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ositive andency rangeere bothrective index

rs becauseotion in

rem. Thee so-called

r a varietyres werearhing inboundary

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s. Asbility

Physics Letters A 351 (2006) 198–204

www.elsevier.com/locate/pla

Transmission spectra in symmetrical Fibonacci superlattices composepositive and negative refractive index materials

Hui He∗, Weiyi Zhang

National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China

Received 9 August 2005; received in revised form 13 October 2005; accepted 25 October 2005

Available online 2 November 2005

Communicated by R. Wu

Abstract

Using the transfer matrix method, we have calculated the transmission spectra of symmetrical Fibonacci superlattices composed of pnegative refractive index materials. In the ideal case where negative refractive index can be approximated as a constant in the frequconsidered, self-similarity behavior and perfect transmission peaks are observed in the spectra. While in the more realistic case whε

andµ are negative, dispersive and modeled by the split-ring resonators (SRR) and cut-wires, both zeron = 0 gap and Bragg reflection gaps aobtained. These transmission spectra are also compared with those of symmetrical Fibonacci superlattices with purely positive reframaterials. 2005 Elsevier B.V. All rights reserved.

PACS: 42.70.Qs; 41.20.Jb; 78.20.Ci; 05.45.Df

Keywords: Photonic band gap; Negative refractive index; Self-similarity; Perfect transmission peaks; Zero-n gap

1. Introduction

The man-made space modulated periodical structures have attracted a great deal of attention in the past twenty yeathey offer extra length dimensions to control and manipulate the wave propagation in media. In analog to the electronic mperiodic crystal structures, the elementary excitations in periodic media also form band like structures due to Bloch theoexistence of band gaps prohibits the wave propagation in such media for certain frequency range and thus creates thband gap materials[1,2]. Depending on material properties to be modulated, various photonic[3–9], phononic[10–15], as wellas polaritonic[16,17] band gap structures have been proposed and made. These band gap materials pave the way foof technological applications. In addition to the periodically modulated structures, quasi-periodically modulated structualso extensively studied both for reasons of fundamental physics interest[18–23], and for their broader application in non-lineoptical devices. For example, the mere availability of richer reciprocal lattice vectors offer more flexibility for phase matcthe second and high order harmonics generations in non-linear optical crystals. Also the quasiperiodic structures lie at thebetween periodically ordered crystals and disordered systems, the spectra possess the self-similarity pattern which indiclocalization nature. More information on properties of quasiperiodic structures can be found in an excellent review arbook given by Albuquerque and Cottam, where the characteristic spectral properties on superlattices with FibonacciThue-Morse sequence, and double period sequence are analyzed and discussed in details[22,23].

Very recently, the so-called negative refractive indexn materials have received renewed attention due to their novel propertiewas noted in the pioneering study by Veselago[24] in 1968, the negativen materials possess simultaneously negative permea

* Corresponding author.E-mail address: kelly_h2@hotmail.com(H. He).

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2005.10.071

H. He, W. Zhang / Physics Letters A 351 (2006) 198–204 199

d thisel system,nd

erials,

e Bragg

th positiveeakslters. Theore, such.tphase

d.

se

-

ndent.terms of

other side

tice.

and

mbedded

erms of

ight

ε(ω) and permittivityµ(ω), the group velocity of wave propagation in such media is opposite to the phase velocity anmakes the perfect lens possible. Though the negative refractive index materials were originally investigated as a modsuch metamaterials are recently realized by Pendry and his colleagues[25,26] using the split-ring resonators and cut-wires, aby other type of designs. In addition to those novel properties proposed in Veselago’s original paper, Li et al.[27] found thatthere is an intrinsic so-called zeron = 0 band gap in superlattice stacking with positive and negative refractive index matthe similar properties have also been investigated for Fibonacci superlattice[28]. These studies show that the zeron = 0 band gapis independent of the details of crystal structure and thus it is quite different from the usual band gaps resulting from threfraction which is structure dependent.

In this Letter, we have studied the transmission spectra of symmetrical Fibonacci superlattices (SFS) composed of boand negative refractive index materials. The advantage of SFS superlattices is its ability to offer perfect transmission p[29–31] due to its internal coupling between localized modes and propagation modes, thus can be used as ideal optical filocalized mode exists because mirror symmetry structure favors wave interference within the SFS superlattices. Furthermstructure is also very sensitive to the phase compensation effect which is unique in the positive and negativen stacked superlatticesOur study shows that the transmission spectra of SFS superlattices with positive and negativen materials are distinctly differenfrom their counterparts with all positiven materials. In general, the transmission spectra unveil smooth structure due to thecompensation effect. The zeron gap in our case with realistic permeability and permittivity is also investigated and discusse

The rest of the Letter is organized as follows: the transfer matrix method is briefly introduced in Section2; the transmissionspectra for symmetrical Fibonacci superlattices are calculated and discussed in Section3, their results are compared with thocalculated with purely positive refractive index materials; Section4 is our conclusion.

2. Transfer matrix method

The isotropic electromagnetic medium can be generally described by permeabilityε(ω) and permittivityµ(ω), the electromagnetic wave for transverse electric mode satisfies the following wave equation

(1)−Z(x)

n(x)

d

dx

[1

Z(x)n(x)

dE(x)

dx

]=

(ω

c

)2

E(x).

Here,Z(x) = √ε/

√µ andn(x) = √

ε√

µ are the impedance and refractive index at a given frequency, they are layer depec is the light velocity in vacuum. For a given symmetrical Fibonacci sequence, above differential equation can be casted ina transfer matrix which relates the incident electromagnetic wave at one side to the transmitted electromagnetic wave at

(2)

(E(x0)

H(x0)

)=

N∏i=1

M(di)

(E(xN)

H(xN)

),

E(x) andH(x) are the electric and magnetic fields whilex0 andxN denote the coordinates of two surfaces of finite superlatThe transfer matrix for layeri has the form

(3)M(di) =(

cos[κnidi cosθi] (i/√

Zi cosθi)sin[κnidi cosθi]i√

Zi cosθi sin[κnidi cosθi] cos[κnidi cosθi])

.

Note thatκ = ω/c is the wave number in vacuum,Zi , ni , di , andθi are the impedance, refractive index, layer thickness,refracted angle inith layer, respectively. For convenience, one usually usesδi = κnidi cosθi to express the phase shift inith layer.For the transverse magnetic mode, parallel derivation can be made and the expression can be obtained by exchangingE ↔ H andε ↔ µ.

To derive the transmission and reflection coefficients of a given superlattice, let us assume that finite superlattice is ebetween two semi-infinite medium with permeabilityεM and permittivityµM , the matrix product of a give sequence ism and

(4)m =N∏

i=1

M(di) =(

m11 m12m21 m22

).

By expanding the matrix Eq.(2), incident and reflected electromagnetic wave on the left out-surface can be expressed in tthe transmitted electromagnetic wave on the right out-surface as below

(5a)E(x0) = m11E(xN) + m12H(xN),

(5b)H(x0) = m21E(xN) + m22H(xN).

On the left incident surface, the electric field is a sum of incident 1 and reflectedr amplitudes andE(x0) = 1+ r , the correspondingmagnetic field is given byH(x0) = −√

ZM(1 − r)cosθM . ZM = √εM/µM is the impedance of embedded medium; On the r

surface, the electric field and magnetic fields can be similarly expressed in terms of transmission amplitudet . The results are

200 H. He, W. Zhang / Physics Letters A 351 (2006) 198–204

tro-

nd

, the

r re-nces

e

ce

evident

fi-

isfied

finiteand

ittivity can

Fig. 1. The schematic diagram of the 4th generation symmetrical Fibonacci superlattice SFS4. EI , ER andET are the incident, reflected and transmitted elecmagnetic waves, respectively.

E(xN) = t andH(xN) = −√ZMt cosθM . Substituting these expressions into Eq.(5) and solving the equation, the reflected a

transmitted amplitudes can be obtained

(6a)r =√

ZM cosθM(m11 − √ZM cosθMm12) + (m21 − √

ZM cosθMm22)√ZM cosθM(m11 − √

ZM cosθMm12) − (m21 − √ZM cosθMm22)

,

(6b)t = 2√

ZM cosθM√ZM cosθM(m11 − √

ZM cosθMm12) − (m21 − √ZM cosθMm22)

.

The reflection and transmission coefficients are given byR = |r|2 andT = |t |2, respectively. For the transverse magnetic modecorresponding expressions can be easily obtained from Eq.(6) by exchanging the permeability and permittivity.

The symmetrical Fibonacci superlattices[29] can be viewed as a superposition of two Fibonacci superlattices with mirroflection at the center. TheJ th generation of the sequenceSJ can be defined in terms of the two corresponding Fibonacci sequeGJ , G+

J with reverse orderings

(7)SJ = GJ G+J .

GJ andG+J are the usual Fibonacci sequences generated with two componentsB andA with reverse orderings, they obey th

recursive relationsGJ = GJ−1GJ−2 andG+J = G+

J−2G+J−1, the zero-th and 1st generations are specified byG0 = G+

0 = B andG1 = G+

1 = A. This yields the general recursive relation for the symmetrical Fibonacci sequence as following

(8)SJ = GJ−1GJ−2G+J−2G

+J−1.

The sequence ofJ th generation can be viewed as by embedding the sequence of(J −2)th generation in the middle of the sequenof (J − 1)th generation. Thus, the recursive relation for symmetrical Fibonacci sequence cannot be expressed alone withSJ . Fig. 1illustrates the schematic diagram for the structure of 4th generation of symmetrical Fibonacci lattice, mirror symmetry isin the figure. From the construction of symmetrical Fibonacci sequence, the number of layers inJ th generation isSJ = 2FJ . FJ isthe usual Fibonacci number andFJ = FJ−1 + FJ−2 for J > 2 with F0 = 1 andF1 = 1.

For the symmetrical Fibonacci superlattices, the diagonal matrix elementsm11 = m22. The reflection and transmission coefcients are reduced to simpler forms:

(9a)r = −(√

ZM cosθM)2m12 + m21

2√

ZM cosθMm11 − (√

ZM cosθM)2m12 − m21,

(9b)t = 2√

ZM cosθM

2√

ZM cosθMm11 − (√

ZM cosθM)2m12 − m21.

From this equation, one sees that a perfect transmission can occur for certain frequencies if the following condition is sat

(10)m21 = (√ZM cosθM

)2m12.

This happens when the total transfer matrix becomes a diagonal matrix.

3. Numerical results and discussion

In this Letter, the symmetrical Fibonacci superlattices (SFS) is composed of two building blocksA andB, A is the usual positiven material andB can be negative or positiven materials. The finite superlattices is sandwiched either between two semi-inmediaM or between two finite substrate mediaM in touched with air background. Both ideal non-dispersive model systemrealistic system are considered. For the ideal model system an assumption is made that both permeability and permbe approximated by constants in the frequency range of interest. While for the realistic situation, effective permeabilityε(ω) andpermittivity µ(ω) are dispersive and modeled by the split-ring resonators (SRR) and cut-wires medium[25,26].

H. He, W. Zhang / Physics Letters A 351 (2006) 198–204 201

-

,

ossess

eembeddingn spectra

tydium

tiondual

imilar to

fter

d reflectionase shiftSiO

tlyels.

ern in theection

Fig. 2. Transmission spectra of SFS superlattices as a function of phase shiftδ for sequences SFS9 to SFS11. (a) nA = 1.45, dA = 120 nm,nB = −2.3,dB = 76.1 nm, nM = 1.45, anddM = ∞; (b) nA = 1.45, dA = 120 nm,nB = −2.3, dB = 76.1 nm, nM = 1.45, dM = 6.5 mm, and in touch with air background; (c)nA = 1.45,dA = 120 nm,nB = +2.3, dB = 76.1 nm,nM = 1.45, anddM = ∞.

We start with the ideal case. To facilitate the comparison between superlattices with purely positiven components and positivenegativen components, we choose the contradistinctive condition that isnA = 1.45,ZA = 1.45 (SiO2) andnB = −2.3, ZB = 2.3(negative TiO2). The thicknesses of two materials are chosen in such a way so as to satisfy the conditionnAdA = −nBdB , whichgives the very reversed phase shift in the two materials. For simplicity, the incident angle is taken asθM = 0. It is well known boththeoretically and experimentally[18–20]that the transmission and reflection spectra of one-dimensional Fibonacci lattices pthe self-similarity feature around the fixed pointδ/π = (m + 1/2), wherem is an integer. For example, at the fixed pointδ = π/2,the trace of transfer matrices recovers after six generation and fixed points in trace mapping form anti-nodal pairs[19,20]. Thedetailed analysis[29,32]at the fixed points suggests that transfer matrix of sequenceGJ can be classified into three types:GJ=3k+1(k is an integer) corresponds to an off-diagonal matrix which recovers after three generation;G3k+2 is a diagonal matrix, though thdiagonal matrix elements interchange after three generation, the transmission spectra remain the same irrespective to themedium;G3k also corresponds to an off-diagonal matrix which does not recover after three generation, but the transmissiocan be recovered if the embedded medium is typeA. In terms of invariant of the trace mappingI = sin4 δ(R2 − 1)2/4R2 withR = nA/nB [20,29], the scaling parameter is given byα = 2a2 + √

1+ 4a4 with a = √1+ I . The spectra have a self-similari

feature around the central wavelengthλ0 = 4nAdA for two sequences differing by six generation for arbitrary embedded meor for two sequences differing by three generations if the embedded medium is typeA.

For the symmetrical SFS superlattices composed of purely positiven materials, theoretical analysis and numerical simulasuggest that self-similarity feature is still preserved[29], and quasiperiodicity is expected to be most effective when the indivilayers are quarter-wave layers[29,32]. In this case, the matrix for a given sequence can be easily calculated at the fixed pointδ = π/2and it possesses rather simple form. The matrix is either plus or minus unit matrix, it recovers after three generation. Sthe simple Fibonacci sequence, the transfer matrix of a given sequence can also be classified into three types:G3k andG3k+1correspond to minus unit matrixI while G3k+2 corresponds to plus unit matrixI . Since the matrices at fixed points recover athree generations, the transmission spectra also possess the self-similarity feature around the central wavelengthλ0 = 4nAdA fortwo sequences differing by three generation irrespective to the embedded medium. For comparison, the transmission ancoefficients for three types of symmetrical Fibonacci superlattices (SFS) are numerically calculated as a function of phor frequency and presented inFig. 2. Fig. 2(a) illustrates the transmission spectra for SFS composed of building blocks2(nA = 1.45, dA = 120 nm) andnegative TiO2 (nB = −2.3, dB = 76.1 nm) which is embedded within two semi-infinite SiO2(nM = 1.45, dM = ∞); Fig. 2(b) is similar toFig. 2(a) except the embedded mediaM (nM = 1.45, dM = 6.5 mm) is finite withair background;Fig. 2(c) is the reference spectra calculated with normal TiO2 (nB = +2.3) while all other parameters are exacthe same as those inFig. 2(a). Three sequential generations SFS9 to SFS11 of SFS superlattices are investigated for each panSeveral features are worth of mentioning: (1) similar to the usual Fibonacci superlattices, there exists a self-similarity pattspectra with respect to the fixed pointδ = π/2; (2) there exist a number of perfect transmission peaks due to the mirror refl

202 H. He, W. Zhang / Physics Letters A 351 (2006) 198–204

p gradually

due topendentthan their

siont in thenditionitive-anotherhen the

ofxed

enerationatisfactoryers in thethat moreuperlattices

eality, allul-esonanceabricated

l negativez.

Fig. 3. Transmission spectra of SFS superlattices as a function of phase shiftδ. nA = 1.45, dA = 120 nm,nB = −2.3, dB = 76.1 nm,nM = 1.45, anddM = ∞.(a) for SFS9 to SFS11; (b) for SFS12 to SFS14; (c) for SFS15 to SFS17. Notice the different scales used in (a)–(c).

symmetry of the superlattices, and number of such peaks increases with the generation; (3) photonic band gaps develoas the generation becomes large.

Overall features of the spectra inFig. 2(a) and (b) look quite similar except the superimposed fast Fabry–Perot oscillationsthe finite size effect inFig. 2(b), these results show that not only the spectra of SFS superlattices at the fixed points are indeof the embedded medium, but also the overall features of SFS superlattices are less sensitive to the embedded mediumcounterpart in Fibonacci superlattices. The band gaps resulting from quasi-periodicity take place aroundδ/π = 0.4 andδ/π = 0.6which can be compared directly withFig. 2(c). By comparingFig. 2(a) and (c), one notices that the number of perfect transmispeaks inFig. 2(a) is much less than that ofFig. 2(c), this feature comes about because of the phase compensation effecsuperlattices composed ofpositive n andnegative n materials, this reduces the phase space where quasi-phase matching cotakes place. The perfect transmission peaks we observed suggests that the symmetric internal structure consist of posn andnegative-n materials also influences the localization property of electromagnetic waves in a quasiperiodic superlattice. Inword, the initially poorly transmitted wave in a ordinary quasiperiodic structure can become a perfect transmission wsymmetric internal structure is imposed in the quasiperiodic superlattice.

To check the self-similarity behavior of the transmission spectra around the fixed-point(δ = π/2), the transmission spectraSFS12–SFS14 and SFS15–SFS17 are also calculated and plotted inFig. 3(b), (c). These are the expanded views around the fipoint by a factor of 1/α and 1/α2 in order to facilitate a comparison with the transmission spectra of SFS9–SFS11 in Fig. 3(a).It is seen that transmission spectra differing by three generation have self-similarity in the overall patterns when the gis larger than 9. Comparison of the transmission spectra of SFS superlattices in low generation sequences yields less sresult on self-similarity, this is so because the number of modes in a superlattice is determined by the total number of laysuperlattice, the pattern of self-similarity appears only when the number of modes in a superlattice is large enough sodetails can be unveiled when spectra are expanded around the fixed points. The situation becomes more obvious in SFS ssince there are less Bragg peaks due to the phase compensation effect as mentioned above.

The above discussion applies only to the ideal case where both permeability and permittivity are non-dispersive. In rrealized artificial negative-n metamaterials are based on SRRs and cut wires,ε(ω) andµ(ω) are frequency dispersive and are simtaneously negative only within a narrow bandwidth. This is so because negative refractive index originates from strong reffect with the applied magnetic field. Thus, above calculations are valid only under the assumption that the size of the fnegative refractive index material can be as tiny as the normal positive refractive material, also the negative-n metamaterial canapproximately be treated as a negative constant in the frequency region of interest. Since the microstructures of practicarefractive metamaterials are on the order of a few millimeters, their typical frequency region ranges from 1 GHz to 14 GH

For the metamaterial assembled with split-ring resonators and cut-wires, the effectiveε(ω) andµ(ω) can be modeled by[25,26]

(11a)ε(ω) = 1+ 52

2 2+ 102

2 2,

0.9 − ω 11.5 − ω

H. He, W. Zhang / Physics Letters A 351 (2006) 198–204 203

re-haves

resented in

quency.cyases, theal lattice

e, thus the

metricalrefractivectra. While-wires,mmetrical

03). Went Youth

Fig. 4. Transmission spectra of SFS superlattices of 8th generation. (a)nA = 1, dA = 12 mm, anddB = 6 mm; (b)nA = 1, dA = 16 mm, anddB = 8 mm. BlockA

is air and blockB is the metamaterial described by Eq.(11).

(11b)µ(ω) = 1+ 32

0.9022 − ω2.

Hereω is the frequency measured in GHz,ε(ω) andµ(ω) have the forms of resonator types. A close investigation shows thatε(ω)

becomes negative in two small frequency windows immediately after resonating frequencies, they are[0.9,3.636] and[11.5,15.55],respectively.µ(ω) becomes negative only in one frequency window[0.902,3.0]. Therefore, there exists a small overlapping fquency region[0.902,3.0] where bothε(ω) andµ(ω) are negative. This is the frequency range where the metamaterial belike negativen material. Using the above metamaterial as negativen material (blockB) and air as positiven material (blockA),we have calculated the transmission spectra of symmetrical Fibonacci superlattices of 8th generation and results are pFig. 4. To distinguish the usual Bragg reflected band gaps from the zeron band gap[27], two configurations with the samen areconsidered.Fig. 4(a) shows the transmission spectra of SFS8 with dA = 12 mm anddB = 6 mm whileFig. 4(b) with dA = 16 mmanddB = 8 mm. The zeron = 0 gap of superlattices is estimated to be around 2.02 GHz, and a gap is indeed found at that freFor frequency larger than 3.0 GHz, both metamaterial and air have positive refractive indexn, thus those gaps above this frequenresult from the Bragg reflection due to modulation of impedance and refractive index. When the lattice constant increzero-n gap does not change with scaling as it should be, but the gaps originating from Bragg scattering involves reciprocvectors and are scale sensitive, they shift down wards in frequency as expected. This is the very point that the zero-n gap differsfrom the conventional Bragg gap. Note that the plasmon frequency is outside of the frequency range we considered herplasmon modes are not excited here.

4. Conclusion

In summary, using the transfer matrix method we have studied the propagation of electromagnetic wave through symFibonacci superlattices composed of positive and negative refractive index materials. In the ideal case where negativeindex can be approximated as a constant, self-similarity behavior and perfect transmission peaks are observed in the spein the more realistic case where bothε andµ are negative, dispersive and modeled by the split-ring resonators (SRR) and cutboth n = 0 gap and Bragg reflection gaps are obtained. The transmission spectra are also compared with those of syFibonacci multilayers with purely positive refractive index materials.

Acknowledgements

This work was supported in part by the State Key Program for Basic Research of China (Grant No. 2004CB6190wish to acknowledge the partial financial support from the NNSFC under Grant Nos. 10474040, 10334090, and “ExcelleFoundation” [10025419].

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