Quantizing the electromagnetic field near two-sided semitransparent mirrors

This paper models light scattering through flat surfaces with finite transmission, reflection, and absorption rates, with wave packets approaching the mirror from both sides. While using the same notion of photons as in free space, our model also accounts for the presence of mirror images and the possible exchange of energy between the electromagnetic field and the mirror surface. To test our model, we derive the spontaneous decay rate and the level shift of an atom in front of a semitransparent mirror as a function of its transmission and reflection rates. When considering limiting cases and using standard approximations, our approach reproduces well-known results but it also paves the way for the modeling of more complex scenarios.

Figure 1

Schematic view of a semitransparent mirror with light incident from both sides. Depending on the direction of the incoming light, we denote the (real) transmission and reflection rates of the mirror by $t_a$, $t_b$, $r_a$, and $r_b$, respectively. In this paper, the possible absorption of light in the mirror surface is explicitly taken into account. However, for simplicity we assume that the medium on both sides of the mirror is the same.

Figure 2

(a)–(c) Left-traveling Gaussian wave packet (blue dotted line) with $E_{\mathrm{free}}(x,0)=E_0{e}^{-{(x-x_0)}^2/2{\sigma }^2}{e}^{i{k}_{0}x}+\mathrm{c}.\mathrm{c}.$, where $E_0$ and $x_0$ are free parameters and where $k_0{x}_0=-6$, $\sigma =(1/\sqrt{2})x_0$, $t_1=0.89x_0/c$, and $t_2=1.83x_0/c$. (d)–(f) Right-traveling Gaussian wave packet (red dash-dotted line). At $t=0$, (a)–(c) the blue wave packet can be interpreted as a real wave packet, while (d)–(f) the red wave packet constitutes its mirror image. When the wave packets cross over at $x=0$, the red wave packet becomes real, while the blue one becomes the image. Moreover, plots (g)–(i) show the sum of the red and the blue electric-field contribution on the right side of the mirror (black solid line), which evolves like a wave packet approaching a perfectly reflecting mirror.

Figure 3

Schematic view of a semitransparent mirror with light incident from the left. The solid black lines incident on the mirror indicate the direction of the wave vector of the incoming light. To predict the effect of the mirror, we decompose the electric-field amplitudes of incoming wave packets into its parallel and perpendicular components (red lines) with respect to the mirror surface. As illustrated, transmission and reflection reduces these components by a factor which equals the corresponding transmission and reflection rate. Notice that these rates have to be the same for parallel and perpendicular field amplitudes. Otherwise, the electric-field vector would not remain orthogonal to the corresponding wave vector $\mathbf{k}$. However, parallel electric-field components obtain a minus sign upon reflection due to the rearrangement of mirror surface charges.

Figure 4

(a) Spontaneous decay rate ${\mathrm{\Gamma }}_{\mathrm{mirr}}$ and (b) atomic level shift ${\mathrm{\Delta }}_{\mathrm{mirr}}$ of an atom in front of a perfect mirror [see Eqs. (52)] as a function of the atom-mirror distance $x$ for different orientations of the atomic dipole moment ${\mathbf{D}}_{12}$. For $\mu =0$, ${\mathbf{D}}_{12}$ is parallel and, for $\mu =1$, ${\mathbf{D}}_{12}$ is perpendicular to the mirror surface. In all cases, we have ${\mathrm{\Gamma }}_{\mathrm{mirr}}=0$ while ${\mathrm{\Delta }}_{\mathrm{mirr}}$ diverges for $x=0$. Moreover, for $k_{0}x\gg 1$, we have ${\mathrm{\Gamma }}_{\mathrm{mirr}}={\mathrm{\Gamma }}_{\mathrm{free}}$ and ${\mathrm{\Delta }}_{\mathrm{mirr}}=0$, as it should. For (a) the $\mu =0$ case corresponds to the line emanating from 0 and becomes the maximum of the oscillation, whereas the $\mu =1$ case corresponds to the line emanating from ${\mathrm{\Gamma }}_{\mathrm{mirr}}=2{\mathrm{\Gamma }}_{\mathrm{free}}$ is increased and becomes the minimum of the oscillation. For (b) the $\mu =0$ case corresponds to the line on left-hand side of the graph (line emanating from minus infinity), which gives the most noticeable shift in the atom's energy levels, whereas the $\mu =1$ case corresponds to the line on right-hand side, which gives the least noticeable level shift.

Figure 5

(a) Spontaneous decay rate ${\mathrm{\Gamma }}_{\mathrm{mirr}}$ and (b) atomic level shift $\mathrm{\Delta }{\omega }_{\mathrm{mirr}}$ of an atom in front of a symmetric mirror [see Eqs. (54)] as a function of the atom-mirror distance $x$ for different values of $r$. For simplicity we assume here that $r=t$ and $\mu =0$. The case $r=0$ corresponds to a completely absorbing surface, while $r=1/\sqrt{2}$ models a 50:50 beam splitter.

Figure 6

(a) Spontaneous decay rate ${\mathrm{\Gamma }}_{\mathrm{mirr}}$ and (b) atomic level shift ${\mathrm{\Delta }}_{\mathrm{mirr}}$ of an atom in front of a nonabsorbing symmetric mirror [see Eqs. (54)] as a function of the atom-mirror distance $x$ for different values of $r$. Again we assume $\mu =0$, while $t^2=1-r^2$. The case $r=0$ corresponds to free space, while $r=1$ models a perfect mirror. For (a) the $r=0$ case corresponds to the horizontal line emanating from ${\mathrm{\Gamma }}_{\mathrm{mirr}}/{\mathrm{\Gamma }}_{\mathrm{free}}=1$, meaning that the atom decays as in free space when there is no mirror present, whereas the $r=1$ case corresponds to the line emanating from 0, which gives the maximum of the oscillation. For (b) the $r=0$ case corresponds to the horizontal line emanating from ${\mathrm{\Delta }}_{\mathrm{mirr}}/{\mathrm{\Gamma }}_{\mathrm{free}}=0$, meaning that there is no mirror-induced shift in the atomic energy levels. The $r=1$ case corresponds to the most noticeable shift in the atomic energy levels.