Shaping Microwave Fields using Non-Linear Unsolicited Feedback: Application to Enhanced Energy Harvesting
SShaping Microwave Fields using Non-Linear Unsolicited Feedback:Application to Enhanced Energy Harvesting
Philipp del Hougne, Mathias Fink, and Geoffroy Lerosey
Institut Langevin, CNRS UMR 7587, ESPCI Paris,PSL Research University, 1 rue Jussieu, 75005 Paris, France (Dated: October 8, 2018)Wavefront Shaping has emerged over the past decade as a powerful tool to control wave propa-gation through complex media, initially in optics and more recently also in the microwave domainwith important applications in telecommunication, imaging and energy transfer. The crux of im-plementing wavefront shaping concepts in real life is often its need for (direct) feedback, requiringaccess to the target to focus on. Here, we present the shaping of a microwave field based on indirect,unsolicited and blind feedback which may be the pivotal step towards practical implementations.With the example of a radiofrequency harvester in a metallic cavity, we demonstrate tenfold en-hancement of the harvested power by wavefront shaping based on non-linear signals detected at anarbitrary position away from the harvesting device.
PACS numbers: 84.40.-x, 41.20.Jb, 42.25.Dd
As a wave propagates through a complex medium, itsinitial wavefront is completely scrambled due to multi-ple scattering and reflection events occurring inside themedium [1]. Depending on the wave type, very differ-ent environments may be considered complex; a thinlayer of paint, biological tissue or a multimode fiber atoptical wavelengths and cities or disordered cavities formicrowaves are common examples [1–6]. Formerly, thiscomplete scrambling was perceived as absolutely detri-mental to information transfer which in turn is crucialfor imaging and communication applications. More re-cently, various novel techniques emerged that embracethe secondary sources offered by complex media ratherthan considering them an obstacle. Around the turnof the millennium, the information capacity achievablewith MIMO communication systems in complex mediawas shown to outperform that of free space [6, 7] andTime Reversal was developed in acoustics and then alsofor microwaves [8, 9]. A bit later, wavefront shaping wasintroduced in optics [10].Since then, wavefront shaping in complex media hasenabled fascinating demonstrations such as focusing be-yond the Rayleigh limit [11–14], the spatiotemporal refo-cusing of distorted pulses [15–17] and sub-sampled com-pressive imaging [18], to name a few. Furthermore, mea-suring the complex medium’s transmission matrix [19–24]provided information about important statistical proper-ties and the transmission eigenchannels of the medium[25–27], as well as being an open-loop tool in contrast toiterative focusing algorithms.However, ten years after Vellekoop and Mosk’s firstdemonstration in optics [10], focusing by wavefront shap-ing has not yet become an omnipresent technique in com-mercial imaging devices, medical therapy or the telecom-munication industry. A challenging hurdle on the pathfrom academic proof-of-concepts towards real-life appli-cations is usually the need for a feedback signal from thetarget point(s) to focus on, a common characteristic ofall wavefront shaping techniques. For medical applica- tions, a camera cannot be placed inside the biologicaltissue and implanting objects that generate fluorescenceor harmonics might be too invasive. Similarly, improvingsignal reception on a wireless device in an indoor environ-ment by wavefront shaping [28] requires real-time accessto the device’s received signal strength indicator, whichis possible to some extent in WIFI, for instance, but dif-ficult to imagine for low energy IoT devices.These difficulties with direct feedback motivate theidentification of indirect feedback schemes. Indirect so-licited feedback about the target intensity was alreadysuccessfully employed in fluorescence experiments in op-tics [29]; indirect unsolicited feedback has been demon-strated with MR-guided ultrasound focusing in acousticsand by exploiting either the photo-acoustic effect or two-photon fluorescence in optics, all using biological tissue[30–32]. In this Letter, we transpose this concept of wave-front shaping with indirect unsolicited feedback to themicrowave domain. Our target to focus on is a non-lineardevice, a radio-frequency (RF) harvester, that capturesthe ambient microwave signal and rectifies it into a DCoutput. The rectification involving diodes is a non-linearprocess inevitably generating non-linear signals that arere-emitted and constitute our indirect feedback.Incidentally, our work addresses a key challenge of cur-rent RF harvesting set-ups: the harvested voltages aretoo low for real-life applications [33]. Potentially, RFharvesting is a promising technique in the advent of con-cepts such as
Smart Home and
Internet of the Things (IoT). It may enable the wireless and battery-free pow-ering of many low-power sensors, recycling the energy ofthe ubiquitous RF fields in our urban environments andthereby constituting a step towards a greener future.Using simple electronically reconfigurable reflector ar-rays, so-called Spatial-Microwave-Modulators (SMMs)[34], we create constructive interferences of the reflectedwaves at the position of the harvester. Thereby we fo-cus the wave field and enhance the harvested energy thatdepends in a monotonous but non-linear way on the in- a r X i v : . [ phy s i c s . a pp - ph ] S e p cident field intensity. Firstly, we demonstrate in the con-trolled environment of a disordered microwave cavity thesignificant enhancement of the harvester’s voltage outputby optimizing the incident wavefront, using the harvestedvoltage as direct feedback. Secondly, we maximize onceagain the harvested voltage, but this time using an in-direct unsolicited feedback: the strength of non-linearsignatures detected inside the cavity at an arbitrary lo-cation away from the harvester.We use a disordered metallic cavity ( . ; Q =835 ) that constitutes a static, well-controllable complexmedium for our proof-of-concept experiments. In the mi-crowave domain, reverberant media are very common:electromagnetic compatibility tests require reverberationchambers [35–37], open disordered cavities currently at-tract a lot of interest for computational imaging [38–41] and indoor environments trap telecommunication sig-nals [6, 28]. The SMM covers roughly of the cavitywalls with binary elements whose boundary condi-tions can be switched dynamically between Dirichlet andNeumann for frequencies within a
100 MHz bandwidtharound .
47 GHz ; their working principle is outlined inthe inset in Fig. 1 and in Ref. [34]. G E N E R A T O R O S C I L L O S C O P E O S C I LL O S C O P E Frequency / GHz -2π Phase(r)
Single Binary SMM Element Π -state -3π-π Π-state0-state
Phase diff.
FIG. 1. Schematic of experimental set-up. Using a wave gen-erator, a quasi-continuous ambient field is generated insidea disordered metallic cavity. This field can be shaped withSpatial-Microwave-Modulators that partially cover the cavitywalls. Oscilloscope 1 monitors the voltage output of a radio-frequency harvester. The high sampling rate Oscilloscope 2is used to analyze the spectrum at an arbitrary position awayfrom the harvester. A mode-stirrer rotation by ◦ conve-niently realizes disorder. Inset adapted from Ref. [34]. With an arbitrary signal generator (sampling at
10 GHz ), we mimic a continuously excited ambient fieldby emitting a µ s long signal within the . WIFIband; a bandpass filter ( . − .
52 GHz ) cleans the sig-nal before it is emitted into the cavity by a monopoleantenna adapted for WIFI frequencies in free space.The RF harvester is a commercial prototype (cf. ac-knowledgments) that uses a low-power Schottky diodecircuit to rectify the captured microwave signal [42]. Theresults we present stand on their own and are indepen-dent of the harvester’s detailed operating mode. The em-ployed device harvests most efficiently around .
42 GHz .With the low-frequency Oscilloscope 1 ( ;
100 MHz ; 8 bits), triggered by the generator, we monitor the har-vested voltage. As exemplified in Fig. 2(a), it takes afew microseconds for the harvested voltage to rise, and abit longer to decay after the excitation signal stops. Therepetition rate of the generator is chosen such that thecycles do not overlap; over a µ s interval (highlightedin green in Fig. 2(a)), the harvested signal is stationary.In the following, harvested voltage V harv refers to theaverage signal received during this stable interval.For the indirect feedback scheme in the second part,a further WIFI monopole antenna is placed at an arbi-trary location inside the cavity outside the harvester’sline of sight; the high sampling rate Oscilloscope 2 mea-sures the received signal, again triggered by the gener-ator. A µ s interval, sampled at
25 GHz and aver-aged over measurements, is acquired and then Fouriertransformed to quantify the intensity of non-linear sig-nals in the spectrum. Using adapted antennas, appropri-ate filters or lock-in detection would be simple, cheap andwell-established means to improve the acquisition robust-ness and simultaneously remove the need for the costlyOscilloscope 2. Note that the frequency of optimal op-eration varies across our employed equipment (monopoleantennas, SMM, harvester); while this does not hinderthe intended proof-of-concept, quantitatively even betterresults are to be expected with refined equipment.Work with disordered media usually requires averagingover many realizations of disorder to get a representativeidea of the underlying physics. An individual optimiza-tion outcome strongly depends on the initial conditions,e.g. whether the speckle-like field initially has a node oranti-node at the target position. We conveniently realizedisorder with the mode-stirrer indicated in Fig. 1: rotat-ing it by ◦ yields a “new” disordered cavity with thesame global parameters (volume, quality factor, . . . ) buta different geometry, enabling a total of 30 independentrealizations. As the SMM has a large control over thewave field in this set-up [43], the experiment can more-over be repeated several times for each mode-stirrer posi-tion, starting with a different random SMM configurationeach time. Random SMM configurations effectively con-stitute different cavity geometries preserving the globalparameters, too.To begin with, we consider the case of an ambientmonochromatic field that we would like to harvest, us-ing the harvested voltage as direct feedback. To iden-tify the optimum SMM configuration, we use an itera-tive continuous sequential optimization algorithm [44].Element after element it tests which of the two possibleSMM states brings us closer to our objective of maxi-mizing a chosen cost function CF , here CF = V harv .This procedure is summarized in Fig. 2 where we showthe harvester output before and after optimization in (a)and the dynamics of the optimization in (b); (c) and(d) present the same quantities averaged over real-izations of disorder. Note that the number of iterationsrequired until saturation in (b) is about twice the num-ber of SMM pixels. Unlike the first optics experiments (a) (b)(c) (d) Time / s V / m V INITIALFINAL
100 200 300 400 500
Iteration V h a r v / m V INITIALFINAL
100 200 300 400 5000100200300400500 V h a r v / m V Monochromatic Ambient Field
Monochromatic Ambient Field
Time / s Iteration V / m V FIG. 2. Experimental method exemplified for a monochro-matic ambient field using direct feedback. (a) shows the har-vester’s voltage output, monitored on Oscilloscope 1, before(blue) and after (red) optimization. The interval chosen to es-timate V harv is indicated in green, and the evolution of V harv over the course of the iterative optimization is displayed in(b). (c) and (d) show the quantities presented in (a) and (b)averaged over realizations of disorder. that used this iterative method to focus through multi-ply scattering paint layers, we cannot limit ourselves totesting each element only once; instead we have to retestthem several times due to the reverberation that corre-lates the optimum states of different elements.Next, we explore how the harvesting enhancement bywavefront shaping with our setup depends on the ambientmonochromatic field’s frequency and power. The gener-ator’s peak-to-peak voltage V pp is used to alter the am-bient field’s power. Each resulting data point displayedin Fig. 3(a) is the average over realizations. Here, wechose a representation in terms of voltage (rather thanpower), as the minimum voltage requirements, even byDC-DC converters, were identified as limiting factor inRef. [33] for harvesting schemes to be useful in practice.Since our employed equipment’s frequency responses arenot flat, a slight frequency dependence is evident inFig. 3(a). The power dependence may be surprising,since power is not a variable appearing in the theoreticalmodel used to explain traditional monochromatic wave-front shaping experiments in terms of degrees of freedom[43]. This can be understood, however, from the fact that V harv depends in a monotonous but not necessarily linearmanner on the ambient monochromatic field’s intensity | S ( f , r ) | at the harvester’s position r . Here, the meanvoltage enhancements vary between about and , thecorresponding power enhancements thus being on the or-der of to ; the attained enhancement is larger forweaker ambient fields. This power dependence, likely tobe generalizable to most harvesting devices, works in fa-vor of our proposal to enhance harvesting by wavefrontshaping, in particular in the case of (realistic) weak am-bient fields. Frequency / GHz V ha r v / m V FINAL V pp = 1.00 VFINAL V pp = 0.75 V FINAL V pp = 0.50 VINITIAL V pp = 1.00 VINITIAL V pp = 0.75 VINITIAL V pp = 0.50 V f in / MHz V ha r v / m V FINAL (noise)INITIAL (noise)FINAL (mono)INITIAL (mono) f = 2.43 GHz (a)(b) Frequency / GHz ∆ f = 60 MHz in Sample Ambient Field Spectrum |S(f)| / a.u.
Frequency / GHz ∆ f ~ 0 in Sample Ambient Field Spectrum |S(f)| / a.u.
Monochromatic Ambient Field
Polychromatic Ambient Field
FIG. 3. The harvested voltages before and after directfeedback based wavefront shaping, (a) for monochromaticfields of different frequencies and powers (cf. legend), av-eraged over realizations of disorder; (b) for polychro-matic (noise) fields of different bandwidths ∆ f in , centeredon f = 2 .
43 GHz , averaged over realizations.
How well does wavefront shaping based harvesting en-hancement do in a more realistic, polychromatic am-bient field? To explore this question, we work withnoise signals [45], emitted by the generator, of differ-ent bandwidths ∆ f in centered on .
43 GHz . We ob-serve a clear decrease of the achievable voltage enhance-ment from a factor of about to a factor of about ,as ∆ f in is increased. This tendency can be understoodwith traditional wavefront shaping tools. In the case ofa polychromatic ambient field, the harvested voltage isessentially equivalent to incoherent polychromatic focus-ing with unknown weights w ( f ) for different frequencies: V harv ≈ (cid:82) ∆ f in w ( f ) | S ( f ) | d f . Wavefront shaping canrelocate a certain amount of energy that is on averageequally spread across the f in / ∆ f corr independentfrequencies, where ∆ f corr = f /Q is the cavity correla-tion frequency; the literature contains multiple reportsconfirming this experimentally [46–50]. In Fig. 3(b) thedecrease of the attainable enhancement is quite drasticas our highly reverberating cavity has a correlation fre-quency of a few MHz , implying a high number of indepen-dent frequencies inside ∆ f in . Yet in lossier and leakierreal life systems ∆ f corr would be rather on the orderof a few tens of MHz such that real scenarios would staywithin the very upper part of the curve, not experiencingmajor drawbacks from broadband operation.Having demonstrated the viability of wavefront shap-ing to enhance the harvested voltages both in monochro-matic and polychromatic ambient fields using direct feed-back, we now turn to the indirect feedback case. Thediode-based rectifier circuit inside the harvester is intrin-sically a source of non-linearities that are re-emitted intothe cavity by the harvester’s receiving antenna. Approx-imated to first-order, the strength of the non-linear re-emissions increases monotonously as the excitation in-tensity incident on the harvester, | S ( f , r ) | , rises. Theintensity | S ( f NL , r NL ) | of a non-linear signature of fre-quency f NL at position r NL away from the harvester maythus serve as feedback about the excitation intensity in-cident on the harvester that is unsolicited as it is gener-ated naturally and inevitably. Moreover, it constitutes ablind feedback in the sense that we focus the wave fieldon the harvester without any knowledge of its position r in space.Under which circumstances will CF = | S ( f NL , r NL ) | provide a reliable feedback about | S ( f , r ) | ? Changesin | S ( f NL , r NL ) | must occur only in response to changesin | S ( f , r ) | . If there were sources other than theharvester emitting at f NL , the detected magnitude | S ( f NL , r NL ) | of the interference of all those f NL -sources would be sensitive to relative phase differencesbetween the sources. Similarly, if the wave field at f NL was modulated by the SMM, the value mea-sured for | S ( f NL , r NL ) | would heavily depend on theSMM state, as well as on | S ( f , r ) | . Fortunately,neither of those scenarios arises in our setup; other-wise, either or both could be circumvented by work-ing with (cid:104)| S ( f NL , r NL ) | (cid:105) independent r NL , the average of | S ( f NL , r NL ) | over several independent positions r NL .
100 200 300 400 500
Iteration C F = | S ( f N L ) | / a . u . f NL = 2 f Average over 90 realizationsSample Single Realization
100 200 300 400 500
Iteration V h a r v / m V Average over 90 realizationsSample Single Realization f NL / GHz P h a r v f (a) (b)(c) FIG. 4. Wavefront Shaping with indirect, unsolicited, blindfeedback CF = | S ( f NL , r NL ) | . For the monochromatic casewith f NL = 2 f , we display the optimization dynamics of thecost function in (a) and the corresponding harvested voltagein (b), both for a single realization and the average over 90realizations of disorder. The average enhancement of the har-vested power η P harv = η V harv is displayed in (c) for a rangeof different choices of f NL . To demonstrate the feasibility of the indirect feedbackbased harvesting enhancement scheme, we here chose towork with a monochromatic ambient field at .
43 GHz and limit ourselves to realizations, as our set-up isnot optimized in terms of speed. In the top row of Fig. 4we present the example of f NL = 2 f , illustrating both asingle realization and the average over 90 realizations ofdisorder. On the left in Fig. 4(a) we show the evolutionof the non-linear feedback signal, over the course of theiterative optimization. On the right in Fig. 4(b) we dis-play how the harvested voltage at the target position isenhanced. Non-linearities being naturally weak in com-parison to the excitation signal, the individual realizationsuffers notably more from noise than in Fig. 2(b) wherewe used the harvested voltage as direct feedback.The achieved mean enhancement of the harvested volt-age of . , albeit being substantial and corresponding to aten-fold enhancement of the harvested power, is nonethe-less notably lower than the results from direct feedbackseen in Fig. 3(a). This can of course be attributed in par-ticular to the unfavorable dynamic range of our temporalmeasurement of | S ( f NL , r NL ) | .Furthermore, we have tested frequencies other than f to provide indirect feedback, the results being on displayin Fig. 4(c), in terms of the average enhancement of theharvested power η P harv = η V harv = (cid:104) V F INharv (cid:105) / (cid:104) V INITharv (cid:105) .It can be seen that only f and f result in an en-hancement of the harvested voltage, which is significantlystronger in the case of f . This confirms that, as onemight have anticipated intuitively, the best candidate towork with is the second harmonic [51, 52]. At f NL = f ,the quantity | S ( f , r NL ) | has of course been heavily en-hanced but this did not correlate at all with the evolutionof | S ( f , r ) | : the value of | S ( f , r NL ) | is dominated bythe SMM’s state and the emitted excitation signal. Theother tested frequencies were arbitrary, thus not corre-sponding to any non-linear signatures, such that theydid not yield any enhancement either. We have also ver-ified that results similar to the ones presented in Fig. 4are obtained for different r NL and f , but they have beenomitted for clarity’s sake here.To conclude, in this Letter we have started off by prov-ing that shaping an ambient microwave field in a rever-berant medium to concentrate it on a radiofrequency har-vester may constitute an innovative improvement to cur-rent RF harvesting schemes that typically do not harvestsufficiently high voltages. Using the harvested voltageas direct feedback, we demonstrated significant harvest-ing enhancements both for a variety of monochromaticand polychromatic wave fields. Then, we exploited thenon-linear nature of the diode-based harvesting devicethat inevitably causes the re-emission of non-linear sig-natures into the cavity. By measuring the intensity ofthe second harmonic at an arbitrary position away fromthe harvester, we obtained an indirect, unsolicited andblind feedback about the ambient field intensity at theharvester. This enabled a tenfold enhancement of theharvested power with our current setup.Using indirect, unsolicited and blind feedback, remov-ing the need to access the target or its spatial positiondirectly, may be the crucial bridge between an academicconcept of focusing by wavefront shaping and its appli-cation in practice, in many cases. We expect our workto be particularly useful for emerging concepts such as Smart Homes that envisage to populate homes and fac-tories with many low-power sensors. With the aid ofspatial microwave modulators (SMMs), IoT devices andsensors could be powered wirelessly, harvesting the am-bient omnipresent RF fields. Improving the SMM design[34, 53–55], matching the operating bandwidths of SMMand harvester as well as covering more than only ofthe walls with SMMs should easily enable much higherenhancements than reported here and counterbalance thedecrease in wavefront shaping ability of the SMM in lessreverberant (lower Q ) realistic environments [43]. Us-ing non-linear feedback is expected to simultaneouslycompress the impulse response of pulsed [1] A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat.Photonics , 283 (2012).[2] M. Kim, W. Choi, Y. Choi, C. Yoon, and W. Choi, Opt.Express , 12648 (2015).[3] T. Čižmár and K. Dholakia, Opt. Express , 18871(2011).[4] Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen,R. R. Dasari, K. J. Lee, and W. Choi, Phys. Rev. Lett. , 203901 (2012).[5] W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rot-ter, and H. Cao, Opt. Express , 2709 (2017).[6] S. H. Simon, A. L. Moustakas, M. Stoytchev, and H. Sa-far, Phys. Today , 38 (2001).[7] A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sen-gupta, and S. H. Simon, Science , 287 (2000).[8] M. Fink, Phys. Today , 34 (1997).[9] G. Lerosey, J. De Rosny, A. Tourin, A. Derode, G. Mon-taldo, and M. Fink, Phys. Rev. Lett. , 193904 (2004).[10] I. M. Vellekoop and A. P. Mosk, Opt. Lett. , 2309(2007).[11] Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee,R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. , 023902 (2011).[12] E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos,A. Lagendijk, and A. P. Mosk, Phys. Rev. Lett. ,193905 (2011).[13] J.-H. Park, C. Park, H. Yu, J. Park, S. Han, J. Shin, S. H.Ko, K. T. Nam, Y.-H. Cho, and Y. Park, Nat. Photonics , 454 (2013).[14] I. Vellekoop, A. Lagendijk, and A. Mosk, Nat. Photonics , 320 (2010).[15] J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, andA. Lagendijk, Phys. Rev. Lett. , 103901 (2011).[16] O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat.Photonics , 372 (2011).[17] P. del Hougne, F. Lemoult, M. Fink, and G. Lerosey,Phys. Rev. Lett. , 134302 (2016). [18] A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz,G. Lerosey, S. Gigan, L. Daudet, and I. Carron, Sci. Rep. , 5552 (2014).[19] S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boc-cara, and S. Gigan, Phys. Rev. Lett. , 100601 (2010).[20] S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gi-gan, Nat. Commun. , 81 (2010).[21] A. Drémeau, A. Liutkus, D. Martina, O. Katz,C. Schülke, F. Krzakala, S. Gigan, and L. Daudet, Opt.Express , 11898 (2015).[22] H. Yu, T. R. Hillman, W. Choi, J. O. Lee, M. S. Feld,R. R. Dasari, and Y. Park, Phys. Rev. Lett. , 153902(2013).[23] M. Mounaix, D. Andreoli, H. Defienne, G. Volpe,O. Katz, S. Grésillon, and S. Gigan, Phys. Rev. Lett. , 253901 (2016).[24] P. del Hougne, B. Rajaei, L. Daudet, and G. Lerosey,Opt. Express , 18631 (2016).[25] M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park,and W. Choi, Nat. Photonics , 581 (2012).[26] I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. ,120601 (2008).[27] A. Peña, A. Girschik, F. Libisch, S. Rotter, and A. Cha-banov, Nat. Commun. (2014).[28] N. Kaina, M. Dupré, G. Lerosey, and M. Fink, Sci. Rep. (2014).[29] I. Vellekoop, E. Van Putten, A. Lagendijk, and A. Mosk,Opt. Express , 67 (2008).[30] B. Larrat, M. Pernot, G. Montaldo, M. Fink, andM. Tanter, IEEE Trans. Ultrason., Ferroelect., Freq.Control , 1734 (2010).[31] T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy,and S. Gigan, Nat. Photonics , 58 (2014).[32] O. Katz, E. Small, Y. Guan, and Y. Silberberg, Optica , 170 (2014).[33] V. Talla, B. Kellogg, B. Ransford, S. Naderiparizi, S. Gol-lakota, and J. R. Smith, in Proceedings of the 11th ACM
Conference on Emerging Networking Experiments andTechnologies (ACM, 2015) p. 4.[34] N. Kaina, M. Dupré, M. Fink, and G. Lerosey, Opt.Express , 18881 (2014).[35] D. A. Hill, Electromagnetic Fields in Cavities: Deter-ministic and Statistical Theories , Vol. 35 (John Wiley &Sons, 2009).[36] J.-B. Gros, U. Kuhl, O. Legrand, F. Mortessagne,E. Richalot, and D. Savin, Phys. Rev. Lett. , 224101(2014).[37] J.-B. Gros, U. Kuhl, O. Legrand, and F. Mortessagne,Phys. Rev. E , 032108 (2016).[38] T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub,C. Decroze, D. Carsenat, and D. R. Smith, Appl. Phys.Lett. , 194104 (2015).[39] T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith,Phys. Rev. Applied , 054019 (2016).[40] M. F. Imani, T. Sleasman, J. N. Gollub, and D. R. Smith,J. Appl. Phys. , 144903 (2016).[41] J. Gollub, O. Yurduseven, K. Trofatter, D. Ar-nitz, M. Imani, T. Sleasman, M. Boyarsky, A. Rose,A. Pedross-Engel, H. Odabasi, et al. , Sci. Rep. (2017).[42] W. Haboubi, H. Takhedmit, J.-D. Lan Sun Luk, S.-E.Adami, B. Allard, F. Costa, C. Vollaire, O. Picon, andL. Cirio, Progress In Electromagnetics Research , 31(2014).[43] M. Dupré, P. del Hougne, M. Fink, F. Lemoult, andG. Lerosey, Phys. Rev. Lett. , 017701 (2015).[44] I. Vellekoop and A. Mosk, Opt. Commun. , 3071(2008).[45] Bandpass filtered long series of numbers generated with MatLab’s uniform random number generator.[46] H. P. Paudel, C. Stockbridge, J. Mertz, and T. Bifano,Opt. Express , 17299 (2013).[47] D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon,and S. Gigan, Sci. Rep. , 10347 (2015).[48] E. Small, O. Katz, Y. Guan, and Y. Silberberg, Opt.Lett. , 3429 (2012).[49] F. Van Beijnum, E. G. Van Putten, A. Lagendijk, andA. P. Mosk, Opt. Lett. , 373 (2011).[50] C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, andH. Cao, Phys. Rev. Lett. , 223901 (2015).[51] M. Frazier, B. Taddese, B. Xiao, T. Antonsen, E. Ott,and S. M. Anlage, Phys. Rev. E , 062910 (2013).[52] M. Frazier, B. Taddese, T. Antonsen, and S. M. Anlage,Phys. Rev. Lett. , 063902 (2013).[53] T. H. Hand and S. A. Cummer, IEEE Antennas WirelessPropag. Lett. , 70 (2010).[54] S. V. Hum and J. Perruisseau-Carrier, IEEE Trans. An-tennas Propag. , 183 (2014).[55] H. Yang, X. Cao, F. Yang, J. Gao, S. Xu, M. Li, X. Chen,Y. Zhao, Y. Zheng, and S. Li, Sci. Rep. , 35692 (2016).[56] J. Aulbach, A. Bretagne, M. Fink, M. Tanter, andA. Tourin, Phys. Rev. E , 016605 (2012).[57] D. R. Smith, V. R. Gowda, O. Yurduseven, S. Larouche,G. Lipworth, Y. Urzhumov, and M. S. Reynolds, J. Appl.Phys. , 014901 (2017).[58] B. Xiao, T. M. Antonsen, E. Ott, and S. M. Anlage,Phys. Rev. E , 052205 (2016).[59] S. K. Hong, V. M. Mendez, T. Koch, W. S. Wall, andS. M. Anlage, Phys. Rev. Applied2