Abstract: Linear photonic band gap structure with defects is analyzed using the one-step alternating direction implicit finite difference time domain method (one step ADI-FDTD). This method is independent of Courant condition and consumes less computational resources and time compared to the conventional ADI-FDTD. The quality of the transmission spectrum of defect mode frequency is studied by the varying the index contrast of one dimensional photonic crystal.
Introduction
Photonic crystals are artificially created dielectric materials in which the index of refraction varies periodically between high-index region and low-index region. This periodic dielectric material offers a great control to manipulate the propagation of the light. Such a photonic crystal exhibits a frequency band gap in which a band of frequency of light is forbidden. When the defect layer is introduced in the linear Photonic Band Gap (PBG) structure with symmetry, it would produce a narrow resonance frequency in the band gap, which is known as defect mode frequency (frequency components other than the defect modes will be reflected by the PBG structure). Recently the simulation of wave propagation through one dimensional symmetric photonic band gap structure with defect layer was extensively investigated by theoretical and numerical methods (Joannopoulos et al., 1995; Lixue et al., 2002).
To explore the band-width of defect mode frequency, one may change the optical thickness of the defect layer either by changing the width or refractive index of the defect layer. If the optical thickness of the defect layer is non integer multiples of λ0/4, it gives rise to acceptor and donor modes as in the solid state electronics (Stanley et al., 1993). When the band width of the defect mode frequency is narrow, higher will be the value of the quality factor (Fan et al., 1995).
In this study, we apply the one-step alternating direction implicit finite difference time domain method (one-step ADI-FDTD) (Shao-Bin and San-Qiu, 2004) to study the essential properties of the linear optical cavities. In the conventional ADI-FDTD method (Namiki, 1999), the time step advances from nth step to (n+1)th step in two time steps, but in one-step ADI-FDTD, it advances from nth time step to (n+1)th time step in a single time step. However, both the time domain methods are free from the numerical stability factor (Courant number S = cΔt/Δz). The most attractive factor in this numerical method is that it computes with high speed and consumes less computational memory compare to the conventional ADI-FDTD method.
| Fig. 1: | Schematic diagram of 1D-PC with defect |
PBG Structure with Defect
The one dimensional PBG structure consist of alternating layers with high
refractive index nH and low refractive index nL. Figure
1 shows the defect layer which is flanked by the one dimensional photonic
crystal (1D-PC) with symmetry. The refractive index of defect layer is nD.
The optical thickness of every layer must be λ0/4, where λ0
is wavelength of incident pulse in free space. The thickness of the high and
low layers are λ0/(4nH) and λ0/(4nL),
respectively.
This structure is simply denoted as
One-step ADI-FDTD Method
The time dependent Maxwell’s equations for one dimensional periodic
photonic crystal with dispersionless, loss-free and source-free stratified linear
isotropic media with z-directed and x-polarized TEM mode are given as:
| (1) |
| (2) |
where ε and μ are the position dependent permittivity and permeability and σz and σ*z are position dependent conductivities of electric and magnetic field, respectively. The Maxwell’s equations are discretized for the electric field as (Shao-Bin and San-Qiu, 2004),
| (3) |
where αE and βE are coefficients of the discretized electric field update equation,
and for the magnetic field,
| (4) |
where αH and βH are coefficients of the discretized magnetic field update equation,
where the indices n and k denotes the discrete time step at t = nΔt and the spatial point on the grid z = kΔz in z-direction, respectively.
Using expression of
| (5) |
This is an implicit discrete equation which updates the electric field
The Maxwell’s equation is spatially discretized in time domain whose spatial discretization must be restricted. In order to terminate the outgoing waves, an absorbing boundary condition has been introduced for FDTD and ADI-FDTD (Taflove and Hagness, 2000). The computational spatial domain is especially designed with a matched medium at the ends. This perfectly matched layer would absorb the outgoing waves without reflection. To have such a matched medium, the condition σz = σ*z, must be satisfied at the computational boundaries.
Results and Discussion
The numerical scheme derived in the previous section is applied to study the band-width of the defect mode frequency. The value of the Courant number S = 2.5. In general, the incident pulse Ex,inc is a monochromatic Gaussian wave which is attached to computational domain using the Total Field Scattered Field (TFSF) formulation (Taflove and Hagness, 2000),
where ω0 is the frequency of the incident pulse, w is the pulse width of the incident pulse.
| Fig. 2: | Transmission spectrum of defect mode frequency for the different optical thickness of the defect layer (a) nD = 1.0, (b) nD = 2.0, (c) nD = 2.5. Dotted line is for Δn = 0.5, dashed line is for Δn = 1.0 and bold line is for Δn = 1.5. The points which is marked with ‘o’ are from one-step ADI-FDTD method with Courant number S = 2.5 |
The influence of the optical thickness of the defect layer and total number of layers of PCs surrounding the defect layer changes the band width of the defect mode frequency which are all well known results (Lioxue et al., 2002). Here, the band-width of defect mode frequency varies as a function of index contrast values of the alternating layers of the PC and optical thickness of the defect layer. The contrast value is the difference between the refractive index of alternating layers in the 1D-PC.
In the analysis of spectrum of defect mode frequency in 1D-PC with defects,
the resonating structure is in the form of
For the numerical analysis, the refractive index value of L layer is nD = 1.5 and index contrast of the 1D-PC structure is studied only by varying the refractive index of H layer, so that index contrast value varies as Δn = 0.5, Δn = 1.0 and Δn = 1.5.
In the Fig. 2a, the refractive index value of the defect layer is nD = 1.5. As the value of index contrast goes on increasing, the width of the defect mode frequency decreases. So, confinement of defect mode frequency increases as the index contrast value of the 1D-PC increases in the resonating structure.
In Fig. 2b, the value refractive index of the defect layer is increased to nD = 2.0. The band width of the defect mode frequency is reduced further as the index contrast increases. When the optical thickness of the defect layer increases, confinement of the localized defect would also increase and hence the width of the defect mode frequency is reduced.
The same analysis is carried out for the refractive index of the defect layer nD = 2.5 as shown in Fig. 2c. In this case, even though the refractive index contrasts variation are the same as before, the increment the optical thickness of the defect layer causes further reduction in the width of the defect mode frequency. For higher optical thickness, the confinement of the defect mode frequency is large compared to the lower optical thickness.
| Fig. 3: | Band-width of defect mode frequency for the different optical thickness of the defect layer nD = 1.0 (dotted line), nD = 2.0 (dashed line) and nD = 2.5 (bold line) |
| Table 1: | Comparison of FDTD and ADI-FDTD |
Here we varied the index contrast of the periodic structure along with optical thickness of the defect layer. The results given by Lixue et al. (2002) show how the influence of the optical thickness of the defect layer affects band width of the defect mode frequency. When the refractive index of the defect layer increases, the band-width of the defect mode frequency becomes narrower. In Fig. 3, although the refractive index of the defect layer is low, due to the increase in index contrast, FWHM of the defect mode frequency is reduced.
One-step ADI-FDTD considerably reduces the computation time. Numerical simulations are carried out using 3.0 GHz processor with 1GB RAM. Table 1 gives the comparison of conventional FDTD with one-step ADI-FDTD.
Conclusions
In summary, our numerical analysis demonstrates the spectrum of the defect mode frequency in one-dimensional resonating structure as the function of index contrast values and increase in the value of the optical thickness of the defect layer. At lower optical thickness of the defect layer, decrease in the band-width of defect mode frequency is achieved by increasing the index contrast value. In Future, this work could be extended to study optical bistability in the PBG structure by introducing nonlinearity in the layer.