Mid-Infrared Polarization Beam Splitter based on Square/circular Hybrid Air Holes with Wide Bandwidth and Ultrashort Length

: A novel mid-infrared polarization beam splitter (PBS) based on GaS is proposed. The high birefringence is achieved by using the cladding structure of alternating arrangement of square and circular air holes as well as introducing double elliptical air holes. The finite element method (FEM) is utilized to investigate the mode coupling characteristics in the proposed PBS. The results show that the highest extinction ratio of 115 dB and shortest length of only 40 μm can be realized at a wavelength of 4 μm. A wide bandwidth of 200 nm ranging from 3.9μm to 4.1μm is obtained.


Introduction
The light polarization management is a crucial issue in optical fiber communication, sensing and other optical systems because many linear and nonlinear optical effects are polarizationdependent [1]. The most prominent feature of light polarization is that it is very vulnerable to environmental factor variations including temperature fluctuation, mechanical vibration, and tension change. Moreover, many optical devices are greatly polarization-dependent. Therefore, the random variation of the light polarization in the optical transmission or measurement system will give rise to various adverse optical effects, and eventually lead to the performance degradation of the optical system. In order to realize polarization independent operation of optical system, various polarization manipulation devices or polarization diversity scheme have been put forward to construct a polarization transparent system [2][3]. Polarization beam splitter (PBS) is one of the most basic polarization handling devices, which can separate input light with random polarization states into two orthogonal polarization components along different pathways propagation [4]. It plays a very crucial role in coherent optical communication, optical-fiber sensing, photopolarimeters, passive mode-locked fiber laser and quantum optics information system [5][6][7][8][9][10].
As early as the 1990s, several PBSs have been fabricated by using conventional optical fiber, but their outstanding disadvantage is that the length of PBSs are too long to meet the requirements of integration and miniaturization of photonic system [11][12]. Recently, quite a few PBSs based on silicon-on-insulator (SOI), indium phosphide (InP) and lithium-niobate-on-insulator (LNOI) platforms are reported [13][14][15]. Some new structures with high performance have been proposed and demonstrated including symmetrical/asymmetric directional couplers [16][17], partially etched multimode interference couplers [18], Mach-Zehnder interferometers (MZIs) [19], tapered waveguides [20], coupled plasmonic waveguide arrays [21] and sub-wavelength grating (SWG) [22]. Nevertheless, most of them have relatively complex structures. Moreover, the rigorous fabrication tolerance ultimately hinders their engineering application in large scale photonic integration circuits (PICs).
More recently, PBSs based on photonic crystal fiber (PCF) have attracted a lot of attention due to the advantages of broad bandwidth and compatible with current optical fiber communication systems. Moreover, because of the flexible air hole structure and extraordinary properties of PCF [23], PBS based on PCF are more efficient and compact that cannot realized by conventional optical fibers. Several PCF-based PBSs with short length have been reported. For example, in 2005, a dualcore PCF based PBS was reported by uniformly distributed elliptically-shaped air holes in the cladding of a PCF [24]. The bandwidth of 5.1 nm at 1.55μm and coupling length of 15.4mm was obtained. In 2013, Lu proposed an 84.7-mm long PBS with 300 nm-bandwidth based on a modified three-core PCF [25]. In 2015, Jiang obtained a 4.036-mm long PBS with 430 nm-bandwidth at 1.55μm based on a square-lattice dual-core PCF [26]. In 2018, Wang realized a 78-μm long PBS with 44 nm-bandwidth at 1.55μm based on a liquid-filled dual-core PCF [27]. In 2019, Zhao proposed a 6.75-cm long ultra-broadband PBS with 310 nm bandwidth based on dual hollow-core anti-resonant PCF [28]. However, we can see all previous reported PBSs mainly works in the near infrared (near-IR) waveband. In recent years, the mid-infrared (mid-IR) waveband (located between near-IR and far-IR wavelengths) has attracted great interest because of its broad atmospheric window and unique gas molecular absorption peaks in this region [29]. The mid-IR band has been widely used in optical communication, biotechnology, spectroscopy, food safety, environmental monitoring, and other fields [30]. But up to now, PBS working in mid-IR region is rarely reported, which is not suitable for the vigorous development of mid-IR technology.
In this paper, a novel mid-IR PBS with wide bandwidth and ultrashort length based on GaS dual core PCF is proposed. By introducing a pair of elliptical air holes into fiber core and using square and circular hybrid holes cladding, efficient polarization splitting with a bandwidth of 300 nm in the mid-IR region is obtained with device length of only 40um. The maximum polarization extinction ratio (ER) of 115 dB is realized at 4μm for the proposed PBS. The proposed PCF based PBS is fully compatible with popular optical fiber system and will play an important role in mid-IR photonics. The white regions denote air holes. The blue regions are substrate material. In view of its wide transparent window and excellent mechanical durability in mid-IR region, GaS is used as fiber substrate material. The proposed PBS is based on a triangular lattice structure which is composed of circular and square air holes. The whole air hole array is rectangular. Two circular air holes and two square air holes are missing in the center of the PBS to form a fiber core, and two elliptical holes arranged along the vertical direction are inserted to divide the fiber core into left and right dual core structures. The cladding air holes are alternately arranged by circular air holes and square air holes. The radius of circular hole in cladding is d1=0.4 um, and the side length of square air hole is d2=1um.The transverse distance between the center of the circular air hole and the center of the square air hole is e1, and the longitudinal distance is e2. The long half axis of the central elliptical air hole is a = 0.6um, and the short half axis is b = 0.22um. In this structure, on the one hand, the high birefringence is increased by using the alternate cladding structure of square air hole and circular air hole; on the other hand, the structure symmetry of the PCF is destroyed by introducing the elliptical air holes in the core, so that the birefringence is increased further, which is convenient for the complete separation of x and y polarized light. To optimized the PBS geometric structure and evaluate its performance, the finite element method (FEM) combined with rectangular perfect matching layer (PML) absorption boundary is used to analysis the mode coupling properties. In the numerical calculation, the mesh resolution is λ/12. The refractive index of the air hole is 1, and the refractive index of GaS is obtained by using semllmeier equation [31].
When light with arbitrary polarization injected into PCF, two orthogonal polarization components are coupled with each other in two cores. The coupling length of the dual core PCF in x and y-polarization direction can be obtained through following equation Here λ is wavelength of incident light in vacuum, n is effective refractive index, the superscript i refers to x and y-direction polarization, the subscripts o and e describes odd mode and even mode. Due to the asymmetry of fiber structure, the even and odd modes in x and y-directions have different effective refractive indexes. When the transmission length is L = mLx = nLy, the two orthogonal polarization components will be separated completely. For clarification, a crucial parameter of coupling length ratio (CLR) is defined by following equation The shortest PBS length can be obtained when CLR is 2 (Lx < Ly) or 1/2 (Lx > Ly). When the incident light launched into port A of proposed PBS, it can transform energy between core A and core B periodically along transmission length. The output power at core A can be obtained by Here Pin are the input optical power. Li denotes coupling length in x-and y-direction, respectively. The polarization ER is defined as the ratio of the optical power of one specific polarization to the optical power of another vertical polarization in the same fiber core at the PBS output end. It can be calculated through following equation 10 10 log Fig.1 The schematic cross section of the proposed PBS with square and circular air holes in the PCF cladding.

Structure optimization and discussion
From the viewpoint of practical applications such as optical communication, toxic gases monitor and so on, PBS with large bandwidth and short device length is highly desirable. According to waveguide optics and coupling mode theory, the guided wave properties of PCF are dependent on the air hole size, shape, and lattice constants. These geometrical parameters will ultimately determinate the mode properties of PCF. Then, by suitable designing the PCF cross-section through optimizing structural parameters, the efficient mode field coupling can be realized between even and odd modes in x-and y-polarized direction, respectively. Therefore, in the following section, to obtain a PBS with broad bandwidth and ultrashort length. We will investigate the influence of PBS geometric parameters (including a, b, d1 d2 e1 and e2) on the mode coupling properties by using FEM numerical method. The dependence of CL (including Lx and Ly), CLR and PBS structural parameters are analyzed carefully.
Firstly, we investigate the dependence of Lx, Ly and CLR on the structure parameters a. In this simulation, d1=1um，d2=0.4um，e1=2.02um，e2=1.5um，b=0.22um. a is increased from 0.56µm to 0.64µm. The results are shown in Fig.2(a). Obviously, with the increase of a, the coupling length Lx is almost unchanged, Ly increases slowly, and CLR increases significantly. But for all a, Ly increases faster than Lx. It can be explained that with the increase of a, the asymmetry of PBS core increases, which increases the birefringence effect and compresses the core mode field. On the other hand, we can find that CLR increases with the increase of a in the wavelength range of 0.56 μm to 0.64 μm. This is because when a increases, the birefringence in x-polarization direction is stronger than that in y-polarization direction. It can be found that when a is 0.6um, the CLR is close to 2. Therefore, a = 0.6μm is selected as the best value of the long axis radius.
Secondly, we investigate the dependence of Lx, Ly and CLR on the structure parameters b. In this simulation, d1=1um，d2=0.4um，e1=2.02um，e2=1.5um，a=0.6m. b is increased from 0.18µm to 0.26µm. The results are shown in Fig.2(b). It can be observed from Fig.2(b) that when the parameter b increases, Lx and Ly increase obviously. At the same time, CLR decreases slightly with the increase of b. It can be explained that when b increases, the asymmetry of the two cores of the PBS increases, which makes the birefringence effect increase. Moreover, the increment of birefringence is different in x and y directions.
Third, the dependence of Lx, Ly and CLR on the structure parameters d1 and d2 are investigated. In this simulation, d1 increases from 0.96 to 1.04um while other parameters are fixed. The results are given in Fig 3(c). Obviously, with the increase of d1, the coupling lengths Lx, and Ly decrease. However, the evolution trend of CLR is decreasing first and then increasing. When d1 increases from 0.96um to 1.01um, the enhancement of birefringence in x-direction is smaller than that in ydirection, so CLR decreases with the increase of d1. When d1 increases from 1.01um to 1.04um, the enhancement of birefringence in x-direction is larger than that in y-direction, so CLR increases. Fig  3(d) describes the dependence of Lx, Ly and CLR on the structure parameters d2. Clearly, the evolution trend of the curves is like that of Fig 3(c). Lx and Ly increase with the increase of d2, but CLR first decreases and then increases. It can be found that when d1 and d2 are 1um and 0.4um, respectively, the CLR is close to 2. So d1 =1um and d2=0.4um are selected as the optimal values of the circular hole radius and square hole side length.
Finally, the dependence of Lx, Ly and CLR on the structure parameters e1 and e2 are investigated. Fig 3(e) describes the dependence of Lx, Ly and CLR on the structure parameters e1. In this simulation, e1 increases from 1.98 to 2.06um while other parameters are fixed. Obviously, with the increase of e1, the coupling lengths Lx, and Ly increase slowly while CLR decreases. It is due to the fact that birefringence variation in x-direction is smaller than that in y-direction. Fig 3(f) describes the dependence of Lx, Ly and CLR on the structure parameters e2. In this simulation, e2 increases from 1.46 to 1.54um while other parameters are fixed. Obviously, the Lx, Ly and CLR are almost invariable with the increase of e2. The reason is that when e2 increases, it has little impact on the asymmetry between the two cores of the PBS, and the birefringence effect is not obvious. It can be found that when e1 and e2 are 1um and 0.4um, respectively, the CLR is close to 2. So e1 =1um and e2=0.4um are selected as the optimal values of the transverse distance and longitudinal distance. Based on the above analysis, a set of optimized parameters of a=0.6um, b=0.22um, d1=1um, d2=0.4um, e1=2.02um and e2=1.5um are selected as the optimal PBS structure parameters.
(a) (b) Fig.2 The Lx, Ly and CLR as a function of wavelength for proposed PBS parameters of (a) a, (b) b, For the optimized PBS, the relationship between effective refractive indices and wavelength is investigated. The results are shown in Fig.3. Obviously, the effective refractive indices of the four super-modes decrease with the increase of wavelength. Nevertheless, due to the asymmetry of the cross-section structure of optimized PCF-PBS, the effective refractive index of the x-polarization mode is slightly higher than that of the y-polarization mode. In addition, Fig.5 illustrates the electrical field distributions of two symmetric modes and two asymmetry modes for proposed GaSbased PCF-PBS at wavelength of 4 μm. The arrows with same direction denote the even mode, including x-polarization even mode (see Fig.5(a)) and y-polarization even mode (see Fig.5(b)); In contrast, the arrows with opposite direction denote odd mode, including x-polarization odd mode (see Fig.5(c)) and y-polarization odd mode (see Fig.5(d)). It can be seen clearly that the light is well confined in the core for all the four super-modes.   Fig.5. Obviously, the two CLs decrease with the increase of wavelength. The reason is that with the increase of the wavelength, the mode field gradually expands to the cladding, and the increase of the mode field makes the core coupling effect strengthen, resulting in the decrease of the fiber coupling length. The CL of y-direction is larger than that of the x-direction from 3.5 to 4.5um. For example, the CL of y-polarization (Ly) decreases from 64μm to 28μm, while the CL of x-polarization (Lx) decreases from 32μm to 15μm. In addition, the difference of Lx and Ly decreases gradually with the increase of wavelength. This can be fully explained by the fact that the effective refractive index difference in x-direction is greater than that in y-direction in Fig.5.

Fig.5 Coupling length as a function of wavelength
In order to explain the energy coupling process of different polarization components of incident light in the proposed PBS, according to the coupled mode equation, we plot the evolution curves of x and y-polarized light powers along the propagation distance. In the calculation, it is assumed that the incident light is in mid-IR region with a wavelength of 4 μm, which enters the one core of the proposed PBS. Fig.6 describes the calculated normalized output powers of x and y-polarization along the propagation distance. Clearly, in the process of transmission, the optical powers of x and y-polarization components change periodically along the propagation distance. The out power of xpolarization is largest while the y-polarization power is close to zero at a propagation distance of 40 μm. Therefore, the two polarization modes are completely separated at the splitter length of 40 μm. Fig.6 Normalized output powers of x and y-polarization lights In generally, when the ER of PBS is better than -20 dB, the incident light for the x and y polarizations are regarded to be separated. Therefore, the ER determines the available bandwidth of a PBS. For our designed dual-core GaS-based PBS, Fig. 7(a) shows the ER as a function of the wavelength when the device length is 40μm. Fig. 7(b) is the 3D view to illustrate the dependence of the ER with wavelength and transmission length. It can be found that the ER is higher than 20 dB in the wavelength range from 3.9 to 4.1μm, so the corresponding bandwidth is as wide as 200 nm. In particular, the ER is even as high as 117 dB at a wavelength of 4 μm. These results reveal that the proposed PBS can obtain an ultrabroad bandwidth of 200 nm in mid-IR region with ultrashort length of only 40 μm. Fig.7 (a) The ER of proposed PBS as a function of wavelength (b) 3D view of ER versus wavelength and propagation distance. Finally, we compare the performance of the proposed PBS in this work with that reported in previous literatures, and the results are shown in Table 1. We focus on two important parameters of PBS including CL and bandwidth. As shown in Table.1, the device lengths of other PBS are all longer than 100 μm, while the device length of our proposed PBS is only 40 μm, which is the shortest compared with other PBS mentioned in Table 1. This characteristic make our proposed PBS has great potential for using in photonic integrated system.  technology.