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Abstract
The present study proposes a new type of wavelength division multiplexing as an optical demultiplexer with four, eight, and sixteen output channels. The proposed structure consists of a two-dimensional square array of dielectric rods. The wavelength selection is performed by two defect rods and one defect with a rectangular cross-section. The values of the parameters related to each defect are selected so that the resonant wavelength associated with each output channel is located in the third telecommunication window (1.55μm). It is observed that the proposed demultiplexers demonstrate a high transmission efficiency and low crosstalk. As it is wanted to see just one sharp wavelength at the output of each channel, the quality factor needs to be maximized. Based on the results, the maximum values for demultiplexers with four, eight, and sixteen channels are 19863.89, 14264.57, and 32304.46, respectively. Due to the device’s small size, it can be used for wavelength division multiplexing (WDM) systems in photonic integrated circuits.
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1. Introduction
The investigation of photonic crystals has grown significantly in the last three decades, and it has become a vast field of research for scientists in various sciences, including telecommunications, computers, medicine, chemistry, pharmacy, and astronomy, from an anonymous technology. Photonic crystals are structured materials or artificial media on a micrometer scale periodically composed of dielectric or metal-dielectric nanostructures. The periodicity of these materials in one, two, and three dimensions respectively form one-dimensional, two-dimensional, and three-dimensional photonic crystals. The periodic feature of these crystals leads to the creation of specific frequency or wavelength ranges called photonic band gaps (PBGs), and photons whose frequency is in this forbidden frequency range cannot enter the structure [1]. It can be said that the photonic bandgap acts as an optical insulator in a specific frequency range. Hence, this photonic bandgap feature makes it possible to guide light waves by creating defects in the structure. It is the result of the formation of an eigenmode of the defect within the bandgap. Photonic crystals can control photons, or in other words, electromagnetic waves, and this control of photons is realized by creating defects in the structure [2]. In photonic crystals, point, linear, and even surface defects can be created by varying structural parameters such as refractive index, rods radius, and lattice constant. Depending on the type of defect, the localized modes within the photonic crystal structure can be controlled. Point and linear defects have the most applications to design and fabricate optical devices based on photonic crystal media in photonic integrated circuits.
Better light confinement, efficient photonic bandgap calculation, spontaneous emission control, relatively simple fabrication, and even easy integration are the features that make the two-dimensional type of photonic crystals unique in the design of optical devices. Although the triangular type has a broader bandgap of the two lattice structures, square lattice-based optical devices offer effective light confinement, easy control of propagated modes, and easy fabrication due to their simple geometry [1]. Today, photonic crystals are among the most important candidates for designing and fabricating optical devices such as filters [3,4], (de)multiplexers [5–8], logic gates [9,10], optical switches [11], encoders and decoders [12], analog-to-digital and digital-to-analog converters [13,14], and power splitters [15].
Wavelength division multiplexing has been adopted as a new optical technology for optimal fiber-optic capacity to increase communication channels. It is done by combining specific frequencies, transferring them from one end of the fiber to the other, and finally, optical demultiplexers are used to separate these frequencies [16]. The demultiplexer is an essential element in wavelength division multiplexing systems used to select a channel with a specific wavelength. The multiplexing operation makes it possible to take a frequency or wavelength from one waveguide and transmit it to another waveguide [17]. In optical demultiplexers, various functional parameters are essential to exploit the total capacity of optical fibers. These design parameters are low channel spacing, low crosstalk, high-quality factor, and high transmission efficiency. Since the advent of photonic crystals, various topologies have been proposed to design demultiplexers, such as linear defect waveguides, coupled-cavity waveguides, direct coupling, and ring resonators. Mehdizadeh and Soroush [18] recently designed a four-channel demultiplexer based on photonic crystals with a hexagonal lattice of air pores inside the dielectric material, which resonant regions are defective resonant cavities. They have shown that the desired wavelengths can be separated by selecting the appropriate values for the radius of defects within the resonant cavity. The maximum transmission efficiency, crosstalk, and average quality factor are $97\%$, $-27.6$ dB, and $5039$, respectively. The total footprint of this proposed structure is $201.6\mu m^2$. Balaji et al. [19] have proposed an eight-channel multiplexer that includes a bus waveguide, a drop waveguide, and a Parallelepiped Resonant Cavity (PRC) based on a triangular lattice. The PRC consists of a parallel resonator with a nano-ring cavity. In their work, the maximum transmission efficiency is $100\%$, the maximum quality factor is $7789.5$, and the worst crosstalk value is $-20$ dB. The size of this demultiplexer is about $434.56\mu m^2$. Mohammadi and Seifouri [20] have proposed a four-channel demultiplexer based on photonic crystals in a square lattice with silicon rods in the air bed using three cascaded ring resonators. They demonstrated that the desired wavelengths could effectively be separated by selecting appropriate values for the radius of the inner rods, scattering rods, coupling rods, and the coupling length between the bus and the drop waveguides. The mean quality factor and the transmission efficiency of their device are computed to be $7358.5$ and $99.25\%$, respectively. The proposed demultiplexer has a minimum and maximum crosstalk values of about $-46.68$ dB and $-9.79$ dB, respectively. In addition, the footprint of the device is $294.4 \mu m^2$. In this work, we have tried to suggest a structure that improves the value of some functional parameters, inspired by the past work of other authors [3,18,21–23]. Our proposed structure is far from complex in terms of compatibility with construction and efficiency. The resonant region has almost simple geometry. The wavelength selection is only performed by changing a physical parameter of the defect such as the radius of the rod or the rectangular cross-section.
The rest of this paper is organized as follows: In Section 2, the band structure is calculated, and the design of the proposed demultiplexer structures is undertaken. Section 3 deals with the simulation results of demultiplexers and concludes with Section 4 of the paper.
2. Proposed structures
2.1 Calculate the band structure
The photonic crystal-based structure considered in this paper for the proposed demultiplexers consists of a square array of dielectric cylindrical rods with a circular cross-section located on the x-z plane. Silicon dielectric rods have a refractive index of $n_{rod}=3.43$, situated in the air bed with a refractive index of $n_{air}=1$. The radius of the rods is $R_{rod}=0.111\mu m$, and the distance between two adjacent rods or lattice constant for this structure is equal to $a=0.555\mu m$.
The first step before designing demultiplexers is to determine the band structure, which determines the forbidden frequency regions. The band diagram is plotted by the plane wave expansion (PWE) method [24]. PWE is a method used to calculate the propagation of electromagnetic modes in a periodic or non-periodic structure. These modes follow Maxwell’s equations in photonic crystal analysis:
Fig. 1. The band structure in the TE and TM polarization for a two-dimensional photonic crystal where the lattice constant is $0.555\mu m$ and the radius of dielectric rods is $0.111\mu m$.
2.2 Design of demultiplexers
Depending on the number of channels a demultiplexer has, it is necessary to design a region called the resonance or wavelength selection and place it in the path of the waves emitted from the input to the output of each channel to select different wavelengths from that of light. This study proposes three demultiplexer structures, and the same resonant region is considered for all three proposed demultiplexers. Figure 2 shows this resonant region. This region consists of two parts, each of which plays a role. The first part consists of a defect with a rectangular cross-section of width $W_{i}$ and length $L_{i}$, whose counter $i$ is the number of each channel, and its refractive index $n_{rec}=n_{rod}$ is the same as the refractive index of the rods of the fundamental structure. The second part consists of two defects with a circular cross-section with a radius $R_{d}=1.5R_{rod}$ and a refractive index of $n_{d}=2.8924$ (corresponding to AlAs at $1.55 \mu m$). The thickness of the photonic crystal is $5 \mu m$.
Fig. 2. The resonant region corresponding to proposed demultiplexers with cylindrical defects of radius $R_d$ and refractive index $n_d$ and rectangular defects of width $W_i$ and length $L_i$ and refractive index $n_{rec}$.
Now, in the continuation of this discussion, the design of demultiplexers will be discussed. The first design is a demultiplexer with four output channels, each output filtering a specific wavelength. This design is shown in Fig. 3. The proposed structure consists of a two-dimensional square lattice with $31$ rods along the x-axis and $13$ rods along the z-axis. By removing $28$ rods in the horizontal direction, a waveguide is formed to enter the light waves into the structure, and the removal of $5$ rods in four different places of the structure in the vertical direction forms the output waveguides. The resonant region shown in Fig. 2 is located in the path between the input waveguide and the output waveguides. The length $L_{i}$ of a rectangular defect is $1.5\mu m$ for all four channels, but the value of the width $W_{i}$ for channels one to four is $W_{1}=0.432\mu m$, $W_{2}=0.434\mu m$, $W_{3}=0.436\mu m$, and $W_{4}=0.438\mu m$. The overall size of this structure is $113.5\mu m^2$. The second design under consideration is a demultiplexer with eight output channels, shown in Fig. 4. This structure consists of a two-dimensional square lattice with $31$ and $23$ dielectric rods arranged in the direction of $x$ and $z$, respectively. The proposed demultiplexer with waveguides and related resonant regions is shown in Fig. 4. There is a rectangular defect in the resonant regions whose width $W_{i}$ is equal to $W_{1}=0.396\mu m$, $W_{2}=0.4\mu m$, $W_{3}=0.394\mu m$, $W_{4}=0.406\mu m$, $W_{5}=0.392\mu m$, $W_{6}=0.404\mu m$, $W_{7}=0.398\mu m$, and $W_{8}=0.402\mu m$ for channels one to eight, respectively. For the length $L_{i}$ of this defect, a value equal to $1.5\mu m$ is selected. The overall size of this structure is $206.5\mu m^2$. As shown in Fig. 5, the final design is a sixteen-channel demultiplexer created by a two-dimensional square lattice with $55$ and $23$ rods on the x-z plane. The design process and type of resonator are the same as the previous two designs. The length $L_{i}$ of the rectangular defect for all channels is $1.5\mu m$, and the width $W_{i}$ for this defect for channels one to sixteen is $W_{1}=0.424\mu m$, $W_{2}=0.41\mu m$, $W_{3}=0.422\mu m$, $W_{4}=0.416\mu m$, $W_{5}=0.42\mu m$, $W_{6}=0.414\mu m$, $W_{7}=0.418\mu m$, $W_{8}=0.412\mu m$, $W_{9}=0.392\mu m$, $W_{10}=0.4\mu m$, $W_{11}=0.394\mu m$, $W_{12}=0.406\mu m$, $W_{13}=0.396\mu m$, $W_{14}=0.404\mu m$, $W_{15}=0.398\mu m$, and $W_{16}=0.402\mu m$, respectively. The overall size of the structure is $370.6\mu m^2$.
Fig. 3. Four-channel demultiplexer with four resonant regions and four output waveguides based on a two-dimensional structure consisting of rods with a circular cross-section with radius $R_{rod}$ and refractive index $n_{rod}$ in a square lattice with lattice constant $a$.
Fig. 4. Eight-channel demultiplexer with eight resonant regions and eight output waveguides based on a two-dimensional structure consisting of rods with a circular cross-section with radius $R_{rod}$ and refractive index $n_{rod}$ in a square lattice with lattice constant $a$.
Fig. 5. Sixteen-channel demultiplexer with sixteen resonant regions and sixteen output waveguides based on a two-dimensional structure consisting of rods with a circular cross-section with radius $R_{rod}$ and refractive index $n_{rod}$ in a square lattice with lattice constant$a$.
3. Simulation results and discussion
After completing the design process of demultiplexers based on photonic crystals, the behavior of electromagnetic light waves inside them is investigated. For this purpose, the finite difference time domain (FDTD) method [25], which is one of the most powerful numerical methods, is used to solve the Maxwell equations governing the light propagation within the proposed demultiplexers. The perfectly matched layers as boundary conditions are assumed for all boundaries. The very fine mesh is also selected to obtain accurate results. By applying a light source with TM polarization in the wavelength range of band C to the input of the structure, each of the monitors calculates its power at the corresponding output. For a linear, isotropic, non-magnetic, and free-charge medium, the Maxwell time-dependent equations are written as follows:
where $\mu (\vec {r})$ and $\varepsilon (\vec {r})$ are the magnetic permeability and electrical permittivity of the medium, and are a function of position [26]. In the two-dimensional case, the fields can be divided into two transverse polarized modes, TE and TM. In this study, TM polarization is used. These equations can be discretized in both time and space using the Yee algorithm [27]. The mesh size of the FDTD method is selected as $\Delta x=\Delta z=\frac {a}{16}=0.0346875\mu m$. The Courant condition, defined as $\Delta t\le 1/c\sqrt {{\frac {1}{\Delta x^2}}+\frac {1}{\Delta z^2}}$, gives the relationship between the time step and the space step. This condition is the numerical stability of this algorithm and is applied to prevent divergence of responses. Hence the time step is $\Delta t\le 0.0817592215747 fs$. The stop time for running this simulation in order to achieve acceptable results is $100,000$.The FDTD method is used to analyze the transmission of the electromagnetic waves of the demultiplexer by taking a Fast Fourier Transform (FFT) from the field distribution to extract the transmission spectrum. At the beginning of the input waveguides of all three demultiplexers and to excite them at the polarization of the TM, a Gaussian pulse source is launched at a central wavelength of $1.55\mu m$. At the end of all output waveguides are placed time monitors that receive the density of the transmitted spectral power. All transmitted spectral power densities are normalized to the spectral power density of the incident light wave. To ensure that the wave is not reflected, perfectly matched layers (PMLs) are placed around the entire structure as an absorbing boundary condition that artificially translocates the boundary of the numerical solution region to infinity. The wave leaves the boundary without reflection [28]. The thickness of this PML and its reflection coefficient are considered equal to $0.5 \mu m$ and $10^{-8}$, respectively.
The performance of an optical demultiplexer depends on the ability of the channels to separate from each other. The functional parameters must be considered in designing a demultiplexer, including central wavelengths, transmission efficiency, quality factor, bandwidth, channel spacing, and crosstalk. First, the optical properties of the four-channel case are investigated. Figure 6 shows the normalized transmission spectrum of a demultiplexer. This demultiplexer can separate four channels with central wavelengths of $\lambda _{1}=1.5692475\mu m$, $\lambda _{2}=1.572931\mu m$, $\lambda _{3}=1.576607\mu m$ and $\lambda _{4}=1.580412\mu m$. Only at these wavelengths, a coupling can occur between the input waveguide mode and the resonant resonator modes. The transmission of most of the power of light waves from the input waveguide to the output waveguide is done using a resonant tunneling process. One of the most critical parameters in designing a demultiplexer that is evaluated and given special attention is the quality factor. The quality factor, which is defined as the ratio of center wavelength to bandwidth (Full Width at Half Maximum), can be written as follows:
Low coupling losses and a very small FWHM are required to obtain a high-quality factor. A long coupling distance also improves light confinement and thus increases the quality factor. The maximum quality factor for this proposed four-channel demultiplexer is achieved at the wavelength of $\lambda _{1}$, which is 19836.89. The bandwidth and transmission efficiency that lead to this quality factor are $0.000079\mu m$ and $71.5\%$, respectively. More details about this structure are listed in Table 1. Another vital parameter in the design of optical demultiplexers is crosstalk. Crosstalk is the unwanted energy of a channel in adjacent channels, measured on a dB scale [29]. The lower the numerical value of this parameter, the better the resolution for the output channels, or in other words, the less interference between the channels. The crosstalk between the channels is calculated using the equation
which $P_{out}$ and $P_{in}$ show the output power and input power of the demultiplexer, respectively. According to Fig. 7 and Table 2, the crosstalk level for the proposed structure varies from $-13.62$ to $-43.64$. In this proposed design, the wavelength selection mechanism is based on selecting different sizes of rectangular defect widths in the resonance region.Fig. 6. Demonstration of the output spectrum on a linear scale related to the proposed four-channel demultiplexer at central wavelengths of $\lambda _{1}$ to $\lambda _{4}$ for rectangular defects with widths of $W_{1}$ to $W_{4}$, similar lengths of 1.5 m and refractive index $n_{rec}$, and cylindrical defects with radius $R_d$ and refractive index $n_d$
Fig. 7. Demonstration of the output spectrum on a dB scale related to the proposed four-channel demultiplexer at central wavelengths of $\lambda _{1}$ to $\lambda _{4}$ for rectangular defects with widths of $W_{1}$ to $W_{4}$, similar lengths of 1.5 m and refractive index $n_{rec}$, and cylindrical defects with radius $R_d$ and refractive index $n_d$.
Table 1. Simulation results of the central wavelength, bandwidth, quality factor, and transmission efficiency parameters related to the proposed four-channel demultiplexer.
Table 2. Absolute values of crosstalk between different channels for a proposed four-channel demultiplexer structure on the dB scale.
For a better understanding, Fig. 8 shows the optical field distribution at the central wavelengths $\lambda _{1}$, $\lambda _{2}$, $\lambda _{3}$, and $\lambda _{4}$. When the light wave enters the horizontal waveguide, it faces the photonic crystal lattice at the sides. The frequency of this light wave is in the photonic band gap, so the only permissible path of its movement will be this waveguide, and the wave propagates in this waveguide. Light waves drop from the input waveguide to the output waveguide based on the coupling of the guided modes of the waveguide and the resonator modes. When a part of the resonator is in the adjacency of the input waveguide, it traps photons at a specific wavelength and emits them to the output waveguide, which leads to the filtering operation. In other words, some electromagnetic modes (which central frequency is the same as the frequency of the resonant region) are coupled to the resonator during wave propagation. Over time, coupling occurs between the resonant region and the vertical waveguide, and certain modes are directed toward the output.
In the following, the results of the demultiplexer with eight output channels are analyzed. The normalized transmission spectrum of this proposed design is shown in Fig. 9, and the details of the results are given in Table 3. The maximum quality factor value at the central wavelength of $1.5243835 \mu m$ of Ch-4 is equal to $14246.57$, which has a transmission efficiency of $93.32\%$ and a bandwidth of $0.000107 \mu m$. The diagram for the crosstalk analysis is shown in Fig. 10, and its numerical values are given in Table 4. The level of crosstalk for the proposed structure of eight channels varies from $-15.34$ to $-56.22$.
Fig. 9. Demonstration of the output spectrum on a linear scale related to the proposed eight-channel demultiplexer at central wavelengths of $\lambda _{1}$ to $\lambda _{8}$ for rectangular defects with widths of $W_{1}$ to $W_{8}$, similar lengths of 1.5 m and refractive index $n_{rec}$, and cylindrical defects with radius $R_d$ and refractive index $n_d$.
Fig. 10. Demonstration of the output spectrum on a dB scale related to the proposed eight-channel demultiplexer at central wavelengths of $\lambda _{1}$ to $\lambda _{8}$ for rectangular defects with widths of $W_{1}$ to $W_{8}$, similar lengths of 1.5 m and refractive index $n_{rec}$, and cylindrical defects with radius $R_d$ and refractive index $n_d$.
Table 3. Simulation results of the central wavelength, bandwidth, quality factor, and transmission efficiency parameters related to the proposed eight-channel demultiplexer.
Table 4. Absolute values of crosstalk between different channels for a proposed eight-channel demultiplexer structure on the dB scale.
The latest analysis of the results is related to the 16-channel demultiplexer. The normalized transmission spectrum of this structure is shown in Fig. 11, and the resulting numerical values are listed in Table 5. The wavelength of $1.550614\mu m$ related to channel Ch-3 with a bandwidth of $0.000048\mu m$ and transmission efficiency of $60.64$ has a maximum quality coefficient of $32304.46$. The diagram for crosstalk is shown in Fig. 12, and its numerical values are given in Table 6. The level of crosstalk for this proposed sixteen-channel structure varies from $-15.12$ to $-89.1$. The physical structure proposed in this paper has been selected so that each of the demultiplexer channels has a narrow bandwidth and the distance between the two adjacent channels is as large as possible to prevent cross-talk and inter-channel interference. Table 7 compares the proposed structures with previous structures in which demultiplexers with different outputs are given.
Fig. 11. Demonstration of the output spectrum on a linear scale related to the proposed four-channel demultiplexer at central wavelengths of $\lambda _{1}$ to $\lambda _{16}$ for rectangular defects with widths of $W_{1}$ to $W_{16}$, similar lengths of 1.5 m and refractive index $n_{rec}$, and cylindrical defects with radius $R_d$ and refractive index $n_d$.
Fig. 12. Demonstration of the output spectrum on a dB scale related to the proposed four-channel demultiplexer at central wavelengths of $\lambda _{1}$ to $\lambda _{16}$ for rectangular defects with widths of $W_{1}$ to $W_{16}$, similar lengths of 1.5 m and refractive index $n_{rec}$, and cylindrical defects with radius $R_d$ and refractive index $n_d$.
Table 5. Simulation results of the central wavelength, bandwidth, quality factor, and transmission efficiency parameters related to the proposed sixteen-channel multiplexer.
Table 6. Absolute values of crosstalk between different channels for a proposed sixteen-channel demultiplexer structure on the dB scale.
Table 7. Comparison between proposed structures and previous work.
4. Conclusion
In this paper, demultiplexers with four, eight, and sixteen output channels based on two-dimensional photonic crystals are designed, consisting of a square lattice with dielectric rods. The quality factor, bandwidth, and transmission efficiency at the central wavelength of $1.5692475\mu m$ are obtained for four-channel demultiplexer as $19863.89$, $0.000079\mu m$, and $71.5\%$, respectively. For an eight-channel demultiplexer, at the central wavelength of $1.5243835\mu m$ the values are obtained equal to $14246.57$, $0.000107\mu m$, and $93.32\%$, respectively. Finally, for sixteen-channel demultiplexer, at the central wavelength of $1.550614\mu m$, the values of $32304.46$, $0.000048\mu m$, and $60.64\%$ are obtained, respectively. According to investigations, the filtering operation of channels is done by wavelength shift. This wavelength shift is caused by a variation in the width of the rectangular defect in the resonance region. Due to the dimensions of the structure, these demultiplexers can be used as a primary element in wavelength division multiplexing systems in photonic integrated circuits.
Funding
Deputy for Research and Technology of Shahid Beheshti University (600/5290).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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