Imaging cold atoms with shot-noise and diffraction limited holography

We theoretically develop and experimentally demonstrate a holographic method for imaging cold atoms at the diffraction and photon shot noise limits. Aided by a double point source reference field, a simple iterative algorithm robustly removes the twin image of an $^{87}$Rb cold atom sample during the image reconstruction. Shot-noise limited phase shift and absorption images are consistently retrieved at various probe detunings, and during the laser cooling process. We consistently resolve less than 2 mrad phase shift ($0.4~\%$ attenuation) of the probe light, outperforming shot-noise limited phase-contrast (absorption) imaging by a factor of 4 or more if the same camera is used without pixel saturation. We discuss the possible extension of this work for precise phase imaging of dense atomic gases, and for off-resonant probing of multiple atoms in optical lattices.

Since the achievement of Bose-Einstein condensation [1], many-body physics in textbooks has been reproduced with beautiful experiments in laser cooling labs. Owing to its controllability and precision, the field of ultra-cold atoms holds unique promise for both answering important questions in condensed matter physics, and to generate new physics. Breakthroughs in cold atom research are often accompanied with improved imaging techniques [2]. Recently, in situ imaging [3,4] has been developed to probe the shortest length scales of the confined, sub-µK quantum gases and is referred to as quantum gas microscopy.
In situ imaging of high density atomic gases suffers from detrimental effects related to resonant interactions mediated by photons. For example, state-of-the-art in situ florescence detection [3][4][5] cannot image an optical lattice site with more than one atom occupancy without losing the extra atoms in pairs. Similarly, absorption imaging is perturbed by resonant dipole interactions, and the atomic density information cannot be faithfully retrieved via the Beer-Lambert law [6]. To mitigate the effect one may saturate the optical transition with a strong probe, which results in images with excessive photon shot noise [7,8]. Off resonant imaging [9][10][11][12] provides a solution to the density dependent line-broadening problem, since the variation of probe light phase shift due to line shape changes vanishes at large detuning. However, the magnitude of the phase shift due to a single atom reduces with detuning, while the phase shift sensitivity is limited by the photon shot noise as δφ ∼ 1/ N p , where N p is the average number of probe photons detected by a pixel. Since N p < N max , the maximum pixel count, it is difficult with phase-contrast imaging to achieve a sensitivity below 5 mrad using standard cameras, and the amount of detuning allowed for off-resonant detection of cold atoms is correspondingly limited.
The present work demonstrates diffraction limited holographic imaging of cold atoms with photon shot-noise limited sensitivity [13].
Holographic mi- * brynsob@yahoo.co.uk † saijun.wu@swansea.ac.uk croscopy [14] reconstructs a complex wavefront E s from a hologram H that contains the interference pattern between E s and a known spherical wavefront E r (Fig. 1A). Holographic imaging can be free from lens aberrations at large numerical aperture (NA) [15], which is particularly useful for imaging cold atoms at long working distances [2][3][4][5]. It is known that an image reconstructed in an inline holographic geometry (Fig. 1A) is contaminated by an out-of-focus twin image [14,16]. We introduce a hybrid geometry (Fig. 1B) to solve the twin image and DC noise problems. By taking advantage of point source holographic recording, we achieve ∼ 2 mrad phase shift and ∼ 0.4% absorption sensitivities, beyond the photon shot noise limit imposed by N max in standard imaging [17]. The sensitivity is near the single atom level with resonant probing. By improving the spatial resolution of x res ∼5.2 µm to the wavelength limit, which would not require precision high-NA optics [15], we expect dramatic improvement of the detection sensitivity, allowing off-resonant single atom imaging. With these features, we discuss the method for off-resonant imaging of high density ultra-cold gases, and for non-destructive probing of optical lattices with multiple atom occupancy per site.
Our microscope is depicted in Fig. 1B, together with the traditional inline setup [14] in Fig. 1A. To gener-ate the reference wave E r = E r,1 + E r,2 in the hybrid setup, we add a second, "off-axis" point source E r,2 at r 2 = (d, 0, 0) displaced from the inline source E r,1 at r 1 = (0, 0, 0) with light power P 2 = ηP 1 , resulting in an interference pattern at the camera plane z = L. In both setups, the known wavefront E r interferes with E s , the elastically scattered light from the sample at z = z 0 . The light intensity is recorded as the hologram H = |E r + E s | 2 . With H 0 the hologram taken without the sample, an approximation of the 2D wavefront E s (L) [18] is written as Using the angular spectrum method [19] we propagate yF , k = 2π/λ andF is the 2D Fourier transform with E(k x , k y , z) = FE(x, y, z). In addition to E s focusing at z = z 0 , the 2nd and 3rd terms in Eq. (1) focus at z ≈ −z 0 and z = 0 as the twin and DC images respectively. Optics between the sample and camera can be modeled for aberration correction. Without aberrations, the resolution of E s (z 0 ) is diffraction limited to x res = λ w 2 /4 + (L − z 0 ) 2 /w, with w the camera width.
To isolate the real image E s (z 0 ), we notice that E s (z 0 ) is in focus, while the twin and DC terms are spread out at z = z 0 (Fig. 1). With an aperture operatorP that sets the data outside of the sample area with diameter a to zero, the energy of the out-of-focus images can be removed iteratively with an algorithm similar to ref. [20], HereÛ 0 ≡Û(z 0 − L), soÛ 0 andÛ −1 0 transform the wavefronts between the camera and sample planes.Ĉ is a conjugation operatorĈE = E * E r /E * r , that converts any wavefront at the camera plane into that of its twin. With E is easily shown to converge near zero [21] (Fig. 2A). Specifically, the residual energy fraction r = |E ( Here is the fraction of energy of the out-of-focus twin imageÛ 0ĈÛ −1 0P I, that is contained within the aperturê P at z 0 (I(x, y) = 1 is a 2D uniform wavefront soPI is the image of the aperture itself.). The apertureP must be larger than the sample, thus can characterize the overlap between the real and twin images. To improve the convergence speed one needs to reduce the overlap.   For the inline geometry ( Fig. 1A) a large sample needs an aperture with a Fresnel number F = a 2 /2z 0 λ > 1, leading to a reduced diffraction effect and thus ≈ 1. In the hybrid geometry the twin image is split into multiple copies displaced by r 2 − r 1 (Fig. 1B). It is easy to show that the twin image that is inline with the sample takes a fractional energy of (1 − η) 2 for η < 1. Thus even for F 1, ≈ (1 − η) 2 can still be small. In Fig. 2A we plot the residual r vs iteration number for a simulated phase object. For the hybrid scheme with η = 0.16, the series converges quickly with N 0 = 2.8 ≈ −1/ log (1 − η) 2 for both F = 3.8, 28. The non-zero final residual in the simulation is mainly due to the boundary artifacts in the FFT. In contrast, the inline scheme fails to converge within 10 3 iterations even for F = 3.8.
So far, we have discussed reconstructing E s from noiseless holograms with a perfect E r . Major sources of noise are imperfect subtraction H − H 0 , aberrations and speckle noise in E r and E s , and photon shot noise in H and H 0 . The subtraction is critical for reaching the shot noise limit, for which we developed an optimization algorithm [22]. Photon shot noise in H and H 0 is proportional to N p at each pixel with area A p . Assuming an equal shot noise level in H and H 0 , the root mean square (rms) E (n) s shot noise level is found to be 2hω/A r Qτ [22] (hω, τ , Q and A r = x 2 res are the photon energy, exposure time, quantum efficiency, and resolution area respectively.). We found the noise penalty to the twin image removal is negligible except when both a/z 0 NA and η 1, a regime easily avoided in the hybrid geometry [22]. The signal-to-noise ratio (SNR) of the converged |E (n) s | is thus shot-noise limited to N s /2 with N s = |E s (z 0 )| 2 A r Qτ /hω [13]. Remarkably, the SNR of the reconstructed image is not sensitive to a change in L, and is only a factor of two less than the SNR → √ 2N s [23,24] in standard absorption or phasecontrast imaging where effectively L ≈ z 0 within the depth of view. By taking holograms at L z 0 to reduce the intensity of the point-source reference fields, the phase shift sensitivity, defined at SNR=1 within A r , is Addressing atomic transitions requires a laser with a long coherence length. Thus holograms of cold atoms can have significant speckle noise (Fig. 3A), which is typically due to distant point-like scatterers. The speckle noise E speck compromises our knowledge of E r = E r,1 + E r,2 + E speck . Ignorance of E speck leads to multiple copies of E s (z 0 ) shifted by the corresponding distances between the distant scatterers and r 1,2 , that blur the reconstructed E s (z 0 ) image (Fig. 2B). In Fig. 2B we plot the SNR of the converged |E (n) s | vs N s /2, simulated in the presence of both speckle and photon shot noise. As expected, the reconstruction can be shot-noise limited if the speckle noise induced blurring of E s (z 0 ) is weak.
We demonstrate the hybrid microscope with a simple experiment: A 87 Rb magneto-optical trap (MOT) [25] is formed at the center of a vacuum glass cell. The atoms are cooled to ∼ 80 µK and holographically imaged by a camera (PCO pixelfly, 1392×1040 pixels, x cam = 6.45 µm pixel pitch) placed outside the cell. The reference fields E r,1 and E r,2 , detuned by ∆ from the D2 hyperfine F = 2 -F = 3 cooling transition (λ=780.2 nm), are formed by focusing two parallel, linearly polarized laser beams to diffraction limited spots of 3.5 µm, and 5.2 µm respectively. With the focal points r 1,2 displaced by d=387 µm, an interference pattern with fringe periodicity 96 µm is captured by the camera at L=47 mm. With z 0 = 3 mm, we adjust the atoms location so they are only interrogated by the inline source E r,1 . To control the sam-ple size, we vary the loading time and ramp the magnetic field gradient from 10 G/cm up to 60 G/cm before imaging. The cooling laser detuning is at −3 Γ (Γ = 2π × 6.1 MHz is the linewidth of the D2 transition). To test the spatial resolution, a 1D lattice 32 Γ detuned is pulsed to write a 20 µm period atomic density grating along y. We image during and after cooling, at various ∆, τ , and probe intensities I.
Typical holograms and reconstructed images of atom samples are displayed in Fig. 3. The hologram H with a small atom sample and H 0 in the absence of the sample are nearly identical (Fig. 3A). Careful subtraction reveals the interference fringes (Fig. 3B), albeit barely visible due to the photon shot noise. We extract E r,2 and ϕ 1,2 = arg[E r,1 + E r,2 ] from H 0 = |E r | 2 , by assuming a spherical wavefront for E r,1 [22]. Our final estimation E r ≈ √ H 0 e iϕ1,2 partly accounts for the E speck effect in the E r amplitude [26]. We also obtain the probe field E r,1 (z 0 ) =Û 0 E r,1 (L) (Fig. 3C) [22], with E speck ignored in this work [26], to simultaneously retrieve the phase Figure 3D shows a phase shift image probed at ∆ = 30 Γ, where an SNR < ∼ 1 for the subtracted hologram on the camera plane (Fig. 3B) is remarkably enhanced to SNR ≈ 7, due to the numerical focusing of E s (z 0 ). The resolution δφ ≈ 1.7 mrad is a factor of 4 smaller than the minimal noise level in phase-contrast imaging. The absorption α ≈ Γ ∆ φ in Fig. 3E is below the noise floor δα ≈ 0.4% and is hardly detected. Such δφ and δα sensitivity levels are consistently reached under various experimental conditions (Figs. 3D-G,J,K,M,N) when the speckle noise induced blurring is below the shot noise level (Figs. 2, 4), and if N p approaches N max = 2 14  i∆)|/ Γ 2 at low probe intensities, and is minimized at ∆ = 0. We plot the sample column density ρ in Fig. 3K (∆ = 0 Γ, τ = 800 µs). The rms atom number noise level is δN atom = δρA r ≈ 0.8. To see atom shot noise, we need to fix the atoms within A r during imaging, and improve δN atom further. The sample in Fig. 3L is continuously probed in the MOT with I = 0.6 W/m 2 and τ =10 ms (with saturation intensity I s = 36 W/m 2 , absorption cross-section σ = 0.14 µm 2 for the unpolarized atoms). Using such a weak probe, the noise level in Fig. 3L is twice the photon shot noise level due to poor subtraction of light scattered from the cooling beams.
Imaging during cooling allows us to observe the effect of light shift on the atoms by the cooling beams. For two samples of similar size, we display absorption and phase shift data for holograms taken during (Figs. 3M,N) and after (Figs. 3F,G) cooling. With ∆ = 1.9 Γ, the absorption is relatively increased when imaged while cooling, from which we infer a light shift of δ ≈ 4 MHz to the cooling transition so the dressed probe detuning∆ = ∆ − δ is closer to the resonance.
We test the resolution of the microscope by imaging atoms subjected to the 1D "writing" lattice. Figures 3H,I are phase shift images of the density modulated sample with ∆ = 1.9 Γ and ∆ = −7.1 Γ respectively. The opposite sign of the detunings leads to advanced and retarded phase shift, as expected. The 20 µm fringe period is clearly resolved with the diffraction limited resolution.
The images reconstructed in Fig. 3 are with both twin and DC images removed. The reconstruction follows Eq. (1) and a modified form of Eq. (2) that converges slower, but stably removes the DC noise [21]. Ther − n plots in Fig. 4A characterize such a convergence. In contrast to the simulated results in Fig. 2A where E s (z 0 ) is known, we calculate the residualr defined by the difference between E (n) s and E (n=250) s (nearly invariant un-der iteration.). Comparing with the simulated results in Fig. 2A, we notice that for η > 1, the hybrid geometry achieves nearly full real and twin image separation (r ≈ 5 × 10 −5 ) with a one-step reconstruction, a situation similar to the "off-axis" geometry [27], but retains the convenience of measuring forward scattering at large NA in the inline geometry. The tiny difference converges rapidly within 10 iterations. Due to the modification of Eq. (2) [21], the convergence for η = 0.17 is not as oscillatory and is twice as slow as the simulated case ( Fig. 2A).
Finally, we calculate the shot noise level in H and H 0 and plot the SNR of the reconstructed |E s | vs the ratio between the signal to the estimated noise level in |E s | due to both shot noise and camera readout noise [22] (Fig. 4B). We achieve photon shot-noise limited reconstructions for samples with SNR < ∼ 30. Images with larger SNR in this experiment are deteriorated by the speckle noise induced blurring, similar to the simulated situation in Fig. 2B. The few data points with SNR < ∼ 30 that are not photon shot-noise limited are caused by changes in the speckle pattern, and relative phase fluctuation in E r,1,2 between the recording of H and H 0 , which are not taken into account in the subtraction [22].
We have demonstrated diffraction limited holographic imaging of cold atoms with photon shot-noise limited sensitivity. The phase shift and absorption sensitivities are beyond those in standard imaging. The simplicity of the setup is in contrast to phase-contrast imaging [9], and our method can image an extended atomic sample where phase-contrast imaging is likely to be deteriorated by artifacts [28]. The technique can be integrated to atom chips [29] where nano-fabricated pinholes are backilluminated to generate the reference fields. The method of increasing the complexity of E r , as in the hybrid geometry, may also be useful in X-ray and electron holography [30]. As for limitations, the lengthy numerical process in this work may be sped up with advanced algorithms [31]. By including speckle noise E speck in the E r phase estimation, the SNR for a sample with large optical depth can be improved [26].
The atom number noise level δN atom ∝ x 2 res /σ reduces with improved imaging resolution. The moderate spatial resolution x res = 5.2 µm can be improved using a camera with a larger width w, or by magnifying the camera with lenses (Fig. 1). The aberration in the latter case is self-suppressed due to the common optical paths for E s and E r when z 0 L, and can be further corrected numerically. With a factor of 10 increase of NA to 0.7 and using a camera with Q=0.8 (Q=0.1 in this work), we expect δN atom ≈ 0.08 with τ = 100 µs resonant detection. By reducing z 0 for a condensed sample, the N max -limited sensitivity δφ can be improved further, even if the camera is magnified by lenses. With a spinpolarized sample to increase σ, δN atom = 0.2 should be achievable for ∆ = 5 Γ, allowing off-resonant imaging with single atom sensitivity. This could be a scenario highly favored for precise imaging of cold atoms at high density [6].
We conclude by mentioning the possibility of nondestructive probing of multiple atoms in single sites of an optical lattice [3][4][5]. The idea is based on optical shielding [32]: Blue detuned light enhances the repulsion between symmetrically excited atom pairs. In the proposed scheme, atoms are trapped in single lattice sites with strong x, y confinement. A blue detuned probe and auxiliary cooling beams, all with polarization in the x−y plane, switch on adiabatically so that atoms confined in single sites find new equilibrium positions separated along z. The blue-detuned molasses should allow sub-Doppler cooling of the atom array confined in the lattice site for continuous, holographic detection of the probe light phase shift/absorption as in this work. More investigations are needed to confirm the applicability of this method, which may substantially extend the range of observables in quantum gas microscopy [3,4].

I. Extracting the reference field.
To extract the reference field E r , we only need to estimate ϕ = arg[E r ] so that E r = √ H 0 e iϕ . With large L and tightly focused E r,j (j = 1, 2), we assume the wavefronts E r,j (R) ∝ H j e ik|R−rj | with R = (x, y, L) on the camera plane to be well-approximated by spherical waves from the focal points r j (j = 1, 2). Here H 1 and H 2 are intensity images taken individually when the inline E r,1 or off-axis E r,2 is on respectively. In the following we describe the procedure to determine the location of the focal points r 1 and r 2 , and additional adjustments to estimate E r (L) for image reconstruction, as well as to estimate E r (z 0 ) for phase shift φ and absorption coefficient α retrieval.
We decide the origin x = y = 0 on the camera plane by considering the projection of r 1 onto the camera. For this we adjust the H 1 = |E r,1 | 2 intensity pattern recorded on the camera so that it is centered and appears as a symmetric Gaussian profile. We then fit H 1 with a 2D Gaussian H 1,f and set the center of the fit as x = y = 0. With an estimate of the r 1 source-camera separation L, the location r 2 is retrieved by numerically propagating the hologram H 0 with E = H 0 /E * r,1 to its focal plane (with knowledge of the r 1,2 relative position, the twin image at −r 2 is easily identified). The relative displacement |r 2 − r 1 | = d = 387.0 µm, measured before the installation, is used to calibrate the L parameter in E r,1 .
The holographic reconstruction of the E r,2 focal point not only allows us to decide L and r 2 , but also provides an estimate of the relative phase φ r between E r,2 and E r,1 . We therefore reach an estimate E (0) r (a 1 , a 2 , φ) = a 1 H 1,f e ik|R−r1| + a 2 H 2,f e ik|R−r2|+iφr on the camera plane with a 1 = a 2 = 1 (H 2,f is a 2D fit of the H 2 intensity pattern.). To improve the accuracy of the estimation, we numerically minimize the difference H 0 − |E (0) r (a 1 , a 2 , φ r )| 2 to achieve the optimal E (0) r,opt = a 1,opt H 1,f e ik|R−r1| + a 2,opt H 2,f e ik|R−r2|+iφr,opt . Finally, we use ϕ 1,2 = arg[E (0) r,opt ] to approximate ϕ = arg[E r ] in this work so that E r (L) ≈ √ H 0 e iϕ1,2 . From the estimated E r (L) it is straightforward to calculate E r (z 0 ) =Û 0 E r (L). The result, however, is contaminated by the boundary artifacts during fast Fourier transforms. To avoid the artifacts we expand the fitted H 1,f and H 2,f onto a grid twice as large as the camera, so that both fitted intensity patterns are well-contained within the grid. Similar to the E r (L) estimation, the approximation E r (z 0 ) ≈Û 0 E (0) r,opt ignores speckle noise. II. Iterative twin and DC removal.
The algorithm in Eq. (2) assumes a negligible DC term in Eq. (1) (i.e. |E s | 2 |E r | 2 everywhere on the camera). To remove the DC term the iterative twin image removal algorithm in Eq. (2) is modified. Using our estimation of E s at each iteration E In Eq. (A.1) a parameter ν is introduced to control the convergence speed. In the original algorithm (Eq. (2)) ν = 1. Due to the additional step to remove the DC term, the iterative algorithm becomes nonlinear, and we find it can be unstable when E s is large. Adjusting the parameter ν can improve the stability and we set ν = 0.5 in this work. The convergence speed is correspondingly a factor of two slower (Fig. 4A), while in addition the oscillatory feature ( Fig. 2A) in the convergence is suppressed.
III. Optimal background subtraction. Practically, the holograms H and H 0 , and the intensity patterns H 1 and H 2 , are affected by power and phase fluctuations of the laser, as well as background ambient light. For retrieving E r using H 0 , H 1 and H 2 as in section I., we simply take a background image B 0 with the same exposure time as H 0 and subtract it away directly. The subtraction H − H 0 in Eq. (1), however, requires a higher precision. We have developed a two-step optimization procedure for the subtraction as following, both using a Matlab optimization program "fminsearch".
First, by taking advantage of the characteristic two point interference on the camera, we optimize the subtraction H  , where three spot images are formed. We isolate the spot corresponding to the off-axis source at r 2 = (d, 0, 0) and calculate its residual power dP (b) (The other two spots correspond to the inline source and the twin image of the off-axis source.). These steps are repeated until dP (b) is minimized at an optimal choice b = b opt . To ensure that the presence of the atomic sample in H does not affect our choice of b opt , we apply a mask that blocks the data in both H and H 0 where we expect the geometric shadow of the sample, before the optimization starts.
Such optimization for b = b opt based on the interference fringes cannot account for any relative power fluctuation between E r,1 and E r,2 . In addition, any fluorescence from the atoms would cause a uniform background f bk . We account for these effects, together with the background ambient light, using a four parameter optimization process with parameters c j , j = 1, 2, 3, 4.
Here P f = H f,1 − H f,2 accounts for power fluctuation between E r,1 and E r,2 , f bk = 1 and B is the ambient background image with the same exposure time as H.
The complex field E(c j ) = H R (c j )/E * r , (j = 1, 2, 3, 4) is then propagated viaÛ 0 to z = z 0 where we expect the in-focus atom image E s (z 0 ) as well as the out-of-focus twin and DC images (as in Fig. 1). We use the mask 1 −P to exclude E s . The leftover atom twin images are converted into a single real image and refocused to z = z 0 using theÛ 0ĈÛ −1 0 operation as in Eq. (2), before it is also excluded via 1 −P. The two-step exclusions ensure that the final image is nearly free of atomic signal, from which we calculate the rms noise level N (c j ) to be minimized at c j,opt and thus the final H R,opt which is considered as the "H − H 0 " to obtain E H in Eq. (1). This is the most time-consuming part of our reconstruction process, and takes approximately 10 mins with a PC (Intel Core i5-2400 CPU, 3.1 GHz).
IV. Propagation of shot noise.
The rms shot noise level in the holograms, being proportional to the pixel count N p = HA p Qτ /hω, is given by n H = H/ HA p Qτ /hω and similarly n H0 = H 0 / H 0 A p Qτ /hω. The associated shot noise δE H of the complex field E H in Eq. (1), defined at each pixel with area A p , has a rms level of: Here we assume n H = n H0 . To derive the rms level of the shot noise field δE a at the atoms location (z = z 0 ), we simply evaluate the intensity contribution |Û 0 δE H | 2 from each camera pixel on which δE H has a random but uniform amplitude. The contribution is then summed over all the pixels to give the intensity of δE a . More specifically, we consider the field δE H through each pixel of the camera, with wavefront area restricted by the tiny pixel area A p , as part of spherical waves propagating toward the focal points r 1 and r 2 . It is thus straightforward to calculate the noise level δE 0 near the point sources, r = O(d), via Kirchhoff's diffraction theorem and further a 2D integration of the intensity contribution over all the pixels. The final result is: Here NA = w/ √ w 2 + 4L 2 is the numerical aperture of the point sources spanned by the camera. With z 0 L the noise level near the atom location n E,a ≈ n E,r=O(d) . WithÃ r ≈ A r , we reach an estimate of the rms level of shot noise δE a near the atom location: s + δE a , while the complex field E H is modified asẼ H = E H + δE H . As the iteration series converges toẼ s = E s (z 0 )+δE s , instead of δE s → 0 in the noiseless case, it is easy to show that the following relation holds (here for simplicity we consider |E s | 2 |E r | 2 as in Eq. (2) so that the DC term can be ignored): The final noise δE s , as a functional of the shot noise δE a specified by Eq. (A.4), is difficult to calculate analytically. However, ifPδE s is, as verified numerically, a uniform shot noise pattern filtered by the apertureP, while further, the twin image termÛ 0ĈÛ −1 0 PδE s is not correlated with the shot noise δE a in Eq. (A.6), then the intensities of the two terms on the right add up: Here ξ is the ratio of the rms level betweenÛ 0ĈÛ −1 0P δE s and δE s . The final result of the shot noise level becomes: For the traditional inline geometry ξ inline = 1/(1 + 2w/L a/z0 ) (z 0 L), which can be derived by considering the area ratio of the apertured shot noise given byP and its NAlimited shadow at the twin image plane. For the hybrid geometry ξ ≈ (1−η)ξ inline for η < 1. In this work ξ ∼ 0.1, so the increase of the shot noise level from n Ea to n Es due to the twin-image removal is less than 1%.
VI. Shot-noise limited SNR, and N max -limited sensitivity With the noise level of δE s specified by Eq. (A.8), the SNR = |E s (z 0 )/n Es | 2 over a particular resolution area A r , is given by SNR = N s /2 with N s = |E s (z 0 )| 2 A r Qτ /hω as in the main text (we consider ξ 2 /2 1.). On the other hand, for a camera with a maximum count N max , the relation |E r (L)| 2 A p Qτ /hω < N max needs to be obeyed for most of pixels on the camera. Thus by considering the SNR = 1 limit, the minimum detectable E s must obey Es(z0) Er(L) > 2Ap NmaxAr , or, with κ = |E r (L)| 2 /|E r (z 0 )| 2 . Equation (A.9) also provides the minimum detectable phase shift δφ, as well as half of the minimum detectable absorption coefficient δα. VII. SNR and shot noise level in the image data To obtain the SNR for the experimental images, we define the signal as the mean value of the reconstructed |E s | above 60% of its maximum. The noise level is given by |E s | 2 D , where D is the area of an annulus surround-ingP with ring width equal to a.
The shot noise level in the reconstruction is simply estimated via the formula n H = H/ HA p Qτ /hω and n H0 = H 0 / H 0 A p Qτ /hω for the holograms H, H 0 respectively and then follows Eq. (A.3) to obtain n EH . A ratio A p /A r is then multiplied according to Eqs. (A.3)(A.5) to obtain the level n Ea ≈ n Es according to Eq. (A.8). The camera readout noise at 8 counts/pixel for both H and H 0 is added to the photon shot noise quadratically. We also include the shot noise in the ambient background B and B 0 , which are averaged over 9 images and contribute negligibly to the overall noise level.
VIII. Sub-pixel resolution, field and depth of view.
To achieve sub-pixel resolution, we simply split each pixel in the camera-recorded hologram into M 2 new pixels, each with new pixel sizex cam = x cam /M ≤ x res .
An important advantage of the holographic microscope is its ability to reconstruct the image of an object within its volume of view, given by the ability of the camera to record the interference fringes determined by the Whittaker-Shannon sampling theorem [19], thereby enabling a 3D volumetric view of sparse objects.