Covariant photon wave mechanics of evanescent fields

A photon wave mechanical theory describing evanescent electromagnetic fields, as these appear in $p$-polarized total internal reflection from a flat dielectric-to-vacuum interface, is presented. In this first-quantized field theory, the Lorenz gauge four-potential relates to the wave functions of the transverse (T), longitudinal (L), and scalar (S) photons. For a homogeneous medium, the source domain (rim zone) of the L photons is shown to coincide with the longitudinal vector-field parts of an electron sheet current density located at the interface. The identical rim zones of the T and L photons exhibit one-dimensional (1D) exponential confinement with a spatial decay constant equal to the magnitude of the incident field's wave vector along the interface plane. The S-photon 1D localization is complete (has $\delta$-function character). Dynamical equations are established for the L- and S-photon variables in the wave-vector–time domain. These Hamiltonian-like equations are easily upgraded to the second-quantized level. The link to the wave mechanics of near-field photons is made. When extended to the quantum electrodynamic level, the present theory is an alternative to the pioneering Carniglia-Mandel (CM) triplet-photon description. It is argued that our theory provides one with an improved physical understanding of the basic role of T photons in evanescent fields. Wave mechanics of evanescent fields give additional insight in the T-photon localizability problem. The particular difficulties arising for $s$-polarized external fields are addressed referring to the rim zone of quantum wells.

Figure 1

Wave-vector diagram in TIR. The wave vectors of the incident $\left({\mathbf{q}}_i\right)$ and reflected $\left({\mathbf{q}}_r\right)$ homogeneous waves are real, have equal components $\left(q_{\parallel }{\mathbf{e}}_x\right)$ parallel to the interface plane ($xy$ plane, here), and the same magnitude $\left(\right|{\mathbf{q}}_i|=|{\mathbf{q}}_r\left|\right)$. The wave vector of the transmitted inhomogeneous wave is complex, with a real part $\left(q_{\parallel }{\mathbf{e}}_x\right)$ along the interface and an imaginary part $\left({\kappa }_{\perp }^0{\mathbf{e}}_z\right)$ normal to it. In TIR, $\omega /c<q_{\parallel }<(\omega /c){\varepsilon }^{1/2}$.

Figure 2

Diagram indicating the linear polarization states of the incident $\left({\mathbf{E}}_i\right)$ and reflected $\left({\mathbf{E}}_r\right)$ electric fields and the clockwise elliptical polarization of the transmitted electric field $\left({\mathbf{E}}_t\right)$ in TIR. The polarization states are normalized so that they have the same $z$ component, $q_{\parallel }$ [corresponding to a $\pi /2$ rotation of the related (complex) wave vectors].

Figure 3

Top: Cycle-averaged momentum densities in TIR. Bottom: Vectorial components of the $TT$ and $LT$ parts of $\langle {\mathbf{g}}_t\rangle$. The oppositely directed $z$ components have the same magnitude. The transmitted longitudinal electric field $\left({\mathbf{E}}_t^L\right)$ lies in the scattering plane and is clockwise circular polarized, as indicated.

Figure 4

Top: In our model of $p$-polarized TIR, the bulk current density is divergence free $(\mathbf{J}={\mathbf{J}}_T)$. In the vacuum half space, ${\mathbf{J}}_T+{\mathbf{J}}_L=\mathbf{J}=\mathbf{0}$. The T-photon source domain is exponentially $[\sim \exp (-i{q}_{\parallel }z)]$ confined, as indicated. Bottom: Exponentially confined source domain $(\sim |{\mathbf{J}}_L\left|\right)$ of T photons originating in a current density sheet located at $z=0$.

Figure 5

Schematic illustration showing that the sheet fields ${\mathbf{A}}_L(z;\left[{\kappa }_{\perp }^0\right])$ and $A_0\left(z\right)$ have the same spatial decay constant $\left({\kappa }_{\perp }^0\right)$. The decay constant $\left(q_{\parallel }\right)$ for ${\mathbf{A}}_L(z;\left[q_{\parallel }\right])$ is larger. The elliptical polarizations of ${\mathbf{A}}_L(z;\left[{\kappa }_{\perp }^0\right])$ and ${\mathbf{E}}_t\left(z\right)$, and the circular polarization of ${\mathbf{A}}_L(z;\left[q_{\parallel }\right])$, in the two half spaces $(z<0,z>0),$ are indicated in the lower part of the figure.

Figure 6

Using a normalization constant $N={\mu }_0\left|J_{B,z}\left(0\right)\right|/\left(q_0^2\sqrt{2}\right)$, the quantities $|{\mathbf{A}}_L(z;\left[q_{\parallel }\right])|/N=\exp (-m{q}_0\left|z\right|)$ [full line], and $\sqrt{2}|A_0\left(z\right)|/N={(m^2-1)}^{-1/2}\exp [-{(m^2-1)}^{1/2}{q}_0\left|z\right|]$ [dashed line] are plotted as a function of $q_{0}z$ for two values of $m=q_{\parallel }/q_0$.

Figure 7

Normalized (with $M={\mu }_0{\mathcal{J}}_0/q_0^2$) plots of the scalar ($A_0$), longitudinal ($A_L$), and near-field ($i\sqrt{2}{A}_{\mathrm{NF}}$) photon variables as functions of $Q/q_0$. Note that the NF variable is not singular at the vacuum wave number, i.e., for $Q=q_0$.