Luneburg lens composed of sunflower-type graded photonic Crystals

Xiao-Hong Sun a,n, Yu-Long Wu a, Wei Liu a, Yu Hao a, Liu-Di Jiang b

a Henan Key Laboratory of Laser and Opto-electric Information Technology, Zhengzhou University, Henan 450052, Chinab Engineering Materials Research Group, Faculty of Engineering and the Environments, University of Southampton, SO17 1BJ, United Kingdom

a r t i c l e i n f o

Article history:Received 28 August 2013Received in revised form25 October 2013Accepted 12 November 2013Available online 27 November 2013

Keywords:Sunflower-type photonic crystalLuneburg lensGraded PC

a b s t r a c t

Sunflower-type graded photonic crystals (GPCs) are investigated and used to design the Luneburg lensfor transverse electric (TE) and transverse-magnetic (TM) polarizations. Our investigation suggests thatthese novel structures present better focusing characteristics as well as wider transmission bandwidthfor TM polarization than that for TE polarization. It is envisaged that these sunflower-type GPCs can bepotentially used in optical system where compact and powerful focusing elements are required such asthe Luneburg lens.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

@Gradient index (GRIN) structures have attracted considerableinterest in the recent years due to their potential applicationsranging from invisibility cloaks [1,2] and optical black holes [3],to planar devices such as Luneburg lens [4,5]. Recent introduction ofGPCs has demonstrated significant potentials in focusing and lightguiding devices [6–9] as results suggest that two-dimensional (2D)GPCs with one-dimensional lattice gradient can bend the lightpath[10,11]. As such, the possible realization of isotropic GRINmedia using 2D GPCs in metamaterial regime was reported[12,13]. It was proved that these isotropic GRIN media composedof 2D GPCs can lead to promising devices which are particularlydesigned to control electromagnetic fields [14,15]. Nonetheless,all these previous studies focus on realization of particular functions.To the best of our knowledge, there is little study on the effect oflattice geometry on functionalities. Furthermore, only GPCs withhigh periodicity such as square or hexagonal lattices and quasi-periodicity like dodecagonal quasicrystal lattice [16–18] werereported.

Compared with periodical and quasi-periodical photonic crystals(PCs), sunflower-type PC is non-periodic with systematic arrange-ment of dielectric rods, which exhibits six-fold rotational symmetry.The dielectric rods are arranged in the form of concentric circles.It can provide isotropic optical properties due to its high symmetry inthe wave vector space [19,20]. These characteristics are advantageous[19] for the design and fabrication of focusing devices such asLuneburg lens.

In this work, we investigate novel designs of GRIN media whichconsist of sunflower-type GPCs. In particular, Luneburg lens usingsunflower type GPCs have been studied in detail. The transmissioncharacteristics of these lenses have been calculated. Finite elementmodeling (FEM) has been employed to study the time-averagedintensity distributions at different frequencies. The results showthat these novel structures present better focusing characteristicsas well as wider transmission bandwidth for TM polarization thanthat for TE polarization.

2. Transmission analysis of sunflower type PCs

A typical scheme of a 6-fold sunflower-type PC is shown in theFig. 1. Spatial lattice positions on the x-y plane are given by

xN;l ¼ aM cos ð2lπ=6MÞ; yN;l ¼ aM sin ð2lπ=6MÞwhere a is the lattice constant which is defined as the radialperiodicity of the adjacent concentric rows, integer M is the radialindex, and l¼1, 2,…, 6 M is the angular index of the rods. Thedielectric rods are arranged in the form of concentric circles withradial distance a.

For fixed value of rod radius r, a filling fraction of the rods canbe obtained by using the formula f¼Nπr2∕S, where S is the area ofphotonic structures, N is the number of the rods. For instance, for atriangle indicated in Fig. 1, it has an area S¼ 0:5a2 sin ðπ=3Þ andN¼0.5. Thus, the filling fraction can be denoted asf ¼ πr2=a2 sin ðπ=3Þ.

In this work, to facilitate the investigation, a typical sunflower-type PC with M¼14 and a¼0.5 μm are adopted, which consistsSiO2 dielectric rods (εr¼2.4) embedded in air media (εhost¼1).The transmission spectra of such a PC are calculated at different filling

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Optics Communications

0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.optcom.2013.11.022

n Corresponding author.E-mail address: iexhsun@zzu.edu.cn (X.-H. Sun).

Optics Communications 315 (2014) 367–373

fractions for TE and TM polarizations, respectively. Fig. 2 showsthat the transmission bandwidth decreases with the increase offilling factor r/a in both TE and TM modes while the change ismore notable for the TE mode.

3. Design of Luneburg lens using 2D sunflower type GPCs

3.1. Design principles

A Luneburg lens is a spherically symmetric lens which focusesincoming electromagnetic field from any direction to the point atthe opposite lens side or transforms a radiation of a point source atthe lens surface into parallel rays at the opposite lens side. Thelens is very attractive as a receiver and transmitter in manyantenna applications and as a field concentrator for focusing.Standard realization is based on spherical dielectric shells withconstant permittivity. A Luneburg lens [21] is a GRIN element withrefractive index distribution:

nðρÞ ¼ n0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�ðρ=RÞ2

qð1Þ

where n0 is the refractive index outside the lens region, ρ is theradial polar coordinate within the lens region and R is the radius ofthe lens. The functionality of the lens is strongly dependent on therefractive index n(ρ) with different coordinate ρ.

From the Maxwell–Garnett effective medium theory, the effectivepermittivity of GPCs with rods embedded in a host medium is [22]

εx ¼ εy ¼ εplane ¼ εhostþf εhostðεrods�εhostÞ

εhostþ0:5ð1� f Þðεrods�εhostÞð2Þ

while εz can be obtained from

εz ¼ ð1� f Þεhostþ f εrods ð3Þwhere f is the filling fraction of rods, εhost is the permittivity of thehost medium, εrods is the permittivity of dielectric rods. The effectiverefractive indexes of the GPCs are nte ¼ ffiffiffiffiffiffiffiffiffiffiffiεplane

p and ntm ¼ ffiffiffiffiffiεz

p.

Thus, in the case of TE polarization, the radius rte can beobtained from Eq. (2):

rte ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSπN

ðεhost�n2ÞðεhostþεrodsÞðεhostþn2Þðεhost�εrodsÞ

sð4Þ

And for TM polarization, the radius rtm can be obtained from Eq. (3):

rtm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSðεhost�n2Þ

πNðεhost�εrodsÞ

sð5Þ

Eqs. (4) and (5) can be put in Eq. (1) to provide evaluation of the lensand thus provide the fundamental guidance for the designs of planarLuneburg lens using graded sunflower-type PCs. which is depicted inthe following sections.

3.2. Structure design considerations

When a Luneburg lens is placed in air (n0¼1), the refractiveindex within the Luneburg lens can range from nmin¼1 tonmax¼1.41. The radii of the rods in the sunflower type GPCs atdifferent circles can be obtained from Eqs. (1), (4) and (5). Theradius of R¼ 14a¼ 7 μm is considered for the Luneburg lens.The calculated radii of rods are shown in the Tables 1 and 2 forTE and TM polarization, respectively. Fig. 3a and b shows the TEand TM mode spectra, respectively, of the designed Luneburglens using sunflower-type GPCs. It is clearly shown that therods of the TE mode lens are more intensive than the TMcounterpart.

3.3. Evaluation of transmission characteristics

In order to evaluate the properties of the designed lens,transmission coefficients for TE and TM polarization plane wavesas a function of the frequency ω were calculated using FiniteDifference Time Domain (FDTD) method based software RSOFT.Fig. 4 shows their respective spectra as a function of theFig. 1. Characteristics of sunflower-type photonic crystal.

Fig. 2. Transmission spectra of (a) TM and (b) TE mode for sunflower type PCs with different filling factors.

X.-H. Sun et al. / Optics Communications 315 (2014) 367–373368

dimensionless units of ϖ¼ωa/2πc, where a is the distancebetween the adjacent circles and c is the speed of light in vacuum.It is observed that the TM transmission curve is smoother and haswider transmission bandwidth as in comparison with the TEresults. Initial reflection is observed for the TE mode spectrum inthe frequency range of approximately 0.1–0.175. Strong reflectionand the rapid decrease of the transmittance are clearly seen for

frequency above 0.325. This increase in the reflection from TEmode can be explained by the fact that the incident electromag-netic plane waves don't perceive lens structures as a locallyhomogeneous medium further more. In the range of frequency0.376–0.40, the transmittance of TE mode has a little increase.Relatively speaking, TM mode has higher transmission efficiencyin the range of 0.31–0.42.

From Tables 1 and 2, we can see that the rods radius of TE lensis larger than that of TM lens for the same radial index. That is tosay, the effective refractive index of TE lens is larger than that ofTM lens in the same location from the center. With the increase offrequency, the Bragg reflection condition of TE lens was fulfilledfirst. The focusing characteristics became worse firstly. So TM lenshas wider transmission bandwidth and better focusing thanTE lens.

3.4. Distributions of field and intensity of different Luneburg lens

The electromagnetic field distribution and time-averagedintensity are calculated using FEM method in Comsol Multi-physics package. Fig. 5 shows the distribution of Hz and Ez fieldsfor the original Luneburg lens, the lens of TE and TM polariza-tions, respectively, at a specific frequency of 0.376. Scatteringboundary condition is used in the FEM. In comparison with theTE mode, the lens of TM polarization performs better in focusing

Fig. 3. Schematic of Luneburg lens based on the sunflower-type GPCs for (a) TE and (b) TM modes.

Table 1Radii of rods at different concentric circles for TE mode.

M (radial index) 0 1 2 3 4Radii of rods (μm) 0.236197 0.235795 0.234574 0.232497 0.229495M (radial index) 5 6 7 8 9Radii of rods (μm) 0.225465 0.220255 0.213649 0.205332 0.194841M (radial index) 10 11 12 13 14Radii of rods (μm) 0.181452 0.163949 0.140006 0.103851 0.

Table 2Radii of rods at different concentric circles for TM mode.

M (radial index) 0 1 2 3 4Radii of rods (μm) 0.221869 0.221302 0.219593 0.216715 0.21262M (radial index) 5 6 7 8 9Radii of rods (μm) 0.207237 0.20046 0.192144 0.182077 0.169948M (radial index) 10 11 12 13 14Radii of rods (μm) 0.155276 0.137246 0.11428 0.082348 0.

Fig. 4. Transmission coefficients for TM (solid) and TE polarization (dashed).

X.-H. Sun et al. / Optics Communications 315 (2014) 367–373 369

as the plane wave from the left side is focused onto the oppositeright side as shown in Fig. 5f. At the same time, the TM lens morerealistically imitates the focusing characteristics of the originalLuneburg lens as shown in Fig. 5a and b. We observe strongscattering and reflection for the lens with TE polarization, asshown in Fig. 5c. It is thus envisaged that higher efficiency can beachieved for the TM lens due to the low scattering and loss ofenergy.

Fig. 6 shows the comparison results of the time-averagedintensity distributions along the longitudinal (x) direction and

the transverse (y) direction in the image plane. The intensity ofTE lens is magnified 105 times in Fig. 6 to facilitate the comparison.The full width at half-maximum (FWHM) of the intensity obtainedfor TM lens can be calculated to be FWHM¼1.4a¼0.526λ, whereasthe axial focus size is FWHM¼3.9a¼1.466λ. In addition, nopronounced focus is observed at ϖ¼0.376 for the TE lens. Forcomparison, FWHM of the TE lens at ϖ¼0.35 was estimated.The focus size of the TE graded lens is FWHM¼1.33a¼0.466λ andthe axial focus size is FWHM¼3.64a¼1.27λ. For the originalLuneburg lens, it is estimated that the axial and transverse focus

Fig. 5. Distributions of Hz field for the original Luneburg lens (a) and TM lens (e), and Ez field for TE lens (c). Distributions of the time-averaged magnetic field intensity forthe original lens (b) and TM lens (f), and electric field intensity for TE lens (d). In all cases, the frequency is ϖ¼0.376.

X.-H. Sun et al. / Optics Communications 315 (2014) 367–373370

size of TM mode is FWHM¼3.4a¼1.28λ and FWHM¼1.25a¼0.47λ,respectively.

The time-averaged intensity distributions at frequenciesϖ¼0.3, 0.325, 0.35 and 0.393 are shown in Figs. 7 and 8. Similarfocusing effect is observed at ϖ¼0.3, 0.325, 0.35, while weakerintensity is obtained for the TE lens. Nonetheless, for ϖ¼0.393,strong scattering happens in the central area for the TE lens. It isthus concluded that TE lens can performs good focusing with

frequency up toϖ¼0.36 while the upper limit of frequency for theTM lens is ϖ¼0.4. It is important to note that, in contrast toLuneburg lens based on conventional periodical GPCs[12], theherein designed lens using sunflower type GPCs present improvedfocusing capability for the frequencies above 0.3. This is attribu-table to the high rotational symmetry and absence of periodicity insunflower type GPCs which prevents the occurrence of the Braggreflection at specific frequencies.

Fig. 6. Time-averaged field intensity distribution along (a) the longitudinal (x) direction and (b) the transverse (y) direction in the image plane for the original lens (dashed),TE graded lens (solid) and TM graded lens (dotted).

Fig. 7. Distributions of the time-averaged intensity for TM graded lens at different frequencies.

X.-H. Sun et al. / Optics Communications 315 (2014) 367–373 371

3.5. Further modeling optimization for the TM lens

From Figs. 6a and 7, it can be clearly observed that the focusingsize is large (FWHM¼3.9a and FWHM¼1.4a for the axial andtransverse direction at ϖ¼0.376) and the focus point also shiftstowards the outside of the TM lens. To overcome this, the Eq. (5) ismodified as r′tm ¼ 1:09rtm ¼ 1:09

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSðεhost�n2Þ=πNðεhost�εrodsÞ

p.

That is to say, the filling factor is modified asf 0 ¼Nπr2=S¼ 0:5πr2=ðπa2=6Þ ¼ 3r2=a2. Based on these equations,the radii of the rods are calculated and results are listed inTable 3. The intensity distributions using this modified equationare shown in Fig. 9. It is clearly observed that the optimizedmodeling has brought the intensity distributions and focusingeffect much closer to the original lens shown in Fig. 5b.The focusing is also much narrower (FWHM¼3.45a and

FWHM¼1.2a at ϖ¼0.376). All leads to improved performanceof the lens.

4. Conclusions

Novel Luneburg lens using sunflower-type GPCs has been investi-gated. Transmission characteristics of the designed lenses have beenevaluated in detail and the spectra of TE and TM modes have beencompared and analyzed. In particular, it is observed that the TMlens can lead to better focusing characteristics and larger trans-mission bandwidth as in comparison with those from the TE lens.Through in-depth studies of the field and intensity distributions,upper frequency limits for the best focusing performance have alsobeen estimated, which can be potentially applied for the lens

Fig. 8. Distributions of the time-averaged intensity for TE graded lens at different frequency.

Table 3Radii of rods at different concentric circles for TM mode.

M (radial index) 0 1 2 3 4Radii of rods (μm) 0.243975 0.243352 0.241473 0.238308 0.233805M (radial index) 5 6 7 8 9Radii of rods (μm) 0.227885 0.220433 0.211289 0.200219 0.186881M (radial index) 10 11 12 13 14Radii of rods (μm) 0.170747 0.15092 0.125666 0.0905522 0

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designs. Further optimization of the calculation model has alsobeen achieved which led to much improved performance of theTM lens. The overall results from this work suggest that Luneburglens using sunflower-type GPCs leads to promising functionalitiesof the lens.

Acknowledgments

The work has been fully sponsored by the National NaturalScience Foundation of China (11104251) and the Henan higherschool funding scheme for young teachers (2009GGJS-012).

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Fig. 9. The modified intensity distributions.

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