1. Introduction

The diffraction of light during propagation in free space is a fundamental and generally unavoidable physical phenomenon. Diffracting light beams do not maintain their intensity distribution in the plane transverse to the propagation direction, unless belonging to a particular class of non-diffracting (Bessel) beams [1]. In nonuniform media, waveguiding is possible for specific spatial modes [2, 3], or equivalently arbitrary images may revive after a certain self-imaging distance [4]. However in such waveguides, the suppression of diffraction for multimode profiles is not trivial, as each transverse mode propagates with a different propagation constant or group velocity, resulting in spatial dispersion of the profile.

Recently, a mechanism was suggested and demonstrated for manipulating the diffraction of arbitrary images imprinted on a light beam for arbitrary propagation distances [5,6]. The technique is based on electromagnetically induced transparency (EIT) [7] in a thermal atomic gas. Unlike other methods utilizing EIT [2–4], [8–17], which rely on spatial non-uniformity, this technique prescribes non-uniformity in k_⊥ space. Here, k_⊥ denotes the transverse wave vectors, i.e., the Fourier components of the envelope of the field in the transverse plane, which is the natural basis for paraxial diffraction. The technique of Refs. [5,6] relies on the diffusion of the atoms in the medium and on the resulting diffraction-like optical response. However, the resolution limit of such motional-induced diffraction in currently available experimental conditions is on the order of 100 μm, preventing it from being of much practical use. Higher resolution requires a denser atomic gas, in which strong absorption is unavoidable due to imperfect EIT conditions. Very recently, Zhang and Evers proposed to circumvent the absorption by generalizing the model of motional-induced diffraction to a four-wave mixing (FWM) process in combination with EIT [18]. The FWM process further allows the frequency conversion of the image and increases the available resolution.

In this paper, we propose a scheme to manipulate diffraction using FWM [19–21] without the need for motional-induced diffraction. The mechanism we study originates from phase matching in k_⊥ space and does not require a gaseous medium; it is therefore directly applicable to solid nonlinear media. For our model to be general and to accommodate motional-broadening mechanisms (not important in solids), we here still concentrate on describing atomic gases and validate our model against relevant experiments. The inherent gain of the FWM process allows us to improve the spatial resolution by working with relatively higher gas densities while avoiding loss due to absorption.

In Sec. 2, we introduce a microscopic model of FWM in a A system, based on Liouville-Maxwell equations, similar to the one used in Ref. [22]. In Sec. 3, we compare the model to recent experimental results of FWM in hot vapor [22,23]. We use our model in Sec. 4 to show that, with specific choice of frequencies, the k_⊥ dependency of the FWM process can be used to eliminate the diffraction of a propagating light beam. We also present a demonstration of negative diffraction, implementing a paraxial version of a negative-index lens [24], similar to the one in Ref. [6] but with positive gain and higher resolution. Finally, we analyze the resolution limitation of our scheme and propose ways to enhance it. We show that, for cold atoms at high densities (∼10¹² cm⁻³), diffraction-less propagation of an image with a resolution of ∼10 μm can be achieved.

2. Theory

2.1. Model

We consider an ensemble of three-level atoms in a Λ configuration depicted in Fig. 1(a). The atomic states are denoted as |u〉, |l〉, and |r〉, for the up, left, and right states, and the optical transition frequencies ω_ul and ω_ur are assumed to be much larger than the ground-state splitting ω_lr = ω_ul − ω_ur. The atom interacts with a weak ’probe’ and two ’control’ electromagnetic fields, propagating in time t and space r,

 

Fig. 1 (a) Four-wave mixing in a three-level Λ system (|u〉, |l〉, and |r〉). Ω_i with i = c,p,s are the Rabi frequencies of the fields. The phase-matching conditions are shown for (b) collinear and (c) non-collinear propagation. The phase mismatch scalar is $2k_{\mathrm{\Delta }}=(2k_c-k_p-k_s)\widehat{\mathbf{z}}$.

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To simplify the formalism, the same dipole moment is assumed for the two optical transitions μ = μ_ul = μ_ur, and the two control fields differ only in their polarizations. Here ω_i are the frequencies of the ’probe’ (i = p) and the ’control’ fields (i = c); $k_0^i={\omega }_i/c$ are the wave vectors in the case of plane waves, otherwise they are carrier wave-vectors; and Ω_i(r, t) are the slowly varying envelopes of the the Rabi frequencies, satisfying $|{\partial }_t^2{\mathrm{\Omega }}_t(\mathbf{r},t)|\ll |{\omega }_i{\partial }_t{\mathrm{\Omega }}_t(\mathbf{r},t)|$ and $|{\partial }_z^2{\mathrm{\Omega }}_t(\mathbf{r},t)|\ll |k_i^0{\partial }_z{\mathrm{\Omega }}_t(\mathbf{r},t)|$. The polarization vectors of the fields are ε_p, ε_cl, and ε_cr. The strong control and weak probe fields stimulate a weak classical ’Stokes’ field (or ’conjugate’) at a frequency ω_s = 2ω_c − ω_p,

To further simplify the analysis, we assume a single relaxation rate Γ between the excited and ground levels and define the complex rates γ_cr = Γ − i (ω_c − ω_ur) and γ_cl = Γ − i (ω_c − ω_ul) for each of the optical transitions. Within the ground state, we consider a population relaxation with symmetric rates Γ_l↔r and a decoherence with a rate Γ_lr. In a frame rotating with the control frequency ω_c, the equations of motion for the local density-matrix ρ(r, t) are better written in terms of the slowly-varying density-matrix R(r,t), where $R_{u,j}(\mathbf{r},t)={\rho }_{u,j}(\mathbf{r},t)e^{i{\omega }_{c}t-i{k}_0^{c}z}$ for j = l,r and $R_{\alpha ,{\alpha }^{\prime }}(\mathbf{r},t)={\rho }_{\alpha ,{\alpha }^{\prime }}(\mathbf{r},t)$ for all other matrix elements,

Here

Finally assuming non-depleted control fields, constant in time and space Ω_c (r,t) = Ω_c, we complete the description of the atom-field interaction with the propagation equations under the envelope approximation for the probe field

2.2. Steady-state solution

The evolution of the fields is described by a set of non-linear, coupled differential equations for the density matrix elements $R_{\alpha ,{\alpha }^{\prime }}$ and the weak fields Ω_p and Ω_s [Eqs. (3)–(5b)], which require further approximations in order to be solved analytically. Most importantly, we assume the proximity to two-photon resonance, such that δ_ω, is on the order of the ground-state frequency splitting ω_lr and much larger than any detuning, Rabi frequency, or pumping rate in the system. Other assumptions are detailed in the appendix, where we derive the steady state of the system to first order in the weak fields,

Here α_j = g(n_l/γ_jl + n_r/γ_jr) are the linear absorption coefficients of the probe (j = p) or Stokes (j = s) fields, with $n_i\equiv R_{i,i}^{(0)}$ the populations of the i = r,l levels. ${\beta }_p=g(n_l/{\gamma }_{pl}+n_r/{\gamma }_{cr}^{*})$ and ${\beta }_s=g(n_r/{\gamma }_{sr}^{*}+n_l/{\gamma }_{cl})$ are two-photon interaction coefficients, γ_jk = Γ − i(ω_j − ω_uk) [j = p,c,s;k = l,r] are complex one-photon detunings, and ${\gamma }_0={\mathrm{\Gamma }}_{lr}+|{\mathrm{\Omega }}_c{|}^2/{\gamma }_{sr}^{*}+|{\mathrm{\Omega }}_c{|}^2/{\gamma }_{pl}-i({\delta }_{\omega }-{\omega }_{lr})$ is the complex two-photon detuning. Eqs. (7) are similar to those obtained by Harada et al. [22] but here including the diffraction term $\pm i{\nabla }_{\perp }^2/(2q)$, required to explore the spatial evolution of the fields.

We start with the simple case of a weak plane-wave probe ${\mathrm{\Omega }}_p(\mathbf{r})=f(z)e^{i{k}_{\perp }^p\cdot r_{\perp }}{e}^{i(k_z^p-k_0^p)z}$ directed at some small angle $\theta \approx k_{\perp }^p/k_0^p\ll 1$ relatively to the z axis (Fig. 1). The generated Stokes field is then also a plane wave ${\mathrm{\Omega }}_s(\mathbf{r})=g(z)e^{i{k}_{\perp }^s\cdot r_{\perp }}{e}^{i(k_z^s-k_0^s)z}$. Substituting into Eqs. (7), the phase-matching condition ${\overrightarrow{k}}_{\perp }^s=-{\overrightarrow{k}}_{\perp }^p$ is readily obtained, and the resulting equations for f and g are [21]

Assuming f(0) = 1 and g(0) = 0, we follow Ref. [21] and find along the medium

In the limit where |B| and |C| are much smaller than |A| and |D|, the solution is governed by independent EIT for the probe and Stokes fields with little coupling between them. In the opposite limit, the fields experience strong coupling, and the real part of the eigenvalues λ_1,2 can be made positive and result in gain.

3. Comparison with experiments

To verify our model, we have calculated the probe transmission as a function of the two-photon detuning and compared it to the data published in Refs. [22, 23]. The Doppler effect due to the motion of the thermal atoms is taken into account by averaging the FWM coefficients, Q = A,B,C,D in Eq. (8), over the Doppler profile [26]. Assuming nearly collinear beams, the mean coefficients are

Figure 2 shows the transmission spectrum in (a) rubidium vapor and (b) sodium vapor (cells length L ≃ 5 cm). Here and in what follows, we characterize the resonance conditions by the one-photon detuning Δ_1p = ω_c − ω_ur and the two-photon detuning Δ_2p = ω_p − ω_c − ω_lr [see Fig. 1(a)]. Our model reproduces the experimental spectra, including the Doppler-broadened absorption lines and the gain peaks for both rubidium and sodium experiments. The missing peak in Fig. 2(b) is due to anti-Stokes generation not included in the model.

 

Fig. 2 Transmission spectra of FWM in (a) rubidium vapor and (b) sodium vapor for a weak probe as a function of the two-photon detuning. The red circles are experimental data from (a) Ref. [23] and (b) Ref. [22]. The black line is calculated from Eqs. (10)–(11) with the following parameters. For the rubidium experiment: ω_lr = −3 GHz, Γ_rl = 5 MHz, Ω_c = 165 MHz, Γ = 5.7 MHz, N = 1.9 × 10¹² cm⁻³, Δ_1p = ω_c − ω_ur = 0.8 GHz, L = 12.5 mm, T = 150° C, and k_⊥/q = 5.2 mrad. For the sodium experiment: ω_lr = 1.777 GHz, Γ_rl = 1 MHz, Ω_c =45 MHz, Γ = 5 MHz, N = 4.4 × 10¹¹ cm⁻³, Δ_1p = ω_c − ω_ur =2 GHz, L = 45 mm, T = 165° C, k_⊥/q = 4.5 mrad.

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4. Diffraction manipulation by FWM

We now concentrate on a specific choice of frequency detunings, for which the phase dependency of the FWM process can be used to manipulate the diffraction of the propagating probe and Stokes. To this end, we study the evolution of an arbitrary image F (r) imprinted on the probe beam, with the boundary conditions Ω_p (r_⊥, 0) = F (r) and Ω_s (r_⊥, 0) = 0. Our prime examples shall be the propagation of the image without diffraction or with reverse diffraction, both of which while experiencing gain.

4.1. Image propagation

We start by solving Eqs. (7) in the transverse Fourier basis,

We choose $|e^{{\lambda }_{2}z}|\gg |e^{{\lambda }_{1}z}|$ and obtain Ω_p(k_⊥,z) = Ω_p(k_⊥,0)e^Z, where

4.2. Suppression of paraxial diffraction

In general, the minimization of both the real and the imaginary k_⊥-dependencies of Z is required in order to minimize the distortion of the probe beam. To better understand the behavior of Z, we expand it in orders of $k_{\perp }^2$. Taking the limit

The k_⊥-dependency, governed by Z⁽²⁾, can be controlled through the FWM coefficients $\overline{A},\overline{B},\overline{C}$ and $\overline{D}$ given in Eq. (8), by manipulating the frequencies of the probe and control fields (ω_p, ω_c), the control amplitude Ω_c, and the density N.

We demonstrate this procedure in Fig. 3, using for example the experimental conditions of the sodium experiment, detailed in Fig. 2. First, we observe the gain of the probe and the Stokes fields in Fig. 3(a) and 3(b), as a function of the one- (Δ_1p) and two- (Δ_2p) photon detunings. The gain is achieved around the two-photon resonance (Δ_2p ≈ 0), either when the probe is at the one-photon resonance (Δ_1p ≈ 0) or the Stokes (Δ_1p ≈ ω_lr, here ≈ 2 GHz); the latter exhibits higher gain, since the probe sits outside its own absorption line. The real and imaginary parts of Z⁽²⁾ are plotted in Figs. 3(c) and 3(d). When ReZ⁽²⁾ = 0 (dashed line), the gain/absorption is not k_⊥-dependent, whereas when ImZ⁽²⁾ = 0 (solid line), the phase accumulation along the cell is not k_⊥-dependent. When both happen, Z⁽²⁾ = 0, and a probe with a spectrum confined within the resolution limit k_⊥ ≪ k₀ propagates without distortion. The exact propagation exponent Z as a function of k_⊥ for the point Z⁽²⁾ = 0 (Δ_2p ≈ 0.4 MHz, Δ_1p ≈ 0) are plotted in Fig. 3(e). As expected, both real (blue solid line) and imaginary (red dashed line) parts of Z are constant for k_⊥ ≪ k₀ (deviation of 1% within k_⊥ < k₀/2 and 0.1% within k_⊥ < k₀/4). In the specific example of Fig. 3, the probe’s gain is ~1.4, the Stokes’ gain is ~ 4, and k₀ ≈ 40 mm⁻¹.

 

Fig. 3 Numerical search for the detuning values that yield suppression of paraxial diffraction and positive gain. This example uses the conditions of the sodium system in Fig. 2. The colormaps as a function of the one- and two- photon detunings are: (a) the Probe’s gain, (b) Stokes’ gain, (c) ImZ⁽²⁾ of Eq. (18), and (d) ReZ⁽²⁾. The contour ImZ⁽²⁾ = 0 is plotted in solid line in (c). The contour ReZ⁽²⁾ = 0 is plotted in dashed line in (d). (e) The exact propagation-exponent Z [Eq. (15)] for the case Δ_2p ≈ 0.4 MHz and Δ_1p ≈ 0 [red square in (c) and (d)]. At these detunings, ReZ⁽²⁾ = ImZ⁽²⁾ = 0, and both the real and imaginary parts of Z are constant for k_⊥ ≪ k₀ = 40 mm⁻¹.

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To illustrate the achievable resolution, we shall employ a conservative definition for a characteristic feature size in the image in area units a = (2π/k_⊥)² [For example: for a Gaussian beam, a^1/2 shall be twice the waist radius, and, for the field pattern E = 1 + cos(k_⊥x) cos(k_⊥y), the pixel area is a². The Rayleigh length is qa²/8]. Figure 4 presents numerical calculations of Eqs. (13) in the conditions found above for a probe beam in the shape of the symbol (R) with features of a ≈ 0.025 mm² (corresponding to k_⊥ = k₀ = 40 mm⁻¹). The propagation distance is L = 45 mm, equivalent to ≲ 2 Rayleigh distances as evident by the substantial free-space diffraction. Indeed when Z⁽²⁾ = 0, the FWM medium dramatically reduces the distortion of the image due to diffraction. Note that the image spectrum (black dashed-dotted line) lies barely within the resolution limit and that the Stokes distortion due to diffraction is also reduced. We emphasize that direct numerical solutions of Eqs. (7) give exactly the same results.

We note that atomic motion plays a negligible role in the parameter regime considered here, because the typical timescales for it to be relevant (hundreds of ns) are much longer than the dynamical timescale of the system. The latter is governed by the power-broadened gain linewidths, resulting in group delays of a few ns. These relatively short delays (large broadenings) characterize the systems considered [22,23]; the required control-field intensity is on the order of 0.1–1 W for beams larger than the typical size of the image.

 

Fig. 4 Simulations demonstrating the suppression of paraxial diffraction by FWM. (a) incident image, (b) after propagating in free space, (c) probe image after propagating in the FWM medium, and (d) generated Stokes image. The diffraction terms of the Probe (blue dashed line) and Stokes (green circles) fields are compared with free-space diffraction (red line). The k_⊥ spectrum of the image is given for comparison (black dash-dot line). The probe propagates in the cell with very little diffraction, and the Stokes’ distortion due to diffraction is reduced. The calculation is carried out in the conditions highlighted in Fig. 3 [detunings Δ_2p = 0.4 MHz and Δ_1p = 0 and other parameters as in Fig. 2(b)]. The image is about 1-mm wide (features area 0.025 mm²) and the propagation distance 45 mm (equivalent to ≲ 2 Rayleigh lengths).

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4.3. Negative paraxial diffraction

Another interesting application of diffraction manipulation is imaging by negative diffraction, similar to the one proposed in Ref. [6]. Using the same tools as above, one can find the conditions for the reversal of paraxial diffraction, namely when ReZ⁽²⁾ vanishes and ImZ⁽²⁾ = 1 (free space diffraction is equivalent to Z⁽²⁾ = −i).

At these conditions, as demonstrated in Fig. 5, the FWM medium of length L focuses the radiation from a point source at a distance u < L to a distance v behind the cell, where u + v = L. The mechanism is simple: each k_⊥ component of the probe accumulates outside the cell the phase $-i{k}_{\perp }^2(u+v)/(2q)=-i{k}_{\perp }^{2}L/(2q)$ and inside the cell the the phase $i{k}_{\perp }^{2}L/(2q)$, summing up to zero phase accumulation. The probe image thus ’revives’, with some additional gain, at the exit face of the cell.

 

Fig. 5 Demonstration of paraxial lensing with FWM. The negative-diffraction medium of length L focuses an image located at a distance u < L to a distance v behind the cell, where u + v = L. In this example, u = L = 45 mm and v = 0. The probe gain is 1.5. The parameters (for which Z⁽²⁾ = i) are Δ_2p = 0.4 MHz, Δ_1p = −1.7 GHz, and N = 4×10¹² cm⁻³; other parameters are as detailed in Fig. 2(b).

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5. Conclusions and discussion

The suggested mechanism for manipulating the paraxial diffraction of light utilizes the k_⊥-dependency of the four-wave mixing process and is thus fundamentally different than that suppressing the diffraction of spatial solitons in nonlinear media. The inherent gain of the FWM process allows one to take advantage of high optical-depths while avoiding absorption and, by that, achieving higher resolution than with previous EIT-based schemes [5,6]. As oppose to a recent proposal incorporating FWM [18], our scheme does not require atomic motion and is expected to work even more efficiently in its absence. We have introduced a microscopic model for the FWM process, based on Liouville-Maxwell equations and incorporating Doppler broadening, and verified it against recent experimental results. The conditions for which the FWM process suppresses the paraxial diffraction were delineated. We have demonstrated the flexibility of the scheme to surpass the regular diffraction and reverse it, yielding an imaging effect while introducing gain. Our proposal was designed with the experimental limitations in mind, and its demonstration should be feasible in many existing setups.

The resolution limit $a^{-1}\propto k_0^2$ of our scheme (and thus the number of ’pixels’ S/a for a given beam area S) is proportional to the resonant optical depth. In practice, the latter can be increased either with higher atomic density N or narrower optical transitions. For example, using a density of N = 5 · 10¹² cm⁻³, 10 times higher than in the sodium setup of Ref. [22], the limiting feature area would be 250 μm² (k₀ ≈ 125 mm⁻¹). As long as NL = const, the other parameters required for the suppression of diffraction remain the same. At the same time, the reduced Doppler broadening in cold atoms media and in solids would substantially increase the resolution limit. Assuming cold atoms with practically no Doppler broadening (and ground-state relaxation rate Γ_lr = 100 Hz), the same limiting feature of 250 μm² can be obtained at a reasonable density of 10¹² cm⁻³. Finally, we note that the best conditions for suppression of diffraction are not always achieved by optimizing Z⁽²⁾ alone (first order in $k_{\perp }^2/k_0^2$); In some cases, one could improve significantly by working with higher orders. As demonstrated in Fig. 6, combining the aforementioned methods for resolution enhancement with N = 10¹² cm⁻³ cold atoms, a resolution-limited feature area as low as 100 μm² with unity gain can be achieved. Going beyond this resolution towards the 1 – 10 μm²-scale for microscopy applications requires the lifting of the paraxial assumption in the analysis.

 

Fig. 6 Calculations for cold sodium atoms at a density of N = 10¹² cm⁻³ and negligible Doppler broadening. The image size is 0.1 mm with 10-μm features. The propagation distance is 0.45 mm (∼1.5 Rayleigh length).

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The FWM process conserves quantum coherence on the level of single photons, as was shown theoretically [27] and experimentally [28] by measuring spatial coherence (correlation) between the outgoing probe and Stokes beams. An intriguing extension of our work would thus be into the single-photon regime. Specifically, the main limitation in the experiment of Ref. [28] was the trade-off between focusing the beams to the smallest spot possible while keeping the ’image’ from diffracting throughout the medium. Our scheme circumvents this trade-off by maintaining the fine features of the image for much larger distances.