Diffractionless image propagation and frequency conversion via four-wave mixing exploiting the thermal motion of atoms

A setup to frequency-convert an arbitrary image encoded in the spatial profile of a probe field onto a signal field using four-wave mixing in a thermal atom vapor is proposed. The atomic motion is exploited to cancel diffraction of both signal and probe fields simultaneously. We show that an incoherent probe field can be used to enhance the transverse momentum bandwidth which can be propagated without diffraction, such that smaller structures with higher spatial resolution can be transmitted. It furthermore compensates linear absorption with nonlinear gain, to improve the four-wave mixing performance since the propagation dynamics of the various field intensities is favorably modified.

Figure 1

Schematic setup proposed to realize diffractionless image reproduction via four-wave mixing in a thermal atom vapor. (a) In the considered five-level model, the lower three states in a $\Lambda$ configuration form the basic electromagnetically induced transparency setup for the probe propagation. The fourth state together with an additional control beam leads to a four-level double-$\Lambda$ system, which enables four-wave mixing to generate an additional signal field. Finally, a two-way incoherent pump field is applied to the fifth auxiliary state. In effect, the single-photon absorptions for both the probe and signal fields are reduced, and the four-wave mixing process is enhanced. (b) Colinear propagation of the four coherent fields in the thermal atomic medium to satisfy the phase matching condition. Due to atomic motion, the paraxial diffraction of arbitrary profiles of both probe and signal fields can be eliminated over a certain bandwidth of the transverse beam profile. Eventually, the incident image encoded onto the probe is copied to the FWM signal field, which also propagates without diffraction.

Figure 2

Motion-induced linear susceptibilities for the probe (a) and signal (b) fields in momentum space. The figures show the linear dispersion without incoherent pump field (black dotted line) and with incoherent pump field (solid blue curve). It can be seen that the dispersion is essentially unaffected by the pump and remains approximately quadratic in the central region ${\mathrm{k}}_{\perp }\ll {\mathrm{k}}_1$. In contrast, the linear absorption with (dashed green curve) and without (dot-dashed red curve) are strongly modified by the incoherent pump. Parameters are ${\lambda }_p=795\mathrm{nm}$, ${\lambda }_s=780\mathrm{nm}$, ${\Gamma }_{\mathrm{D}1}=2\pi \times 5.75\mathrm{MHz}$, ${\Gamma }_{\mathrm{D}2}=2\pi \times 6.07\mathrm{MHz}$, ${\Gamma }_{31}={\Gamma }_{\mathrm{D}1}/4$, ${\Gamma }_{32}={\Gamma }_{\mathrm{D}1}/6$, ${\Gamma }_{41}={\Gamma }_{\mathrm{D}2}/4$, ${\Gamma }_{42}={\Gamma }_{\mathrm{D}2}/6$, ${\Gamma }_{51}={\Gamma }_{\mathrm{D}1}/12$, ${\Gamma }_{52}={\Gamma }_{\mathrm{D}2}/2$, ${\gamma }_{21}=0.001{\Gamma }_{31}$, $T=300$ K, ${\mathrm{v}}_{\mathrm{th}}=240\mathrm{m}/\mathrm{s}$, $\Delta \mathrm{k}=22.8{\mathrm{m}}^{-1}$, ${\Omega }_{c1}=1.55{\Gamma }_{32}$, ${\Omega }_{c2}=1.43{\Gamma }_{42}$, ${\Delta }_{c1}=0$, ${\Delta }_{c2}=0$, and ${\Delta }_p=\Delta$. With incoherent pump field, $n_0=1.32\times {10}^{18}{\mathrm{m}}^{-3}$, ${\gamma }_c=1600\Delta \mathrm{k}{\mathrm{v}}_{\mathrm{th}}$, and $p=0.7{\Gamma }_{31}$. Without incoherent pump field, $n_0=6.2\times {10}^{17}{\mathrm{m}}^{-3}$, ${\gamma }_c=30000\Delta \mathrm{k}{\mathrm{v}}_{\mathrm{th}}$, and $p=0$. In both cases, parameters are chosen such that the diffraction for both probe and signal fields can be eliminated, and that the transverse momentum scales with and without pump are comparable, ${\mathrm{k}}_0\approx {\mathrm{k}}_1$.

Figure 3

Motion-induced nonlinear susceptibilities related to the forward (a) and backward (b) FWM processes. The figures show the nonlinear dispersion without incoherent pump field (black dotted line) and with incoherent pump field (solid blue curve). As for the linear susceptibilities, the dispersion is essentially unaffected by the pump and remains approximately quadratic in the central region ${\mathrm{k}}_{\perp }\ll {\mathrm{k}}_1$. In contrast, the nonlinear absorption with (dashed green curve) and without (dot-dashed red curve) is strongly modified by the incoherent pump. Parameters can be chosen such that the nonlinear and the linear absorption cancel for a suitable choice of the pump field. Parameters are as in Fig. 2.

Figure 4

(a) Zeroth-order populations ${\rho }_{ii}^{\left(0\right)}(i=1,2,3,4)$ and (b) coherences ${\rho }_{ij}^{\left(0\right)}(i\ne j)$ for a motionless atom as a function the intensity of the two-way incoherent pump field. At zero pump intensity, the atoms are in the ground state $|1\rangle$. As the intensity of the incoherent pump increases, atoms are gradually redistributed among the other four states, and atomic coherences arise between $|2\rangle$, $|3\rangle$, and $|4\rangle$ due to interactions with the two coherent control fields ${\Omega }_{c1}$ and ${\Omega }_{c2}$. Parameters are as in Fig. 2.

Figure 5

(a) Transverse intensity profile of a two-dimensional Gaussian probe beam after propagation over one Rayleigh length. The figure shows the one-dimensional slice $y=0$. The different curves compare the profile of the probe beam in free space with those of the probe and FWM signal beams in the thermal medium. The intensities of all beams are scaled such that the peak intensity of the input probe field also shown in the figure is normalized. It can be seen that the power of the input beam is distributed between the probe and FWM signal beam, and that in the thermal medium, diffraction is significantly reduced compared to the free space case. (b) and (c) show the normalized power and width of the probe and FWM signal as a function of the propagation distance. Note that the starting point for $z$ in (c) is $z=0.03z_R$, as at small propagation distance, the signal field is very weak as shown in (b), such that the width cannot easily be extracted. Parameters are the same as in Fig. 2.

Figure 6

The normalized output powers (a) and widths (b) of a Gaussian probe and the generated FWM signal field. Results are shown after propagation over one Rayleigh length in the thermal medium, as a function of the amplitude of the two-way incoherent pump $p$. If can be seen that, already after a short propagation distance, the output probe and FWM signal fields widths and powers approach each other. Afterwards, the powers increase slowly over the residual propagation distance due to overall gain. Throughout the whole propagation, the widths of the two beams remain essentially unchanged, as can be seen from the small scale on the $y$ axis of (b). Parameters are the same as in Fig. 2.

Figure 7

Diffractionless propagation and frequency conversion of arbitrary images. The original two-dimensional image encoded onto the transverse profile of the probe beam is shown in (a). In free space, the image is severely distorted after propagation over one Rayleigh length, as shown in (b). In our suitably tailored thermal medium, the probe beam profile is preserved; see (c). Additionally, the same profile is converted onto the signal field via the FWM process as given in (d). All axis labels are given in units of $100\mu \mathrm{m}$. Other parameters are as in Fig. 2.