Multiple-camera defocus imaging of ultracold atomic gases
A. R. Perry, S. Sugawa, F. Salces-Carcoba, Y. Yue, I. B. Spielman
MMultiple-camera defocus imaging of ultracoldatomic gases
A. R. P
ERRY , S. S
UGAWA , F. S
ALCES -C ARCOBA , Y. Y UE , AND
I. B. S
PIELMAN Joint Quantum Institute, National Institute of Standards and Technology, and University of Maryland,Gaithersburg, Maryland, 20899, USA Honeywell Quantum Solutions, 303 S. Technology Ct. 80021 Broomfield, Colorado, USA Institute for Molecular Science, National Institutes of Natural Sciences, Myodaiji, Okazaki 444-8585,Japan SOKENDAI (The Graduate University for Advanced Studies), Myodaiji, Okazaki 444-8585, Japan LIGO Laboratory, California Institute of Technology, MS 100–36, Pasadena, CA 91125, USA * [email protected]://ultracold.jqi.umd.edu Abstract:
In cold atom experiments, each image of light refracted and absorbed by an atomicensemble carries a remarkable amount of information. Numerous imaging techniques includingabsorption, fluorescence, and phase-contrast are commonly used. Other techniques such asoff-resonance defocused imaging (ORDI, [1–4]), where an in-focus image is deconvolved from adefocused image, have been demonstrated but find only niche applications. The ORDI inversionprocess introduces systematic artifacts because it relies on regularization to account for missinginformation at some spatial frequencies. In the present work, we extend ORDI to use multiplecameras simultaneously at degrees of defocus, eliminating the need for regularization and itsattendant artifacts. We demonstrate this technique by imaging Bose-Einstein condensates, andshow that the statistical uncertainties in the measured column density using the multiple-cameraoff-resonance defocused (MORD) imaging method are competitive with absorption imaging nearresonance and phase contrast imaging far from resonance. Experimentally, the MORD methodmay be incorporated into existing set-ups with minimal additional equipment. © 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ultracold atoms exist in isolation, enshrouded in ultrahigh vacuum, so that nearly everymeasurement on them relies on their interaction with electromagnetic fields. The most commonmeasurements use a probe laser beam that is attenuated and phase shifted by the atoms to recovertwo-dimensional images of the integrated density–the column density–of the atoms. Whether thetechnique be absorption imaging (AI), or phase-contrast imaging (PCI), the spatially resolvedcolumn density of the atomic cloud is recovered; from this, physical information regarding theatomic ensemble can be extracted.In this paper, we describe and demonstrate an extension to off-resonance defocused imaging(ORDI) pioneered in Refs. [1–4]. ORDI uses information from both the absorbed and phase-shifted probe laser light; by contrast, absorption or phase-contrast imaging rely only on theabsorption or phase shift signal, respectively. Both the absorption and phase-shift are proportionalto the quantity of interest, the column density. When the laser detuning from atomic resonance 𝛿 is large, the fractional absorption ∝ / 𝛿 , while the phase-shift ∝ / 𝛿 : thus for sufficientlylarge detunings the phase-shift is more significant than absorption. Typically AI [5] is usedfor clouds of low to medium column density using a resonant probe beam (no phase-shift),and PCI [6] is used to image clouds of high column density using a far-detuned probe beam(negligible absorption). In AI and PCI, intensity images are recorded by a detector at the focus ofthe imaging system. In ORDI, a image is taken by a detector positioned away from the focus: a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b . . . . k ⊥ ( µ m − ) − − CT F , ˜ h ( k ⊥ ) (a) 0 . . . . k ⊥ ( µ m − ) − − − I n v e r s e C R F , ˜ h − ( k ⊥ ) (b) Fig. 1. Contrast transfer function (a) and its inverse (b), as discussed in Sect. 2.3, usingeqns. (18) and (22). These are computed as a function of transverse wave number 𝑘 ⊥ , for a probe laser beam with wavelength 𝜆 =
780 nm, detuning ¯ 𝛿 =
1, and defocusdistance 𝑧 = 𝜇 m. The black curves denote ˜ ℎ ( 𝑘 ⊥ ) and ˜ ℎ − ( 𝑘 ⊥ ) where appearance ofrepeating zeros in the CTF and the associated divergences in its inverse indicate spatialfrequencies where all information about the initial density profile is lost. The red curveshows the regularized inverse ˜ ℎ − ( 𝑘 ⊥ ) with regularization parameter 𝜂 = .
2, whichtracks ˜ ℎ − ( 𝑘 ⊥ ) until it exceeds a threshold value and then returns to zero. remarkably, Ref. [2] showed that given full knowledge of the atoms’ complex susceptibility itis possible to digitally refocus intensity images of atoms by inverting contrast transfer function[CTF, shown in Fig. 1(a)] without knowing the phase of the underlying optical field. Still,ORDI was beset with unavoidable imaging artifacts resulting from lost information at somespatial frequencies in defocused images where the inverse CTF diverges [Fig. 1(b)]. Here wedemonstrate a technique to reconstruct defocuse images of ultracold atoms with greatly reducedartifacts. In this technique, multiple-camera off-resonance defocused (MORD) imaging, wesimultaneously use cameras placed at different defocused distances and show that suitably placedcameras allow for essentially artifact-free reconstruction of the atomic column density. Wecompare this technique to conventional imaging techniques and show that its signal to noise ratio(SNR) is comparable to AI near atomic resonance and comparable to PCI far from resonance.This paper is organized as follows; in Sec. 1 we discuss the solution to the vector wave equationunder the paraxial approximation after interacting with a thin, dilute atomic cloud. In Sec. 2,we describe absorption and phase-contrast imaging; derive expressions for ORDI and MORDimaging; and conclude with a theoretical comparison of MORD to absorption and phase-contrastimaging. In Sec. 3, we describe the experimental implementation of the MORD method withthree detectors, and the procedure we used to prepare Rb Bose-Einstein condensates (BECs) ofvariable column density. In Sec. 4, we present our experimental MORD results.
First we introduce the theoretical problem, starting with the propagation of an electromagneticwave defined at all positions r = 𝑥 e 𝑥 + 𝑦 e 𝑦 + 𝑧 e 𝑧 . For our neutral atomic systems interrogated bya monochromatic probe laser beam with wavelength 𝜆 and wave number 𝑘 = 2 𝜋 / 𝜆 , the evolutionof the laser beam’s electric field in the presence of atoms with complex susceptibility 𝜒 isdescribed by a pair of scalar wave equations ∇ E 𝑖 ( r ) + 𝑘 [ + 𝜒 ( r )]E 𝑖 ( r ) = , (1).e., Helmholtz equations, one for each polarization component E 𝑖 . These equations are validprovided that the susceptibility 𝜒 changes slowly with respect to 𝜆 [allowing the term ∇(∇ · E ) in the vector wave equation to be neglected]. For each polarization component, the Helmholtzequation then has the formal solution E ( r ⊥ , 𝑧 + Δ 𝑧 ) = exp (cid:110) ± 𝑖 Δ 𝑧 (cid:2) ∇ ⊥ + 𝑘 + 𝜒 ( r ⊥ ) 𝑘 (cid:3) / (cid:111) E ( r ⊥ , 𝑧 ) , (2)provided 𝜒 ( r ) has no dependence on 𝑧 . Here r ⊥ = 𝑥 e 𝑥 + 𝑦 e 𝑦 and ∇ ⊥ = 𝜕 / 𝜕 𝑥 + 𝜕 / 𝜕 𝑦 are thetransverse position and Laplacian, respectively. E ( r ⊥ , 𝑧 ) is the field before propagation, and E ( r ⊥ , 𝑧 + Δ 𝑧 ) is the field that has propagated an infinitesimal distance Δ 𝑧 through the medium.This solution becomes straightforward to evaluate numerically even for arbitrary displacementsprovided that 𝜒 has no spatial dependence – such as the case for propagation in a homogenousisotropic medium – by working in the Fourier domain.In our simulations, we only used eq. (2) to evolve the electric field in free space where 𝜒 = 𝜒 ( r ⊥ , 𝑧 ) dependent on both r ⊥ and 𝑧 , a situation where eq. (1) is inconvenient to manipulate analytically.For these cases, we make the paraxial approximation that renders both numerical and analyticaltreatments straightforward. We write the field E ( r ⊥ , 𝑧 ) = exp ( 𝑖𝑧𝑘 ) 𝐸 ( r ⊥ , 𝑧 ) making explicit thenominal propagation axis e 𝑧 . In addition, we assume that the transverse spatial structure is slowlyvarying compared to the optical wavelength, i.e., | 𝜕𝐸 / 𝜕 ( 𝑘 𝑧 )| (cid:29) (cid:12)(cid:12) 𝜕 𝐸 / 𝜕 ( 𝑘 𝑧 ) (cid:12)(cid:12) . Together theseapproximations lead to the paraxial wave equation 𝑖 𝜕𝜕𝑧 𝐸 ( r ⊥ , 𝑧 ) = (cid:20) − ∇ ⊥ 𝑘 + 𝑘 𝜒 ( r ⊥ , 𝑧 ) (cid:21) 𝐸 ( r ⊥ , 𝑧 ) . (3)In this manuscript, the paraxial wave equation offers two primary benefits. Firstly the free spacepropagator ℎ p ( r ⊥ , Δ 𝑧 ) = exp (cid:18) 𝑖 Δ 𝑧 ∇ ⊥ 𝑘 (cid:19) , that gives 𝐸 ( r ⊥ , 𝑧 + Δ 𝑧 ) = ℎ p ( r ⊥ , Δ 𝑧 ) 𝐸 ( r ⊥ , 𝑧 ) , is Gaussian and is therefore straightforward to manipulate analytically. Secondly, we parametrizepropagation through a medium localized at 𝑧 = 𝛼 ( r ⊥ ) anda phase shift 𝜙 ( r ⊥ ) . We relate the electric field 𝐸 − ( r ⊥ , 𝑧 = ) just prior to the medium to thefield 𝐸 + ( r ⊥ , 𝑧 = ) just after the medium with 𝐸 + ( r ⊥ , 𝑧 = ) = exp (cid:20) 𝑖 𝑘 ∫ 𝜒 ( r ⊥ , Δ 𝑧 ) 𝑑𝑧 (cid:21) 𝐸 − ( r ⊥ , 𝑧 = ) = exp [− 𝛼 ( r ⊥ ) + 𝑖𝜙 ( r ⊥ )] 𝐸 − ( r ⊥ , 𝑧 = ) . (4)This expression is valid provided the medium is thin compared to the depth of field 2 𝑘 / 𝑘 ,where 𝑘 max is the largest wave-vector of any significance present in 𝐸 ( r ) , or more pragmatically 𝑘 max might be the largest wave-vector resolvable by the experimental detector. In effect, eq. (4)neglects diffraction as light propagates through the atomic layer, and is equivalent to droppingthe Laplacian in eq. (3).For two-level atoms, we characterize the absorption and phase shift via the complex suscepti-bility 𝜒 ( r ) = 𝜎 𝑘 (cid:20) 𝑖 − 𝛿 + ¯ 𝐼 ( r ) + 𝛿 (cid:21) 𝜌 ( r ) , (5)for an ensemble of atoms with density 𝜌 ( r ) illuminated by a probe laser of wavelength 𝜆 . Eq. (5) isvalid for dilute clouds 𝜌 ( r ) (cid:28) 𝑘 such that collective effects can be neglected. Here 𝜎 = 𝜆 / 𝜋 s the resonant cross-section, ¯ 𝐼 ( r ) = 𝐼 ( r )/ 𝐼 sat is the optical intensity 𝐼 ( r ) = 𝑐𝜖 | 𝐸 ( r )| / 𝐼 sat , and ¯ 𝛿 = 𝛿 / Γ is the detuning 𝛿 from atomic resonance in unitsof the natural line-width Γ . Here, 𝑐 is the speed of light and 𝜖 , once known as the permittivity offree space, is the now ill-named electric constant.The two-level model for the susceptibility gives coefficients 𝛼 ( r ⊥ ) = OD ( r ⊥ )/ , and 𝜙 ( r ⊥ ) = − ¯ 𝛿 OD ( r ⊥ ) . (6)Where the optical depth OD ( r ⊥ ) ≡ − ln (cid:20) 𝐼 + ( r ⊥ ) 𝐼 − ( r ⊥ ) (cid:21) = 𝛼, (7)expresses exponential attenuation of light by the atoms and is defined in terms of the intensityjust before 𝐼 − ( r ⊥ ) and just after 𝐼 + ( r ⊥ ) interacting with the atoms. The column density 𝜌 ( r ⊥ ) = ∫ 𝜌 ( r ⊥ , 𝑧 ) d 𝑧 can be derived from the optical depth given both the detuning and the intensity using [7] 𝜎 𝜌 ( r ⊥ ) = ( + 𝛿 ) OD ( r ⊥ ) + ¯ 𝐼 − ( r ⊥ ) (cid:104) − 𝑒 − OD ( r ⊥ ) (cid:105) . (8)This leads us to two essential messages for this section: (I) according to eq. (6) the completeimpact of the atomic ensemble on the electric field can be parametrized in terms of OD ( r ⊥ ) alone; and (II) once OD ( r ⊥ ) is obtained, the column density can be reconstructed using eq. (8),independent of what measurement technique was employed to obtain the optical depth.As a result, any imaging technique will first find the optical density and then obtain the columndensity using eq. (8). Therefore in the following discussion we will compare imaging techniquesin terms of their ability to reconstruct the optical depth.
2. Imaging techniques
In our experiments, the detectors – CCD cameras – are sensitive to the field’s intensity, not itsamplitude. The imaging techniques discussed here use image pairs, each consisting of a 2Darray of pixels giving the optical intensity with atoms present ¯ 𝐼 ( 𝑖 )+ ( r ⊥ ) resulting from the field 𝐸 + ( r ⊥ , 𝑧 ( 𝑖 ) ) , and the intensity with the atoms absent ¯ 𝐼 ( 𝑖 )− ( r ⊥ ) resulting from the field 𝐸 − ( r ⊥ , 𝑧 ( 𝑖 ) ) .The superscript ( 𝑖 ) indicates that for MORD several cameras simultaneously recordthe intensityat different positions 𝑧 ( 𝑖 ) . We characterize the impact of the atoms in terms of the fractional dropin intensity 𝑔 ( 𝑖 ) ( r ⊥ ) = − 𝐼 ( 𝑖 )+ ( r ⊥ ) 𝐼 ( 𝑖 )− ( r ⊥ ) . (9)Where it is clear, we will omit the camera index 𝑖 and pixel coordinate r ⊥ . Evidently this detectionprocess has no direct sensitivity to the fields’ phase, nor the phase shift imprinted by the atoms.Each acquired image has a noise contribution 𝛿𝐼 ( 𝑖 ) ( r ⊥ ) that we model as a classical randomprocess from shot noise on the CCD detector with (cid:104) 𝛿𝐼 ( 𝑖 ) ( r ⊥ )(cid:105) = , and (cid:104) 𝛿𝐼 ( 𝑖 ) ( r ⊥ ) 𝛿𝐼 ( 𝑖 (cid:48) ) ( r (cid:48)⊥ )(cid:105) = 𝛿 𝑖,𝑖 (cid:48) 𝛿 r ⊥ , r (cid:48)⊥ 𝐼 (cid:104) 𝐼 ( 𝑖 ) ( r ⊥ )(cid:105) , (10)where 𝐼 = ℏ 𝜔 / 𝜂 𝐴 Δ 𝑡 is the intensity required to generate a single photo-electron. Here ℏ 𝜔 = 𝑐 ℏ 𝑘 is the single-photon energy, 𝜂 is the detector quantum efficiency (shot noise is really the shotoise of the photo-electrons not of the photons), 𝐴 is the area of a single pixel, and Δ 𝑡 is themeasurement time. This noise model neglects other physical sources of noise such as CCDreadout noise, or excess noise from dark current.Since the fractional absorption contains the full information about the atomic ensemble, wetransfer the noise in the images to the noise in the fractional intensity which has (cid:104) 𝛿𝑔 ( 𝑖 ) ( r ⊥ )(cid:105) = (cid:104) 𝛿𝑔 ( 𝑖 ) ( r ⊥ ) 𝛿𝑔 ( 𝑖 (cid:48) ) ( r (cid:48)⊥ )(cid:105) = 𝛿 𝑖,𝑖 (cid:48) 𝛿 r ⊥ , r (cid:48)⊥ 𝐼 𝐼 ( 𝑖 )− ( r ⊥ ) (cid:104) − (cid:104) 𝑔 ( 𝑖 ) ( r ⊥ )(cid:105) (cid:105) . (11)In what follows, we describe the AI and PCI techniques, and then derive the ORDI and MORDmethods. For each technique, we compute the expected uncertainty in optical depth resultingfrom photon shot noise, and conclude with a comparison of these uncertainties across all cases. AI proceeds directly from the intensities recorded by a single camera placed in-focus (effectivelyrecording the intensity as it was in the object plane at 𝑧 = ( r ⊥ ) = − ln [ − 𝑔 ( r ⊥ )] without further approximation. AI is most straightforward forresonant ( ¯ 𝛿 =
0) low intensity ( ¯ 𝐼 − (cid:28)
1) imaging, giving an optical depth OD ( r ⊥ ) = 𝜎 𝜌 ( r ⊥ ) proportional to the column density. In this limit, the repeated scattering of resonant photonsstrongly perturbs the atomic ensemble after a single instance of imaging. As a result, AI is notgenerally used to perform weakly destructive measurements .Using the formalism in eqn. (11) along with eqn. (7), we find the noise for AI to be (cid:104) 𝛿 OD ( r ⊥ )(cid:105) = , and (cid:104) 𝛿 OD ( r ⊥ ) 𝛿 OD ( r (cid:48)⊥ )(cid:105) = 𝛿 r ⊥ , r (cid:48)⊥ (cid:104) 𝛿𝑔 ( r ⊥ ) (cid:105)[ − (cid:104) 𝑔 ( r ⊥ )(cid:105)] . (12)This shows that the noise is still 𝛿 -correlated in space and diverges at large optical depth. This isexpected because at large OD the vast majority of the probe is absorbed, leading to an increasedfractional contribution of photon shot noise. PCI is often used for weakly destructive measurements, and readily allows the same cloud to beimaged repeatedly. Like AI, PCI is performed with a single in-focus camera, but unlike AI, therecorded intensity of a phase-contrast image contains phase information from which the columndensity is extracted. After traversing the layer of atoms, the electric field 𝐸 + = 𝐸 − + Δ 𝐸 maybe expressed as a sum [5] of its unscattered and scattered parts 𝐸 − and Δ 𝐸 . PCI is typicallyimplemented by adding a 𝜃 = 𝜋 / so that 𝐸 − → 𝐸 − exp ( 𝑖𝜃 ) .PCI is typically applied in the far detuned limit where absorption can be neglected, i.e., 𝛼 (cid:28) 𝜙 ,giving the PCI intensity pattern 𝑔 ( r ⊥ ) = [ cos 𝜙 ( r ⊥ ) − sin 𝜙 ( r ⊥ ) − ] ≈ − 𝜙 ( r ⊥ ) = 𝛿 OD ( r ⊥ ) (13)resulting from the interference of the the phase-shifted light with the light refracted by the atoms.The final approximation in Eq. (13) is valid for small phase shifts (requiring a combination oflarge detuning), and in this limit the PCI signal is linear in 𝜙 , therefore proportional to the opticaldepth [6]. Reference [8] describes a weakly destructive absorption technique called partial transfer absorption imaging (PTAI)where only a small fraction of a large atomic ensemble is transferred to a internal state sensitive to a probe laser, whilethe majority of the atoms were kept in a dark state. In practice this phase shift is created using a phase plate, with a retarding spot slightly larger than the focused beamspot-size, in the imaging system (Fig. 4). The 𝜋 / 𝜃 = 𝜋 ,the Taylor expansion is dominated by even functions, so that 𝐼 ∝ 𝜙 ). ext, we find the uncertainty in the recovered optical depth for PCI using eqn. (13) to be (cid:104) 𝛿 OD ( r ⊥ )(cid:105) = , and (cid:104) 𝛿 OD ( r ⊥ ) 𝛿 OD ( r (cid:48)⊥ )(cid:105) = 𝛿 r ⊥ , r (cid:48)⊥
14 ¯ 𝛿 (cid:104) 𝛿𝑔 ( r ⊥ ) (cid:105) . (14)This shows that for large detuning (where the approximations leading to PCI are valid) andlow optical depth ( 𝑔 (cid:28) 𝛿 compared to that of AI. The AI technique introduced in Sec. 2.1 is best implemented on resonance, where there are nophase shifts. On the other hand, PCI works best off-resonance when absorption is minimal.The ORDI technique [1–4] is a method that works best in the intermediate regime where bothabsorption and phase shift are important. ORDI and MORD build from an invertible relationbetween the observed intensity and the optical depth that recovers all but a small range of spatialfrequencies.ORDI relies on several simplifying assumptions, the first of which is the paraxial approximationto the electric field that has propagated through an atomic cloud in Sec. 1.1. We assume that theelectric field did not diffract as it traveled through the cloud (i.e., that it was thin compared to thedepth of field). Both AI and PCI require these same approximations.Going forward we introduce the Fourier transform (FT) of a two-dimensional function 𝑓 ( r ⊥ ) as ˜ 𝑓 ( k ⊥ ) = ∫ ∞−∞ 𝑓 ( r ⊥ ) exp (− 𝑖 k ⊥ · r ⊥ ) d r ⊥ .Using the paraxial propagator and the convolution theorem, we readily obtain the Fresneldiffraction integral 𝐸 ( r ⊥ , 𝑧 ) = 𝑖𝜆𝑧 ∫ ∞−∞ 𝐸 ( r ⊥ , 𝑧 = ) exp (cid:20) 𝑖𝑘 𝑧 (cid:12)(cid:12) r ⊥ − r (cid:48)⊥ (cid:12)(cid:12) (cid:21) d r (cid:48)⊥ . (15)Using eqn. (15) and the electric field just after it traversed the cloud in eqn. (4), the normalizedintensity at a detector placed an arbitary distance from the atomic ensemble is˜ 𝑔 ( k ⊥ , 𝑧 ) = ∫ ∞−∞ (cid:26) − exp [− 𝛼 ( r ⊥ + 𝑧 k ⊥ / 𝑘 ) − 𝛼 ( r ⊥ − 𝑧 k ⊥ / 𝑘 )+ 𝑖𝜙 ( r ⊥ − 𝑧 k ⊥ / 𝑘 ) − 𝑖𝜙 ( r ⊥ + 𝑧 k ⊥ / 𝑘 )] (cid:27) × exp (− 𝑖 r ⊥ · k ⊥ ) d r ⊥ . (16)To derive the ORDI technique we further required that the phase is slowly-varying: | 𝜙 ( r ⊥ + 𝑧 k ⊥ / 𝑘 ) − 𝜙 ( r ⊥ − 𝑧 k ⊥ / 𝑘 )| (cid:28)
1, and that absorption is small: 𝛼 ( r ⊥ ) (cid:28)
1. To the lowestorder in 𝜙 and 𝛼 , the resulting normalized intensity˜ 𝑔 ( k ⊥ , 𝑧 ) ≈ ∫ ∞−∞ [ 𝛼 ( r ⊥ + 𝑧 k ⊥ / 𝑘 ) + 𝛼 ( r ⊥ − 𝑧 k ⊥ / 𝑘 ) (17) − 𝑖𝜙 ( r ⊥ − 𝑧 k ⊥ / 𝑘 ) + 𝑖𝜙 ( r ⊥ + 𝑧 k ⊥ / 𝑘 )] × exp (− 𝑖 r ⊥ · k ⊥ ) d r ⊥ , is a Fourier integral of 𝜙 and 𝛼 . Defining the FTs of 𝜙 and 𝛼 as ˜ 𝜙 and ˜ 𝛼 , respectively, we find˜ 𝑔 ( k ⊥ , 𝑧 ) = 𝛼 ( k ⊥ ) cos ( 𝑧𝑘 ⊥ / 𝑘 ) − 𝜙 ( k ⊥ ) sin ( 𝑧𝑘 ⊥ / 𝑘 ) , (18)where 𝑘 ⊥ = 𝑘 𝑥 + 𝑘 𝑦 . Using eq. (6) we arrive at the explicit expression˜ 𝑔 ( k ⊥ , 𝑧 ) = [ cos ( 𝑧𝑘 ⊥ / 𝑘 ) + 𝛿 sin ( 𝑧𝑘 ⊥ / 𝑘 )] ˜OD ( k ⊥ ) ≡ ˜ ℎ ( k ⊥ , 𝑧 ) ˜OD ( k ⊥ ) , (19)hat uses a CTF ˜ ℎ ( k ⊥ , 𝑧 ) to provide a linear relation between the Fourier transformed opticaldepth ˜OD ( k ⊥ ) and the fractional change in intensity ˜ 𝑔 ( k ⊥ , 𝑧 ) defined in eq. (9).We plot a representative CTF and its inverse in Fig. 1 showing singularities at some spatialfrequencies 𝑘 ⊥ , the locations of which depend upon the camera position 𝑧 and the detuning¯ 𝛿 . When ˜ ℎ ( k ⊥ , 𝑧 ) − is applied, measurement noise is amplified near the divergences, so thatno useful information can be extracted from these spatial frequencies. The ratio 𝛼 / 𝜙 = − / 𝛿 along with the sign of 𝑧 determines the quality of information at low spatial frequencies. Whensign ( 𝑧 ) 𝛼 / 𝜙 <
0, the inverse CTF has a divergence at a low spatial frequency, and informationmay be lost for the spatial structure of interest. There are two mathematically equivalent cases toachieve the “good” condition: ¯ 𝛿 < 𝑧 < 𝛿 > 𝑧 >
Here we compute the anticipated diverging uncertainty at the zeros of the CTF quantitativelyand introduce a regularization parameter to resolve these divergences. In general the spatialstructure in eq. (11) would lead to correlated noise in (cid:104) 𝛿 ˜ 𝑔 ( k ⊥ ) 𝛿 ˜ 𝑔 ( k (cid:48)⊥ )(cid:105) , however, for simplicitywe assume that the spatial dependence is weak (as would be the case for an extended system). Weare reminded that random variables that are uniform and 𝛿 -correlated spatially are also uniformand 𝛿 -correlated in the Fourier basis, giving (cid:104) 𝛿 ˜ 𝑔 ( k ⊥ ) 𝛿 ˜ 𝑔 ( k (cid:48)⊥ )(cid:105) = 𝛿 k ⊥ , k (cid:48)⊥ (cid:104) 𝛿 ˜ 𝑔 ( k ⊥ ) (cid:105) .The anticipated variance of the reconstructed optical depth is therefore (cid:104) 𝛿 OD ( k ⊥ ) 𝛿 OD ( k (cid:48)⊥ )(cid:105) = 𝛿 k ⊥ , k (cid:48)⊥ ˜ ℎ ( k ⊥ , 𝑧 ) − (cid:104) 𝛿 ˜ 𝑔 ( k ⊥ ) (cid:105) , divergent at the zeros of ˜ ℎ ( k ⊥ , 𝑧 ) .Here we describe the process of regularization used to mitigate the amplification of noise neardivergences in the inverse CTF. The basic idea is to include a “default” (i.e., a Bayesian prior)optical depth OD ( k ⊥ ) with uncertainty quantified by a known 𝛿 -correlated random variable 𝛿 OD ( k ⊥ ) , and we use the weighted average of the reconstructed and prior images˜OD ( k ⊥ ) = 𝑎 ( k ⊥ ) ˜ ℎ ( k ⊥ , 𝑧 ) − ˜ 𝑔 ( k ⊥ , 𝑧 ) + 𝑏 ( k ⊥ ) OD ( k ⊥ ) 𝑎 ( k ⊥ ) + 𝑏 ( k ⊥ ) (20)as our reconstructed optical depth. We determine the coefficients 𝑎 ( k ⊥ ) and 𝑏 ( k ⊥ ) by minimizingthe variance (cid:104) 𝛿 OD ( k ⊥ ) 𝛿 OD ( k (cid:48)⊥ )(cid:105) = 𝛿 k ⊥ , k (cid:48)⊥ 𝑎 ( k ⊥ ) ˜ ℎ ( k ⊥ , 𝑧 ) − (cid:104)( 𝛿 ˜ 𝑔 ⊥ ) (cid:105) + 𝑏 ( k ⊥ ) (cid:104) 𝛿 OD ( k ⊥ ) (cid:105)[ 𝑎 ( k ⊥ ) + 𝑏 ( k ⊥ )] , for each wavevector. This gives standard expression for Wiener deconvolution [9]˜OD ( k ⊥ ) = ˜ ℎ ( k ⊥ , 𝑧 ) ˜ 𝑔 ( k ⊥ , 𝑧 ) + 𝜂 ( k ⊥ ) OD ( k ⊥ ) ˜ ℎ ( k ⊥ , 𝑧 ) + 𝜂 ( k ⊥ ) (21)with regularization constant 𝜂 ( k ⊥ ) = (cid:104) 𝛿 ˜ 𝑔 ( k ⊥ ) (cid:105)/(cid:104) 𝛿 OD ( k ⊥ ) (cid:105) . In practice we typically selectthe prior OD ( k ⊥ ) =
0, essentially assuming no prior information, and take 𝜂 to be k -independent.This results in the Tikhonov regularized inverse CTF˜ ℎ R ( k ⊥ , 𝑧 ) − = ˜ ℎ ( k ⊥ , 𝑧 ) ˜ ℎ ( k ⊥ , 𝑧 ) + 𝜂 (22)that blocks the divergence of statistical noise, but introduces artifacts. . . . . k ⊥ ( µ m − ) − − CT F , ˜ H ( k ⊥ ) (a) 0 . . . . k ⊥ ( µ m − ) − − − I n v e r s e C R F , ˜ h − ( k ⊥ ) (b) Fig. 2. Forward (a) and inverse (b) CTFs computed as a function of transverse wavenumber 𝑘 ⊥ , for detuning ¯ 𝛿 = 𝜇 m (black), − 𝜇 m(blue), and 87 𝜇 m (orange). This shows that for a generic selection of defocus distancesthe CTFs do not have coincident zeros Since the ORDI technique is implemented in the Fourier basis, we find the noise-correlationfunction in coordinate space (cid:104) 𝛿 OD ( r ⊥ ) 𝛿 OD ( r (cid:48)⊥ )(cid:105) = (cid:104) 𝛿𝑔 ( r ⊥ ) (cid:105) 𝑁 ∑︁ k ⊥ ˜ ℎ ( k ⊥ , 𝑧 ) (cid:2) ˜ ℎ ( k ⊥ , 𝑧 ) + 𝜂 (cid:3) 𝑒 𝑖 k ⊥ · ( r ⊥ − r (cid:48)⊥ ) (23)where we explicitly expressed the discrete FT in terms of a sum over a total of 𝑁 pixels. UnlikeAI or PCI, ORDI reconstruction of the optical depth has correlated noise. We showed in Sec. 2.3 that with the ORDI method, spatial frequencies exist for which we retrieveno information (where ˜ ℎ → 𝑧 , the position in the objectplane where the intensities are recorded (Fig. 2). This suggests that by adding cameras to thesystem at different 𝑧 , we might recover information at all spatial frequencies. In the multiplecamera method, the images are combined and processed in a way that minimizes the uncertaintyin the recovered optical depth.As shown in Sec. 2.3, each CTF taken independently provides a linear relation between ameasurement and the estimated optical depth [eqn. (19)]. Our task is to find the best estimateof the optical depth for MORD using the measurements taken in the laboratory [eqn. (9)] andtheir corresponding CTFs. We model shot-noise, the only source of noise we consider, as noisethat is spatially uncorrelated and uniform over all spatial frequencies (i.e., additive white noise).Following the argument in Sec. 2.3.1, we minimize the noise in the reconstructed optical depthand find ˜OD 𝑘 = − (cid:205) ( 𝑖 ) ˜ ℎ ( 𝑖 ) 𝑘 ˜ 𝑔 ( 𝑖 ) 𝑘 (cid:205) ( 𝑖 ) | ˜ ℎ ( 𝑖 ) 𝑘 | + 𝜂 . (24)For 𝜂 = − − S N R (a) AIPCIMORD10 − Negative detuning − ¯ δ − − − O D (b) 10 − Detuning ¯ δ Fig. 3. Simulations. (a) Computed signal to noise ratio for AI (black markers), PCI (bluemarkers), and MORD (red markers) for both red and blue detunings plotted on separatelog scales. The numerical simulations and analytical solutions (see Sects. 2.1 and 2.2)are shown as symbols and solid curves, respectively. These simulations combined threedefocus distances 60 𝜇 m, − 𝜇 m, and 87 𝜇 m and set the regularization parameter 𝜂 =
0. (b) Optical depth obtained for AI, PCI, and MORD compared to the true value(black curve). Both MORD and AI show a systematic shift at small OD resulting fromnoise rectification when converting from measured intensities to OD.
We compare the signal to noise ratio (SNR) of three-camera MORD to those of AI and PCI inFig. 3(a). The solid symbols are the result of a numerical simulation of our full imaging process,while the curves for PCI and AI plot the expected SNR given the expressions in eqns. (12) and(14), thereby confirming the performance of our numerical model. Except for the resonant case,the MORD imaging technique gives a larger SNR than AI at the same detuning. Further, forpositive detuning [right panel to Fig. 3(a)] MORD method shows the same 𝛿 − scaling in theSNR as PCI, but with about 2 × lower SNR. The performance at negative detuning is slightlyreduced owing to two of the cameras having a near-zero 𝑘 node in their CTF as compared to justone for positive detuning.Next Fig. 3(b) quantifies systematic uncertainties in the MORD calculated optical depth. Herethe black curve plots the true optical depth, and the symbols plot the outcome of our numericalsimulation. At very low optical depth both AI and MORD begin to diverge from the true signal.These systematic shifts result from the non-linearity in the logarithm used to convert fromfractional absorption to OD, where detector noise is rectified, leading to excess signal.
3. Experimental techniques
Our optical geometry, schematically depicted in Fig. 4, consisted of a standard two-lens Keplerianmicroscope. The first lens (focal length 𝑓 ) was positioned a distance 𝑓 from the object (a BECin our experiments); and the second lens (focal length 𝑓 ) was placed a distance 𝑓 + 𝑓 fromthe first lens. Typically the detector would be placed at the focus of the second lens, and wouldhave a magnification 𝑀 = 𝑓 / 𝑓 . This system forms the conceptual basis of the imaging system ig. 4. A simplified schematic of a two-lens Keplerian imaging system with magnifica-tion 𝑀 ≈ . 𝑓 =
75 mm and 𝑓 =
250 mm. In MORD, the three detectorplanes 𝑧 ( 𝑖 ) are located away from the focus. This is equivalent to having three detectorsnext to BEC, as shown. The imaging resolution is set by the diameter of the aperturesplaced in the imaging system. A representative aperture, placed near the Fourier plane,is shown above. described in this manuscript.We implemented the MORD technique using a two-lens imaging system with magnification 𝑀 ≈ . 𝑓 =
75 mmand 𝑓 =
250 mm separated by 325(10) mm. In such an imaging system, a point ( 𝑥 , 𝑦 , 𝑧 ) inthe object space is imaged to the point (− 𝑀𝑥 , − 𝑀 𝑦 , 𝑀 𝑧 ) in the image space. We placed a18 ( ) mm aperture close to the imaging system’s Fourier plane to minimize spherical aberrations.This gave a numerical aperture NA ≈ .
12. After the final lens, roughly equal fractions of thelight was directed to each of our three detectors (Fig. 5) using non-polarizing beam splitters(NPBSs) with reflection to transmission (R:T) ratios 70:30 and 50:50. We detected this light oncharge-coupled device (CCD) cameras with 648 ×
488 square pixels with width 5.6 𝜇 m. Eachcamera was on a translation stage, so that the set-up could be used for both the defocused andstandard absorption imaging methods. Here we describe the experimental procedure used to acquire the images for the MORD technique.We collected about 2 × Rb atoms in a vapor-fed six-beam magneto-optical trap, performedsub-Doppler cooling, and then trapped the atoms in the | 𝑓 = , 𝑚 𝐹 = (cid:105) state in a sphericalquadrupole trap. We used magnetic transport [10] to move the resulting cloud about 42 cmvertically in 2.2 s, giving an ensemble at ≈ 𝜇 K with about 1 × atoms. We thenevaporated to degeneracy in the combined magnetic/optical technique described in Ref. [11].During evaporation, we performed a microwave transfer between the ground hyperfine states | 𝑓 = , 𝑚 𝐹 = (cid:105) to | 𝑓 = , 𝑚 𝐹 = − (cid:105) , giving 100 × atom BECs in a cross-optical dipole trapevery 15 s. The resulting trap frequencies were ≈
110 Hz, 75 Hz, and 50 Hz along e 𝑥 , e 𝑦 , and e 𝑧 respectively. To achieve the desired optical depth to test our imaging technique, we performed apartial ≈
10 % transfer to the | 𝑓 = , 𝑚 𝐹 = (cid:105) state with a short ( (cid:28) 𝜋 /
2) microwave pulse [8].We then used a probe beam with 𝜆 ≈
780 nm detuned a variable ¯ 𝛿 from the 𝑓 = 2 to 𝑓 (cid:48) = 3cycling transition (without a repumping laser) to image the transferred 𝑁 ≈ × atoms after a6 ms time of flight (TOF) with the three cameras simultaneously. In Sec. 4.1, we present theresults of this experiment for ¯ 𝛿 = . ( ) . We used a probe beam with intensity 𝐼 ≈ . 𝐼 sat , where 𝐼 sat ≈ .
67 mW / cm [12] for a circularly polarized probe beam. ig. 5. Schematic of our MORD setup. The two-lens imaging system described byFig. 4 is followed by two non-polarizing beam splitters that nominally split the probelaser beam into three equal intensities. Each image is then recorded by its designatedcamera. Each camera takes a defocused image at position 𝑀 𝑧 ( 𝑖 ) from focus, wherethe red circles denote the image planes.
4. Measurement and analysis
Fig. 6(a) shows the fractional absorption 𝑔 ( 𝑖 ) recorded on each camera as described by eqn. (9).In practice we obtained a third image with neither the atoms nor the probe light. This backgroundimage was subtracted from 𝐼 ( 𝑖 )+ and 𝐼 ( 𝑖 )− to remove dark counts and stray light from the images.The camera 1 image was defocused a distance 𝑧 ( ) = 1 . 𝑧 ( ) = 3 . 𝑧 ( ) = 7 . 𝑛 ( 𝑥 ) = 𝑛 (cid:2) − ( 𝑥 − 𝑥 ) / 𝑟 (cid:3) / . (25)Figure 6(c) plots cross-sectional cuts along e 𝑥 through the reconstructions with black symbolsalong with fits to the Tomas-Fermi model with red curves. We see that the data is in goodagreement with the expected behavior.The defocus distances 𝑧 ( 𝑖 ) used in the ORDI and MORD reconstructions shown in Fig. 6 werefirst measured in the lab with ≈ y p o s i t i o n ( µ m ) Camera 1(a) Camera 2 Camera 3 − y p o s i t i o n ( µ m ) ORDI 1(b) ORDI 2 ORDI 3 MORD0 . . . A b s o r p t i o n , g . . . . O D −
100 0 100 x position ( µ m)0 . . . O p t i c a l d e p t h , O D (c) −
100 0 100 x position ( µ m) −
100 0 100 x position ( µ m) −
100 0 100 x position ( µ m) Fig. 6. Raw and refocused data. (a) Raw data with defocus distances of 𝑀 𝑧 ( ) = . 𝑀 𝑧 ( ) = . 𝑀 𝑧 ( ) = . e 𝑥 (black markers) along with fits (red curves) to the expected“Thomas-Fermi” distribution. Ref. [14] to fine tune the displacements 𝑧 ( 𝑖 ) .
5. Conclusion
We experimentally demonstrated an improvement to the single-camera ORDI technique by usingthree cameras simultaneously to eliminate the divergences that arise in any single CTF. Westudied the systematic uncertainties of the MORD method, and theoretically compared multipletechniques using simulated data to understand the array of imaging techniques on equal footing.We showed that in the regime of far detuned probe beam and low optical depth, the MORDmethod is comparable to PCI. Therefore, in experiment, the easier to implement MORD set-upmay be preferable to PCI.Because the first low-spatial-frequency divergence of a CTF may occur at length scales of tensof micrometers, ORDI has been limited to imaging relatively large objects. By eliminating thespatial CTF divergences, MORD offers a complementary technique to in-situ AI and PCI imagingof quantum gases. The SNR of MORD imaging is about 1.5 times below both AI and PCI intheir respective optimal operating conditions: resonant imaging for AI and far-detuned imagingfor PCI. As a result MORD can switch between near optimal strong (destructive) measurementsand weak (minimally destructive) measurements with no hardware modifications. In addition,MORD eliminates the need to refocus the imaging systems, for example in TOF experimentshere the object plane shifts depending on the TOF.A final noteworthy outcome of our noise analysis is that even with spatially uncorrelateddetector noise, the ORDI and MORD methods introduce correlations into the reconstructed signal,meaning that without added calibration, these techniques would not be of use in experimentsstudying noise-correlations between atoms [16].
Funding
This work was partially supported by the AROs Atomtronics MURI, by the AFOSRsQuantum Matter MURI, NIST, and the NSF through the PFC at the JQI.
Acknowledgments
We thank D. Barker and A. Putra for their careful and thorough read-ing of this manuscript. We appreciate the efforts of E. Altuntas, R. P. Anderson, Q.-Y. Liang,D. Trypogeorgos, and A. Valdés-Curiel for employing defocus imaging in their experiments, andmotivating the completion of this manuscript.
Disclosures
The authors declare no conflicts of interest.
Data Availability Statement
Data underlying the results presented in this paper are not publiclyavailable at this time but may be obtained from the authors upon reasonable request.
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