An Ultrashort Wavelength Multi/Demultiplexer via Rectangular Liquid-Infiltrated Dual-Core Polymer Optical Fiber

We propose a rectangular liquid-infiltrated dual-core polymer optical fiber (POF) for short-range communication systems by the beam propagation method (BPM). The POF multi/demultiplexer (MUX/DEMUX) at the wavelengths of 0.52/0.65-μm, 0.57/0.65-μm, and 0.52/0.57-μm are devised. The simulation results demonstrate that the ultrashort length of three ultrashort POF couplers are 183.6 μm, 288 μm, and 799.5 μm. Compared with the conventional optical fiber couplers, these results could have significant applications in the miniaturization of optical devices for visible light communication.


Introduction
In recent years, with the rapid expansion of optical communication technology, application demands of users for optical transmission networks and systems are growing exponentially. In order to relieve the huge pressure of bandwidth for an optical communication network and system, all kinds of measures have been proposed to increase the capacity of optical networks [1][2][3][4][5][6][7][8]. Wavelength division multiplexing (WDM) is a key technology in advanced optical communication networks. The use of WDM technology not only significantly increases the capacity of the existing optical communication networks without increasing the number of fibers, but also possesses advantages in flexible services, network provision, and network management [9]. The wavelength division multi/demultiplexer (MUX/DEMUX) is an important passive component in the WDM system, which splits/combines lights with different wavelengths into different outputs [10]. All sorts of wavelength MUX/DEMUXs using silica-PLC [2], InGaAs/InP avalanched photodiodes [11], Chirped fiber Bragg grating [12], 2D/3D photonic crystal [13,14], polymer photonic structure [15], and photonic crystal fiber [4] have been demonstrated. At present, the wavelengths of most wavelength division MUX/DEMUXs are located in near-infrared wavelengths [2,4,[11][12][13][14][15]. There are very few wavelength division MUX/DEMUXs in the visible wavelengths. Therefore, the design and application of such optical devices in the visible wavelengths are of great significance to develop the visible short-distance communications systems.
Photonic crystal fibers (PCFs) are a special class of optical fibers characterized by a periodical arrangement of microcapillaries that form the fiber's cladding around a solid or hollow defect core [16][17][18]. Since then, various fiber types such as honeycomb PCF [19], triangular PCF [20], rectangular PCF [21][22][23][24], D-shape PCF [25], side-polished PCF [26], and metal-coated PCF [27] Figure 1) and the cladding region are organized in rectangular formation across the PMMA backdrop. In contrast to a glass optical fiber, the PMMA is a popular material for optical fibers due to the low polymerization temperature, low cost, high transparency, and high mechanical flexibility [57,58]. With the cladding, the green air holes have a diameter of d. In the core region, d 1 is the diameter of the blue air hole filled with benzene (n = 1.366). Benzene is a highly toxic carcinogen. How to quickly and easily detect benzene in the environment and food is very important [69]. Therefore, the sensing characteristics of polymer devices based on filled benzene will be considered in future work. The distance of hole to hole can be expressed as period Λ, the air-filling ratio is d/ Λ. The refractive index of background material is set as 1.49. work. The distance of hole to hole can be expressed as period Λ, the air-filling ratio is d/Λ. The refractive index of background material is set as 1.49. The vector wave equation, which is the basis of BPM [66][67][68], can be expressed by  The vector wave equation, which is the basis of BPM [66][67][68], can be expressed by

Design Principle and Theoretical Modeling
where k ≡ ω √ µε. These two equations are known as the Helmholtz equations. The electric field E(x, y, z) can be separated into two parts: the fast change term of exp(-ikn 0 z) and the envelope term of φ(x, y, z) of slow change in the axial direction, n 0 is a refractive index in the cladding. Then, E(x, y, z) is stated as Substituting Equation (3) in Equation (1) results in where n is a refractive index in the fiber core. The relative refractive index is an important parameter to describe the constrained optical field capability of fiber which is expressed as when ∆ < 0.01, ∆ n−n 0 n 0 is called the weakly guiding condition. Assuming the weakly guiding condition, we can approximate n 2 − n 2 0 2 n 0 (n − n 0 ). Then Equation (4) can be rewritten as A similar expression can be written for H. We find that n n 0 if the fields vary in the transverse direction to propagation. Light propagation in various kinds of waveguides can be analyzed by the above method.
The birefringence is an important index to evaluate the performance of polarization maintaining, which is expressed as [51][52][53]70] B = n x − n y where n x and n y represent the effective refractive index of x-polarization and y-polarization, respectively. There are four modes of dual-core PCF on the basis of the principle of coupling mode, namely, even-mode of x-polarization, odd-mode of x-polarization, even-mode of y-polarization, odd-mode of y-polarization. The coupling length can be defined as [4] where n x,y even , n x,y odd denote the effective indexes of even-mode of x-polarization, odd-mode of x-polarization, even-mode of y-polarization, odd-mode of y-polarization, respectively.
When L λ1 and L λ2 satisfy the following Equation (9) or (10), a polymer coupler can separate two wavelengths λ 1 and λ 2 transmitted in a core [4].
L λ 1 : L λ 2 = even : odd (9) or L λ 1 : L λ 2 = odd : even (10) Assuming that the incident power is emitted to a certain core, the output power of xor y-polarized light in the core can be expressed [71].
where the transmission distance is denoted by z.
The confinement loss of PCF is calculated from the imaginary part of the effective refractive index, using the following equation [36],

Simulated Results and Analysis
First, we analyze the coupling lengths as a function of period Λ for different air-filling ratio d/Λ, where d 1 = 0.4 µm as shown in Figure 2. It is observed that the coupling length is increased when period Λ is increased for a constant air-filling ratio d/Λ. Moreover, the coupling length of y-polarization is higher than the coupling length of x-polarization. Since the x-axis is parallel to the core A and core B, the coupling length of the y-polarization is smaller than that of the x-polarization. Furthermore, we can clearly see that coupling length is decreased when air-filling ratio d/Λ is increased for the same value of period Λ. This is because the restriction of the outer cladding to the light wave is enhanced as the air-filling ratio increases. For the coupler with excellent performance, not only the strong coupling effect between core A and core B but also the good extinction ratio should be considered [4]. Based on the above considerations, we decided to use the y-polarization for the polymer optical fiber couplers.
light wave is enhanced as the air-filling ratio increases. For the coupler with excellent performance, not only the strong coupling effect between core A and core B but also the good extinction ratio should be considered [4]. Based on the above considerations, we decided to use the y-polarization for the polymer optical fiber couplers. High birefringence can not only maintain the linear polarization state in the fiber but also increase the difference in coupling length of x-polarized mode and y-polarized mode of PCF. Figure  3 shows the birefringence as a function of period Λ. We found that the birefringence of PCF increases with the increase of air-filling ratio d/Λ, which results in stronger coupling strength between the two cores for a shorter coupling length of the polymer optical fiber. Based on the high birefringence of the fiber, we chose the air-filling ratio of 0.9. Additionally, when d1 = 0.4 μm, d/Λ = 0.9, the coupling length of y-polarized mode is shown as a function of period Λ in Figure 4, in which it is observed that the coupling length is increased if period Λ is increased. As the period increases, the coupling between the cores becomes weaker. Meanwhile, the coupling length of y-polarization decreases with increasing operating wavelength. High birefringence can not only maintain the linear polarization state in the fiber but also increase the difference in coupling length of x-polarized mode and y-polarized mode of PCF. Figure 3 shows the birefringence as a function of period Λ. We found that the birefringence of PCF increases with the increase of air-filling ratio d/Λ, which results in stronger coupling strength between the two cores for a shorter coupling length of the polymer optical fiber. Based on the high birefringence of the fiber, we chose the air-filling ratio of 0.9.
light wave is enhanced as the air-filling ratio increases. For the coupler with excellent performance, not only the strong coupling effect between core A and core B but also the good extinction ratio should be considered [4]. Based on the above considerations, we decided to use the y-polarization for the polymer optical fiber couplers. High birefringence can not only maintain the linear polarization state in the fiber but also increase the difference in coupling length of x-polarized mode and y-polarized mode of PCF. Figure  3 shows the birefringence as a function of period Λ. We found that the birefringence of PCF increases with the increase of air-filling ratio d/Λ, which results in stronger coupling strength between the two cores for a shorter coupling length of the polymer optical fiber. Based on the high birefringence of the fiber, we chose the air-filling ratio of 0.9. Additionally, when d1 = 0.4 μm, d/Λ = 0.9, the coupling length of y-polarized mode is shown as a function of period Λ in Figure 4, in which it is observed that the coupling length is increased if period Λ is increased. As the period increases, the coupling between the cores becomes weaker. Meanwhile, the coupling length of y-polarization decreases with increasing operating wavelength. Additionally, when d 1 = 0.4 µm, d/ Λ = 0.9, the coupling length of y-polarized mode is shown as a function of period Λ in Figure 4, in which it is observed that the coupling length is increased if period Λ is increased. As the period increases, the coupling between the cores becomes weaker. Meanwhile, the coupling length of y-polarization decreases with increasing operating wavelength.         Table 1 shows the optimal parameters of three different wavelength couplers. We can clearly see that the length of coupler 1 is shorter than coupler 2 and 3. This phenomenon is probably related to the difference between the coupling length at 0.65 μm and 0.52 μm. Meanwhile, the length of couplers is much shorter than optical couplers in the References [4,72,73]. The main reasons for the ultrashort coupler are related to the design of fiber structure (rectangular lattice structure), the selection of background materials (PMMA), and the filling of functional materials (benzene).   Table 1 shows the optimal parameters of three different wavelength couplers. We can clearly see that the length of coupler 1 is shorter than coupler 2 and 3. This phenomenon is probably related to the difference between the coupling length at 0.65 µm and 0.52 µm. Meanwhile, the length of couplers is much shorter than optical couplers in the References [4,72,73]. The main reasons for the ultrashort coupler are related to the design of fiber structure (rectangular lattice structure), the selection of background materials (PMMA), and the filling of functional materials (benzene). In order to analyze the influence of liquid filling on the transmission performance of couplers, we study the relationship between liquid filling and birefringence and confinement loss of couplers. Figure 6 shows the relationship between the birefringence and filling material for d 1 = 0.48 µm, Λ = 0.9 µm, and d/ Λ = 0.9. It is observed that birefringence of coupler filled with liquid is higher than birefringence of coupler without liquid. Figure 7 shows the variation of confinement loss with filling material for d 1 = 0.48 µm, Λ = 0.9 µm, and d/ Λ = 0.9. It can be seen that the confinement loss of the coupler without liquid is higher the confinement loss of the coupler with filled liquid. Therefore, the coupler with filled liquid has lower confinement loss and higher birefringence than the coupler without liquid. In order to analyze the influence of liquid filling on the transmission performance of couplers, we study the relationship between liquid filling and birefringence and confinement loss of couplers. Figure 6 shows the relationship between the birefringence and filling material for d1 = 0.48 μm, Λ = 0.9 μm, and d/Λ = 0.9. It is observed that birefringence of coupler filled with liquid is higher than birefringence of coupler without liquid. Figure 7 shows the variation of confinement loss with filling material for d1 = 0.48 μm, Λ = 0.9 μm, and d/Λ = 0.9. It can be seen that the confinement loss of the coupler without liquid is higher the confinement loss of the coupler with filled liquid. Therefore, the coupler with filled liquid has lower confinement loss and higher birefringence than the coupler without liquid.   We also demonstrate that three couplers can separate λ1 and λ2 according to the simulation results by BPM. The fundamental modes of y-direction at λ1 and λ2 are imported into the core A or core B in Figure 1. Figure 8 shows the propagation distance dependence of the normalized power. We observed that the separation of two wavelengths of λ1 and λ2 for couplers 1 to 3 are achieved at the distances of 183.6 μm, 288 μm, and 799.5 μm, respectively shown as the blue line in Figure 8. Obviously, the three polymer optical fiber couplers can operate as wavelength MUX/DEMUX at the wavelength of 0.52/0.65-μm, 0.57/0.65-μm, and 0.52/0.57-μm, respectively. In order to analyze the influence of liquid filling on the transmission performance of couplers, we study the relationship between liquid filling and birefringence and confinement loss of couplers. Figure 6 shows the relationship between the birefringence and filling material for d1 = 0.48 μm, Λ = 0.9 μm, and d/Λ = 0.9. It is observed that birefringence of coupler filled with liquid is higher than birefringence of coupler without liquid. Figure 7 shows the variation of confinement loss with filling material for d1 = 0.48 μm, Λ = 0.9 μm, and d/Λ = 0.9. It can be seen that the confinement loss of the coupler without liquid is higher the confinement loss of the coupler with filled liquid. Therefore, the coupler with filled liquid has lower confinement loss and higher birefringence than the coupler without liquid.   We also demonstrate that three couplers can separate λ1 and λ2 according to the simulation results by BPM. The fundamental modes of y-direction at λ1 and λ2 are imported into the core A or core B in Figure 1. Figure 8 shows the propagation distance dependence of the normalized power. We observed that the separation of two wavelengths of λ1 and λ2 for couplers 1 to 3 are achieved at the distances of 183.6 μm, 288 μm, and 799.5 μm, respectively shown as the blue line in Figure 8. Obviously, the three polymer optical fiber couplers can operate as wavelength MUX/DEMUX at the wavelength of 0.52/0.65-μm, 0.57/0.65-μm, and 0.52/0.57-μm, respectively. We also demonstrate that three couplers can separate λ 1 and λ 2 according to the simulation results by BPM. The fundamental modes of y-direction at λ 1 and λ 2 are imported into the core A or core B in Figure 1. Figure 8 shows the propagation distance dependence of the normalized power. We observed that the separation of two wavelengths of λ 1 and λ 2 for couplers 1 to 3 are achieved at the distances of 183.6 µm, 288 µm, and 799.5 µm, respectively shown as the blue line in Figure 8 Figure 9 shows odd-and even-mode of x-polarization and y-polarization for the coupler, when d1 = 0.48 μm, Λ = 0.9 μm, and d/Λ = 0.9. It shows the mode-field distribution of the odd-and evenmode in two vertical directions. Moreover, the difference propagation constants and phase difference change of odd-and even-mode in transmission results in a power transfer between two cores [74].  Figure 9 shows odd-and even-mode of x-polarization and y-polarization for the coupler, when d 1 = 0.48 µm, Λ = 0.9 µm, and d/ Λ = 0.9. It shows the mode-field distribution of the odd-and even-mode in two vertical directions. Moreover, the difference propagation constants and phase difference change of odd-and even-mode in transmission results in a power transfer between two cores [74].

Conclusions
Three ultrashort couplers based on rectangular liquid-infiltrated POF have been demonstrated by BPM. The POF couplers for 0.52/0.65-μm, 0.57/0.65-μm, and 0.52/0.57-μm wavelength multi/demultiplexer (MUX/DEMUX) are designed by manipulating structural parameters. The numerical results demonstrate that the lengths of three ultrashort POF couplers are 183.6 μm, 288 μm, and 799.5 μm for the wavelength multiplexing. Compared with the conventional optical fiber couplers, the POF couplers in the visible wavelengths have an ultrashort length, which is important for the application in the miniaturization of optical devices in short-range telecommunication networks [75,76].

Conflicts of Interest:
The authors declare no conflicts of interest.

Conclusions
Three ultrashort couplers based on rectangular liquid-infiltrated POF have been demonstrated by BPM. The POF couplers for 0.52/0.65-µm, 0.57/0.65-µm, and 0.52/0.57-µm wavelength multi/ demultiplexer (MUX/DEMUX) are designed by manipulating structural parameters. The numerical results demonstrate that the lengths of three ultrashort POF couplers are 183.6 µm, 288 µm, and 799.5 µm for the wavelength multiplexing. Compared with the conventional optical fiber couplers, the POF couplers in the visible wavelengths have an ultrashort length, which is important for the application in the miniaturization of optical devices in short-range telecommunication networks [75,76].

Conflicts of Interest:
The authors declare no conflicts of interest.