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Optimally focused cold atom systems obtained using density-density correlations
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22.Moreover, the gradient term ∇ln ε(r) in Eq. (1) cannot be safely neglected for systems where |χ(r)| is large or sufficiently rapidly varying, although this would generally imply a breakdown of the paraxial approximation as well.
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24.In our SSFM simulation of light traversing this medium, we used a Δz = 1 μm step size.
25.The Wiener-Khincin theorem states that the spectral decomposition of the autocorrelation function is equal to the power spectral density.
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Resonant absorption imaging is a common technique for detecting the two-dimensional column density of ultracold atom systems. In many cases, the system's thickness along the imaging direction greatly exceeds the imaging system's depth of field, making the identification of the optimally focused configuration difficult. Here we describe a systematic technique for bringing Bose-Einstein condensates (BEC) and other cold-atom systems into an optimal focus even when the ratio of the thickness to the depth of field is large: a factor of 8 in this demonstration with a BEC. This technique relies on defocus-induced artifacts in the Fourier-transformed density-density correlation function (the power spectral density, PSD). The spatial frequency at which these artifacts first appear in the PSD is maximized on focus; the focusing process therefore both identifies and maximizes the range of spatial frequencies over which the PSD is uncontaminated by finite-thickness effects.
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