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Abstract
The linear polarized (LP) mode multiplexer based on the inverse designed multi-plane light conversion (MPLC) has the advantages of low insertion loss and low mode crosstalk. However, the multiplexer also requires the fabrication and alignment accuracy in experiments, which have not been systematically analyzed. Here, we perform the error tolerance analysis of the MPLC and summarize the design rules for the LP mode multiplexer/demultiplexer. The error tolerances in the fabrication process and experimental demonstration are greatly released with proper parameters of the input/output optical beam waist, the pitch of optical beam array, and the propagation distances between the phase plane. To proof this design rule, we experimentally demonstrate the LP mode multiplexer generating LP01, LP11a, LP11b, LP21 modes and coupling to the few mode fiber, with the insertion loss lower than -5 dB. The LP modes are demultiplexed by MPLC, with the crosstalk of different mode groups lower than -10 dB. LP modes carrying 10 Gbit/s on-off keying signals transmit in a 5 km few mode fiber. The measured bit error rates (BER) curves of the LP01, LP11a, LP21 modes have the power penalties lower than 12 dB.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the explosive growth of network traffic, single-mode fiber is approaching its capacity limit [1,2]. Traditional multiplexing communication technologies, such as amplitude, phase, polarization, and wavelength division multiplexing will be exhausted in the near future. There is much interest in utilizing mode division multiplexing (MDM) as a multiplexing technique to further increase the capacity of optical fiber communication systems [3–5]. MDM uses multiple orthogonal modes to increase the parallelism of the communication channels. Each orthogonal mode carries an independent data channel, thus greatly increase the capacity of the fiber without increasing the number of fibers [6]. There are several sets of orthogonal modes used in the MDM, such as linear polarization (LP) mode [7–9], orbital angular momentum (OAM) mode [10–13] and cylindrical vector beam (CVB) mode [14–16]. The LP modes are widely used in the few-mode fiber communication system, while the OAM and CVB modes need specially designed ring-core fiber. For the large channel number of mode multiplexing, the LP modes have more degenerated modes than the OAM or CVB (4 degenerated modes for all the orders), which require complex digital signal processing resources in the communication system. For MDM, one of the key elements is the multiplexer that can convert multiple independent input optical beams (Gaussian beams) into different coaxially propagating orthogonal modes, and the demultiplexer that performs the inverse operation. MDM multiplexer/demultiplexer should have low insertion loss and high channel isolation for optical communication [17–19].
In the past few years, some developments have been made in MDM multiplexing/demultiplexing. The phase plates or q-plate were used to generate or detect a single LP, OAM and CVB mode [20]. Each mode is generated by a specific phase plate, and the coaxial modes needs to be generated by a series of beam splitters with low efficiency. The vortex Dammann gratings were developed to generate or detect multiple OAM and CVB simultaneously [21]. However, the Dammann grating has many diffraction orders, which lead to low efficiency of the multiplexing. To further improve the efficiency of OAM demultiplexing, Gregorius et al. proposed an OAM sorting method based on geometric transformation [22–25]. By using the liquid crystal structures with birefringence, anisotropic optical geometric transformation has also been demonstrated in CVB demultiplexing [26,27]. The drawbacks of the optical geometric transformation are mainly the requirement of a 4f optical system and mode crosstalk. For the MDM multiplexer/demultiplexer, mode crosstalk, efficiency and miniaturization are the key concerns. Labroille et al. proposed a new type of multi-channel spatial mode multiplexer, based on multi-plane light conversion (MPLC), with low loss and high mode selectivity [28,29]. Fontaine et al. proposed an improved MPLC to achieve 210 mode multiplexing by 7 phase planes loaded on the spatial light modulator [30]. It is possible to optimize a LP multiplexer/demultiplexer with low mode crosstalk, high efficiency using MPLC [31,32].
In this paper, we optimize the LP mode multiplexer/demultiplexer performance by error tolerance analysis, and experimentally demonstrated the LP mode multiplexing optical communication based on inverse designed MPLC. Using an analytical method, we analyze the error tolerance of different parameters, including the input and output beam waist diameters, the distance between light source and the first phase plane, the spacing of phase planes, the shift of beam, and the shift of each phase planes. All these parameters are related to the LP mode conversion efficiency, mode crosstalk, and mode purity. We select the optimal parameter design and fabricate the LP mode multiplexer/demultiplexer. In the experiment, 4 Gaussian beams are converted into coaxial LP mode based on MPLC, with the device loss lower than -3 dB and the mode crosstalk lower than -21 dB. The total insertion loss of the coaxial LP modes coupled to the few mode fiber is lower than -5 dB, including the device loss and the coupling loss of the few mode fiber. In the communication experiment, we demonstrate 3 channels multiplexing each carrying 10 Gbit/s OOK signal in a 5 km few mode fiber using the LP mode multiplexer/demultiplexer. The LP mode multiplexer/demultiplexer based on the inverse design MPLC shows potential application in the next generation high capacity optical fiber communication.
2. Principle and error tolerance analysis of MPLC
The mode multiplexer converts multiple channels of independent input Gaussian beams into corresponding high order coaxial LP modes, and the demultiplexer performs in the inverse way. There is no analytic function expression for MPLC design using the wavefront matching method. The wavefront matching algorithm means that the forward propagating light fields match with the backward propagating light fields at any position in the space. Therefore, the phase distributions of the phase plane equal to the superposition of the conjugate input light fields and the output light fields in the phase plane. The input light fields and output light fields can be obtained by the angular spectrum algorithm. The phase plane is updated by the superposition of the conjugate incident light fields and the output light fields, and the iterative calculation is repeated until the algorithm converges to obtain the phase distribution of the phase plane. Figure 1 (a) shows the schematic of LP mode multiplexer based on the inverse designed MPLC. The light beams from the fiber array are collimated by the microlens array. Four collimated Gaussian beams input to the phase masks from different positions, and convert into four different orders of LP modes for coaxial transmission. The beam waist diameter in the first phase plane is defined as the input beam waist (win), and the beam waist diameter in the last phase plane is defined as the output beam waist diameter (wout).
Fig. 1. (a) Schematic of LP mode multiplexer based on the inverse designed MPLC; FA, fiber array; MA, microlens array; Δm1, Δm2, Δm3, Δm4, The displacements of input beam; win, the waist diameter of input beam; wout, The waist diameter of output beam; d1, The distance between light source and the first phase plane; d2, d3, d4, The distance between the phase planes; Δx1, Δx2, Δx3, Δx4, The displacements of the phase planes. (b) Calculated intensity profiles of the LP modes, and (c) the phase of LP modes generated by the inverse design MPLC. (d) Calculated intensity profiles of the LP modes, and (e) the phase of LP modes generated by the inverse design MPLC with the propagation distance errors for d2, d3, d4 of 1 mm and alignment error for Δm1, Δm2, Δm3, Δm4 of 20 μm.
To design the LP mode multiplexer/demultiplexer based on the inverse designed MPLC, we need to determine the input light fields and the output light fields, and then choose the parameters properly including: (1) The beam waist diameter of the input Gaussian beam (win); (2) The input fiber array pitches; (3) The number of phase planes (P1, P2, P3, P4); (4) The distance between light source and the first phase planes (d1); (5) The spacing of the phase planes (d2, d3, d4); (6) beam diameter of the output beam (wout). We choose the initial spacing of the phase planes to match the propagation distance for the input beam diverging to the output beam. Therefore, the phase planes do not need to have additional lens phase for beam focusing. Based on this initial value, we further optimize the spacing parameters to improve the performance with the algorithm. In the inverse design, we calculate the forward and backward propagation of the beams according to the above parameters and obtain the optimized phase distribution of multiple phase planes by the wavefront matching algorithm. We firstly calculate the forward propagation of the light field from the input mode into output mode through random phase distribution of multiple phase planes. Then, we calculate the backward propagation of the light field, and obtain the corresponding phase distribution of multiple phase planes. Then the phase planes are updated by the superposition of the conjugate forward propagation of the light field and the backward propagation light field. Repeat this iterative calculation until the phase distribution of the phase planes converge. Figure 1 (b) and (c) show the calculated intensity and phase profiles of the LP modes generated by the inverse designed MPLC. The LP multiplexer/demultiplexer require high fabricating accuracy and experimental alignment accuracy. With the error of imperfect fabrication and misalignment, the generated LP modes intensity and phase will show worse qualities compared with error free case. Figure 1 (d) and (e) show the intensity profiles and phase of the LP modes, when the errors of d1, d2, d3, d4 are 1 mm, and the errors of Δm1, Δm2, Δm3, Δm4 are 20 μm. Therefore, it is necessary to improve the error tolerance of the MPLC in the design.
We then analyze the error tolerance of the parameters in the LP mode multiplexer design. Insertion loss is defined as the average power loss of all the generated modes by the MPLC device. For each LP mode generated from MPLC, the overlap ${\delta _{xx}}$ between the normalized produced mode $w_p^{}(\overrightarrow r )$ and a theoretical mode ${w_{L{P_{xx}}(\overrightarrow r )}}$ is given by [28]:
The overlap between the generated LP mode and the theoretical LP mode results in mode purity, while the overlap with the other theoretical LP modes of the MPLC results in mode crosstalk. We calculate the correlation value between the generated LP modes and the theoretical LP modes, and extract the insertion loss and mode crosstalk. When the input beam waist diameter is from 60 μm to 160 μm with a fixed output beam waist diameter of 400 μm, we calculate the insertion loss (Fig. 2 (a)) and mode crosstalk (Fig. 2 (b)) depending on the propagation distance error of d1. As shown in Fig. 2, the input beam with a large waist diameter has better error tolerance in the insertion loss and crosstalk. The input beam with a small beam waist diverges fast with propagation, therefore the propagation distance error will induce significant misalignment between the incident beam and the phase mask. As a result, the MPLC with a small input beam waist is sensitive to the propagation distance parameters, and suffers the error induced insertion loss and crosstalk. The beam waist diameter of the input beam is determined by the focal length of the microlens array. When the input beam diameter is fixed, we should choose the output beam waist as the calculated beam diameter after propagating through the MPLC.
Fig. 2. The error tolerance analysis for the input beam with different diameters from 60 μm to 160 μm. With a fixed output beam of 400 μm, we calculate the (a) insertion loss and (b) mode crosstalk depending on the propagation distance error of d1.
We choose the input beam waist diameter as 110 μm and the output beam waist diameter as 400 μm. The distance between the light source and the first phase plane is set as 16 mm and the spacing between the 4 phase plane is set as 20 mm. The input 4 Gaussian beams array with a pitch of 250 μm are converted into coaxial output modes of LP01, LP11a, LP11b and LP21. After calculation, we get multiple phase planes for the LP mode multiplexing. In the experiment, the alignment error of the beam and the phase plane will increase the insertion loss and mode crosstalk. The mode dependent loss is defined as the largest loss different among all the LP modes, which is typically between the LP01 and LP21. Figure 3 (a) shows the variation of insertion loss, mode dependent loss and mode crosstalk under different propagation distance errors of d1. Figure 3 (b) shows the variation of insertion loss, mode dependent loss and mode crosstalk with the error of d2, d3 and d4. In the design of LP mode multiplexer, we mainly consider the insertion loss and mode crosstalk. Therefore, we set the light source position offset (Δm) and calculate the insertion loss (Fig. 3 (c)) and mode crosstalk (Fig. 3 (d)). The simulation results show that the MPLC performances are not sensitive to the light source position offset within 10 μm. The change of the phase plane position (Δx1, Δx2, Δx3, Δx4) will also induce additional insertion loss and mode crosstalk. We calculate the insertion loss (Fig. 3 (e)) and mode crosstalk (Fig. 3 (f)) of the LP mode when there is an error in one of the four phase planes spacing. The calculation results show the MPLC can still achieve an insertion loss less than -3 dB and a crosstalk better than -20 dB, when there is a phase plane position offset within 30 μm. We also calculate the insertion loss (Fig. 3 (g)) and mode crosstalk (Fig. 3 (h)) of the LP modes when there are errors in two of the four phase planes. It is important to do the error tolerance analysis of the MPLC design to ensure the performance in the experiments.
Fig. 3. The error tolerance analysis of the light propagation and alignment parameters. The insertion loss, mode purity, mode dependent loss and crosstalk with a propagation distance error of light source (a) and phase plane (b). The insertion loss (c), crosstalk (d) and mode purity (e) with misalignment of the light source. The insertion loss (f), crosstalk (g) and mode purity (h) with misalignment of a phase plane. The insertion loss (i), crosstalk (j) and mode purity (k) with misalignment of two phase planes.
3. Experiment results
We experimentally demonstrate the LP mode multiplexing based on inverse design MPLC devices. Figure 4 (a) shows the schematic of the LP mode multiplexer. We use a reflective devices and mirror to build the MPLC setup. The input consists of a fiber array and a microlens array to generate collimated four Gaussian beams. Four Gaussian beams incident to the first phase plane with a small angle. By carefully controlling the incident angle and the positions of the phase mask and mirror, the light beams aligned with the four phase planes through the propagation between the phase mask and the mirror. Four Gaussian beams are converted to multiple coaxial LP mode, and efficiently coupled to the few mode fiber by graded index lens (G-lens). Figure 4 (b) show the calculated images of the phase planes. The phase mask is fabricated on the silicon wafer using photo lithography and dry etching. By 3 times of photo lithography and dry etching, we fabricate the phase mask with 8 phase levels between 0 and 2π for 1550 nm wavelength reflective devices, corresponding to a etch depth between 0 and 775 nm. Figures 4 (c) show the corresponding microscopic images of the fabricated devices. The pixel size of the phase mask is 8 μm. The surface of the phase masks is coated with gold to improve the reflectivity.
Fig. 4. (a) The LP mode multiplexer. (b) Calculated images of the phase masks. (c) Microscopic image of the fabricated phase masks. (d) The intensity profiles of the LP modes generated by the MPLC in the experiment. (e) LP mode interferes with Gaussian mode. (f) The intensity profiles of LP modes are restored by off-axis holography. (g) The phase of LP modes are restored by off-axis holography. (h) Measured mode crosstalk matrix for the LP modes.
In the implementation of MPLC mode multiplexer, the reflective devices have the advantages of low loss and more compact size compared with the transmitting devices. Table 1 shows the error tolerance analysis for the reflective MPLC devices. We consider the error of the etching depth for the 8 steps phase mask in the fabrication process. The misalignments of the optical components are estimated by the parameters of the translational stages used in the experiments. The reflection loss of the thin film gold coating on the devices are also included. By the tolerance analysis of all the parameter errors in the practical fabrication facility and experiment setup, the designed MPLC is proved to be robust for the real demonstration with good performance.
Table 1. Error tolerance analysis for MPLC
Figure 4 (d) shows the intensity profiles of the LP modes generated by the MPLC in the experiment. In the experiment, the insertion loss of Gaussian mode converted to LP01, LP11a, LP11b, LP21 mode are measured as -2.70 dB, -2.64 dB, -2.87 dB and -2.55 dB, respectively, which includes the loss in the phase masks and the mirror. From the intensity profiles of the LP modes, the experimental results are consistent with the calculation. We reconstruct the LP mode phase by off-axis holography and analyze the LP modes crosstalk. Figure 4 (e) shows the LP mode interference patterns with Gaussian beams. Figure 4 (f) and (g) shows the intensity and phase profiles of LP modes reconstructed by off-axis holography. The phases of the LP modes are consistent with the calculated phase of the LP mode. The amplitude and phase information of each mode output from MPLC is obtained by digital holography method. Then, correlation values are extracted by calculating the coupling relationship between the modes. The correlation values of different modes are defined as mode crosstalk [28,30]. By the correlation among all the generated LP modes, we extract the LP mode crosstalk matrix for the LP modes as shown in Fig. 4 (h). The mode crosstalk of LP modes generated by MPLC are lower than -21 dB, mainly from the imperfect fabrication and misalignment in the experiment. The phase masks of the MPLC with more steps of phase can further reduce the mode crosstalk of LP modes. In the experiment, we couple the LP mode to the few-mode fiber by the G-lens. The total losses of LP01, LP11a, LP11b and LP21 modes are -4.03 dB, -4.30 dB, -4.15 dB and -4.21 dB, respectively, including the device loss and coupling loss.
MPLC can also be used in LP mode demultiplexing. MPLC is an inverse device and can convert the input LP modes into separated Gaussian modes. In our communication system, we directly used the same MPLC as demultiplexing. In the fiber transmission, the LP modes will couple to degenerated modes and induce additional loss and crosstalk at the demultiplexing part compared with the multiplexer part. In the follow work, we will study the LP mode group demultiplexer based on MPLC. The multiplexed coaxial LP mode is output from the G-lens and then demultiplexed by the MPLC. Figure 5 (a) shows the experimental setup of the LP modes multiplexing-based fiber communication using the MPLC. A 1550 nm laser beam is modulated to carry a 10 Gbit/s on−off keying signal from a pattern generator, and then split into three branches by a coupler. There is a 10 m length difference between the three branches to ensure the three signals are unrelated. Then the three channels are converted to multiplexed coaxial LP modes with orders LP01, LP11a and LP21 using the MPLC. The multiplexed coaxial LP mode is coupled to a commercial step-type few-mode fiber for 5 km transmission. The core diameter of the commercial few-mode fiber is 18.5 μm, which supports the transmission of LP01, LP11, LP02 and LP21. The 5 km transmission distance is limited by the available few mode fiber length in our lab. The FMF fiber indeed introduce extra crosstalk among the LP modes and will finally limit the transmission distance. The phase of the LP modes output from the few-mode fiber is matched with the phase of the demultiplexer, by adjusting the polarization controller. The LP modes are demultiplexed by the MPLC and coupled into single mode fiber array for bit error rates detection. The coaxial LP mode is demultiplexed by MPLC with the insertion loss of -5.59 dB, -6.84 dB, -6.75 dB, and -7.22 dB for LP01, LP11a, LP11b, and LP21 respectively. Figure 5 (b) shows the mode crosstalk matrix of the LP modes multiplexing/demultiplexing based on the inverse designed MPLC. The crosstalk of Fig. 5(b) includes multiplexing crosstalk, optical fiber transmission crosstalk and demultiplexing crosstalk. The crosstalk performance can be improved by using weakly coupled few mode fiber and mode group demultiplexer. The crosstalk between the adjacent LP modes is around -10 dB, which mainly comes from side lobes. LP11a and LP11b are degenerated modes in the same mode group with similar effective refractive index and strong coupling. Therefore, we choose LP01, LP11a, LP21 modes as three data channels for LP modes multiplexing-based fiber communication. Figure 5 (c) shows the measured BER curves of the LP01, LP11a, LP21 modes with a power penalty of 12 dB. Figure 5 (d) to (f) show the eye diagrams of LP01, LP11a, and LP21, respectively.
Fig. 5. (a) Experimental setup of LP mode multiplexing-based fiber communication using the MPLC, SMFA, single mode fiber array. (b) Mode crosstalk matrix of the LP modes demultiplexing based on the inverse designed MPLC. (c) Measured bit error rates of the multiplexed coaxial LP modes channels in few mode fiber communication. The eye diagram of (d) LP01, (e) LP11a, (f) LP21.
4. Conclusion
In summary, the LP mode multiplexer based on the inverse design MPLC calculates the forward and backward transmission of the beam according to the wavefront matching algorithm, and obtains the optimized phase masks after multiple iterations. We have carried out error tolerance analysis on LP mode multiplexer/demultiplexer, which has guiding significance for device design based on inverse design. The LP mode multiplexer/demultiplexer is fabricated with optimized parameters and shows excellent performance in terms of insertion loss and mode crosstalk, which greatly increases the fiber capacity in the communication experiment. The MPLC device design method based on inverse design is also suitable for the optimization of OAM and CVB multiplexer/demultiplexer. In the follow work, we will change the output mode to OAM/CVB, and obtained OAM/CVB multiplexer/demultiplexer through parameter optimization. The error tolerance analysis of the MPLC and inverse design provide new method for the design and research of mode multiplexer/demultiplexer, which provide the optimal solution for overcoming the limitation of fiber capacity.
Funding
National Key Research and Development Program of China (2018YFB1801801); National Natural Science Foundation of China (11774240, 61935013, U1701661, U2001601); Science, Technology and Innovation Commission of Shenzhen Municipality (KQTD2015071016560101, KQTD20170330110444030).
Acknowledgments
The authors would like to acknowledge the Photonics Center of Shenzhen University for technical support.
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 reasonable request.
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