rm (p)} \Vert^{2} + \cos^{2} \alpha_{0}\ \Vert w_{+}^{\rm (p)} \Vert^{2}$  
    $\displaystyle \qquad = 1+\rho_{02}^{\rm (s,p) 2}\ {\rm e}^{- \tau_{w}} + 2 \rho... ...rm e}^{- \tau_{w}/2} \cos \left(\delta_{w}^{\rm (p)}\right) \cos (2 \alpha_{0})$  
    $\displaystyle \textrm{recalling that }\quad \Delta^{\rm (s,p)}(h)\ = \ 2\delta_... ...mbda} n_{0} h \cos \alpha_{0} + \delta_{01}^{\rm (s,p)}+\delta_{02}^{\rm (s,p)}$  
    $\displaystyle \delta_{w}^{\rm (s,p)} = 2\ \delta_{0}(H-h) + \delta_{1}(H-h) + \... ...=\quad \frac{4\pi}{\lambda} n_{0} (H-h) \cos \alpha_{0}+\delta_{02}^{\rm (s,p)}$  
    $\displaystyle \tau_{w} = 2\ \tau_{0}(H-h) \quad =\quad \frac{8\pi}{\lambda} \kappa_{0} \frac{H-h}{\cos \alpha_{0}},$  

where $\Vert t_{01}^{\rm (s,p)}\Vert^{2}$, $\rho_{ij}^{\rm (s,p)}$ and $\delta_{ij}^{\rm (s,p)}$ are expressed in the next section. The variable $M^{\rm (s,p)}$ gives the influence of the multiple-beam interference, and $W^{\rm (s,p)}$ is due to the wide angle interference.

B.4.3 Radiant intensity observed in medium 1

Until now, we have not being interested in the wavelength dependence of the emission. The observed intensity depends on $\lambda $ because each dipole radiates with a power $P(\lambda, h)$ that depends on the wavelength through an emission profile ${\rm Em}(\lambda)$ (in nm-1) proper to the material of the film. In the case of photoluminescence, the emission of each dipole will also result in an excitation absorbed by the film ( ${\rm Abs}(h)$ is the power absorbed in unit of length (W.m-1)) that can depend on the film depth h. The emission also results in the photoluminescence yield $\eta_{\rm E}$[*].

The specific power emitted by unit of film length (W nm-1 m-1) is

$\displaystyle L_z(h)\ P(\lambda,h) = {\rm Abs}(h) \ \eta_{\rm E} \ {\rm Em}(\lambda).$     (B.16)

The total power (W) between the wavelengths $\lambda_{\rm min}$ and $\lambda_{\rm max}$, emitted by the film in all directions in space is
                           $\displaystyle P_{\rm tot}$ = $\displaystyle \int_{\lambda_{\rm min}}^{\lambda_{\rm max}} \int_{0}^{H} L_z(h)\ P(\lambda,h)\ {\rm d}h\ {\rm d}\lambda$ (B.17)
$\displaystyle P_{\rm tot}$ = $\displaystyle \eta_{\rm E} \ \int_{0}^{H} {\rm Abs}(h)\ {\rm d}h \int_{\lambda_{\rm min}}^{\lambda_{\rm max}} {\rm Em}(\lambda)\ {\rm d}\lambda.$ (B.18)

Moreover, because the whole film is radiating, the electric dipoles $\vec{p}$ are located at all the film depths h between 0 and H, with a constant distribution on this interval. Thus, to obtain the specific radiant intensity that is observed in medium 1, we must integrate the expression given by the Eq. (B.15) on the interval [0,H]:

    $\displaystyle I_{1}^{\rm (s,p)}(\alpha_0, \lambda) = \int_{0}^{H} I_{1,h}^{\rm (s,p)}(\alpha_0, \lambda)\ {\rm d}h$ (B.19)
    $\displaystyle I_{1}(\alpha_0, \lambda) = I_{1}^{\rm (s)}(\alpha_0, \lambda) + I_{1}^{\rm (p)}(\alpha_0, \lambda).$ (B.20)

The total power between the wavelengths $\lambda_{\rm min}$ and $\lambda_{\rm max}$, received on a detector D that ``sees'' an solid angle $\Omega_{D}$ is
$\displaystyle P_{1, D}(\alpha_0) = \int_{\Omega_{D}} \int_{\lambda_{\rm min}}^{\lambda_{\rm max}} I_{1}(\alpha_0, \lambda)\ {\rm d}\lambda\ {\rm d}\Omega.$   (B.21)

Appendix C: Fresnel coefficients

In this appendix, we express the Fresnel coefficients for an absorbing medium (meaning with a complex refractive index).

C.1 Reflection

The Fresnel reflection coefficients are given by the equations:

$\displaystyle r_{ij}^{\rm (s)} = \frac{\tilde{n}_{i}\cos \alpha_{i} - \tilde{n}... ...\alpha_{j}}{\tilde{n}_{i}\cos \alpha_{i} + \tilde{n}_{j}\cos \alpha_{j}} \qquad$   $\displaystyle \qquad r_{ij}^{\rm (p)} = \frac{\tilde{n}_{i}\cos \alpha_{j} - \t... ... \alpha_{i}}{\tilde{n}_{i}\cos \alpha_{j} + \tilde{n}_{j}\cos \alpha_{i}} \cdot$ (C.1)

The angles $\alpha_{i,j}$ are here complex angles given by the Descartes-Snell law of refraction for absorbing media, i.e., with complex refractive indexes (Kovalenko 2001):
$\displaystyle \tilde{n}_{i}\ \sin \alpha_{i} = \tilde{n}_{j}\ \sin \alpha_{j}.$   (C.2)

The variables $\rho_{ij}^{\rm (s,p)}$ and $\delta_{ij}^{\rm (s,p)}$ are defined by $r_{ij}^{\rm (s,p)} = \rho_{ij}^{\rm (s,p)} \cdot {\rm e}^{{\rm i}\ \delta_{ij}^{\rm (s,p)}}$.

C.2 Transmission

The Fresnel transmission coefficients (ratio of wave amplitudes) are given by the equations:

\begin{displaymath}t_{ij}^{\rm (s,p)} = \frac{E_{j}^{\rm (s,p)}}{E_{i}^{\rm (s,p)}} \end{displaymath} (C.3)


$\displaystyle t_{ij}^{\rm (s)} = \frac{2 \tilde{n}_{i}\cos \alpha_{i}}{\tilde{n}_{i}\cos \alpha_{i} + \tilde{n}_{j}\cos \alpha_{j}} \qquad$   $\displaystyle \qquad t_{ij}^{\rm (p)} = \frac{2 \tilde{n}_{i}\cos \alpha_{i}}{\tilde{n}_{i}\cos \alpha_{j} + \tilde{n}_{j}\cos \alpha_{i}} \cdot$ (C.4)

The Fresnel transmittances (ratio of powers) are given by the equations
             $\displaystyle T_{ij}^{\rm (s,p)}$ = $\displaystyle \frac{P_{j}^{\rm (s,p)}}{P_{i}^{\rm (s,p)}}\ =\ \frac{{\rm Irr}_{j}^{\rm (s,p)} A_{j} }{{\rm Irr}_{i}^{\rm (s,p)} A_{i}},$ (C.5)

where Ai,j is the surface section of the light beam in medium i or j, and Pi,j and ${\rm Irr}_{i,j}$ are the incident and transmitted power (in Watt) and the irradiance (in ${\rm W~m^{-2})}$:
                $\displaystyle T_{ij}^{\rm (s,p)}$ = $\displaystyle \frac{{\rm Irr}_{j}^{\rm (s,p)} \cos \alpha_{j} }{{\rm Irr}_{i}^{\rm (s,p)} \cos \alpha_{i} }$ (C.6)
$\displaystyle T_{ij}^{\rm (s,p)}$ = $\displaystyle \frac{n_{j} \cos \alpha_{j} }{n_{i} \cos \alpha_{i} }\ \Vert t_{ij}^{\rm (s,p)}\Vert^2.$ (C.7)

Remark: In the same way, the reflectance is also defined as $\qquad R_{ij}^{\rm (s,p)} = \frac{P_{i, {\rm reflected}}^{\rm (s,p)}}{P_{i}^{\rm (s,p)}}\ =\ \Vert r_{ij}^{\rm (s,p)}\Vert^2$
$\displaystyle R_{ij}^{\rm (s,p)}\ +\ T_{ij}^{\rm (s,p)} = 1.$   (C.8)

We can also define another transmission factor $\hat{T}_{ij}$ that takes the modification of the solid angle due to refraction at the interface between media i and j into account:
                $\displaystyle \hat{T}_{ij}^{\rm (s,p)}$ = $\displaystyle T_{ij}^{\rm (s,p)} \frac{{\rm d}\Omega_{i}}{{\rm d}\Omega_{j}}$ (C.9)
$\displaystyle \hat{T}_{ij}^{\rm (s,p)}$ = $\displaystyle T_{ij}^{\rm (s,p)} \frac{n_{j}^2 \cos \alpha_{j}}{n_{i}^2 \cos \alpha_{i}}$ (C.10)
$\displaystyle \hat{T}_{ij}^{\rm (s,p)}$ = $\displaystyle \Vert t_{ij}^{\rm (s,p)}\Vert^2 \frac{n_{j}^3 \cos ^2\alpha_{j}}{n_{i}^3 \cos ^2\alpha_{i}} \cdot$ (C.11)

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