Topic area: AOS
Holographic imaging of cold atoms L. D. Turner, K. F. E. M. Domen† and R. E. Scholten School of Physics, University of Melbourne, Melbourne, Australia † Department of Physics, Eindhoven University of Technology, Eindhoven, The Netherlands e-mail of corresponding author:
[email protected] Introduction We describe a novel holographic method of imaging, which retrieves high-resolution images of objects from diffraction patterns, and apply our method to lensless and minimally-destructive imaging of cold atoms. Unlike traditional holography, the image is retrieved by a deconvolution algorithm rather than optical reconstruction (or its digital equivalent). The algorithm is restricted to weakly-absorbing monomorphous objects for which the phase-shift and absorption are proportional to the object column-density. The solution is more numerically stable if the object advances the phase of the incident wave; this is equivalent to having a refractive index with real part less than one. Conventional optical materials do not meet this criteria. Two exceptions are x-ray imaging, where the refractive index is usually less than one [1], and imaging atomic gases with light detuned slightly to the blue of an atomic resonance. For these situations, the algorithm retrieves quantitative images of column density from a single diffraction pattern. No lenses, beam splitters or other optics are required and the retrieval is fast, stable and devoid of the twin image contamination inherent to conventional inline holography. Connecting diffraction patterns and column density The object-plane wavefield after an optically thin object can be written f (r) = f0 exp(−µ(r) + iφ(r)),
(1)
where µ(r) is the absorption and φ(r) the phase-shift due to the object at transverse coordinate r and I0 = | f0 |2 is the intensity of the illuminating plane wave. This wavefield f (r) may be propagated through a distance z by the Fresnel transform +∞ i iπ 2 f (r, z) = exp(ikz) f (r) exp (2) |r − r | dr λz λz −∞ where λ = 2π/k is the wavelength of the illuminating plane wave. The Fresnel approximation agrees closely with rigorous diffraction theory except for propagation at large angles to the axis or within a few wavelengths of the object. Optical detectors measure intensity I(r, z) and it can be shown that the Fourier transform F of the diffracted intensity can be expressed in terms of the object-plane wavefield f (r) as [3]
F [I(r, z)] =
Presenting author’s name:
+∞ −∞
f ∗ (r + λzu/2) f (r − λzu/2) exp(−2πir·u) dr,
L. D. Turner
(3)
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Topic area: AOS in which u is the spatial frequency conjugate to r. Substituting Eq. (1) into this expression yields
F [I(r, z)] = I0
+∞ −∞
exp {−µ(r + λzu/2) − µ(r − λzu/2) + i [φ(r − λzu/2) − φ(r + λzu/2)]}
× exp(−2πir·u) dr.
(4)
Assuming both real and imaginary parts of the exponential are small, we expand, noting the Fourier transforms:
F [I(r, z)]/I0 = δ(u)−F [µ(r + λzu/2) + µ(r − λzu/2)] +iF [φ(r − λzu/2) − φ(r + λzu/2)].
(5)
Applying the Fourier shift theorem F [ f (r − a)] = exp(−2πia · u) F [ f (r)] to each term and rearranging recovers F [I(r, z)] = I0 δ(u) − 2 cos(πλzu2)F [µ(r)] + 2 sin(πλzu2 )F [φ(r)] , (6) which is a linear expression relating absorption and phase-shift to the intensity of the diffraction pattern, in Fourier space. Retrieving column densities from diffraction patterns If the object is monomorphous (made of a single material) then both absorption µ and phase-shift φ are proportional to the column density ρ, µ(r) = kβρ(r) and
φ(r) = kδρ(r),
and then Eq. (6) can be solved for the column density by 1 I(r, z) −1 F −1 . ρ(r) = F 2k (δ sin(πλzu2) − β cos(πλzu2)) I0
(7)
(8)
The linearising assumption made in obtaining Eq. (5) is that 2µ(r) 1
and
|φ(r + λzu/2) − φ(r − λzu/2)| 1.
(9)
The object must be weakly absorbing, but it need not be non-absorbing. The phase-shift should not vary too rapidly, but weak phase-shifts are not required. The phase condition in Eq. (9) may always be met at small z, but phase objects of many radians thickness may require impractically small propagation distances and phase shifts of order 1 radian are preferable. Directly implementing Eq. (8) is not possible as the denominator vanishes at certain spatial frequencies. The inverse problem may be regularised by, for example, the Tikhonov method [4]. For entirely transparent objects (β = 0), there remains a fundamental catastrophe at u = 0, but this is avoided by even minor residual absorption. Retrievals are best for phase-advancing objects (δ < 0). The column-density may be retrieved for phase-retarding objects (δ > 0), however the focusing action of the phase-shift cancels the absorption at low spatial frequencies leading to lower quality retrievals.
Presenting author’s name:
L. D. Turner
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Topic area: AOS Off-resonant imaging of atoms Imaging ultracold clouds of atoms such as Bose-Einstein condensates by resonant absorption usually destroys the cloud: the trap is shallower than a photon recoil. Detuning the probe beam from resonance reduces the absorption and the cloud becomes a phase object. For clouds optically thick on resonance, detuning until this phase-shift is of order one radian will reduce the absorption sufficiently that many images may be taken before the cloud is appreciably heated. This transparent object must then be rendered visible from its phase-shift. Several phase imaging techniques have been demonstrated for cold atom objects: dark ground imaging [5], Zernike phase contrast [6] and off-axis image holography [7]. We show here that an off-resonant column density image of cold atoms can also be retrieved from a single diffraction pattern. For a cold atom cloud, the absorption and phase coefficients in Eq. (10) are β=
σ0 1 2k 1 + 4∆2
and
δ=−
σ0 2∆ 2k 1 + 4∆2
(10)
where ∆ is the detuning in natural linewidths and σ0 is the resonant cross-section. Substituting into the retrieval expression Eq. (8) yields 1 I(r, z) 1 + 4∆2 −1 ρ(r) = − F F −1 , (11) σ0 2∆ sin(πλzu2) − cos(πλzu2 ) I0 where again the expression must be regularised in a numeric implementation. Atom imaging results Figure 1 shows the apparatus used to record the diffraction pattern of a cold atom cloud, in this case a 87 Rb MOT. A bare fibre end produces a diverging beam which passes through the MOT and onto a bare CCD chip. No optics (lenses, beamsplitters) are used. The diverging beam magnifies the diffraction pattern by M = (R1 + R2 )/R1 . It can be shown that the diffraction pattern is the magnified version of the plane-wave illuminated pattern, but at an effective propagation distance zeff = R1 R2 /(R1 + R2 ) [2]. Hence magnification is achieved without the use of lenses and aberrations are obviated completely. Vacuum vessel Viewports
CCD camera CCD
Atom cloud
20mm
9mm
Fibre-coupled laser
R1=125mm
R2=155mm
Figure 1: Recording the diffraction pattern of a cloud of cold atoms in a MOT. The two relevant apertures which set the diffraction-limited resolution are indicated. Presenting author’s name:
L. D. Turner
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Topic area: AOS
1.3
I/I0 (average of 5 rows)
1 0.7
x/µm) 1 9 Figure 2: At left, the recorded diffraction pattern. The lineout below was taken through the centre of the pattern. At right is the retrieved column-density from a regularised implementation of Eq. (11). White corresponds to a column density of 8 × 1012 atoms/m2 . Figure 2 shows the recorded diffraction pattern, after the standard background subtraction and flat-fielding process has been applied. The probe laser was detuned +Γ from resonance. The pattern appears holographic with no detail discernable. The column density retrieved from this diffraction pattern is also shown. Parameters to the retrieval are the propagation distances, detuning and the pixel size of the camera. The retrieved column density image had fewer artefacts and was clearer than both on-resonance absorption and fluorescence images of the MOT, while heating of the MOT was a factor of five lower. It is not necessary to know the propagation distances accurately. The image may be focused at retrieval time, i.e. after the diffraction image has been recorded, by varying the propagation distance used in the retrieval algorithm. Objects at widely separated distances may be brought into focus, demonstrating the holographic depth-of-field of the technique. Conclusion This novel technique greatly simplifies off-resonant imaging of cold atoms. Phase plates and interferometers are not required, and lenses (and aberrations) are avoided. The imaging system does not require physical focusing with focal adjustments being performed in the computer. We hope that this technique will lead to the wider use of minimally-destructive imaging of Bose-Einstein condensates. This work was supported by the Australian Government International Science Linkages programme and by the Australian Research Council. L. D. Turner, B. B. Dhal, J. P Hayes et al., Opt. Express 12, 2960 (2004). A. Pogany, D. Gao and S. W. Wilkins, Rev. Sci. Instrum. 68, 2774 (1997). J.-P. Guigay, Optik 49, 121 (1977). A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems (V. H. Winston, Washington D.C., 1977). [5] M. R. Andrews, M.-O. Mewes, N. J. van Druten et al., Science 273, 84 (1996). [6] M. R. Andrews, D. M. Kurn, H. J. Miesner, Phys. Rev. Lett. 79, 553 (1997). [7] S. Kadlecek, J. Sebby, R. Newell and T. G. Walker, Opt. Lett. 26, 137 (2001).
[1] [2] [3] [4]
Presenting author’s name:
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