Experimental demonstration of superresolution of partially coherent light sources using parity sorting
S. A. Wadood, Kevin Liang, Yiyu Zhou, Jing Yang, M. A. Alonso, X.-F. Qian, T. Malhotra, S.M. Hashemi Rafsanjani, Andrew N. Jordan, Robert W. Boyd, A. N. Vamivakas
aa r X i v : . [ phy s i c s . op ti c s ] F e b Superresolution of partially coherent light sources using parity sorting
S. A. Wadood,
1, 2
Yiyu Zhou,
1, 2
Jing Yang,
2, 3
Kevin Liang,
1, 2
M. A. Alonso,
1, 2, 4
X.-F. Qian, T. Malhotra,
2, 3, ∗ S. M. Hashemi Rafsanjani, Andrew N. Jordan,
2, 3, 7
Robert W. Boyd,
1, 2, 3, 8 and A. N. Vamivakas
1, 2, 3, 9, † The Institute of Optics, University of Rochester, Rochester, New York 14627, USA Center for Coherence and Quantum Optics, University of Rochester, Rochester, New York 14627, USA Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA Aix Marseille Univ, CNRS, Centrale Marseille,Institut Fresnel, UMR 7249, 13397 Marseille Cedex 20, France Department of Physics and Center for Quantum Science and Engineering,Stevens Institute of Technology, Hoboken, NJ 07030, USA Department of Physics, University of Miami, Coral Gables, Florida 33146, USA Institute for Quantum Studies, Chapman University, Orange, California 92866, USA Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada Materials Science, University of Rochester, Rochester, NY 14627, USA (Dated: February 5, 2021)Analyses based on quantum metrology have shown that the ability to localize the positions oftwo incoherent point sources can be significantly enhanced through the use of mode sorting. Herewe experimentally investigate the effect of partial coherence on the sub-diffraction limit localizationof two sources based on parity sorting. When the sources are fully coherent or anti-coherent, theachievable Fisher information is found to be lower than that for the incoherent case. Higher Fisherinformation is obtainable with the prior information of a negative and real degree of coherence. Ourexperiment addresses the recent debate on the effect of partial coherence [Optica 5, 1382 (2018) andOptica 6, 400 (2019)] in point source localization.
I. INTRODUCTION
The resolution of imaging systems is limited by the sizeof the diffracted-limited point spread function (PSF) [1].To quantify this resolution, the Rayleigh criterion hasbeen widely used [2]. Recently, the analysis of opticalresolution has been recast in terms of Fisher Information(FI) [3–5], which quantifies the precision of measurementsand is inversely proportional to the parameter estima-tion error. Generally, the FI of the estimation of sepa-ration δ between two spatially incoherent point sourcesdepends on the type of measurement performed on theimage plane field. In the case of direct detection of imageplane intensity, the FI goes to zero as δ −→
0, an effecttermed as Rayleigh’s curse. In their seminal work [3],Tsang et al. showed that Rayleigh’s curse can be over-come if the optical field is detected by an appropriatespatial mode demultiplexer (SPADE), given prior knowl-edge of two equal intensity and incoherent point sourcesversus a single emitter. The FI for such a scheme is con-stant as δ −→
0, as has been verified experimentally[6–10].There has been a debate [11–14] on whether SPADEcan overcome Rayleigh’s curse if the two sources are spa-tially partially coherent. The work on superresolution ofpartially coherent sources was initiated in [11]. There,Larson and Saleh (LS) predicted that Rayleigh’s cursecannot be broken for partially coherent sources. How-ever, the Tsang–Nair (TN) model presented in [12] ar-gued that SPADE can break Rayleigh’s curse and showed ∗ Currently with Facebook Reality Labs, Redmond, WA, USA † [email protected] that a more physical reparameterization of the model in[11] is needed. More importantly, the proposal in [12]showed how an experiment can be done to show super-resolution for partially coherent light. The work in [12]motivated a reparameterization of the LS model in [13],which also then predicted a breaking of Rayleigh’s curseusing the quantum Fisher Information metric. We useTN to refer to the model of [12] and LS to refer to thereparameterized model in [13]. We think all of theseworks have advanced the understanding of partial coher-ence in superresolution measurements. Our work distillsthe different elements of these works and combines themto show the breaking of Rayleigh’s curse for partially co-herent light sources.The FI predictions for the LS and TN models using aparity sorter are shown in Fig. (1). We emphasize thatthese FI predictions are only with respect to δ , and as-sume a known degree of spatial coherence γ . Specifically,there are two different predictions for the case of δ −→ • When the degree of spatial coherence γ =-1, the TNmodel [12] predicts a maximum FI, while the LSmodel [13] predicts that FI −→ • When γ <
0, the FI for TN model remains boundedbelow the maximum FI for γ = −
1. For the LSmodel, the FI can grow larger than the bound pre-dicted by the TN model.Note that both models agree in their predictions of FIexactly for γ = 0, and qualitatively for 0 < γ . Themain result of this paper is to experimentally demon-strate that Rayleigh’s curse can be broken by SPADEfor γ <
0, while the curse resurges for | γ | = 1. Ourresults agree with the LS prediction. We also show that F i s he r I n f o r m a t i on / ( N / (cid:1) ) F i s he r I n f o r m a t i on / ( N / (cid:0) ) / =-1=-0.75=-0.5=0=0.5=1 (a) (b) FIG. 1. Expected FI for Parity Sorter plotted versus δ/σ .A higher FI corresponds to a lower estimation error. a: LSprediction using modal weights given by Eqs. (6,7). b: TNprediction, calculated using the modal weights in (6,7), butweighted by a factor of (1+ γd ); the weighting factor multipliesthe modal weights, not the final FI. In Fig. (1a), the γ = 1curve is exactly on top of the γ = − γ = 0 curves (green line) are same in both figures. after proper normalization, the TN model agrees with theLS prediction. The model is verified experimentally bymeasuring the FI for a parity sorter. Section II explainsthe model used, the generation of spatial coherence, theuse of SPADE to measure the FI for our setup, and adiscussion of our results. Section III discusses the rela-tion of our experiment to previous works, and Section IVsummarizes the results. II. EXPERIMENTA. Model and Spatial Coherence Generation
We use a parity sorter to perform SPADE on twospatially partially coherent sources. To generate par-tial coherence, we rely on the coherent mode decompo-sition (CMD) [15]. For our problem, the simplest choiceof modes is to decompose the mutual coherence in thesymmetric (in phase) and antisymmetric (out of phase)combinations of the two sources. The mutual coherence
FIG. 2. Experimental Setup: A 795nm Gaussian beam isconverted to either a symmetric ( φ ) or antisymmetric ( φ )mode, shown in the inset, via a mode converter consisting oflinear optical transformations. The mode amplitudes are setto √ N i = √ N t P i , with P i given by Eqs. (6,7), to generatethe mutual coherence function given by Eq. (1). Polarizationoptics and attenuators, not shown in the figure, are used tocontrol the power of the beam. At any given time, one of thecoherent modes is sent to a Michelson type image inversioninterferometer, which separates the even and odd componentsof the input field. One arm has a 4 f system, which acts as anidentity operator after the beam double passes it. The otherarm has a 2 f system, implemented by a convex mirror, andan extra quadratic phase, not shown, to cancel the defocusdue to diffraction. This arm implements the transformation( x, y ) −→ ( − x, − y ). The combined beams from both arms areimaged onto a bucket detector. The power in the even andodd modes can be measured by setting the phase difference θ to 0 and π respectively. In the experiment, all modes usedare symmetric about the y axis such that E ( x, − y ) = E ( x, y ).The interferometer then works as a parity sorter in the x -direction. Γ( x , x ) in the image plane isΓ( x , x ) = N t A X k =1 p k φ ∗ k ( x ) φ k ( x ) , (1)where N t = 2 N such that N is the average image planephoton number emitted by each point source, φ k ( x ) = f + ( x ) − e ikπ f − ( x ) are the symmetric ( k = 1) and anti-symmetric ( k = 2) coherent modes, f ± ( x ) = f ( x ± δ/ δ - theparameter to be estimated, A is a normalization factorsuch that the total photon number in the image planeis R dx Γ( x, x ) = N t , p k is a real number such that0 ≤ p k ≤
1, and p + p = 1. In what follows, the termseven and odd modes are used interchangeably with sym-metric and antisymmetric modes respectively. These co-herent modes can be generated by a mode converter usinglinear optical transformations, as shown in Fig. (2). Weuse Gaussian PSFs of width σ such that f ( x ) = e − x / σ (2 πσ ) / is the field PSF. The average number of photons at point x is given by the intensity Γ( x, x ) asΓ( x, x ) = N t A ( p | φ ( x ) | + p | φ ( x ) | ) , = N t A (cid:18) | f + ( x ) | + | f − ( x ) | + 2 γf + ( x ) f − ( x ) (cid:19) , (2)where γ = p − p is an effective degree of spatial co-herence between the two sources, and we have used p + p = 1. Physically, such a CMD means that thespatial coherence at the input plane to the SPADE setupcan be engineered by incoherently mixing appropriatelyscaled symmetric and antisymmetric modes. This canbe realized by adding a path difference between coherentmodes that is larger than the laser coherence length. Al-ternatively, we can ‘switch’ between the modes in time,with the switching time longer than the laser coherencetime, and add the recorded intensities digitally [16, 17].The CMD therefore allows us to generate spatial co-herence ‘offline’, by performing the intensity summationelectronically. To generate an intensity distribution cor-responding to a specific γ in Eq. (2), we can post-selectfrom a set of recorded intensities of φ , modes. This al-lows a great simplification of the experiment with respectto the precise control of γ . Note that we are not changingthe temporal coherence properties; all the beams used arequasimonochromatic and therefore temporally coherent. B. SPADE using parity sorting
After generating partial coherence, the next step is toperform SPADE on the field described by Eq. (1). For γ = 0 and δ ≪ σ , it has been shown that a measurementof the even and odd projections of the input field hasan FI that converges to the quantum optimal FI [7, 9].Similarly we implement SPADE using an image inversioninterferometer that sorts the input field based on its par-ity, as shown in Fig. (2). Parity sorting falls under thescheme of binary SPADE (BSPADE), which is a fam-ily of measurements that simplifies SPADE at the costof losing large-delta ( δ > σ ) information [3]. A Gaus-sian beam with σ = 327 ± µm is converted into eithera symmetric or antisymmetric mode using linear optics,which includes a spatial light modulator. The beam fluxcan be adjusted using polarization optics. The mode ispresented to a Michelson type interferometer. The toparm, which includes a 2 f imaging system and an extraquadratic phase implemented to cancel the defocus due todiffraction, implements the transformation ( x ) −→ ( − x )and the arm with the 4 f system images the field withunity magnification, after two reflections. Experimentaldetails of the interferometer are described in [18]. Forparity sorting, we set α = π in Eqs. (1-3) of [18]. The field at the output of the interferometer is E out ( x ) = √ N i φ i ( x ) + e iθ φ i ( − x )) , (3)where i = 1 , θ is the global phase difference betweenthe two arms of the interferometer, N i is the photon num-ber in the input mode φ i , and each φ i is spatially coherentin Eq. (3). Note that the coherent modes used are sym-metric in y , so the 1D analysis is valid for the experiment.To project onto the even and odd components of thefield, we can choose θ = 0 , π . As explained in sectionII A, we send only one of the coherent modes φ i at a giventime. To generate mutual coherence functions for a given γ , we add the measured intensities offline. For θ = 0( π ),all of the symmetric (antisymmetric) mode power will bedirected to the bucket detector, while the antisymmet-ric (symmetric) mode will destructively interfere at thedetector. For θ = 0( π ) the output is called as the even(odd) port. We use a bucket detector to measure the pho-ton number in a port, which is just N t R dxp i | φ i ( x ) | /A for each port.The normalization factor A is given as A = Z ∞−∞ dx X k =1 p k | φ k ( x ) | = 2( p + p + γd ) , (4)= 2(1 + γd ) , (5)such that R dx Γ( x, x ) = N t , d = R dxf + ( x ) f − ( x ) = e − δ / (8 σ ) , and we used p + p = 1. The normalizedmodal weights in the two ports are P ( δ, γ ) = N t R dxp | φ ( x ) | /A R dx Γ( x, x ) = (1 + γ )(1 + d )2(1 + γd ) , (6) P ( δ, γ ) = N t R dxp | φ ( x ) | /A R dx Γ( x, x ) = (1 − γ )(1 − d )2(1 + γd ) , (7)where the subscripts 1 , image plane modal weights. In imaging partially coher-ent point sources with a fixed δ , the total number of de-tected photons can change with γ , as pointed in [12, 19]. P , represent the probabilities of detecting a click in theeven or odd ports for a particular γ , δ and σ , given a sin-gle photon in the image plane, irrespective of the numberof photons in the object plane. For a total of N t im-age plane photons, N i = N t P i photons would then clickthe i th port. In the Supplement, we explain the requiredmodifications to obtain the object plane model that takesinto account the change of detected photons versus γ .Equations (6,7) can be used to calculate the FI for aparity sorter using the expression F ( δ, γ ) = X i =1 , P i ( δ, γ ) (cid:18) ∂P i ( δ, γ ) ∂δ (cid:19) . (8) FIG. 3. Measured modal weights for the parity sorter. a-c: Modal weights for γ = − , , − .
75. Blue and red color indicatesthe odd and even modal weights respectively. Solid lines indicate the theoretically expected modal weights with zero cross-talk.The dashed lines indicate the expected modal weights for 8% cross-talk. The circles indicate the measured values. All themodal weights are normalized by N t , the total number of photons in the interferometer. Each point on the graphs representsa mean of 10 measurements, while the error bars are too small to be noticed on the graph. Note that for | γ | = 1, the modalweights are constant versus δ . Plugging Eqs.(6,7) in Eq. (8) gives the FI for the paritysorter F ( δ, γ ) = δ d (1 − γ )16 σ (1 − d )(1 + γd ) , (9)Fig. (1a) shows F ( δ, γ ) plotted for different γ values,which matches with the FI prediction of the LS model in[13]. In particular, F ( δ, | γ | = 1) = 0 for all δ .The TN prediction for FI does not follow directly fromusing Eqs.(6,7) in Eq. (8). This is because the modalweights used in the TN model are not normalized; es-timation theory requires a normalized probability dis-tribution. The FI plotted in Fig. (1b) is derived us-ing P i (1 + γd ) instead of P i in Eq. (8), even though P i =1 P i (1 + γd ) = 1. We show in the supplementarymaterial that the properly normalized TN model predictsthe same FI as the LS model. C. Results
We now present experimental results of the FI mea-surement, which is a twofold process. On one front, ourgoal is to measure the modal weights P , as a function of δ and γ , and fit our measured data points to Eqs. (6,7).The fit is used in Eq. (8) to calculate the FI. On the sec-ond front, we compare the variance in the δ estimate for100 measurements with the FI prediction. This processis repeated for different γ values.For the first type of FI measurement, the procedure isas follows. For a fixed δ , we prepare the correspondingantisymmetric mode and record its photon number N at the output of the parity sorter. We then prepare a symmetric mode for the given δ , and record its flux atthe output of the parity sorter over a wide range of in-put powers. For a particular γ and δ , the correspondingphoton number N of the symmetric mode is given as N = KN , where K = (1 + γ )(1 + d ) / (1 − γ )(1 − d ).The modal weights are normalized by the total photonnumber in the interferometer N t = 2 N = N + N suchthat P , = N , / ( N + N ). We then post-select valuesfor the desired γ . For example, assume δ = 0 . σ , and1 photon in the antisymmetric mode, i.e., N = 1. For N = { , , , } photons in the symmetric mode, the γ would correspond to {− , − . , , . } respectively.Figure (3) shows the measured modal weights for dif-ferent γ values. The solid red and blue lines in Fig. (3a-c)are theoretically expected plots of odd and even modalweights respectively. The dashed lines are the theoret-ically expected plots incorporating effects of cross-talk.The circles indicate the mean values of data recorded for10 measurements. For | γ | = 1, all the optical power isdirected into a single port; P = 1 (solid red line) and P = 0 (solid blue line) for γ = 1. Similarly P = 0 and P = 1 for γ = −
1. If the optical power in either portdoes not change as a function of δ , which is the parame-ter to be estimated, one would expect no information tobe gained by parity sorting. The error bars on all figuresare too small to be visible on the graph.Finally, Fig. (3 b) shows the modal weights measuredfor γ = − .
75. The modal weights change rapidly with δ ,and therefore one would expect a higher FI for γ = − . γ = 1 , −
1. It can be seen from Eqs. (6-9) that as γ becomes more negative, | ∂P , /∂δ | increases at δ ≪ σ ,and the modal weights become more sensitive to a smallchange in δ . .When processing the measured optical powers for the FIG. 4. a: FI calculated from the fit to Eqs. (6,7). Note that FI curves for γ = 1 , − δ . b: Estimatedshift ˆ δ/σ for different γ values using MLE on the measured modal weights. The estimated shifts are all below the Rayleighlimit ( δ = σ ). The MLE estimates for | γ | = 1 are all constant because the modal weights are constant versus δ . Each pointrepresents the mean MLE of 100 measurements. The error bars are too small to be noticed on the graph, but are still boundedby the CRB as shown in Fig. (5). Note that the γ = 0 estimates are not biased; to distinguish the two data sets, we introducea vertical offset between the γ = 0 and the γ = − .
75 estimates. Both γ = 0 and γ = − .
75 estimates are in good agreementwith the expected shifts.
FI measurement, we have subtracted any cross-talk,which was 8% on average, between the two ports. Thecross-talk could be attributed to intensity mismatch ofthe two beams, finite coherence time, and fluctuationsof path length and polarization in the two arms. Thisbackground subtraction is allowed because we are gen-erating coherence offline by adding post-selected intensi-ties, and only one of the coherent modes is present at agiven time. Better alignment of the system can reducethe cross-talk. However, this experimental complexity isirrelevant to the analysis of partial coherence and thus isavoided in our proof-of-principle experiment by using of-fline coherence generation. The modal weights for othervalues of γ are shown in the supplementary material.After fitting the cross-talk subtracted modal weightsto Eqs. (6,7), we can use Eq. (8) to estimate the FI forthe parity sorter. The results are shown in Fig. (4a). InFig. (4a), the FI for | γ | = 1 is zero for all δ , whereasthe maximum FI for δ −→ γ = − . δ forregions of δ < σ . To estimate δ , we use maximum likeli-hood estimation (MLE) on the measured modal weights.The model of probability distribution used is that de-scribed by Eqs. (6,7). The estimated ˆ δ is shown inFig. (4b). Note that all the estimated δ ’s are below theRayleigh limit ( δ = σ ). The MLE estimates for | γ | = 1are all constant! This explicitly shows that parity sort-ing does not work for localization of completely coherentsources, and thus explains the zero FI for | γ | = 1. Incontrast, the MLE works very well for γ = 0 , − . γ = 0 , − .
75. Formally, Var[ˆ δ ] ≥ ( N t F ) − ,where Var[ˆ δ ] is the variance in the MLE estimator ˆ δ ,and F is the FI as given by Eq. (9) and shown in Fig.(4a). For δ in the interval [0 . − σ (in increments of0 . σ ), we take 100 images each of the symmetric andantisymmetric modes, thus getting 100 MLE estimatesand the corresponding variance. We have not observedany bias in the estimates, as evident in Fig. (4b), wherethe mean of the estimates are equal to the true value of δ/σ . Figure (5a) shows the normalized Mean Square Er-ror (MSE) = N t Var[ˆ δ ] as a function of δ and two valuesof γ = 0 , − .
75. More importantly, Fig. (5a) shows thatthe MSE for γ = − .
75 is below the CRB for the γ = 0case. In other words, not only is Rayleigh’s curse avoidedfor γ = − .
75, the estimation is more precise than theincoherent case of γ = 0. Fig. (5b) shows the measuredFI, which is given by (MSE) − . Note that the MSE (themeasured FI) are still offset from the CRB (theoreticalFI). To truly saturate the CRB, the system must be shotnoise limited. The dominant noise in our system are thephase fluctuations in the interferometer when it is biasedat θ = 0 or π (See Fig. (2)). Furthermore, the MSEfor γ = 0 , − .
75 might appear correlated, for exampleat δ = 0 . , .
3. This is because the same set of imagesare used for CMD of both γ = 0 , − .
75, and hence both γ = 0 , − .
75 MSE’s will be affected by the same phasefluctuations; if the γ = 0 MSE is higher, so will be the γ = − .
75 MSE. For the completely coherent case of | γ | = 1, Fig. (4b) showed that the ML estimates have FIG. 5. a: Measured MSE for γ = 0 (green triangles) and γ = − .
75 (red crosses). For each data point, MLE estimatesfrom 100 trials were used to calculate the variance. Note thatfor a given δ/σ , the MSE for γ = − .
75 is consistently lessthan the MSE for γ = 0. The dashed green and solid redlines indicate the CRB for γ = 0 , − .
75 respectively. TheCRB is given by the inverse of Eq. (9). b: FI measured fromthe inverse of the measured MSE’s given in (a). For a given δ/σ , the measured FI for γ = − .
75 is higher than the FI for γ = 0. The dashed green and solid red lines indicate the FI forthe respective γ = 0 , − .
75 respectively, calculated from Eq.(9). Technical noise factors causing the discrepancy betweentheory and experiment are explained in the main text. zero precision, and hence the measured FI for | γ | = 1 iszero. The details of image processing, CMD, the photonnumber in Fig. (5) versus δ , and mode generation aregiven in the Supplement. III. RELATION TO PREVIOUS WORKS
The LS [11, 13] and TN [12] models deal with the im-age plane FI. We have shown that with proper normal-ization, the TN model agrees with the LS model, whichpredicts a resurgence of Rayleigh’s curse for | γ | = 1; thisprediction was verified experimentally. Our experimentalso shows that per image plane photon, γ < γ = 0, as predicted by both LSand TN models. Our image plane FI results also exactlymatch the predictions of [19, Fig. (1) ], if one substitutesthe (real) γ with cos ( ψ ) in our model, where ψ is therelative phase between the two shifted Gaussian PSFs. Of course, for γ −→ − δ −→
0, the image planepower also vanishes. In this case, one needs to pumpharder at the source end to maintain a constant photonnumber at the image plane. Hradil et al. in [19] correctlyargue that this ‘cost’ of generating coherence should betaken into account when discussing the FI of partiallycoherent sources; we call this cost-accounted FI as theobject plane FI, which assumes a constant photon num-ber at the object plane. In the supplement, we show thatthe object plane FI for γ < γ = 0case, but cannot be higher than twice the γ = 0 case. Therange of δ for which the γ < γ = 0FI grows smaller, however, as γ −→ −
1. These results, ac-tually implicit in [19], do not contradict the analysis ofHradil et al. since our source preparation model is differ-ent. Specifically, in the notation of [19], we assume thesource | φ i = | Φ i |↑ z i can be prepared directly withoutderiving it as a subensemble of a classically entangledstate, as done in [19, Eq. (7)]. This mimics more closelythe scalar theory problem of two point sources emittingin the same polarization, an assumption made in all ofthe preceding papers on superresolution and SPADE forincoherent [3] or partially coherent [11, 12] sources. Theobject plane FI also does not suffer from the ‘divergence’predicted in the image plane FI for the case of vanishingpower for γ −→ − , δ −→ γ itself can be unknown andneeds to be estimated jointly with δ . Other potentiallyunknown parameters include the phase of γ , the centroid,and the intensity ratio of the two sources, and Rayleigh’scurse is expected to persist in general. Finding the quan-tum FI and the optimal measurement scheme for thisgeneral scenario is an active area of research [20]. De-pending on the measurement scheme used, the experi-mentalist may or may not have access to source planephoton number, or a ‘knob’ to tune the source parame-ters. In the special case of when only δ is unknown, ourwork shows how the lowerbound on the Var[ˆ δ ] changeswith γ , for a given photon budget in the image or objectplane. IV. CONCLUSION
We performed parity sorting on two Gaussian PSFswith varying degrees of spatial coherence. Our resultsshow that partial anticorrelation of the two sources in-creases the FI of δ estimation. For perfect coherence( | γ | = 1), the FI is zero and SPADE does not offer anyadvantange for sub-diffraction limit localization. Theprincipal difference between the predictions of [11, 13]and [12] is due to normalization of modal weights. Af-ter proper normalization, the TN model gives the sameFI predictions as the LS model. Therefore, Rayleigh’scurse can be avoided for partially coherent sources. Ouranalysis assumes a real, known value of γ . Further stud-ies could include concurrent estimation of δ and γ , forwhich [11] predicts a vanishing FI with δ −→
0. The nat-ural extension of the current work is to consider the morerealistic case of multiparameter estimation of a complex γ , and the centroid and intensity ratio of the two sources[20]. FUNDING INFORMATION
A.N.V and S.A.W acknowledge support from theDARPA YFA
ACKNOWLEDGEMENTS
S.A.W acknowledges Prof. J. R. Fienup and WalkerLarson for useful discussions.
DISCLOSURES
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