Space-division multiplexing (SDM) technology is capable of effectively improving the transmission capacity of fibers and has attracted more attention since it was first proposed in the 1980s^[1–8]. SDM technology can be divided into core-division multiplexing, mode-division multiplexing (MDM), and their combination^[4,6]. Among them, MDM systems based on few-mode fibers (FMFs) have been considered a promising candidate^[9–14]. MDM systems can be achieved using strongly or weakly coupled FMFs. The former must use a complicated multiple-input multiple-output (MIMO) equalizer to compensate for any linear mixing between modes^[12]. However, in weakly coupled MDM systems, the scheme of MIMO-free or a simple MIMO equalizer is desirable for low modal crosstalk (XT)^[15]. In addition, the MIMO-free scheme is attractive for applications compatible with the currently available optical transport networks (OTNs)^[16,17].

In commercial OTN-based MDM transmission systems, polarization-dependent loss (PDL) has a great influence on coherent receiver sensitivity^[17]. The additional PDL introduced by the MDM system is prone to exceeding the compensation range of digital signal processing (DSP) algorithms in dual-polarization (DP) coherent receivers^[18,19]. Therefore, it is very necessary to reduce the PDL of MDM systems. In our previous work^[17], we proposed the PDL measurement method for MDM systems but did not point out how to effectively reduce the PDL, particularly from a theoretical perspective.

In this paper, we investigate the PDL performance of the fundamental MDM system by theory and experiment. We aim to identify the origin of PDL in the MDM systems and optimize the transmission performance of DP signals. First, in order to find a polarization-insensitive MDM link state, we examine the PDL of the LP11-mode channel with a few-mode polarization controller (FMPC). Then, according to the modal pattern characteristics of the LP11-mode DP signal, we analyze the variation of the polarization power difference ΔPxy with the relative orientation angle β of the modal field; the theoretical results are in agreement with the experimental data. Finally, the influence of intra- and inter-modal XT on the ΔPxy−β curves for the DP signal is taken into account. The results presented here can provide a guide to the optimal design of MDM systems with DP signals.

A fundamental MDM system consists of a pair of mode multiplexer/demultiplexer (MMUX/MDEMUX) and an FMPC, as shown in Fig. 1. MMUX or MDEMUX is used for the mode conversion between the fundamental modes in different single-mode fibers (SMFs) and the higher-order modes in the FMF, such as mode-selective couplers (MSCs) and mode-selective photonic lanterns (MSPLs)^[20–23]. The FMPC is made by winding the FMF onto three paddles and is used to control the spatial distribution and state of polarization (SOP) of the LP mode field. Each paddle of the FMPC can be adjusted in the range of −π/2≤θ≤π/2.

For each mode channel, the relationship between the output and input fields can be expressed through the following transfer matrix: Eout=TMDEMUX·TFMPC·TMMUX·Ein,(1)where TMDEMUX, TFMPC, and TMMUX are the transfer matrices of the MDEMUX, FMPC, and MMUX, respectively.

Then, we give the transfer matrix of the three-paddle FMPC, which is dependent on the rotation angle of each paddle and the corresponding fiber loop count. Here, we focus on the noncircularly symmetric LPlm (l≠0) mode, in which the subscripts l and m denote the transverse and radial indices, respectively. Taking the LP11 mode as an example, we analyze the influence of the three-paddle FMPC on the output spatial distribution and SOP. According to the four orthogonal mode bases of the LP11 mode^[24–26], the transfer matrix of the three-paddle FMPC is as follows: AoutT=M3M2M1AinT,(2)where AinT and AoutT are, respectively, the input and output complex amplitudes of the LP11 mode on four orthogonal mode bases (Aex, Aox, Aey, Aoy), and Mj (j=1, 2, 3) is the transfer matrix of each paddle. Specifically, Mj=R−1(θ)D(φ)R(θ), in which R(θ) is the mode rotation matrix of one paddle, R−1(θ) is the inverse matrix of R(θ), and D(φ) is the corresponding phase shift matrix, as follows^[24,25]: R(θ)=[cosθcosθsinθcosθcosθsinθsinθsinθ−sinθcosθcosθcosθ−sinθsinθcosθsinθ−cosθsinθ−sinθsinθcosθcosθsinθcosθsinθsinθ−cosθsinθ−sinθcosθcosθcosθ],(3)D(φ)=e−jφ4[e−jΔφ10000e−jΔφ20000e−jΔφ300001],(4)where Δφi (i=1, 2, 3) is the phase shift difference between four mode bases, and the common factor e−jφ4 can be omitted.

For an ideal LP11 MSPL demultiplexer, according to coupled-mode theory^[27], the even-mode (LP11e) and odd-mode (LP11o) components of the LP11 mode are, respectively, demultiplexed to the outputs of Ports B (LP11e-port) and C (LP11o-port).

In what follows, we first simulate the PDL of the LP11-mode channel through a homemade FMPC, whose states depend on the phase shift difference and rotation angle of each paddle. Strictly speaking, PDL is defined as the maximum to minimum transmission ratio (Tmax/Tmin) of a device or system when the input SOPs are ergodic, which is also given in dB, as follows^[18]: PDL=10log(Tmax/Tmin).(5)

For the MDM system based on LP11 modes, the PDL can be rapidly obtained by changing the linear polarization angle α of the input light. The homemade FMPC consists of three paddles, and the diameter of each paddle is 27 mm. The FMF used here has the core and cladding diameters of 19.4 and 125 µm, respectively, with a relative refractive index difference of ∼3.45×10−3. Thus, the effective refractive indices (neff) of the four LP11 mode bases in the bending FMF can be calculated by COMSOL Multiphysics software. The calculation results are listed in Table 1, in which the effective refractive index difference (Δneff) of each mode base relative to the LP11oy mode base is also given. Thus, the phase shift difference (Δφ) between four LP11 mode bases can be calculated by^[28]Δφ=2πLΔneff/λ,(6)where L represents the bending length of the FMF on each paddle and λ is the wavelength of 1550 nm.

Thus, the phase shift differences Δφi induced by one FMF loop are 8.44π, −0.17π, and 8.28π for the first three LP11 mode bases, respectively. The numbers of the FMF loops on the three paddles are, respectively, 2, 4, and 2 for the homemade FMPC. Then, we can adjust the rotation angles of the three paddles to change the percentage of these mode bases. Here, two typical FMPC states with the rotation angles of θA=[π/3,−4π/9,−5π/18] and θB=[π/6,−4π/9,π/4] are considered. In our investigation of the fundamental MDM system, the MMUX and MDEMUX pairs are always aligned in the spatial domain. In other words, the introduction of the FMPC gives rise to the PDL.

Under the two typical FMPC states of θA and θB, as the linear polarization angle α input to the FMPC is changed, the output modal pattern varies in turn, as shown in Figs. 2(a) and 2(b). The corresponding output power percentages of the LP11 mode bases are given in Figs. 2(c) and 2(d). The LP11 mode is then converted to the output ports (LP11e- and LP11o-ports) through the MSPL demultiplexer, and the power transmission dependent on α is shown in Figs. 2(e) and 2(f). From Fig. 2(a), with θA, the spatial orientation of the output LP11 modal pattern varies significantly with α, and the orientation angle change is up to ∼84°. In contrast, for the case with θB, the spatial orientation of the LP11 modal pattern has a small dependency on α, with a maximum orientation angle change of ∼7°. According to the power transmission curves shown in Figs. 2(e) and 2(f), we can obtain the corresponding PDL. For the case with θA, the output PDLs of the LP11e- and LP11o-ports are 9.93 and 8.58 dB, respectively. Accordingly, those of the case with θB are, respectively, 3.19 and 1.06 dB, and the transmission power of the LP11o-port is greater than that of the LP11e-port. It is shown that the PDL of the LP11-mode channel can be reduced by appropriately adjusting the FMPC in the MDM system.

Then, we build up a fundamental MDM system with two MSPLs connected by the homemade FMPCs to measure the PDL of the LP11-mode channel at 1550 nm, as shown in Fig. 3. FMPC1 and FMPC2 are, respectively, used to control the LP11 modal pattern and optimize the performance of the MSPL demultiplexer. The detailed PDL measurement process is as follows:

(1) A 1550 nm continuous-wave (CW) beam generated by the tunable laser source (TLS) is input to the electric polarization controller (EPC) (PSY-101, General Photonics, USA) to produce the linearly polarized light with an azimuthal angle of 0°; (2) the linearly polarized light is input to the LP11o-port of the MSPL multiplexer (LPMUX3, Modular Photonics, Australia) to excite an LP11 mode, and then FMPC1 is used to ensure the desired LP11 modal pattern. The output modal pattern can simultaneously be observed by the near-field beam profiler (CinCam CMOS-1201 IR, CINOGY, Germany); (3) the LP11 mode is input to the MSPL demultiplexer through FMPC2, and then the output power of the LP11e- and LP11o-ports of the MSPL demultiplexer is measured using an optical power meter (OPM), in which FMPC2 acts as a compensation device for the MSPL demultiplexer to guarantee the maximum power output from the LP11o-port; (4) for a given FMPC1 state, we only adjust the input linear polarization angle α via the EPC, and record the modal pattern output from FMPC1 and the output power of the MSPL demultiplexer; (5) repeat steps (2) and (4) for different FMPC1 states to analyze the PDL of the LP11-mode channel.

Here, we present two groups of experimental data for the LP11 modal pattern under two different FMPC states, as shown in Fig. 4. From Figs. 4(a) and 4(c), for the first state (θ1), the spatial orientation angle change is up to ∼80°, and the power fluctuations (i.e., PDL) of the LP11e- and LP11o-ports are 13.59 and 15.07 dB, respectively. As for the second state (θ2), there is a maximum spatial orientation angle change of ∼25°, and the PDLs of the LP11e- and LP11o-ports are 0.67 and 9.46 dB, respectively. The experiments show that the PDL of a mode channel is associated with the orientation angle of the modal pattern; namely, it is important for achieving low PDL to keep a fixed modal pattern, which can be achieved by appropriately adjusting the FMPC in the MDM system.

We investigate the transmission of a DP signal composed of two orthogonal polarization components in the fundamental MDM system by simulation and experiment. The generation of PDL in the transmission link will incur the polarization power imbalance and/or degrade the orthogonality of polarization^[18,19]. Here, we focus on the polarization power difference (ΔPxy) between two components of the DP signal after MDM transmission, which is defined as the power ratio between the x- and y-polarization components as follows^[18]: ΔPxy=|10log(Px/Py)|,(7)where Px and Py represent the powers of the x- and y-polarization components, respectively. The modal pattern change in the MDM system may lead to the fluctuation of ΔPxy, which has a great influence on the DSP demodulation of the DP signal.

Here, we assume that the DP signal is always of polarization orthogonality in the MDM system, and the ΔPxy is mainly determined by the PDL performance of the mode channel. Figure 5 simulates the power transmission of the x- and y-polarization components output from LP11o-port of the MSPL demultiplexer under the two typical FMPC states (θA and θB). From Fig. 5, the x- and y-polarization components have a contrary transmission characteristic. Under the two typical FMPC states of θA and θB, the maximum ΔPxy is, respectively, up to 8.58 and 1.06 dB, equal to the PDL of the mode channel.

The above analysis shows that the output from the MSPL demultiplexer is dependent on the relative orientation angle of the modal pattern. In other words, ΔPxy is related to the relative orientation angle β between their corresponding modal patterns. Thus, we can make use of the relationship between ΔPxy and β to experimentally verify the simulation results. Assuming that the x-polarization component always corresponds to the standard LP11e mode (represented by LP11ex), we can adjust the relative orientation angle β of the modal pattern corresponding to the y-polarization component (represented by LP11βy). In this case, the modal field of LP11βy can be regarded as the combination of the LP11ey and LP11oy modes, as follows^[23]: LP11βy=LP11eycosβ+LP11oysinβ.(8)

When the LP11e-port of the MSPL demultiplexer is used as the target output, only the power component on the LP11e mode can be output. Thus, according to Eq. (7), the ΔPxy dependent on β can be expressed as ΔPxy=10log(1/cos2β).(9)

Figure 6 shows the power transmission of x- and y-polarization components and the corresponding ΔPxy−β curve for the LP11e-port of the MSPL demultiplexer. It is clear from Fig. 6 that β should be sufficiently small to obtain a low ΔPxy.

On the basis of the experimental setup with the generation of DP light, as shown in Fig. 3, we measure the curve of ΔPxy dependent on β at 1550 nm to compare with the corresponding simulation. The detailed test process is as follows:

(1) Two 1550 nm CW lasers are used to generate DP light through variable optical attenuators (VOAs), single-mode polarization controllers (SMPCs), and a polarization beam combiner (PBC). VOAs are used to equalize or switch the powers of two polarization components output from PBC; (2) the generated DP light is input to the LP11o-port of the MSPL multiplexer to excite the LP11 mode, and then the FMPC1 state is adjusted to obtain the desired output modal pattern corresponding to two polarization components. The orientation angle of each modal pattern is determined by the line linking two light spots of the LP11 mode, and then the relative orientation angle β between two modal patterns is obtained; (3) the DP light is output from the LP11o-port of the MSPL demultiplexer, and then the optical powers of two polarization components are measured to further calculate ΔPxy, in which the power transfer matrix of the MSPL demultiplexer can be optimized by FMPC2 to align with the modal pattern of the x-polarization; (4) repeat steps (2) and (3) to obtain different relative orientation angles β and the corresponding ΔPxy.

Figure 7(a) illustrates two groups of LP11 modal patterns for the DP light (DP1 and DP2) and corresponding x- and y-polarization components when β≈4°, 30°, 45°, 65°, and 86°, respectively. The two groups of modal pattern diagrams are obtained by interchanging the x- and y-polarization components input to the MSPL multiplexer. It can be observed from Fig. 7(a) that the mutual exchange has no influence on β. Then, we measure the corresponding ΔPxy output from the LP11o-port of the MSPL demultiplexer, as shown in Fig. 7(b). For comparison, the theoretical curve of ΔPxy=10log(1/cos2β) is also plotted in Fig. 7(b). It can be clearly seen that the measured ΔPxy is very close to the theoretical curve in the range of ΔPxy≤6.04dB (β≤65°) of interest in practice. Seen from Fig. 7, when β is ∼86°, the measured ΔPxy are 11.83 and 10.42 dB for DP1 and DP2, respectively, which are lower than the theoretical values because the nonideal MSPL is used in the experiment. Therefore, it is very necessary for MDM transmission of DP signals to ensure their modal pattern consistency.

We discuss the PDL characteristic of the MDM system with a nonideal MSPL demultiplexer. For a nonideal MSPL demultiplexer with modal XT, the power transfer matrix between the output mode ports [PaPbPc]outT and input modes [PaPbPc]inT can be expressed in the form as follows: [PaPbPc]out=[ηaaηabηacηbaηbbηbcηcaηcbηcc][PaPbPc]in,(10)where ηij is the transfer efficiency between input mode (j=a, b, c) and output mode ports (i=a, b, c).

When a single LP11-mode DP signal with the relative orientation angle β is input to the nonideal MSPL demultiplexer, the input normalized power matrices of the x- and y-polarization can be, respectively, expressed as [010]inT and [0cos2βsin2β]inT. From Eq. (10), the normalized powers output from Port B of the MSPL demultiplexer for the x- and y-polarization are ηbb and ηbbcos2β+ηbcsin2β, respectively. In this case, Eq. (7) can be modified as ΔPxy=10log[1/(cos2β+ηbcηbbsin2β)],(11)where the factor of XT11=ηbc/ηbb represents the contribution of intra-LP11 modal XT. For an ideal MSPL demultiplexer, ηbb=1 and ηbc=0, Eq. (11) reduces to Eq. (9). For the nonideal MSPL demultiplexer with XT11=−12.55dB, the corresponding ΔPxy−β curve is also illustrated in Fig. 7(b). It can be seen that our experimental data are closer to the theoretical results with XT11. That is, the nonideal MSPL demultiplexer leads to the reduction of ΔPxy for large β.

When two DP signals of LP01 and LP11 modes are input to the MSPL demultiplexer, their input power matrices can be represented by [110]inT and [1cos2βsin2β]inT, respectively. In this case, the powers output from Port B of the MSPL demultiplexer include the extra contribution of the LP01 mode due to the LP01 to LP11 crosstalk, XT10=ηba/ηbb, and the ΔPxy dependent on β can be further modified as ΔPxy=10log[(1+ηbaηbb)/(cos2β+ηbcηbbsin2β+ηbaηbb)].(12)

Figure 7(b) plots the case with XT11=XT10=−12.55dB. It can be seen that ΔPxy at high β will further decrease in the presence of modal XT. According to the above analysis, the intra- and inter-mode XT will reduce PDL. On the other hand, the introduction of modal XT must degrade the performance of the DP signal. So, in practice, the bit error rate (BER) of a DP signal cannot be determined only through the PDL parameter of the MDM transmission system^[17,19].

In summary, the PDL performance of the fundamental MDM system is studied theoretically and experimentally. The impact of the FMPC state on the PDL of the LP11-mode channel is investigated. It is shown that the PDL of a mode channel is associated with the orientation angle change of the modal pattern, which can be characterized by the relationship between the polarization power difference ΔPxy and the relative orientation angle β of the modal pattern for the DP signal ΔPxy=10log(1/cos2β). In other words, two polarization components of a DP signal should have as identical a modal pattern as possible in the MDM link for low PDL. The influence of modal XT for the nonideal MSPL demultiplexer on PDL characteristics is also studied, which shows that ΔPxy can be reduced in the cases of intra- and inter-mode XT. Our theoretical and experimental results can guide PDL optimization of DP signals in MDM transmission.