High Efficient Far-Field Nanofocusing with Tunable FocusUnder Radial Polarization Illumination

Lin Cheng & Pengfei Cao & Yuee Li & Weijie Kong &

Xining Zhao & Xiaoping Zhang

Received: 12 July 2011 /Accepted: 17 October 2011 /Published online: 4 November 2011# Springer Science+Business Media, LLC 2011

Abstract We design a new nanofocusing lens for far-field practical applications. The constructively interfer-ence of cylindrical surface plasmon launched by thesubwavelength metal l ic structure can form asubdiffraction-limited focus, which is modulated by thedielectric grating from the near field to the far field.The principle of designing such a far-field nanofocusinglens is elucidated in details. The numerical simulationsdemonstrated that nanoscale focal spot (0.12λ2) can berealized with 3.6λ in depth of focus and 4.5λ in focallength by reasonably designing parameters of the grating.The focusing efficiency can be 7.335, which is muchhigher than that of plasmonic microzone plate-like lenses.A blocking chip can enhance the focusing efficiencyfurther as the reflected waves at the entrance would berecollected at the focus. By controlling the number of thegrooves in the grating, the focal length can be tuned easily.This design method paved the road for utilizing theplasmonic lens in high-density optical storage, nano-lithography, superresolution optical microscopic imaging,optical measurement, and sensing.

Keywords Subwavelength metallic structure .

Superfocusing . Far field . Focusing efficiency .

Tunable focus

Introduction

Plasmonic lens (PL) [1–6], which consists of metal anddielectric, can excite surface plasmon polaritons (SPPs)and can be used for subwavelength focusing, imaging,and beam shaping in recent years. Researches onmanipulating SPPs are mostly focused on specificspatial distribution of subwavelength metallic structures,such as annular rings [6], curved chains of nanoparticlesor nanoholes [7], plasmonic microzone plate (PMZP)-like or chirped slits [8, 9], and so on. When thepolarization state of incident light matches the geometrystructure of plasmonic lenses, the optimum SPPsfocusing effect will be obtained [10]. Since the axialsymmetry of annular rings and multiconcentric struc-tures, SPPs can be excited by radially polarization beamsand reach the focal points of the geometry due to theconstructive interference to achieve high-energy focus-ing with superresolution [11–14].

Since surface plasmon polaritons is bound to a metalsurface, it is an evanescent field in the vertical directionwith exponential decay in amplitude and cannot reachthe far field. It must take effect action to couple theSPPs into the radiating form and let higher spatialcomponent contribute to superposition at the far-fieldfocal region to produce subwavelength focal spot withdesired dimensions. D. Zhou et al. [15] took advantageof tapered dielectric grating to couple SPPs launched bytwo subwavelength metallic slits, therefore it can obtainthe far-field focus by improving the irradiative field.However the parameters of tapered grating should beelaborately engineered; especially when the focal lengthvaries, the tapered grating should be redesigned totally.

L. Cheng : P. Cao :Y. Li :W. Kong :X. Zhao :X. Zhang (*)School of Information Science and Engineering,Lanzhou University,Lanzhou 730000, Chinae-mail: zxp@lzu.edu.cn

Plasmonics (2012) 7:175–184DOI 10.1007/s11468-011-9291-7

Zhang et al. [16] present a simple plasmonic lenscomposed of an annular slit and a single concentricgroove. The subwavelength groove can scatter the SPPsand constructively interfere to get a far-field focal spot.The focal length can be adjusted by changing the groovediameter. However, the intensity at the focal spot is veryweak because the scattering loss results in low focusingefficiency.

In this paper, we present a simple method to realizefar-field nanofocusing by utilizing dielectric surfacegrating upon the circular plasmonic lens. The periodicgrooves can modulate the cylindrical surface plasmonwaves launched by annular metallic slits, and then partof the evanescent components can be coupled intopropagation waves, which can be eradiated to the far-field region and constructively interfered to cause thesuperfocusing effect. The principle of design is given indetails, including the slits width, the inner radius of thering slits, the filling material in the rings, and the periodof the dielectric grating. The full wave simulationsillustrate that a nanoscale focal spot with full width ofhalf maximum (FWHM)=0.38λ can be obtained in farfield with depth of focus about 3.6λ. It is very flexible toadjust its focal length by altering the number of thegrooves. At the same time, the focusing efficiency is muchhigher than that of PMZP. In order to depress the energyfocusing at the incident space, a blocking chip is used tooptimize the far-field plasmonic lens, and then thefocusing efficiency is enhanced further. Our far-fieldnanofocusing scheme can supply for the requirement ofpractical applications such as high-density data storage,superresolution optical microscopy, nanolithography, opticalmeasurement and sensing, etc.

Principle of Nanofocusing

The cross section of our far-field nanofocusing lens isdepicted in the r–z plane, as shown in Fig. 1. The lower

part is an ordinary circular plasmonic lens with severalslit rings milled into metal film upon the glass substrate.A dielectric layer with high refractive index (nd) isadjacent to the metal film, smoothing the metallicsurface and protecting the metallic thin film fromsulfuration or oxidation as well. At the same time, thedielectric layer functions as a Fabry–Perot-like resonatorand can enhance the focal intensity at a specificthickness [17]. In order to prolong the focal length, i.e., extend the subwavelength focus into far field, acircular dielectric grating is added topside, acting as amodulation device.

In the following paragraphs, we will discuss theprinciple of our structure in details. Firstly, the metallicfilm is uniformly deposited on the quartz substrate.Usually silver, gold, or aluminum is selected forplasmonic lenses due to their relative less intrinsic loss.However, in our new structures, high-intrinsic-loss metalsuch as titanium (Ti) or chrome (Cr) is used, as it isbenefit for the subwavelength annular slits excitingdiffracted cylindrical waves to launch the cylindricalsurface plasmon (CSP) predominantly. Here the thick-ness of metallic film (d1) is much greater than the skin

depth of SPP in the metal (dm ¼ l2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijRe "mð Þjþ"d

"2m

q, where εm,

εd denote the permittivity of metal and adjacent dielectric,respectively), so that no direct transmitted light wouldreach the focal region, hence hindering the interference ofthe propagation waves reradiated by the surface plasmons.Multiple annular slits are milled into the metallic film; theradius and the width of each slit are denoted as ri and wi,respectively. In order to excite the CSP effectively, thewidth of slits should be less than half of the incidentwavelength.

The incident light is the radial polarization wave,impinging from the bottom of the substrate. Since thepolarization symmetry of a radially polarized illumina-tion matches to the rotational symmetry cylinderstructure, the entire incident beam is TM polarized with

Fig. 1 Schematic of thefar-field nanofocusingplasmonic lens in the cylindricalcoordinate system is plotted asthe r–z cross section, showingthe geometrical parametersdefining the structure, as well asthe half of the three-dimensionalmodel is depicted at the topright corner

176 Plasmonics (2012) 7:175–184

respect to the annular slit rings, enabling surfaceplasmon excitation from all directions [18]. Part of theincident light is diffracted by the sharp edge of the slit,and then CSP waves are excited by the diffracted lightfor an extra wave vector in the direction along the filmsurface is obtained. The effective wavelength of the SPwave can be expressed with incident wavelength λ0 asfollows:

lsp ¼ l0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiReð"mÞ þ "dReð"mÞ � "d

sð1:1Þ

The excited CSP at each slit share the same phase,and the constructive interference of the SP wavesemanating from different slits leads to an intense focal

spot in near field when the distance d between the slitsshould be an integer number of plasmon wavelength,i.e.,

2pdlsp

¼ 2pm; ðm ¼ 1; 2; 3 . . .Þ ð1:2Þ

Since surface plasmons interfere constructively, theenergy is captured on the metal surface and guideduntil it is absorbed by the metal. This energy is lostand will not be transmitted by the slits. While surfaceplasmons interfere destructively, only little energy iscaptured by the metal and the rest energy will betransmitted pronouncedly by the slits. On the conser-vation of energy, the destructive interference of SPPs ismore favorable to get more energy in the far-field focal

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-5

0

5

10

15

20x 106

kr/k0

Tra

nsm

issi

on [a

.u.]

Bessel-Fourier transformation of Circ(r/rcd)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1

0

1

2

3

4

5x 105

kr/k0

Tra

nsm

issi

on [a

.u.]

Bessel-Fourier transformation of ring: Circ(r/p)-Circ(r/r0)(a)

0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

kr/k0

Tra

nsm

issi

on [a

.u.]

|T(Kr)|

rcd=808nm

rcd=1212nm

rcd=1616nm

rcd=2020nm

rcd=2424nm

(c)

(b) Fig. 2 The Fourier spectralprofiles of the circular stepfunction with rcd=2.424 μm (a)and a ring with inner and outerradius are 404 and 151.5 nm,respectively (b); c is theamplitude of the transmissionfunction T(kr) with different ra-dius of aperture stop

Plasmonics (2012) 7:175–184 177

region. Then the distance between the slit rings (ri+1−ri)should be an odd number times the quarter of plasmonwavelengths.

2plsp

ðri � ri�1Þ ¼ 2pmþ p; ðm ¼ 1; 2; 3 . . .Þ ð1:3Þ

In order to acquire a focal spot at the far field with afocal length f, the inner radius of each slit rings shouldsatisfy the relation as below:

2pl

ffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2 þ r2i

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2 þ r21

q� �þ $8 i ¼ 2p � N ; ðN ¼ 1; 2; 3 . . .Þ

ð1:4Þ

For protecting the metallic thin film from sulfurationand oxidation, especially in the case of metal lamellaexposing to the air, a dielectric film is coated upon themetal. Hence the slits are filled with the same materialand this dielectric layer can smooth the metallic surfacefurther, which is quite beneficial to integrate with otherinstruments. The thickness of dielectric layer d2 willaffect the performance of the plasmonic lens [17]. Also thevarious materials of dielectric layer (i.e., different refrac-tive index nd and "1 ¼ n2d) will result in different intensitydistribution at focal region.

In order to adjust the focal length, a periodic concentricdielectric grooves is added topside, which can be treated asan annular surface grating limited by a convergentdiaphragm, acting as a modulation device. Because theannular surface grating with a period of Λ can modulate the

waves by changing the transverse wave numbers asfollows:

kr ¼ kri þ 2p0

n; ðn ¼ 0; � 1; . . .Þ ð1:5Þ

It is obviously that part of the evanescent componentscan be transferred into propagation waves, which canradiate and reach the far field. If we decrease the gratingperiod, the figures of merit of the focal spot will be greatersince more components will be converged into the focalregion.

Fig. 4 The normalized intensity profile along the optical axis(r=0).Solid line without grating, dashed line with grating. The grating canadjust the focal spot from near field into far field

Fig. 3 Surface plot ofnormalized intensity distribu-tion: a without grating, the focusis formed on the center of exitplane, i.e., near-field focusing; bwith grating, the focus is formedfar away from the exit plane, i.e., far-field focusing. Here theperiod of grating is 402 nm,line/space=202/200 nm,depth=180 nm

178 Plasmonics (2012) 7:175–184

On the other hand, we can use an aperture stop tocontrol the contribution of the surface grating tofocusing. This can be explained by Fourier optics, aswe define the transmission function T(kr) of the grating

with finite radius rcd in wave vector space as

TðkrÞ ¼ Bfcircðr=rcd Þg � Gratingðkr Þ ð1:6Þ

Fig. 5 a–f The surface plots of normalized intensity distribution with different grooves number. The focal length is increased by more groovesparticipating into focusing

Plasmonics (2012) 7:175–184 179

Here step function circðr=rcdÞ ¼1 r < rcd1=2 r ¼ rcd0 else

8><>: ,

Bf�gdenotes Bessel–Fourier transformation, i.e., the

Fourier transformation in cylindrical coordinate system

and Bfcircðr=rcd Þg ¼ 2prcdkr

J1 ðkr � rcd Þwith J1(·) the first

order Bessel function, as shown in Fig. 2a. The gratingfunction grating(kr)can be represented as the superimpo-sition of concentric rings with period 0.

GratingðkrÞ ¼X1n¼0

2pn0 J0ðn0 krÞ � 2pkr

0 J1ð0 krÞ � r0 J1ðr0krÞf g

¼X1n¼0

2pkr

ð1þ nÞ0 J1½ð1þ nÞ0 kr� � ðr0 þ n0 ÞJ1½ðr0 þ n0 Þkr�f gð1:7Þ

Then the Eq. 1.6 can be rewritten as

TðkrÞ ¼ 2prcdkr

J1ðkr � rcdÞ �X1n¼0

2pn0 J0ðn0 krÞ � 2pkr

0 J1ð0 krÞ � r0J1ðr0krÞf g

¼ 2prcdkr

J1ðkr � rcdÞ �X1n¼0

2pkr

ð1þ nÞ0 J1½ð1þ nÞ0 kr� � ðr0 þ n0 ÞJ1½ðr0 þ n0 Þkr�f gð1:8Þ

Obviously when the circular aperture stop isnarrower (i.e., rcd decreases), the curve of the transmis-sion function will be damped more gently with kr (wavevector along the r-axis) as shown in Fig. 2c. It meansthat the wave vector along z-axis (kz) will decrease

according to the relation k2r þ k2z ¼ "d2pl

� �2when kr

increases, especially if rcd=0, then it degenerate intothe situation of no grating, then the focus spot appearson the exit surface. The subsequent simulation willtestify this point.

Fig. 5 (continued)

180 Plasmonics (2012) 7:175–184

Simulation and Discussion

We use COMSOL Multiphysics 3.5a and MATLAB 7 totestify the performance of our far-field nanofocusing lens.As the structure is rotating symmetry and the incident lightis radially polarization, we just use a two-dimensional axissymmetry model, which is the half of the cross sectiondepicted in Fig. 1, to simplify such three-dimensionalproblem.

Only two subwavelength slit rings were milled into the200 nm thickness Ti film deposited on quartz substrate, theslits are 228 nm in width, and the inner radii satisfy Eq. 1.3and 1.4. The working wavelength is 532 nm, the dielectricfunction [19] of titanium is −5.8450+10.2354i and the

dielectric is α-SiO2 with εd=2.4.With radially polarizationlight impinging on the substrate, a brilliant focal spot isformed on the center of exit plane as shown in Fig. 3a dueto the constructive interference of the cylindrical surfaceplasmon. While a grating is adherent on the exit plane, thefocus is shifted to far field, as shown in Fig. 3b, in thecondition of the circular grating having a period of 402 nmwith 202 nm line width. Here the dielectric grating is madeof zinc oxide, which has a high refraction index of 2.16.Obviously, the strongest intensity appears at 6.7 μm alongthe z-axis, which is far away from the exit plane comparedwith the corresponding result of no grating. The normalizedintensity profile along the optical axis(r=0) is shown inFig. 4 clearly.

Fig. 6 The performance of the optimized far-field nanofocusing lens. a Surface plot of the normalized intensity distribution; the profile of the focib vertical to the optical axis and c along the optical axis

Table 1 The focal length and FWHM vary with the number of grooves through numerical simulation

Grooves number 11 10 9 8 7 6 5 4 3 2

f (μm) 2.94 2.34 2.15 1.99 1.85 1.62 1.55 1.40 1.24 1.01

FWHM (nm) 210 204 204 203 203 203 206 204 203 205

The period of grooves is 404 nm, and the duty ratio is 3/8

Plasmonics (2012) 7:175–184 181

Since the dielectric grating plays an important role inour far-field nanofocusing, we discuss further theinfluence of subwavelength period of the grating. It isfound that there exists an optimum value of the period,i.e., 404 nm under the aforementioned structure, which

can get much stronger intensity at the focal spot withthe smallest focal spot.

By utilizing the aperture stop to control the groovesof the dielectric grating, we can adjust the focal lengthflexibly as shown in Fig. 5. When the grooves number

Fig. 7 The surface plot of intensity distribution a without and b with blocking chip, c the intensity profile is compared at z=2.28 μm

182 Plasmonics (2012) 7:175–184

decreases, the focal spot approaches the exit plane,while the FWHM value of the spot is nearly 204 nm witha slight change. The numerical results are listed inTable 1.

In order to evaluate the performance of our far-fieldnanofocusing lens, three figures of merit are exploited:the spot size, the depth of focus (DOF), and the energyfocusing efficiency. The spot size is measured as thearea of the focal region vertical to the optical axis withrespect to the FWHM, and the DOF can be consideredas the FWHM along the optical axis. With theoptimized lens structures, the area of spot size is0.12λ2 and the DOF is 3.6λ. The energy focusingefficiency is defined as the ratio of the flux density ofthe focused beam to that of incident on the whole lens,expressed as follows.

h ¼ XXjEFocj2dsFocSFoc

=XXjEIncj2dsInc

SIncð1:9Þ

According to Eq. 1.9, the energy focusing efficiency inFig. 6 is 733.5%. While the focusing efficiency of theFresnel zone plate (FZP) is around 20–80% in general,even the best of the plasmonic microzone plate lenses isless than 100% [20]. The reported concave lens withmetamaterials such as photonics crystal at most has anintensity gain of 6.3 with TM mode illumination [21]. In amanner, our far-field nanofocusing lens has a higherenergy focusing efficiency than FZP-liked lens andmetamaterials lens due to the cylindrical surface plasmonof our designed structure under the condition of radialillumination.

In order to suppress the energy focusing at the incidentspace, we use a blocking chip to reflect the evanescentwaves excited by the metallic annular slits’ entries, asshown in Fig. 7b. It is found that the blocking chipenhances the maximum intensity at the focal region morethan twofold from 161 to 346, while the profile of the focusdoes not change at all, which is depicted obviously inFig. 7c since the zero points are at the same position alongthe radial axis. Totally, the focusing efficiency increasesnearly 50% when the blocking chip is added beneath themetal layer.

According to the principle of nanofocusing aforemen-tioned, we had also performed the simulations by a higherintrinsic loss metal Cr (εm=−9.8514+23.1960i, at 532 nmoperation wavelength [19]) taking the place of Ti. It wasfound that the results of Cr are very similar, except themaximum value of focus intensity is less than that of Ti,because the intrinsic loss of Cr is much higher and thereflection of Cr layer is much stronger than those of Ti. Bythe length limit, the results of Cr are not exhibiting any

more, yet Ti is provided as the exemplification in ourmanuscript.

Conclusion

In conclusion, we design a highly efficient nanofocusinglens capable of subwavelength focusing in far field underradial polarization illumination matched with the axissymmetry structure. The parameters of the structure havebeen discussed and numerically simulated. The focal lengthcan be adjusted by controlling the number of grooves thatparticipate in the focalization. The blocking chip can beused to enhance the focusing efficiency further. Our highlyefficient far-field nanofocusing lens is potential for high-density information storage, nanolithography, superresolu-tion imaging, optical measurement and sensing, highlyintegrated nanoscale photonic devices, and related applica-tions. On the other hand, the difficulty of nanometer-scaleengineering and processing needs to be conquered in thefuture works to turn our highly efficient far-field nano-focusing lens into a real one.

Acknowledgments This work was partly supported by the NaturalScience Foundation of Gansu Province (No.096RJZA063) and theFundamental Research Funds for the Central Universities (lzujbky-2011-k02 and lzujbky-2010-88).

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