Diffraction pattern with mechanical grating to produce square wavepackets and focusing light wave at z = z grating + L T / 2. The density plots are in 

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... x i is the half–width of the initial beamlet. Includ- ing the effect of aberration, the central peak of the diffraction pattern can still be well described by an Airy pattern. With the parameter set of Fig. 4, where the focal length is around 0 . 002 L T , ∆ x f as determined from an Airy–pattern fit to the focus is 0 . 0054 λ , while (A.2) yields ∆ x = 0 . 0049 λ . The focal length of the atom lenslets at the nodes of the focusing standing light wave depends on the intensity and detuning of the light wave and on the interaction time. The detuning has to be large enough that spontaneous emission from the upper atomic level of the transition does not play a significant role. In the limit of large laser intensity (saturation parameter s 1), this condition can be evaluated to ...

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... plane wave (on the right); with the incident atoms restricted by square apertures to the nodes of the light field (width of the apertures ∆ x = 0 . 5 d ) (center); and with incident “Gaussian beamlets” centered at the nodes with an rms width 0 . 088 d (left). The density plots of the two-dimensional probability distribution (with the propagation direction of the atomic wave shown vertically) show the characteristic “chan- nel” or “quantum carpet” pattern discussed extensively in the literature [16, 17]. Cross-sections through the distribution at z = z f and z = z f + L T / 20 (see the dotted lines in the figure) are shown at the bottom and top of the figure. It is obvious that, while the quality of the focus does not differ markedly, the quality of the high-order Talbot image is dramatically im- proved with the use of apertures. Both hard-edged square and Gaussian apertures can be realized experimentally. For square apertures, freestanding nanostructure transmission gratings with a period of 213 nm and slit widths around 100 nm , as would be needed for the chromium example, can be produced with relative ease at a number of facilities (e.g., the Cornell Nanofabrication Fa- cility or the MIT Nanostructures Laboratory). To obtain “soft- edged” Gaussian apertures physical transmission gratings are of course of no use. However, for metastable rare gas atoms, Gaussian apertures can be approximated with good accuracy by letting the incident atomic beam pass first through a separate standing light wave [18–20]. In this standing wave, tuned to an optical pumping transition, the atoms are excited from the metastable state to an upper level which decays prefer- entially to the atomic ground state. Thus, the atoms end up in the ground state except when they pass the standing wave near the nodes. Rare gas atoms in the inert ground state do not contribute to exposing the resist layer and hence can be neg- lected. The width of the “wavepackets” of metastable atoms which pass the nodes can be adjusted with the intensity of the light wave and the interaction time. The transverse probability of the wavepackets can be made close to Gaussian, as is illustrated in Fig. 3 where calculated and fitted Gaussian probability distributions are shown. The calculation is based on a similar numerical integration of the one-dimensional wave equation as before, including the spontaneous emission from the upper level of the transition to the ground state as a damping term. The resonant light wave used for the optical pumping will be relatively weak (just enough to saturate the transition). Be- cause of this, the non-dissipative part of the interaction of the metastable atoms with the light wave is weak and does not cause appreciable focusing effects. For the focusing, an intense light wave can be used which is far detuned with respect to the same optical transition. By detuning, optical pumping can be suppressed while inducing strong focusing. For example, the parameters used in Figs. 1 and 2 can be achieved for ( 360 m / s axial velocity) metastable argon atoms on the λ = 706 . 9 nm transition with a laser power of 35 mW in a Gaussian beam with a waist of 18 μ m and a detuning of 23 GHz . The probability for optical pumping will then be less than 2 . 5 %. In principle, the difference in wavelength between the resonant optical pumping light field and the off–resonant focusing field causes a mismatch between the periods of the incident array of Gaussian beamlets and the focusing array. However, even at 23 GHz detuning the relative mismatch is only 8 × 10 − 5 . Hence, a few thousand periods of the focusing field can be supplied with Gaussian beamlets positioned at the nodes with acceptable tolerance. In practice, positioning a mechanical grating immediately in front of the focusing light field is difficult. Since after the grating, the initially square wavepackets spread quickly because of diffraction, this means that the atoms cannot be restricted to the nodes of the focusing field. For the optical pumping approach, the same problem exists: if the distance between the optical pumping field and the focusing field is too short, overlap between the light fields results in unwanted interference effects. The problem can be solved by separating grating and focusing field (or the two light field) by an axial distance z = L T / 2. At this distance, a Talbot image of the original wavepackets is produced which has the same periodicity as the original, but is shifted by half a period. The full diffraction patterns with this approach when using a mechanical transmission grating and when using optical pumping are shown in Figs. 4 and 5 respectively. The positions of the light waves and gratings are indicated in the figure. The area near the focus ( z z f 1 004 L T 2) and an area near z z f L T / 20 (indicated by brackets) are depicted enlarged at the bottom of each figure, the latter region illustrating the fractional Talbot images at L T / 20. For Fig. 4, the parameters used for the calculation can be achieved, e.g., by focusing 360 m / s (axial velocity) metastable argon atoms with a laser beam with a Gaussian waist radius w l = 25 μ m near–resonant with the λ = 811 . 5 nm two–level transition, but with a larger laser power and detuning than in Fig. 1 ( 14 mW at ∆ ν = 44 GHz ). This leads to somewhat stronger focusing. For Fig. 5, where optical pumping is used to produce near-Gaussian beamlets, the weaker optical transition at λ = 706 . 9 nm requires even larger power and detuning ( 260 mW at 45 GHz as well as a smaller laser beam ( w l = 25 μ m ). The dashed lines in the figures indicate the positions of the two laser beams (or the mechanical transmission grating and the laser beam) at z = 0 and z = L T / 2. The probability distribution as a function of x in the focus plane and at z = z f + L T / 20 is shown at the right in the figures. For a discussion of the width of the focal spot, and the influence of residual aberrations on this width, see Appendix A. In a real experimental setup, the perfectly monochromatic, plane wave, assumed in the calculations above, cannot be realized. The atomic beam originates from a finite sized source and is collimated with apertures. As a result, the beam can be described by an incoherent superposition of plane waves, incident under all angles allowed by the collimation ratio of the beam. After passing through a regular structure, the Talbot image of each plane wave is simply projected under the angle of incidence of the wave and the associated probabilities have to be summed. In order for the high-order Talbot image to be retained under the summation, the spread in position of the individual images in the image plane has to be smaller than the distance between the peaks in the image. We will first look at the visibility of the image at L T / 2. With L T = 2 d 2 /λ dB ( d = λ/ 2 and λ dB = h / m v z , with m the atomic mass and v z the (axial) velocity of the atoms in the beam), we can also express the Talbot length ...