Rigorous theory of thin-vapor-layer linear optical properties: The case of specular reflection of atoms colliding with the walls

The theory of the thin-vapor-layer linear optical properties is presented for the case of specular reflection of atoms colliding with the walls. The effects of light absorption and the shift in the resonance frequency are taken into account by means of self-consistent calculation of the field and polarization in a gaseous medium. The obtained formulas reproduce the complex dependence of the spectral line profile on the gas layer thickness and allow one to determine numerically the exact values of the “blueshift.” It was shown that, despite the low vapor concentration, the resonance shift is on the same order of magnitude as the width of the spectral lines.

Figure 1

Schematic of light reflection (at normal incidence) from a thin gas layer.

Figure 2

Reflectivity of a thin-vapor layer against dimensionless detuning calculated by the Fourier method for eight different layer thicknesses. Dashed curves: (a) $l=\lambda /2$ (the inset: spectral behavior of reflectivity near resonance: $\left|\mathrm{\Omega }\right|\le 0.03$), (b) $l=3\lambda /4$, (c) $l=\lambda$, and (d) $l=5\lambda /4$; solid curves: (a) $l=3\lambda /2$, (b) $l=7\lambda /4$, (c) $l=2\lambda$, and (d) $l=9\lambda /4$. The parameters $\mathrm{\Gamma }$, $n_1$, $n_2$, and $m$ were taken equal to 0.01, 1.5, 1.5, and 0.001, respectively.

Figure 3

(a) Reflectivity of a thick gas layer in the resonance region versus the dimensionless detuning calculated by the Fourier method for $m=0.005$, $\mathrm{\Gamma }=0.01$, $l=500\lambda$, $n_1=1.5$, and $n_2=1.5$. (b) Reflectivity of the sub-$\lambda$-thick vapor layers calculated by the Fourier method for $m=0.001$, $\mathrm{\Gamma }=0.01$, $n_1=1.5$, and $n_2=1.5$: dashed curve: $l=\lambda /\phantom{\lambda 4}4$; solid curve: $l=\lambda /\phantom{\lambda 8}8$; and dotted curve: $l=\lambda /\phantom{\lambda 13}13$. (c) Comparison of the spectral line profiles of the reflectivity calculated by the Fourier method (solid and dot-dashed curves) with the perturbation theory solution (dashed and dotted curves) at various atomic vapor densities: solid and dashed curves: $m=0.001$; dotted and dot-dashed curves: $m=0.002$ for $\mathrm{\Gamma }=0.01$, $n_1=1.5$, and $n_2=1$.

Figure 4

The blueshift normalized to the Doppler width as a function of $m$ for $m\le \mathrm{\Gamma }=0.01$ and $l=\lambda /\phantom{\lambda 2}2$. The dashed line represents the linear fit $\mathrm{\Delta }\mathrm{\Omega }=3.54m$.

Figure 5

The origin of the sub-Doppler features illustrated on the example of atomic polarization near an isolated boundary situated at $\xi =0$. The difference between transient and steady-state polarizations of an atom departing from the surface is given for the particular absolute value of atomic velocity $\left|\nu \right|=1$ and the following set of parameters: $\mathrm{\Omega }=0$, $\mathrm{\Gamma }=0.1$, and $E_0=1$. All complications connected with the self-consistent solution of Eqs. (6) are omitted. The real and imaginary parts of $\sigma$ are presented in subfigures (a) and (b), respectively. Dotted curve: steady-state polarization of an atom arriving at the surface with velocity $\nu =-1$; solid curve: steady-state polarization of an atom departing from the surface with velocity $\nu =1$; dashed curve: transient polarization of the departing atom after quenching collision with the surface for $\nu =1$ $\left[\sigma \right(\xi =0,\nu >0)=0]$; dot-dashed curve: transient polarization of the departing atom after specular collision with the surface for $\nu =1$ (the real part of this solution coincides with the steady-state polarization of the departing atoms represented by the solid curve). Although the steady-state polarization implied in the conventional dispersion theory leads to the Doppler broadened spectral lines, transient polarizations initiated by specular as well as by quenching collisions lead to sub-Doppler linewidths.