Optical cloning of arbitrary images beyond the diffraction limits

Cloning of arbitrary images from the spatial profile of a laser beam onto that of a second beam is theoretically investigated. The two fields couple to each other while propagating in an atomic $\Lambda$ medium displaying coherent population trapping in the case where probe and control fields have comparable strength. Our method is suitable to clone arbitrary images as demonstrated in numerical simulations where the three letters “CPT” are encoded in the control field profile. The cloned structures have features reduced in size by about a factor of 2, when compared to the initial control images, and are consistent with a recent related experiment.

Figure 1

Schematic setup for the cloning of arbitrary images encoded in the spatial profile of a control field onto the spatial profile of a probe beam. Both fields copropagate through a rubidium vapor cell and couple to the atoms on the $|3\rangle \leftrightarrow \langle 1|$ (probe field with frequency ${\omega }_1$) and $|3\rangle \leftrightarrow \langle 2|$ (control field with frequency ${\omega }_2$) transitions, respectively. Our analysis includes the case in which both fields are equally strong.

Figure 2

Probe beam susceptibility as a function of the transverse coordinate $x$. The control field has a Gaussian $(m=0,n=0)$ beam profile. Panel (a) shows the imaginary and (b) the real component of the susceptibility. The three different curves show probe field detunings ${\Delta }_1=-0.005$ (blue dotted line, blue dashed line), ${\Delta }_1=0.0$ (green solid line, green dot-dashed line), and ${\Delta }_1=0.005\gamma$ (red dashed line, red dashed double-dot line) for two data sets of probe amplitudes $g=0.015\gamma$ and $g=0.15\gamma$, respectively. The other parameters are $y=0.005$ cm, $w_p=150\mu$m, $w_c=400\mu$m, ${\Delta }_2=0$, $\Gamma =0.001\gamma$, and $N=5\times {10}^{11}$ atoms/cm${}^3$.

Figure 3

Real part of the control field susceptibility as a function of transverse position $x$. The control field has a Gaussian $(m=0,n=0)$ shape. The three different curves show probe field detunings ${\Delta }_1=-0.005$ (blue dotted line, blue dashed line), ${\Delta }_1=0.0$ (green solid line, green dot-dashed line), and ${\Delta }_1=0.005\gamma$ (red dashed line, red dashed double-dot line) for two data sets of probe amplitudes $g=0.015\gamma$ and $g=0.15\gamma$, respectively, with different scales of the axes. The other parameters are the same as described in the legend of Fig. 2.

Figure 4

Probe field susceptibility as a function of the transverse spatial coordinate $x$. The control field has a doubly peaked spatial profile characterized by $m=1$ and $n=0$ in Eq. (13). The central part around $x=0$ is magnified in the inset in the top part of the figure. The other parameters are $y=0.005$ cm, $G_0=1.5\gamma$, $g_0=0.015\gamma$, $w_p=400\mu$m, and $N=7.5\times {10}^{11}$ atoms/cm${}^3$.

Figure 5

Normalized intensity profile of the propagating probe beam against the transverse coordinate $x$ for $y=0$. Panel (a) shows the beam profile at different propagation distances $z$. In (b), the transmitted probe beam intensity at the output of a 4-cm-long medium is shown for different detunings. All other parameters are chosen as described in the legend of Fig. 2.

Figure 6

Spatial intensity profile of the control field as a function of the transverse coordinate $x$ after propagation through a 4-cm-long atomic medium. Results are shown for different field parameters. The profile is shown in the $y=0$ plane. All other parameters are the same as described in the legend of Fig. 5.

Figure 7

Transverse spatial intensity profile of the probe and control fields after propagation through an atomic vapor cell of length 4 cm. The initial control field profile is chosen as a Hermite-Gaussian doubly peaked profile. Panel (a) shows the probe field; panel (b) shows the control field. The parameters are as described in the legend of Fig. 4.

Figure 8

Transverse spatial intensity profile of the control and probe fields. Panel (a) shows the initial profile of the control field at the medium entry. Panel (b) shows the control field profile after propagation through a vapor cell of length 3 cm filled with rubidium atoms. Panel (c) shows the probe field profile after the propagation. Initially, the probe field is chosen as a plane wave. The parameters are $g=0.15\gamma$, $G=1.5\gamma$, ${\Delta }_2=0$, ${\Delta }_1=0.002\gamma$, $\Gamma =0.001\gamma$, and $N={10}^{12}$ atoms/cm${}^3$.