Controlling electromagnetic fields with graded photonic crystals in metamaterial regime

Controlling electromagnetic fields withgraded photonic crystals in

metamaterial regime

Borislav Vasic,1,∗ Goran Isic,1,2 Rados Gajic,1 and Kurt Hingerl3

1Institute of Physics, Pregrevica 118, P. O. Box 68, 11080 Belgrade, Serbia2School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, UK3Zentrum fur Oberflachen- und Nanoanalytik, Universitat Linz, Altenbergerstr. 69, A-4040

Linz, Austria∗[email protected]

Abstract: Engineering of a refractive index profile is a powerful methodfor controlling electromagnetic fields. In this paper, we investigate possiblerealization of isotropic gradient refractive index media at optical frequen-cies using two-dimensional graded photonic crystals. They consist ofdielectric rods with spatially varying radii and can be homogenized in broadfrequency range within the lowest band. Here they operate in metamaterialregime, that is, the graded photonic crystals are described with spatiallyvarying effective refractive index so they can be regarded as low-loss andbroadband graded dielectric metamaterials. Homogenization of gradedphotonic crystals is done with Maxwell-Garnett effective medium theory.Based on this theory, the analytical formulas are given for calculations of therods radii which makes the implementation straightforward. The frequencyrange where homogenization is valid and where graded photonic crystalbased devices work properly is discussed in detail. Numerical simulationsof the graded photonic crystal based Luneburg lens and electromagneticbeam bend show that the homogenization based on Maxwell-Garnett theorygives very good results for implementation of devices intended to steer andfocus electromagnetic fields.

© 2010 Optical Society of America

OCIS codes: (160.5298) Photonic crystals; (160.3918) Metamaterials; (260.2065) Effectivemedium theory; (080.2710) Inhomogeneous optical media; (230.0230) Optical devices.

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1. Introduction

Metamaterials are artificial electromagnetic structures consisting of unit cells whose size a ismuch smaller then the wavelength λ so they can be regarded as media with effective param-eters. In most of reported realizations, the ratio Ω = a/λ was not negligible but the effectiveparameters were still well defined. This transitional regime between the effective medium andthe photonic crystal regime is denoted as the metamaterial regime [1].

Graded matematerials (GMs) [2] consist of spatially varying unit cells and enable realizationof various gradient refractive index (GRIN) media for controlling electromagnetic fields. TheGRIN media were realized using GMs with metallic inclusions in the cases of electromagneticcloak [3–5], carpet cloak [6], field rotator [7] and bend structure [8] or using structured metallicsurfaces capable for focusing and guiding of surface waves [9–11].

However, in order to fabricate broadband and low-loss GRIN media at optical frequencies,dielectric media should be used instead of metallic inclusions. This has been recently reportedfor carpet cloak [12–15] and beam steering and focusing devices [16]. The basis of the deviceswas a dielectric slab structured with holes of variable relative distances [13] or with rods/holesof variable radii [15,16] and such slab can be modeled as medium with spatially varying refrac-tive index. This way for realizations of GRIN media suggests that they could be implementedby inhomogeneous photonic crystal (PC) with gradually varying unit cells.

Guiding of light in inhomogeneous PCs using not the existence of a photonic band-gap buta well designed spatially dependent dispersion was proposed in Refs. [17, 18]. The concept ofgraded photonic crystals (GPCs) was introduced in Refs. [19, 20] where it was shown how tobend a path of light by two-dimensional (2D) GPC with one-dimensional (1D) lattice gradient.Similar GPCs were then used for a design of focusing and guiding devices [21–23]. The previ-ous studies on 2D GPCs were focused on realizations of particular functions, but they did notanswer how to determine the geometry of unit cells in GPCs which would implement desiredGRIN media. Also, the 2D GPCs were graded in one direction giving way for realization ofGRIN media with a 1D gradient only. Graded subwavelength gratings could be also regardedas 1D GPCs and they were used for for achieving highly efficient diffractive components [24]and for realization of focusing lens [25] and graded-index waveguide [26].

In this paper we give a general procedure for implementation of specified isotropic GRINmedia with 2D GPCs. The GPCs are described as effective media and they operate in meta-material regime. The effective refractive index of the GPCs is controlled by varying the radiiof dielectric rods in both directions which enables realization of GRIN media with a 2D gradi-ent. This extends the scope of applicability of GPCs and enables realizations of more complexGRIN devices. Homogenization of GPCs is approximately done with Maxwell-Garnett (MG)effective medium theory (EMT). Using this approximation, we give analytical formulas to de-termine the rods radii. The GPCs are low-loss and enable implementation at optical frequencies


(a) (b) (c)

x

y

xi

yjnij

rod�

host�

nij

2rijy y

x x

Fig. 1. (a) Refractive index profile with 2D gradient and (b) its discrete approximation. (c)Implementation of the profile with the GPC.

as well. The proposed method is verified using numerical simulations of GPC based Luneburglens and electromagnetic beam bend where we focus on investigation of the frequency rangewhere the homogenization procedure is applicable.

2. Graded photonic crystals in metamaterial regime

Consider a 2D device with a GRIN medium whose refractive index profile n(x,y) in xy-planeis shown in Fig. 1(a). In order to realize this profile, it is firstly approximated with discrete oneas shown in Fig. 1(b). Each i j-cell is a square of side a, the coordinates of its central point arexi and y j and the cell refractive index, ni j, is equal to n(xi,y j). The ray path through the GRINmedium is governed by equations of Hamiltonian optics [27]. Hamiltonian for the i j-cell is aplane wave dispersion

k = ni jωc

, (1)

where c is speed of light in vacuum and k is modulus of wave vector in xy-plane.Implementation of the refractive index profile from Fig. 1(b) requires use of many materials

with appropriate discrete values of refractive index. If the refractive index distribution is slowlyvarying, simpler realization can be achieved by GPC with square lattice whose cross sectionin the xy-plane is shown in Fig. 1(c). Dielectric rods of only one material are used while theirradii are spatially varying. εrod is the permittivity of dielectric rods and εhost is the permittivityof the host medium. The field propagation can be treated by equations of Hamiltonian optics[17,18] if the field can be approximated with a plane wave locally. This is fullfiled in the lowestband except in the vicinity of the band edge [28]. Here equifrequency contours (EFCs) canbe approximated with circles and the GPC can be modeled with isotropic and homogeneousmedium locally [29], whose dispersion is

k =(nα

e f f (k))

i j

ω(k)c

, (2)

with(

nαe f f (k)

)

i jbeing the frequency dependent effective refractive index obtained from dis-

persion curves of a PC whose unit cell is the i j-cell whereas α stands for transverse magnetic(TM, magnetic field is in xy-plane) and transverse electric mode (TE, electric field is in xy-plane). Equations (1) and (2) have to be equivalent in order that the GPC from Fig. 1(c) wouldimplement GRIN medium from Fig. 1(b), which gives the condition to determine a radius ofrod, ri j, of the i j-cell:

(nα

e f f (k))

i j= ni j. (3)


Ω

n

Ω

n

Ω

n

Ω

n

(a1) (a2)

(b1) (b2)

0 0.1 0 0.1 0.20.2 0.3 0.4 0.30.5 0.4 0.5

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

1.5

2

2.5

3

3.5

1 1

1.5

2

2.5

3

1.2

1.4

1.6

1.8

2

1

1.2

1.4

1.6

1.8

2

1

Fig. 2. Refractive index calculated from dispersion curves of a PC as a function of normal-ized frequency Ω = a/λ for the PC with (a) SiO2 and (b) Si rods. On the left side, (x1), thevalues for TM mode are shown and on the right side, (x2), the values for TE mode. Straightred lines represent values of refractive index calculated with EMT. Ratios r/a are changedfrom 0.1 to 0.5: �(r/a = 0.1), ◦(r/a = 0.2), ♦(r/a = 0.3), �(r/a = 0.4), ∇(r/a = 0.5).

The ratio between unit cell size a and wavelength λ , Ω = a/λ , (introduced at the begin-ning of the paper) will be hereafter denoted as normalized frequency. In this paper we studiedGPCs with SiO2 (ε = 4.5) and Si (ε = 11.8) rods whose EFCs can be approximated withcircles locally up to even Ωmax ≈ 0.44 in the lowest band. This means that the GPCs can behomogenized [29] and work in metamaterial regime up to Ωmax which is a decisive reason forbroadband work of GPC based devices. In Fig. 2 are shown calculated values nα

e f f (k) (bluelines) of a PC for five different values r/a. The dispersion curves of the PC were calculatedusing COMSOL Multiphysics by considering single unit cell of the PC with periodic boundaryconditions. The eigenvalue problem was then solved numerically while the effective refractiveindex was calculated from Eq. (2). Due to spatial dispersion, the effective refractive index startsto differ from the value in the long-wavelength limit nα

e f f . But for the normalized frequenciesΩ � 0.25, we can adopt the approximation nα

e f f (k) ≈ nαe f f , so Eq. (3) reads

(nαe f f )i j = ni j. (4)

In the long-wavelength limit, PCs can be regarded as effective homogeneous media [30–32]with a diagonal effective dielectric tensor in principal set of axes. Denote principal dielectricpermittivities with εκ(κ = x,y,z). In a general case, the PCs are biaxial, but in the consideredcase of circular cilynders in a square lattice, they are uniaxial, εx = εy = εplane, that is, theyare isotropic in xy-plane [32]. In this paper we use analytical formulas for homogenization ofPCs in the long-wavelength limit since it significantly simplifies design of GPC based devices.Homogenization of periodic structures using analytical formula has been done long ago [33]whereas here we use the MG theory [30, 34, 35] so the effective permittivity εplane of uniaxialPC is given as [36]

εplane = εhost +f εhost(εrod − εhost)

εhost +Lplane(1− f )(εrod − εhost), (5)


while εz is obtained using the exact formula, that is, as the volume average of εrod and εhost

εz = f εrod +(1− f )εhost . (6)

Here f stands for the filling fraction of rods, f = r2π/a2 and Lplane = 1/2 is the depolariz-ing factor of rods in the long-wavelength limit for xy-plane. The effective refractive indicesbased on EMT theory are nT E

EMT = √εplane and nT MEMT =

√εz. Their values are represented withstraight, red lines in Fig. 2. As can be seen, in the long-wavelength limit, there is a completeagreement with values obtained from dispersion curves for TM mode while for TE mode, ac-ceptable agreement is observed for r/a < 0.4. This is in accordance with the results reportedin Ref. [34]. For TE mode, the electric field is normal to the rods in PCs and the main assump-tion in the derivation of MG theory is that a rod is placed in the local field of homogeneouslypolarized matter [36]. For low filling fraction, this is a good approximation, but with increas-ing radius of rods, the field within the rod becomes more perturbed and the approximation ofhomogeneously polarized environment fails. This is the reason for the significant differencebetween the effective index for TE mode obtained from MG theory and the one calculated fromdispersion curves for r/a > 0.4 even in the long-wavelength limit. In the case of TM mode, theelectric field is parallel to the rods so the PC is homogenously polarized even for large fillingfractions and field averaging give exact value in the long-wavelength limit.

Since the effective index nαe f f can be expressed using EMT, the condition from Eq. (4) now

reads (nαEMT )i j = ni j. Radii of rods in the GPC from Fig. 1 (c) can be determined by putting this

condition in Eqs. (5) and (6) with spatially dependent filling fraction, fi j = r2i jπ/a2. Finally, in

the case of TE polarization, the radius ri j is obtained from Eq. (5) as

ri j = a

√√√√ (εhost −n2

i j)(εhost + εrod)

π(εhost +n2i j)(εhost − εrod)

, (7)

whereas for TM mode, the radius is obtained from Eq. (6) as

ri j = a

√n2

i j − εhost

π(εrod − εhost). (8)

In the previous discussion we neglected frequency dispersion of rods permittivity so Eqs. (7)and (8) should be used in frequency range where this approximation is valid. The use of onlydielectric inclusions makes possible realization of low-loss devices but limits values of effectiverefractive index achievable with GPCs. The lower bound is 1 while the upper bounds are deter-mined with a maximum allowed filling fraction. In the case of TM mode, the maximum fillingfraction is 0.78 because of the touching rods whereas in the case of TE mode, the maximumfilling fraction is 0.5 due to the condition r/a < 0.4.

The analytical formulas [Eq. (7) and (8)] greatly simplify the design compared to that basedon extracting effective index from numerical simulations of unit cells and adjusting their ge-ometries to fit the specified refractive index. The main assumption was the approximationnα

e f f (k) ≈ nαEMT for Ω � 0.25 which is consistent with the result reported in Ref. [37]. The

approximation is applicable due to approximately linear dispersion curves of PC in the lowestband. In the next section, it will be shown that this approximation could give good results inimplementation of GRIN media even for frequencies Ω � 0.25, although here the differencebetween nα

e f f (k) and nαEMT is more pronounced. By calculating nα

EMT using extended MG the-ories [38, 39], it would be possible to completely match frequency dependent refractive indexnα

e f f (k) in Fig. 2, not only its value in the long-wavelength limit but the effective index nαEMT

would be then Ω dependent. On the other hand, Eqs. (7) and (8), enable frequency independentdesign applicable in broad frequency range.


-1 0 1 0 1

(a1) (a2)

(b1) (b2)

Fig. 3. FEM simulation results of TE mode for (a) the original lens and (b) the GPC basedLuneburg lens, Ω0 = 0.14 whereas Ω ∈ (0.14,0.2). On the left side, (x1), the z-componentof magnetic field is shown and on the right side, (x2), the magnetic field intensity.

3. Device implementation by graded photonic crystals in metamaterial regime

The proposed method was validated by numerical simulations of GPC based Luneburg lensand electromagnetic beam bend with finite element method (FEM) based software COMSOLMultiphysics. Since refractive index in the devices changes from minimal nmin, to maximalvalue nmax, common characteristic for all cells in GPCs which implement devices is free-spacewavelength λ0, while wavelength λ = λ0/n and the normalized frequencies Ω = a/λ varyfrom cell to cell. Therefore, the excitation frequency in simulations will be expressed in termsof Ω0 = a/λ0, while Ω will lie between nminΩ0 and nmaxΩ0.

3.1. 2D Luneburg lens

The Luneburg lens [40] is a spherically symmetric lens which focuses incoming electromag-netic field from any direction to the point at the opposite lens side or transforms a radiation ofa point source at the lens surface into parallel rays at the opposite lens side. The lens is veryattractive as a receiver and transmitter in many antenna applications and as a field concentratorfor focusing. Standard realization is based on spherical dielectric shells with constant permit-tivity. Using of PCs for optical devices such as lens was proposed in Ref. [41] whereas theimplementation of sliced Luneburg lens with drilled holes was presented within the frameworkof metamaterials in Ref. [42]. Recently, 2D Luneburg lens has been experimentally realized atmicrowave frequencies using metamaterials based on non-resonant metallic inclusions [43].

Here we considered the implementation of 2D cylindrically symmetric Luneburg lens withGPCs as an example for testing the proposed method for TE polarization. The lens refractiveindex is given by

n =

√

2−(ρ

R

)2, ρ < R, (9)

where R is the lens radius and ρ is modulus of radius vector of the lens internal points. Inorder to implement this permittivity profile (nmin = 1, nmax = 1.41), SiO2 rods (ε = 4.5) wereused whereas the host medium was vacuum. Radii of rods were calculated using Eq. (7). Thesimulation results for the original lens with refractive index profile given by Eq. (9) are shown


reflection

(dB

)

normalized frequency =a/� �0 0

0.1 0.15 0.2 0.25 0.3 0.35 0.40.05

0

-20

-40

-60

-80

-100

-120

-140

original lens

GPC lens

Fig. 4. (Reflection (S11 parameter) from the original (blue) and GPC (red) Luneburg lens.

in Fig. 3(a) whereas the results for the GPC lens are given in Fig. 3(b). Excitation frequency inthe simulated case was Ω0 = 0.14 while Ω ∈ (0.14,0.2). As can be seen the GPC lens worksas well as the original one, the plane wave from the left side is focused onto the opposite, rightside, while the field within the lens behaves as a plane wave locally which indicates that theeffective medium approximation is valid. To give exact criterion for the description of the lensperformances and to suggest possible way for experimental testing, we calculated reflection infront of the GPC lens as a function of excitation frequency Ω0, Fig. 4, while the reflection fromthe original lens is given for comparison. As can be seen, for Ω0 = 0.14, the reflections fromboth lenses are negligible meaning that the incident field does not resolve periodic structures ofthe GPC lens perceiving it as a locally homogeneous medium such as the original lens.

Simulation results for the GPC lens at lower frequencies, Ω0 = 0.08, Ω ∈ (0.08,0.11), areshown in Fig. 5(a). As can be seen, the lens works fine while the reflection is low, Fig. 4.However, the focus became wider, it was moved to the lens interior and field intensities in thefocus were decreased. These lower performances are because the original lens was designedin the limit of geometrical optics. Nevertheless, simulations showed that the lens worked welleven for Ω0 = 0.067, far beyond the geometrical optics when free-space wavelength was equalto the lens radius. It should be emphasized that existence of the lower frequency limit is notconsequence of the implementation with GPCs.

The GPC lens performances at higher frequency can be explored through the simulationfor Ω0 = 0.3, Ω ∈ (0.3,0.42), where the reflection is still low and at the same level as forthe original lens, Fig. 4. As can be seen from Fig. 5(b), the GPC lens works very fine forthis frequency, focal point is narrower, field intensities are greater, so focusing is even bettersince the work at higher frequencies is closer to the limit of geometrical optics. But for Ω �0.25, the difference between the effective index based on MG theory and the effective indexcalculated from dispersion curves of PC results in slight changes of refractive index profilewithin the GPC lens so the focus point is moved toward the lens interior. Therefore, for thefrequencies Ω � 0.25, implementation of the Luneburg lens with GPCs is still possible sinceit is not strongly sensitive to slight changes of refractive index distribution but for the precisecontrol, the effective index should be calculated from dispersion curves of PCs.

Further increasing of excitation frequency Ω0 above 0.3 leads to strong reflection from theGPC lens, Fig. 4. Since the reflection from the original lens is still negligible, the increasedreflection means that the incident field does not perceive the GPC lens as a locally homogeneousmedium anymore. This can be also observed as a standing wave pattern in a front of and withinthe central part of the GPC lens, Fig. 5(c), as a result of Bragg reflections from the central cells.


(b)(a)

0 02.6 5.1

(c) (d)

Fig. 5. FEM simulation results of the magnetic field intensity for TE mode in the GPCbased Luneburg lens: (a) Ω0 = 0.08, (b) Ω0 = 0.3, (c) Ω0 = 0.33 and (d) Ω0 = 0.38.

The effective refractive index is the largest in the center of the lens, n = nmax, implying thelowest value of λ so this is the place where Ω reaches the Bragg reflection condition firstly. Withfurther increasing of Ω0, the Bragg reflection condition becomes fulfilled in parts of the GPClens with lower value of effective refractive index, so Bragg reflections start to appear from theside cells in the GPC lens as well, Fig. 5(d). Therefore, based on calculation of reflection fromthe GPC lens, we conclude that Ω0 ≈ 0.3 is the maximal excitation frequency and Ω ≈ 0.42 isthe upper frequency limit for proper work of GPC lens.

The maximal excitation frequency for proper work of the GPC lens can be determined withsimple approximative formula. The upper frequency limit Ω ≈ 0.42 after which Bragg reflec-tions appear is very close to Ωmax ≈ 0.44, the maximal normalized frequency for the homoge-nization and work in metamaterial regime for PCs. This means that Bragg reflections firstly ap-pear when Ω starts to overcome Ωmax within the part of a GPC based device with the highest ef-fective refractive index. For the lens, this is the central part. Therefore, for the proper work of thedevice, the excitation frequency have to satisfy the approximative condition Ω0 � Ωmax/nmax.The advantage of this formula is because Ωmax can be simply obtained by calculating EFC forunit cell which implements the highest effective refractive index.

It should be noted that the upper frequency limit depends on resolution of a discrete refrac-tive index profile. When the discrete profile approximates the original one sufficiently well,further decreasing of unit cell side will not significantly improve device performances but itwill enable work at higher frequencies due to scaling law of PCs. Of course, the price will bemore complicated fabrication of smaller inclusions in GPCs. In previous examples, the lens wasimplemented with 15 unit cells per radius, which implied that the change of refractive index perunit cells was 0.027. Simulations of the lens (not shown here) implemented with 10 (approxi-mately 33% decreased) unit cells per radius and the change of refractive index per unit cells of0.041 showed that it worked fine, but the upper frequency limit was decreased by approximately33%.


(a) (b)

x

y

u

v

eu

2eu

1

eu

1

eu

2

u1 u2

0

��

�-complex plane z-complex plane

Fig. 6. The mapping z = exp( w

C

), C = 1, for a design of an electromagnetic beam bend: (a)

the square domain in Cartesian coordinates of complex plane w = u+ iv and (b) its image,the annular segment in curvilinear coordinates of complex plane z = x + iy. The inner andouter bend radii are exp(u1) and exp(u2), respectively, whereas π/2 is the bend angle. Redarrows show propagation directions of an electromagnetic wave.

3.2. Electromagnetic beam bend

The bend is a device intended for steering and guiding of electromagnetic beams. In Ref. [44]we investigated the bend design and implementation in the framework of transformation opticswhereas here we designed it by the conformal mapping [45]. The underlying mapping is

z = exp(w

C

), (10)

which is also used for a design of transformation optics based devices [46]. By this mapping,square domain from straight Cartesian grid in w-complex plane, w = u + iv, is transformed toannular segment in polar coordinates of z-complex plane, z = x + iy, as shown in Fig. 6. Theconstant C can be used for adjusting dielectric profile within the bend. If the square domain isempty that is, if refractive index within square domain is nw = 1, refractive index profile withinthe bend is [45]

n =∣∣∣∣dwdz

∣∣∣∣ =

C√

x2 + y2. (11)

Exhaustive analysis of a Gaussian beam propagation through 2D PCs with slowly varyingnonuniformities was given in Ref. [18] while the experimental realization of the bend withstructured dielectric slab has been recently reported [16]. Here we investigated validity of theproposed method for TM mode. For the chosen bend geometry refractive index varied fromnmin = 1.25 to nmax = 2.86. This is a significantly larger variation of the refractive index com-pared to the Luneburg lens so a denser distribution of rods was required in order to obtain goodbehaviour. The bend was implemented with Si (ε = 11.8) rods in vacuum background, Eq. (8)was used to determine radii of rods while C = 1. The simulation results for the original and GPCbased bend are shown in Fig. 7(a) and 7(b), respectively, with excitation frequency Ω0 = 0.11while Ω∈ (0.137,0.32). As can be seen, the field distribution in the GPC bend matches the fieldin the original one and the field perceives the GPC bend as an effective medium. In order toexactly compare performances of GPC bend and the original one, we calculated transmissionsthrough both of them which could be also appropriate for experimental testing. The calculationresults are shown in Fig. 8 and for Ω0 = 0.11 both bends have the same transmission.

However, both bends suffer from the impedance mismatching at the entrance and the exitwhich leads to the reflection of the incident field which results in decreased transmission, Fig. 8,while the standing wave appears in front of the bend as well as within it, Figs. 7(a2) and 7(b2).


(a1) (a2)

(b1) (b2)

-0.8 01.3 1.3

Fig. 7. FEM simulation results of TM mode for (a) the original bend and (b) the GPC basedbend, Ω0 = 0.11. On the left side, (x1), the z-component of electric field is shown and onthe right side, (x2), the electric field intensity.

The reflection could be decreased by designing appropriate antireflection coatings or by usingnumerical conformal transformations for a design which results in reflectionless devices for TMmode [47]. Implementation of the later approach requires permittivity below one within someparts of the devices so metallo-dielectrics structures have to be used. Also, the question is ifit is always possible to find appropriate conformal/quasiconformal transformation for a devicedesign. In cases where it is not, ray-tracing method based on Eikonal equation could be usedinstead [8, 16, 48].

The design based on conformal mappings is valid in the limit of geometrical optics [45] sorefractive index should vary slowly over the wavelength [27]. Although the gradient of refrac-tive index in the bend was 3.5 times larger than in the Luneburg lens, the simulations showedthat the bending effect worked even in the cases where wavelength was comparable to the benddimensions while the transmissions stay at the same level for both bends, Fig. 8. However, abeam spreading at the lower frequencies was more distinct so the beam exceeded the bend widthsignificantly causing distortion of wave front. Such case is shown in Fig. 9(a) for Ω0 = 0.068.

As can be seen from Fig. 8, the transmissions for the both bends are at the same level up toΩ0 = 0.15 (Ω is between 0.187 and 0.43). The simulation results for this excitation frequencyare shown in Fig. 9(b). The bending effect works well, but there is a little phase delay (abouthalf a wavelength) in the GPC bend compared to the original one (simulation not shown here).This indicates a slight difference between original distribution of refractive index and the oneimplemented with GPC as a result of applied homogenization based on EMT. The bending ef-fect showed robustness to this difference but in cases where refractive index have to be preciselycontrolled, it should be calculated from PC dispersion curves.

Beyond the frequency Ω0 ≈ 0.15 transmission for the GPC bend decreases abruptly while itstays at the same level for the original bend, Fig. 8. This means that the incoming field does not


transm

issio

n(d

B)

normalized frequency =a/� �0 0

original bend

GPC bend

0.06 0.08 0.20.140.120.1-40

0.16 0.18

-30

-25

-20

-15

-10

-5

-35

0

Fig. 8. Transmission (S21 parameter) through the original (blue) and GPC (red) electro-magnetic beam bend. The small oscillations in transmissions are Fabry-Perot resonanceswithin the bends due to impedance mismatching at the bend entrance and exit.

(a) (b)

-0.56 1 -1.8 1.77

Fig. 9. FEM simulation results of the z-component of electric field for TM mode in theGPC based bend: (a) Ω0 = 0.068 and (b) Ω0 = 0.15.

perceive the GPC bend as an effective medium anymore. Bragg reflections start to appear fromthe cells along the inner bend edge because the refractive index is the highest and the wave-length is the lowest here and this is the place where Bragg condition is satisfied firstly. Sincethe bending effect does not work anymore, Ω0 ≈ 0.15 is maximal allowed excitation frequencywhile Ω ≈ 0.43 is the upper frequency limit for proper work of GPC bend. As in the case of theLuneburg lens, the upper frequency limit Ω ≈ 0.43 is very close to Ωmax. Therefore, achievingof that limit can be again interpreted as a result of increasing of Ω above Ωmax for the cellswith the highest effective refractive index in GPC. As a result, the maximal allowed excitationfrequency can be again determined using the approximative formula Ω0 � Ωmax/nmax.

4. Conclusion

In a summary, the proposed method for implementation of isotropic GRIN media using GPCsin metamaterial regime was validated through numerical simulations of the Luneburg lens andthe beam bend. It was shown that they can be implemented with dielectric rods of only one


material by changing their radii. This makes possible fabrication of low-loss devices at opticalfrequencies. Simplicity of the proposed method is a result of approximately linear dispersion ofPCs for wide frequency range in the lowest band. This enables calculation of effective param-eters in the framework of MG theory and analytical formulas for determination of rods radii.As a conclusion, the GPC studied in this paper can be regarded as a class of graded dielectricmetamaterials.

The proposed method enables work in broad frequency range in the lowest band. For normal-ized frequencies Ω � 0.25, the GPC based devices worked as well as the original GRIN devicesso here the proposed method for implementation gave excellent results. At higher frequencies,Ω � 0.25, homogenization of PCs is still possible since EFCs can be approximated with circles.Although effective parameters of PCs then start to differ from the values obtained from MG the-ory, in the examples of the Luneburg lens and the bend structure it was shown that the methodgave good results and it was very robust. The conclusion is that the method is applicable forimplementation of devices intended to guide, steer and focus electromagnetic fields, which areinsensitive to a slight variation of refractive index. But, for Ω � 0.25, in the application whereit is necessary to control refractive index very precisely, it have to be calculated from dispersioncurves.

The upper frequency limit for proper work of GPC based devices was detected with appear-ance of Bragg reflections from GPC devices. This happens when normalized frequency Ω inthe parts of the devices with the highest value of refractive index starts to overcome Ωmax, themaximal normalized frequency for the homogenization and work in metamaterial regime forGPCs. As a result, the maximal excitation frequency for proper work of the GPC devices can bedetermined using approximative formula Ω0 � Ωmax/nmax. Although the original devices withGRIN medium were designed in the limit of geometrical optics, simulations showed that theGPC based bend and lens worked very well outside the limit of geometrical optics.

Acknowledgments

This work is supported by the Serbian Ministry of Science under Project No. 141047. G.I. ac-knowledges support from ORSAS in the U. K. and the University of Leeds. R.G. acknowledgessupport from EU FP7 projects Nanocharm and NIMNIL. K.H. is grateful to the Austrian NIL-meta-NILAustria project from FFG for partial support. We thank Johann Messner from the LinzSupercomputer Center for technical support.


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Controlling electromagnetic fields with graded photonic crystals in metamaterial regime

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