Comparative Study of the One Dimensional Dielectric and Metallic Photonic Crystals ()
1. Introduction
Photonic crystals (PCs) are macroscopic media which arranged periodically with different refractive indices and their periodicities are in the range of the incident light [1]. In such structures the permittivity is a periodic function in space. In this case, the dielectric permittivity function repeats itself in one dimension (1D) the structure called one dimensional photonic crystal (1D-PC), if it repeats itself in 2D or 3D the structure called 2D or 3D PC. The one dimensional photonic crystal (Figure 1) is a multilayered media. It is worthy to mention that, the propagation of photons in the PCs is similar to the propagation of electrons in the semiconductor crystals, where the effect of the periodic dielectric function on the propagating photon in PCs is much like the effect of the periodic potential function on the propagating electron in semiconductor crystal. Consequently, a photonic band is created in PCs similar to the electronic bad gap in semiconductor crystal [2].
On the other hand, when electromagnetic waves (EM) incident on the PCs Bloch states create in the crystal, if the Bloch wave falls in the so called forbidden bands (photonic band gap) such a wave is evanescent and can’t propagate in the crystal. Thus the light energy is expected to be totally reflected, and the crystal acts as a high reflectance reflector for the incident wave. The photonic band gap of the photonic crystal makes us able to control the light even the spontaneous emission [3]. In this paper, we are going to do comparative study between the 1D-DPCs and -MPCs pointing to the general applications of each kind according to its characteristics.
2. Analysis
2.1. Dielectric Photonic Crystals (DPCs)
In the last decades dielectric photonic crystals have attracted much research interest due to their various applications for example, optical filters, waveguides, and optical fibres [4-7]. In this section, we restrict our communications on the characteristics of the 1D-DPC showing its various applications. The reflection of the EM waves through DPCs exhibit resonance reflection very much like the diffraction of x-rays by crystal lattice planes, therefore it’s called Bragg reflector.
We have designed 1D-DPC composed of a low index material (Cryolite = 1.34) and a high index material (Silicon = 3.4) stacked alternatively on a glass substrate. The number of periods, lattice constant, effective refractive index, and the filling factors of the low and the high index materials are taken to be 10, 250, 2.389, 0.6, and 0.4 nm, respectively. The filling factor (f) of a material in a 1D-PC can be given by [8];
Figure 1. Schematic diagram shows a one dimensional photonic crystal.
(1)
where d and 
are the material layer thickness in the unit cell of the PC and the lattice constant (spatially periodic constant), respectively. The transmission spectra of the DPC are displayed in Figure 2. The figure shows that the DPC presents a transmission band for the low frequencies (long wavelengths), and the first band gap is associated with the Bragg condition [2];
(2)
where 
is the centre wavelength of the first band gap, 
is the lattice constant, and n is the effective refractive index. The effective refractive index can be given by [2];
(3)
The term 
represents the filling factor of the low index material multiplied by its permittivity, and the term 
represents the filling factor of the high index material multiplied by its permittivity. The centre wavelength of the first stop band determined by Bragg condition is approximately equal to 1194 nm which is consistent with the value deduced from the transfer matrix method that shown in Figure 2. The photonic band gap of the DPC can be tuneable by varying n or
, and the band gap can be shift to longer (shorter) wavelengths with increasing (decreasing) the lattice constant or the effective index as Bragg condition predict.
It’s known that, in a specified frequency range the band gap width depends on the difference in the refractive indices of the two constituent materials (
). So to show effect of 
we have designed DPC from Cryolite/Silicon dioxide (
). The transmission spectra are displayed in Figure 3, it is obvious that, when the number of periods is equal to ten, 
of Cryolite/Silicon dioxide is small not enough to open deep gap, but when the number of periods become equal to fifty, a narrow deep gap can be open. We have observed that the number of periods doesn’t effect on the position or the width of the band gap. But increasing number of periods enhances the reflectivity of the bad gap and makes the band gap edges steeper.
In Figure 3, the number of the resonance transmission peaks (RTPs) for the PC of fifty periods is larger than number of RTPs for the PC of ten periods. We have noticed that, the RTPs of the DPC is directly proportional to the number of periods, and the RTPs have become closer to each other and sharper as the wavelength decreases and vice versa.
Figure 2. Calculated transmission spectra of a dielectric photonic crystal with n1 = 1.34, n2 = 3.4, d1 = 150 nm, d2 = 100 nm, number of periods = 10, and θ = 0˚.
Figure 3. Calculated transmission spectra of a dielectric PC with n1 = 1.34, n2 = 1.46, d1 = 150 nm, d2 = 100 nm, period = 10, and θ = 0˚.
On the other hand, in order to study the filling factor effect, we have designed two DPCs both composed of Cryolite/Silicon but it is differ in the filling factors of the two constituent materials. The transmission spectra of the two PCs are shown in Figure 4 by increasing the filling factor of the high index material (HIMF) red shift of the band gap occurs, this due to the increase of the effective refractive index of the dielectric stack. It is observed that, the band gap width slightly decreases with increasing the HIMF.
The incidence angle effect on the transmittance of the DPC for Sand P-polarized waves is displayed in the Figures 5 and 6, respectively. When the incidence angle of the electromagnetic waves increase blue shift of the band gaps of the Sand P-polarized waves occur. The band gap of the P-polarized wave shrinks due to Brewster effect at the interface between low and high index layers [9]. But the band gap of the S-polarized wave increases slightly. The forbidden gaps for the two polarizations not coincide due to the loss of the degeneracy. From the Figures 5 and 6, we have observed that, the P-Polarized wave more sensitive to the change of angle than S-polarized wave. 1D-DPCs structure has many applications such as filters [10], omnidirectional reflectors [11-19], polarisers [20-25], antireflection coatings, distributed Bragg reflectors for vertical-Cavity surface emitting lasers (VCSEL), and wavelength division multiplexers/demultiplexers on the basis of fibre Bragg
Figure 4. Calculated transmission spectra of two DPCs with n1 = 1.34, n2 =3.4, Λ = 250 nm, θ = 0˚ and different only in the filling factors of the constituent materials.
Figure 5. Calculated transmission spectra of DPC for S-polarized wave at different incident angles; the DPC with n1 = 1.34, n2 = 3.3, d1 = 150 nm, d2 = 100 nm, and number of periods = 10.
Figure 6. Calculated transmission spectra of DPC for P-polarized wave at different incident angles; the DPC with n1 = 1.34, n2 = 3.3, d1 = 150 nm, d2 = 100 nm, and number of periods = 10.
grating (FBG) [3].
2.2. Metallic Photonic Crystals (MPCs)
We have shown in the previous section that, in order to achieve photonic band gap, the system must has high contrast in the refractive index with negligible the absorption of light. These conditions have restricted the set of dielectrics that exhibit a photonic band gap. One suggestion is to use metals which have large value of dielectric permittivity rather than dielectrics. Accordingly a fewer numbers of periods would be enough to achieve photonic band gap [26,27]. We have designed 1D-MPC composed of Cryolite/Silver with 10 periods, lattice constant = 210 nm, and the filling factors of Sillver and Cryolite are 0.0476, and 0.9634, respectively. The dispersion has been taken into account by using Drude model and then we can alculate the refractive index of metals. The transmission spectra of the MPC are displayed in Figure 7. As shown in the figure, the MPC present like the DPC alternation of transmission bands and band gaps with the same progressive decrease of the transmission contrast. However for low frequency region starting from zero frequency of the spectrum, MPC exhibit plasmonic band gap. This plasmonic gap extends from 309.3 THz (970 nm) to zero frequency. This band gap not originated from the structure but from the bulk silver properties. In addition to the plasmonic band gap, the MPC exhibits structural band gap extends from 420 to 570 nm. The structural band gap follows the first transmission band that extends from 570 to 970 nm. The Plasmonic band gap is followed by a first transmission band whose centre wavelength corresponds to Bragg condition. The situation here turns out to be reversed compared to the case of the DPC, where the same exact relation corresponds to the first band gap. The value of the centre wavelength of the first transmission band determined from Bragg condition (750 nm) nearly consistent with the value deduced from the transfer matrix method shown in Figure 7. The first transmission band or the band gaps of the MPC can be tuned by varying n or 
as in the DPC.
Figure 8 shows the transmission spectra of the previous designed MPC at the number of periods equal to five periods. By decreasing number of periods, no change in