Ultra-thin, entirely flat, Umklapp lenses

Gregory J. Chaplain1 and Richard V. Craster1,21 Department of Mathematics, Imperial College London, London SW7 2AZ, UK

2 Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UK

We design ultra-thin, entirely flat, dielectric lenses using crystal momentum transfer, so-called Umklappprocesses, achieving the required wave control for a new mechanism of flat lensing; physically, these lensestake advantage of abrupt changes in the periodicity of a structured line array so there is an overlap betweenthe first Brillouin zone of one medium with the second Brillouin zone of the other. At the interface betweenregions of different periodicity, surface, array guided waves hybridise into reversed propagating beams directedinto the material exterior to the array. This control, and redirection, of waves then enables the device to emulatea Pendry-Veselago lens that is one unit cell in width, with no need for an explicit negative refractive index.Simulations using an array embedded in an idealised slab of silicon nitride (Si3N4) in air, operating at visiblewavelengths between 420− 500THz demonstrate the effect.

I. INTRODUCTION

Inspired by the ability to create flat lenses, such as thePendry-Veselago lens1, there has been a drive towards devel-oping flat optical devices to manipulate light2,3. Advancesin metasurface-based technologies have paved the way forthese planar devices, with realisations in a variety of settings,operating by several different modalities. These operationalregimes include plasmonic waveguide structures4, i.e. bulkmaterials composed of alternating metal-dielectric layers5,or all dielectric inhomogeneous layered lens antennas6. Al-ternate flat lens devices focus on controlling abrupt phasechanges endowed to incident wavefronts upon transmission7,8,taking advantage of gradient index structures9, or by manu-facturing protruding subwavelength elements, enabling gen-eralised refractive laws to be observed2.

Less exotic materials which achieve anomalous refractivephenomena are photonic crystals (PCs). These structures ex-ert their control over electromagnetic radiation through ef-fects, such as Bragg scattering, which arise due to their pe-riodic structure, allowing all angle negative refraction to beachieved10 without the existence of simultaneously negativeeffective permittivity and permeability, which is the case inmetamaterials1. All-angle negative refraction, together withrestoring evanescent wave components, is the hallmark of theperfect lens1. Either or both aspects have been used in PCdevices to enable super-resolution11–13. Unlike conventionalmetalens or diffractive lenses14, here we present a new, novel,flat dielectric lens antenna (DLA) device based on simple one-dimensional periodic structures, with positive permittivity andpermeability, operating in narrow frequency bands over thefrequency range 420 − 500THz. These devices operate bypromoting Umklapp scattering15 at a designed region, and of-fer a new, unconventional, modality of flat lensing, as shownin Fig. 1. The proposed structures act to emulate negative re-fractive effects; no super-resolution is achieved as we do notreconstruct any evanescent wavefields or use resonant mate-rials. As such the proposed devices mimic only one propertyof the perfect lens, and can be considered a new type of ‘poorman’s superlens’.

A thorough review of anomalous refractive effects in two-dimensional photonic crystals16 outlines the techniques in

in-coupled guided mode

out-coupled Umklapp beam

guided mode

Source

1

2

2

Figure 1: Umklapp scattering mechanism yielding negativerefraction and focusing. A source incident on region 1 of the

device excites an in-coupled (leaky) guided mode (redarrow). This propagates to regions 2, where the periodicity isabruptly changed. Similar guided modes are excited in these

regions (blue arrows), along with transfer of crystalmomentum via Umklapp scattering, resulting in out-coupled,

backwards-bended, Umklapp beams.

analysing isofrequency contours, or wavevector diagrams, ofperiodic media, enabling the design of simple photonic crys-tals with remarkable refractive properties, taking use of Umk-lapp refracted beams. We will focus on this scattering mecha-nism throughout the article, elucidating on nuances that arisewhen interpreting periodic media, particularly in terms ofcrystal momentum transfer. The advantages, if any, of study-ing simpler one-dimensional (1D) periodic structures are notimmediately apparent as compared to their two-dimensional(2D) counterparts; many effects in 2D photonic crystals arisedue to the periodicity in both directions. For 1D structures,

2

analysis has only been concerned with cases that have inter-faces chosen along the stacking direction (the direction of pe-riodicity), which is the only symmetry direction for the 1Dgrating system16,17. A clear advantage of 1D devices is thatthey are substantially thinner, but would require the ability tomanipulate waves propagating with the periodic direction. Inthis article we show that not only is this possible, but it is alsoeffective in creating new unconventional lensing applications,through promoting crystal momentum transfer.

When dealing with periodic media, analysis in reciprocalspace via the Brillouin Zone (BZ)18 is invaluable. To un-derstand, predict and interpret the existence and direction ofany refracted beams at the interfaces involving periodic me-dia, isofrequency contours (wave vector diagrams) are used19;these contours offer simplistic yet powerful insight into inter-actions within periodic media. An important concept is thatthe Bloch wavevectors in a periodic structure are defined upto a reciprocal lattice vector, G = 2π/a, with a the widthof the unit cell. Band folding interpretations are commonlyused to infer higher order interactions within periodic media,as a direct consequence of the Floquet-Bloch nature of thewaves; in periodic media, wavevectors define a superpositionof plane waves through Bloch’s expansion. As a consequencethe phase of the envelope wave, which carries fast oscillatorysolutions, lies within the first Brillouin zone16,20,21. Conserv-ing the sum of reduced wave vectors leads to descriptions ofsuch interactions by the addition of a phase factors, or phasematching conditions22, due to the periodicity of the mediumand nature of the Bloch solutions it supports. Throughoutthis article we will promote Umklapp scattering to convertsurface guided waves into backward-bent propagating bulkmodes by physically exploiting anharmonicity within the lat-tice; we design an abrupt transition in periodicity within a one-dimensionally periodic dielectric waveguide. Using this tech-nique allows focusing of energy to be achieved, and as suchwe term the presented devices Umklapp lenses.

There is a long history of controlling waveguide modes in1D periodic structures, particularly through anomalous refrac-tion and, notably, light propagation in planar waveguides isextensively investigated in23. The structures we consider, andsubsequent analysis, is similar to that presented in23, in thatwe design 1D periodic waveguides using wave vector dia-grams. There are however, some important distinctions be-tween previous work and the new devices we design. To high-light these we compare our device to the 1D periodic cor-rugated plane waveguide in23, with the distinctions outlinedin Fig. 3. Firstly, typical 1D periodic devices are extendedin the orthogonal direction to the periodicity (in the plane ofthe device), providing a long interaction length for so calledphase matching conditions to occur. In the devices we con-sider, the orthogonal direction is very much finite with respectto the direction of the periodicity i.e. there are no propagat-ing modes in the orthogonal direction. Secondly, to ensureefficient coupling tapered, or graded, regions are used in cor-rugated waveguides. Contrary to this we have no grading. Weare in fact utilising, indeed advocating, Umklapp scatteringat a region of abrupt transition in periodicity. Thirdly, thecorrugated guide has only one periodicity, the region in re-

1

2

2

(a)

(b)

x

y

z

x [m]

y [m]

x [m]

y [m]

x

y

z

Figure 2: Dielectric thin flat lens (2D simulation): logarithmof Electric field log(|Ez|) (arbitrary units), perpendicular tothe page, excited by current line source, marked at white star

operating at frequency 484THz. (a) Array centred source,giving reversed conversion via the Umklapp mechanism. (b)

Flat lensing by source placed at −8λ, producing image atopposite focal spot. The array is shown on the left: device

length is L = 12.3µm, with width w = 500nm, and infinitein z. The maximum amplitude in each focal spot in (a) is

∼ 10% of the source amplitude.

ciprocal space where coupling to directed waves is efficientis local to the band edge (of the first Brillouin Zone), andis only achieved by means of Bragg scattering. By design-ing, as we do here, two regions of differing periodicity weare able to move further away from the edge of the BZ inreciprocal space; coupling between different order frequencybands of the geometrically distinct regions permits operationat several frequencies. This is achieved by the transfer of crys-tal momentum, emphasising the attribution to the Umklappmechanism. Fourthly, and perhaps most pertinent, is that in

3

conventional periodic structures “cross coupling is a specialproperty of wave propagation at oblique angles with respectto the grating, it cannot exist for waves at normal incidenceto the grating lines.23” We show here, that our structures havemaximal coupling with back-bended propagating modes forguided waves which propagate at normal incidence to the grat-ing lines/periodic direction.

The ability to excite counterpropagating modes in coupledperiodic waveguides by phase matching arguments is wellestablished24, as is leveraging momentum transfer for negativerefraction effects25. However, the Umklapp devices presentedhere possess a certain novelty in that the conversion takesplace not only in the opposite direction, but from a guidedsurface mode to a propagating beam, all through promotingUmklapp scattering at a desired region.

The design of structures capable of supporting guided sur-face modes is not limited to the field of optics. Recent designparadigms for adiabatically graded arrays have resulted in aremarkable level of wave control, and phenomena being ob-served, in multiple disciplines within wave physics stretchingbeyond electromagnetism, from elastic vibrations to acous-tics. The inspiration of many such designs stem from therainbow trapping effect, originating in electromagnetism26,whereby the speed and phase of localised array guided modesis manipulated by graded geometrical changes of the arraycomponents. Elementary resonant, often sub-wavelength, de-vices have been proposed27,28, and built29, for elastic mediaenabling trapping, and mode conversion transferring energyfrom surface to body waves, effects for array guided surfacestates for applications to energy harvesting30 from vibration.Recently, a reversed conversion phenomena which emulatesnegative refraction by a line array31 has been developed usinga counter-intuitive effect hybridising both trapping and con-version; this relies upon band crossings and phase-matchingall within the first Brillouin zone and uses adiabatic gradingof an array all in the setting of elastic waves.

In this article we propose a much more versatile de-sign paradigm employing Umklapp ”flip-over” processes toachieve flat lensing. Despite its origins in thermal transport,we can readily adapt the Umklapp mechanism for our dielec-tric structures since they are periodic; these processes are in-herent to any system permitting Floquet-Bloch waves. Theoperation rests on the segregation of the device into structuredregions of different periodicities, thereby creating two differ-ent Brillouin zones in reciprocal space, and the subsequentanalysis of the corresponding dispersion curves. Contrary tothe adiabatically graded arrays considered in31, there is nograding between the regions but instead an abrupt change inthe array periodicity; undesirable scattering of the field at suchan interface is anticipated, but by carefully engineering the de-sign we can recapture the scattered field by promoting Umk-lapp processes thereby providing a remarkably simple way toachieve flat lensing, without the need for exotic materials orinhomogeneities.

We consider the full, three component time harmonic elec-tric fields, E, using the Finite Element Method (FEM) with

Figure 3: Comparison between conventional 1D periodicgrated structures, with differing refractive indices shown asni, as in23 (a) and the presented Umklapp devices (b). (a)

Shows a 1D periodic corrugated device, with the periodicity,a, shown. Prior to this region is a tapered region. This is notpresent in (b), showing a portion of the Umklapp lens, as an

abrupt transition region is used between the two differingperiodicities, a1, a2 - (The simulations throughout the article

are considered for a similar device, but infinite in thez−direction). Further to this, in the orthogonal direction tothe periodicity, the spacing ∆y, in the grating case is such

that ∆y � a to provide a long interaction length. This is notthe case for the Umklapp devices, where ∆y ∼ a. Finally,

the red arrow in (a) shows the direction, perpendicular to thegrating interfaces, in which no refractive effects can be

achieved23. The same direction is shown in (b) in blue, wherethe effects are maximally achieved.

COMSOL multiphysics32 to solve the equation

∇× (µ−1r ×E)− k20

(εr −

jσ

ωε0

)E (1)

where µr and εr are the relative permeability and permittiv-ity tensors respectively, ε0 is the permittivity of vacuum, k0is the free space wavenumber, σ the conductivity and ω theangular frequency. For the purely dielectric devices consid-ered throughout, µr = 1 and σ = 0 . We focus on localisedguided electromagnetic waves confined to a dielectric slab inair, choosing silicon nitride Si3N4 as the dielectric as it iswidely used in applications due to its high relative permit-tivity (εr = 9.7), ubiquity in integrated circuitry33 and trans-parency over the visible range in nanoengineered structuresboth with low contrast (n ≈ 2) and loss34. Throughout thisarticle we will work with an idealised slab of Si3N4, whoseproperties are assumed to be frequency independent, as suchn =

√εrµr, since the important characteristics arise from

the periodicity of the structure (and not its composition). Thesupported guided states manipulated throughout are neithersurface plasmon polaritons, or their spoof counterparts35,36,since we do not deal with opposing signs of permittivities orstructured metallic interfaces. Indeed the proposed structur-ing takes place within the device, leaving its edges completelyflat (Fig. 4). As such we are free to model these devices ef-ficiently in air operating in the terahertz frequency range; the

4

corresponding device dimensions are achievable in waveguidetechnologies37,38. We do not exploit resonance effects, andso the imaging is not subwavelength, yet the proposed DLAdevices, are ideal for demonstrating the efficacy of Umklappscattering.

Our analysis of where (spatially) and how Umklapp mech-anisms can be exploited to produce back-bended reversedbeams is, theoretically, independent of the material and de-vice size and rests upon exploiting the dispersion curves andisofrequency contours of the different periodicities within thedevice, as detailed in Section II. The performance of the de-vice as a flat lens however, is strongly dependent on the sizesof the constituent regions, as demonstrated in Section III. Themethodology can be readily extended for dielectric coatings orsubstrates, i.e. the device does not have to have identical me-dia on either side. Using this, we then outline how to harnessthis promoted scattering effect to generate a new mechanismfor flat lensing, and investigate the effects of losses in the de-vice. Suitable scaling to other materials, and sizes, permitsthe tunability of the frequency bandwidth, leading to a newoperational capacity for dielectric substrates.

II. DESIGN METHODS

When analysing periodic media it is naturally convenient todisplay the dispersion diagrams of such materials within theirreducible Brillouin zone (IBZ)18. Conventionally, we con-centrate upon modes below the dispersionless light line of freespace waves, that is, within the first BZ; Umklapp processes,arising due to the transfer of crystal momentum from higherBZs, are often thereby ignored. We demonstrate that thereare advantages in using these processes by considering tworegions of differing periodicities within the same dielectric ar-ray, see Fig. 7, such that there is an overlap between the firstBZ of one periodic region and the second BZ of the other. Atthe abrupt transition between these regions Umklapp scatter-ing is dominant and reversed conversion can be achieved andutilised for flat lensing.

The Umklapp mechanism, first hypothesised by Peierls in192915 is conventionally used to describe thermal transportand resistivity of metals at high temperatures, and is nowprevalent in the quantum theory of transport39,40. It is a man-ifestation of the transfer of crystal momentum within the sys-tem, and exploits the fact that in periodic media wavevectorsare defined up to a reciprocal lattice vector, G ≡ 2π/a, witha being the periodicity of the unit cell. Thus in the case ofscattering two initial wavevectors, say, k1,k2 then if the re-sultant k3 lies beyond the first Brillouin zone it experiencesthe Umklapp, or ‘flip-over’ mechanism via crystal momen-tum transfer20. Two types of scattering processes are defined:normal (N-processes) and Umklapp (U-processes) through

k1 + k2 − k3 =

{0 N-process,G U-process.

(2)

This mechanism has recently been observed to cause excessresistivity in graphene42 and used to explain the coupling of

0

0.8

0.4

a1

a2

w

10w

1

2

Figure 4: Typical electric fields, of similar mode symmetry,as |E| for each regions 2(1) with unit cells a2(1). Elliptical

inclusions have semi-major and minor axes ra and rbrespectively, detailed in Table I. FEM method and boundary

conditions are shown in Supplemental Fig. S341.

acoustic and optical branches in crystals43. We apply it, forthe first time, in a flat lensing scenario, circumnavigating anyambiguities considered with Umklapp scattering20, by adopt-ing the conventional description in equation (2), analysingU-processes for scattered wavevectors lying outside the firstBZ; this is a natural way to distinguish between N- and U-processes and is critical in this design process.

For the application of the Umklapp mechanism to localisedguided electromagnetic (EM) waves for flat lensing, we par-tition a dielectric slab into two regions of differing periodici-ties, see Fig. 1. To set the context, we first analyse a perfectlyperiodic medium of consisting of a slab of Si3N4 in air struc-tured with a periodic array of elliptical inclusions with pitcha1, with geometrical parameters given as region 1 in Table Ias shown in Fig 4. Then consider the same dielectric materialbut with the array having a larger periodicity, a2, and differ-ent inclusion size (region 2 in Table I). The two regions aredefined with unit cells of length a1, a2 respectively, such thata2 > a1. Consequently the first BZ of region 2 is smaller thanthat of the region 1. Thus, the edges of the first BZ for the re-gions, Xi ≡ π/ai, satisfy X2 < X1 and are offset from eachother. The dispersion curves within this overlapping regionare calculated using the FEM software Comsol multiphysicsand shown in Fig. 5. The frequencies where overlap betweenthe dispersion curves of similar mode shapes are where U-processes take effect efficiently; the excited mode in region 1must be able to excite a mode that exists in the region 2, oth-erwise an effective hard boundary is reached at the transitionregion, resulting in undesirable scattering; typical fields show-ing similar modal symmetries in the two regions are shownin Fig. 4. Whilst not a requirement for the transfer of crys-tal momentum, designing the band crossings to have oppo-site signs of group velocity enhances the coupling betweenthe surface guided modes and the propagating waves excitedby Umklapp scattering20,40. This effect is achieved for severalof the overlapping bands shown in Fig 5, ranging between420−500THz, and therefore has potential for broadband per-formance. A further example of the reversed conversion effectis shown in supplemental Fig. S141.

We now consider the implications of structuring a finite slabof Si3N4 into two distinct regions, one with the periodicity andunit cell structure of region 1, in Table I, that then transitionsabruptly to incorporate the parameters associated with region2. Conversion of the array guided mode occurs into a beam in

5

Region 1 Region 2

a 300nm 330nmw 500nm 500nmra 110nm 80nmrb 170nm 130nm

Table I: Parameters withindielectric slab; region 1(2)

coloured red(blue) in Fig 7(g)(z−direction taken to be

infinite).

Figure 5: Dispersion curves for region 1(2) in red(blue),plotted from X2 to X1 i.e. in the second BZ of region 2

whilst in the first of region 1. The dashed line represent thedispersionless light line, with the ‘folded’ light line also

shown. Matching mode shapes promotes U-processes. Fulldispersion curves, and domain probe method used to

eliminated spurious modes (green curves) are shown insupplemental Fig. S241

κ

κn

(a) (b)

Figure 6: Conventional mode coupling diagrams, as in22. (a)shows a schematic of an incident wavevector (blue arrow)coupling to a waveguide mode (red arrow) through some

interaction with a periodic medium, thanks to the transfer ofcrystal momentum. This is shown in the wavevector diagram

in (b), due to phase matching, which is really a result ofcrystal momentum transfer.

the exterior medium that is backward-bent; the angle of the re-versed conversion is predicted by inspecting the isofrequencycontours of the two regions. Shown in Fig. 7 are simplisticisofrequency contours for the respective components of thedielectric slab; we show for clarity the isofrequency contourof each medium as a circle, acknowledging that it in fact cor-responds to a point on the κx axis when projecting the disper-sion curves into wavevector space. This is how wavevectordiagrams are historically used when the device dimensionsin the orthogonal direction to the periodicity are large23, assuch the periodicity acts to modulate the free-space wavevec-tor diagrams. Despite our devices not having any propagating

-a1

-a2 a2

a1G

n0

a1 a2

(a) (b)

(c) (d)

(e) (f)

(g)

n0

a1 a2

n0

G

-a1 a1

Γ-πa2

πa2

-πa1

πa1

Γ-πa2

πa2

-πa1

πa1

Γ

-πa1

πa1

Γ-πa2

πa2

-πa1

πa1

Γ-πa2

πa2

Figure 7: Step-by-step wavevector diagrams. (a,b) showschematics of the two periodic regions, with the

corresponding wavevector diagrams below. Each has thesame length of incident wavevector, which lies in different

BZs depending on the periodicity, shown by the BZboundaries ±π/a1,2 respectively. (c) Shows a band folded,or Irreducible, picture by translating points outwith the first

BZ of region 2 by a reciprocal lattice vectorG = 2π/a2. (d)Shows the resultant wavevector after Umklapp scatteringtakes place through the transfer of crystal momentum. (e)

shows the conserved wavevector by phase matching with theisocircle of the free space surrounding the device. (f) shows

the previous 5 wavevector diagrams superimposed into acompact diagram which allows for the prediction of the

angles of the reverse converted waves. (g) shows a schematicof the process.

components in the orthogonal direction to the periodicity, weadopt this picture for familiarity and clarity. Exploiting thecoupling at a sharp change in periodicity is inspired by con-ventional diffractive mode coupling theory22; incident radia-tion of wavevector κ in a media with refractive index n0, iscoupled to a waveguide mode of vector κwg through phasematching by incorporating the periodicity, such that

κwg = κn0 sin θ + Λ, (3)

where, for integer m, Λ = 2mπ/a. This is shown in Fig. 6.

6

Using this mode coupling picture we can interpret the hybridi-sation mechanism from the ‘contours’ of the medium com-posed of differing periodicities; whilst the following resem-bles this diffractive theory, the mechanism is fundamentallyachieved through Umklapp processes by crystal momentumtransfer. The coupling into the waveguide by gaining mo-menta of multiples ofG is therefore distinct to Umklapp pro-cesses, in which momenta is subtracted, which leads to theout-coupling into back-bended beams. Understanding bothmomentum transfer mechanisms is important for the perfor-mance of the device in terms of exciting the guided modes onthe device and their conversion.

In Fig. 2(a), the array is excited with a source to generatea guided wave with wavevector in region 1. This is shownschematically in Fig. 7(a), with the corresponding wavevectordiagram below. Despite being 1D periodic we show this as ahalf circle as is customary in these interpretations22. The cor-responding ‘contour’ is defined within the first BZ such thatκ ∈ [−π/a1, π/a1] ≡ [−X1, X1]. At a designed spatial posi-tion, we then abruptly change the periodicity to that of region2, marked by the schematic of the blue region in Fig. 7(b). Be-low this shows the corresponding wavevector diagram wherethe length of the wavevector is the same as in Fig. 7(a) aswe are operating at the crossing of two supported dispersioncurves; the material properties have not changed, only the ge-ometry has altered. At this frequency, the wavevector nowlies in the second BZ of region 2 since X2 < κ < X1. Thisis highlighted in Fig. 7(c), by drawing a ‘band folded’ ver-sion of the contour describing this wave vector. This is ac-tually formed by translating every point outside the first BZby G = 2π/a2. The critical observation is that this initialwavevector, marked by the red arrow, experiences a trans-fer of crystal momentum via the Umklapp effect, resultingin the translation of a collinear reciprocal lattice vector, G,denoted by the folded isofrequency contours. The resultantflipped vector is shown in Fig. 7(d). Shown in Fig. 7(e) is thesuperimposed isofrequency contour of the exterior medium(in this case, air) that surrounds the array (yellow circle).Phase-matching, by conserving the tangential component ofthe flipped vector, with this contour gives the resultant scat-tered wavevector that, in turn, predicts the reversed conversionangles as in Fig. 2(a). Since the device is surrounded by air oneither side, there are two beams shed with equal angle, high-lighted in the compact diagram in Fig. 7(f), with a schematicshown in (g). The wavevector components of Fig. 2(a) areseen in the Fourier spectrum shown in Fig. 8, corroboratingthe adopted wavevector diagram analysis.

We stress that this methodology is completely general, andas such the device can have different media on either side ofthe interfaces, raising the possibility of different angles of re-versed conversion on either side. As such there is a broadrange of applications for these structures as dielectric coat-ings, capable of mimicking generalised lenses44. The angleis explicitly predicted from mode coupling analysis by rear-ranging equation (3) to incorporate the effect of the secondperiodic region; in this setting Λ = 2π/a2 > κwg . Thus wecan generalise conventional mode coupling techniques for awave confined to the array arriving at a region with altered

κwg

κwg-G

κx

κy

X1

X2

-X2

-X1

Γ

Figure 8: Fourier Transform of Fig. 2(a), normalised tomaximum Fourier amplitudes (arbitrary units). The

periodicity is understood to be in the direction of κy shown.We show one wavevector component of the initially excited

guided mode, namely the vertical wavevector κwg , lyingbetween X2 and X1. This is flipped by Umklapp scattering

to the negatively pointing κwg −G, with respect to Γ.Conserving the tangential component of this hybridises the

guided surface waves counter propagating bulk waves, shownby the bright spots lying on the isocircle of the free space

waves; any wavevector components which lie off the verticalcyan dashed-dotted line are understood to be propagatingwaves off the array.. The centre of the Brillouin Zones is

shown (Γ), with the BZ boundaries of region 1(2) highlightedby the dashed lines ±X1(2). This diagram is the

scattering-simulation counterpart of Fig. 7(f) obtained fromthe dispersion relation.

periodicity and notably U-processes provide coupling to thefirst negative diffractive order of mode coupling theory22. By(a)symmetrically introducing abrupt changes in periodicityabout a central point, we can image a line source on the ar-ray to two focal spots on either side of the slab, as in Fig. 2(a),and tune the position of these focal points.

III. UMKLAPP LENSING

Our arguments also generalise for excitation by a sourceremoved from the array. We again use the Umklapp and thereversed conversion mechanisms at interfaces between differ-ent regions of periodicity, but now introduce reciprocity togenerate focusing from one side of the slab to the other. Anisotropic current line source is placed at one focus of Fig. 2(a),and an image is produced at the other side of the array, show-ing the device is capable of emulating negative refraction. Thefocusing response is due to the horizontal wave componentigniting the surface guided wave, which propagates along thedielectric array to the regions of altered periodicity, seen in

7

(b) (c)(a)

Figure 9: Performance of device with respect to horizontaldistance of source. (a) shows schematic of the setup where

the source position (star) is gradually placed further from thedevice, along the centre line. The performance, P , is

measured by integrating the electric field |E| over a regionencapsulating the focal spot predicted from Fig. 2

(rectangular box). (b) shows the power in this region for threetypes of material: the Umklapp lens (green), a dielectric slabpatterned only with Region 1 (blue) and a blank slab (red),normalised to the maximum performance of the Umklapp

lens. The optimal focal distance away from the device edge(shown as grey region) for the Umklapp lens corresponds tothe focal length. (c) shows schematics of the tested devices.

Fig. 2. Upon reaching these altered regions, Umklapp scat-tering takes place and the reversed conversion effect acts torefocus this point source on the opposite side of the device,as clearly shown in Fig. 2(b). The device then acts as a lensin the sense that it can focus incoming radiation at a givenfocal point. Unlike conventional lenses, changing the sourceposition will not alter the focal spot position; the flexibility inthe positioning of the focal spots is however achieved throughaltering the position of the transition regions, the relative pe-riodicities, frequency of excitation and symmetry. This effectonly takes place due to the transfer of crystal momentum andas such we term these devices Umklapp lenses.

The position of the source position does however, unlike inclassical lenses, affect the performance of the structure. Thiscan be seen in Fig. 9, where we analyse the performance ofthe Umklapp lens compared to two other structures; a plain di-electric slab and a dielectric slab which is only patterned withthe periodicity and inclusions of Region 1. The position of thesource is altered along the line which perpendicularly bisectsthe device. The performance is analysed by integrating |E|over a region surrounding the focal spot predicted from thedevice size and angles of the reversed conversion (Fig. 9(a)).Exciting very close to the device couples most efficiently thenear vertical wavevector components, parallel to the direc-tion of the periodicity, contrary to other 1D periodic gratingdevices23. Increasing the distance of the source from the cen-tre line of the device initially decreases the performance, untilthe source is placed near the focal length of this frequency.In the vicinity of the focal length, focussing is enhanced onthe opposite side of the device (Figs. 2(b), 9) due to the addi-tional coupling of the wavevector components incident on thetransition region, by reciprocity. As such, unlike conventional

‘poor man’s superlenses’, these devices have scope for emu-lating negative refraction in the far field, as this focal lengthfor this frequency is ∼ 8λ45.

Past this point, a critical angle is reached such that there isvery little coupling between the propagating wavevector com-ponents of the source with Region 1. As such, the focussingcapabilities of the device are limited, depending on the sourcepositionand the length of the central Region 1.

Considering the fabrication of such devices atop otherstructures, as is conventional in dielectric waveguides37, alsopermits tunability of the focal lengths on either side of thedevice31. This in turn would effect the optimum source posi-tions for focussing within each of the media. Improvementsin the limitation of the device can be made by the periodicstructuring of the lens surface, at the cost of it no longer beingentirely flat; more propagating wavevector components cancontribute to the guided surface modes by an initial transferof momentum from the periodic structuring46.

The validity of this concept for devices is further tested bythe introduction of losses within the material, through the ad-dition of an imaginary component to the permittivity. An ex-ample is shown in Fig. 10, where the field strength along thecentre of the devices is compared with, and without, loss forboth configurations in Fig. 2. For a lossy material, there isweaker propagation along the array and so the reversed con-version is not as pronounced; the maximum amplitude at thefocal spots is now only ∼ 2% of the maximum source ampli-tude. This can be mitigated by designing the position of thetransition region to lie closer to the point of excitation, andwill influence the materials and dimensions chosen to circum-navigate any losses. For decaying surface waves the effect canstill be achieved by incorporating the decay length, which canbe predicted through homogenisation techniques27,31. Despitethis, the point spread function of the focal point in the losslessand lossy case remains practically identical.

IV. CONCLUDING REMARKS

We develop the concept of Umklapp lensing by designingan array that requires operation at wavevectors outside the firstBrillouin zone, and then manipulating confined surface wavesby utilising crystal momentum transfer. The resulting elec-tromagnetic radiation generates focal points, using high per-mittivity dielectrics that are both entirely flat, and ultra thin:Negative refractive effects can therefore be emulated with bya line array with thicknesses of just one unit cell.

Similar to classical ‘poor man’s superlenses’ the devices fo-cussing capabilities are optimal for near field excitation, how-ever there is scope for far field focussing capabilities due to theadditional coupling from wavevector components when thesource is placed at a focal length. Unlike for true superlensesno evanescent components are reconstructed and as such theimaging is not subwavelength.

Considering lossy materials motivates the design of thetransition region between differing periodicities, which is keyto employing the Umklapp effect. So too is choosing the peri-odicities in such a way as to achieve maximal overlap of mode

8

(a)

(b)

x

y

z

x [m]

y [m]

x [m]

y [m]

x

y

z

Figure 10: Same configurations and frequency as in Fig. 4,showing logarithm of electric field, log(|Ez|), for the case

with loses introduced with εr = 9.7 + 0.2i. Side panels showcomparisons of normalised electric field norm between

lossless (black) and lossy (blue) media, plotted along thedashed white lines. Local periodicity, a, as a function ofposition is shown (red). Losses reduce the field strength

along the array in (a), whilst showing that the lossy lens (b)preserves the point spread function.

shapes within the differing Brillouin zones. This methodol-ogy can be trivially extended to finite 3D devices, although anextra degree of freedom presents itself in an electromagneticsetting as the relative polarisations of the guided leaky modesbecome important. As such these devices have the capabilityof acting as polarisation selective Umklapp lenses. At presentthe proposed devices are purely passive, but there is scope toachieve active components by utilising piezoelectric materialsto alter the periodicities and as such, in theoretically demon-strating this new lensing mechanism, we envisage motivationtowards experimental verification.

ACKNOWLEDGEMENTS

The authors thank the UK EPSRC for their support throughProgramme Grant EP/L024926/1, grant EP/T002654/1, anda Research Studentship. R.V.C acknowledges the support ofthe Leverhulme Trust and of the European Union FET Open,grant number 863179, Boheme. The authors also thank Yao-Ting Wang for helpful conversations.

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