Plano-concave microlenses with epsilon-near-zero surface-relief coatings for efficient shaping of nonparaxial optical beams

1.7 MB·

by  Mahin Naserpour

Plano-concave microlenses with epsilon-near-zero surface-relief coatings
ffi
for e cient shaping of nonparaxial optical beams
MahinNaserpoura,b,CarlosJ.Zapata-Rodr´ıgueza,∗,MahdiehHashemic
aDepartmentofOpticsandOptometryandVisionScience,UniversityofValencia,Dr.Moliner50,Burjassot46100,Spain
bDepartmentofPhysics,CollegeofSciences,ShirazUniversity,Shiraz71946-84795,Iran
cDepartmentofPhysics,CollegeofScience,FasaUniversity,Fasa7461781189,Iran
Abstract
7
1Epsilon-near-zero (ENZ) materials, including artificial metamaterials, have been advanced to mold laser beams and antenna-
0mediatedradiatedwaves. HereweproposeanefficientmethodtocontrolOhmiclossesinherenttonaturalENZmaterialsbythe
2assemblyofsubwavelengthstructuresinanonperiodicmatrixconstitutinganENZmetacoating. Implementedoverplano-concave

ntransparent substrates whose radius can be of only a few wavelengths, ENZ surface-relief elements demonstrate to adequately
ashape a plane wave into highly localized fields. Furthermore, our proposal provides an energy efficiency even higher than an
J
ideally-losslessall-ENZplano-concavelens. Ourprocedureissatisfactorytogenerateaberration-freenonparaxialfocusedwaves

1andacceleratingbeamsinminiaturizedspaces.
1
Keywords: Nonparaxialbeamshaping,Epsilon-near-zeromaterials,Subwavelengthstructures.

]
s
c
ti1. Introduction increasedefficiencyprovidedthattheslabthicknessremainsin
p
thescaleofthewavelength[20,21]. Nevertheless,theirplanar
o Dielectric microlenses used for imaging and focusing in structurecircumscribesthefocalarchitecturetolowandmod-
.
sminiaturized optoelectronic devices are currently prospects to eratenumericalaperturesthusrestrictingtheachievableresolu-
c
ibe substituted by metalenses and metasurfaces exhibiting an tionpoweroftheopticaldevice.
sextraordinaryopticalperformancewithinnotablyreducedvol- Here we propose the use of epsilon-near-zero materials ad-
y
humes. Suchengineeredmaterialsincludeelementaryscatterers heredtotheterracedexitsurfaceofplano-concavetransparent
psuchasnanoslits[1,2,3]andnanoholes[4,5,6],split-ringres- substratestoefficientlyshapeaplanewaveintoanaberration-
[onators [7, 8], and V-shaped nanoantennas [9, 10], to mention freenonparaxialfocusedwave. Ourprocedureisalsoadequate

1 afew,enablingthesimultaneouscontrolofthephasefrontand togeneratenonparaxialacceleratingbeamswithBesselsigna-
vpolarizationofanincomingplanewave. Infact,beamshaping tureinminiaturizedspaces[22].Wedesignsurface-reliefstruc-
0based on holographic metamaterials may be further applied to turesusingbothmetalsanddopedsemiconductorsasENZma-
5ageneralclassofwavefields[11]. terialsspanningthemid-IR,visibleandUVspectralbands.
8 In particular, epsilon-near-zero (ENZ) materials have been
2
proposed theoretically [12, 13, 14] and demonstrated exper-
0
imentally [15, 16] for far-field light focusing due to its en- 2. ENZplano-concavelensestofocusandaccelerateoptical
.
1hancedwavedirectivityexperiencedatboundarieswithdielec- beams
0
tricenvironments. SincenophasedelayoccursinsidetheENZ
7 Aplano-concaveENZlenscantransformaplanewaveinto
1medium, one can manipulate the phase radiation pattern of a
a cylindrical converging wave field, as depicted in Fig. 1(a).
:given impinging phase front and transform it into a desired
v Here we adopted a two-dimensional geometry where the con-
shape by properly sculping the exit surface of the ENZ slab.
i cavesurfacehasanaxisofrevolutionatC;notethataconverg-
XForphotonicapplications,however,theintrinsiclossesofnatu-
ingsphericalwavefrontcanbeobtainedbyusingconcavesur-
rralDrudematerialsatitsplasmafrequencyandofartificialma-
aterialsdesignedtoundergoazeroeffectivepermittivityseverely faceswhosecenterofsymmetryissetatthefocalpoint. Asim-
pleinterpretationcanbefollowedbyusingopticalrays. When
restrictstheiruseatpropagationdepthsofseveralwavelengths
acollimatedbeamimpingesnormallytothefrontplanesurface
[17]. Evenusingall-dielectriczero-indexphotoniccrystals,the
oftheENZlens,theopticalraysdoesnotundergoanydeviation
lens performance can be notably undermined due to radiative
initstrajectory. Ontheotherhand,lightemergesfromanENZ
losses, interface losses, and imperfections in the nanofabrica-
material forming a zero angle of refraction with respect to its
tion [18, 19]. Zoned plates are then good candidates with an
normal line. This can be straightforwardly deduced by apply-
ing the Snell’s law over the exit surface of the plano-concave
∗Correspondingauthor: lens. Therefore, an ENZ material bounded by a surface with
Emailaddress:[email protected](CarlosJ.Zapata-Rodr´ıguez) centerofcurvatureatthepointC willrefractlightindirection
PreprintsubmittedtoOneJournal January12,2017
Figure1: Plano-concavefocusingdevicesofhighnumericalaperturemadeof
ENZmaterials.(a)Perfectlysphericalexitsurfaceenablestocreateaberration-
freenonparaxialfocalfields. (b)Angular-resolvedcurvaturetogeneratecon-
centriccausticfields.
tosuchapoint. Smalldeviationsofthecurvaturewillproduce
aberratedfocalfieldsthat,ingeneral,areundesirable.
For simplicity, let us consider light rays propagating in the
incidence plane, that is the xy plane, which is perpendicular
to the axis of the cylindrical concave surface of the lens. If
the function R = R(θ) represents the curve at the exit surface
in polar coordinates, as drawn within the incidence plane, the
trajectory of a light ray is driven by the normal vector to such
curve, N = −rˆ+θˆR(cid:48)/R, whererˆ andθˆ areunitaryvectorsand
R(cid:48) =∂R/∂θisafirstderivativeevaluatedatthepointofinterest
Figure2:Time-averagedenergydensitygeneratedbyaplano-concavelossless
overthecurve. Definingδastheangleofthetrajectoryofthe
ENZlensimmersedinair,providedthattheincidentplanewaveispolarized
lightraywithrespecttotheunitaryvector−rˆ,thelattergiving along the y axis, that is TM-polarized, at a wavelength λ = 326 nm. The
theraytrajectoryinthecaseofaperfectcylindricalsurfaceas exitsurfaceis(a)perfectlycircularwithR(cid:48) = 0,wherethesymbol×denotes
illustratedinFig.1(b),wefinallyobtainthat thecenterpointC;anditisshapedbythefunctiongiveninEq.(2)with(b)
R(cid:48) =R0/10,(c)R(cid:48) =R0/5,and(d)R(cid:48) =R0/2.InallcasesR0 =10λ.Insetin
|R(cid:48)| (a):Apodizingtime-averagedenergydensityalongtheconcaveexitsurface.
tanδ= . (1)
R
An analogous interpretation can be given in terms of wave
PositivevaluesofR(cid:48) willmakethelightraystopassbelowthe
optics. The phase shift accounted in the optical path of rays
pointCthusformingashadowregion.
travelingfromtheconcavesurfacetothefocusCisgivenby
Controlled curvature engineering can give, as a result, pat-
terned fields near the focal point. As an example illustrated exp(−ikR)=exp(−ikR )exp(cid:0)−ikR(cid:48)θ(cid:1), (3)
0
in Fig. 1(b), let us consider an exit surface with azimuthally-
varyingradiusofcurvatureRgivenby where k = 2π/λ is the wavenumber in vacuum. The
term (−ikR ) is related with the phase shift produced by an
0
R(θ)=R +R(cid:48)(θ−π/2), (2) aberration-free converging lens. Assuming that R(cid:48) (cid:28) R ,
0 0
exp(−ikR(cid:48)θ)canbesetasalinearphaseshiftexp(−iklR(cid:48)/R ),
0
where R stands for the initial radius at θ = π/2 as measured wherelrepresentsanspatialcoordinateasthearclengthmea-
0
from the curvature centerC and R(cid:48) is here a constant term in- suredalongthereferencesurfaceofradiusR . Thusthewave-
0
dicatingtherateofangularvariationofthesurfaceradius. As front emergingfrom the vicinitiesof a point P of theexit sur-
notedabove,suchcurvaturedeviationoftheexitsurfaceleads face will cross the reference surface obliquely, undergoing an
toaprismaticeffectonalltheemerginglightrays. Letusfirst angular deviation δ = −R(cid:48)/R . This is in agreement with the
0
assume a modest deviation of the perfectly circular symmetry factthatδalsodenotestheangulardeviationofthenormalline
so that |R(cid:48)| (cid:28) R . Now it is clear that a ray emerging from a totheexitsurfacewithrespecttothesegmentPC. Iffinallywe
0
givenpoint Poftheexitsurfacewillundergoanangulardevi- includethegiventermoftheformexp(imθ),denotingthefield
ationδ=|R(cid:48)|/R withrespecttothesegmentconnectingPand in the reference surface, in a diffraction integral (for instance
0
C. Asaconsequence,therayspropagatinginthe xy-planewill theDebyediffractionintegral[23])toestimatethefielddistri-
generateacausticcurvearoundthefocalpointC,whichiscir- butionnearthefocalpointC, whereinourcasetheparameter
cularwithcenteralsoatC andwitharadiusofcurvaturegiven m = −kR(cid:48), it might be demonstrated that the amplitude of the
by|R(cid:48)|. diffracted field results in direct proportion to exp(imθ)J (kr)
m
2
nearthecausticcurve,whereristheradialcoordinateandJ (·)
m
istheBesselfunctionofthefirstkindandorderm;afulldemon-
strationcanbefoundinRef.[24]. Suchwavefunctionhasan
intensitywithmaximumaroundC atadistanceapproximately
givenbyr =|m|/k,thatis|R(cid:48)|.Thelatterisconsistentwithour
m
previousdescriptiongivenintermsoflightrays.
Toillustratethecapabilitiesofaplano-concaveENZlensto
focus and accelerate an incident plane wave, we numerically
estimate the scattered wave field by means of a commercial
full-wavesolveroftheMaxwell’sequations(COMSOLMulti-
physics)basedonthefiniteelementmethod(FEM).InFig.2(a)
we show the spatial distribution of the time-averaged energy
densitywhenaplanewavethatislinearlypolarizedalongthey
axisimpingesnormallyontheflatentrancesurfaceoftheENZ
lens. We consider a wavelength λ = 326 nm for which the Figure3: Focaldistributionofthetime-averagedenergydensityproducedby
dielectric constant of silver has its real part dropping to zero. aplano-concavesilverlensofradiusR0 = 10λatλ = 326nm. Theelectric
fieldoftheincidentplanewaveisorientedalongtheyaxis. In(a)thereisno
In this case, the concave surface is perfectly circular, that is
modulationoftheradius,R(cid:48) = 0,andin(b)themodulationisdeterminedby
R(cid:48) = 0, with a radius R0 = 10λ. For numerical purposes, we R(cid:48) =R0/10. Colorscalinginthefocalregionreferstoanenergyrangelower
firstconsideraslightlylossyENZmaterialwithrelativepermit- thaninFig.2tobetterdepictthefieldpattern.
tivity(cid:15) =0+i0.001. Inthefigureweobservetheformationof
alightspotcenteredatCwithaFWHMof184nminthetrans-
materialwillmakeadifferenceasdiscussedbelow.
versedirection,thatistheyaxis,andawidthof504nmalong
the optical axis. This means that the effective numerical aper-
ture(NA)isapproximately0.90,inagreementwiththefactthat 2.1. ApplicationtonaturalENZmaterials
theeffectiveangularsemi-apertureoftheplano-concavelensis
64◦[25].Suchdecrementoftheangularsemi-aperturefromthe Letusapplythepreviousconceptsofbeamshapingtoare-
optimalπ/2radians, asshownintheinsetofFig.1(a), should alistic ENZ material. For that purpose we consider silver at a
beattributedtoasignificantreductionofthelenstransmittance wavelength of λ = 326 nm, for which its permittivity has a
atangles|θ| ≈ π/2. Asaconsequence,focalwavesundergoan vanishingrealpart. However,theimaginarypartoftherelative
apodizingeffectwheresidelobesarenotablyreduced,however, permittivityofsilvercannotbeneglected,whichisexperimen-
remaining its central peak with an slightly greater width than tally estimated as Im((cid:15)) = 0.7 [17]. In Fig. 3(a) we show the
potentiallyexpected. profile of the time-averaged energy density of the electromag-
InFig.2(b)wemodulatetheradiusR(θ)ofthelensexitsur- neticfieldfocusedbyaplano-concavesilverlenswhoseradius
facesuchthatR(cid:48) = R0/10. Asdiscussedabove, wemayform of curvature is R0 = 10λ. In comparison with Fig. 2(a), we
anacceleratingbeamwithcirculartrajectoryaroundacircum- firstobserveacleardecreaseofthetransmittedfield,achieving
ferenceofradius∼ λ. Notethattheaccelerationoftheoptical areductionof58%atthefocalpoint,afactthatisattributedto
beamcanbeproducedabovethepointC providedthatR(cid:48) < 0. absorptioninthelossymetal. Infact,themaximumconcentra-
However, the trajectory of the beam is shorter than expected; tionofenergydensityisnotlocatedatC butdisplacedtowards
notethatacausticcurvewithangulardistributionofπradians thelens,aneffectthatiscommonlycoinedasfocalshift[26],in
wouldhypotheticallybeformed,asinferredfromthegeometric agreementwiththefactthattheeffectivenumericalapertureof
picturegiveninFig.1(b). Thedecreaseoftheeffectiveangular thelenshasbeenreducedto0.085(semi-apertureangleofonly
apertureoftheplano-concavelensisobviouslyaffectingtothe ∼5◦),asillustratedintheinsetofFig.3(a). Ontheotherhand,
trajectorylengthoftheacceleratingbeam[24]. theresolutionpowerhasdroppedconsiderably; theFWHMin
Wepointoutthattheaccelerationofthefocusedfieldisalso thetransversedirectionisnow751nm.Wemightconcludethat
produced even when the condition R(cid:48) (cid:28) R is not satisfied. thelimitofresolutionisnotdeterminedbytheangularaperture
0
In Fig. 2(c) and (d) we represent the intensity distribution of of the concave exit surface, but the metallic losses which are
themagneticfieldscatteredbyaplano-concavelenswithhigh evidencedinthethicklens.
valuesoftheparameter(c)R(cid:48) =R /5and(d)R(cid:48) =R /2. From In Fig. 3(b) we analyze the scattered field produced by a
0 0
ourFEM-basednumericalsimulations,theradiusofthecaustic plano-concave lens with an angular-resolved curvature given
curveforthefirstcaseisapproximately2.0λ,whereasitsradius by R(cid:48) = R /10. Essentially, we cannot recognize a different
0
increases to 5.1λ for the case (b), which roughly are equal to responseincomparisonwiththecase showninFig.3(a). The
|R(cid:48)|. Theseexamplesdemonstratetheeffectiveimplementation expected acceleration of the field is fundamentally carried out
oftherequiredmanipulationofthephasefront. atoff-axisanglesθ (cid:44) πwherethelenspresentsahigherthick-
Finallywepointoutthatthepolarizationoftheincidentplane ness, and therefore dissipative effects are largely manifested.
waveisreflectedinaneglectingdeviationofthespatialdistri- Althoughnotshowninthefigures,highervaluesofR(cid:48) indicat-
bution of the focal waves shown in Figs. 2(a)-(d). However, ingfastermodulationofthesurfacecurvatureprovidebarelya
a nonvanishing imaginary part of the permittivity of the ENZ similarresult.
3
Figure4:Schematicrepresentationofthemulti-prismENZnanostructurecoat-
ingtheterraced-concavesurfaceofthedielectricthicklens, whoseindexof
refractionisn. Inset: Zoom-inviewaroundasingleENZsubwavelengthele-
mentcomposingtheultrathincoating. Here,theincidentwavefieldinteracts
attheentrance(vertical)surfaceandthebeamemergesfromtheexitcurved
Figure5: Time-averagedenergydensitygeneratedbyaplano-concave(PC)
surface.
lenswithsurface-reliefsilvercoatingsculpturedwithanangular-resolvedcur-
vatureR0 = 10λand(a)R(cid:48) = 0,(b)R(cid:48) = R0/10. Again,theincidentplane
waveisTM-polarizedatawavelengthλ=326nm.
IntheFEM-basednumericalsimulationsshowninFig.3(a)
and (b) we considered TM-polarized plane waves, that is the
electricfieldisorientedalongtheyaxis, whichareimpinging
wavefront within the focal region. Finally, a reduced Ohmic
overtheplano-concavemetalliclens.TE-polarizedplanewaves
lossinthetransitofthewavefieldthrougheachENZelement
particularlyprovideinferiorfocalizationandaccelerationdueto
issupportedbyitssubwavelengthwidth.
a lower transmittance. Such a distinct behaviour is ultimately
caused by the nonvanishing imaginary part of the permittivity For the sake of illustration, we first consider a FEM-based
characterizingtheENZmaterial. numerical simulation showing the scattered field of a surface-
Finally, alternate ENZ materials like noble metals at the relief ENZ metacoating containing 32 elements, which is de-
plasma frequency (and anisotropic metamaterials) can be ana- positedontheterracedexitsurfaceofadielectricplano-concave
lyzedfortheuseofplano-concavelensestogeneratenonparax- lens. If the radius of the device is R0 = 10λ, the index of re-
ial focused waves and accelerating beams at different wave- fractionoftheglasssubstrateisn = 1.5,andtheENZmaterial
lengths. However, the inherent lossy characteristics of them issilver(weuseawavelengthλ=326nmwhereIm((cid:15))=0.7),
makesthetargetedbeamshapingimpracticable,evenforalens theresultanttime-averagedenergydensityofthefocusedwave
thicknessofafewwavelengths. Forthatreason,itispreferably field is shown in Fig. 5(a). Therefore, each prismatic element
tousealternatearchitecturesincludingultrathinENZlayersto of the ENZ silver assembly has a width of only 217 nm. We
activelymoldthewavefrontoftheincidentplanewave. point out that a higher number of pieces can be used to com-
pletely cloak the terraced-concave surface of the lens, but in
practical term we may restrict such number with a negligible
3. ImplementationofultrathinENZmetacoatings
loss of electromagnet energy due to the inherent apodization
We propose to substitute the ENZ plano-concave lens by a effectofhigh-numericalaperturegeometries, asshownabove.
lossless dielectric plano-concave lens, which inherently is di- In the numerical simulation we observe the formation of a fo-
vergent, also including a surface-relief ENZ ultrathin coating. cused field with a FWHM of 198 nm in the geometrical focal
The latter will perform the necessary steering of the incident planeand512nmalongtheopticalaxis,demonstratinganear-
plane wave to generate either the focused field or the acceler- optimalresolutionperformance. Wepointoutthatamoderate
atingbeam. Sinceinadditionthethickdielectriclenswillnot focalshiftisobservedwithanincreasedenergydensityof53%
absorbtheelectromagneticenergyoftheimpingingwavefield, andareducedtransverseFWHMof175nm. Furthermore,the
weareabletooptimallymaintaintheeffectivenumericalaper- focalenergydensityismultipliedbyafactorof∼19whenitis
tureofthefocusingarchitecture.AsdepictedinFig.4,theENZ comparedwiththefocusingarrangementcomposedofasilver-
elements of the coating have a straight (vertical) side where only plano-concave lens. Surprisingly, we also compared the
the plane wave is coupled by normal incidence, whereas they energyefficiencybynumericallydisregardingthemetallosses
also exhibit a designed curved side where the beam emerges ofthesilver-onlyplano-concavelens;insuchacasetheenergy
to be focused and accelerated, thus producing a prismatic ef- density is multiplied by a factor of ∼ 1.9. Therefore, our pro-
fect. Furthermore, thewidthofeachENZelementisgivenby posalprovidesahigherenergyefficiencythanalosslessplano-
λ/n,wherendenotestherefractiveindexoftheplano-concave concavelens. Finally,asoccursabove,ourresultobtainedfora
dielectric lens. In this case, the incident field will undergo a TM-polarizedincidentplanewaveisnotablebetterthanassum-
phaseshiftof2πradianswhenitiscoupledinadjacentENZel- ingaTE-polarizedimpingingfield(notshowninthefigure).
ements, enabling in-phase interference of the divided incident Tofurtherinspectthebehaviorofthecoatedmicrolenseswe
4
ofFig.7.Forthesakeofillustration,oncemoreweconsiderthe
plano-concavelenscoatedbyn-CdO,whichbecomesENZata
wavelengthofλ=1870nm. Asaresult,thedividedwavefront
willbeindividuallybentineachENZelementwithadifferent
angle, thus constructively interfering in the focal region along
the caustic curve. However, such caustic curve is now a reg-
ularpolygonapproachinganincompletecircle,whichleadsto
afractionalTalboteffect[28]. Notethatinthenumericalsim-
ulations shown in Fig. 7, the plano-concave dielectric lens is
marginallythinnerincomparisonwiththatshowninFig.6(c),
insuchawaythatthegeometricalnumericalapertureisslightly
reduced in order to account for the ENZ elements making the
terraced-concavesurfaceopticallyactive. Again,bycomparing
thedistributionofthetime-averagedenergydensityinthefocal
Figure7:Spatialdistributionofthetime-averagedenergydensitygeneratedby
aplano-convacelenscoatedbyn-CdOmicroprisms(aszoomed-inintheinset) region of the original architecture shown in Fig. 6(c) and the
atawavelengthλ=1870nm. simplified model depicted in Fig. 7 using prismatic elements,
it is evident that the pattern differences on the generated non-
paraxialacceleratingbeamsarenegligible.
design the ENZ elements to form a nonparaxial accelerating
beam. Again,thewidthoftheseelementsareλ/nandthever-
4. Conclusions
ticalentrancesurfaceenablestheincidentTM-polarizedplane
wavetoperfectlycoupletothesilverassembly. Inthiscase,we Insummary,wenumericallyinvestigatetheefficientgenera-
modulate the exit surface following the equation given in (2),
tionofnonparaxialfocusedwavefieldsandacceleratingbeams
provided that R(cid:48) = R /10, and again R = 10λ. Figure 5(b)
0 0 withBesselsignatureusingENZmaterials.ENZ-enabledbeam
clearly shows the acceleration of the beam around the central shaping is inherently based on a surface effect and therefore
pointC, contrarilytowhatoccursforfull-ENZplano-concave
bulk lossy devices are here substituted by transparent mate-
lenses(seeFig.3(b)).
rials which in addition include an ENZ subwavelength coat-
In Fig. 6(a) we show the focal distribution of the time-
ing. We demonstrate that the locally steering of the beam
averaged energy density of the scattered field for an increased
canbestraightforwardlyexecutedbyanassemblyofsubwave-
spatialacceleration. NowtheFEM-basednumericalsimulation
length ENZ prisms with tailored hypotenuse surface, suitably
is carried out for a silver assembly which is patterned by fol-
depositedontheterraced-concavesurfaceofahigh-numerical-
lowingaradialmodulationR(cid:48) =R /2. Ourresultsdemonstrate
0 aperture lens made of a lossless dielectric. In practical terms,
the feasibility of the procedure in good agreement with those
thecurvedsurfaceoftheENZsubwavelengthprismsconstitut-
obtained with a lossless ENZ microlens, previously shown in
ingthesurface-reliefcoatingoftheplano-concavelenscanbe
Fig.2(d).Furthermore,suchanstrategycanbeimplementedby
flattened for manufacturing purposes, still providing satisfac-
usingothersortofENZmaterials.LetusnowconsiderAl:ZnO,
toryresults.Inthisstudyweusednoblemetalsanddopedsemi-
whichisadopedsemiconductorwhosepermittivityhasavan-
conductorsasENZmaterialsintheUV,visibleandnear-IR,but
ishingrealpartatλ=1355nm;inthiscaseIm((cid:15))=0.31[17].
artificialENZmetamaterialscanbeusedtoextendtheapplica-
Whenthemodulationoftheradiusofcurvatureisagaingiven
bilityoftheproposedarrangementinabroaderelectromagnetic
by R = 10λ and R(cid:48) = R /2, keeping as n = 1.5 the index
0 0 spectrum[29,30,31]. Wepointoutthatourproposalprovides
of refraction of the dielectric thick lens, the resultant acceler- anenergyefficiencywhichishigherthanthatalosslessall-ENZ
ating beam is represented in Fig. 6(b). Roughly speaking, the plano-concave lens can offer. As concluding remark, our pro-
beam shaping is mimetically reproduced as expected from the
cedurecanalsobeusedtogenerateAirybeamsintheparaxial
scalingpropertiesoftheMaxwellequations[27]. Tocomplete
regime[32],acceleratingnonparaxialbeamswithBesselsigna-
ournumericalanalysis,wefinallyoperatewiththeENZn-CdO ture[33],andevennondiffractingBesselbeamwithasymmetric
occurring at a wavelength of λ = 1870 nm, where in addition
transverseprofile[34].
Im((cid:15)) = 0.127 [17]. Due to the reduced Ohmic losses of the
dopedsemiconductor, theenhancedqualityofthenonparaxial
Acknowledgments
acceleratingbeamisevident.
The robustness of our procedure to focus and accelerate a
This work was supported by the Spanish Ministry of Econ-
plane wave within an area of a few µm2 will be demonstrated
omyandCompetitivenessundercontractTEC2014-53727-C2-
next. Forthatpurpose,wesubstituteeachENZcurvedelement
1-R.
ofthemetacoatingbyarectangularprism,insuchawaythatthe
baseagainhasalengthofλ/n,aswellasthelateralfacewhere
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Figure6: Generationofhighly-acceleratingbeamsusingplano-concavelenseswithsurface-reliefcoatingcurvaturegivenbyR0 = 10λandR(cid:48) = R0/2. Inthe
FEM-basednumericalsimulations,thetime-averagedenergydensityatthefocalregionisevaluatedbyusingENZmaterials:(a)Agatλ=326nm(Im((cid:15))=0.7),
(b)Al:ZnOatλ=1355nm(Im((cid:15))=0.31),and(c)n-CdOatλ=1870nm(Im((cid:15))=0.127).
7