OPTICS Ultraslow waves on the nanoscale ON OUR WEBSITE · REVIEW OPTICS Ultraslow waves on the nanoscale Kosmas L. Tsakmakidis,1† Ortwin Hess,2 Robert W. Boyd,3 Xiang Zhang1,4,5

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OPTICS

Ultraslow waves on the nanoscaleKosmas L. Tsakmakidis,* Ortwin Hess, Robert W. Boyd, Xiang Zhang*

BACKGROUND: The past 10 to 15 years havewitnessed substantial advances in our abilityto controllably adjust, slow down, or acceleratethe speed of light signals propagating throughdispersive optical media. In the fields of opto-electronics and photonics, we have for decadesbeen accustomed to using propagating orguided light waves with speeds only slightly lessthan the speed of light in a vacuum, c—usuallyby a factor of 2 to 4. We now know that thegroup velocity of light waves entering a highlydispersive medium can be reduced by a factorof millions, down to the “human” scale, by ex-ploiting judicious interference effects at theatomic scale of thematerial. Light decelerationsby a factor of a few hundred, with low losses,can be attained in periodic dielectric structures,such as photonic crystal and coupled-resonatoroptical waveguides. Enabled applications in-

clude all-optical tunable delays for routers anddata synchronization, optical buffers, enhancedlight-matter interaction and nonlinear effects,and miniaturized photonic devices, such asmodulators and interferometers. However, suchatomic- or dielectric-media–based “slow light”is still fundamentally limited by thewavelengthof light, l, to spatial dimensions larger than~l/2(~300 nm for visible light)—i.e., it is still diffrac-tion limited and cannot reach true nanoscopicdimensions (e.g., below~30nm). It is at this pointthat media featuring negative electromagneticparameters, such as negative-permittivity plas-monicmedia or negative–refractive indexmeta-materials, come to the rescue.

ADVANCES: Light deceleration in these mediaarises from the presence of a negative (real partof) electric permittivity, e, or refractive index,

n, leading to antiparallel power flows in thenegative-parameter medium and the surround-ing dielectric host—thereby giving rise to energyand group velocities that can be reduced evendown to zero for a suitable choice of optogeo-

metric parameters. Thismechanism for slowingdown light is thus non-resonant, and as such itcanbebroadband,withtyp-ical bandwidths being onthe order of ~1 to 10 THz.

Most notably, these structures support ultra-slow surface-plasmonor surface-phonon polar-itons that can be concentrated tightly into thenanoscale, at nanovolumes (at least) thousandsof times smaller than l3, upon suitable adia-batic tapering. This leads to large local fieldenhancements, of the order of 102 to 103, oversmall nanovolumes that can be exploited for ahost of useful applications, including nano-imaging, biosensing and nanoscale chemicalmapping, high-density magnetic data-storagerecording, light harvesting, nanolasing, andnanoscale quantum and nonlinear optics. Onemust be mindful, though, of targeting applica-tions (such as the ones mentioned above) forwhich dissipative losses, which are normallyhigher in these media compared to their di-electric counterparts, are not a prohibitivefactor—i.e., applications for which energy ef-ficiency is not the key figure-of-merit.

OUTLOOK: Thus far, thismethod of nanoscale,broadband slow light has relied on the use ofuniform or nanostructured metals (plasmonicmedia), but continued advances in materialsdesign and synthesis have recently enabled atransition to alternativemedia, such as dopedsemiconductors, graphene, hexagonal boronnitride, transition-metal dichalcogenides (suchas TaS2), van der Waals crystals, and hetero-structures. There are important ongoing ef-forts to push the response of these alternativemedia, which is currently mainly in the mid-and near-infrared regimes, into the visible re-gion as they allow for considerably lower lossesand enhanced controllability (e.g., simply byapplication of a gate voltage). With these newmaterial platforms, opportunities for newapplications emerge, too, particularly thoserequiring nanoscale (and even atomic) con-finements and ultrahigh field enhancements,such as low-threshold single-photon nonlinear-ities and applications relying on the genera-tion andmanipulation of nonclassical light, atambient conditions.▪

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Tsakmakidis et al., Science 358, 319 (2017) 20 October 2017 1 of 1

The list of author affiliations is available in the full article online.*Corresponding author. Email: [email protected](K.L.T.); [email protected] (X.Z.)Cite this article as K. L. Tsakmakidis et al., Science 358,eaan5196 (2017). DOI: 10.1126/science.aan5196

Broadband subdiffraction ultraslow and stopped light. (Left) Light beams incident (redarrows) on a Si film bounded symmetrically by a negative-permittivity (plasmonic) medium(er < 0) excite guided light pulses that can have zero group velocity, vg, decaying with time(dashed arrows) exactly at their injection points rather than propagating along the Si film. (Right)Surface plasmon polaritons propagating along an adiabatically tapered plasmonic nanoguide(white prism) are slowed down over a broad range of frequencies (“colors”), accumulating at adeep-subdiffraction spot (d << l/2) at the tip of the guide and leading to large local fieldenhancements.

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Ultraslow waves on the nanoscaleKosmas L. Tsakmakidis,1*† Ortwin Hess,2 Robert W. Boyd,3 Xiang Zhang1,4,5*

There has recently been a surge of interest in the physics and applications of broadbandultraslow waves in nanoscale structures operating below the diffraction limit. They rangefrom light waves or surface plasmons in nanoplasmonic devices to sound waves inacoustic-metamaterial waveguides, as well as fermions and phonon polaritons ingraphene and van der Waals crystals and heterostructures. We review the underlyingphysics of these structures, which upend traditional wave-slowing approaches based onresonances or on periodic configurations above the diffraction limit. Light can now betightly focused on the nanoscale at intensities up to ~1000 times larger than the output ofincumbent near-field scanning optical microscopes, while exhibiting greatly boosteddensity of states and strong wave-matter interactions. We elucidate the generalmethodology by which broadband and, simultaneously, large wave decelerations, wellbelow the diffraction limit, can be obtained in the above interdisciplinary fields. We alsohighlight a range of applications for renewable energy, biosensing, quantum optics,high-density magnetic data storage, and nanoscale chemical mapping.

More than100yearsago, in 1914,Sommerfeld,Brillouin, and Lorentz started investigat-ing situations inwhich the group velocityvg of light signals exceeds the vacuumspeed of light, c, leading to what they

initially thought were apparent inconsistencieswith Einstein’s special theory of relativity (1–3).The group velocity of a wave packet propagatinginside a dispersive medium of refractive index nis given by vg = c/ng, where ng = n + w(dn/dw) isthe group refractive index of the medium, with wbeing the angular frequency, from which it isseen that in the region of anomalous dispersionin a normal absorbing medium, where dn/dw <0, vg may exceed c—but with high losses (4–7).More recently, experimental observations of super-luminal group velocities have been reported in thepropagation of mode-locked pulse trains in reso-nant absorbers (8) and in photon tunneling frompotential barriers (9, 10). Furthermore, distortion-less superluminal pulse propagation withminimalamplitude change has been observed in gaindoublets (11), together with evanescent-wavesuperluminal group velocities at the cutoff ofmicrowave waveguides (12), and superluminaleffects arising solely from the nature of the

propagating field, such as in the propagationof Bessel beams (5, 13). None of these situationsviolates relativistic causality as in all cases thefront velocity—the velocity with which a dis-continuity in a waveform or one of its derivativespropagates—is smaller than c (5, 9).The opposite phenomenon, inwhich the group

velocity of a light wave can becomemuch smallerthan c, was also identified and studied (3–6, 14).For instance, within a hollow waveguide, thewave vector b along the guide is reduced belowthe free-space value, leading to a phase velocityvφ = w/b greater than c. But within such a struc-ture, the product of the phase and the groupvelocities is given as (15) vφvg = c2, thereby re-sulting in a group velocity along the guide thatcan become considerably smaller than c. A fur-ther intuitive example is epsilon-near-zeromedia(16), in which the electric permittivity e(w0)→0at a frequencyw0. In suchmedia, the wave vectork = (w/c)

ffiffiffiffiffiffiffiffiffiffieðwÞp

becomes zero at w0, hence thephase velocity vφ becomes infinite, but the group ve-locityvg = (dk/dw)

−1 = c½ ffiffiffiffiffiffiffiffiffiffieðwÞp þ w=ð2 ffiffiffiffiffiffiffiffiffiffi

eðwÞp Þ��1

becomes zero at w = w0 (17, 18)—a property thatcan be exploited for boosting light-matter inter-actions and nonlinear effects (16).Much of the impetus for the contemporary

interest in such “slow” and “fast” light phenomenawas provided by the 1999 experiment by Hauet al. (19), which demonstrated that the speed oflight in atomic electromagnetically induced trans-parent (EIT) media could be reduced to the “hu-man” scale of ~17 m/s—albeit over very narrowbandwidths (20), typically of the order of a fewtens of kilohertz. Thereafter, a range of solid-state systemswere proposed and investigated forslowing down the speed of guided light signals,including photonic crystal (PhC) waveguides(21, 22), coupled resonator optical waveguides(CROWs) (23), and standard optical fibers (24, 25).

These slow-light structures exhibit fascinatingfundamental physics and have enabled a rangeof useful applications underpinned by the strong(and broadband) light-matter interactions thatthey entail (26–28). However, they are funda-mentally limited by the wavelength of light—i.e.,they are diffraction limited, and cannot reach truenanoscopicdimensions, e.g., scales less than~30nm.More recently, media exhibiting negative electro-

magnetic or optical parameters (such as nega-tive permittivity and/or refractive index) (29–31)have emerged as a new means of allowing largeand broadband light decelerations—but, thistime, well below the diffraction limit, right at thenanoscale, where a host of new material prop-erties, functionalities, applications, and oppor-tunities emerge. These media typically featurehigher losses compared with their dielectriccounterparts; thus, one has to bemindful of target-ing applications that can be more loss tolerant(32, 33). We will focus on wave-decelerationmethods that allow—on the basis of the sameunderlying mechanisms in all cases—for simul-taneously broadband operation and small groupvelocities, below the diffraction limit, so that thenanoscale canbe reached, in nanophotonicmedia,metamaterials, graphene, van derWaals crystals,and heterostructures. We detail the broadbandmechanisms, not involving bulk-material param-eter (permittivity, refractive index) resonances,by which subdiffraction waves of various kindscan be slowed down in the aforementioned struc-tures, their unique characteristics, and how theseare exploited in a host of applications across thefields of optics and photonics, acoustics, and two-dimensional (2D) condensed matter.

Physics of subdiffraction slow andstopped light waves

Almost all types of nanoscale guiding structuresinherently feature slow waves of some kind—light/photons, guided Dirac fermions, surfaceplasmon polaritons (SPPs), or surface phononpolaritons. This section reviews the physical mech-anisms by which extremely slow light or SPPscan be obtained over broad spectral rangesthrough use of suitably engineered and nano-photonic media that confine the light field todimensions smaller than the conventional dif-fraction limit (34–36).Unlike atomic schemes (19, 37, 38) or periodic

dielectric structures (21–23, 26–28), which areusually diffraction limited (34, 35), this routedoes not entail bulk-material resonances, dis-order, or periodic back reflections to slow orstop light. The deceleration can occur even inlongitudinally uniform (nonperiodic) structures,and away from intrinsic material resonances—e.g.,in a regime where the real part of the permittivity,e, of a heterostructure’s layer is negative, followinga nonresonant Drude variation (39). Such struc-tures can be either metamaterial or plasmonicwaveguides (36,39–62) (Fig. 1,A toC). For transversemagnetic (TM) polarization, the time-averagedcomplex Poynting vector in the longitudinal zdirection of a metamaterial waveguide (Fig. 1A),�Sz, is given by�Sz ¼ 1

2 RefE�H�gz ¼ b2we0ei

jHyj2;

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Tsakmakidis et al., Science 358, eaan5196 (2017) 20 October 2017 1 of 11

1NSF Nanoscale Science and Engineering Center (NSEC),University of California at Berkeley, 3112 Etcheverry Hall,Berkeley, CA, USA. 2The Blackett Laboratory, Department ofPhysics, Imperial College London, London SW7 2AZ, UK.3Department of Physics and School of Engineering andComputer Science, University of Ottawa, Ottawa, OntarioK1N 6N5, Canada. 4Material Sciences Division, LawrenceBerkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA94720, USA. 5Department of Physics, King AbdulazizUniversity, Jeddah 21589, Saudi Arabia.*Corresponding author. Email: [email protected](K.L.T.); [email protected] (X.Z.) †Present address:Bioengineering Department, EPFL, 1015 Lausanne, Switzerland,and Department of Solid State Physics, National and KapodistrianUniversity of Athens, Panepistimioupolis, GR - 157 84, Athens, Greece.



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where b is the longitudinal propagation con-stant, ei is the layers’ dielectric permittivities,and i is the core layer or cladding layer (43).Thus, in the negative-permittivity (and negative–refractive index) core layer of this symmetric,lossless waveguide, �Sz;co < 0; whereas in thepositive-permittivity dielectric cladding layers,�Sz;cl > 0:Hence, the total time-averaged power

flow, P ¼ ∫1

�1

�Szdx; in the +z direction will be re-

duced owing to the negative contribution from�Sz;co; and can even become zero when j�Sz;coj ¼2�Sz;cl: But in a lossless dispersive medium (orwaveguide), the energy velocity, vE, is equal tothe group velocity, vg, and is given by vE = P/U,withU being the energy density (5, 6, 63). There-fore, as a result of the antiparallel power flows inthe core and cladding layers of this waveguide,both its energy and group velocities can be re-duced to zero. As shown in Fig. 1A, the structureof the instantaneous Poynting vector S on the xzplane of the guide takes on a vortex form at eachinterface, with energy flowing back and forthacross each interface but, on average, with no netenergy being transported across each interface.Indeed, a straightforward calculation shows thatthe curl of S, which characterizes its vorticity,is nonzero and is given by∇� S ¼ ðrs=e0ÞHyy0;where rs is the surface polarization charge den-sity. Thus, the curl of the Poynting vector field iszero in both the negative–refractive index coreand dielectric cladding layers, and is nonzero (andmaximum) only at the two interfaces, as inferreddirectly from Fig. 1A.Figure 1B illustrates a further example of light

slowing down at the nanoscale, this time in anadiabatically tapered plasmonic nanoguide (41, 42).SPPs guided along this nanocone are progressivelydecelerated until their group velocity becomes zeroat the structure’s tip, where they accumulate (41).The SPPs are longitudinally and laterally com-pressed as they reach the tip, with the electricfield being largely enhanced in that region, deepbelow the diffraction limit. Both of these nano-scale slow-light mechanisms—the double-vortexstructure in the power flux vector and the use ofa tapered structure to accumulate light in a sub-diffraction spot—underpin many of the under-lying physics and applications that this type oflight-deceleration features.Slow light in these media is typically broad-

band, with Fig. 1C illustrating a tapered negative-index metamaterial waveguide capable of haltinga “rainbow” of light wavelengths at predefinedcritical points (43). Here, a light ray is forced bythe negative electromagnetic “environment” thatit experiences in the core layer of the guide tomake, in each half period, a forward “step” fol-lowed by a negative phase-shift step. At the zero–group-velocity point, the negative Goos-Hänchenphase shifts bring the light ray precisely backto its initial position. The Goos-Hänchen phaseshift arises when a finite-width beam (i.e., not alaterally infinitely extended plane wave) is totallyinternally reflected at the interface of two media(e.g., from a high-, negative-permittivity e medium“1” to a lower-, positive-permittivity medium “2,”


Fig. 1. Broadband ultraslow light in metamaterials and nanoplasmonics. (A) Structure of thePoynting vector field in a symmetric waveguide made of a negative–refractive-index core layer andpositive-index claddings, showing the opposite directions of the power flux in the core (Pco < 0) andcladding (Pcl > 0) layers. Owing to the characteristic double-vortex structure of the power flux, the totaltime-averaged power flow in the +z direction (Ptot = Pco + Pcl) is reduced, leading to correspondinglyreduced energy and group velocities (40). (B) Adiabatic nanofocusing of surface plasmon polaritons(SPPs) guided along a tapered plasmonic nanoguide. The group velocity of the SPPs progressivelyreduces as they propagate, becoming zero at the nanostructure’s tip, thereby leading to substantialspatial compression in the longitudinal direction and to a correspondingly large local field enhancement(here, of the order of ~103) (41). (C) Snapshot of the propagation of a monochromatic light wave alongan adiabatically tapered waveguide with negative–refractive-index core and positive–refractive-indexcladdings.The light wave progressively slows down until it stops at a “critical” thickness, where the electricfield builds up. The top-right and top-left insets associate the wave propagation with the correspondingzigzag ray picture at different guide widths. Note the negative Goos-Hänchen phase shifts (denotedwith dotted white arrows) each time the ray hits an interface (43). (D) (Lower panel) A near-infrared(NIR, ~800 nm) SPP propagating along a tapered conical plasmonic nanoguide after being excitedby a NIR laser pulse.The SPP slows down while propagating toward the exit aperture, where it eventuallyconverges to a subwavelength high-intensity local spot. The hollow plasmonic (Ag) nanoguide is filledwith xenon atoms, which are ionized by the high-intensity local field, producing high-harmonic extremeultraviolet (EUV) pulses that are radiated through the exit aperture.The upper panel shows the measuredhigh–harmonic-generation spectrum, spanning from the 15th to the 43rd harmonic (68). (E) SlowSPPs propagating azimuthally (in the q direction) toward the singular (“kissing”) point of the crescentnanostructure shown in the inset, after being excited by a plane wave. Shown is the amplitude of the xcomponent of the electric field at an arbitrary 2D cross section of the nanocrescent. Note that the field-enhancement peaks at an angle other than 180°. The upper inset shows a cross-sectional plot of theamplitude of the x component of the electric field normalized by the incident field (polarized along x).The color scale in this inset is linear and restricted to [–5, 5], but the field magnitude is notably much largeraround the singularity (“kissing point”) of the two touching cylinders, as shown in the main figure (64).

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with e1 < 0 and e2 > 0, but with |e1| > e2) (45, 46).If k is the component of the wave vector k per-pendicular to the media interface, and g is thedecay constant of the field into medium “2,” thenthe Goos-Hänchen phase shift is given by d =2tan−1[(e1g)/(e2k)] and becomes negative for e1 < 0(and e2 > 0) (43). This negative phase shift—aninterference effect of multiple “plane waves” intowhich the incident beam can be analyzed, eachimpinging on the interface with a slightly dif-ferent angle of incidence—gives rise to a negative(backward) spatial displacement of the (center ofthe) beam, as shown schematically in the top in-sets of Fig. 1C. These backward “steps” in the prop-agation of a light ray along the guide can alsobe understood by comparison with the negative(backward) power flow in the core layer of thenegative-index waveguide shown in Fig. 1A, andprovide an intuitive, pictorial (yet accurate evenat the level of Maxwell’s equations) view of thelight deceleration attained herein. These back-ward Goos-Hänchen steps arise from the presenceof a bulk, volume-averaged negative refractiveindex or permittivity in one or more of the layers.As a result, the effect of possible bulk disordersand fluctuations in the layers of the heterostructureis naturally averaged out (contained in the defi-nition of the macroscopic effective refractive indexor permittivity) (29). It is only the negativity in thereal part of the effective refractive index or per-mittivity that enables the deceleration of light,even in uniform structures containing layers made

of completely amorphous (meta)materials (44).The associated structures may also be longitudi-nally uniform plasmonic media, where disorderis not an inherent issue. Detailed studies haveshown that in the presence of nanometer-scalesurface roughness, very large slowdown factorsare still attainable—e.g., of the order of ~107 (39).A universal characteristic of such negative-

parameter wave-slowing structures—that one mayalso encounter for acoustic (61, 62) ultraslow-wavestructures and 2Dmaterials—is that they allow forfield accumulation and substantial field enhance-ment around the critical light-stopping point,deep below the diffraction limit, which can beexploited for a host of nanoscale applications. Thisis in contrast to, e.g., slow light in atomic media,where, although the energy density U is indeedenhanced in the slow-light regime (U = P/ug, withP being the power density, and ug→0), theelectric-field amplitude E is not enhanced becausemost of the compressed energy is stored in theatomic polarization—and not in the electric field(63). Thus, although a light pulse is compressed inspace (in the longitudinal direction of propagation)upon entering such an atomic ultraslow-light me-dium, and the energy density does increase, there isusually no enhancement in the local electric field.The situation is different in the subdiffraction-

limited nanophotonic structures considered here.Assuming a parabolic-like ug profilewith distance(with ug = 0 at the critical point), it is straight-forward to show that the time it takes for the light

wave to exactly reach the critical point in anadiabatically tapered structure diverges logarith-mically (41, 64); hence, that critical point (whereug = 0) is, in theory, never reached because theeffective optical distance, Lng—where L is theguide’s physical length—diverges (ng → 1); it isas though a guided light wave propagates towardone end of an infinitely long waveguide (65–67),never quite reaching it. Accordingly, a sinusoidallight signal keeps accumulating, with a progres-sively smaller group velocity, near the “critical”zero-ug point in this singular, idealized structure(Fig. 1, B and C). In practice, material imperfec-tions lead to the critical point being reached,resulting in a back reflection, which togetherwith material losses, provide a finite limit tohow much the field can locally be enhanced.Nevertheless, the large spatial compression ofthe field in the slow-light regime of such sub-diffraction nanostructures leads to correspond-ingly large field-enhancement factors—of theorder of 350 or more—being realistically attain-able, and over ultrabroad bandwidths [e.g., fullwidth at half maximum (FWHM) of 100 nm inthe near-infrared] (68). The back reflection can berigorously suppressed if nonreciprocity is broken,e.g., by application of a magnetic field in gyro-electric semiconductors (such as InSb) (55, 56, 69)or by deploying PT-symmetric structures (58), inwhich case rigorous light-stopping canbeobtained.The ultimate degree of slow-light–aided field en-hancement is set by the nonlocal effects in plas-monic structures and by the size of the unit cells inmetamaterial structures (spatial dispersion), whichlimit how much a light field can be spatially com-pressed (70–72). These nonlocal (spatial dispersion)effects usually arise at scales less than ~1 to 5 nm(~30 to 50 nm), and in that deep-subwavelengthregime, the induced spatial frequencies (i.e., wavevectors) are large, to the point of potentially allow-ing for “direct diagonal” s-band absorption (i.e.,Landaudamping) (32)—therebypotentially provid-ing an additional loss channel that should beavoided. Such “structural slow-light” field enhance-ments are also obtained in photonic structuresabove the diffraction limit, as in photonic crys-tals and coupled-resonator optical waveguides(4–6, 21–28). However, the compression andfocusing of light in those cases is limited bythe wavelength of light and cannot reach thetruly nanoscopic dimensions (and correspondingnanoscale applications) targeted herein.

Robustness in the presence ofdissipative and scattering channels

A key issue to be addressed when attempting toslowdownor stop guided light signals innanoscalestructures is how dissipative losses (39, 49–53) andback-scattering channels (54–56, 69) might affectthe light-deceleration ability of these devices.Because the slowing-down of light is due to thepresence of negative electromagnetic parameters(Fig. 1, A to C), which normally entail losses anddispersion, the modes of these structures will ingeneral be complex (53), and it is important toexamine whether the zero-vg points in the asso-ciated dispersion diagrams are still preserved


Fig. 2. Robust stopped light in the presence of absorption and scattering channels. In (A) and(B), the results displayed were obtained on the basis of exact, analytic solutions of Maxwell’sequations. (A) Dispersion diagram of the two classes of complex TM2 modes, complex-w (blue) andcomplex-b (red and green), supported by a lossy plasmonic (ITO)–silicon–plasmonic (ITO) planarwaveguiding heterostructure. Solid lines show the real part of the frequency [fr = Re{w}/(2p)] and ofthe longitudinal propagation constant (br = Re{b}), while the dashed lines show the correspondingimaginary parts (fi = Im{w}/(2p), bi = Im{b}). The imaginary part of the frequency, fi, provides thetemporal losses associated with the complex-w mode, while the imaginary part of the propagationconstant, bi, provides the spatial losses associated with the two complex-b modes. Note how only thecomplex-w mode retains a zero group velocity, at point “a,” where the band’s slope locally becomes zero(see inset). The shaded gray region indicates the free-space light cone (39). (B) Band diagram of agyroelectric-semiconductor (InSb)–silicon–plasmonic (Ag) heterostructure for the case where anexternal magnetic field B0 = 0.2 T is applied in a direction perpendicular to the structure. Shown, both forpositive and negative longitudinal wave vectors k, are the dispersion curve of the supportedsemiconductor–dielectric–metal surface magnetoplasmon (SMP), the surface wave at an InSb/Ag interface(SW), and the region of the bulk modes in InSb. The area shaded in cyan indicates the band region wherecomplete unidirectional propagation (CUP) of the supported surface magnetoplasmon is attained (69).


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when Joule losses are present. Furthermore, be-cause the light-stopping points are usually reachedand accessed with an adiabatic taper (Fig. 1, B andC), it is also crucial to check whether the tapermight introduce back-scattering channels throughwhich the trapped (stopped) light energy mightescape.Both of these issues have been extensively inves-

tigated and addressed. Exact solutions toMaxwell’sequations in lossy plasmonic-dielectric-plasmonicheterostructures (39) reveal that there are twoclasses of modes in these structures (39, 49–53)(Fig. 2A). The first class is characterized by areal frequency f and complex longitudinal wavevector b = br + ibi (red and green lines in Fig.2A), and does not preserve the zero-vg points inthe presence of losses (49). These modes can beexcited by sinusoidal steady-state sources be-cause they are characterized by a well-definedreal frequency (and complex wave vector owingto the losses). They are not the modes of interestin the stopped-light regime, because in thatregime where light is trapped, spatial losses (com-plex wave vectors) lose their physical meaning(53)—a stopped light pulse simply decays withtime at its localization point, rather than decayingin space (because it does not propagate); hence,spatial losses and the search for complex-b solu-tions cease to be physically meaningful in the

stopped-light regime. The second class of modes,by contrast, is characterized by a complex fre-quency f = fr + ifi and real longitudinal wavevector b (blue lines in Fig. 2A) (39, 51–53). These“complex frequency” modes are somewhat sim-ilar to modes excited inside lossy cavity resonatorsand are the ones to be sought after in the stopped-light regime. They can be excited by short pulsesthat, e.g., have fr as their central frequency anda bandwidth (38) Dw = fi. Notably, the temporallosses characterizing these modes increase byonly a small factor when entering the stopped-light regime (e.g., from ~2.4 × 1013 s−1 at point “b”to ~3.6 × 1013 s−1 at the zero-vg point “a” in Fig. 2A,an increase by a factor of only 0.5—and corre-sponding to lifetimes at the stopping point of~30 fs, which are typical nanoplasmonic ones)(39, 51–53).A forceful way to solve the second potential

issue and suppress back-scattering channels is toinduce a unidirectionality mechanism, e.g., eitherby application of a static magnetic field in gyro-electric or gyromagnetic structures (55, 56, 69),nonlinear or refractive-index modulationmech-anisms (57), PT-symmetric media (58), or pho-tonic topological insulators (59, 60). For instance,Fig. 2B shows that the first of the aforemen-tioned schemes, when applied to a (gyroelectric)semiconductor-dielectric-metal heterostructure,

leads to a region in the band diagram wherecomplete unidirectional propagation (CUP) isallowed. A guided pulse with a bandwidth fallinginside this CUP region can neither scatter tobackward modes (k < 0 in Fig. 2B) nor to bulkmodes in the semiconductor (orange-coloredregions in Fig. 2B) nor to surface waves (SWs)at the semiconductor-metal interface of such ametal-terminated waveguide—nor can it be per-turbed by surface roughness and geometric im-perfections, given that it is a photonic analog of aquantum Hall state (55, 69). Thus, with such uni-directional schemes suppressing back-scatteringchannels, robust slow and stopped light can beattained, unaffected by spatial variations in theguiding structure.

Applications of subdiffraction ultraslowlight and acoustic waves

An example of such a broadband ultraslow-lighteffect, below the diffraction limit, and its appli-cation is illustrated in Fig. 1D, where an adiabat-ically tapered plasmonic nanoguide capable ofdecelerating and accumulating at its tip SPPsover a broad band (~100 nm) (compare Fig. 1B)is used as a means of generating ultrashort,extreme-ultraviolet (EUV) pulses from near-infrared incident pulses (centered at ~800 nm)(68). The SPPs in this tapered hollow nanocone


Fig. 3. Broadband ultraslowlight at the meta-atomiclevel, and its application inphotovoltaics and biosens-ing. (A) Schematic illustra-tion of a solar cell based onan upconverter systemwhere above-bandgap light isabsorbed by the cell, whilesub-bandgap light is initiallyabsorbed in the upconverterlayer, converted to above-bandgap light, and sent backto the cell where it is har-vested. The upconverterconsists of slow-light nano-crescent meta-atoms, whosecore is doped with theupconverting material(86). (B) The upconvertersubstantially increasesthe efficiency of an idealsingle-junction solar cell.As shown, the relativeincrease is greatest whenthe upconverter absorptionefficiency and the cellbandgap are high. The black-line inset shows the absoluteefficiency of an ideal solarcell, both with and without an ideal upconverter (UC) (86). (C) Calculationof the upconverted power toward the solar cell (in W/m2). At a wavelengthof 644 nm, the power flow into the cell is enhanced by more than 100 timescompared with the case (not shown here) where there is no crescent in theupconverter (86). (D) Conceptual illustration of a SERS substrate made ofbroadband slow-light nanocrescent moons. The gold surface is functionalized

with biomolecular linkers to recognize specific biomolecules. Also shown onthe right is a transmission electron microscope image of an actual nano-crescent moon, with an inner diameter of 300 nm and a bottom thickness of100 nm (87). (E) Comparison between the SERS spectra of 1 mM R6Gmolecules sitting on top of the slow-light nanocrescent moons and on 60-nmstandard colloidal Au nanospheres (87).


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are markedly slowed down near its exit point,accumulating in an area with a diameter of~240 nm. Atomic gases enter the device at theinput port and are controlled by changing thepressure between the input and output ports.Despite the plasmonic losses, optimized struc-tures lead to intensity enhancements exceeding20 dB (i.e., a factor of ~350), enough to triggeratomic ionization with moderate input inten-sities of ~1 × 1011 W cm−2. This leads to high-harmonic generation, up to the 43rd harmonic(Fig. 1D) and deep into the UV regime (photonenergy ~67 eV). In addition to not requiring phasematching, as the field concentration is deep-subwavelength, an advantage of this means ofnanolocalizing and enhancing the intensity oflight compared to bowtie or nanorod elements

is that the large field enhancements occur oversubstantially larger nanovolumes (with a diameterof ~240 nm and a length of ~450 nm), therebyenhancing the efficiency of the nonlinear genera-tion. For instance, the photonnumbermeasured forthe 15thand 17thharmonicswas~1× 109photons/s,corresponding to a conversion efficiency of ~1 ×10−8, which is about 10 to 100 times higher thanthat measured for a 150-bowtie-nanoantennaarray (73). We note that under similar conditions,noncoherent multiphoton or high-field fluores-cencemay also dominate the spectra (74), becausethese effects too are favored below the ionizationthreshold by the large field enhancements andthe linear interaction-length scaling.This type of broadband slow-light structure

need not be planar (39, 65–67) or linearly tapered

(41–43, 73–82), with two examples being the cy-lindrically (64) or spherically (83) crescent nano-structures (Figs. 1E and 3). Surface plasmonmodesexcited by a uniform electric field of wavelengthl ~375 nm travel with a gradually slower speedtoward the singular “kissing” point of the twocylinders (of external radius ~20 nm) where theyaccumulate, having a gradually smaller wavelengthand increased energy density (Fig. 1E). On the basisof a transformation-optics analytic approach, itwas shown that this effect is ultrabroadband—across the entire visible spectrum—and, despitethe plasmonic loss, leads to large field enhance-ments, of the order of 103, at the critical stoppingpoint (64). The role of surface roughness andunavoidable geometric bluntness and imperfec-tions has also been studied (84, 85), and it wasfound that the structures’ fundamental behav-ior remained unaltered. We note that similarplasmonic nanostructures have been reported,even ones with subnanometer features, in theregime where nonlocal effects start playing adominant role (70–72).An example of how such a broadband slow-

light–aided field enhancement can be exploitedto improve the performance of photovoltaic solarcells is shown in Fig. 3A. A layer containing spheri-cal nanocrescents composed of an upconverter-doped dielectric core and a crescent-shapedmetallic shell is placed behind the solar cell (86),capable of transforming over broad spectral andspatial regions transmitted low-energy photonsto higher-energy, above-bandgap photons thatcan be absorbed by the solar cell. Detailed cal-culations showed a 100-fold increase in above-bandgap power emission toward the solar cell(86) and a greater than 10-fold absorption en-hancement for a broad range of sub-bandgapwavelengths (Fig. 3, B and C), leading to single-junction solar cell power-conversion efficien-cies that can exceed the Shockley-Queisser limit(from ~30 to >44%). Such nano-engineered struc-tures provide one with the ability to exploitbroadband slow-light and large near-field intensity-enhancement effects right at the meta-atomiclevel, i.e., not only the macroscopic overall struc-ture but also its constituent metamolecules (nano-antennas) can be designed to harness broadbandslow light—a unique feature compared with othermacroscopic slow-light-structure designs.These slow-light “nanocrescent-moon” meta-

molecules can also be used as an ultrasensitivesurface-enhanced Raman scattering (SERS) sub-strate (Fig. 3D) (87). The gold nanocrescents(300-nm inner diameter and 100-nm bottom thick-ness) are functionalized with biomolecularlinkers to identify specific biomolecules. Whena single gold nanocrescent moon is excited by a785-nm diode laser, the broadband hotspot thatis generated is used to detect the SERS spectraof rhodamine 6G (R6G) molecules, with an en-hancement factor larger than 1010. Comparingthe SERS spectra of 1 mM R6G molecules on asingle gold nanocrescent moon and on 60-nm col-loidal nanospheres (Fig. 3E) shows substantialimprovement when the former structure is used—both in the number and the relative intensity of


Fig. 4. Broadband slow, conventional and “spoof” surface plasmon polaritons, and theirapplication in spectrum demultiplexing and color sorting. (A) Schematic illustration of theexperimental setup used for slowing and bringing to the nanoscale TM-polarized light (magnetic fieldparallel to the z axis). The incident vacuum wavelength was 1532 nm (80). (B) (Left) An SEMimage of two subwavelength-width (~100 nm) V-groove output slits and a large reference hole asseen from the back side (along x). (Right) Output signal for TM polarization of the incident lightfield.When the incident-light polarization is transverse electric, no plasmon mode is excited at the twosubwavelength slits (80). (C) Dependence of intensity (|E|2) enhancement with the width of theV-groove slit, as determined experimentally and numerically (80). (D) Numerical simulations of slow“spoof” SPPs propagating along a tapered plasmonic grating. SPPs of broadly different frequencies(in the telecommunications regime) are stopped and trapped at correspondingly different pointsalong the nanoguide (76). (E) Experimental demonstration, at visible frequencies, of the scheme in(D). Here, visible light of different colors is incident on a nanoslit from the bottom of the structure(the nanoslit extends horizontally, and is perpendicular to the arrows at their starting points),acquiring enough momentum to couple to spoof SPPs. The SPPs excited in this way propagateslowly in the direction indicated by the arrows along the tapered grating (shown here from above),with each color eventually being stopped and trapped at a different groove depth (“rainbowtrapping”). The white scale bar shown in the top left panel corresponds to 10 mm (97).


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the resolved Raman peaks. These broadband slow-light nanophotonic crescent moons exhibit Ramanenhancement factors (for an 830-nm laser excita-tion source) exceeding 1010, as compared with 103

to 104 for on-resonance Au nanospheres (using514-nm laser excitation) reported previously (88).The broadband field-enhancement in the

ultraslow-light regime of tapered metamaterialand nanoplasmonic waveguides has been ex-ploited in a variety of further applications, in-cluding broadband thin-film infrared absorbers(89), spontaneous-emission enhancement of single

quantum dots and nonclassical light sources (90),cavity-free stopped-light nanolasing, and activeon-chip nanophotonic devices (34, 91–94). How-ever, several of the reported 3D subdiffractionslow-light structures, such as the linearly taperedcylindrical nanoguides in Fig. 1, B and D, arestandalone devices, with a challenging on-chipimplementation. A first quantitative study towardovercoming this issue was reported in (80), whichdemonstrated nanoscale 2D slowing of light inthe form of SPPs (Fig. 4, A to C). Here, light wasadiabatically slowed down and squeezed (with

an efficiency of ~50%) through a 45 nm by 2 mmV-groove slit aperture, resulting in a measuredintensity enhancement of ~10—the defining fea-ture of so called “structural slow light” (4–6) inguiding structures, both below and above thediffraction limit. The full extension of this nano-scale slow-light scheme to three dimensionswas reported recently (95), giving rise to experi-mentally estimated intensity enhancements of~400 within a mere 14 nm by 80 nm cross-sectional area and allowing for efficient on-chipslow light that can be readily integrated with


Fig. 5. Adiabatic waveslowing-down and focusingfor enhanced acoustic sens-ing and nanooptical heat-assisted magnetic recording(HAMR). (A) Schematic ofhow the pressure field of asound wave is spatially com-pressed and enhanced whiledecelerated inside an acousticmetamaterial, before beingdetected by a sensor (61).(B) Photonic-acoustic meta-material hybrid sensing sys-tem.The device is constructedfrom an array of stainless-steel plates, spaced by airgaps, and has a progressivelyincreased width (61). (C) Thesignal-to-noise ratio (SNR)measured in the metamaterialcompared with that obtainedin free space. The inset showsthe input Gaussian pulsesfrom the speaker in timedomain (left) and frequencydomain (right). The diamondsare the experimental dataand the solid line is the fittedcurve with a slope of 114,indicating an ~20-dB SNRimprovement by using theultraslow–acoustic-wave–enhanced sensing system. Inthe gray highlighted zone, themeasured free-space SNRis smaller than 1, indicatingthat the input signal is belowthe detection limit (61).(D) Time-domain andfrequency-domain pulsesignal measured in freespace (top) and in themetamaterial (bottom). TheSNR measured in the metamaterial device is 32.7, compared with themeasured free-space SNR of 0.27, which corresponds to point A in (C) (61).(E) A HAMR head records data to a disk medium of high thermal stability.Here, a guided wave excited by a laser source propagates slowly along aparabolically tapered metal-clad waveguide [shown in (F)], adiabaticallystopping and focusing at the cutoff-tip of the guide, and eventually coupling toa judiciously shaped plasmonic nanoparticle (nanoantenna). The wave’sadiabatic slowing and stopping is similar to that shown in Fig. 1, B to E, andFig. 3, A and D, but for a parabolic taper. The resulting high-intensity local

field is used to heat the recoding medium locally, assisting the recordingprocess by temporarily lowering its resistance to magnetic polarization(101). (F) Schematic illustration (not to scale) of a side view of the HAMRrecording head. In the upper part of the recording head, a gratingcouples laser-diode light to the parabolically tapered metal-clad opticalwaveguide. At the bottom of the tip, there is a plasmonic nanoantenna(“near-field transducer,” NFT) on which light energy is evanescentlycoupled (red arrows) (101). (G) Magnetic force microscope image of arecorded track. The track width is ~70 nm (scale bar, 300 nm) (101).


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other on-chip nanophotonic components. Thespatial compression achieved in the slow-lightregime of plasmonic slot nanoguides was ex-ploited in (95) to reduce the length and overallfootprint of a plasmonic electro-optic phase mod-ulator (96), giving rise to large bandwidths(~120 nm around 1.55 mm), high modulationspeeds (~40 Gbit s−1), compactness (a lengthof only ~29 mm and a width of ~140 nm), andenergy-efficient operation (energy consumption~60 fJ bit−1).This type of ultrabroadband light deceleration

can be attained in almost any regime of the elec-tromagnetic spectrum, from microwaves (66, 67)to the terahertz (55, 75, 77), near-infrared (76), and

visible regime (78–82, 97, 98), using suitably en-gineered plasmonic structures. Two examplesof this are shown in Fig. 4, D [theory, at tele-communication wavelengths (76)] and E [exper-iment, at visible wavelengths (97)]. Figure 4Eshows so-called “spoof” (engineered) SPPs beingexcited at a nanoslit (extending horizontally, fromleft to right, in the upper panels, perpendicularlyto the arrows) and propagating along the direc-tion of the arrows. The SPPs propagate along atapered plasmonic grating structure, similar tothat discerned in Fig. 4D, in the direction in-dicated by the arrows. In this adiabatically taperedstructure, the period and the width of the groovesare fixed (~475 and 150 nm, respectively), but

the groove depth increases with distance slowlyfrom ~6 to 100 nm. As they propagate along theadiabatically tapered structure, the SPPs (in thevisible-wavelength region, 500 to 700 nm) dis-perse, with each visible color eventually beingtrapped and localizing at a correspondinglydifferent point along the tapered structure—with the larger (red) wavelengths localizing atthe deepest grooves. These experimental results,therefore, provide confirmation in the visible re-gime of the spatial demultiplexing and trappingat different positions of the spectrum of a broad-band pulse (“rainbow trapping”) propagatingalong a tapered plasmonic waveguide—similarto the effect outlined theoretically in Fig. 1, B


Fig. 6. Broadband ultraslow guided Diracelectrons, plasmons, and phonon-polaritonsin graphene, van der Waals crystals, and het-erostructures. (A) (Top) Potential profile ofn-doped graphene, bounded symmetrically bytwo p-doped regions. (Bottom) Illustration of thereflections of the two pseudospin Dirac-electroncomponents at the two p-n interfaces, along withthe associated negative quantum Goos-Hänchenphase shifts. The two zigzag-propagating rays,shown in red and purple, trace the movementof the center of a guided slow electron beamon the two graphene sublattices (102).(B) Schematic of graphene encapsulated byh-BN, and the associated s-SNOM measurementsetup for the excitation of a guided slow Diracplasmon. The simulated Ex field-component ofthe plasmon in the out-of-plane direction (20 nmfull-width at half-maximum) is shown on theright (104). (C) Calculated dispersion diagram ofthe slow Dirac plasmons of (B), for a carrierdensity of ns = 7.4 × 1012 cm−2. The crosses showexperimental data, and the shadowed regionsshow the h-BN frequency bands where propa-gating phonon polaritons can exist (shadedorange) and the electronic intraband Landaudamping region (shaded green). The redbackground color shows the imaginary part ofthe reflection coefficient of evanescent waves,evaluated at the top h-BN surface using thetransfer-matrix method for a vacuum–

SiO2(285 nm)–h-BN(46 nm)–graphene–h-BN(7 nm)–vacuum heterostructure (with thedips in the reflection coefficient coincidingwith the excited modes in the heterostructure)(104). (D) Schematic of structure and the setupfor exciting and observing ultraslow negative–phase-velocity hyperbolic phonon polaritons ina thin h-BN slab. The polaritons are launchedwith the aid of the metal (Au) edge, and thethickness of the h-BN layer is 135 nm. Type Ipolaritons are found in the spectral region 22.8 THz (13.6 mm) to 24.7 THz(12.1 mm), whereas type II polaritons exist between 41.1 THz (7.3 mm) and48.3 THz (6.2 mm) (109). (E) (Left) Numerical calculation of the fielddistribution (on a logarithmic scale) for the structure of (D), at w = 1563 cm−1

(46.9 THz; 6.4 mm). The propagation of the type II fundamental hyperbolicpolariton mode is shown as rays propagating in a zigzag fashion alongthe slab, being reflected from the top and bottom media interfaces—similarlyto the zigzag ray propagation shown in Fig. 1C (although here the phasevelocity of the type II mode is positive and the Goos-Hänchen phase

shifts are not clearly discerned). (Right) Field-intensity profiles of thefirst three guided hyperbolic slow modes (109). (F) Infrared near-fieldimage of the interference between the polaritons launched by the goldedge and those by the tip [see (D)] upon being reflected by the h-BNedge (109). (G) Comparison between the experimentally (color) andnumerically (white solid) determined dispersion curves of the funda-mental type II (left) and type I (right) hyperbolic polariton mode. Thewhite dashed lines show numerically calculated higher-order guidedmodes (109).


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and C, for tapered plasmonic and negative-indexwaveguides, respectively. The resulting spectrumdemultiplexing can, in this scheme, find interest-ing applications for on-chip spectroscopy (82),filtering, and color sorting (78, 99).As an essentially interference phenomenon

(Fig. 1, A and C), the wave deceleration in adi-abatically tapered metamaterial waveguides canalso be exploited for acoustic waves (61, 62). Here,the concept of metamaterials (judiciously engi-neered media with desired wave responses)is especially fitting as it allows the design—inthe deep-subwavelength, true–effective-mediumregime—of structures with broadband slow-waveacoustic responses that are unattainable withnatural media. An example is illustrated in Fig. 5,A to D. Here, a tapered acoustic metamaterialwaveguide brings an input pressure field (soundwave) to a halt at a critical point inside the wave-guide, where the acoustic field can then build upand be locally enhanced (compare Fig. 1, B andC, for the photonic analog), so that it can be de-tected with ultrahigh sensitivity by an acousticsensor at the guide’s end (Fig. 5A) (61). Theacoustic-metamaterial device is constructed byan array of stainless plates, spaced by air gaps,and behaves as a continuous medium. Excitedby a speaker, the acoustic wave (frequency ~7 kHz)propagates with a progressively slower speedalong the structure, and is therefore spatiallycompressed and enhanced in amplitude, until itis eventually detected at its stopping point by afiber-optic microphone integrated at the wide,open end of the device (Fig. 5B). More than 10-fold pressure amplification is achieved in thisway. The broadband acoustic-wave enhancementin this type of a system is so large that it can beused to successfully recover very weak acoustic-pulse signals far below the detection limit ofconventional acoustic-sensing systems. Figure 5, Cand D, shows how a series of Gaussian-modulatedacoustic pulses with a center frequency of 9.3 kHzand a bandwidth of 0.5 kHz are effectively am-plified in the slow-wave acoustic-metamaterialdevice, with a signal-to-noise ratio (SNR) en-hanced by more than 100 times (that is, morethan 20 dB) compared with that obtained infree space. This SNR enhancement enabled themetamaterial slow-wave system to detect weakacoustic signals below the detection limit of aconventional acoustic sensor. Indeed, Fig. 5D(top) shows that whereas the measured free-space signal is completely overwhelmed by thenoise (i.e., SNR = 0.27), the same weak acousticpulse is successfully recovered (i.e., SNR = 32.7)in the slow-wave metamaterial region (Fig. 5D,bottom).Finally, Fig. 5, E to G, highlights what we be-

lieve is one of the most exciting and technolog-ically promising applications of slow-light–aidednanoscale field-enhancement: heat-assisted mag-netic recording (HAMR) for data storage. Since2012, the density growth rate of hard disk drives(HDDs) has begun saturating owing to funda-mental physical limits reached by perpendicularmagnetic recording—in particular, because therequired writing field in HDD electromagnets

must be higher than that of the magnetic fluxsaturation in a typical CoFe-based electromagnet(100). A promising avenue that the industry iscurrently pursuing to overcome this limitation isHAMR, which exploits the sudden drop of amagnet’s coercivity at the critical Curie temper-ature. To achieve the required high local heating(~400°C), typically ~80 mW of laser light froman inexpensive semiconductor diode (operatingat ~830 nm, although blue-ray laser diodes canoperate at ~400 nm) must be efficiently broughtto the nanoscale to heat (from only ~5 nm abovethe recording head) a volume of ~50 nm by15 nm by 10 nm of a FePt ferromagnet. To putthis into a context, the required power densityfor HAMR is ~1000 times larger than that re-sulting from standard Au-coated tapered opticalfibers used in near-field scanning optical mi-croscopy (NSOM), which typically have an op-tical transmission efficiency of less than 10−5.Figure 5, E and F, shows an example, reportedin 2009 by Seagate Technology, of how slow-light nanoguides can be used in a HAMR record-ing head to solve the above challenges (101).Here, diode-laser light at 830 nm is coupled toa gold-coated parabolically tapered Ta2O5 slab.As it propagates along the metal-clad taperedwaveguide (45), the light signal is gradually de-celerated, spatially compressed, and enhancedin amplitude, until it reaches the guide’s cutoffthickness where it momentarily stops [vg = 0 atthe cutoff (45)]—a behavior fundamentally simi-lar (64, 83–85) to that described for the struc-tures of Figs. 1 and 3, except that the taper isnot linear (Fig. 1, B and D), cylindrical (Fig. 1E),or spherical (Fig. 3, A and D) but rather para-bolic. It then evanescently couples with goodefficiency to a gold nanoantenna, producing anintense localized field, well below the diffractionlimit, and only a few nanometers above the harddisk—enough to induce the required local heat-ing. For optimized structures, Challener et al.(101) reported maximum field-enhancements of~2000, and a maximum coupling efficiency (at~870 nm) of ~8%—orders of magnitude greaterthan the coupling efficiency of conventional aper-tures with diameters of the same dimensionsas the nanoantenna. This method has therebyenabled local heating of a magnetic track only~70 nm wide (Fig. 5G), giving rise to a data-recording areal density of ~375 Tb m−2—muchhigher than previously reported storage densities(100, 101). Important remaining challenges forthis technique concern the efficient integration ofmaterials that will be even more resilient to ther-mal damage (upon repeated heating cycles), moreefficient heat sinks, and dedicated nanometer-gap–regulation techniques between the writinghead and the recoding disk.

Broadband ultraslow electron andphonon-polariton waves in 2D materials

The nonresonant mechanisms for the decelera-tion of guided light waves in nanostructurescontaining media with negative optical parame-ters (Fig. 1, A to C) also play a critical role in theslowing-down of electron and phonon-polariton

waves in 2D nanoguides and heterostructures.Here, an additional advantage is the conveniencewithwhich one can control the velocity of guidedmatter waves, e.g., by smoothly varying a gatevoltage applied on a 2D graphene quantum-wellstructure, thereby allowing for the possibility ofvariable delays—an important aspect of coherentbuffers and memories (4–6).The quantumGoos-Hänchen effect in the ultra-

relativistic limit ofmassless electrons, relevant forgraphene, considered the propagation ofmasslessspin-½ particles in two dimensions, in a p-n-psingle-layer graphene junction with a voltage(potential barrier) U0 applied in the p-dopedregions (102) (Fig. 6A, upper panel). It was foundthat the Goos-Hänchen effect at the p-n interfacesdepends on the pseudospin (sublattice) degreeof freedom of themassless Dirac fermions, and,moreover, that it changes sign at an angle ofincidence as = arcsin{(sinac)

1/2}, where ac is thecritical angle for total reflection, for Fermi en-ergies EF < U0. For angles of incidence smallerthan as, the shift becomes negative (Fig. 6A,lower panel), just as with guided light waves inmetamaterial and plasmonic waveguides shownin Fig. 1C. In that regime, the group velocity v ofthe guided quasiparticles can become zero. Inparticular, as the normalized width (U0W/ħv) ofthe central n-doped channel is progressively re-duced to around unity, i.e., as one enters the fullquantum-mechanical regime, the dispersion rela-tion of the lowest channel mode features twopronounced minima (where the group velocityreduces to zero, v = 0), each contributing to theconductance a quantum of e2/ħ per spin and valleydegree of freedom, thereby making a total contri-bution to the conductance of 8e2/ħ—amacroscopicconductance step that is observable in these bi-polar junctions. Similar effects can be observedin bilayer graphene junctions, too, where it ispossible to coherently decelerate, stop, and re-accelerate the guided quasiparticles, allowingfor interesting coherent-manipulation function-alities in graphene-based integrated devices (103).In addition to slowly propagating guided

Dirac fermions in doped graphene slots of suit-able geometry, there is currently much interestin the interaction of infrared light with Diracplasmons or phonons, particularly in the slow-wave regime of these nanostructures. In thisregime, the longitudinal wave vector of the guidedwave is large and the effective wavelength small,leading to volume confinements that can bemillions of times smaller than in free space,with relatively low losses—thereby allowing forultrastrong light-matter interactions. An interest-ing example of this is illustrated in Fig. 6B, whichshows a graphene sheet encapsulated by twofilms of hexagonal boron nitride (h-BN)—thewhole structure assembled by a polymer-freevan derWaals (vdW) assembly technique (104, 105).Hexagonal boron nitride is an interesting op-tical material for nanoscale slow-light applica-tions as it exhibits natural hyperbolic behavior(106), i.e., the in- and out-of-plane componentsof the permittivity tensor have opposite signsin the reststrahlen frequency bands. Figure 6C



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presents the dispersion diagram (energy/wavelengthversus wave vector) of propagating Dirac plas-mon modes in the structure shown in Fig. 6B.In Fig. 6C, the crosses are the experimentallyextracted data, indicating group velocities vg =(dReqp/dw)

−1, with qp being the plasmon wavevector, as small as ~106 m s−1 (group refractiveindex ~300), and corresponding to observed life-times (not shown directly in this figure), tp =(Im qp)

−1/vg, as long as ~500 fs (104)—more thanan order of magnitude longer than those of puresurface plasmons in silver, which are typically~20 fs (36). Furthermore, the wavelength lp of theDirac plasmons is as low as 70 nm—~150 timessmaller than the free-space wavelength of theincident light. With an even further improve-ment of these plasmons’ lifetimes, such a struc-ture should be ideal for slow-wave applicationswhere ultrahigh field confinement and tun-ability are desirable (104, 107), particularly forachieving low-threshold single-plasmon (photon)nonlinearities (108).Still larger propagation lengths, in atomic

vanderWaals crystals (e.g., h-BN), can be achievedwhen light couples to pure optical phonons, givingrise to slowly propagating phonon polaritons.In this case, one of the dominant loss channels,electronic losses, is absent. This leads to propa-gation lengths of up to 50 mm (for incident free-space wavelengths l0 ~5 to 6 mm), as comparedto ~2 mm (for a typical l0 ~10 mm) in graphene(107). Figure 6D illustrates howultraslownegative-phase-velocity phonon polaritons are excitedand detected in a thin h-BNwaveguide exhibitinghyperbolic dispersion and deep subwavelength-scale field confinement (109). Owing to its layeredcrystal structure, h-BN is a natural material inwhich phonon polaritons exhibiting hyperbolicdispersion exist—and there are two types ofguided slow modes: type I, when Re{ez} < 0and Re{ex} > 0; and type II, when Re{ez} > 0and Re{ex} < 0. Here, incident infrared light(from free space; see Fig. 6D) of frequency w =1563 cm−1 (l ~6.4 mm) couples to and excitesguided hyperbolic polariton modes in the h-BNslab, whose transverse (along z) spatial profileis shown in the right panel of Fig. 6E. Specifically,the Au film shown in this figure serves as a broad-band optical antenna, converting the incidentp-polarized field (Ein) into strongly spatially con-fined (high momentum) near fields at the film’srightmost edge, thereby providing the necessarymomentum for coupling the free-space Ein fieldinto the guided modes of the h-BN slab. Basedon time-domain interferometric and scattering-type near-field microscopy measurements (Fig. 6,D and F), it was found that the group velocity ofthe fundamental type-I mode was as small as~0.002c (group index ~500) and with a large life-time of ~1.8 ps—more than an order ofmagnitudelarger than that of SPPs inAuat visible frequencies.Furthermore, the phase velocity of thismodewasfound to be negative (antiparallel to the forward-longitudinal direction of the group velocity) (109).Both of these results were found to be in excellentagreement with theory (see, e.g., Fig. 6G). Theoperation of these structures is currently con-

strained at longer (infrared) wavelengths, and im-plementation in the visible regime is challenging.Furthermore, in these “phononic”materials, mostof the energy is stored in the kinetic and potentialenergy of the oscillating ions—and not in theelectric field (32)—which limits the maximumfield enhancements that can be attained.

Conclusions and outlook

We have examined the underlying physical prin-ciples enabling broadband, ultraslow waves insubdiffraction-limited nanostructures and pres-ented exemplary technological applications in thefields of nanophotonics, metamaterials, acoustics,and 2D materials. The structures deployed areusually geometrically smoothly varying (Fig. 1B),and theunderlyingwave-decelerationmechanismsinherently do not involve material-parameter(permittivity, refractive index) resonances—typically relying on antiparallel power flows inthe negative- and positive-permittivity regions(Fig. 1A) or, equivalently, to negative Goos-Hänchenphase shifts at the interfaces of these media(Fig. 1C). Ultimately, material and group-velocitydispersion set a limit to the bandwidth perform-ance that can be attained (25, 63). However, itshould be noted that the bandwidths attained inthese structures can still, in the optical regime, belarge, typically of the order of ~1 to 10 THz—orders of magnitude larger than, e.g., theiratomic-media counterparts (which feature sim-ilar order-of-magnitude decelerations). Further-more, upon suitable design, e.g., in multilayerheterostructures, the dispersion band can (for acontinuous range of wave vectors) become veryflat, i.e., group-velocity dispersion can be mini-mized (39)—thereby leading to enhanced band-width performance. This last feature—the abilityto enhance and engineer bandwidth “by design”at the nanoscale—is a key characteristic of the(ultra)slow-light structures reviewed here.Specific applications range from enhanced and

more efficient nonlinear effects (68, 73, 74, 108),to light-harvesting (64, 83, 89), biosensing (87, 88),nanoimaging (80, 110, 111), optical and acousticspectral demultiplexing (41–43, 55, 56, 61, 62, 64,68, 75–81, 112), on-chip spectroscopy (82), non-classical light sources (90, 91), cavity-free plasmonicnanolasing (92–94), enhanced acoustic sensorsoperating beyond thenoise-threshold limit (61, 62),and tunable, deep-subwavelength,ultraslowguidedDirac fermions (102, 103), plasmons, and surfacephonon-polaritons in atomically thin crystals andheterostructures (104–109, 113). Broadband slow-light effects are also attained in other structuresabove the diffraction limit, including photoniccrystals and CROWswhere broadband slow lightis usually obtained with group indices of ~30 to100 (21–28), and PT-symmetric structures, whichcan be broadband and with the light speed re-ducing to zero at the exceptional point (58, 114).Further related schemes include broadbandmeta-material and nanophotonic analogs of EIT (115),or suitably engineered metasurfaces to inducenegative Goos-Hänchen phase shifts and achievethe same light-stopping effect but without theuse of bulk negative-parameters media (116–118).

The broadband, ultraslow-wave regime insubdiffraction-limited nanostructures is essentiallya new regime for nanophotonics, acoustics, and2Dmaterials, as in all three fields we have beenaccustomed to using fast propagating waves fordiverse operations and devices. Many surprisesare waiting to be revealed and further excitingapplications to emerge in this new regime.

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ACKNOWLEDGMENTS

We acknowledge useful discussions with J. Pendry ontransformation-optics slow-light structures, and with E. Yablonovitchon the slow-light physics of the Seagate Technology HAMRdevices. This work was primarily funded by the U.S. Departmentof Energy, Office of Science, Office of Basic Energy Sciences,Materials Sciences and Engineering Division under contract no.DE-AC02-05-CH11231 (Subwavelength Metamaterials Program,KC12XZ). It was further supported by the Office of Naval Research(ONR) MURI program under grant no. N00014-13-1-0678, theLeverhulme Trust, the UK Engineering and Physical SciencesResearch Council (EPSRC) under grants EP/L024926/1 andEP/L027151/1, the Canada Excellence Research Chairs Program,and the Eugen Lommel fellowship of the Max Planck Institute for theScience of Light, Erlangen, Germany.

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Ultraslow waves on the nanoscaleKosmas L. Tsakmakidis, Ortwin Hess, Robert W. Boyd and Xiang Zhang

DOI: 10.1126/science.aan5196 (6361), eaan5196.358Science

, this issue p. eaan5196Scienceeffects of interest in signal processing and optoelectronic communication.and fabricated structures. They show how the combination of slow light and nanotechnology gives rise to a number of

review how the speed of light can be controlled using designed materialset al.for practical applications, Tsakmakidis exceptional circumstances, light can be slowed to a walking pace or even stopped momentarily. Exploring approaches

When light passes through an optical material, its speed is reduced by the refractive index of that material. UnderSlow light on the nanoscale

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OPTICS Ultraslow waves on the nanoscale ON OUR WEBSITE · REVIEW OPTICS Ultraslow waves on the nanoscale Kosmas L. Tsakmakidis,1*† Ortwin Hess,2 Robert W. Boyd,3 Xiang Zhang1,4,5*

OPTICS Ultraslow waves on the nanoscale ON OUR WEBSITE · REVIEW OPTICS Ultraslow waves on the nanoscale Kosmas L. Tsakmakidis,1† Ortwin Hess,2 Robert W. Boyd,3 Xiang Zhang1,4,5