Reflective metasurface lens with an elongatedneedle-shaped focusMEHDI VEYSI, CANER GUCLU, OZDAL BOYRAZ, AND FILIPPO CAPOLINO*Department of Electrical Engineering and Computer Science, University of California, Irvine, California 92697, USA*Corresponding author: f.capolino@uci.edu

Received 21 October 2016; revised 7 December 2016; accepted 7 December 2016; posted 15 December 2016 (Doc. ID 279188);published 18 January 2017

Novel multifocal flat metasurface (MS) lenses are developed using two techniques: (i) polarization diversity, and(ii) annular segmentation of the lens aperture. The polarization-diversity technique enables overall lens aperturereuse, thus doubling the number of foci through the simultaneous focusing of two orthogonal linearly polarizedincident beams at two distinct foci using the lens aperture. The annular-segmentation technique, on the otherhand, is independent of incident beam polarization and is only based on dividing the lens aperture into concentricannular segments that converge different portions of the illuminating beam at different foci. The total number offoci can be further increased by combining the polarization-diversity and the annular-segmentation techniques.Subsequently, the concept of multifocality is further extended to design a novel flat lens with an overall singleneedle-like focal region with elongated depth of focus (DOF) without loss of lateral resolution. To this goal, wedesign a multifocal lens with overlapping profiles of foci superposed into a single elongated needle-shaped focalregion. Using the combination of polarization-diversity and annular-segmentation techniques, we develop a novelMS flat lens made of Y-shaped nanoantennas, whose polarization-dependent reflection phase and amplitude canbe controlled independently via their geometrical parameters. Via numerical calculations, we demonstrate thatthe proposed MS lens has an overall single focal region with an extremely long DOF of about 74.1 λ, a lateralfull width at half maximum varying in the range of 1.37 λ to 2.8 λ, and a numerical aperture of about 0.26 (con-sidering the center of the focal region as the effective focal point). Here, the MS lens’s capability to synthesizeextremely long DOF is conceptually demonstrated without resorting to time-consuming and complicatedwavefront synthesis methods. The engineering of focal intensity profiles with flat MSs may introduce significantadvancement in nanoimaging, many areas of microscopy, and ophthalmic applications. © 2017 Optical Society of

America

OCIS codes: (160.3918) Metamaterials; (250.5403) Plasmonics; (050.6624) Subwavelength structures.

https://doi.org/10.1364/JOSAB.34.000374

1. INTRODUCTION

Increasing depth of focus (DOF) without losing lateral spatialresolution has been a classical challenge in optical systems suchas data recording systems and microscopy [1–9]. Particularly, ashallow DOF limits the capability of the imaging systems forthick specimens. Therefore, such systems usually use an “opti-cal sectioning” technique [4,5,10] by moving the specimenalong the beam axis. Different extended DOF algorithms arethen employed to restore a single image of the specimen from arange of images taken at different positions of the specimenalong the beam axis [4,5,11]. Another approach to overcomethe shallow DOF barrier is to use a variable focus hologramtogether with a restoring extended DOF algorithm while keep-ing the specimen position fixed on the beam axis [6,7]. To re-duce the complexity of such focal scanning systems, a particular

class of lenses with a single focal region featuring a narrow lateralwidth and an elongated DOF is employed for data recordingsystems and microscopy [1,12–16]. Such lenses with a longDOF provide a longer interaction range between the focusedbeam and the specimen in optical systems and therefore remedythe problems caused by the specimen being out of focus. In thisregard, the use of optical power absorbing apodizers has beenthoroughly examined to increase the DOF in optical systems[15,16]. It has been also demonstrated that upon focusing op-tical vector beam modes with annular intensity shape possess-ing radial [17–20] and azimuthal [21,22] polarizations througha conventional lens, a focal region is achieved with a very longDOF and narrow lateral full width at half maximum (FWHM).The narrower the annular width of the incident radiallypolarized beam, the longer the DOF [17,20]. However, these

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methods usually suffer from a decrease of the optical power inthe focal plane due to the partial blockage of the illuminatingbeam. Recently, a method based on optical “super-oscillations”(see Ref. [23] for more details) has been developed to generate aneedle-shaped focus with a spot size smaller than the diffractionlimit and a DOF of ∼15 λ [24]. However, this techniqueusually suffers from high intensity side lobes surrounding thefocal region.

In this paper, we demonstrate that a single elongated focalrange with a very narrow lateral FWHM is obtained by gen-erating multiple overlapping foci using, for the first time to theauthors’ knowledge, a multifocal metasurface (MS) lens. Thetechniques described here together with its realization usingMS technology (which provides independent control of phase,amplitude, and polarization of the illuminating beam) wouldresult in an extended DOF with reduced power loss. The onlysource of the power loss in our proposed method is the metallosses, which cannot be neglected when operating in the visiblespectrum or at infrared when nonnoble metals are used. Notethat a multifocal lens by itself is an important optical compo-nent and is at the heart of many optical devices, such as multi-focal plane microscopy [25–27], dual field-of-view opticalimaging systems [28], DVD drivers [29,30], optical coherencetomography imaging systems, and intraocular contact lenses[31]. A conventional bifocal three-dimensional (3D) lens isrealized by dividing the lens aperture into two sublenses: (i)a central zone with focal distance f 1 and (ii) a peripheral con-centric zone with focal distance f 2 greater than f 1 [32], inwhich the incident beam illuminates the central and peripheralzones of the lens with a specific power proportion. In addition,conventional multifocal lenses are usually designed based on adiffractive optics approach and have found their use in manypractical applications [33,34].

The novel multifocal MS lenses proposed here have a flat,compact configuration and therefore are easier to fabricateand integrate in nanophotonic systems as compared to theirconventional bulky counterparts [28–34]. In addition, theyprovide more functionality by allowing independent tailoringof not only the local phase and amplitude but also the polari-zation of the beam [35,36]. In order to design the proposedmultifocal MS lenses, two techniques are investigated here:(i) polarization diversity [simultaneous focusing of twoorthogonal linearly polarized incident beams at two distinctfoci, as shown in Fig. 1(a)] that uses the overall lens aperture,and (ii) annular segmentation [dividing the lens aperture intomultiple concentric sublenses that converge the incident beamat distinct foci, as shown in Fig. 1(b)]. We also describe a novelcombination of these two techniques either with spatially sep-arated foci [see Fig. 1(c)] or with a single needle-like focal re-gion with an elongated DOF. Recently, with the availability ofplanar fabrication technologies, various flat optics devices havebeen developed and have received increasing attention inmodern optics and photonics [37–44]. In particular, flat mono-focal MS lenses comprising nanoantennas have been extensivelyinvestigated in the literature [39,45–47]; however, such designsusually have a shallow DOF. It has been recently demonstratedthat a flat MS lens forms a single focal range with a DOF ofabout 21 λ and a lateral FWHM of 1.8 λ (at λ � 425 μm) byproperly tailoring the local phase and amplitude of the scattered

field from the MS lens through an iterative phase-amplitudeoptimization algorithm [48]. However, due to the large numberof elements on the MS lens, more than hundreds of elements, itwould be difficult and time consuming to optimize the phaseof reflection/transmission coefficient at all MS elements usinglocal or global search algorithms.

Here, we employ the flat MS lens technology due to itsflexibilities in the engineering of the focal intensity profileby spatially tailoring the local phase, amplitude, and polariza-tion state of the incident beam. Note that, in principle, MSscan be realized with either metallic or dielectric nanoantennas.First, we design a novel bifocal flat MS lens based on the firsttechnique, the polarization-diversity technique illustrated inFig. 1(a). A slant-polarized incident beam (here representedas a superposition of two x 0-polarized and y 0-polarized beamssuch that the projections of the incident electric field on thex- and y-axes are of equal amplitude and phase) illuminatesthe bifocal MS lens. The angle of incidence, defined as theangle between the beam axis z 0 and the MS normal z, is α. Thelens focuses the two orthogonally polarized linear components

(a)

(b)

(c)

Fig. 1. Schematic of representative multifocal MS lenses based on(a) polarization-diversity technique, (b) annular-segmentation tech-nique (in this representative example, the lens aperture is divided intotwo concentric annular segments, central and peripheral segments,focusing the illuminating beam onto two well-separated foci), and(c) combined polarization-diversity and annular-segmentation tech-nique. The illuminating beam is defined in the primed coordinate sys-tem, and the x 0-axis is aligned with the x-axis while the angle betweenthe z-axis and the z 0-axis is α. The subscripts x and y refer to the elec-tric field polarization direction, and the superscripts c and p refer to thecentral and peripheral MS segments, respectively.

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of the incident beam at two spatially separated foci whoseseparation is determined by the design. Upon choosing theproper separation distance of these two orthogonally polarizedfoci, a needle-like intensity profile forms along the axis ofpropagation with a very fine lateral resolution and an extendedDOF. The simple design procedure makes this method prom-ising for applications which are sensitive to intensity rather thanpolarization, such as microscopy and data storage devices.

Next, the multifocal lens concept is further developed viathe second technique discussed here, the annular-segmentationtechnique as illustrated in Fig. 1(b), in which the lens aperturedivides into separate concentric annular sub-MSs. Differentportions of the incident beam illuminate different sub-MSs andthus are focused into different closely spaced/separated foci. Inconventional multifocal diffractive lenses [33], foci are symmet-rically distributed and equally spaced on the beam axis aroundthe zeroth diffractive order focus, which is the focus corres-ponding to the rays diffracted according to the conventionalSnell’s law. In contrast, the foci locations and separations forthe flat multifocal circular lens proposed in this paper can bechosen arbitrarily in three-dimensional space. Furthermore,a lens with an extraordinarily elongated DOF is realized withthe combination of the two techniques just described, polari-zation diversity and annular segmentation, in such a way thateach sub-MS exhibits two foci for two orthogonal linearly po-larized components of the illuminating beam [see Fig. 1(c)].Simulation results show that the proposed multifocal lens de-sign procedure, albeit simple and time-effective, is a great can-didate for the design of various versions of multifocal lenses andmonofocal lenses with elongated DOF. Note that although wehere utilize the reflection-type geometry due to its high effici-ency [49–52] and polarization conversion efficiencies [53], thetechniques proposed in this paper (see Fig. 1) can be straight-forwardly extended to the transmission-type geometry usinghighly efficient metallic and dielectric MSs [40,54–58]. Further-more, the techniques proposed in this paper for elongating theDOF of the lens may also be beneficial for other focusing typessuch as antiresolution focusing [59] (i.e., generating an emptylight capsule) and require further exploration in a future study.

2. POLARIZATION-DIVERSITY TECHNIQUE

First, the concept of the polarizing lens introduced in Ref. [46]is extended here to the investigation and development of bifocallenses. Let us denote the coordinates of the center of MS cellswith rmn � r00 � max� nby, where m and n are integersthat denote MS cells, and a and b denote the unit cell periodsalong the x and y directions, respectively. In order to refract anincident ray propagating with the wave vector ki onto a point-like spot at rf , the required local phase of the reflection coeffi-cient at themnth MS cell, denoted by ϕmn, can be found as [46]

ϕmn − �k0jrmn − rf j � rmn · ki � � 2sπ � ψ ; (1)

where s � 0;�1;… is an arbitrary integer and ψ is an arbitraryphase offset, constant with varying n and m. Note that thisequation holds for both x and y polarizations, and differentrf can be chosen depending on different polarizations. Theproposed flat MS lens consists of anisotropic Y-shaped nano-antennas. The advantages of these Y-shaped nanoantennas

compared to the previously employed V-shaped designs asin Refs. [40,45] consist in having an extra degree of freedom(stub length) that helps in balancing the reflection intensities ofx- and y-polarized waves and having two different and indepen-dent reflection phase profiles for two orthogonal linear polar-izations. The Y-shaped nanoantennas are made of Al andpatterned on one side of a silica substrate with thickness of400 nm, while a sufficiently thick aluminum layer is depositedon the other side of the substrate to act as a ground plane. Inthis paper, the operating wavelength of the MS lens is 4.3 μm.The complex relative permittivities of the Al and SiO2 substrateare also provided in the caption of Fig. 2. Figure 2(a) showsthe geometry of the Y-shaped nanoantenna together withthe current paths for symmetric and antisymmetric resonancemodes that correspond to y- and x-polarized electric fields, re-spectively. While tuning the arm angle and the arm length l 1changes both the reflection coefficients of x- and y-polarizedwaves, tuning the stub length l2 mainly changes the y-polarizedwave’s reflection coefficient and has a slight effect on thex-polarized wave’s reflection coefficient (antisymmetric mode)[46]. In order to focus the slant-polarized incident beam (i.e.,the projections of the incident electric field on the x- and y-axesare of equal amplitude and phase) onto two foci at locationsrf x

and rf yupon reflection, the physical parameters of the

Y-shaped element at the mnth cell’s location on the MS mustbe properly tuned. It is required that the simulated reflectionphases from the mnth element for both principal polarizations(ϕsim

x and ϕsimy ) meet the required phases of the reflection co-

efficients at the mnth location (ϕmnx and ϕmn

y ) from Eq. (1).Owing to the three physical parameters (l 1, l 2, andΔ) of the

Y-shaped nanoantenna, we have enough degrees of freedom tocontrol the reflection coefficients of orthogonally polarized in-cident waves almost independently. Figures 2(b) and 2(c) showthe range of phases of the x- and y-polarized waves’ reflectioncoefficients (ϕsim

x and ϕsimy ) as a function of the Y-shaped nano-

antenna’s design parameters (l 1, l 2, and Δ) at λ � 4.3 μm. Thefrequency domain finite-element method (commercially avail-able in HFSS by Ansys Inc.) is used to simulate the Y-shapednanoantennas in a fully periodic arrangement with normal in-cidence illumination, applying the local periodicity assumption[39,46,62–64]. The bifocal lens is subsequently designed towork under a beam incidence angle of α � 30° (i.e., the beamis coming at an angle of α � 30° from the MS normal, z-axis)as shown in Fig. 1(a). In this figure, the slant-polarized beamimpinges on the lens with an electric field vector having twoorthogonal x- and y-polarized components with identicalamplitude and phase. Upon reflection from the lens surface,the x- and y-polarized components of the incident beam arefocused into two spatially separated spots at rf x

� f x z andrf y

� f y z, respectively. Based on the element characterizationprovided in Figs. 2(b) and 2(c) and using Eq. (1), an illustrativeflat bifocal circular lens of radius 25 λ (wavelength is λ �4.3 μm) is designed to have foci at f x � 42.8 λ (numericalaperture, NA, ofNAx � 0.5 ) and f y � 89.5 λ (NAy � 0.27).The transverse scattered electric field within the focal range isnumerically computed from the transverse electric field onthe MS by using the plane wave spectrum decomposition asin Ref. [21]. The corresponding longitudinal component of

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the scattered electric field is also found by applying Maxwell’sdivergence equation in the spectral domain followed by an in-verse Fourier transform (see Ref. [65], Chapter 12, page 705).The Fourier and inverse Fourier transform integrals in planewave spectral calculations [see Eqs. (24) and (25) in Ref.[21]] are implemented numerically via a 2D fast Fourier trans-form algorithm where the spatial domain size and spatialresolution are 102.4 λ × 102.4 λ and λ∕20, respectively. Thetransverse field on the MS lens is stepwise approximated,and the reflected electric field at each MS cell is assumed uni-form and equal to the reflected electric field calculated via thesimulated reflection coefficient of the element multiplied by thetransverse incident field at the center of the cell.

The electric field is assumed null outside of the overall MSequivalent aperture. A linearly polarized Gaussian beam, whoseelectric field and power density at the beam center are 1 V/mand 1.32 mW∕m2, respectively, illuminates the MS lens withα � 30° angle of incidence. The total power carried by thebeam is 246.8 pW, which is calculated by the surface integralof its longitudinal Poynting vector. The beam waist of the in-cident Gaussian beam in the MS plane is chosen to be threetimes larger than the radius of the MS (w0 � 75 λ) such thatincident beam has a plane-wave-like wavefront over the entireaperture of the lens. The intensities of the x- and y-polarizedscattered fields in the longitudinal plane (x-z plane) are shownin Figs. 3(a) and 3(b). The slant-polarized incident beam isfocused onto two spatially separated foci having different polar-izations, that is, x- and y-polarized. Note that the actual focicenters observed in Fig. 3 (obtained by PWS calculations) oc-curring at 42.5 λ and 89.1 λ slightly deviate from the designedgeometrical point-like spot at 42.8 and 89.3 λ [used in Eq. (1)].

The larger the lens diameter, the smaller the difference betweenthe designed geometrical focal distances and the actual ones.Note that such a shift occurs in all the reported cases in thispaper. Since the bifocal lens has a higher NA for the x-polarizedcomponent of the incident field, the x-polarized spot (locatedcloser to the lens) has a narrower lateral FWHM and shorterDOF compared to the y-polarized spot (the farther spot). Thedesigned bifocal lens also works as a monofocal lens for an inci-dent beam that is polarized purely along either the x 0- or y 0-axis.

Fig. 3. Simulation results for an illustrative bifocal MS lens shownin Fig. 1(a) (with two well-separated foci of orthogonal polarizations)of radius 25 λ, placed at the z � 0 transverse plane, upon plane waveincidence from an angle α � 30° in the x-z plane, at λ � 4.3 μm. Theprojections of the incident electric field on the x- and y-axes are ofequal amplitude and phase. The intensity (normalized to its maxi-mum) of (a) the x-polarized scattered field and (b) the y-polarized scat-tered field, in the x-z longitudinal plane: f x � 42.8 λ (NAx � 0.5),f y � 89.5 λ (NAy � 0.27).

Fig. 2. (a) Geometry of a Y-shaped nanoantenna together with its current distribution for symmetric (top left) and antisymmetric modes (topright), and the 3D view of each MS cell (bottom). (b) Reflection phase ϕx of the antisymmetric mode, upon x-polarized incidence, for a MSmade ofidentical Y-shaped nanoantennas at 4.3 μm as a function of the arm length, l 1, and the angle between the two arms, Δ, for l2 � 200 nm. Results forother values of the stub length l 2 ranging from 50 nm to 1100 nm provide analogous trends not shown here for simplicity. (c) Reflection phase ϕy ofsymmetric mode, upon y-polarized incidence, for a MS made of identical Y-shaped nanoantennas at 4.3 μm as a function of the arm length l1 andstub length l2 forΔ � 80°. In Fig. 2(c) we keep the arm angle fixed atΔ � 80°; however, the reflection phase of the symmetric mode also changes ina similar manner when using other values of the arm angle Δ, which is not shown here for brevity. The lateral width and thickness of each arm arefixed at 200 nm and 125 nm, respectively. The complex relative permittivities of the Al and SiO2 at 4.3 μm are -1601.3-j609.4 [60] and 1.9-j0.018[61], respectively.

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In the next step, the farther spot at f y is placed very close tothe closer spot at f x , aiming at increasing the DOF. The designparameters of the Y-shaped nanoantennas on the MS arelocally modified based on the aforementioned algorithm suchthat now we have two foci designed to be at f x � 42.8 λ(NAx � 0.5) and f y � 47.7 λ (NAy � 0.46).

Under the slant-polarized incident beam illumination withthe incidence angle of α � 30° with respect to the MS normal,we plot the intensity maps of the x- and y-components of thescattered electric field in Figs. 4(a) and 4(b), respectively.Moreover, the intensity of the total scattered electric field vec-tor, accounting also for the longitudinal component, is reportedin Fig. 4(c). The foci of different polarizations are at distinctlocations whereas the focal range for the total field intensityextends over the two foci. The overall focal region has auniform intensity distribution with a lateral FWHM of lessthan λ and a DOF, axial FWHM, of about 10.1 λ (extendsfrom z � 40.4 λ to 50.5 λ). The proposed bifocal lens has asingle elongated focal region [Fig. 4(c)] with a DOF which is1.6 times longer than that of a simple monofocal lens of thesame aperture size [Figs. 4(a) and 4(b)], while the lateralFWHM is kept almost the same. In other words, the bifocallens designed based on the polarization-diversity technique in-creases the DOF without compromising the lateral resolutionand aperture efficiency of the monofocal lens of the sameaperture size. Although the polarization diversity techniqueis used here to generate linearly polarized, spatially separatedfoci with orthogonal polarization vectors, such a techniquecan be straightforwardly used with other pairs of orthogonalpolarizations, such as right- and left-hand circularly polarizedbeams, by employing a proper MS element. However, thisshould be further investigated in a future publication.

3. ANNULAR-SEGMENTATION TECHNIQUE

In order to further extend the DOF of the lens, an annular-segmentation technique is proposed such that the circularMS lens aperture is divided into a few annular regions, eachregion associated to an individual focal spot. In contrast tothe polarization-only technique proposed in the previous sec-tion, which only offers two degrees of freedom (bifocal lenses),

the annular-segmentation technique simply offers an increasein the degrees of freedom. By allowing extra foci, we designmultiple overlapping foci that collapse into a needle-like focalregion with elongated DOF. Here, we need to make an impor-tant remark regarding the superposition of fields scattered bydifferent sub-MSs along the focal axis. In contrast to the polari-zation-diversity technique, the annular-segmentation techniqueleads to foci mainly with the same polarization. The copolarizedfields scattered from different sub-MSs are not necessarily inphase all along the focal range. Therefore, when using onlythe annular-segmentation technique, it is difficult to avoid de-structive interference, which would inhibit a single elongatedfocal region with uniform intensity distribution by bringingmultiple foci close to each other. To demonstrate this, a rep-resentative circular flat lens consisting of two concentric sub-MS regions of outer radii rc and rp, where superscripts c and prefer to the central and peripheral sub-MS regions, respectively[see Figs. 1(b) and 5(a)], is designed to work under y 0-polarizedincident beam only. Therefore, the electric field at the foci ismainly y-polarized. The Y-shaped MS elements are oriented asin Fig. 5(a). The electric field scattered only from the central(peripheral) sub-MS is calculated by setting the field amplitudeon the peripheral (central) sub-MS to zero. The outer radius ofthe peripheral sub-MS and the geometrical focus for the centralsub-MS are fixed at rp � 17.5 λ and f c � 30 λ, respectively.The outer radius of the central sub-MS and the geometrical

Fig. 5. (a) Geometry of a circular flat MS lens operating at λ �4.3 μm that consists of two annular sub-MSs and (b) normalized totalscattered field intensity from the MS lens illuminated by a y 0-polarizedincident beam with the incidence angle of α � 30° with respect toMS normal [see Fig. 1]. The electric field intensity plot is normalizedto the maximum intensity within the focal range. (c) Magnitude of theelectric field scattered from the central, the peripheral, and the overallMSs along the focal axis (z-axis). (d) Phase difference between the elec-tric fields scattered from the central and peripheral sub-MSs along thefocal axis. The MS is located at z � 0 transverse x-y plane and theouter radii and the designed focal distances corresponding to centraland peripheral sub-MSs are set as rc � 10.3 λ, rp � 17.5 λ; f c �30 λ, f p � 53 λ.

Fig. 4. Simulation results for an illustrative bifocal MS lens (withtwo overlapping foci of orthogonal polarizations) of radius 25 λ fea-turing a single elongated focal region and operating at λ � 4.3 μmwith f x � 42.8 λ, f y � 47.7 λ. The illuminating beam has slant pol-arization, and its incidence angle is set at α � 30°. Intensity (normal-ized to its maximum) of (a) x-polarized scattered field, (b) y-polarizedscattered field, and (c) total scattered field in the x-z longitudinalplane.

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focus for the peripheral sub-MS are then tuned such that themaximum magnitudes of the scattered fields from the centraland peripheral sub-MSs are almost equal at their individualfocal distances. Furthermore, field amplitudes are designedto drop to around half of their peak field amplitude betweenthe two foci (here at the distance of z � 38.3 λ ). For construc-tive superposition, it is important for these two contributions tobe in phase at z � 38.3 λ. Figure 5(b) shows the scattered fieldintensity (normalized to its maximum) from the overall lensaperture on the x-z longitudinal plane. It is observed thatthe scattered field intensity from the overall MS lens dropsby about 5.6 dB at the distance z � 46.3 λ away from theMS with respect to the maximum scattered field intensity alongthe focal z-axis that occurs at z � 27.5 λ. The magnitude ofthe electric fields scattered from the central and peripheralsub-MSs together with the magnitude of the electric field scat-tered from the overall MS aperture are plotted in Fig. 5(c). Weobserve that the fields due to the central and peripheral subMSs are in comparable amplitude; thus the dip in the totalfield is attributable to a degree of destructive interference.Note that although the maximummagnitudes of the focal fieldsfocused by the central and the peripheral sub-MSs are almostequal (see Fig. 5), the axial FWHM (i.e., the DOF) of thefocal field for the peripheral sub-MS is longer than that forthe central sub-MS. This is attributed to the fact that theperipheral sub-MS lens has a lower NA (i.e., a wider focus)and a higher incident illumination power (i.e., higher scatteredfield power) as compared to the central sub-MS lens. The phasedifference between the electric fields scattered from the centraland peripheral sub-MSs is also plotted in Fig. 5(d). This plotclearly shows that the fields scattered from the central andperipheral sub-MSs possess a varying phase difference alongthe z-axis; therefore it is not easy to guarantee constructive in-terference between the fields scattered from two sub-MSs anduniform focal field amplitude along the z-axis. This fact isclearly noticed at around z � 48 λ, marking the destructiveinterference since the phase difference is 180°. Based on thisremark, in the next section we design the multifocal lens withthe combination of the polarization-diversity and the annular-segmentation techniques to eliminate the destructive interfer-ence and achieve a flat intensity profile along the extendedDOF. Note that for a given MS lens radius, the number offoci can be increased by increasing the number of annularsub-MSs in Fig. 5(a). However, an important limitation mustbe kept in mind: the annular width of each sub-MS must belarge enough such that the local periodicity assumption[39,46,62–64] remains valid (note that nanoantennas’ dimen-sions might change considerably from one sub-MS to anothersince each sub-MS has a distinct focal distance). Note that thescattered field polarization upon reflection from the elements ismainly the same as the incident field polarization when thelatter is polarized along the symmetry axis (y-axis) or alongthe asymmetry axis (x-axis) of the Y-shaped elements [46](similarly to the V-shaped elements [40]). Moreover, sincethe plane of incidence (yz) coincides with the symmetry planeof the Y-shaped elements, the MS lens is also symmetric withrespect to the incidence (yz) plane. Due to these symmetryproperties, the cross-polarized scattered field is minimized.

4. LENS WITH A SINGLE ELONGATED DOF

In this section, we combine the polarization-diversity and an-nular-segmentation techniques to design a lens with an overallsingle focal region with an elongated DOF. This is achieved byexploiting the degrees of freedom offered by the Y-shapednanoantennas. The proposed lens aperture consists of twoconcentric sub-MSs, each with two separate foci of differentpolarizations, which leads to a total of four foci. Importantly,consecutive foci along the focal axis are always chosen fromorthogonal polarizations, such that the focal fields with thesame polarization and scattered from separate sub MSs arefarther separated to minimize destructive interference whereverthey are in comparable amplitudes. Next, we optimize thex- and y-polarized focal distances and the outer radius of thecentral sub-MS to achieve a single focal region with uniformintensity distribution and elongated DOF. Considering thedependence of the spot size and DOF on the ratio of focal dis-tance to lens diameter (i.e., the NA), the peripheral sub-MS(which has a larger diameter) is set to obtain the focus withthe longer focal distance such that an overall focal region withuniform transverse resolution is achieved. The outer radius ofperipheral sub-MSs is also fixed at 17.5 λ. The radius of thecentral sub-MS and the distances of the corresponding x- andy-polarized foci for each sub-MS are selected by minimizing theerror function, defined as

error � maxfjE�0; 0; z�j2g −minfjE�0; 0; z�j2gmeanfjE�0; 0; z�j2g ; (2)

where z ∈ �30 λ; 80 λ�, f cx < f c

y < f px < f p

y (in which sub-scripts x and y denote the polarization), E�0; 0; z� is the totalelectric field vector on the z-axis (focal axis), and mean functionmeanf·g represents the average over z ∈ �30 λ; 80 λ�. In thisprocedure, we use the genetic algorithm technique to minimizethe error function in Eq. (2), where the number of samplingpoints along the z-axis is 51. The field E�0; 0; z� is evaluatedusing plane-wave spectrum decomposition where the field onthe MS is step-wise approximated as explained in Section 2 ofthis paper and in Section 4 of Ref. [21]. The optimal design inthe end consists of two annular regions of outer radii 10.7 λ and17.5 λ whose expected geometrical foci are allocated, respec-tively, at f c

x � 40.2 λ, f cy � 50.5 λ, f p

x � 71.4 λ, andf py � 78.4 λ. The scattered field intensity (normalized to its

maximum) on the x-z longitudinal plane and along the z-axisare reported in Figs. 6(a) and 6(b), respectively. In the end, aneedle-shaped focal region with axial FWHM (DOF) of about74.1 λ (extending from z � 27.7 λ to 101.8 λ) is formed.

A very shallow dip of about −0.58 dB (with respect to themaximum scattered field intensity) along the focal axis (z-axis)is observed at distance z � 50 λ away from the MS [marked asIII in Fig. 6(b)]. The NA of the lens, considering the center ofthe focal region as the effective focal point, is about 0.26. Thereported elongated DOF of ∼74.1 λ is much longer than thetheoretical DOF of a conventional lens (i.e., nλ∕NA2, in whichn is the refractive index of the host medium [66]) which is∼14.8 λ. In Figs. 6(c)–6(g) we report the field intensity mapson different x-y transverse planes at z locations within the focalrange, marked by roman numerals in Fig. 6(b). Moreover, theFWHM of the focus field in the x-y transverse plane (lateral

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resolution) is shown in Fig. 7 varying the transverse planelocation within the focal region. It is observed that the lateralFWHM slightly changes from ∼1.37 λ to ∼2.8 λ along the fo-cal range extended from z � 27.7 λ to 101.8 λ. The absoluteefficiency of the MS lens is also ∼28% at 4.3 μm. In order todetermine the absolute efficiency [58], we first calculate thewaist of the scattered field in the transverse focal plane. Herethe transverse plane located in the middle of the focus range

(extending from z � 27.7 λ to 101.8 λ) is considered as the fo-cal plane, which is the z ≈ 65 λ transverse plane. The waist ofthe scattered field (in the focal plane) is defined as the full widthof the normalized intensity at 1∕e2 and is evaluated by fittingthe scattered field intensity distribution at the focal plane to aGaussian function [58]. The absolute efficiency is then definedas the total power flowing within the waist at the above definedfocal plane, divided by the incident power illuminating theMS [58].

5. CONCLUSION

We have shown a novel concept and design for flat MS multi-focal lenses, which provides multiple foci with arbitrary loca-tions and separations. Instead of forming only a single spot,the proposed lens can focus the incident beam into multipleclosely spaced/spatially separated foci. Thanks to the proposedY-shaped antenna elements that add a degree of freedom com-pared to the V-shaped elements used in other publications, bi-focal lenses are implemented based on the polarization-diversitytechnique in which each polarization contributes to one focalspot. Furthermore, the flat MS lens consisting of multiple con-centric annular regions provides the possibility of focusing theincident beam into multiple well-separated foci. Subsequently,a flat lens with a single elongated focal region with DOF ofabout 74.1 λ and lateral FWHM of about 1.7 λ is successfullyimplemented by combination of the polarization-diversity andthe annular-segmentation techniques. The implementation ofmultifocal lenses and monofocal lenses with elongated DOF bythin flat MS would significantly reduce cost, volume, opticalloss, and system complexity in integrated optics.

Funding. National Science Foundation (NSF) (NSF-SNM-1449397).

Acknowledgment. The authors are grateful to ANSYS,Inc., for providing HFSS that was instrumental in the design.

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