Control of beam propagation in optically written waveguides beyond the paraxial approximation

Beam propagation beyond the paraxial approximation is studied in an optically written waveguide structure. The waveguide structure, which leads to diffractionless light propagation, is imprinted on a medium consisting of a five-level atomic vapor driven by an incoherent pump and two coherent spatially dependent control and plane-wave fields. We first study beam propagation in a single optically written waveguide and find that the paraxial approximation does not provide an accurate description of the probe propagation. We then employ coherent control fields such that two parallel and one tilted Gaussian beams produce a branched waveguide structure. The tilted beam allows selective steering of the probe beam into different branches of the waveguide structure. The transmission of the probe beam for a particular branch can be improved by changing the width of the titled Gaussian control beam as well as the intensity of the spatially dependent incoherent pump field.

Figure 1

The five-level type scheme considered in the analysis. The probe field couples to transitions $|1\rangle \leftrightarrow |4\rangle$ and $|4\rangle \leftrightarrow |5\rangle$, and the two coherent control laser fields ${\Omega }_s,{\Omega }_c$ are far detuned from respective transition frequencies in order to split states $|4\rangle$ and $|5\rangle$ into suitable dressed states. An incoherent pump control field is applied to $|1\rangle \leftrightarrow |4\rangle$ to achieve a population inversion. The control fields modify the probe field coupling to achieve high refractive index contrast with low absorption.

Figure 2

Real (blue solid line) and imaginary (red dashed line) parts of the linear susceptibility obtained from Eq. (9) as a function of the position in the presence of a control laser field ${\Omega }_s$ with a Gaussian intensity profile. (a) Results for a constant incoherent pump rate; (b) results for spatially dependent pumping as explained in the main text. The inset in (b) depicts the detailed structure of the linear susceptibility in the central area, which is similar to that in (a). Note that the profile of the real part in the central region forms a waveguide structure. Parameters are chosen as $N=1.4\times {10}^{27}$ m${}^{-3}$, ${\lambda }_p=813.2$ nm, ${\gamma }_{54}^d=0.8$ GHz, ${\gamma }_{41}^d={\gamma }_{42}^d=0.3$ GHz, ${\gamma }_{43}^d=0.2$ GHz, ${\gamma }_{52}^d={\gamma }_{21}^d={\gamma }_{32}^d=0$, ${\Gamma }_{41}={\Gamma }_{42}=45$ Hz, ${\Gamma }_{54}=15$ Hz, ${\Delta }_{p1}=-{\gamma }_{41}$, ${\Delta }_{p2}=19.7$ GHz, ${\Delta }_s=10.0$ GHz, ${\Delta }_c=18.0$ GHz, ${\Omega }_{s0}=\sqrt{2{\gamma }_{41}{\Delta }_s}$, ${\Omega }_c=5.2$ GHz, $w_s=1.0$ $\mu$m, and $p=45.9943$ Hz.

Figure 3

Normalized intensity of the probe field as a function of propagation distance $z$. (a) The free-space case, (b) the propagation in a medium prepared with a control field and evaluated beyond the paraxial approximation, and (c) the corresponding results evaluated in paraxial approximation. The maximum propagation distance is $z_0=20$ $\mu$m. The initial width of the probe is $w_p=1.0\mu$m at $z=0$. Other parameters are as in Fig. 2.

Figure 4

(a) Intensity profile of the propagating probe field at the medium entrance (dashed black line) and output after a propagating distance $z_0=20\mu$m. The dot-dashed green line shows propagation in free space, the dotted red line propagation in the medium with control field evaluated in paraxial approximation, and the solid blue line corresponding results beyond paraxial approximation. (b) Transversal width of the probe field as a function of propagation distance $z$ evaluated within (dashed red) and beyond (solid blue) paraxial approximation. Parameters are chosen as in Fig. 3.

Figure 5

Sketch of the branched waveguide structure. (a) The spatial profile of ${\Omega }_s$ for $k=1$ and the corresponding position-dependent (b) real and (c) imaginary parts of the linear susceptibility in the medium. The labels $i\in \{1,2,3\}$ in (a) indicate the three output ports corresponding to the $i$th control beam ${\Omega }_{si}$. The size of the medium is chosen as 60 $\mu$m $\times$ 200 $\mu$m. The other parameters are as in Fig. 2.

Figure 6

Transmitted powers (a) $T_1$, (b) $T_2$, and (c) $T_3$ at the three output ports indicated in Fig. 5a after the propagation through the branched waveguide medium. (d) The visibility $V=(T_1-T_2)/(T_1+T_2)$ for the switching of light between output ports 1 and 2. All results are plotted against the relative width of the tilted control beam $k$. The blue solid (red dashed) line shows results for tilting angle $\tan \theta =0.18$ ($\tan \theta =0.26$). The probe beam initially has a Gaussian shape with $w_p=1.0$ $\mu$m. Other parameters are as in Fig. 2.

Figure 7

Spatial field configuration in the plane transverse to the propagation direction at the output of the medium at $z_0=200$ $\mu$m. The red solid line shows results without a tilted control beam, whereas the blue dashed line shows results including a tilted control with relative width $k=4$. The vertical green dashed lines indicate the positions of the three output ports. The tilting angle is chosen as $\tan \theta =0.18$. Other parameters are as in Fig. 6.

Figure 8

Visibility $V=(T_1-T_2)/(T_1+T_2)$ as a function of the tilting angle of the control field for three different relative widths of the tilted beam. The top red solid line shows $k=2$, the middle blue dashed line shows $k=3$, and the bottom green dotted line shows $k=4$. Other parameters are as in Fig. 6.

Figure 9

The effect of the incoherent pump on the output of the probe. The angle and width of the probe are chosen as $\tan \theta =0.18$ and $k=4$, respectively. Other parameters are as in Fig. 6.