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Abstract
We propose an all-fiber mode-selective power splitter (MSPS) for non-circular-symmetric LPlm (l = 1, 2, …) modes, which is suitable for multicasting and optical performance monitoring in mode-division multiplexing optical fiber networks. The MSPSs are asymmetric two-core few-mode directional couplers composed of a few-mode fiber and a two-mode fiber. We theoretically studied the three conditions required by the MSPSs. By carefully choosing the core-to-core distance and coupling length, the MSPS can achieve arbitrary splitting ratio regardless of the modal field orientation of the input non-circular-symmetric LP mode. By using an asymmetric structure, the MSPS can ensure the power splitting only happens on the target non-circular-symmetric LP mode when the phase matching condition is satisfied. In addition, we designed and numerically simulated LP31 MSPSs with four kinds of splitting ratios, among which the one with 90/10 splitting ratio was fabricated based on tapering and polishing method. The fabricated LP31 MSPS is characterized and the results show that its splitting ratio is much more stable than regular LP31 mode-selective coupler.
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1. Introduction
Mode division multiplexing (MDM) optical transmission systems and networks employing linear polarized (LP) modes of the few-mode fibers (FMF) has been investigated for more than ten years [1–3]. In MDM systems, few-mode optical fiber couplers (FM-OCs) are indispensable components which enable a wide variety of functions. For examples, mode-selective fiber couplers (MSCs) are frequently-used FM-OCs by matching the propagation constant of the fundamental mode in the SMF with a higher-order mode in the FMF. The MSCs for each spatial mode can be cascaded orderly to form a mode multiplexer [4]. A pair of MSCs helps with the add and drop function in MDM optical switching networks [5]. FM-OCs consisting of two identical FMFs can combine or split the optical power for non-circular-symmetric LP modes at arbitrary coupling ratios regardless of the modal orientation [6,7]. Nonetheless, some MDM optical network scenarios require FM-OCs with new functions for implementation.
Multicasting and optical performance monitoring (OPM) are two crucial functions in optical fiber transmission systems and networks [8,9]. Assume a typical MDM optical network scenario with several FM-OCs as shown in Fig. 1. The signals carried by mode B need to be multicast to user1, 2 and 3; The optical power in mode C need to be separated out a small portion for OPM; In the meantime, mode A cannot be affected when passing through these FM-OCs. As the excited LP modes propagate in the FMF, the orientation of their modal field will be randomly rotated by imperfect fiber fabrication and external perturbations. For circular-symmetric LP0n (n = 1, 2, …) modes, their fields are invariant under such rotations, so regular all-fiber LP0n MSCs which are composed of an FMF and a SMF can achieve the functions required in the given scenario by setting a proper coupling length according to the desired splitting ratio. However, for non-circular-symmetric LPlm (l = 1, 2, …) modes, regular MSCs cannot ensure stable splitting ratio because of the random rotation of the modal fields [10].
Several FM-OCs have been proposed so far to avoid the influence of the rotation of the non-circular-symmetric LP modes. For instance, a three-core MSC was employed to convert the two degenerate modes of the non-circular-symmetric LP modes into the fundamental modes of two SMFs respectively, and then the optical powers from the two SMFs are combined to eliminate the null situation [10,11]. However, the two-step processing (converting and combining) requires more devices. Besides, the two optical paths must have strictly identical length during the combination to avoid the temporal misalignment. As a novel two-core FM-OC, the orientation-insensitive few-mode fiber couplers are able to achieve arbitrary splitting ratios for non-circular-symmetric LP modes and can be used in the given scenario when the number of the exited modes are 1 and 2 [6,7]. But when the number of the modes are larger than 2, the drop FMF will extract energy from unwanted modes due to the lack of modal selectivity. A degenerate mode-selective fiber coupler is proposed for the issue of degenerate-mode-reception [12]. However, almost all optical power is demultiplexed by the drop two-mode fiber (TMF) in which LP01 mode and two degenerate LP11 modes are sustained, so the multicasting and OPM functions cannot be satisfied.
In this paper, we propose a FM-OC named mode-selective power splitter (MSPS) for each non-circular-symmetric LP mode. MSPSs are asymmetric fiber couplers composed of a FMF and a TMF. The principles are investigated analytically and three necessary conditions are discussed. LP31 MSPSs with four kinds of splitting ratios (90/10, 70/30, 50/50, and 30/70) are designed and simulated to verify the principles. Then we fabricated a 90/10 LP31 MSPS utilizing a double-ring assist FMF and a step-index TMF based on the taper-polish method. The characterization results shows that the splitting ratio of the fabricated LP31 MSPS is much more stable than regular LP31 MSC. The splitting ratio and modal selectivity are also evaluated over the C band.
2. Principles of MSPSs for non-circular-symmetric LP modes
In order to meet the demands of the given scenarios shown in Fig. 1, a MSPS for non-circular-symmetric LPlm (l = 1, 2, …) mode should satisfy three conditions: (1) The LPlm MSPS should be insensitive to the modal field orientation of the LPlm modes so as to ensure the splitting stability; (2) The LPlm MSPS must be highly mode-selective so that other LP modes will not be affected when passing through it; (3) The LPlm MSPS should achieve arbitrary splitting ratio in order to be applicable to different scenarios.
At a certain coordinate, any non-circularly-symmetric LPlm modes with a rotation angular α can be regarded as the combination of the LPlma and LPlmb modes which are a pair of degenerate modes with identical propagation constant. In order to satisfy condition 1, the proposed LPlm MSPS should have the same splitting ratio for both degenerate LPlma and LPlmb modes. The coupling behaviors between two LP spatial modes of an FM-OC is analyzed in [12]. Since the coupling coefficient between the LPlmb mode of the FMF and the LP01 mode of SMF is zero, the optical power of these two modes cannot couple to each other even the phase matching condition is satisfied [10,13]. Therefore, the drop fiber cannot be a SMF. The LPlm MSPS cannot be a symmetric FM-OC as well, because the phase matching condition is always satisfied between two identical FMFs so that the drop FMF will extract energy from unwanted modes. From the perspective of dimensions, if the optical power in degenerate LPlma and LPlmb modes are converted by the LPlm MSPS into a single drop fiber, the drop fiber must support at least two spatial degrees of freedom. Besides, the drop FMF is usually coupled to a photo detector (PD) in the user side for optoelectronic conversion. Smaller fibers are preferred to couple to high-speed PDs with smaller effective areas. Since every non-circular-symmetric LP mode have and only have the same two-fold degeneracy, it would be useful to choose a TMF as the drop fiber, which possesses enough dimensionality and modal selectivity simultaneously. In this case, condition 2 can be satisfied. Additionally, the signals in LP11 mode from the drop TMF of a MSPS can be detected by the equipment which is spatially coupled or have multimode pigtail fiber such as optical power meter, optical spectrometer, commercial intensity modulation and direct detection transceivers and so on. For the equipment with only single-mode pigtail fiber, the compatibility of the reception for LP11 mode should be considered. Consequently, a MSPS is an asymmetric FM-OC which could split the optical power in the LPlm mode to the LPlm mode of the output FMF and the LP11 mode of the output TMF. The structure of the proposed MSPS is shown in Fig. 2(a), where R (0 < R < 1) is the percentage of the power for the output TMF. The splitting ratio of the MSPS is defined by (1 - R)/R.
We investigate the necessary conditions for the MSPS. Since the LPlma and LPlmb modes can only couple to the same kind if degenerate LP11 mode as shown in Fig. 2(b), the proposed LPlm MSPS functions as two independent FM-OCs. Assuming that the FMF initially carries unit optical power in LPlma mode with no power being exited in the TMF, the power transfer for the FMF and TMF are [14]:
respectively, where Plma(z) is the normalized optical power carried by LPlma mode in the FMF (as functions of the axial distance z along the coupler), P11a(z) is the normalized optical power carried by LP11a mode in the TMF. Fa is the normalized peak power and Ca is the power coupling coefficient [14]:In order to achieve the maximal peak power and suppress the modal crosstalk, the LPlm mode in the FMF should be phase-matched to the LP11 mode in the TMF, which is βlm = β11. In this case, Fa = 1, Ca = κa. The power transfer for LP11a mode can be written as P11a(z) = sin2(κaz). The above analysis is also suitable for the coupling between the LPlmb mode in the FMF and the LP11b mode in the TMF. Similarly, the power transfer for the LP11b mode can be written as P11b(z) = sin2(κbz), where P11b(z) is the normalized optical power carried by the LP11b mode in the TMF. κb is the coupling coefficient between the LPlmb mode in the FMF and the LP11b mode in the TMF. In order to achieve (1 - R)/R splitting ratio for both degenerate modes, the power transfer should satisfy
where L is the coupling length. Note that the absolute values of the coupling coefficients κa and κb decrease along with the increase of core-to-core distance d [10]. Therefore, Eq. (4) can be satisfied by choosing proper d and L. To conclude, equation sin2(κaL) = sin2(κbL) makes condition 1 true; The asymmetric structure of the MSPS meets condition 2. The establishment of Eq. (4) satisfies conditions 3.3. Design of LP31 MSPSs
In this section, four kinds of LP31 MSPSs with different splitting ratios (90/10, 70/30, 50/50, 30/70) are designed to verify the above theoretical analysis. For the sake of simplicity, we choose the fibers which will be used in the following fabrication process for simulation verification. In order to suppress the modal crosstalk, the minimum effective refractive index difference (min|Δneff|) of the FMF in the LP31 MSPS should be as large as possible. A step-index FMF with double-ring areas in the core is utilized whose min|Δneff| is up to 1.49 × 10−3 [15]. Figure 3(a) shows the designed and fabricated index profiles of the FMF. Its core/cladding radius is 8.25/62.5 µm and the core-cladding index difference Δn is 0.748% at 1550 nm. The drop TMF of the LP31 MSPS is a step-index TMF whose designed and measured index profiles are shown in Fig. 3(b). The core/cladding radius of the TMF is 5/62.5 µm and the core-cladding index difference Δn is 0.688% at 1550 nm.
Fig. 3. The index profile and supported LP modes of the (a) double-ring assisted FMF and (b) step-index TMF.
For LP31 MSPS with any splitting ratio, the phase matching condition should be satisfied firstly. Since the effective refractive index (neff) of the LP11 mode in the TMF is slightly larger than the nef of LP31 mode in the FMF, we choose to taper the TMF to reduce its neff. During the tapering process, the core and cladding become thinner in equal proportions. In this case, the structure of the LP31 MSPS is illustrated in Fig. 4(a), whose coupling region consists of intact FMF core and tapered TMF core. Note that this process can be achieved during the fabrication by standard fused biconical taper technique [16]. The neff of the LP31 mode in the FMF and the LP11 mode in TMF as functions of the tapered radius are calculated by COMSOL multiphysics based on the index profiles. The results are shown in Fig. 4(b). The intersection of the dotted lines indicates that the TMF should be tapered to 58.5 µm for the phase matching condition. After the tapering, the LP01 mode of the TMF does not phase match to any modes of the FMF which avoids additional modal crosstalk.
Fig. 4. (a) The structure of the proposed LP31 MSPS using a double-ring assisted FMF and a step-index TMF. (b)The neff of LP31 mode in FMF and LP11 mode in TMF as functions of the tapered radius.
Since the core-to-core distance d determines the coupling coefficients κa and κb which are related to the periods of power transfer for the two degenerate modes, condition 1 can be met by choosing the intersection of the power transfer curves for both degenerate modes. Therefore, desired splitting ratios can be achieved by setting d. Then d and the coupling length L are designed through parameters sweeping utilizing beam-propagation method (BPM). For the LP31 MSPSs with splitting ratio 90/10, 70/30, 50/50 and 30/70, d is set to 4.22, 4.52, 4.07 and 5.32 mm respectively. Only the power transfer of the LP11a and LP11b modes of the TMF versus coupling length are shown in Fig. 5(a)-(d). Since there is minor excess loss under the phase matching condition, the total coupling efficiency is almost 100% and the power transfers of the LP31 modes in the FMF are not shown.
Fig. 5. The coupling efficiencies of the LP11a and LP11b modes of the TMF versus the coupling length of the LP31 MSPS with splitting ratio (a) 90/10, (b) 70/30, (c) 50/50 and (d) 30/70. The black curves are for LP11a modes; The red curves are for LP11b modes; The intersections of the curves for the designed splitting ratio are encircled by green circles.
As shown in Fig. 5(a), if LP31 MSPS requires a stable splitting ratio 90/10, the coupling efficiency of LP11a and LP11b modes of the TMF should be both 10% at a same coupling length L. Then L is set to 15.92 mm which is the intersection of two curves. For LP31 MSPSs with splitting ratio 70/30, 50/50 and 30/70, the coupling length is 24.28, 12.73 and 20.25 mm at each designed d, respectively. So far, three conditions are all satisfied. The modal selectivity of each LP31 MSPS is also verified by injecting LP01, LP11, LP21 and LP02 modes, respectively. The results are shown in Table 1. The modal crosstalk from other four LP modes are all less than - 21.4 dB for each LP31 MSPS owing to the phase matching.
Table 1. Modal crosstalk for the four designed LP31 MSPSs (Unit:dB)
4. Fabrication and characterization
The designed LP31 MSPS with splitting ratio 90/10 is fabricated based on the taper-polish method and the performance is characterized [17]. According to the simulation parameters, the step-index TMF is firstly tapered in order to satisfy the phase matching condition. The tapering process was conducted on a fused biconical taper station. A microscope was utilized to measure the diameter of the tapered FMF to ensure that the radius of the waist region reaches the designed value. Then the FMF and pre-tapered TMF are respectively embedded into quartz substrates with arc grooves and polished on a grinding platform. After polishing, two coupler halves can be mated together to form a LP31 MSPS with splitting ratio 90/10. Condition 1 and 3 are determined by the core-to-core distance d and coupling length L. As shown in Fig. 6(a), the adjustment of d can be achieved by controlling the residual cladding thickness (RCT) during the polishing process or adjusting the lateral offset between the two coupler halves. The coupling length L can be tuned by adjusting the longitudinal offset of two coupler halves. The adjustment was implemented by rotating the micrometer rods on a fixture as shown in Fig. 6(b). After the adjustment, the device was fixed by the optical glue. The fabrication deviations may come from the longitudinal and lateral offsets. The lateral offset should be precisely adjusted before applying the glue, since the performance of the MSPS is more sensitive to the it.
Fig. 6. (a) The schematic diagram of adjusting the core-to-core distance d and coupling length L. (b) The photo of the fixture used to adjust the longitudinal and lateral offsets of the MSPS.
The stability of the 90/10 splitting ratio of the fabricated LP31 MSPS is characterized. The experimental setup is shown in Fig. 7. A tunable continuous-wave (CW) laser with scrambled polarization at 1550 nm was injected into the SMF port of a LP31 MSC. The exited power in LP31 mode then passed through a mode rotator. In this experiment, the mode rotator is actually a three-paddle polarization controller based on stress-induced birefringence produced by wrapping the FMF around a few loops [18]. It can be seen that the spatial orientations of the LP31 mode were randomly adjusted by the mode rotator. Then the LP31 MSPS convert the rotated LP31 mode in the FMF to the LP11 mode in the TMF output. At the same time, the FMF output is still LP31 mode. Then the output power of the FMF and TMF are measured by a power meter. The mode rotator was randomly adjusted 50 times and the measured splitting ratios are calculated and shown in Fig. 8(a). The excess loss of the fabricated LP31 MSPS was about 0.8 dB. As a contrast, the splitting ratios of a LP31 MSC as a power splitter are measured in the same way as shown in Fig. 7. The measured LP31 MSC has a 10% coupling efficiency from SMF input to FMF output. The measured splitting ratios are shown in Fig. 8(a) as well. It can be seen that the splitting ratios of the fabricated LP31 MSPS are much more stable than that of the LP31 MSC whose SMF output often has very little power. The wavelength dependence of the splitting ratio of the fabricated LP31 MSPS are also measured over the C band and the results are shown in Fig. 8(b). It can be seen that the fabricated LP31 MSPS is only optimized for C band. Some recent works have proven that by carefully matching the effective refractive indices of two fibers in the modal coupler, the operating wavelength could span over S + C + L band [19,20].
Fig. 8. (a) The results of 50 times splitting ratio measurements with randomly input spatial orientations at 1550 nm for LP31 MSPS/MSC. (b) The wavelength dependence of the splitting ratio for the LP31 MSPS over the C band. (c) The measured modal crosstalk for the LP31 MSPS over the C band.
In order to characterize the modal selectivity of the fabricated LP31 MSPS. The modal crosstalk was measured by injecting 0-dBm LP01, LP11, LP21 and LP02 modes respectively over the C band. The power at the TMF output was recorded and the modal crosstalk are shown in Fig. 8(c). The largest modal crosstalk came from the LP02 mode. This is because the neff of LP02 mode is closest to the LP31 mode than other LP modes. The modal selectivity could be further improved by designing the FMF profile to enlarge the Δneff between LP02 and LP31 mode [21].
5. Conclusion
In this paper, we propose a MSPS for non-circular-symmetric LP modes whose principles and necessary conditions are analyzed in detail. We designed four kinds of LP31 MSPS with different splitting ratios to verify the theoretical analysis. A 90/10 LP31 MSPS was fabricated using a weakly-coupled FMF and a TMF based on polishing and mating method. The device was characterized and the results show it could be applied to the MDM optical fiber networks. The MSPSs are also expected to be achieved by other waveguide structures, such as femtosecond direct-written integrated couplers and three-dimensional polymer waveguide [22,23].
Funding
National Natural Science Foundation of China (62101009, U20A20160); Peng Cheng Laboratory (PCL2021A04).
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 request.
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