Band edge states of the < n >=0 gap of Fibonacci photonic lattices

Band edge states of the

Š

n

‹

= 0 gap of Fibonacci photonic lattices

A. Bruno-Alfonso,1 E. Reyes-Gómez,2S. B. Cavalcanti,3,4and L. E. Oliveira4,5

1

Faculdade de Ciências, UNESP-Universidade Estadual Paulista, 17033-360, Bauru, São Paulo, Brazil

2

Instituto de Física, Universidad de Antioquia, AA 1226, Medellín, Colombia

3

Instituto de Física, Universidade Federal de Alagoas, Maceió, Alagoas, 57072-970, Brazil

4

Inmetro, Campus de Xerém, Duque de Caxias, Rio de Janeiro, 25250-020, Brazil

5

Instituto de Física, Universidade Estadual de Campinas-UNICAMP, CP 6165, Campinas, São Paulo, 13083-970, Brazil

共Received 14 May 2008; published 4 September 2008兲

The stationary and normally incident electromagnetic modes in Fibonacci lattices with generating layers of positive and negative indices of refraction are calculated by a transfer-matrix technique. It is shown that the condition for constructive interference of reflected waves is fulfilled when the ratio of optical paths in positive and negative media are given by the golden ratio. Furthermore, in the long-wavelength limit, it is demonstrated that the edges of the具n典= 0 gap are the frequencies satisfying the conditions具⑀典= 0 and具␮典= 0.

DOI:10.1103/PhysRevA.78.035801 PACS number共s兲: 42.70.Qs, 42.70.Gi

Conventional band gap materials based on right-handed electromagnetism, i.e., materials in which the phase velocity points in the same direction as the group velocity, have been extensively studied. They exhibit a photonic band gap due to the constructive interference of multiple Bragg scattering and thus, a sensitive gap with respect to changes of the lattice parameter关1兴. Recently, materials exhibiting simultaneously negative permittivity ⑀ and negative permeability ␮ and, therefore, a negative index of refraction n=

冑

⑀

冑

␮, have drawn considerable attention. These have been referred to as metamaterials, or left-handed materials, once idealized by Veselago 关2兴, who deduced that the propagation of electro-magnetic radiation through such media should exhibit back-ward wave propagation and reversed refraction. Lately, ma-terials with ⑀ near zero have also received considerable attention for their unexpected behavior that permits the squeezing and tunneling of radiation关3兴.

Theoretical and experimental studies关4–7兴on the propa-gation of light through one-dimensional共1D兲periodic arrays of alternating layers of positive and negative materials have evidenced the existence of a band gap which is insensitive to lattice parameter changes in contrast with the behavior ex-hibited by Bragg gaps. The physical origin of such a gap in periodic arrangements stems from the fact that there is no net phase change during propagation in the sense that the phase delay experimented in the positive media is reversed in the negative one and, therefore, there is no net phase delay in one unitary cell. This leads to the condition具n典= 0共null av-erage of the refractive index兲, which depends only on the ratio of the layer thicknesses关8,9兴. There are cases where the 具n典= 0 gap is omnidirectional, but this is not a rule关6兴.

Recently, the具n典= 0 gap of 1D periodic arrays of alternat-ing layers of air and a negative dispersive medium with Drude-type behavior has been thoroughly studied 关9–11兴. It has been demonstrated both the robustness of the具n典= 0 gap to changes in the unit-cell size, and the fact that, in the long-wavelength limit, the lower and higher gap edges satisfy the 具⑀典= 0 and 具␮典= 0 conditions. Furthermore, Da et al. 关12兴 have theoretically analyzed the influence of losses in Fi-bonacci lattices containing left-handed material and con-cluded it does not change the characteristic of omnidirec-tional reflection.

In this work we study the 具n典= 0 gap of a periodic struc-ture where the unit cell is a Fibonacci chain of layers Aand B. Hence, the quasiperiodic limit is essentially attained when the number of layers in the unit cell is sufficiently large. The condition for constructive interference of reflected waves, known in periodic systems to define Bragg mirrors, leads to 具n典= 0 for the zeroth-order gap. Furthermore, in the long-wavelength limit, the gap-edge frequencies are shown to cor-respond to null average values of the electric and magnetic response functions.

Let us begin by considering a periodic layered structure with interfaces normal to the zaxis, where each unit cell is composed by layersAandB. The constituent materials inA and B are air and a dispersive metamaterial following the dispersion law used by Li et al.关4兴, respectively. A diagram of this structure is shown in Fig. 1. The width, electric per-mittivity, and magnetic permeability of the layerA共B兲area 共b兲,⑀A共⑀B兲, and␮A共␮B兲, respectively. Figure2displays the ratio between the dispersion laws governing the electric and magnetic responses, as well as the index of refraction, of the materialsAandB. The responses are plotted as functions of the frequency ␯=␻/共_2␲兲_{. The harmonic electromagnetic}

waves which propagate along the z-axis direction with the magnetic field along thex-axis direction are described by the fields

Hជ共z,t兲=eˆx␺共z兲e−i␻t, Eជ共z,t兲=eˆy ic␸共z兲

␻ e

−i␻t

, 共1兲

where␺共z兲and␸共z兲=␺

_⬘

共z兲/⑀共z兲 satisfy

A B A A B A B A A B A A B A B A

0 S5 L5 S5 2L5

z x

... ...

FIG. 1. 共Color online兲 Photonic lattice whose unit cell is the Fibonacci chainS5=ABAABABA.

PHYSICAL REVIEW A78, 035801共2008兲

−␸

_⬘

共z兲=␻2_{␮共z兲␺共z兲/c}2

. 共2兲

Moreover, both ␺共z兲 and ␸共z兲 should be continuous across the interfaces. Let Lbe the size of the lattice unit cell. This means that ⑀共z+L兲=⑀共z兲 and ␮共z+L兲=␮共z兲. Due to the translational invariance of the lattice, the functions␺共z兲may be labeled by the Bloch wave vectorq, i.e., they satisfy the Bloch condition␺q共z+L兲=eiqL␺q共z兲.

The Fibonacci chain ofmth order is obtained according to the concatenation rule Sm=Sm−1Sm−2, with the first two chains as S0=B and S1=A. For instance, the chain of fifth order is S₅=ABAABABA, as shown in Fig. 1. This kind of structure has been theoretically 关4,12兴 and experimentally 关13兴 investigated. The number of layers in Sm is the Fi-bonacci number Fm+1, where F1= 1, F2= 1, and Fm=Fm−2 +Fm−1. A very important property of the Fibonacci numbers involves the quotient␶m=Fm+1/Fm, with

lim m→⬁

␶m=␶=共1 +

冑

5兲/2. 共3兲

The number of layers of type A 共B兲 in Sm isFm共Fm−1兲. Hence, the period of the lattice based on Sm is Lm=Fma +Fm−1b. Furthermore, the optical path along the unit cell is

␦m=FmanA+Fm−1bnB, and the Bragg condition for construc-tive interference of reflected waves is given by

␦m= N␲c

␻ , 共4兲

whereN= 0 for the zeroth-order gap, i.e., the具n典= 0 gap, and N=⫾1 ,⫾2 ,⫾3 , . . . for the so-called Bragg gaps. The aver-aged refractive index is defined by

具n典m=共FmanA+Fm−1bnB兲/Lm, 共5兲

and the具n典= 0 gap occurs around the frequency satisfying the condition具n典m= 0, i.e.,

anA+bnB/␶m−1= 0. 共6兲

In this way, the frequency for the limiting condition in the case of largem is given when the optical paths in mediaA andBare in the golden ratio; that is,

b兩nB兩 anA

=␶. 共7兲

To deal with layered structures, one may use a

transfer-matrix approach. According to Eq. 共2兲 and the continuity conditions at the interfaces, the functions ␺q共z兲 and ␸q共z兲 may be determined from the initial values␺q共z0兲and␸q共z0兲. In fact, for the frequency␻ we arrive at

冉

␺q共z兲

␸q共z兲

冊

=T共␻,z,z0兲

冉

␺q共z0兲

␸q共z0兲

冊

, 共8兲

whereT共␻,z,z0兲is the transfer matrix from z0 toz. Taking the Bloch condition into account, one may connect the val-ues of ␺q and␸q at z= 0 and z=L, with L being the lattice period. Thus one obtains

冉

␺q共L兲

␸q共L兲

冊

=M共␻兲

冉

␺q共0兲 ␸q共0兲

冊

=e

iqL

冉

␺q共0兲

␸q共0兲

冊

, 共9兲

where M共␻兲=T共␻,L, 0兲 is the transfer matrix from 0 toL, for the frequency␻. Therefore, the relation between␻andq is

cos共qL兲=R共␻兲, 共10兲

with R共␻兲 being the semitrace of M共␻兲. Moreover, the al-lowed 共forbidden兲 frequencies are those satisfying 兩R共␻兲兩

艋1 共兩R共␻兲兩⬎1兲.

Let us now consider the elementary unit cell of the lattice consisting ofNhomogeneous layers with refractive indexnl and thicknesses dl, wherel= 1 , 2 , . . . ,N. The left 共right兲 in-terface of the first 共last兲layer will be atz= 0共z=L兲. Hence, the transfer matrix from the left to the right interface of the lth slab is

Tl共␻兲=

冢

cos共qldl兲

⑀l ql

sin共qldl兲

−ql ⑀l

sin共qldl兲 cos共qldl兲

冣

, 共11兲

whereql=␻nl/c. The matrixM共␻兲is a product of the trans-fer matrices of the slabs, namely,

M共␻兲=TN共␻兲· ¯ ·T2共␻兲·T1共␻兲. 共12兲 For the Fibonacci lattice based on the chainSm, the trans-fer matrix M共␻兲, the semitraceR共␻兲, and the periodL, are labeled by the index m. However, instead of Eq. 共12兲, it is better to use the recurrence relation

Mm共␻兲=Mm−2共␻兲Mm−1共␻兲, 共13兲

where M0共␻兲 and M1共␻兲 have the form in Eq. 共11兲, with l being BandA, respectively. The semitrace obeys the recur-rence relation

Rm共␻兲= 2Rm−1共␻兲Rm−2共␻兲−Rm−3共␻兲. 共14兲

Then, the semitrace Rm共␻兲 may be easily calculated from R0共␻兲= cos共qBb兲,R1共␻兲= cos共qAa兲, and

R2共␻兲= cos共qAa兲cos共qBb兲

−1 2

冉

⑀BqA

⑀AqB +⑀AqB

⑀BqA

冊

sin共q_Aa兲sin共q_Bb兲. 共15兲

In Fig.3 the photonic spectrum is depicted for Fibonacci lattices with a= 4 mm, b= 2 mm, and 2艋m艋10. The fre-FIG. 2. 共Color online兲The ratio between the dispersion laws of

materialsAandB.

BRIEF REPORTS PHYSICAL REVIEW A78, 035801共2008兲

quency range is chosen to clearly display the具n典= 0 gap. The profile of this gap is shown in Fig. 4, as a function of the Fibonacci order, for three values of awitha/b= 2. The best agreement with the long-wavelength approximation is ob-tained for the lattice with the thinner layers, i.e., for a = 4 mm. Also, Fig.4suggests a convergence of the gap-edge frequencies, as the ordermincreases.

Let us now assume that 兩qAa兩=␻兩nA兩a/c and 兩qBb兩 =␻兩nB兩b/c are much less than 1. In this regime R0共␻兲

⬇R1共␻兲⬇¯Rm共␻兲⬇1. Hence, there should be gap-edge frequencies satisfying Rm共␻兲= 1, which are associated with gaps atq= 0. Approximate values of those frequencies may be obtained关10,11兴by expandingRm共␻兲up to second order in q_Aa and q_Bb. It may be shown that such an expansion reads,

Rm共␻兲 ⬇1 −␻2L_m2具⑀典m具␮典m/共₂c2兲, 共16兲

where the average permittivity and the average permeability are 具⑀典m=共Fma⑀A+Fm−1b⑀B兲/Lm and 具␮典m=共Fma␮A +F_m−1b␮B兲/L_m, respectively. The edge frequency corre-sponding to具⑀典= 0 satisfies

a⑀A+b⑀B/␶m−1= 0, 共17兲

and that for 具␮典= 0 obeys

a␮A+b␮B/␶m−1= 0. 共18兲

The limiting equations for largemare given by the following relations:

b兩⑀B兩

a⑀A

=␶ and b兩␮B兩 a␮A

=␶, 共19兲

which correspond to null averages of the response functions, i.e.,具⑀典= 0 and 具␮典= 0, respectively. Figure5shows the gap edges, as functions of the ratio a/_{b with a= 4 mm, for m}

= 2 andm= 11. Since the long-wavelength approximation ap-plies, the frequencies satisfying Eqs.共17兲and共18兲, displayed as solid lines in Fig. 5, are in excellent agreement with the numerical solutions of Rm共␻兲= 1 near the具n典= 0 frequency.

To understand Eq. 共16兲, it is useful to write Rm共␻兲⬇1 −␻2_{gm共␻兲/共2c}2_{兲. Since the second term in gm共␻兲 is small,} Eq. 共14兲leads to

gm共␻兲= 2gm−1共␻兲+ 2gm−2共␻兲−gm−3共␻兲. 共20兲

We also have

g0共␻兲=n_B2b2=共b⑀B兲共b␮B兲

=共F0a⑀A+F−1b⑀B兲共F0a␮A+F−1b␮B兲, 共21兲

g1共␻兲=n_A2a2=共a⑀A兲共a␮A兲

=共F1a⑀A+F0b⑀B兲共F1a␮A+F0b␮B兲, 共22兲

and

2 4 6 8 10

Fibonacci order 1

1.5 2 2.5 3 3.5 4

Frequency

GHz

FIG. 3. 共Color online兲The photonic spectrum in Fibonacci lat-tices with a= 4 mm and b= 2 mm, as a function of the Fibonacci order. Consecutive bands are depicted as vertical-line segments of different widths.

FIG. 4. 共Color online兲 The具n典= 0-gap profile of Fibonacci lat-tices with a/b= 2, as a function of the Fibonacci order m. The crosses are for the conditions具⑀典m= 0 and具␮典m= 0.

(a)

(b)

FIG. 5. 共Color online兲The具n典= 0-gap profile of Fibonacci lat-tices witha= 4 mm, as a function of the ratioa/_{b. In}共_a兲_and共_b兲_{, the}

Fibonacci order equals 2 and 11, respectively. The solid lines共dots兲 are for具⑀典m= 0 or具␮典m= 0关the solutions ofRm共␻兲= 1兴.

BRIEF REPORTS PHYSICAL REVIEW A78, 035801共2008兲

g2共␻兲=n_A2a2+n_B2b2+

冉

⑀BnA ⑀AnB

+⑀AnB ⑀BnA

冊

nAanBb

=共a⑀A+b⑀B兲共a␮A+b␮B兲

=共F2a⑀A+F1b⑀B兲共F2a␮A+F1b␮B兲. 共23兲

Then, using Eqs.共20兲–共23兲, one arrives at

gm共␻兲=共Fma⑀A+Fm−1b⑀B兲共Fma␮A+Fm−1b␮B兲

=L_m2具⑀典m具␮典m. 共24兲

Note thatFmFm−1andFm 2

satisfy the same recurrence relation as gm共␻兲does, namely, Eq.共20兲.

One may wonder whether the solutions of Eqs.共17兲 and 共18兲 are the edges of a frequency gap. To answer this, we assume that in the frequency range of interest共i兲⑀A⬎0 and

␮A⬎0,共ii兲⑀Band␮B may take any negative value, and共iii兲

⑀Band␮Bare monotonic functions of the frequency共see Fig. 2兲. This guarantees that Eqs. 共17兲 and共18兲have the unique solutions ␯_⑀m and␯_␮m, respectively. By using gm共␻兲=F_m2共a⑀A +b⑀B/␶m−1兲共a␮A+b␮B/␶m−1兲, one may show that R共␻兲⬎1 for ␯_⑀m⬍␯⬍␯_␮m, which corresponds to a range of forbidden frequencies.

The lower and upper frequencies of the gap are denoted as ␯+

m and␯−

m

, respectively. To see whether␯− m

is either␯_⑀mor␯_␮m, it is useful to observe Fig. 2, where the ratios ⑀B/⑀A and

␮B/␮A are given as functions of the frequency. Since Eqs. 共17兲 and 共18兲 are equivalent to ⑀B/⑀A= −␶m−1a/b and

␮B/␮A= −␶m−1a/b, respectively, the frequencies ␯⑀

m

and ␯_␮m may be graphically found by drawing a horizontal line at the height −␶m−1a/b of the ratio. The case a/b= 2 and m= 2 is depicted in Fig. 2, where it is apparent that␯−

m

=␯_␮mand ␯+ m

=␯_⑀m. It is important to note that the 具n典= 0 frequency is within the considered gap. This frequency is denoted as␯0

m , and satisfies Eq.共6兲, i.e.,n_B/n_A= −␶m−1a/b. The rationB/nA, as displayed in Fig. 2, is the geometric mean of the ratios

⑀B/⑀Aand␮B/␮A. Therefore, it is clear that␯0 m

is between␯_␮m and␯_⑀m.

We may wonder why, in the long-wavelength limit, the conditions具⑀典= 0 and具␮典= 0 should apply at the edges of the 具n典= 0 gap. The reason is that for sufficiently low frequen-cies, the unit cell of the lattice behaves as an effective optical medium whose response is determined by the averaged val-ues of permittivity and permeability. Considering the disper-sion relations in Fig.2, those averaged values are increasing functions of the frequency. From this, we obtain the follow-ing three cases: 共i兲 if ␯⬍␯_␮m, then 具⑀典m⬍0 and 具␮典m⬍0, 共ii兲 if ␯_␮m⬍␯⬍␯_⑀m, then 具⑀典m⬍0 and 具␮典m⬎0, and 共iii兲 if␯⬎␯_⑀m, then具⑀典m⬎0 and具␮典m⬎0. Hence, in cases共i兲and 共iii兲the wave propagates along the effective medium without absorption. This corresponds to the allowed frequencies. In-stead, in case 共ii兲 the refractive index of the effective me-dium is imaginary. Hence, the effective meme-dium does not allow the propagation of the wave, and the corresponding frequency range should be a gap of the electromagnetic band structure.

Summing up, the propagation of light through a 1D lattice composed by layers of optical mediaAandB, of positive and negative refractive indices, arranged according to a concat-enation rule of the Fibonacci type, has been studied. We have proven that a robust zeroth-order gap arises. Moreover, when the Fibonacci order is high, the gap occurs around the fre-quency for which the ratio of the optical paths is close to the golden ratio. In the long-wavelength limit, analytical expres-sions that define the gap edges around the zeroth-order gap were obtained, and the corresponding frequencies were shown to satisfy the具⑀典= 0 and 具␮典= 0 conditions.

The authors would like to thank the Brazilian Agencies CNPq, FAPESP, FAPERJ, MCT-Millenium Institute for Quantum Information, and MCT-Millenium Institute for Nanotechnology, as well as the Scientific Colombian Agency COLCIENCIAS, and CODI-University of Antioquia for par-tial financial support.

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