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IJMOT-2009-00-000 © 2009 ISRAMT

INTERNATIONAL JOURNAL OF MICROWAVE AND OPTICAL TECHNOLOGY,

VOL. , NO. , MONTH 2009

Stop Band Filter by Using Hybrid Quasi-periodic One Dimensional Photonic Crystal in Microwave Domain

N.Ben Ali*, Y.Trabelsi, and M.Kanzari

Photovoltaic and Semiconductor Materials Laboratory, El-Manar University-ENIT PO Box 37, Le belvedere 1002-

Tunis, Tunisia Tel: 0021697713158; E-mail: [email protected]

Abstract-Hybrid quasi-periodic multilayer structures constructed from high-low indices-contrast (Roger and Air) quarter-wavelength layers are used in novel multilayer systems. The Quasi-periodic hetero-structures, formed by concatenating hybrid quasi-periodic (generalized Fibonacci structures GF (m,n) and generalized Thue-Morse structures GT-M(m,n); m and n are integer) sequences of different order have properties markedly different from classic systems. In this study, the transmission properties of hybrid quasi-periodic photonic crystals with defects are studied by using a Matrix method approach (MM). Results are presented for normal incident wave with TE polarizations. By increasing the parameter n and m of thus structures, the corresponding transmission properties exhibit interesting properties in microwave domain. We propose a narrow stop band polychromatic filter that covers the frequency range of Global System for Mobile Communication (GSM). Index Terms-Stop band filters, hybrid quasi-periodic structures, generalized Fibonacci class, generalized Thue-Morse class, microwave.

I. INTRODUCTION Photonic band gap (PBG) materials permit electromagnetic field propagation in certain frequency bands, but not in others, their crucial feature being the periodic arrangement of high contrast electromagnetic properties. Recently, some novel periodic structures, such as photonic band gap structure (PBG) and electromagnetic band gap structure (EBG), have been applied popularly to microwave applications [1–2]. Dielectric photonic crystal structures [2, 3] have recently attracted great attention thanks to their

property of forbidding the propagation of light at fixed wavelengths and of the intriguing possibility to realize an all-optical integrated circuit which is the basic element of the optical computer [4]. The simplest form of a photonic crystal is the one-dimensional (1D) periodic structure. It consists of a stack of alternating layers having a low and high refractive indices, whose thicknesses satisfy the Bragg condition: nLdL=nHdH=�0/4 where �� 0 is the reference wavelength. It is known as Bragg Mirror [3, 4]. But this periodic structure has only one photonic band gap centered in reference wavelength which excited the outside face of multilayer structure surrounding the transmission peak. The quasi-periodic system presents one solution which can give several pseudo band gaps in whole spectral domain [5]. Recently, not only periodic structures or Photonic Crystal (PC) with defects have become significant and interesting physical properties, but also quasi-periodic systems (quasi-crystals). Perfectly quasi-crystal does not have a geometrical periodicity but is still deterministically generated cover only the two extremes of the rich spectrum of complex dielectric structures [6, 7]. Quasi-periodic multilayer systems can be considered suitable models to describe the transition from perfect periodic structures to random structures [8, 9]. Built according to determinist order quasi-periodic multilayer system, the spectra of this multilayer stacks show very interesting physical properties like the

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existence of a photonic band gap (PBG) for different frequency regions. We distinguish two famous quasi-periodic sequences: the generalized Fibonacci sequence [10, 11] and the generalised Thue-Morse sequence [12]. Based on this type of structure many applications are made in different domains such as optical frequency-selective filters based on thin-film technology which have been investigated for a broad range of applications [13, 14]. In this paper transmission by Thue-Morse multilayer structure in microwave region is investigated. We extracted the transmission versus the parameter and the variable of Thue-Morse sequence. Multi narrow band gap cover all frequency domain: 0-50 GHz without the perturbation and peak of transmission outside this succession of band can easily be designed. Furthermore, the number of PBGs of the hybrid quasi-periodic structures can be manipulated by varying the parameter n of the generalized structures. Based on the proposed analysis; multi stop band filters can easily be designed. In this work we employ the Matrix Method [15], including components of the refractive index, to extract the transmission properties and consider their sensitivity to material and geometrical variation. We investigated the transmission through hybrid generalised quasi-periodic multilayer structures and found that the corresponding transmission spectra exhibit interesting properties depending on the parameter n. we found also that it can cover the frequency range of the GSM. So, in this paper we describe a novel procedure approach to determining succession of multi-narrow stop band filters using hybrid quasi-periodic structures. This succession of band forms a wide band gap which covers much of the portion of the microwave domain.

II. QUASI-PERIODIC MODEL

A. Generalized Thue-Morse Multilayer Structure

Thue-Morse sequences are multilayer structures consisting of two different materials. The number of layers depends on the order and parameters of the Thue-Morse sequence. Each layer is labelled either H or L. H denotes the material with the higher refractive index, and L denotes the one with the lower refractive index. The generalized Thue-Morse GTM (m, n) multilayer is recursively constructed as: m

kn

kk SSS )()(1 =+

with HS =1 and LS =1 and arranged according to an inflation rule �T-M: H � HmLn, H � LmHn

[12]. B. Generalized Fibonacci Multilayer Structure

The interest in generalized Fibonacci GF (m, n) quasi-periodic structures has increased since it appears that the physical properties of the Fibonacci quasi-crystal may not, in some fundamental aspects, be genetic [16]. The GF (m, n) sequences are a class of aperiodic lattices generated by the substitution rules

mm LHH → and mHL → , where m and n are

all positive integers [17]. Starting with a S0=L, S1=H, the GF (m, n) sequences are expressed by the following recursion relation n

lmll SSS 11 −+ = for

l� 1. The number of layers depends on the order of the Fibonacci sequence. Each layer is labelled either H or L. H denotes the material with the higher refractive index, and L denotes the one with the lower refractive index, respectively. Based on the characteristics of the construction of Fibonacci sequences, we consider the matrices of wave propagating through the Fibonacci multilayer of jth generation (Sj), which is sandwiched by two media of material of type L and H.

III. PROBLEM FORMULATION

For the calculation of system transmission, we employed the Matrix Method (MM). This method permit particularly to extract and solve the standard problem of the photonic band

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structures (transmission, reflection and absorption) spectrum. It is based on Abeles method [15] to calculate the reflection and transmission spectrum. Abeles showed that the relation between the amplitudes of the incident wave +

0E , reflected wave −0E and

transmitted wave ++1mE is expressed as follows:

��

�

�

��

�

�=

��

�

�

��

�

�−

+

++

+

+−

+

1

1

1321

1321

0

0

......

m

m

m

m

E

EttttCCCC

E

E (1)

jC is the propagation matrix of each layer :

( ) ( )( ) ( ) ��

��

�

−−

=−−

−−

11

11

expexpexpexp

jjj

jjjj iir

iriC φφ

φφ (2)

Where jt and jr are the Fresnel transmission

and reflection coefficients. The Fresnel coefficients jt and jr can be expressed as

follows by using the complex refractive index jjj iknn +=ˆ and the complex refractive angle jθ .

For parallel polarization (P):

11

11

cosˆcosˆcosˆcosˆ

−−

−−

+−

=jjjj

jjjjjp nn

nnr

θθθθ

(3)

11

11

cosˆcosˆcosˆ2

−−

−−

+=

jjjj

jjjp nn

nt

θθθ

(4)

Moreover, for perpendicular polarization (S):

jjjj

jjjjjS nn

nnr

θθθθ

cosˆcosˆ

cosˆcosˆ

11

11

+−

=−−

−− (5)

jjjj

jjjS nn

nt

θθθ

cosˆcosˆ

cosˆ2

11

11

+=

−−

−− (6)

The complex refractive indices and the complex angles of incidence obviously follow Snell’s law:

-1 -1ˆ ˆsin sinj j j jn nθ θ= (j=1, 2…, m+1).

The values 1−jφ in equation (2) indicate the

change in the phase of the wave between ( )thj 1− and thj interface and are expressed by the equation: 00 =φ (7)

and 1111 cosˆ2−−−− = jjjj dn θ

λπφ (8)

except for j = 1. � is the wavelength of the incident light in vacuum and 1−jd is the thickness

of the thj )1( − layer. By putting 11 =−+mE , Abeles

obtained a convenient formula for the total reflection and transmission coefficients, which corresponds to the amplitude reflectance r and transmittance t, respectively, as follows:

ac

E

Er ==

+

−

0

0 (9)

attt

E

Et mm 121

0

1 ... ++

++ == (10)

The quantities a and c are the matrix elements of the all product matrix:

��

��

�=∏ dc

baC j

j

(11)

By using Eq. (9) and (10), we can easily obtain the energy reflectance R as: 2rR = (12) for (S) and (P) polarizations, the energy transmittance T as:

2

00

11

cosˆcosˆ

Re Smm

S tn

nT �

��

��

�= ++

θθ

(13)

2

00

11

cosˆcosˆ

Re Pmm

P tn

nT �

��

��

�= ++

θθ

(14)

Re indicates the real part of PT

IV. RESULTS AND DISCUSSION

In the following numerical investigation, we chose Roger (H) and Air (L) as two elementary layers, with indices of refraction nH= 3.134 and nL= 1 respectively. The thicknesses dL,H of the two materials, (layer L) and (layer H), have been chosen to satisfy the Bragg condition, nLdL=nHdH=λ0/4, λ0=12mm is the reference wavelength of the structure. Here we study the optical transmission properties in the microwave domain, for normal incidence through the juxtaposition of quasi-periodic dielectrics multilayer. A. Periodic Sequence To illustrate the property of the transmission spectra in microwave domain of the hybrid quasi-

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periodic multilayer structures we begin at once to study the transmission coefficient of the periodic structure in this part and the transmission coefficient of the quasi-periodic structures in the next parts. Fig. 1 shows the simplest case of transmission coefficient of photonic crystal made of alternating layers of low (nL) and high (nH) refractive index, whose thicknesses satisfy the Bragg condition: nL.dL+nH.dH=�0/2. According to these conditions, dH = 0.957 mm and dL =3 mm. For this periodic structure, there is a range of wavelengths (PBGs) centered around the reference wavelength λ0 for which the structure is almost totally reflective (that is, no propagating photon modes are allowed through the structure).

Fig.1. Transmission spectrum versus frequency (0-50 GHz) for one dimensional periodic photonic structure (32 layers). B. Generalized Fibonacci sequence Fig. 2 shows the results of numerical calculations of the transmission T as function of frequency with the variation of the parameter m (m=1, 2, 3, 4). It is seen from the figures that in the same case of the spatially varying number of layers there exists two PBG of size and depth variations according to parameter m. Moreover, the variation and the distribution of layers lead to the displacement of the area of forbidden gap and to the significant widening of the band width; the bands are centered respectively around 28 GHz and 33 GHz.

The first great band gap is bounded between 16.5 Ghz and 26.5 GHz whereas the second band gap is located between 27.5 GHz and 39 GHz. The width of two bands varies respectively versus the parameter m of Fibonacci sequences FC (m, 1). This variation indicated the relation between the number of layer which is fixed by putting parameter m and large of band. The existence of more band gaps is due to the quasi-periodicity of sequence where we detect some local periodicity in global structure. These periodicities produce the original band in each frequency (see figure 1). So, the number of band can be controlled by use the parameter m and n of the Fibonacci sequence.

Fig. 2. Transmission spectra as function of frequency (GHz) for m=1, 2, 3, 4 and n fixed to 1.

C. Generalized Thue-Morse Sequence In this part we study the variation of the transmission spectrum according to the permittivity and the degree of multiplicity for a generalized Thue-Morse sequence. On increasing the order of T-M sequence, the PBG splits and very narrow transmission peaks appear. Also the centre of each pseudo band varies according to the parameters puts m or n and the distribution of the layers H and L forming the system. The number of multi-narrow bands can be controlled by varying m and n, fig. 3 represents an example of transmission spectra versus frequency in the microwave domain.

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Fig. 3. Transmission spectrum as function of frequency (GHz) for Thue-Morse quasi-periodic structure.

C.1. Effect of Varying the Value of Permittivity of High Refractive Index

From Fig. 4 we can see that the high frequency term in the spectra depends on class order and the value of permittivity (εr) of the layer with higher refractive index. The depth of photonic band gap increases gradually according to the contrast of index (to increase the value of permittivity of the layer with higher refractive index and to preserve that of low index equal to 1). In this case we obtained respectively 3 PBGs covering the entire spectral field. The stacking of these PBGs characterized by its centre frequency and its width. The absences of the electromagnetic wave propagation in this frequency domain is covered respectively by these three band gaps. Thus, we use this device as a multi stop band filter which can be ordered by selecting respectively rejection band and controlled by varying the parameter of system.

(a)

(b)

Fig. 4. Transmission spectrum as a function of the frequency (GHz) for the second generation of the generalized Thue-Morse sequence for different value of permittivity (εr): (a) (εr =10); (b) (εr =70)

C.2. Effect of Varying the Degree of Multiplicity k From fig. 5 we note a stacking of several and similar photonic band gaps (PBG) covering all the spectral fields. The number of PBG increases gradually with the degree of multiplicity k that binds the parameter m to n but their sizes are reduced forming selective stop band filters. Each PBG inhibits the propagation of electromagnetic wave and forms a stop band filter by detecting its band of rejection and its frequency centre. The degree of multiplicity fixed the number of

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forbidden gaps and their sizes and gives a distribution of high and low index layer allowing to control the staking of the forbidden gaps. The number of the multi narrow bands can be controlled by varying the degree of multiplicity k. Figs. 4(d), 5(a, b and c) show a stop bands filter that covers all the frequency range of Global System for Mobile Communication (GSM) where the frequencies range are between [0.88-0.96] GHz. Table 1: Frequencies transmitted by GT filter for the different values of k

Table 2: The transmission values of frequencies transmitted by the GT filter for the different values of k.

From tables 1 and 2 we notice that the frequency fs=2.5 1010 Hz is always present in a PBG and that all picks of transmissions are symmetrical in relation to this frequency. For example, for k=3 Tmax1 and Tmax4 have the same value also Tmax2 and Tmax3 having the same value (fs is between f2 and f3). According to the same tables, we note also that, for every value of k the widths of PBGs are equal.

(a)

(b)

(c)

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(d)

Fig. 5. Transmission spectrum as a function of the frequency (GHz) for the second generation of the generalized Thue-Morse sequence for different value of degree of multiplicity k: (a) k=2;(b) k =3 ;(c) k =4;(d) k =5 D. Hybrid periodic quasi-periodic photonic band gap D.1. Stop Band Filters with GF (m, n) /Air/ GF (m, n) In this part we will study two generalized Fibonacci sequences separated by 24 mm of air. For construction of the generalized Fibonacci sequence, two basic elements named H and L are necessary. Here, H and L are defined as homogeneous ideal dielectric layers with different width and the index of refractions. These parameters are degrees of freedom for manipulation of the reflection and transmission characteristics. From Fig. 6 we can see a several consecutive stop bands which form a selective stop band filter with this sample distribution. The transmission spectrum varies versus parameter m or n of Fibonacci multilayer. For m=2, n=5m and l=5 of 5th Fibonacci sequences we can obtain 10 bands gaps for different frequency ranges between 0-50 GHz. However, if we take n=10m, the number of bands increases the peak of transmission deleted in whole spectral range.

(a)

(b) Fig. 6. Transmission spectra versus frequency for GF (m, n) /Air/ GF (m, n) structure with (a) m=2, n=10 and (b) m=2, n=20 D.2. Stop band filters with GF (m, n)/GTM (m, n)/GF (m, n) Fig. 7 shows one-dimensional hybrid quasi-periodic-class multilayer stack at normal incident wave constructed by using generalized Thue-Morse structure intercalated between two generalized Fibonacci sequences. Here the number of iteration for the GF sequences is l=5 and for the GTM sequence is l=2. Similarly, we obtain the same number of stop band of the previous structure (Fib/air/Fib) but are not centred around the same frequencies. We can deduce that the position of frequency center of corresponding band gap is controlled by the type of distribution. We noted in this class that the distribution of Fibonacci partition determines

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the number of band which composes the stop band filter (the number of these bands is equal to the value of parameter n of GF sequences).

(a)

(b)

Fig. 7. Transmission spectra versus frequency for GF (m, n)/GTM (m, n)/GF (m, n) structure with (a) m=2, n=10 for GF and GTM and (b) m=2, n=20 for the GF and m=2, n=10 for the GTM D.3. Stop band filters with GTM (m, n)/Air/ GTM (m, n) The transmission spectrum varies versus frequency range of system formed by two generalized Thue-Morse sequence separated by 24 mm of air. We can note in this case (fig. 8) that the number of band gap which exists in whole frequency range is equal to the value of parameter n. Consequently, the choice of parameter n fixes the number of band gap. Then if we know the n parameter, multi stop band filter

can easily designed by the distribution above. Compared to the previous system, the transmission peaks are increased and the same number of bands is found with lower iteration of Thue-Morse system (3rd iteration).

(a)

(b)

Fig. 8. Transmission spectra versus frequency for GTM (m, n)/Air/ GTM (m, n) structure with (a) m=2, n=10 for the GTM and (b) m=2, n=20 for the GTM D.4. Stop band filters with GTM (m, n)/GF (m, n)/GTM (m, n) In this final part, we studied the hybrid quasi-periodic structures. The global structure is constructed by two generalized Thue-Morse sequences separated by generalized Fibonacci. Here the number of iteration (l) for the two sequences (GTM and GF sequences) is not the same. From the fig. 9 and 10 it is noticed that only the parameter n of the generalized Fibonacci

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structure defines the number of PBGs; from the fig. 11 we can see that the number of these bands is equal to the value of parameter n of GF sequence. Also, we noted that by this sample distribution, the second band gap covers the all frequency range of the GSM band.

(a)

(b)

Fig. 9. Transmission spectra versus frequency for GTM (m, n)/GF (m, n)/GTM (m, n) structure with (a) m=2, n=10 and l=6 for GF and m=2, n=10 and l=2 for GTM and (b) m=2, n=10 and l=6 for the GF and m=2, n=20 and l=2 for the GTM

(a)

(b)

Fig. 10. Transmission spectra versus frequency for GTM (m, n)/GF (m, n)/GTM (m, n) structure with (a) m=2, n=20 and l=6 for GF and m=2, n=20 and l=2 for GTM and (b) m=2, n=20 and l=6 for the GF and m=2, n=10 and l=2 for the GTM

Fig. 11. Variation of the number of the PBGs according to the parameter n of GF sequence

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V. CONCLUSION In summary, we have studied stop band filter by using hybrid quasi-periodic photonic crystals. Although quasi-periodic systems have attracted a lot of interest due to their unusual physical properties, the transmission spectrum is characterized by several band gaps separated each by transmission peak at normal incident. Increasing the parameter of hybrid quasi-periodic sequence, the number of PBG increases and the transmission peaks is deleted. Again, some stop band filters can cover the frequency range of the GSM bands.

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