Study transmission characteristics of graded photonic crystal as low pass filter

Here, we investigate propagation characteristics of graded thickness one-dimensional photonic crystal structure made of silicon and air. The optical transmission is obtained by transfer matrix method (TMM) and studied impact of graded index and refractive index variation on the transmission spectra. The optical transmission of suggested structure drops as frequency increases and falls to zero more rapidly as graded index increases. The graded index variable can be used to tune the suggested structure’s cut off frequency, which corresponds to half of the transmission value. So, such device has useful application as tunable optical low-pass filter.


Introduction
A periodic array of dielectrics known as photonic crystals (PCs) displays forbidden bandgaps for photons, also referred as photonic bandgaps (PBG) [1].PBG can be used to modulate the flow of light in PCs, which is similar to electronic bandgap in semiconductor [1].Such PBG is responsible for many potential applications of PCs as optical filter, optical sensor, switch etc. [2][3][4][5][6][7].As per the periodicity of dielectric in space, PCs are classified in three different kinds as one-dimensional (1-D) PC [8], twodimensional (2-D) PC [9] and three-dimensional (3-D) PC [1].The simplest geometry is 1-D PC, whose optical properties can be controlled by the width and refractive index of dielectric layers [10].The various materials as dielectric, plasma, chalcogenide, superconductor etc. are investigated as 1-D PC materials by researchers [11][12][13][14].
Modification in structural parameters of PC, graded photonic crystal (GPC) is also in trend from last decades.GPC is defined as gradual variation in refractive index and thickness of layers.GPC is investigated to achieve large PBG compared to regular 1-D PC, which can be tuned using graded parameters.Singh et al. have investigated GPC structure as various device application like multichannel filter and sensor [15].Ankita et al. have used 1-D GPC to improve the optical sensitivity for a PC based optical biosensor device to detect malaria [16].
Here, we study optical properties of 1-D photonic crystal with linearly defined graded thickness structure.The investigation is extended to obtain effect of graded index (GI) and refractive index of optical transmission.

Model and Numerical Method
In this section, a 1-D GPC structured as  1  1 / 2  2 / 3  3 / … . .    is illustrated in Figure 1.1-D GPC structure contains periodic layers A as dielectric and B as air.The geometrical parameters of first layers A1 and B1 are considered as d1 and d2, respectively.For 1-D GPC, the thickness of n th layer is considered as dn = di (1+ n*GI), where i = 1, 2 for dielectric and air, respectively and GI is graded index (GI).

Figure 1. Schematic diagram of 1-D GPC
The fields of electromagnetic wave can be connected using transfer matrix on the both side of layer.Such matrix related to any single layer is given as [17] sin cos M sin cos where δj = (2π//λ)njdj, dj is width of j th layer and nj is refractive index, λ is wavelength.The characteristic matrix of whole 1-D GPC structure is expressed as 11 12 1-D GPC's transmission can be described as [17]     (3) where ni,s are refractive index corresponding to incoming and emerging media.

Numerical Results and Analysis
We consider about the 1-D GPC configuration, which are graded thickness multi-layered structure of alternative layers of dielectric and air.The initial thickness of dielectric and air are considered as 80 nm and 980 nm, respectively.For the initial study, silicon is considered as dielectric layer of refractive index 3.4 in THz frequency range.The transmission characteristic of 1-D GPC structure with zero graded index is measured by equation ( 3) and plotted as depicted in Fig. 2. As can be seen in Fig. 2 that photonic bandgap obtains at higher frequency region and good transmission shows at lower frequency region.Such a transmission characteristic can be utilized to optical limiter devices and optical low pass filters (optical LPF).Here, the variation in the transmission values shows more fluctuation near PBG region, which can limit utilization as optical LPF.Therefore, we are interested to investigate the impact of graded index (GI) of 1-D GPC structure.

Effect of Graded Index
Here, we study the impact of GI on transmission characteristic of 1-D GPC structure as illustrated in Fig. 3.In this case, GI values changes from 0.01 to 0.015 for both silicon and air.From Figure 3, it is observed that high transmission region shifted towards lower frequency region with GI.It is also observed that the variation of transmission values at low frequency region decreases compared to regular structure.The frequency corresponding to 50% transmission is known as cut-off frequency, which are tabulated as below.It is observed from table 2 that cut-off frequency shifted toward low frequency region with increasing GI.

Effect of Refractive Index
The effect of refractive index is studied for 1-D GPC by altering the higher dielectric medium's refractive index.The transmission spectra for varying refracting index from 2.2 to 3.8 with interval of 0.4 of layer A with GI of both layers as 0.02 is plotted as illustrated in Figure 4.

Conclusion
Here, the optical transmission characteristic of 1-D GPC configuration studied using TMM.The transmission of proposed structure gave the high transmission region at lower THz frequency region and zero transmission at high frequency regime.The cut-off frequency of high transmission shifted with GI and high dielectric layer's refractive index.The high transmission region reduced fluctuation for low refractive index and GI values.Such configuration can be utilized in optical communication as an optical limiter or low pass filter device.

Figure 3 .
Figure 3. 1-D GPC's transmission characteristic with graded index for both layer.

Figure 4 .
Figure 4. 1-D GPC's transmission characteristic with refractive index variation of layer A. It is observed from Figure 4, the high transmission region gradually moved towards higher frequency region.The fluctuation in high transmission reduces with decrease in refractive index values.The cutoff frequencies are tabulated as below.

Table 1 .
Cut-off frequency with GI

Table 2 .
The cut-off frequency with Refractive Index It is clear from table 2, the cut-off frequency shifted from 63.99 THz to 42.77 THz as refractive index increases from 2.2 to 3.8.Therefore, GI value and refractive index can be used to tune the cut-off frequency and smooth the transmission values at lower frequency region.Such transmission characteristics can be applied to optical limiter or low pass filters in optical communication devices.