We propose and design a new kind of all-fiber polarization beam splitter and rotator (PBS and PBR) based on vector-mode-assisted coupling. By embedding a high-contrast-index ring-core between two cores of the conventional fiber couplers, being a three-core coupling structure, the state of polarization (SOP) of fiber-guided modes can be availably controlled, such as polarization splitting and rotating, by transitional coupling through TM01 or TE01 vector mode. Furthermore, the SOP of coupled mode can be rotated with arbitrary angle under different three-core layouts. In particular, by exploiting HE21-assisted coupling case, we can realize full-dimensional SOP rotation for arbitrary polarization input. We give the numerical simulation for the proposed all-fiber PBS and PBR, and investigate the corresponding polarization extinction ratio and polarization rotating purity in detail. The calculation results manifest a favorable performance on SOP management of fiber-guided modes. This vector-mode-assisted coupling might be expected to find potential applications in the polarization-based optical signal processing, multiplexing, and sensing system.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Polarization is one of the most salient features of light that has of great value in fundamental and applied optics [1–3]. In free space, the state of polarization (SOP) of light can be controld by using birefringent passive elements, such as wave plates (WPs) and polarization beam splitter (PBS). Flexible controlment for modal polarization states in optical waveguides is indispensable to advance the development of photonics integrated circuits (PICs), fiber optics and their related applications [4, 5]. The silicon-based waveguided eigenmodes manifest as transverse-electrical (TE) and transverse-magnetic (TM) modes, of which the SOP can be split and rotated by directionally coupling a specific SOP through special waveguide structural design . However, it is still a great challenge to control SOP using all-fiber coupling devices. As is well known, the usual optical fibers belong to weakly guiding waveguides where fiber eigenmodes are degenerated into linear polarization (LP) modes, therefore, giving rise to the polarization-insensitivity of the conventional fiber couplers. There are previous reports about the methods of achieving fiber-based PBS by means of birefringent-fiber coupler and photonic crystal fibers [7–10]. However, it is unlikely that they can be expanded to arbitrary SOP control and vector mode operation, for example, rotating SOP with arbitrary angles and adding/dropping a special fiber vector mode. What is more, as for the photonic-crystal-based BPS, there is no good compatibility with the existing fiber communication networks.
Since optical vector fields aroused much interest recently and have developed the fundamental researches and new functionalities of light [2, 3, 11–13], vector modes as fiber eigenmodes have received more and more attention, and so is the high-contrast-index optical fiber because of its ability of lifting modal degeneracy [14–16]. For example, hollow ring-core fibers (RCFs) are usually used to support and transmit the separated vector modes with large effective refractive index differences. The degree of lifting modal degeneracy is largest for the first vector mode group including TM01, TE01, and HE21 modes. Vector mode coupling in optical fibers is characterized by polarization-dependence and -isolation features . Based on these, in this paper, we propose a new kind of all-fiber PBS and polarization beam rotator (PBR) by embedding a high-contrast-index core within the conventional fiber coupler. When the TM01 or TE01 vector mode serves as middle assisted mode, a specific SOP of fundamental mode can be split or/and rotated when coupling from one core to another, which can be used to design all-fiber PBS and PBR for a specific SOP. When HE21 vector mode serves as middle assisted mode, arbitrary SOP can be rotated and thus be designed as the full-dimensional PBR. Furthermore, the proposed all-fiber structural design can be expanded to add/drop special fiber vector modes (or SOP) for future fiber-based full-vector mode (or polarization) multiplexing system. We believe that a new controllable degree of freedom for all-fiber SOP can be achieved by the proposed design scheme, compared to the previous control of modal power, wavelengths and modal forms using fiber couplers [18–21].
In this paper, firstly, we exhibit the polarization dependence and rotation feature of vector-mode-assisted coupling in the three-core coupling system, and we supply the three-mode coupled theory and give its analytical solution that is the theoretical basis of the proposed vector-mode-assisted three-core coupling system. Additionally, we briefly introduce a kind of fabrication process of these three-core couplers, and then focus on the detailed parameter design for the embedded high-contrast-index ring-core. As a typical SOP controlable, polarization splitting, we design a TM01-assisted all-fiber PBS and give the simulation results. The polarization-dependent coupling transmission and polarization extinction ratios for this PBS are calculated and discussed under different core distances. Furthermore, we present the designed TM01-assisted all-fiber PBR, and numerically investigate its purity for polarization rotation. Especially, the case of full-dimensional polarization rotation is also simulated and presented by designing the HE21-assisted all-fiber PBR. At last, we discuss the expansibility and expectation for the vector-mode-assisted three-core coupling design, and summarize the all-fiber SOP controllable design based on the proposed vector-mode-assisted coupling system.
2. Coupling principle and fiber design
The proposed all-fiber PBS and PBR design is characterized by three-core coupling structure. A high-contrast-index ring-core is embedded between two cores of the conventional single-mode fibers (SMFs). This ring-core can lift the polarization mode degeneracy and thus support separated vector modes , such as the exploitable cylindrical vector modes TM01, TE01, and HE21. When the TM01 or TE01 vector mode supported in ring-core is matched to mode coupling within the conventional SMF couplers, it allows coupling between specific polarized modes, and isolates coupling for its orthogonal counterparts . What is more, this matched vector-mode-assisted coupling can arbitrarily rotate modal polarization orientation when two SMF cores are placed at different relative azimuthal positions around the ring-core. For example, as shown in Figs. 1(a)-1(c), the bottom fundamental mode HE11 can be coupled to different azimuthal positions around the middle assisted vector modes along the indicated directions of arrows, interestingly, at the same time, rotating SOP with corresponding angles. Because directional coupling between two modes strongly depends upon the polarization uniformity on the overlap region of two coupled modes . Based on this polarization isolation and uniformity of vector-mode-assisted coupling, we can implement two typical all-fiber SOP management, i.e. polarization beam splitting and rotating.
The evolution of the electrical fields in this three-mode coupling system can be described by the coupled-mode equationsFig. 1(d). To get uniform analytic expression and discussion for the three-core coupling system here, we also make a reasonable assumption about for a vertical core layout in Fig. 1(e), even though that the relative core distance between core1 and core3 becomes small in this case. Nonetheless, the direct coupling from core1 to core3 has been taken into consideration when numerically calculating the coupling performance of polarization splitting and rotating below. Assuming that core1 and core3 have identical sizes and refractive index distribution, it makes . Under the boundary conditions of , , , one can get the analytic solutions to Eqs. (1)-(3), as follows,
A feasible way to fabricate this kind of fiber couplers is to bond three independent side-polished fibers with refractive index matching liquid or glue that is close to the fiber cladding [22, 23]. We present the possible fabrication process of all-fiber PBS and PBR as shown in Figs. 1(d) and 1(e), respectively. An additional method may be adopted to fabricate them by means of fused tapering that has been successfully used to selectively radially and azimuthally polarized modes, as well as orbital angular momentum (OAM) mode [24, 25], but it should be noticed that the relative locations with respect to three cores must be well preserved, especially for the vertical core layout to get polarization rotation. In our design, two cores on its each side correspond to cores of single-mode fibers (SMFs) with core refractive index . Three cores are packed with a mutual cladding of SiO2 material. The ring-core has a high step index of , corresponding to , and the inner layer is air, i.e., . In Fig. 2, we present the effective index variation of the fundamental mode HE11 in the independent SMF and several high-order vector modes in the independent RCF as a function of core radius at the wavelength of 1550nm. One can get the effective refractive indices map by numerically solving the eigenmode equations of two and three layers of waveguide models that correspond to SMF and RCF [26, 27], respectively. The size design and refractive index distribution of SMF and RCF are indicated as the insets. The inner and outer radii of RCF are conditioned to , and we adjust the inner radius to make the effective index of desired higher-order vector modes in RCF equal with that of HE11 in SMF. One can see from Fig. 2 that the effective refractive indices of the supported first (TM01, TE01 and HE21) and second (HE31 and EH11) vector mode groups are completely segregated in the high-contrast index ring-core. The segregation degree of the first one is higher than the second one, and the effective refractive indices difference between any two of them is more than . Under the fixed core radius of SMF, if one wants to get coupled TM01 mode in RCF, the inner radius of RCF needs to be fabricated as or tapered to the marked value of nm to satisfy the resonance condition of vector mode coupling near the wavelength of 1550 nm. If getting HE21 mode, it corresponds to nm.
3. All-fiber PBS
In this section, we numerically simulate the designed TM01-assisted all-fiber PBS and calculate its polarization extinction ratio under different core distances. The sketch of all-fiber PBS is depicted in Figs. 3 to visualize the polarization splitting mechanism. Three core layout of this structure with being in a line corresponds to Fig. 1(d). In this case, the TM01 mode serves as the middle assisted mode to split the incident x-polarization component of HE11 mode, shown as red dotted line. In contrast, its y-polarization component remains uncoupled. The insets indicate power flow among different modes. The thick red arrows represent the dominant mode coupling, while the thin yellow arrows indicate the undesired mode coupling as crosstalk.
The power evolution of TM01-assisted PBS is simulated by means of the simulation software of finite-difference time-domain (FDTD) mode solutions, and the results are shown in Fig. 4. In this simulation, the core distance between SMF and RCF as indicated in Fig. 1(d) is designed as μm, and the coupling length is about 8.6 mm. In Fig. 4(a), the incident x-polarized HE11 from core1 can be fully coupled to core3 through TM01 mode in ring-core2. In this full coupling case, mode coupling among three modes in Fig. 4(a) follows Eqs. (7)-(9). As shown in Fig. 4(b), however, the y-polarized HE11 propagation is uncoupled with this system and keeps propagation along the core1. Obviously, the TE01 mode can also serve as the assisted mode provided that the phase-matching condition is satisfied between HE11 mode in core1 and TE01 mode in core2 by means of adjusting core radius of middle ring-core according to Fig. 2. In this case of TE01-assisted coupling, the split SOP would be reversed, in other words, it is the y-polarized component that is split into core3, instead of x-polarized component under the TM01-assisted case, due to the polarization-dependence of vector mode coupling.
In the designed structure, there are many design parameters to affect the polarization spitting efficiency, such as core distance, ring-core size, and refractive index contrast of RCF. As for the well matched SMF and RCF according to Fig. 2, the core distance is the primary parameter that determines the coupling efficiency and polarization splitting degree. We calculate the polarization-dependent power evolution along the core1 and core3 under the x-polarization and y-polarization input from core1, respectively, when the core distance is designed to different values. As plotted in Figs. 5(a)-5(c), corresponding to core distances of μm, 8 μm, and 10 μm between SMF and RCF, respectively, the results show that the coupling crosstalk for y-polarization input from core1 to y-polarization output from core3 that is indicated as the yellow arrows in the bottom inset of Fig. 3 can be sharply reduced by increasing the core distance. Meanwhile, the efficiency of targeted coupling for x-polarization input from core1 to x-polarization output from core3 indicated as the red arrows in the top inset of Fig. 3 can be enhanced on the condition of the longer coupling length. Because the weakened coupling strength due to larger core distance needs to be compensated by increasing the coupling length.
To further present the work performance of the designed all-fiber PBS, we next investigate the extinction ratios for polarization-dependent power output from both core1 and core3, which can be expressed by
Before giving the polarization extinction ratios, we calculate the coupling efficiency of the x-polarized HE11 from core1 to core3 over a large wavelength range under three core distances f μm, 8 μm, and 10 μm, corresponding to the full coupling length of 4.0 mm, 8.6 mm, and 16.2 mm, respectively, and the results are shown in Fig. 6(a). One can see that the polarization splitting bandwidth could be decreased when the core distance is enlarged. The calculated polarization extinction ratios under these three core distances are exhibited in Figs. 6(b) and 6(c), corresponding to the polarization output from core1 and core3, respectively. From the results, one can see that the absolute values of extinction ratios for polarization output from core1 are generally larger than these from core3 at the central wavelength under the same core distances. This is because the three-core coupling structure substantially couples x-polarization component from core1 to core3, while rarely couples y-polarization component as coupling crosstalk at the central wavelength. This unequal coupling efficiency naturally makes the extinction ratios different according to Eqs. (10) and (11). Interestingly, these extinction ratios can be efficiently enhanced by enlarging the core distances, but, at the cost of reducing the bandwidth because of the longer coupling lengths, which may be potentially used to polarization-based optical signal processing and sensing. Additionally, the central wavelength shown in Figs. 6(a) and 6(b) is shifted to the longer wavelength when enlarging the core distance due to the weakened coupling crosstalk. It should also be noted that there may be possible wavelength- and bandwidth-sensitivity to radii size and refractive index contrast of ring-core, due to the effect of waveguide dispersion on vector-mode-assisted coupling.
4. All-fiber PBR
The sketch of three core layout for all-fiber PBR is depicted in Fig. 7, of which the section view corresponds to Fig. 1(e). When the TM01 or TE01 mode serves as the middle assisted mode, the PBS just can rotate a specific SOP. As for the TM01-assisted case, as shown in Fig. 7, the split x-polarized HE11 mode can be rotated about 90 degrees into y-polarized HE11 mode, relative to the incident SOP. Similar to the TM01-assisted all-fiber PBS design, the core distance as indicated in Fig. 1(e) is also set as μm, and the coupling length is about 8.6 mm. The power map with SOP rotation is presented in Figs. 8(a) and 8(b). Figure 8(a) shows the map at the x-z cross section for the x-polarization input from core1 being coupled into TM01-assisied mode in core2. Figure 8(b) shows that at the y-z cross section for y-polarization output from core3 being coupled from the TM01-assisied mode in core2.
We calculate the polarization-dependent power evolution along the core1 and core3 of the designed TM01-assisted all-fiber PBR with core distance of μm under the x-polarization input, as shown in Fig. 9(a). As for the x-polarization input, the coupled modal SOP into core3 for PBR becomes the dominated y-polarized HE11 mode instead of the x-polarized one for PBS shown in Fig. 5(b), because of the different core layouts. Noticeably, the vertical core layout for PBR may give rise to larger direct coupling of the x-polarized HE11 from core1 to core3, relative to the straight core layout for PBS. Consequently it may affect the purity of polarization rotation that can be quantified as
In Fig. 9(b), we give the calculated purity of polarization rotation over a large wavelength range under three different core distances, corresponding to the full coupling lengths of 4.0 mm, 8.6 mm, and 16.2 mm, respectively. It shows that the polarization rotation has high purity, especially at the central wavelength, nearly approaching 1. To clarify the effect of core distances on the polarization rotation, we further calculate this polarization rotating purity at the central wavelength as a function of the core distances, and the results are exhibited in the left y-axis in Fig. 9(c). It shows that the rotated polarization purity can reach above 97%, and can be further enhanced by increasing the core distance. In this case, the required full coupling lengths are also correspondingly given in the right y-axis in Fig. 9(c).
Finally, we discuss and simulate the case of HE21-assisted all-fiber PBR. Although the TM01 or TE01-assisted PBR can be used to split and simultaneously rotate a specific SOP, it does not work for its orthogonal counterpart. To get full-dimensional polarization rotation, we can utilize the HE21 mode as the assisted-mode. Since that odd and even HE21 modes in the circularly symmetric fibers have nearly the same effective refractive indices due to the modal degeneracy, modal polarization rotation can work for both x-polarization and y-polarization input. We simulate the HE21-assisted full-dimensional PBR and exhibit the results in Fig. 10. In this case of fiber design, the inner radius size of ring-core is slightly reduced to to match HE21 mode serving as assisted mode to the three-core coupling system. The core distance is designed as μm, and the coupling length is about 9 mm. The simulation results show that the incident SOP in these four cases is all rotated about 90 degrees through the HE21 assisted-mode, and has high coupling efficiency. As for the arbitrary polarization input, it can be divided into the x-polarization and y-polarization components as a set of orthogonal basis. So it is obvious that the incident arbitrary polarization orientation could be rotated with 90 degrees through the HE21-assisted all-fiber polarization beam rotator, because both the x-polarization and y-polarization components are rotated with 90 degrees with the same phase changes. Here it's unnecessary to repeat the detailed numerical analysis about polarization purity of the HE21-assisted full-dimensional PBR, because of its similarity to the case of TM01-assisted PBR.
5. Discussion and conclusion
We have presented the TM01-assisted all-fiber PBS and PBR for x-polarization orientation, and especially HE21-assisted PBR for full-dimensional SOP. It needs to be emphasized that the higher-order vector-mode-assisted cases, such as the group of HE31 and EH11 modes, or HE41 and EH21 modes in ring-core, etc., they cannot be used to SOP split, because their even and odd vector mode components are degenerated, but can be utilized to SOP rotation. As for SOP split, apart from TM01 and TE01-assisted cases supported in high-index-contrast ring-core or step-index fibers, any high-contrast-index fibers can be utilized as long as they can lift the modal degeneracy between even and odd vector mode components, for instance, the elliptical-core fiber due to its remarkable modal birefringence effect. In practice, to make SOP control with high efficiency, the assisted vector modes need to be fully matched to the polarization-dependent coupling in the three-core system by strictly designing the middle high-contrast-index core, and the coupling length also need to be well controlled to realize full coupling. It is promising that our proposed all-fiber SOP splitting scheme might be extended to design all-fiber polarization (or vector modes) multiplexer/demultiplexer in polarization (or vector mode) multiplexing system instead of by means of the complex multi-input and multi-output (MIMO) technology .
In summary, we have proposed a new kind of all-fiber vector-mode-assisted PBS and PBR, and have given detailed numerical simulation and performance analyses. The high polarization extinction ratios for PBS and high polarization purity for PBR show a feasible way to control fiber-guided SOP based on the vector-mode-assisted coupling method. Compared to previous mode control of optical power, wavelengths and modal forms using fiber couplers, this new kind of polarization-sensitive couplers can realize an additional controllable degree of freedom for fiber-guided modal SOP and vector modes. It is expected that the presented all-fiber SOP controllable method might find wide applications in the polarization-based optical signal processing, multiplexing and sensing system.
Natural Science Foundation of Hubei Province of China (2018CFA048); Royal Society-Newton Advanced Fellowship; National Natural Science Foundation of China (NSFC) (61761130082, 11574001, 11774116); National Basic Research Program of China (973 Program) (2014CB340004); National Program for Support of Top-notch Young Professionals; Program for HUST Academic Frontier Youth Team.
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