ic

ssiofed tdese-diithl toen t

eived increasing attention innas caation,ntennupper mave sodancetical bically aielectr

ber and so on; the quantum theory is not used here. The EMequations are considered sufcient in the analysis of electromag-netic elds in nano-material which is characterized with theparameters mentioned previously. In [10], the dielectric waveguidetheory was used to discuss the effective length of rod opticalantennas. It was found that the effective length was shorter thanthe physical length of the rods. The equivalent circuit theory

derived from the classic EM theory was also used to analyze theelectromagnetic elds in nano devices [1113].

In this paper, the electromagnetic property of dielectricmaterials

x20 x2 cx

er2x K cxx20 x22 cx22

where v is electric susceptibility, K is a constant determined byphysical parameters of atoms and electrons in the material, and cis a damping rate. In (2) and (3) x0 is the resonant frequency ofelectrons within the material, and is generally greater than the fre-quency of visible light. It is noted from (2) and (3) that the relativepermittivity in microwave band has an effective real part with a

Corresponding author. Tel.: +852 3400 3602; fax: +852 2765 7198.

Optical Materials 34 (2011) 126130

Contents lists available at

M

.e lE-mail address: ivanwang@live.com (I. Wang).metal in optical frequency actually works as solid plasma, havingits own plasma frequency, collision frequency, damping and soon, which result in a complex permittivity with a negative realpart. The dielectric, which cannot be described by the Drude model[9], has a frequency-dependent complex permittivity in opticalfrequency.

Nano antennas can be analyzed using the classical electromag-netic (EM) theory with the parameters of permittivity, wave num-

2. Dielectric material in optical band

Dielectric materials show unique characteristics in optical band.Note that a dielectric material is characterized with relative per-mittivity er(x) = er1 + ier2. The real part er1 and the imaginary er2are generally expressed by [6]

er1x 1 v K x20 x2

2 2 11. Introduction

Recently nano antennas have recnanotechnology research. Such antencopy, spectroscopy, data-communicharvesting [13]. It is known that ametal or a dielectric material. In thetric antennas are favored as they hwider bandwidth, less loss and avoipared to metal antennas [4]. In opdielectric antennas are utilized practerties exhibit in the metallic and d0925-3467/$ - see front matter 2011 Elsevier B.V. Adoi:10.1016/j.optmat.2011.07.027n be applied in micros-and even solar energyas can be made with aicrowave band dielec-

me advantages such asof surface waves, com-and, both metallic ands unique material prop-ic materials [58]. The

in optical band is rst described. Using the classical electromagnetictheory, the excited eld modes in nano-optical disc antennas arethen discussed. The mode with the maximal direct eld enhance-ment is identied. Simulation examples are given to illustrate thecoupling effect of discs in disc antenna arrays. An investigation intothe effect of eld enhancement in disc antenna arrays is presented.Optimal disc spacing for the maximal direct eld enhancement inthe disc antenna array is discussed.Directional eld enhancement of dielectr

Ivan Wang , Y. DuDept. BSE, The HongKong Polytechnic University, Hong Kong

a r t i c l e i n f o

Article history:Received 28 December 2010Received in revised form 7 July 2011Accepted 21 July 2011Available online 9 September 2011

Keywords:OpticalDisc antennaNano antennaDirectivityField enhancement

a b s t r a c t

This paper presents a discuoptical band. The propertyare discussed. It is identidirectivity than other moachieved by using multipldisc antenna array varies wing is approximately equaimproved signicantly wh

Optical

journal homepage: wwwll rights reserved.nano optical disc antenna arrays

on on the directive eld enhancement of dielectric disc antenna arrays indielectric material is addressed, and eld modes in a cylindrical resonatorhat the fundamental mode of HE11d generates the far eld with a higher. More effective eld enhancement in the radiation direction could besc antenna arrays. Simulation examples indicate that the directivity of athe disc spacing. The maximum directivity is observed when the disc spac-the half of the vacuum wavelength. The maximum directivity can be

he disc number is increased. 2011 Elsevier B.V. All rights reserved.

SciVerse ScienceDirect

aterials

sevier .com/locate /optmat

nearly zero imaginary part. When the frequency is increased to theinfrared, optical or ultraviolet band, the imaginary part is non-triv-ial as the frequency is close to the resonant frequency of the mate-rial. The relative permittivity tends to be a frequency-dependentcomplex number with both real and imaginary parts being positive.This has been veried in the measurement [7],

It is known that the real part er1 represents the capability that ofthe material holds energy, and the imaginary part er2 representsthe capability that of the material generates the loss of energy.Clearly a dielectric material in optical band loses some energywhen it is excited by an EM eld source. As the real part of relative

parallel eld lines for the mode HE11d, radial eld lines for themode TM01d, or circular eld lines for the mode TE01d. The eldpeak appears in the center for the mode HE11d, and appears at

(a)HE11 (c)TE01(b)TM01E-field H-field

1-disc 2-disc 4-disc 9-disc

I. Wang, Y. Du /Optical Materials 34 (2011) 126130 127permittivity is positive, the dielectric material in optical band canalso be used as a resonator. In contrast, plasmonic material (metalin optical frequency) which is characterized by the Drude modelhas a negative real part. Therefore, an EM wave cannot penetrateinto the metal. The following mode analysis for a dielectric wave-guide is then inapplicable to metal in optical frequency. Note againthat the real part er1 in optical frequency is larger than that inmicrowave frequency. The effective wavelength of a dielectric an-tenna is usually shorter in optical frequency.

3. Analysis of electric eld distribution

In microwave band, antennas made of dielectric can effectivelyradiate EM waves [14]. These antennas work as a resonator. As aresonator is a section of a waveguide, the electromagnetic eldaround the resonator can be analyzed using the dielectric wave-guide theory. Similarly, a dielectric disc antenna in optical bandworks as a radiative cylindrical dielectric resonator, as illustratedin Fig. 1. The wave propagation within a dielectric waveguide canbe supported. The electric eld distribution within the dielectricwaveguide is determined by the boundary conditions. The equiva-lent magnetic current yields on the dielectric material as a result ofthe inner electric eld. This current radiates the electromagneticeld from the disc antenna like a traditional metal antenna [11,12].

It is noted that in a dielectric waveguide propagation constant cis determined from [6],

c2 k2c k2 3where wavenumber k 2pfl0ere0p and cutoff wavenumberkc = (k/kc)k. Cutoff wavelength kc can be computed by enforcingthe boundary conditions on the waveguide surface [16]. It is deter-mined by geometry and material properties of the waveguide, andeld mode in the waveguide. The complex propagation constantcan be expressed by c = a + jb. Both wave attenuation constant aand wave phase constant b are given by

b 2pf le0er1p 1 k=kc2q 4a per2=er1= k

1 k=kc2

q 5

r

out

Waveguider

out

(b) Disc Resonator/Antenna

L

L(a) Cylindrical Rod (a section of cylindrical rod)Fig. 1. Cylindrical rod and disc resonator/antenna.For the dielectricmaterial in optical band, both real part er1 and imag-inary part er2 are positive. Wave propagation in the dielectric wave-guide is supported if the wavelength of the electromagnetic (EM)eld is less than the cutoff wavelength of the waveguide, as shownin (4). The fundamental eld mode in a cylindrical dielectric wave-guide is found to be the HE11 mode, not the TM01 mode appearingin a traditional metallic waveguide in microwave band. Fig. 2 illus-trates theelddistributions ofrst threeeldmodes in the cross-sec-tion area of a cylindrical waveguide or resonator, which are alsoapplicable to a dielectric circular rod [15]. The numerical analysisin [15] indicates that E-eld lines for the mode HE11 run in parallelon the cross-section area. The eld reaches the maximum value atthe central point of the cross-section, and declines when movingaway from the central point. For the mode TM01 or TE01, the E-elddistribution is rotationally symmetric on the cross-section area.

In a circular dielectric waveguide [16], if the attenuation con-

stant of a cladding medium q b2 2pf 2leout

qhas a complex

value (eout is the permittivity of the medium cladding around thedielectric material, Fig. 1), the EM eld can be effectively radiatedfrom the dielectric material to the outer medium. When parameterq has a positive value, the EM wave is conned within the wave-guide. No radiation is generated from the waveguide or resonatorin this case.

Normally, a cylindrical resonator is considered as a segment ofthe cylindrical waveguide with the length of k/2, as illustrated inFig. 1. When the cylindrical resonator works as a disc antenna forradiating an EM wave, segment length, which is often dened asd, is much smaller than k/2. The E-eld distribution on the crosssection area is similar to those given in Fig. 2. The eld distributiondetermines the equivalent magnetic current M E bn on the sur-face of a disc antenna [4], which is treated as the radiating source.The disc antenna made of a dielectric material may support themode HE11d [16] while the other modes (TE01d and TM01d) may alsobe generated. As seen in Fig. 2, the E eld has quasi-straight

Fig. 2. EM eld distributions on the cross-section area of a cylindrical dielectricwaveguide or resonator: (a) the fondamental mode of HE11 (b) the 2nd mode ofTM01, and (c) the 3rd mode of TE01.16-disc 25-disc

Fig. 3. Congurations of dielectric nano optical disc antennas arrays.

mode of HE11d can be further enhanced.

cited in the antenna array at the frequency of 500 THz. As a result,the discs have the radius of 76 nm. The disc thickness is set to be50 nm in all cases.

The far eld pattern generated by a single disc antenna is com-puted rst using the FDTD method. The disc antenna is excited tocreate three basic modes HE11d, TE01d and TM0d1 in turn for exam-ining the effect of directional eld enhancement. The simulationresults of directivity in the E- and H-planes are shown in Fig. 4.It is noted that the directional eld enhancement at the modeHE is the best. The directivity of the disc antenna is 2.82dBi

nd

128 I. Wang, Y. Du /Optical Materials 34 (2011) 1261304. FDTD simulation of dielectric disc antenna arrays

FDTD (nite-time nite-difference) simulations have been per-formed for several examples of dielectric disc antenna arrays toillustrate their directional eld enhancement. Fig. 3 shows the con-gurations of disc antenna arrays with different disc number.These discs are made of Silicon with the relative permittivity of15.598 + j0.21464 at 500 THz [6]. The disc radius is set to be thehalf wavelength k/2 of the source eld, which is given by

k k0=er1

p 6the half of disc radius for the mode TE01d or TM01d. For a single discantenna, the mode HE11d can effectively radiate directionally in thedirection perpendicular to the top surface of the disc. As the eldlines are rotationally symmetric for the modes TM01d and TE01d,the disc antenna operating at these modes has the maximum radi-ation in radial directions on a plane. The radiation pattern, there-fore, is less directive.

When several disc antennas are assembled to form an antennaarray, the effect of mutual coupling between elements should betaken into consideration [17]. The coupling between adjacent discsdetermines scattering of the eld which nally has an effect on thedirectivity of the array. The directivity of the antenna array at the

Fig. 4. Directivity in the E-plane awhere k0 is the vacuum wavelength. For dielectric material withcomplex permittivity, the real part is used only in (6). Disc spacing,however, varies from one tenth of the wavelength to the full

Fig. 5. Comparison of the effect of mutuwavelength for the discussion of maximal coupling among discsin the antenna array.

In the simulation the disc array is excited by an exciting sourceat one circular end of the discs, and radiates the electromagneticeld from another circular end (top surface) of the discs. The fareld pattern of the disc antenna array is computed for discussion.To illustrate the directional eld enhancement, a single mode is ex-

H plane of a single disc antenna.

Table 1Directivity of disc arrays with different disc number and spacing (unit: dBi).

Spacing (s0) k0/10 k0/4 k0/3 k0/2 2k0/3 3k0/4 k0

2-disc 4.3 5.3 5.9 6.3 6.1 5.8 5.34-disc 5.2 4.1 4.7 5.7 6.1 5.7 5.49-disc 9.1 14.1 15.3 12.4 11.7 12.7 14.916-disc 14.5 19.4 27.4 28.9 27.9 22.8 22.125-disc 22.1 24.3 37.9 59.5 59.9 41.7 45.911d

for the mode HE11, 1.41dBi for the mode TE01 and 1.28dBi for themode TM01. The radiation pattern for the mode TE01d or TM0d1 isless directive as shown in Fig. 4.

al coupling in 2-disc antenna array.

aterI. Wang, Y. Du /Optical MFig. 5 shows the simulation results for a two-disc antenna arraywith the disc spacing of k0/2 at the mode HE11d. To investigate the

(a.1) n=4 s0= 20/3, E-Plane

(b.1) n=9 s0= 0/3, E-Plane

(c.1) n=16 s0= 0/2, E-Plane

(d.1) n=25 s0= 20/3, E-Plane Fig. 6. Directional eld enhancement of thials 34 (2011) 126130 129coupling effect of two discs, the E- and H-planes of the antenna ar-ray without any coupling are also presented in the gure. It is

(a.2) n=4 s0= 20/3, H-Plane

(b.2) n=9 s0= 0/3, H-Plane

(c.2) n=16 s0= 0/2, H-Plane

(d.2) n=25 s0= 20/3, H-Plane e n-element array at the mode HE11d.

noted that there is a difference of the eld pattern in these twocases. The directivity is 6.3dBi for the case with mutual coupling,

5. Conclusions

This paper presented a discussion on the directive eldenhancement in a disc antenna array in optical band. The propertyof dielectric materials was addressed, and eld modes in a cylindri-cal resonator were discussed. It was identied that the fundamen-tal mode of HE11d generated the far eld pattern with higherdirectivity than other modes. More effective eld enhancementin the radiation direction could be achieved by using a multiple-disc antenna array. Simulation examples indicated that the direc-tivity of the square array varied with the disc spacing, and the

Table 2Comparison of directivity in different modes (unit: dBi).

Mode HE11 TM01 TE01

2-disc@k0/2 6.3 5.12 2.74-disc@2k0/3 6.1 5.06 2.539-disc@k0/3 15.3 3.75 4.6816-disc@k0/2 28.9 8.25 11.425-disc@2k0/3 59.9 23.1 8.37

130 I. Wang, Y. Du /Optical Materials 34 (2011) 126130and 5.64dBi for the case without mutual coupling. This is mainlycaused by the interaction of the electromagnetic elds withintwo discs. This coupling enhances the directivity of the far eldpattern, and leads to a narrower main lobe and a higher directivity,as illustrated in Fig. 5.

The directional eld enhancement is a multiple-disc antennaarray is then investigated numerically by varying the disc num-ber and spacing. In all cases the mode HE11d is excited in thediscs. Fig. 2 shows the disc array with the number n of 2, 4, 9,16 or 25. The disc spacing between any two adjacent discs (fromedge to edge) remains same, but its value is taken to be k0/10,k0/4, k0/3, k0/2, 2k0/3, 3k0/4 or k0 in the stimulation. k0 is thevacuum wavelength of the eld at 500 THz, and is equal to600 nm.

Table 1 shows the directivity of disc antenna arrays at themode HE11d with different disc number and spacing. It is notedthat the directivity of the disc array generally increases withincreasing disc number. The maximum directivity of an n-elementarray is 6.1dBi for n = 4, and reaches 59.9dBi for n = 4. Fig. 6 showsthe far-eld pattern of the n-element array (n = 4, 9, 16 and 25)with the spacing for maximum directivity being achieved. Notethat the main lobe becomes narrow when the element numberincreases.

When the disc number is xed, the directivity of the disc arrayvaries with the disc spacing. The maximal directivity is observed ifthe disc spacing is approximately equal to half of the free-spacewavelength k0. This indicates that there is an effective mutual cou-pling among the disc elements. When the disc number increasesthe disc spacing for the maximum directivity varies in the rangeof k0/32k0/3.

A comparison of the directivity against other modes is also con-ducted. Table 2 shows the directivity of disc antenna arrays withxed spacing at three different modes HE11d, TE01d and TM0d. Thedisc spacing is selected in such a way that the directivity at themode HE11d reaches the maximal value. It is noted that the direc-tive eld enhancement at other modes is not as strong as that atthe mode HE11d. The mode HE11d radiates the electromagnetic eldwith the maximal directivity. .maximum directivity was observed when the disc spacing wasapproximately equal to the half of the vacuum wavelength. Themaximum directivity could be improved signicantly when thedisc number was increased.

Acknowledgments

The work leading to this paper was supported by grants fromthe Research Committee of the Hong Kong Polytechnic University,and the Research Grants Council of the Hong Kong Special Admin-istrative Region (Project No. 516008).

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Directional field enhancement of dielectric nano optical disc antenna arrays1 Introduction2 Dielectric material in optical band3 Analysis of electric field distribution4 FDTD simulation of dielectric disc antenna arrays5 ConclusionsAcknowledgmentsReferences