INTRODUCTION
Photonic Band Gap (PBG) materials have been intensively investigated experimentally and theoretically during the past two decades (Bowden et al., 1993). Conventional dielectric structures with a periodically modulated refractive index exhibit forbidden frequency band gaps. Photons or electromagnetic waves with energy or frequencies within this band gap cannot propagate through these media (Knight et al., 1998). PBG structures can be used as a reflector for photon or electromagnetic waves and hence is employed for devices, such as lasers, waveguides, stop band filters and switches (Rhodes, 2003). The band gaps occur due to the Bragg scattering in a periodical dielectric structure with period comparable to the incident photon wavelength (Li and Zhang, 2000). Such Bragg band gaps strongly depend on the incident angle and the polarization of the light (Nusinsky and Hardy, 2006).
Recently left handed materials have been fabricated and the results are in good agreement with the theoretical predictions (Jakšiæ et al., 2006; Shelby et al., 2001). Left handed (LH) or Negative Refraction Index (NRI) materials must have both negative electric permittivity and magnetic permeability in the same range of wavelengths (Smith et al., 2000).
A stake of layers with alternating conventional dielectric and negative refraction
materials leads to a new type of PBG corresponding to a zero average refractive
index (<n> = 0) gap (Li et al., 2003; Yuan et al., 2006).
A number of unique properties of the zero average refractive index gap on the
beam shaping effect has been studied (Shadrivov et al., 2003). Such a
novel gap is quite different from the conventional Bragg reflection gap and
it appears due to a different physics of wave reflection. This gap is independent
of scaling and insensitive to the disorder (Wu et al., 2003). It has
been demonstrated that the edges of the zero averaged refractive index gap in
a special structure is insensitive to the incident angle for different wave
polarizations, leading to an omnidirectional gap (Jiang et al., 2003).
When the periodicity is broken by introducing a defect into the left handed
material, a localized defect mode will appear inside the Bragg gap or zero averaged
refractive index gap, which depends on the defect parameters. Defect modes in
zero averaged refractive index band gap, contrary to the Bragg ones are weakly
dependent on the angle and polarization of the incident wave (Jiang et al.,
2003; Shadrivov et al., 2004).
In previous investigations the models for the simulation of constitutive parameters of the materials were simply the same. But since the origins of phenomena to create these negative parameters are different, therefore their model must be different. The DrudeLorentz model describes very well the behavior of these parameters.
In this study, variations of the transmission characteristics of the real dispersive one dimensional left handed photonic band gap structure with defects are studied by Finite Difference Time Domain (FDTD) method. The effects of variations of the parameters of the layers and defects of these materials, by assuming the DrudeLorentz model for the frequency dependent permittivity and permeability are analyzed. A onedimensional band gap structure that has the same response to both TE and TM polarizations is proposed. The properties of the defect modes inside the zero averaged index band gap structure are investigated. Based on these theoretical results, a multi channel filter is introduced.
Analysis of photonic crystals with constant permittivity and permeability: Some periodic structures have band gaps for certain ranges of propagation directions and/or TE or TM modes. If a structure has band gaps for all ranges of propagation directions, it demonstrates a complete band gap or omnidirectional gap.
In this study, the propagation of the electromagnetic waves in the structure
of Fig. 1, an infinite onedimensional periodic structure
consisting of two different layers of materials with periodicity in the z direction,
is studied. For TE polarized electromagnetic waves, which the electric field
component is parallel to the layers, the wave equation is given by:
where:
is the periodic refractive index with period of unit cell width, a.
Propagation of electromagnetic waves in photonic crystals obey the Blochs conditions
and are characterized by the Bloch wave number, k_{b}, expressed as
(Li et al., 2003):
where:
k_{b} 
= 
The Bloch wave number. 
k_{x} 
= 
The xcomponent of wave number. 
k_{x = 0} 
= 
The normal incident. 

Fig. 1: 
Structure of a onedimensional photonic crystal consisting
of two different alternative layers 
Now, the oblique monochromatic electromagnetic fields, as the solutions of
Eq. 1 in different layers of the structure of Fig.
1 are derived. By satisfying the boundary conditions, at the interface of
media 1 and 2 and considering the Bloch's condition, the relation for the TE
polarization case is given by Li et al. (2003):
where, a is the width of the unit cell are the z component of the wave vector in the first (j = 1) and the second (j
= 2) medium and is the wave number in each media with refractive index of n_{j},
ω is the wave angular frequency and c is the speed of light in free space.
Similarly, the dispersion relation of the TM polarized wave is derived by replacing
μ by ε in Eq. 3. Solutions of this dispersion relation
demonstrate the behavior of the structure. If k_{x} and k_{b}
are real, the Bloch wave propagates through the periodic structure, whereas
if k_{x} or k_{b} are imaginary, the wave propagation is not
possible and there will be a band gap in the transmission spectrum of the structure.
It has been shown that the gap width of a one dimensional photonic crystal strongly
depends on the angle of incident light. Moreover, the band gap is sensitive
to polarization. A complete band gap occurs if for all real k_{x}, k_{b}
remains complex, which the wave propagation in this structure is inhibited (Li
and Zhang, 2000).
Now, if media 2 is considered as a LHM, the only difference in the derivation
of Eq. 3 is that the sign of the wave vector in this medium
must be reversed. This structure may have particular property with new band
gap when the condition of zero average refraction indices of Eq.
4 is satisfied:
In particular, the frequency corresponding to the zero average refraction index gap is independent of the width of the unit cell, while all Bragg gaps are scaled with the width of unit cell. Based on the results derived in (Shadrivov et al., 2005), a one dimensional periodic structure composed of unit cells with two layers of conventional and LH material with particular parameters will possesses a complete band gap for one polarization. However, there is no complete band gap for the other polarization.
Shadrivov et al. (2005) have shown that in a three layer periodic structure with cell property of two LHM layers with the same thickness and dual constitutive parameter (ε_{1,2} ↔ μ_{2,1}) surrounding a conventional dielectric, a complete band gap for both polarizations exits.
Analysis of dispersive left handed photonic crystals: Most of the early researches were concentrated on the LHPBG structures consisting of materials with special frequency dependent parameters. In this study, real dispersive materials with frequency dependent parameters are considered.
The left handed materials have structures, which are usually printed or drilled
onto appropriate substrates (for example split ring resonators and fine wires).
Their characteristic size is small compared to the wavelength. The material
behaves with averaged effective properties describable by ε and μ.
It is evident that these parameters directly depend on the shape, kind and type
of the materials of the structure (Jakšiae et al., 2006). The inherent
feature of these materials is their frequency dependent dispersion and losses.
Materials with negative refractive index, i.e., simultaneously negative ε
and μ are dispersive. The DrudeLorentz model describes very well the constitutive
parameters of these materials. In frequency domain, the effective permittivity
ε(ω) and permeability μ(ω) can be expressed as (Ramakrishna,
2005; Smith et al., 2000):
where:
ω_{pe} and ω_{pm} 
= 
The effective electric and magnetic plasma frequencies, respectively. 
ω_{mo} 
= 
Magnetic resonance frequency. 
ω 
= 
The angular frequency of the wave. 
These models include losses represented by the parameters Γ_{e}
and Γ_{m} and can also be modified for implementation of the nonlinear
effects (Zharov et al., 2003). From Eq. 5, it is clear
that these materials have a limited frequency bands in which their constitutive
parameters are negative, therefore, some new methods are required for analysis
of these structures. An analytical computation of the band structure and transmission
spectra of LHPBG with dispersive parameters and defect is difficult. The FDTD
method is a powerful method that can simulate the complex structures.
First, the PBG structure of Fig. 1 consisting of two cascaded
negative and positive refractive index slabs is considered. The one dimensional
FDTD method is used to simulate the behavior of the structure. At the beginning
we have selected the same Drude model for constitutive parameters which means
ω_{mo} = 0 (Ziolkowski, 2003; Young and Nelson, 2001). At both
ends of simulation windows, absorbing boundary conditions are used. The spatial
steps is set at Δz = λ/100, where λ is the free space wavelength.
The time step Δt is set at Δt = 0.95Δz/c, where c is the speed
of light. The input wave is a narrow Gaussian pulse with duration of λ/10
in order to obtain the impulse response of the structure, while 1000 spatial
steps and 30000 time steps are considered.
The transmittance of the structure is shown in Fig. 2. The
dashed line gives the transmittance of a stack of eleven unit cells with Γ_{e}
= Γ_{m} = 3.75x10^{4} ω_{pe, pm} and solid
line gives it for Γ_{e} = Γ_{m} = 0.
The <n> = 0 condition occurs at about 2.72 GHz and the band gap width for both cases is 1.25GHz. For the low loss structures, as are fabricated recently, the loss has a very weak effect on the band gaps. Therefore, in the next simulations, the values of Γ_{e} and Γ_{m} are assumed to be zero.
Deeper band gaps are induced for higher number of unit cells. The number of
ripples in the pass band is equal to the number of unit cells. These ripples
are the frequencies in which the complete transmissions occur. Therefore, to
obtain a flat pass band and deeper band gap, the number of unit cells must be
increased. The effects of different parameters of effective permittivity and
permeability on the band gap of the structure under study are shown in Fig.
3.
It is clearly shown that the edges of zero average refractive index band gap
is a function of magnetic resonance, electric and magneticplasma frequencies.
The width of bandgap depends on the difference of electric and magnetic plasma
frequencies. Variations of magnetic resonance frequency, ω_{mo},
shift the upper edge of the band gap. Since ω_{mo} is much lower
than ω_{mp}, we have used the same Drude model for the constitutive
parameters ε and μ (Ramakrishna, 2005). When effective electric and
magnetic plasma frequencies are equal, all the band gaps are disappeared due
to the matching of two layers.
Now, a band gap structure constituting of Munit cells, (Fig.
4) is selected to be analyzed. Each cell is composed of three layers. One
of these layers is a conventional dielectric (air) which is sandwiched by other
two layers. These two layers have a left handed nature with dual constitutive
parameters. All the layers have the same thickness d_{1} except the
conventional dielectric in the middle which is d_{2}. The defect at
the center of the structure is a dielectric layer, as d epicted in Fig.
4. The width of this defect layer is d_{3} = d_{2} (1Δ),
where Δ is the normalized defect size. Due to the symmetric parameters,
this structure has the same response to both polarizations.
A normal incident Gaussian beam is launched to the structure for both polarizations.
The structure parameters are ε_{d} = μ_{d} = 1, ε_{1}(ω)
= 15^{2}/ω^{2}, μ_{1}(ω) = 13^{2}/ω^{2},
ε_{2} = μ_{1} μ_{2} = ε_{1},
d_{1} = λ/20 and d_{2} = λ/10, with λ = 10 cm.

Fig. 2: 
Transmittance of eleven unit cells with alternate layers of
air (12 mm thick) and LHM material (6 mm thick) with effective ε(ω)
and μ(ω) given by Eq. 5 (f _{pe} = ω _{pe}/2π
= 3GHz, f _{pm} = ω _{pm}/2π = 5GHz) 

Fig. 3: 
Variation of zero averaged refractive index band gap of seventeen
unit cells with alternate layers of air (12 mm thick) and LHM (6 mm thick)
with effective ε(ω) and μ(ω) given by Eq.
5 for different values of magnetic plasma and magnetic resonance frequencies.
(a) f _{mo} = 0GHz, (b)f _{mo} = 0.5 Ghz and (c) f _{mo}
= 1GHz, with f _{pe} = 3GHz 

Fig. 4: 
A 1D complete band gap left handed structure consisting of
layers of LHM, conventional dielectric or positive handed material (PHM)
and one defect layer in the middle 
RESULTS AND DISCUSSION
The FDTD method predicts the electric field intensities at all points of the
simulation window. The transmittance spectra of the structure of Fig.
4 with seven unit cells, with and without defect are demonstrated in Fig.
5. Three possible kinds of band gap are shown in Fig. 5.
One is a Bragg band gap that generally appear in photonic crystals. The other
is zero averaged (<n> = 0) band gap, which is particularly created due
to layers of alternating conventional dielectric and lefthanded materials.
Another band gap exists in the spectrum, which is called ‘zeroorder
stop band’ and appeared just above zero frequency (Fig.
5). Existence of this band gap is determined by appropriate selection of
the parameters of the structure (Ye et al., 2005). Also defect mode has
been created inside the zero averaged and Bragg band gaps. Comparison of
Fig. 5 and 6 approves that the defect mode inside the
zero averaged band and the edges of the zero averaged band gap is independent
of the scaling. Also, it is shown that it has a weak dependence on the incident
angle of the input wave (Jiang et al., 2003).
Deeper band gaps are induced for higher number of unit cells, as depicted in
Fig. 7. Consequently, induced band gaps over a wide range
of frequency can be achieved, which the effect can be applied for the perfect
reflectors for both polarizations.
Design of bandreject filters are accessible by these structures, the description
of which will be published in a new paper. Variation of the normalized defect
size, Δ, shifts the defect mode. With special parameters the defect mode
appears in the zero averaged refractive index or Bragg gap, but it does not
always appear simultaneously in all gaps. Variations of transmittance spectra
and shift of the defect mode with different normalized defect size, Δ are
shown in Fig. 8.
The width and peak frequency of the defect mode inside the zero averaged gap
vary with number of cells of LHPBG structure (Fig. 9). When
the number of unit cells increases, the defect mode becomes less visible.

Fig. 5: 
Transmittance of seven unit cells of Fig. 4,
for the structure with normalized defect size of Δ = 0.8 (solid line)
and without defect (dashed line) 

Fig. 6: 
Transmittance of seven unit cells of Fig. 4,
for the structure with normalized defect size of Δ = 0.8 (solid line)
and without defect (dashed line) that unit cell is scaled by 3/4 

Fig. 7: 
Transmittance of eleven unit cells of Fig. 4,
for structure with normalized defect size of Δ = 0.8 (solid line) and
without defect (dashed line) 

Fig. 8: 
Frequency spectrum of the defect modes for seven unit cells
of Fig. 4 versus the normalized defect size Δ 
The
defect mode weakly depends on the scaling and polarization. The defect mode
is useful for design of high quality filters.
The simulation can be extended to the study of the properties of multiple defect
layers. By increasing number of defect in the structure of Fig.
4, the number of defect modes in the <n>=0 gap increases.
Figure 10 compares the defect modes of structures with one
and two defect layers. This phenomenon may be useful in the design of multiple
channel filters.
CONCLUSIONS
In this study, the transmittance spectrum of the one dimensional left handed
photonic band gap structure with multiple defects has been analyzed by dispersive
FDTD.

Fig. 9: 
Half maximum width and peak frequency of the defect mode inside
the zero averaged gap versus the number of unit cells of Fig.
4 

Fig. 10: 
Defect modes inside <n> = 0 gap for 17 unit cells of Fig. 4, with d _{1} = 8 mm, d _{2} = 16
mm and d _{defect} = 26 mm. Position of the defect layer for single
defect structure is in the 9th and those of double defects are in 9th and
11th cells 
The band gap spectra of the one dimensional multi layer slabs of conventional
dielectric and left handed materials, with and without defect were derived.
Three different band gaps of zero order, Bragg and zero average refractive index
can be displayed in the spectrum of the multi layer slabs with particular parameters.
The width of the zero average refractive index band gap depends on the difference
between the electric and magnetic plasma frequency, not their direct values.
Deeper band gap and flatter pass band are induced for higher number of units.
The edges of the zero average refractive index band gap are function of effective
permittivity and permeability parameters and less dependent on the scaling of
the layer thicknesses. Increasing the magnetic resonance frequency, ω_{mo}
increases the upper edge of the band gap. Since ω_{mo} is much
lower than magnetic plasma frequency, ω_{pm} the models without
ω_{mo} can also give acceptable results. When effective electric
and magnetic plasma frequencies are equal all band gaps are closed due to the
matching of layers. Properties of the defect modes are demonstrated. Depending
on the defect size, the defect mode may appear in zero average index or Bragg
band gap. The edges of the band gaps and the defect modes depend weakly on frequency
scaling. A particular structure may have the same response for both polarizations.
We have studied properties of single and multiple defects. When the total number
of layers of a structure with defect is more than 10, the defect mode width
and the defect peak frequency remain constant. Multiple defect modes are produced
when multiple defect layers are periodically placed in the structure. As the
periods of the defect layers increase, the resonance transmission modes locate
symmetrically in the band gap region. The properties of these structures provide
possible applications, such as multiple channel omnidirectional filtering.
ACKNOWLEDGMENT
The authors would like to thank Iran Telecommunication Research Center (ITRC) for the financial support of this project.