Abstract
Any image, imprinted on a wave field and propagating in free space, undergoes a paraxial diffraction spreading. The reduction or manipulation of diffraction is desirable for many applications, such as imaging, waveguiding, microlithography and optical data processing. As was recently demonstrated, arbitrary images imprinted on light pulses are dramatically slowed^{1,2} when traversing an atomic medium of electromagnetically induced transparency^{3,4} and undergo diffusion due to the thermal atomic motion^{5,6}. Here we experimentally demonstrate a new technique to eliminate the paraxial diffraction and the diffusion of slow light, regardless of its position and shape^{7}. Unlike former suggestions for diffraction manipulation^{8,9,10,11,12}, our scheme is linear and operates in the wavevector space, eliminating the diffraction for arbitrary images throughout their propagation. By tuning the interaction, we further demonstrate acceleration of diffraction, biased diffraction and induced deflection, and reverse diffraction, implementing a negativediffraction lens^{13}. Alongside recent advances in slowlight amplification^{14} and image entanglement^{15}, diffraction control opens various possibilities for classical and quantum image manipulation.
Main
Electromagnetically induced transparency (EIT) is a coherent twophoton interaction between light and atoms^{16}. It involves an atomic medium and two light fields, usually a strong ‘pump’ and a weak ‘probe’, resonantly coupling two of the atomic levels to a common excited level. When the Raman resonance condition is satisfied, namely when the twophoton frequency detuning, Δ, is within the atomic spectral width, Γ, the atoms are driven towards a ‘dark’ state, which substantially reduces the absorption in the medium. A short probe pulse propagates in the medium with a reduced group velocity, owing to the steep dispersion inside the narrow transparency window^{4,17,18,19,20,21}. A travelling atomiccoherence field accompanies the probe, and the combined light–matter excitation is termed a darkstate polariton^{22}. If an image is imprinted on the probe field in the plane normal to the propagation direction, the complex amplitude of the darkstate polariton follows the amplitude and phase of the image. Freespace diffraction, being essentially a geometric effect, occurs for slow images precisely as it would in free space.
In an EIT medium of hot vapour, the thermal motion of the atoms affects the propagation of images. The addition of a buffer gas attenuates the thermal motion, which becomes diffusive and can be characterized by a single diffusion coefficient, D. As a result, on EIT resonance, the atomic part of the darkstate polariton undergoes diffusion while propagating slowly in the medium^{5}. The polariton thus experiences both diffraction and diffusion. Here, we exploit the atomic motion to influence the diffraction, as illustrated in Supplementary Fig. S1. Altering the Raman detuning provides control of the polariton’s coupling with atoms moving in a desired direction. We therefore counterbalance diffraction by ‘Doppler trapping’ the outwardsconfronting light components with inwardsmoving atoms or, alternatively, force diffraction in a preferable direction.
The freespace diffraction of an optical field envelope E(x,y;z) travelling in the z direction is described in the transverse Fourier plane (Fig. 1a) as , where is the transverse wavenumber, q=2π/λ and λ is the optical wavelength. In a vapour EIT medium, the propagation of a weak probe depends on its angular deviation from the pump due to the Doppler–Dicke effect^{23,24}. For a wide homogenous nondiverging pump, finitepump effects such as transverse intensity variation^{25} and Ramsey narrowing^{26} are made negligible, and the dynamics of the probe becomes^{5} , where is the linear susceptibility and
Here, 2α is the absorption outside the EIT window, Γ_{p} is the powerbroadening width, proportional to the pump intensity, and D k^{2} is the Doppler–Dicke width. The latter has a simple physical interpretation: both the residual Doppler broadening and the Dicke narrowing are linear in k, which corresponds to the angle between the probe and pump, resulting in a combined quadratic effect^{24}. At a given Δ, different k components of a probe image experience different EIT spectra, and consequently the image is altered. When Δ=0 (red lines in Fig. 1b), χ_{EIT} is purely imaginary and induces a lowpass absorption filter in k space, with a halfwidth of . For k≪k_{0}, the filter is quadratic in k and corresponds to standard diffusion, accompanied by the freespace diffraction. To manipulate diffraction, a nonzero Raman detuning should be introduced. The case Δ=±Γ is of special importance: in this case, the leading quadratic term becomes purely real, inducing diffraction without diffusion,
with v_{g}=Γ^{2}/(α Γ_{p}) being the slow group velocity. Here, the induced diffraction is continuous, in contrast to the diffraction manipulation explored in periodic systems^{27,28,29}. When Δ=−Γ the induced diffraction negates the freespace diffraction, and if v_{g}=D q they are cancelled altogether. As demonstrated in Fig. 1b (black lines), for k≪k_{0}, the susceptibility curves are flat and both diffraction and diffusion are eliminated. In contrast, when Δ=+Γ, the induced and the freespace diffraction add together, increasing the overall diffraction. Note that for given D and q the amount of induced diffraction is determined exclusively by the group velocity, which is easily controlled by the pump power and the atomic density. A major difficulty of the scheme is the substantial absorption at Δ≠0. Fortunately, this absorption is uniform both in real and k spaces and may be compensated for by any linear gain mechanism^{7,14}.
The experimental setup is depicted in Fig. 1c. A glass vapour cell of length L=50 mm is airheated to 72 ^{∘}C and filled with ^{87}Rb and 10 torr of neon buffergas. The rubidium diffusion coefficient is D=1,100 mm^{2} s^{−1}, satisfying the diffractionelimination condition with . For our optical depth (∼6), we arrive at these conditions with a pump intensity of 660 μW mm^{−2}, yielding an EIT linewidth of Γ=70 kHz and transmission at Δ=±Γ of about 1%. In each shot, an image is imprinted on a weak probe pulse (2 μW mm^{−2}) and projected onto the entrance facet of the cell. The transmitted probe at the exit facet is recorded with a CCD camera.
A demonstration of image propagation without diffraction is presented in Fig. 2a with an image of the symbol ‘®’. The 100μm features in the image are significantly distorted after 50 mm of freespace diffraction. On the EIT resonance (Δ=0), the image covers this distance in 5.75 μs, during which it diffracts and diffuses. However, for Δ=−Γ, both diffraction and diffusion are clearly suppressed. The corresponding calculations verify that the observed minor spreading for Δ=−Γ is due to subdiffractive^{29} and subdiffusive terms of fourth order in k. Indeed, the k spectrum of the image extends beyond the k≪k_{0} region (k_{0}=20 mm^{−1}), as seen in Fig. 1a (Experiment I). For Δ=+Γ, and as predicted in equation (2), the image does not diffuse but rather undergoes substantial diffraction, of effectively twice the distance travelled. The presented calculations are done numerically by taking the Fourier transform of the twodimensional incident field, multiplying by , and taking the inverse Fourier transform. This computation does not require the paraxial k≪k_{0} approximation and is hereafter denoted as the numerical calculation.
A slow image of a line grating, shown in Fig. 2b, provides a quantitative measurement of the actual diffraction and diffusion. This image has the property that it reappears periodically after propagating a distance of L_{T}=2a^{2}/λ, known as the Talbot selfimaging distance, where a is the grating period. At L_{T}/2, a reciprocal grating is created; the original lines disappear and new lines appear in the originally dark areas. In our experiment, the cell length was ∼L_{T}/4, at which the amplitudes of the original and reciprocal gratings are equal, resulting in a grating with a period of a/2.
The results for Δ=−Γ,0,+Γ are presented in Fig. 2b. Evidently, the image does not change for Δ=−Γ, and it is essential to note that, in contrast to a selfimaging effect, the image is maintained throughout its propagation. In the Δ=0 image, diffusion broadens the lines and erases the grating. For Δ=+Γ, we observe double diffraction, with the reciprocal grating dominating the original, in correspondence to L_{T}/2. To quantify the amount of diffraction, we measure the intensities at the positions of the original and reciprocal gratings, 〈I_{o}〉 and 〈I_{n}〉 respectively, and define the contrast C=(〈I_{o}〉−〈I_{n}〉)/(〈I_{o}〉+〈I_{n}〉), shown in Fig. 2b. C=1 (−1) at integer (halfinteger) Talbot distances, and C=0 when the original and reciprocal gratings are comparable. The maximum value of C is obtained at Δ=−Γ and is about 0.85. Notice that C=0 also when the gratings are indistinguishable owing to diffusion, and quantitative indicators that separate the diffusion and diffraction effects are given in the Supplementary Information.
In addition to the magnitude of diffraction, we can use the atomicmotion mechanism to control the diffraction directionality. By setting a small angle θ_{pump} between the pump and the z axis, as illustrated in Fig. 3a, we superimpose a transverse phase grating, exp(i x q θ_{pump}), on the pump–probe interference^{5}, replacing D k^{2} in equation (1) with (see Supplementary Equation S8). For Δ=±Γ and θ_{pump}≪(k_{0}/q), a term proportional to , which inflicts a directional deflection on the probe, at an angle , is added to the dispersion in equation (2). Similarly to the walkoff phenomenon in birefringence crystals, the beam is deflected whereas the carrier wavevector stays parallel to the z axis (equalphase surfaces maintain their original orientation). Hence, the beam keeps its original direction on exiting the cell.
We have performed deflection experiments in the conditions for diffraction elimination, v_{g}=D q and Δ=−Γ. Figure 3b presents the measured θ_{probe} versus the applied θ_{pump}, showing that θ_{probe}=θ_{pump} as long as the paraxial approximation holds (q θ_{pump}≪k_{0}). This striking phenomenon, that the probe takes the direction of the pump while in the cell regardless of the incident angle, provides another explanation for the elimination of diffraction. As illustrated in Fig. 3b (inset), all diverging k components of a focused beam are made to propagate in the axial (pump) direction while maintaining their phase relation, thus detaining the diffractive divergence. Supporting measurements and an analytic derivation are given in Supplementary Information.
Diffraction vanishes when the induced diffraction, D/(2v_{g}), is equal in size and opposite in sign to the freespace diffraction, 1/(2q). However, if the induced diffraction is strengthened, for example by lowering the group velocity, the overall diffraction will become negative. Such a medium reverses the diffraction of slow images and can undo diffraction that has already taken place. As far as spatial diffraction is concerned, a medium with negative diffraction behaves similarly to a negativerefractionindex medium, within the limits of the paraxial approximation. The effective index is n_{eff}=(1−q D/v_{g})^{−1}. When v_{g}=q D/2, n_{eff}=−1 and the overall diffraction becomes exactly the opposite of freespace diffraction.
A fascinating application of negativeindex materials is an unusual lens^{13}, demonstrated in Fig. 4. This slabshaped lens of length L and n_{eff}=−1 focuses the radiation coming from any point source, located at a distance u<L, to a distance v behind the lens, where u+v=L. Parallel light rays, however, are not focused and continue in their original directions. Indeed, the images in the experiment are reconstructed by the lens regardless of the distance u.
As a geometrical effect, paraxial diffraction is manifested through k dependence of the dispersion curve. The optimal approach to eliminate it is to superimpose dispersion with the same form and the opposite sign that is uniform in real space. An EIT vapour medium with a buffer gas and a uniform pump manifests these properties. The pump provides the anisotropy needed to single out the transverse wavevector, and Dicke narrowing accounts for the quadratic k dependence. The strength, frequency and orientation of the pump control the magnitude, sign and direction of the resulting diffraction. The possibilities may be further extended by introducing a nonuniform pump. The phase gradient of the pump then acts as a vector potential for the probe, in an effective Schrödinger dynamics of a charged particle. Currently, inherent loss limits the effectiveness of the scheme, for example in terms of resolution. Resolving this limitation by combining linear gain, probably by using more elaborate Raman processes, would clear the way to a vast variety of applications, in microscopy, lithography, switching and more.
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Acknowledgements
We thank D. Yankelev for assistance with the experiments and Yoav Erlich for technical support. We thank R. Pugatch for discussions and suggestions.
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O.F. conceived the idea with significant input from A.R. and N.D. O.F. and A.R. carried out the theoretical modelling. The experiment was conducted by O.F. with the assistance of P.L. and M.S., and all authors contributed to the data analysis. The paper was written by O.F., M.S. and N.D.
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Firstenberg, O., London, P., Shuker, M. et al. Elimination, reversal and directional bias of optical diffraction. Nature Phys 5, 665–668 (2009). https://doi.org/10.1038/nphys1358
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