Keywords

Photonic Crystal, demultiplexer, filter, channel spacing, quality factor, crosstalk 
INTRODUCTION

Photonic crystal is a periodic arrangement of different refractive index materials and this arrangement can be one, two or three dimensional known as 1D, 2D or 3D photonic crystal respectively [1]. Defect in these crystals create a gap of frequency band and photons which have the frequency within this forbidden gap are not allowed to pass through the crystal structure and hence this property of photonic crystal allows us to guide the light along the defect efficiently in photonic communication [27]. Light in dielectric medium travels much faster than electrons in metal and this gives the optical communication the ability to transmit data at a very high speed. Apart from that significant bandwidth is offered by the dielectric material as compared to its electrical counterpart, making the high bit rate data transmission possible. As far as energy loss is concern, the optical communication outclasses the electrical communication on this point as well, because of less interaction of light particle as compare to electrons [8]. 
State of the art integrated circuits have very high density integration and the IC manufacturing companies have a big challenge ahead to further enhance the performance of ICs. Increment in number of transistors results in the increment in heat dissipation and consequently vulnerability of the ICs. The speed of these electronic circuits can be increased at the cost of complexity and reliability. Use of photonic crystals in optical communication has revolutionized the communication totally by providing much faster computing, massive parallel processing, and a new step in digital age [9, 10]. 
Wavelength division multiplexing is the key component in modern optical communication systems. In this article, a novel concept for a wavelength demultiplexer is proposed and investigated. It utilizes the filtering properties of photonic crystal structure and based on a shift of the cutoff frequency of the photonic band gap mode in a planar photonic crystal waveguide. The rest of this article is organized as follows. Section 2 concentrates on numerical modeling techniques of photonic crystal design. Section 3 concentrates on the modeling and simulation results of different photonic crystal demultiplexers for wavelength division multiplexing systems. Finally, some conclusions are drawn in Section 4. 
NUMERICAL MODELING TECHNIQUES

There are various numerical modeling techniques for the modeling of photonic devices such as Beam Propagation Method, FiniteElement Method, FiniteDifference TimeDomain (FDTD) Method, FiniteVolume TimeDomain Method, Finite Integration Technique, Multiresolution TimeDomain Method, Plane Wave Expansion (PWE) Method, etc. However, only few techniques have shown great popularity in the area of research like FDTD and PWE method [11]. In this research work OptiFDTD, design tool of Optiwave Systems Inc., has been used, which utilizes the concept of finitedifference timedomain (FDTD) algorithm. 
A. PLANE WAVE EXPANSION (PWE) Method 
In this research work the location and size of the band gap are calculated by Plane wave expansion method. For calculating band structure the plane wave expansion method uses an eigen formulation of the Maxwell's equations. Hence solved for the eigen frequencies for every propagation direction of the wave vectors. The solution directly provides the dispersion diagram. 
The Maxwell equation in a transparent, source free, timeinvariant, and nonmagnetic medium can be represented by the following equation: 
(1) 
where is the dielectric function dependent on space vector, c is velocity of light in vacuum, and s the optical magnetic field vector of a certain frequency ω which has the dependency on time given by e^{iwt}. This equation is sometimes referred as master equation [12], and represents a Hermitian eigenproblem, which would not be applicable if this wave equation would have been derived in terms of the electric field. 
According to the Bloch theorem the infinite periodicity causes the magnetic field to take the form [11]: 
(2) 
where for all combinations of lattice vectors. Thus, master equation in operator form will be represented by following equation: 
(3) 
This represents the fundamental equation, which should be solved. By solving the magnetic field employing simple plane waves in a finite domain, this fundamental equation turns into a finite problem. To solve the final discretized problem different approaches can be explored, e.g. Blockiterative frequencydomain methods [13] and Simple plane wave implementation for photonic crystal [14]. The dispersion relationship between the modes frequencies and the wave vector is given by the solution of discretized problem, which is also known as band diagram. 
SIMULATION AND RESULTS

The proposed wavelength demultiplexers are designed by using wavelengthselective filtering property of photonic crystal structures. These proposed wavelength demultiplexers are designed and optimized by finite difference time domain (FDTD) method and the band gap of proposed demultiplexers are calculated by using plane wave expansion (PWE) method. 
A. Design of filter 
We have designed 2D Photonic Crystal of square lattice with identical rods. The refractive index of rods is 3.47 (Si rods) and the radius of rods are taken as 0.18a (where ‘a’ is lattice constant). Optical filters are the basis of proposed wavelength demultiplexers. Some point defects are created in the photonic crystal structure to generate these optical filters. 
The size of the local point defect is determined very carefully for the propagation or transmission of a specific wavelength light signal. Analysis of Gap map diagram gives the idea about the radii of point defects. Gap map diagram is drawn with RSoft Photonics CAD Suite. The gap map diagram of twodimensional square lattice photonic crystal with identical Si rods (nsi =3.47) in air, is shown in the Figure 1. 
It can be seen from Figure 1 that photonic crystal with the radius of Si rods r = 0.18a has photonic band gap at the frequency range of 0.2983 ~ 0.4409(a/λ). The value of lattice constant ‘a’ is chosen appropriately to get the required normalized frequency range. If ‘a’ is taken as 0.52μm, the normalized frequency corresponding to two wavelengths 1.31μm and 1.55μm are given by 0.3969(a/λ) and 0.3355(a/λ), respectively. These normalized frequencies lie within the photonic band gap and thus cannot propagate through the photonic crystal until the defect is created. 
In Figure 2, we can see that for filter with radius of defect rods r = 0.25a, wavelength 1.55μm is in the band gap and wavelength 1.31μm is out of band gap. So wavelength 1.31μm can propagate through the filter while 1.55μm wavelength is restricted to propagate [15]. Filter constructed with radius of defect rods r = 0.36a, passes 1.55μm wavelength and restricts 1.31μm wavelength. This can be seen from Figure 2. Now, a filter is constructed emerging both the defect rods with radius r = 0.25a and r = 0.36a into a single filter is shown in Figure 3. Wavelength 1.31μm is obtained at output port 1 and wavelength 1.55μm is obtained at output port 2. 
Figure 4 (a) and (b) shows the power at output port 1 and 2 respectively for the input power 1. 
In the Gap Map shown in Figure 5 we can see that at radius r1 the frequency λ1’ is out of gap map. So it will propagate through the photonic crystal structure of r1 radius of rods. At radius r1 frequency λ2’ and λ3’ is in the Gap Map. So these frequency will not propagate through the Photonic crystal structure with radius of rods r1.Now decreasing the radius of rods to r2, frequency λ2’ and λ1’ are out of Gap map. So these frequency will propagate through the photonic crystal of rods having radius r2.Again at lowest radius r3, frequency λ3’, λ2’ and λ1’ are out of Gap map. So these frequencies will propagate through Photonic crystal structure of radius r3. 
The dispersion curves of photonic crystal (r = 0.18a) for different radius of defect were calculated by plane wave expansion method and are shown in Figure 6. 
We can observe that by reducing the radius of defect rods, modes move towards higher frequencies. So for radius of defect rod r=0.16a, wavelength λ1 will propagate through the filter. As we decrease the size of defect rod from 0.16a to 0.14a, wavelength λ1 and λ2 both will propagate through the filter. For the filter with defect rod radius r=0.12a, wavelength λ1, λ2, λ3 will propagate through filter as shown in Figure 7 [16, 17]. 
With the decrease in radius of rods at border of defect we can shift the modes at higher frequency and can stop the lowest frequency to propagate through further ports. It is shown in the Figure 8 given below [17, 18]: 
For border rod radius r=0.16a, wavelength λ1 will not propagate through it as shown in Figure 10. With further decrease in border rod radius to r=0.14a, wavelength λ2 will not propagate through it. Thus decreasing border rod radius will decrease effective refractive index of the waveguide, which in turn shift the modes at higher frequencies and we can stop lower frequency to transmit through the waveguide. 
If we decrease the diameter of the border rods, the lower edge of the transmission band shifts to higher frequency as the amount of higher refractive index material is decreased in the waveguide. This effect can be used to prevent the propagation of a narrow wavelength range near the lowfrequency transmission edge by decreasing the size of the border rods in consecutive sections along the photonic crystal waveguide. The restricted narrow wavelength band finds its guiding path along nearly placed waveguide and can be collected in that waveguide. Thus, a wavelength channel is demultiplexed to the adjacent photonic crystal waveguide from the incident signal. In this way we can design a multichannel optical demultiplexer by modifying border rods in consecutive sections which block some specific wavelength band and that can be collected at nearly placed waveguides. 
B. Design of demultiplexer for three different optical windows wavelength 
We have proposed demultiplexer for wavelength corresponding to three different optical windows. Three wavelengths 1.31, 1.55 and 1.625μm are considered which correspond to second, third and fourth optical window respectively [19]. The proposed 2D Photonic Crystal structure consists of square lattice with identical rods. The refractive index of rods is 3.47 (Si rods) and the radius of rods are taken as 0.18a (where ‘a’ is lattice constant). 
It is seen from Figure 11 that photonic crystal with the radius of r = 0.18a has a band gap in frequency range (ω/2πc) = 0.573679 to 0.847886, 1.41463 to 1.457 for TE mode. Thus, if the lattice constant ‘a’ is taken as 0.52μm, the first PBG is at the normalized frequency range of 0.2983 ~ 0.4409(a/λ). For a=0.52μm, three wavelengths 1.31μm, 1.55μm and 1.625μm correspond to the normalized frequency 0.3969(a/λ), 0.3355(a/λ) and 0.32(a/λ) respectively, and as these frequencies lying in the PBG, thus cannot propagate through the photonic crystal until the defect is created in the structure. Three Gaussian input planes are used to launch the frequency corresponding to wavelength 1.31μm, 1.55μm and 1.625μm simultaneously at input port. 
Radius of defect rod corresponding to output port 1, 2 and 3 are taken r1=0.07176μm (0.138a), r2=0.1872μm (0.36a), r3=0.13μm (0.25a) respectively. Demultiplexer is designed to receive wavelength 1.625μm, 1.55μm, 1.31μm at output port 1, 2 and 3 respectively. Radius of defect rods at output port 1 is taken after optimization of radius with the help of gap map diagram (Figure 1) and FDTD simulations such that it should propagate wavelength 1.625μm and prohibit the propagation of wavelength 1.55 and 1.31μm through it. Radius of the border rods of single line defect waveguide is decreased in section (black box) shown in the Figure 12 to prohibit transmission of wavelength 1.625μm from this section. Radius of border rods in black box section is taken as 0.0728μm (r=0.14a). 
Observation points (green dot) are placed at output port 1, 2 and 3 to observe the power received from the source. Power measured at output ports 1, 2 and 3 after simulation by OptiFDTD (Finite differential time domain) is normalized with respect to the maximum power received at the corresponding port and is shown in Figure 13. 
Quality factor Q=λ0/ΔλFWHM, with λ0 is the center wavelength and ΔλFWHM is the full wavelength width at half maximum, is the important parameter in wavelength division multiplexer which gives the resolution of wavelength selectivity. For wavelength selective applications high quality factor is required [20]. Table 1 gives the quality factor of every channel. 
Another important parameter in designing wavelength division demultiplexer is Crosstalk which is unwanted leakage or bleed of a signal from one or more channels to other channels within a device. Lower value of crosstalk is always desired [20, 21]. In Table 2 crosstalk is named as Xtij, which i and j vary from 1 to 3. Xtij shows the effect of crosstalk of output j in ith output ( i and j indices are row and column in Table 2 respectability). For example Xt12 shows the amount of crosstalk of Output 2 in Output 1. 
C. Design of demultiplexer for CWDM system 
The proposed demultiplexer is designed using two dimensional square lattice photonic crystal structure with identical Si rods (refractive index nSi = 3.47) and radius of rods are taken as 0.375a (where ‘a’ is lattice constant). The band diagram shown in Figure 14 gives the propagation modes in the designed photonic crystal structure, which represents the photonic band gap (PBG) for Transverse Electric (TE) modes alone. It is seen from Figure 14 that photonic crystal with the radius of r = 0.375a has two band gap in frequency range (ω/2πc) = 0.362685 to 0.406143, 0.605351 to 0.691514 for TE mode. Thus, if the lattice constant ‘a’ is taken as 0.589μm, the PBG are at the normalized frequency ranges of 0.2136 – 0.2392(a/λ) and 0.3565 – 0.4073(a/λ). For lattice constant a = 589 nm, wavelengths 1.57, 1.55 and 1.53μm correspond to the normalized frequencies 0.3751(a/λ), 0.38(a/λ) and 0.3849(a/λ) respectively, and as these frequencies lying in the PBG, thus cannot propagate through the photonic crystal until unless the defect is created. 
Three Gaussian input planes are used to launch the frequency corresponding to wavelength 1.57μm, 1.55μm and 1.53μm simultaneously at input port. Radius of defect rod corresponding to output port 1, 2 and 3 are taken r1=0.189μm, r2=0.184μm, r3=0.1815μm respectively as shown in Figure 15. Demultiplexer is designed to receive wavelength 1.57μm, 1.55μm, 1.53μm at output port 1, 2 and 3 respectively. Radius of the border rods of single line defect waveguide is decreased in consecutive section to prohibit transmission of wavelength 1.57μm from the first section and 1.55μm from the second section. Radius of border rods at first section and second section are taken 0.202 and 0.195μm respectively. 
Observation points (green dot) are placed at output port 1, 2 and 3 to observe the power received from the source. Power measured at output ports 1, 2 and 3 after simulation by OptiFDTD (Finite differential time domain) is normalized with respect to the maximum power received at the corresponding port and is shown in Figure 16. 
CONCLUSIONS

In conclusion, we have proposed different novel components for wavelength demultiplexing based on photonic crystal waveguides. First wavelength demultiplexer was designed for splitting wavelength 1.625, 1.55 and 1.31μm by using the filtering nature of photonic crystals. Second wavelength demultiplexer was proposed for splitting 1.57, 1.55 and 1.53μm. With the variation in defect size these demultiplexers provide the way to reach the goal of 20nm channel spacing for CWDM systems defined by ITUT Recommendation G.694.2. In addition, high value of quality factor and low amount of crosstalk has been achieved simultaneously. 
These results can be of interest in design of dense wavelength division multiplexers with very narrow channel spacing, adddrop filters, optical switches and many more applications to give rise of high speed communication. 
Tables at a glance





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