We have guided cold 85Rb atoms in a blue-detuned, high-order hollow laser beam generated by a binary spatial light modulator. The binary holograms written to the modulator produce smooth hollow laser beams with steep intensity walls that can be updated with a 1.6 kHz refresh rate. We analyze the guiding laser beams numerically and experimentally, and show that the atoms are exposed to an average intensity that is ~2% of the maximum intensity of the guide at a laser detuning of 1 GHz and 2 mW of optical power.
©2006 Optical Society of America
In recent years there has been substantial interest in confining and guiding cold atoms using blue-detuned optical fields . Atoms seek regions of low intensity in blue-detuned laser fields, thereby reducing spontaneous photon scattering events, collisional losses, energy level shifts, and required laser powers. Traps based on blue-detuned light are generally more difficult to create than red-detuned traps, which can be formed by focusing standard Gaussian laser beams. Hollow laser beams have shown to be promising guides for cold atoms, and several techniques have been discussed and employed for generating static blue-detuned hollow guides, including axicon prisms  and holographic phase plates [3–4], wedge prisms , and astigmatic mode conversion . Spatial light modulators (SLMs) provide a dynamic holographic technique for producing various optical traps and guides , including hollow beams [8–9]. Chattrapiban et al. have discussed the use of a nematic liquid crystal SLM to generate hollow Bessel beams that can be altered on the time scale of 10 Hz . Daria et al. used a nematic SLM to generate an array of dark optical traps for microspheres . Nematic SLMs have also been used to confine atoms using red-detuned light with standard Gaussian beams , but the ability to adjust the potentials on a rapid time scale would open new avenues for dynamic manipulation of atoms in complex guides and traps. By altering the local intensity with a programmable hologram, the motion of an atom can be controlled spatially, temporally, and in a complex manner with no moving parts. High-speed manipulation of microspheres with Gaussian beams has recently been demonstrated with a ferroelectric liquid crystal (FLC) binary SLM . These high-speed SLMs can generate clean hollow beams  but to our knowledge they have not been used to guide or control cold atoms.
In this paper, we demonstrate, to the best of our knowledge, the first use of a FLC binary SLM to guide cold atoms in a hollow laser beam. The atoms are confined radially in a 0.5 mm diameter beam with a detuning of 1 GHz and only 2 mW of diffracted power. We also describe the propagation properties of these beams through the focus of a lens experimentally and numerically and find that the beams reach maximum intensity prior to the focal plane of the lens. We demonstrate the effect of rapidly switching the intensity pattern on a sub-millisecond timescale. Although binary phase modulation has roughly half the diffraction efficiency of continuous phase modulation, (FLC) binary SLMs have update rates of a few kHz, significantly faster than the nematic SLMs, which are limited to refresh rates on the order of 10 Hz. In this way, FLC SLMs may be useful for dynamic manipulation and smooth spatial control of atoms .
The potential energy U(r) of a two-level atom in an intensity distribution I(r) is given by
where Δ=ωL - ωo denotes the laser frequency (ωL) detuning from the atomic resonance (ωo) in rad/sec . The saturation intensity I o is 1.6 mW/cm2 and the natural linewidth Γ for Rb is 2π × 6.1 MHz. When ΔΓ ≫ [I(r)/Io]1/2, Eq. (1) can be approximated as
Under this condition, the scattering rate Γsp of an atom in this field is approximated as 
By blue-detuning the optical potential, the atoms seek low intensity and the scattering rate is reduced substantially. This benefit has been utilized in a single-beam, blue-detuned optical trap made by a lithographed phase plate [ 4]. In that paper, the scattering rate was reduced to 1/700 of the maximal scattering rate for the trap.
A laser field modified by a transmission function has a far-field intensity pattern proportional to the square of the Fourier transform of the transmitted field. In principle, a transmitted field that spatially averages to zero will have no intensity along the axis in the far field because there is no DC term in the Fourier transform. An azimuthally symmetric hollow laser beam can be formed by modifying a uniform laser field with a phase profile of the form Ψn(ρ,ϕ)=nϕ + αρ, where ρ, ϕ, are cylindrical coordinates, n is an integer, and α is a constant [15, 16]. These profiles are simple to encode using nematic SLMs with continuous phase programmability [–8]. The primary drawbacks of nematic SLMs are low refresh rates of 10 Hz, and interframe artifacts, which limit their usefulness for applications requiring dynamic manipulation. Binary phase modulation is less efficient than continuous phase modulation, but FLC binary SLMs have refresh rates exceeding 1 kHz and can also generate clean hollow beams . The theoretical diffraction efficiency into first order of a phase profile that has been discretized into Q phase levels is (Q/π)2 sin2 (π/Q) or 0.41 for Q = 2 .
2. Hollow beam characteristics
Figure 1(a) shows the setup using a 1280 × 1024 reflective binary SLM (CRL Opto SXGA-R2-H1). Collimated s-polarized light from a polarization-maintaining fiber strikes the SLM, and each 15-micron-pixel acts as an orientable half-waveplate providing either 0 or π relative phase. The incident beam has an intensity profile I(ρ) = I0exp[-2(ρ/ωo)2], where ωo is the beam waist. The p-polarized reflected light is directed through an achromatic lens with focal length f = 300mm after passing through a polarizing beamsplitter cube (PBS). This lens is mounted on a translation stage so that we can align the atoms in the magnetooptical trap (MOT) with different locations along the guiding beam. The phases written to the SLM are discretized to 0 or π: For Ψn(ρ,ϕ) mod 2π > π the pixel value is π, otherwise it is 0. The phase profile for n = 16 is shown in Fig. 1(b); prior to the discretization, we added a linear phase ramp to Ψn(ρ,ϕ) to separate the diffracted light from the zeroth order, undiffracted light.
For this report, we have demonstrated atom guidance for n = 16. It has been shown that for larger n, the intensity of the hollow beam is greater for the same beam diameter . This property allows atom guidance with low laser power. Furthermore, we demonstrate here experimentally and numerically that for n = 16, the hollow beam reaches its highest intensity prior to the focal plane of the imaging lens for finite-sized beams. We have found this property to be true for n > 1 . This is in contrast to Gaussian laser beams and fundamental (n = 1) hollow laser beams, which both reach peak intensity in the focal plane . To model the behavior of the hollow beam through the focal plane, we have integrated the Fresnel diffraction integral numerically with our experimental parameters.
Figure 2 shows the theoretical intensity profiles at various propagation distances from the imaging lens to the focal plane for n = 16, an initial beam waist ωo=2.2mm, and a focal length f = 300mm. Because binary patterns diffract light equally into the 1st and -1st orders, there are two hollow beams generated, each diffracted symmetrically away from the original laser beam axis. In Fig. 2, we have moved the simulation axis to the center of one of these hollow beams. Included in Fig. 2 is an experimental intensity profile taken 3 cm before the focal plane of the lens, which shows good agreement with the numerical simulations. Beyond ~15 cm, the two beams are sufficiently separated and have little axial intensity. We find that when the +1st and -1st orders are sufficiently separated, the two beams no longer interfere and the beam uniformity approaches that generated by continuous phase modulation . From this point, they continue to propagate through the focal plane with a hollow profile, reaching a maximum intensity near Z=28.5 cm for these beam parameters. A more detailed analysis of the propagation characteristics of these beams will be published elsewhere.
We have also plotted the radial intensity profiles of the hollow beams for several propagation distances near the focal plane (Fig. 3). In the focal plane, the beam is hollow, but the wall thickness is large – almost 200 μm. Moving out of the focal plane results in a more clearly defined hollow profile with a steeper intensity gradient, thinner wall, and higher peak intensity. For the beam parameters used in this experiment, the position of maximum intensity occurs before the focus of the lens by about 1.5 cm. We performed our experiments with the MOT located roughly 2 cm from the focus of the lens. At this location, the hollow beam diameter is 510 |jm with peak intensity 3 1I o. The wall thickness of the hollow beam is approximately 40 μm (full width at half maximum). With Δ = 2π × 1000 MHz, the potential depth is ~0.2 ħr, and the peak scattering rate is 50 kHz.
3. Atom experiments
We begin the experiment with 107 85Rb atoms in a standard magneto-optical trap (MOT), using the F=2 - F’=3 transition as a repump; the repump beam is superimposed on all 3 retroreflected MOT beams. Atoms are loaded for 5 seconds and then cooled in optical molasses for 10 ms to ~35 μK, or 0.12hr, measured by the expansion of the cloud with no beams present. One of the diffracted hollow beams in Fig. 2 is aligned with the MOT by first tuning the guide laser to atomic resonance. The hollow beam phase pattern (n = 16) is then replaced with a standard Gaussian beam by setting n = 0. The alignment is optimum when the MOT is most strongly perturbed. The polarization of the hollow beams is linear and the magnetic fields are shut off during the guiding period.
The laser frequency for the hollow beam is derived from the cooling transition laser by an acousto-optic modulator (AOM) capable of shifting the detuning Δ/2π from 400 – 1000 MHz above the F=3 - F’=4 hyperfine transition. This laser frequency is red-detuned from all F=2 -F’ hyperfine transitions, but the repump laser remains on during the experiment so that the atoms spend most of their time in the F=3 hyperfine level. The AOM-shifted light injection locks another diode laser to the same frequency, and 50 mW of guiding light is coupled into polarization-maintaining fiber. The light exiting the fiber is collimated to a beam waist ωo = 2.2 mm so that no aperture effects of the SLM are present. The SLM for this work only has 4% diffraction efficiency into one hollow beam, leaving only 2 mW available for the guiding. Binary SLMs with improved efficiency are now available.
At time T = 0, the F=3 - F’=4 MOT cooling beams are shut off and the hollow beam is switched on. Figure 4 shows images of the atoms at later times with and without the optical guide. At Δ/2π = 1000 MHz, the maximum potential of the guide is 0.2 ħT. This is sufficient to guide most of the 35 μK atoms originating from the MOT. Without the guide present, the atoms expand ballistically and fall under the influence of gravity. With the 2-mW hollow beam present, the atoms are supported against gravity and expand axially along the hollow beam. We are able to use such low diffracted power because of the steepness of the potential wall offered by high charge number hollow beams. The relative number of atoms remaining in the guide, measured by integrating the amount of fluorescence in the 1-cm-long imaging length, decays with a time constant of roughly 10 ms (Fig. 5). This exponential decay time is measured after T=5 ms, when most of the atoms initially outside or on the guide have been pushed away.
The atoms leave the imaging volume through several channels. First, the hollow beam is smaller than the starting size of the atom cloud, so many of the atoms are initially either outside the hollow beam or starting in regions of high intensity. Second, the guide potential depth is approximately the same as the mean temperature of the atoms. Third, the gravitational potential for atoms starting in the middle of the 510 μm-diameter guide is approximately 25 μK, so that many of the atoms sample the highest intensity in the beam and are heated by spontaneous emission events. We repeated the experiment for Δ/2π = 440 MHz, 800 MHz, and 1000 MHz, and the benefit of increased detuning is seen clearly in Fig. 6. At 20 ms, almost none of the atoms remained in the imaging volume at the detuning of 440 MHz. Although the trapping potential increases for smaller detuning, the increased scattering rate heats the atoms and pushes them out of the imaging volume more quickly. Finally, atoms leave the imaging volume through the open ends of the guide. We have not used “plug” beams that would confine the atoms . By using an SLM with higher diffraction efficiency, significant improvements to the guiding efficiency are expected.
The atoms in a blue-detuned guide should scatter photons at a much lower rate than a red-detuned guide of the same trap depth and detuning magnitude. We can extract some information about the effective scattering rate of the atoms in this guide by measuring the location of the center of mass of the atom cloud as a function of time. Our guide is horizontal, and the atoms are released with zero velocity on average but the center of mass is pushed approximately 1 mm in 10 ms, which corresponds to an average scattering rate of 1 kHz. This rate is 2% of the maximum scattering rate for this potential (50 kHz). This may be reduced further depending on the potential configuration, detuning, and atom temperature. Ozeri et al. demonstrated a single-beam, blue-detuned optical trap in which atoms felt an average intensity 1/700 of the maximum light intensity of the trap .
One potential benefit of the binary SLM is the high refresh rate possible for quickly reconfiguring the optical potentials . To demonstrate one use of this high update rate, we have switched the pattern between two different intensity profiles at a 1.6 kHz update rate. The first frame is the “leaky” hollow beam shown in Fig. 7; the second is a similar leaky hollow beam [Fig. 7(b)] rotated by π/2n so that the sum of the two frames is a complete hollow beam [Fig. 7(c)]. These patterns are generated by writing the same phase profile as used earlier, but without the linear grating term. This causes the +1 and -1 orders to interfere and generates an axial hollow laser beam whose circumferential intensity profile has a sin2(nϕ) dependence. As shown in Fig. 7(d), this rapid averaging does help to confine the atoms, but at 35 μK, most of the atoms can still escape the guide. During each 600 μs period, these atoms move ~50 μm comparable to the thickness of the intensity wall we have written (40 μm). These losses can be reduced for atoms with lower transverse velocity, or for situations where the intensity profile is more gradually changed. In practice, one would not perform temporal averaging when a cleaner option is available, but the high update speed may be useful for situations in which a smooth hollow beam or pattern is smoothly scanned or varied in size.
We have demonstrated cold atom guidance in a hollow laser beam created by a binary spatial light modulator. We examined the hollow laser beams both numerically and experimentally as they propagate through the focus of a lens, and showed that they reach the highest intensity away from the focal plane where aberrations are less important. We used this property to create steep intensity walls with only 2 mW of diffracted power. We also demonstrated that the atoms sample on average only 2% of the maximum intensity of the beams.
This work was funded in part by the Office of Naval Research and by the Defense Advanced Research Projects Agency.
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