Deeply Sub-Wavelength Localization with Reverberation-Coded-Aperture
DDeeply Sub-Wavelength Localization with Reverberation-Coded-Aperture
Michael del Hougne, Sylvain Gigan, and Philipp del Hougne ∗ Julius-Maximilians-Universit¨at W¨urzburg, D-97074 W¨urzburg, Germany Laboratoire Kastler Brossel, Universit´e Pierre et Marie Curie,Ecole Normale Sup´erieure, CNRS, Coll`ege de France, F-75005 Paris, France Univ Rennes, CNRS, IETR - UMR 6164, F-35000, Rennes, France
Accessing sub-wavelength information about a scene from the far-field without invasive near-fieldmanipulations is a fundamental challenge in wave engineering. Yet it is well understood that thedwell time of waves in complex media sets the scale for the waves’ sensitivity to perturbations.Modern coded-aperture imagers leverage the degrees of freedom (DoF) offered by complex media asnatural multiplexor but do not recognize and reap the fundamental difference between placing theobject of interest outside or within the complex medium. Here, we show that the precision of local-izing a sub-wavelength object can be improved by several orders of magnitude simply by enclosingit in its far field with a reverberant chaotic cavity. We identify deep learning as suitable noise-robust tool to extract sub-wavelength information encoded in multiplexed measurements, achievingresolutions well beyond those available in the training data. We demonstrate our finding in themicrowave domain: harnessing the configurational DoF of a simple programmable metasurface, welocalize a sub-wavelength object inside a chaotic cavity with a resolution of λ/
76 using intensity-only single-frequency single-pixel measurements. Our results may have important applications inphotoacoustic imaging as well as human-machine interaction based on reverberating elastic waves,sound or microwaves.
Retrieving a representation of an object based on howit scatters waves is a central goal across all areas of waveengineering (light, microwaves, sound, ...) with appli-cations ranging from biomedicine via microelectronicsto astrophysics. Since wave energy cannot, in general,be focused beyond the diffraction limit in free space, awidespread misconception is that sub-wavelength infor-mation can only be accessed via evanescent waves. Thisargument ignores the crucial roles of a priori knowledgeand signal-to-noise ratio (SNR); moreover, many imag-ing schemes do not even rely on focusing. Indeed, givenextensive a priori knowledge, an imaging task can col-lapse to a curve fitting exercise without any fundamentalbound on the achievable precision (e.g. deconvolution mi-croscopy [1, 2]). The advent of deep learning has enabledelaborate demonstrations of such nonlinear function ap-proximations, facilitating deeply sub-wavelength imagingeven with a simple plane wave from the far field [3]. De-spite the resulting frequent absence of any wavelength-induced fundamental resolution bounds, specific physi-cal mechanisms can be useful to boost the practicallyachievable resolution. A common example is the above-mentioned access to evanescent waves either via near-fieldmeasurements [4, 5] or by coupling them to the far-fieldwith near-field scatterers [6–15]. Similarly to the use offluorescent markers [16], these approaches are inherentlyinvasive since they rely on manipulations of the object’snear-field. A further notable idea relies on tailored co-herent far-field illumination to create super-oscillatoryhotspots [17, 18] but suffers from inherently low SNRs.In the wave chaos community [19], it is well known thata wave’s sensitivity to geometrical perturbations [20–23]is directly related to its dwell time in the interaction do-main [24]. This effect can be thought of as a general- ization of the sub-wavelength interferometric sensitivityin phase microscopy [25–27] or high-finesse Fabry-Perotcavities [28]. If the scene to be imaged is enclosed inits far field by a reverberant chaotic cavity, the dwelltime is drastically enhanced. Different scenes can thenbe interpreted as different perturbations of an otherwisestatic complex scattering geometry. This clearly hintsat the potential of chaotic reverberation to significantlylower the resolution limit without any near-field manipu-lation – provided that the complete scrambling of waves(and the information that they carry) inside the complexmedium can be untangled in post-processing.While linear chaotic reverberation as simple route todeeply sub-wavelength resolution has to date remainedunexplored, possibly with the exception of diffusing wavespectroscopy capable of extracting global features (e.g.,scattering cross-section) of moving scatterers in complexmedia [23, 29–31], a rich literature actually exists onimaging and sensing with a complex medium as codedaperture (CA). This research track is driven by the de-sire to achieve imaging with as few detectors and mea-surements as possible. Rather than directly mapping theobject to its image, the spatial object information canbe multiplexed across random configurations of a CAonto a single detector [32] – see Fig. 1(a). Practical im-plementations of CAs often leverage the fact that wavetransmission through a complex medium (multiply scat-tering medium, chaotic cavity, disordered metamaterial)constitutes random multiplexing thanks to the medium’sspectral, spatial or configurational degrees of freedom(DoF) [33–37] – Fig. 1(b) illustrates the former. In otherwords, the transmission matrix of a complex medium nat-urally offers the desired properties of a random multiplex-ing matrix [33, 35]. a r X i v : . [ phy s i c s . a pp - ph ] F e b Within this realm, scenarios in which object and wavesource are embedded within the complex medium, asin Fig. 1(c), have been treated as a simple alternativeway of natural random multiplexing. In photoacous-tics, it was recently suggested to enclose the imagingtarget in an acoustically reverberant cavity [38]. More-over, several schemes for human-machine interaction areinevitably confronted with waves reverberating aroundan object, for instance, object localization with mi-crowaves in indoor environments or with elastic wavesin solid plates [39–42]. However, the benefits of such“reverberation-coded-apertures” (RCAs) go far beyondrandomized multiplexing. If the object is inside (ratherthan outside) the complex medium, the wave interactswith the object not once but countless times, therebydeveloping a much higher sensitivity to sub-wavelengthobject details. Recent efforts to construct optimal co-herent states for sensing in complex media [43, 44] differfrom our problem, besides their requirement for multi-channel excitation, in that they rely on (and are limitedto) small perturbations of the sought-after variable.In this Letter, we introduce RCAs, combined with deeplearning to extract encoded sub-wavelength information,as truly non-invasive route to deeply sub-wavelength res-olution. Previous works on setups that can be consid-ered as RCAs assumed that resolution was inherentlydiffraction-limited [40, 41], or that a non-linear pro-cess resulting in self-oscillations was required for sub-wavelength resolution [45]. Here, we first establish thefundamental link between dwell time, wave sensitivityand resolution in a semi-analytical study of the prototyp-ical example of object localization inside a linear chaoticcavity using spectral DoF. Then, we demonstrate ourfinding experimentally in the microwave domain, usingconfigurational DoF provided by a simple programmablemetasurface.To start, we consider for concreteness a 2D model prob-lem in semi-analytical simulations based on a coupled-dipole formalism (see Refs. [15, 46] and SM). As depictedin Fig. 2(a-c), we consider three scenarios expected tocorrespond to different durations of the scattering pro-cess: free space, a cavity with quality factor Q = 263 anda cavity with Q = 556. In each case, the transmission S between a transmit and a receive port is measuredfor various frequencies in order to localize (using ANN-based data analysis) a non-resonant dipole that couldbe located anywhere along a circular perimeter. Giventhe single-channel nature of our single-detector scheme,we estimate the duration of the scattering process viathe “phase delay time” τ = ∂ arg( S ) /∂ω [47]. The ir-regularly shaped cavity constitutes a complex mediumfor wave propagation in which τ is hence a statisticallydistributed quantity; in Fig. 2(d), we plot the cumula-tive distribution function (CDF) of its magnitude for thethree considered cases, confirming that they correspondto increasing dwell times. Compared to a regular cavity, FIG. 1. (a) Conventional CA. A wavefront is scattered byan object and then multiplexed across different masks onto asingle-pixel detector. (b) Use of a complex medium’s spectralDoF (color-coded) as CA. Wavefronts are scattered by an ob-ject and then propagate through a chaotic cavity such thatspatial information is multiplexed across different frequenciescaptured by a single-pixel detector. (c) Reverberation-coded-aperture: same as (b) except that object and wave source are inside the chaotic cavity. such a chaotic cavity has not only the practical advantageof being easy to implement but also that ergodicity en-sures statistically similar properties [48, 49] irrespectiveof the object position.The dwell time plays a crucial role in mesoscopicphysics because it is related to several other relevantquantities [47, 50–52], some of which happen to also becritical metrics for RCA-based imaging and sensing. Themost obvious quantity is the energy stored in the com-plex medium [47, 53, 54] which is directly related to thereceived signal strength and thereby the measurement’sSNR (assuming detector-induced noise). Of course, mea-surements with higher SNR contain more information.The direct link between | τ | and the stored energy is ap-parent upon visual inspection of Fig. 2(a-c), and indeedthe received signal strength is on average 2.9 [3.9] timeshigher in (b) [(c)] than in (a).The sensitivity to parametric perturbations of the scat-tering system is another important quantity that can berelated to the wave’s dwell time in the interaction do-main [24, 55]. Intuitively, this can be understood as fol-lows: the probability that a tiny perturbation impactsthe evolution of a wave increases with the wave’s lifetimein the interaction domain. Analogous to τ , we define χ = ∂S /∂X , where X denotes the considered param-eter (the object position along the allowed trajectory in FIG. 2. (a-c) Electric field magnitude in 2D semi-analyticalsimulations for (a) free space, a chaotic cavity with (b) Q =263 and (c) Q = 556. All maps use the same color scale.(d) CDF of the dwell time magnitude | τ | distribution in thethree cases. Vertical dashed lines indicate the correspondingmean values. (e) CDF of the parametric derivative magnitude | χ | (with respect to the object position) in the three cases.Vertical dashed lines indicate the corresponding mean values.(f) SV spectra of T ( X, f ) (without any noise) for the threeconsidered cases. (g) Average localization error (cid:15) in termsof the smallest utilized wavelength λ min as a function of the absolute magnitude of the measurement noise for the threecases. Vertical dashed lines indicate the corresponding signalmagnitudes. The horizontal black line indicates the trainingdata resolution. our case). Indeed, in Fig. 2(d,e) we observe a clear corre-spondence between the distributions of | τ | and | χ | , for in-stance in terms of the mean value (dashed vertical lines).The difference between two nearby object positions interms of the corresponding scattering matrices (specifi-cally, S ) is thus on average larger if the dwell time inthe interaction domain is larger. This effect is induced byusing a RCA instead of a conventional CA for which theobject scatters the wave before the wave is multiplexedacross the CA.In the case of a CA (RCA or conventional) leveragingspectral DoF, the amount of information that can be ex-tracted from measurements within a given bandwidth isalso tightly linked to the dwell time. Indeed, the spectraldecorrelation is related to the rate at which S fluctuateswith respect to the frequency. The lower the correlationbetween different measurement modes is, the less redun-dant information is acquired. To illustrate this effect,we consider the singular value (SV) decomposition of a2D matrix T ( X, f ) containing the measured transmissionfor different frequencies and positions along the allowed trajectory; T ( X, f ) later serves as training data for theANN. In Fig. 2(f), we plot the SV spectra of this ma-trix for all three considered cases. Indeed, higher dwelltimes correlate with a flatter SV spectrum, implying thatdifferent frequencies are less correlated.Having established three distinct RCA mechanisms ex-pected to enhance the possibility of physically encoding deeply sub-wavelength information about the object viawave propagation inside the RCA in multiplexed mea-surements, we now tackle the problem of digitally decod-ing this information. In order to approximate an inversefunction of the physical wave scattering process, mappingthe measured data to the object position, we use deeplearning. We deliberately use an artificial neural network(ANN) consisting of several fully connected layers (seeSM) as opposed to more popular convolutional architec-tures because the latter excel at identifying relevant localcorrelations in the data whereas we hypothesize that thecomplete scrambling caused by wave scattering encodesthe relevant features in long-range correlations within thedata [46, 56, 57]. Moreover, our ANN does not solve aclassification problem but predicts a continuous variable:the object’s position.We report in Fig. 2(g) the average localization errorin terms of the smallest utilized wavelength, (cid:15)/λ min , asa function of the measurement noise magnitude. First,we observe that even the free space scenario can achievedeeply sub-wavelength resolution beyond λ min /
10 at lownoise levels, stressing the absence of any fundamen-tal wavelength-induced resolution bounds, similar toRefs. [1–3]. Second, as hypothesized, the longer the dwelltime in the RCA, the higher the achievable resolutionat a given noise level. In our case, we observe resolu-tions beyond λ min / but, as justified above, we re-frain from comparing these absolute resolution values toother works with different a priori knowledge and SNR.Third, remarkably, the achievable resolution can be morethan an order of magnitude better than the resolution ofthe training data, suggesting that beyond being an effi-cient approximator to “seen” data, our ANN also veryfaithfully interpolates between “seen” data points. Deeplearning also offers a remarkable noise-robustness whichsignificantly outperforms a simple multivariate linear re-gression (see SM section D and Fig. S3).One question naturally arises: can we isolate thecontribution of the three identified RCA mechanisms?Specifically, we now evidence the major role of the dwell-time-enhanced sensitivity to tiny perturbations. To thatend, we consider a setting in which the other two factorsdo not impact the localization accuracy: we compare thethree considered scenarios in terms of their SNR (remov-ing benefits due to enhanced signal strength), and oper-ate with a single DoF (removing benefits due to fasterspectral decorrelation). Table 1 summarizes the achiev-able localization accuracies for two rather high values ofSNR; the focus here is not on the absolute localization TABLE I. Average localization error (cid:15)/λ min using a singleDoF for two magnitudes of the measurement noise relative tothe measured signal strength.SNR [dB] no cavity cavity Q = 263 cavity Q = 55630 1.12 0.53 0.2660 0.59 0.31 0.24 precision but on how it compares between the three con-sidered scenarios. The dependence of the achieved local-ization accuracy on the dwell time emerges very clearly,confirming our argument that placing the scene inside aRCA rather than outside a CA boosts the sensitivity tosub-wavelength scene details. Incidentally, even in Ta-ble 1 we observe a resolution of ∼ λ/ . Q = 556 at an SNR of 30 dB.Having investigated fundamental RCA mechanisms,we now report an experimental demonstration in the mi-crowave domain inside a 3D irregularly shaped metal-lic enclosure – see Fig. 3(a). We use configurational in-stead of spectral DoF: we measure the transmission be-tween two antennas (“single-pixel detector”) at a singlefrequency ( f = 2 .
463 GHz) but for a fixed series of ran-dom configurations (parameter c ) of the cavity’s scatter-ing properties. The latter is conveniently implementedwith a simple programmable metasurface [58, 59] con-sisting of an array of individually tunable meta-atomswith two digitalized states mimicking Dirichlet or Neu-mann boundary conditions [60]. Moreover, we now onlyuse the intensity information of the measurements, toillustrate that RCAs enable deeply sub-wavelength reso-lution even without access to phase information, whichrelaxes hardware requirements considerably. We accessdifferent dwell time regimes by tuning the opening of thecavity’s ceiling.Of the above identified three distinct RCA mecha-nisms, two (signal strength and sensitivity) are indepen-dent of the utilized type of DoF; the third (measurementdiversity) turns out to be favorably linked to enhanceddwell times if configurational DoF are used, too, albeitfor a different reason. The amount of information thatcan be extracted from a series of measurements at f with random metasurface configurations is larger if thelatter induce stronger fluctuations of S ( f ). A longerdwell time correlates with a larger standard deviation of | S ( f ) | , as evidenced in Fig. 3(b). This can be under-stood by decomposing the transmission between the twoports into all contributing ray paths. If the metasurfaceis small compared to the cavity surface and the dwelltime is relatively low, only a few rays are affected by themetasurface configuration. We sketch for such a scenariothe cloud of accessible S ( f ) values in the Argand dia-gram in Fig. 3(c); the cloud is not centered on the originbecause many rays are not controlled by the metasurface.The longer the dwell time, the more rays will encounter FIG. 3. (a) Experimental setup: a 3D complex scattering en-closure contains a sub-wavelength metallic object on a preci-sion turntable, two monopole antennas, and a programmablemetasurface in the vicinity of one antenna. The enclosure’sceiling can be open ( Q = 40), partially covered ( Q = 92)or fully covered ( Q = 252) with metal. See SM for tech-nical details. The inset shows the standard deviation ζ a of | S | over 100 random metasurface configurations and iden-tifies the chosen operating frequency f . (b) Dependence ofmean of ζ a over object positions on Q . (c) Sketch of S ( f )distribution for 100 random metasurface configurations. (d)SV spectra of T ( X, c ) for the three considered cases. (e) Av-erage localization error (cid:15) in terms of the utilized wavelength λ as a function of SNR. The horizontal black line indicatesthe training data resolution. some of the metasurface elements such that the radius ofthe cloud increases, as witnessed in Fig. 3(b). To illus-trate that T ( X, c ) contains more information if the dwelltime is longer, we plot the corresponding SV spectra inFig. 3(d). A qualitatively similar trend as in Fig. 2(f) isseen, despite the use of a different type of DoF.The average experimental localization error plotted inFig. 3(e) is consequently significantly lower if the dwelltime is longer. The dependence on the SNR is evalu-ated by adding white noise to the experimentally mea-sured values. Unlike in Fig. 2(g), we plot (cid:15)/λ as afunction of the relative rather than absolute noise magni-tude due to experimental constraints, such that the curvedoes not reflect the first RCA mechanism’s benefits (sig-nal strength). Despite the use of low-cost measurementequipment (see SM) and intensity-only data, we achieveresolutions up to λ/
76 in our experiment. For Q = 252,we observe once again that our ANN decoder achieves aresolution clearly exceeding that of the training data.To conclude, in this Letter, we proved that reverbera-tion in a complex medium efficiently encodes deeply sub-wavelength details in multiplexed measurements withoutany manipulation of the object’s near field. We evidencedthat the wave’s dwell time is directly linked to the achiev-able resolution via three mechanisms, irrespective of theutilized type of DoF: (i) enhanced signal strength, (ii) en-hanced sensitivity, and (iii) enhanced measurement diver-sity. We further showed that ANNs are capable of decod-ing such measurements with unexpectedly high fidelity.In microwave experiments in a chaotic cavity leverag-ing the configurational DoF offered by a programmablemetasurface, we successfully localized sub-wavelength ob-jects with a resolution of λ/ ∗ [email protected][1] J.-B. Sibarita, Deconvolution Microscopy (Springer, 2005).[2] P. 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For the interested reader, here we provide numerousadditional details that complement the manuscript andmay support any efforts to reproduce our work. Thisdocument is organized as follows:A. Semi-Analytical Simulations.B. Experimental Setup.C. Artificial Neural Network.D. Comparison of Linear Multi-Variable Regressionand Deep Learning.E. Further Parameters that Impact the LocalizationPrecision.
Semi-Analytical Simulations
We consider a 2D model of N dipoles in the x − y planewhose dipole moments are oriented along the vertical z axis [1, 2]. The dipole moment p i of the i th dipole isrelated to the local electric field at the dipole’s position r i via the dipole’s polarizability α i : p i ( f ) = α i ( f ) E loc ( r i , f ) . (S1)A Lorentzian model is used for the inverse polarizability, α − i ( f ) = 4 π γ (cid:0) f − f (cid:1) + j k (cid:15) , (S2)where the imaginary part corresponds to radiation damp-ing in accordance with the optical theorem. k denotes thewave vector and (cid:15) is the medium’s relative permittivity.The local field at the i th dipole is the superposition of theexternal field exciting the system and the fields radiatedby the other dipoles: E loc ( r i , f ) = E ext ( r i , f ) + (cid:88) j (cid:54) = i G ij ( r i , r j , f ) p j ( f ) . (S3)Here, G ij ( r i , r j , f ) = − j k (cid:15) H (2)0 (cid:18) πfc | r i − r j | (cid:19) (S4)is the Green’s function between the positions r i and r j with H (2)0 ( . . . ) denoting a Hankel function of the secondkind. We hence arrive at the relation α − i ( f ) p i ( f ) − (cid:88) j (cid:54) = i G ij ( r i , r j , f ) p j ( f ) = E ext ( r i , f ) (S5)which can be solved via matrix inversion at each con-sidered frequency. For our problem, we are interested in measuring the transmission coefficient S ( f ) betweendipoles 1 (excitation) and 2 (detection). This quantitycorresponds to the field measured at port 2 if the excita-tion field is unity at port 1 and zero elsewhere. The res-onance frequency of the dipoles constituting cavity fenceand object are chosen well above the highest consideredfrequency to ensure that the cavity and object propertiesare not heavily frequency dependent.The circle on which the object is allowed to be located(see Fig. 2(a-c) of the main text) has a radius of λ whichis the wavelength corresponding to the central frequencyof the considered frequency interval. The width of theconsidered frequency interval is ∆ f /f = 0 .
22. The cav-ities seen in Fig. 2(b,c) in the main text are created witha dipole “fence” [2] whose density is varied to tune thecavity’s quality factors.
Experimental Setup
Our complex scattering enclosure is a 0 . × . × . metallic enclosure with scattering irregularities inside, asseen in Fig. 3(a) of the main text. The sub-wavelengthobject to be localized, a metallic cube of side length4 . . × π upon reflection, depending on how the meta-atom is pro-grammed. Inside the complex scattering enclosure, wavesare incident from random angles and with random polar-izations. Therefore, the specific electromagnetic responseof the meta-atom is not important here and our techniqueleveraging configurational diversity can be implementedwith any meta-atom design as long as the meta-atom hasat least two digitalized states with distinct electromag-netic responses. The metasurface is placed not too farfrom one of the monopole antennas to ensure a strongimpact on the transmission despite using only 9 pro-grammable meta-atoms. An Arduino microcontroller im-poses the desired bias voltages on each meta-atom’s p-i-ndiode. Artificial Neural Network
Overview of Information Flow
The flow of information in our RCA scheme is sum-marized in Fig. S1 [2]. As in any coded-aperture imager,the desired information (here the object position) is notdirectly measured. Instead, wave propagation acts as aphysical encoder and we measure a variable y that is re-lated to the latent variable of interest via a function f that describes the wave propagation. In order to extractthe desired information from the measurement, a post-processing step is necessary in which the measurementis decoded digitally, in order to return from the mea-surement space to the latent variable space. The digitaldecoder attempts to approximate the inverse function of f . Countless approaches to identifying a suitable decoder f − are in principle conceivable (see also Section D). Inthe present work, we train a simple fully-connected ANNto approximate f − . ANN Architecture
Our ANN consists of four fully connected layers with256, 128, 26 and 2 neurons, respectively, as shown inFig. S2. We use the sigmoid function as activation be-tween different layers. No activation function is used af-ter the last layer. In the case of complex valued inputs,we stack real and imaginary components of our data.We normalize the input such that it has zero mean andunity variance (normalization parameters are based on
FIG. S1. Schematic of the flow of information through thephysical and digital layers. The latent variable of interest,namely the object position, is first encoded through wavepropagation on the physical layer in the measurement space.Then, an ANN is used to approximate an inverse function toreturn to the latent variable space in order to estimate theobject position from the measurements. FIG. S2. Illustration of utilized ANN architecture. N mea-surements corresponding to the use of N DoF are made: ifspectral DoF are used, each measurement is taken at a dif-ferent frequency (left); if configurational DoF are used, eachmeasurement is taken at the same frequency but for a differ-ent metasurface configuration (right). These measurementsare then injected into the ANN’s input layer (stacking realand imaginary components of the measurements in case mag-nitude and phase information is available). Multiple hiddenlayers with sigmoid activation process the information; a sin-gle output neuron without activation predicts the object po-sition. Note that the output variable is continuous (this isnot a classification ANN). the training data). We train the ANN’s weights via er-ror backpropagation using the Adam optimizer [5] with astep size of 10 − . The loss function is defined as the aver-age localization error. We observed that the results didnot significantly depend on the exact choice of hyper-parameters (number of layers and neurons); using theroot-mean-square localization error instead of the meanerror also did not result in significant differences. We un-derline that this ANN architecture differs in importantways (activation function, loss function) from ANNs usedfor classification (e.g. in Ref. [2]) since we intend to pre-dict a continuous variable here.Note that our deep learning strategy is supervised, itrelies hence on “labelled” training data (a series of trans-mission measurements for which the corresponding ob-ject position is known). This terminology potentiallycauses an unfortunate confusion with the use of “unla-beled” in the imaging literature where it refers to theabsence of near-field manipulations seeking to label theobject of interest with a marker (e.g. a fluorophore). Training and Test Data
For the localization results presented in Fig. 2(g) ofthe main text based on the semi-analytical simulations,the training data is a 257 ×
257 matrix of complex-valuedtransmission measurements between the two ports. Thefirst dimension corresponds to 257 equally spaced fre-quencies within a fixed frequency interval, the seconddimension corresponds to 257 equally spaced object lo-cations covering the entire allowed circular trajectory.The raw simulation results can be considered noiseless0(negligible numerical errors) but zero-mean white Gaus-sian noise of appropriate standard deviation is added inTensorFlow before the data enters the ANN. Specifically,this means that the noise realization is different in ev-ery iteration of training the ANN. The test data is a257 × ×
100 matrix of measured transmission magni-tudes between the two ports at the working frequencyof 2.463 GHz. The first dimension corresponds to 100predefined random configurations of the programmablemetasurface, the second dimension corresponds to 100equally spaced object locations covering the entire al-lowed circular trajectory. One fifth of the test data isused as validation data to determine at what epoch theANN training is stopped (to avoid overfitting), the re-maining four fifth are used to compute the reported ac-curacies.
Comparison of Linear Multi-Variable Regressionand Deep Learning
Given the flow of information as illustrated in Fig. S1,a natural question arises: which digital decoding methodshould be used? The goal of our present work is notto identify the best decoder; instead we claim that ourfully-connected ANN from Fig. S2 is a good choice ofdecoder that allows us to prove that useful deeply sub-wavelength information can be extracted from RCA-multiplexed measurements. A thorough discussion of dif-ferent decoding methods is a signal-processing topic be-yond the scope of our present paper in which we focus onthe physics behind the links between dwell time, sensitiv-ity and localization precision. The interested reader mayrefer to Ref. [2] for a comparison of different decodingmethods in a related context (however, Ref. [2] does notconsider the extraction of sub-wavelength information).Nonetheless, given the apparent simplicity of the sens-ing task we consider (localizing a single scatterer asopposed to recognizing more complex scattering struc-tures), it may be tempting to assume that the use ofdeep learning is not really necessary here. While we donot claim that elaborate signal processing methods otherthan deep learning cannot excel at our problem, we illus-trate in this section that a simple linear multi-variableregression does not.In Fig. S3 we compare our results based on deep learn-ing from Fig. 2(g) of the main text with corresponding re-sults obtained using a linear multi-variable regression asdigital decoder. The results suggest that in principle thelatter is also capable of achieving deeply sub-wavelengthresolution (which may (slightly) exceed the training data
FIG. S3. Comparison of deep learning (DL, continuous lines)with linear multi-variable regression (LR, dashed lines) as dig-ital decoder of the multiplexed measurements containing sub-wavelength information. The DL curves are reproduced fromFig. 2(g) of the main text. resolution). However, deeply sub-wavelength resolutionis only observed at noise levels that are six or more or-ders of magnitude smaller. In other words, our ANNdisplays a remarkable robustness to noise. The linearmulti-variable regression can only achieve good resultsat unrealistically low signal-to-noise ratios. Moreover,we note that for the “no cavity” case it does not man-age to extract any information, and for the other casesit appears to saturate at resolutions about an order ofmagnitude worse than those achieved with deep learn-ing at a noise magnitude that is six orders of magnitudehigher. Finally, we note that a similar noise robustnessof ANN decoders was also observed in Ref. [2], albeitfor a problem that did not consider any sub-wavelengthinformation.
Further Parameters that Impact the LocalizationPrecision
In Fig. 2(g) of the main text, we report the localizationprecision as a function of the measurement noise and thequality factor of the cavity. Here, we complement theseresults with three further parameters that were kept fixedin Fig. 2(g) but are now varied one at a time, whilekeeping the others fixed. Overall, these results furtherunderline the absence of any fundamental wavelength-induced bound on the achievable localization precision.By choosing suitable parameters, the performance can beimproved significantly and, as discussed in the introduc-tion of the main text, even trivial setups (like unlabelledfar-field plane wave approaches) can easily achieve reso-lutions well beyond the diffraction limit. An importantinsight is therefore that in reporting a physical mech-anism to improve the achievable resolution, one shouldbenchmark it against a reference case rather than merelyreporting that one achieves sub-wavelength resolution.1
Training Data Resolution
The finer the training data T ( X, f ) resolves the al-lowed trajectory of the object, the more precisely theANN should be able to approximate the mapping frommeasurement vector to object position. Beyond somepoint, however, the measurement noise exceeds the typi-cal difference between two neighbouring sampling pointssuch that they cannot be distinguished anymore. Theparameter T defines the training data resolution as illus-trated in Fig. S4(a).As expected, we observe in Fig. S4(c) for the case ofthe cavity with Q = 263 that lowering T generally resultsin a higher localization error. We note that the achiev-able localization error can be considerably lower thanthe training data resolution for all considered values of T except T = 9. Moreover, we note that even with anextremely coarse training data resolution such as T = 17we achieve a clearly superior performance in the cavitywith Q = 263 than using an extremely fine training dataresolution ( T = 257) in the case without cavity. Frequency Interval Sampling
In principle, the finer a given frequency interval is sam-pled, the more information one can extract. This is gen-erally true as long as the different samples are reason-ably independent. The parameter F defines the mea-surement’s spectral sampling as illustrated in Fig. S4(b).A rule of thumb is that frequency samples should be sep-arated by at least ∆ f corr = f /Q although this definitionof spectral DoF is only approximative. The degree of in-dependence can be evaluated more rigorously based onthe singular value spectrum of the 2D training data ma-trix T ( X, f ) (one dimension being different object po-sitions, the other dimension being different frequencysamples – see Fig. 2(f) in the main text). Variablesresulting from a random process are usually not per-fectly independent but present a finite amount of correla-tions [6, 7]. Therefore, although increasing the frequencyinterval sampling is initially expected to improve the lo-calization accuracy, the marginal improvement from fur-ther increases of the frequency interval sampling eventu-ally tends to zero.As expected, we observe in Fig. S4(d) for the case ofthe cavity with Q = 263 that lowering F generally resultsin a higher localization error. We note that the achievablelocalization error can be considerably below the trainingdata resolution in all considered cases. Moreover, we notethat even using an extremely low value of F = 9 in thecase of the cavity with Q = 263 yields a localization errorthat can be orders of magnitude below that of the casewithout cavity using F = 257. Object Size
The larger the object is, the more it scatters wavesand the easier one should be able to localize it based onthe limited amount of information contained in measure-ments with a given noise level, a given training data res-olution and a given sampling of the considered frequencyinterval. In Fig. 2 of the main text, we considered themost difficult case: a non-resonant point-like object. InFig. S4(e), we compare these results with a larger line-like object composed of two point-like dipoles separatedby 0 . λ . In principle, objects may also be effectively“larger” than others despite the same physical size dueto being resonant or not, resulting in distinct scatteringcross-sections.As expected, we observe in Fig. S4(e) that consideringthe larger object yields a small but notable improvementof the localization accuracy. The improvement is morepronounced for the case of the cavity with Q = 263 thanfor the case without cavity. The Role of Signal Strength Enhancements
We clarified in the main text that the achievable reso-lution depends on various parameters such as the SNR,the dwell time in the interaction domain, and the numberand independence of the utilized measurement modes.In order to clearly illustrate that the signal strength en-hancement is not the only relevant factor in explainingthe improved localization precision as the cavity’s qual-ity factor is increased, we contrast Fig. 2(g) from themain text, reproduced in Fig. S5(a), which visualizes thedependence of (cid:15) on the absolute magnitude of the mea-surement noise with the dependence of (cid:15) on the relative magnitude of the measurement noise, that is, as functionof the SNR, in Fig. S5(b). In other words, the curvesin Fig. S5(b) are independent of any signal strength en-hancement. The vertical dashed lines indicate zero SNR,hence they are distinct in (a) but coincide in (b). Thedifferences between the three scenarios are somewhat re-duced in (b) as compared to (a) since differences in signalstrength are ignored, but the three curves continue to besignificantly different, evidencing that the benefits of thecavity go beyond boosting the signal strength.2
FIG. S4. Further parameters that impact the localization precision. (a) Definition of parameter T to quantify the trainingdata resolution. The allowed trajectory on which the object may move is divided into T segments of equal length R ∆ θ .(b) Definition of parameter F to quantify the frequency interval sampling. A fixed interval B is divided into F segments ofequal size ∆ f . (c) Dependence on T . The average localization error is plotted for the cavity with Q = 263 (Fig. 2(b) in themain text) for different training data resolutions T (color-coded), keeping all other parameters fixed. Horizontal lines indicatethe corresponding training data resolution in terms of the smallest utilized wavelength. For reference, the thick black curverepresenting the case without cavity (Fig. 2(a) in the main text) for T = 257 is indicated. The black and blue thick curvesare the same as the blue and red curves in Fig. 2(g) of the main text, respectively. (d) Dependence on F . Same as (c) butwith the roles of T and F inverted. All plotted curves correspond to the same training data accuracy (black horizontal line).(e) Dependence on the object size. The average localization error is plotted for the point-like object considered thus far in thesimulations (thick lines) as well as for a line object consisting of two scatterers 0 . λ apart (dash-dotted lines) is indicated,both for the case without cavity (black) and for the cavity with Q = 263 (blue). FIG. S5. Average localization error (cid:15) in terms of the smallest utilized wavelength λ min as a function of the absolute (a) or relative (b) magnitude of the measurement noise for the three cases considered in Fig. 2(a-c) of the main text. Vertical dashedlines indicate the corresponding signal magnitudes. The horizontal black line indicates the resolution of the training data. Notethat (a) and Fig. 2(g) of the main text are identical. ∗ [email protected][1] B. Orazbayev and R. Fleury, Far-field subwavelengthacoustic imaging by deep learning, Phys. Rev. X ,031029 (2020).[2] P. del Hougne, Robust position sensing with wave fin-gerprints in dynamic complex propagation environments,Phys. Rev. Research , 043224 (2020).[3] Myriad-RF, LMS7002M Python package (2019).[4] N. Kaina, M. Dupr´e, M. Fink, and G. Lerosey, Hybridizedresonances to design tunable binary phase metasurface unit cells, Opt. Express , 18881 (2014).[5] D. P. Kingma and J. Ba, Adam: A method for stochasticoptimization, arXiv preprint arXiv:1412.6980 (2014).[6] P. del Hougne, M. Fink, and G. Lerosey, Optimally diversecommunication channels in disordered environments withtuned randomness, Nat. Electron. , 36 (2019).[7] P. del Hougne, M. Davy, and U. Kuhl, Optimal multi-plexing of spatially encoded information across custom-tailored configurations of a metasurface-tunable chaoticcavity, Phys. Rev. Applied13