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Abstract
In this paper, a novel full vector numerical simulation method based on the finite element method (FEM) and local coupled mode theory (LCMT) for analyzing the mode transmission characteristics of photonic lantern (PL) with arbitrary input mode field is proposed. Compared with the traditional numerical simulation methods for PL, our method can greatly reduce the computational complexity and ensure high precision. Taking a three-core PL as an example, we verify the validity of our method. The advantages and properties of our method are also discussed in detail and found instructive for optimization design of PL. Through specifically optimizing the geometric parameters of the PL according to the properties, mode selectivity of LP01 and LP11 can be respectively improved up to 44.5 dB and 54.7 dB with more than 95% coupling efficiency.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the transmission capacity of optical fiber communication system based on single mode fiber (SMF) approaching its limits [1], mode division multiplexing (MDM) is regarded as the most effective technique to increase the capacity [2–4]. In MDM system, all-fiber photonic lantern (PL) based multiplexer/demultiplexer has attracted worldwide attention due to its low insertion loss (IL), high mode selectivity and low mode dependent loss (MDL) [5,6]. As a multiplexer/demultiplexer, PL can be divided into two types. The first type is mode selective photonic lantern (MS-PL) with diverse input fibers to excite one spatial mode at the tapered end when one core mode is input. The second type is mode-group selective photonic lantern (MGS-PL), which excites a superposition of degenerate modes within a mode group when one core mode is input. During the transmission through MDM system, the degenerate modes within the same mode group will strongly couple to each other due to fiber bending or twisting. Thus, the MGS-PL is more practical for MDM system to lower the complexity of digital signal processing (DSP) [7,8].
For few-mode fiber (FMF) and multi-mode fiber (MMF), the difference of effective indexes between mode groups is constant and larger than 10−4 through transmission by designing refractive index or geometric parameters of FMF/MMF [9]. But for MGS-PL, the difference of effective indexes between mode groups varies with taper position and will rapidly decrease at some specific positions, which will lead to strong coupling between mode groups and decrease of mode selectivity.
To design MGS-PL with high mode selectivity, low IL and low MDL, we need to analyze mode transmission characteristics and optimize all sorts of parameters in MGS-PL based on optimum core arrangement [10]. The most common numerical simulation method for MGS-PL is beam propagation method (BPM) [10–12]. BPM has great advantages to analyze complex tapered fiber devices such as mode selective coupler or PL. However, MGS-PL usually requires a relatively long taper length L which scales as $L \propto {N^2}$to reach adiabatic criterion, where N represents the number of cores in the un-tapered end [7]. Thus, the simulation result of BPM will accumulate large errors with the increase of taper length. Finite element method (FEM), as a very accurate full vector numerical simulation method, is often used to analyze mode characteristics of MGS-PL such as propagation constant, electromagnetic field distribution, dispersion and so on [5]. But high computational complexity prevents FEM from being applied to three-dimensional PL [13]. Due to the same reason, finite difference time domain (FDTD) method is also limited for simulation of MGS-PL. Another method is transfer matrix method whose computational complexity is relatively low, but the transfer matrixes between every two chips are difficult to get. Therefore, it is only suitable for simplified PL models [14].
In this paper, we propose a novel method based on FEM and local coupled mode theory (LCMT) to analyze mode transmission characteristics of MGS-PL with arbitrary input mode field. With low computational complexity and high precision, we can get various information of mode field at arbitrary position of PL like mode component, propagation constant, amplitude, phase and so on. Taking a three-core MGS-PL as an example, we demonstrate the validity and high efficiency of our method. The mode coupling process becomes clear by analyzing propagation constant and coupling coefficient curves between modes. To the best of our knowledge, it is the first time that the electric field distributions of local modes are found related to the coupling strength between modes, and are specifically adjusted for optimization design of photonic lantern. We believe that the advantages and special properties of our method will contribute to a more comprehensive understanding of the coupling process in photonic lantern.
2. Principles and methods
We take a three-core MGS-PL as an example to show how LCMT solves the transmission process along the PL. The structure of the three-core MGS-PL is shown in Fig. 1(a). Three SMFs in a low-index capillary are tapered adiabatically and spliced to a FMF. One bigger core and two smaller cores are selected to break the degeneracy of the mode groups throughout the entire taper [15]. According to optimum core arrangement [10], three cores should be placed in a triangle pattern. Thus, the cross section of the PL can be approximately regarded as three isolated cores with a circular cladding and a low-index capillary, as shown in Fig. 1(b).
LCMT is usually applied to analyze optical fiber devices whose refractive index varies along the propagation direction [16]. Although there is no orthogonal eigenmodes transmitted stably in PL, LCMT can be used to analyze mode transmission characteristics when it meets adiabatic criterion.
If we excite a local mode p whose amplitude is Ap at the un-tapered end of the PL, then the coupled mode equation between local modes can be described as Eq. (1),
where Cpq is the coupling coefficient between local mode p and q, as shown in Eq. (2),The schematic diagram and flow chart of our method for the PL is shown in Fig. 2. Firstly, we take n evenly spaced cross sections along the taper (un-tapered end and tapered end must be included). Secondly, m eigenmodes are solved by FEM on each cross section, forming a m by n eigenmodes matrix. Then we use Eq. (2) to calculate coupling coefficient between any two eigenmodes and get a m by m by n matrix. Finally, in order to solve Eq. (1) by using the fourth-order Runge-Kutta method with small step size, a proper interpolation method is applied to the coupling coefficient matrix to get continuous coupling coefficient. If a PL meets the adiabatic criterion, the mode field and propagation constant will change slowly and there is no crossing of effective index between modes along the taper, which results in slow and smooth variation of the coupling coefficient [17,18]. Therefore, we can fit the coupling coefficient curves quite accurately by interpolating the discrete coupling coefficient matrix and don’t need to calculate eigenmodes on each step.
In order to ensure the feasibility of the LCMT applied to our PL model, adiabatic criterion is introduced to quantitatively characterize the slow change of the mode field of the taper [16], as shown in Eq. (3),
where $\overline{\delta\beta }_{pq}$ is the cumulative average difference of the propagation constants between local mode p and q, as shown in Eq. (4), As long as the adiabatic criterion is satisfied, the slow change of mode fields and coupling coefficient can be ensured, which guarantees the validity of our method.3. Method validation
In order to verify the validity of our method and explore how many cross sections need to be taken to interpolate accurate coupling coefficient curves, we simulate the PL with certain parameters. The interval between three cores is set to be 42 µm and the cladding diameter is 125 µm. The bigger core with 11 µm diameter corresponds to LP01 mode multiplexing, while the two smaller cores with 7 µm diameters correspond to LP11 mode multiplexing. The refractive indexes of cores, cladding and low-index capillary are 1.4482, 1.444 and 1.4398, respectively. The taper ratio is 0.112 with 2 cm taper length.
Firstly, we take 1000 evenly spaced cross sections on the PL and calculate coupling coefficient between LP01 and LP11b on each cross section as standard values. Then we use different number of evenly spaced cross sections from 20 to 100 and three interpolation methods to interpolate the coupling coefficient to 1000 values. The goodness of fit, also known as R-squared, is shown in Fig. 3(a). The spline interpolation is obviously more suitable for PL than cubic and linear interpolation, which can be attributed to slow and smooth variation of the coupling coefficient. The illustration in Fig. 3(a) shows that the R-squared of interpolated values with spline interpolation only decreases rapidly when the number of cross sections is less than 30. Thus, we can take relatively small number of cross sections but still be able to interpolate the coupling coefficient curve quite accurately. The interpolated values of coupling coefficient with 20 and 100 cross sections are shown in Fig. 3(b). When the coupling coefficient changes dramatically, the interpolated values with 20 cross sections deviate more from the standard values than that of 100 cross sections. The average relative error for interpolated values with 20 cross sections is 0.61%, while only 0.091% for 100 cross sections. Therefore, 100 evenly spaced cross sections are taken to ensure the accuracy of interpolation in the following simulations.
Fig. 3. (a) Goodness of fit (R-squared) of interpolated values with different section number and interpolation method. (b) Coupling coefficient between LP01 and LP11b (Standard values and interpolated values with 100 and 20 cross sections).
After the coupling coefficient curve is obtained, the adiabatic criterion F(z) between LP01 and LP11b is also calculated and shown in Fig. 4(a). F(z) ≪ 1 is satisfied through the whole taper, which ensures the feasibility of LCMT for the PL. Then we calculate the LP11 mode power when excite normalized fundamental mode in core 1, as shown in Fig. 4(b). The result is compared with BPM result which is calculated by the overlap integral of the local mode profile with the BPM simulated transverse electrc field [19]. The results have some differences at peaks, and the mismatch occurs when power leaks from the core into the cladding, i.e., the field ceases to be negligible at the cladding-capillary interface, where there is an appreciable field variation over a large index step. Thus, the results will be different due to different fitting algorithms for the cladding diameter in two methods [20]. But at any other positions, the results are almost the same and prove the accuracy of our method.
Fig. 4. (a) The adiabatic criterion F(z) between LP01 and LP11b. (b) Calculated normalized LP11 mode power with LCMT and BPM.
4. Advantages and properties of the proposed method
Compared with traditional numerical simulation methods such as BPM and FDTD, our method has many advantages and special properties. In this section, we will state the advantages and properties in detail, and finally give the optimization process of the PL parameters based on these advantages and properties.
4.1 Less computational complexity and higher precision
For numerical simulation methods solving 3D waveguides, the grid size is usually required to less than one-tenth of the wavelength. Thus, although FDTD and FEM are accurate full vector numerical methods, a lot of computational resources consumed in the simulation prevent them from being applied to large size 3D models like PL. While BPM using paraxial scalar approximation is widely utilized in most of the references. But for BPM, it is also necessary to carry out computation in the whole cuboid region containing the PL, which results in a lot of redundant parts involved in the computation. On the other hand, with the decrease of the PL features, the initial cross section grid may be inadequate to be applied to the tapered end, which will result in the decrease of precision, as shown in the illustrations of Fig. 5(a). Even though this problem could be solved by interpolation, it would further contribute to the already excess computational time [21]. But for our method, the fine structure of the PL can be better divided by triangular adaptive meshes of FEM, and we can properly vary the computational grid synchronously with the decrease of the features along the PL, as shown in Fig. 5(b). Therefore, the redundancy of the computation is removed and the precision is guaranteed at the same time. Then the iteration of the electric field along the taper is replaced by the iteration of the complex amplitudes of local modes, so we can use very small step size to reduce error and ensure the convergence of the results. Basically, for the 2 cm PL, if we use 100 cross sections to solve 20 eigenmodes on each of them, a total of 2 × 103 eigenmodes need to be solved by FEM. The coupled mode equations can be solved within a few minutes even if we use 1 nm step size. As for BPM, the electric field needs to be iterated at least 2 × 104 times through the whole taper (1 µm step size). In our actual simulation, a solution of an eigenmode with FEM took about 2 times longer than an iteration in BPM, so our method consumed about a quarter of the total time of simulation with BPM. Moreover, because the process of solving eigenmodes on cross sections is independent with the length of the PL, we can quickly simulate PLs with the same cross sections but different lengths after obtaining the eigenmodes.
Fig. 5. The schematic of the grids on un-tapered/tapered end and main computational domain for (a) BPM, and (b) the proposed method.
Especially in the simulation of PL made by step index fibers, we find that the computational complexity can be further reduced. The refractive index changes only at the circular boundary between core and cladding or between cladding and low-index capillary for step index PL. Then the refractive index distribution for the ith boundary can be expressed as Eq. (5),
where nin, Δi and ρi(z) are the refractive index inside the ith boundary, relative index difference and radius for the ith boundary, respectively. H is the Heaviside step function. r is the distance from the center of the ith boundary, where xi(z), yi(z) are the coordinates of the center of the ith boundary.Then the coupling coefficient [Eq. (2)] can be derived as Eq. (7),
4.2 Full vector simulation with arbitrary input mode field
For input mode field with arbitrary polarization, phase and amplitude, it can always be decomposed into a linear combination of eigenmodes on the un-tapered end. When a specific mode field is input, we actually input an array of eigenmodes. For example, if we input a mode consisted of two x-polarized eigenmodes with the same amplitude and different initial phase, as shown in Fig. 6(a), we can get different superposed output mode fields and their phase distributions, as illustrated in Fig. 6(c). The two local modes will accumulate phase difference during propagation due to the separation of their propagation constant curves and evolve into LP11b and LP11a, respectively, as shown in Fig. 6(b). For this PL, the accumulated phase difference is 4π/10.
Fig. 6. (a) Input two eigenmodes with initial phase difference $\Delta \mathrm{\varphi }$. (b) Effective index curves of local modes in the PL. (c) Output mode fields and phase distributions with different $\Delta \mathrm{\varphi }$.
It is already known that a linearly polarized OAM mode can be expressed as the combination of two LP modes [22]:$OAM_{ {\pm} l}^{(x,y)} = LP_{l,1}^{({x,y} )}a \pm iLP_{l,1}^{(x,y)}b$.When the sum of the phase difference accumulated between two local modes during propagation and the initial phase difference is ± (k+1/2) π, the output mode field will be a donut-like OAM mode. If the sum is ± kπ, the output mode field will be LP mode. Otherwise, it will be a mixed state. From Fig. 6(c), we can find that the mode field distributions of OAM modes are not perfectly circular symmetry and the phase distributions of LP modes are not perfectly axial symmetry. This indicates that the output mode is not pure due to the coupling between LP01 and LP11. It should be noted that although all the eigenmodes solved with FEM in this example are linearly polarized, they are not the same as scalar approximations of vector modes in a cylindrical fiber.
4.3 Clear coupling process
From the form of the coupled mode equation [Eq. (1)], the coupling strength between local modes is related to propagation constants and coupling coefficient. Figure 7 illustrates the reciprocal of the difference between propagation constants 1/Δβ(z), coupling coefficient curves C(z) and adiabatic criterion F(z) of LP01, LP11b and LP11a, respectively. A total of 10 local modes from LP01 to LP12 are calculated to ensure the accuracy. Due to the axial symmetry of the PL, electric field distributions of LPmna modes are always symmetrical along y-axis. Thus, coupling coefficient is only nonzero between LP0n modes and LPmnb modes or within LPmna modes. For the simplicity, only non-zero coupling coefficient curves are shown in the figure. The adiabatic criterion for all modes along the taper is far less than 1, indicating the 2 cm taper length is sufficient for our PL model.
Fig. 7. The reciprocal of the difference between propagation constants 1/Δβ(z), coupling coefficient curves C(z) and adiabatic criterion F(z) for (a) LP01, (b) LP11b and (c) LP11a with other modes.
After obtaining the coupling coefficient curves between all modes, the nontarget coupled mode power (crosstalk and loss) when the normalized fundamental mode is excited in core 1 or core 2/3 is calculated and shown in Fig. 8. The coupled LP11/LP01 mode power increases sharply around the 0.4 taper ratio, then drops and oscillate till the tapered end. The results from Fig. 7 and Fig. 8 shows that although 1/Δβ(z) and C(z) between many modes can reach a relatively high level in a certain period of the PL, the coupling between LP01 and LP11 is much stronger than between other modes. The only explain for this phenomenon is both of the 1/Δβ(z) and C(z) between LP01 and LP11b increase sharply around 0.4 taper ratio and result in the strong coupling. While only one of the factors can reach a high level between all other modes at a position of the taper. Thus, 1/Δβ(z) and C(z) are the two vital factors to decide the coupling strength between modes.
Fig. 8. The nontarget coupled mode power (crosstalk and loss) when the normalized fundamental mode is excited in core 1 or core 2/3.
We calculate the coupling efficiency of the PL which is defined as the ratio of output mode power at the tapered end over input mode power at the un-tapered end:
and the ratio of power in the desired mode group i over power in all other mode groups called mode selectivity:4.4 Specific optimization methods
Although we can do the simulation with arbitrary input mode field, the most common input mode for PL as multiplexer are the fundamental mode in one of the initial cores. Thus, the coupling between the first few order modes is essential. Given that the simultaneous increase of 1/Δβ(z) and C(z) will result in an increase of the coupling strength between LP01 and LP11, we can reduce either of them or both to improve the mode selectivity by designing the parameters of the PL. From the form of the coupling coefficient [Eq. (7)], C(z) is also proportional to 1/Δβ(z). In addition to it, C(z) depends on the overlap integral between modes around the refractive index boundary. Thus, we define O(z) as a part of the coupling coefficient C(z), which is calculated by the overlap integral between local modes around the refractive index boundary:
We calculate the 1/Δβ(z), O(z) and C(z) between the first three order modes with different diameter of core 2/3, and show three curves between LP01 and LP11b when diameter of core 2/3 is respectively 10 µm, 9 µm and 6 µm as examples in Figs. 9(a)-(c). We find that the peak positions of 1/Δβ(z), O(z) and C(z) will shift with different diameter of core 2/3. This means the strong coupling area changes with diameter of core 2/3. The maximum of 1/Δβ(z), O(z) and C(z) for different diameter of core 2/3 is shown in Figs. 9(d)-(f). Although the maximum of O(z) between LP01 and LP11b increases with the decrease of diameter of core 2/3, the maximum of 1/Δβ(z) decreases sharply and lead to the decrease of C(z). Thus, we can conclude that the mode selectivity will inevitably increase with the decrease of diameter of core 2/3. One point to be aware of is the maximum of 1/Δβ(z) between LP11 and LP21 increases sharply with the decrease of diameter of core 2/3, which means the coupling between LP11 and LP21 will increase sharply and coupling efficiency will decrease sharply due to LP21 is not supported in the tapered end.
Fig. 9. (a) 1/Δβ(z), (b) O(z) and (c) C(z) between LP01 and LP11b when diameter of core 2/3 is 10 µm, 9 µm and 6 µm, respectively. Maximum of (d) 1/Δβ(z), (e) O(z) and (f) C(z) for different diameter of core 2/3 with markers corresponding to curves in (a)-(c).
The second direction is to reduce O(z) between LP01 and LP11b. To achieve this, we need to analyze their electric field distributions around 0.4 taper ratio, as shown in Fig. 10(a). The electric field intensity is mainly concentrated at core 1 for both modes and result in a large O(z) between them. Therefore, we can design the position of cores in the cladding to change electric field distributions of modes and O(z) between them. We keep the arrangement of three cores as equilateral triangle and set the moving down distance of the three cores as D, as shown in Fig. 10(b). The electric field distributions of LP01 and LP11b when D = ± 20 µm is shown in Fig. 10(c). The results clearly show that if we move the cores upwards, the electric field distributions will become more concentrated, while the downward movement will reduce the overlap of the electric field between two modes. Thus, we can speculate that the downward movement of three cores will improve the mode selectivity.
Fig. 10. (a) Electric mode field of LP01 and LP11b at initial position. (b) Schematic diagram of the downward movement of three fiber cores. (c) Electric mode field of LP01 and LP11b when D = ± 20 µm.
We calculate the 1/Δβ(z), O(z) and C(z) between the first three order modes with different D, and show three curves between LP01 and LP11b when D is respectively 5 µm, 15 µm and 25 µm as examples in Figs. 11(a)-(c). Different from the first method, the peak position of 1/Δβ(z) stays at 0.4 taper ratio while peak positions of O(z) and C(z) shift with D. As we discussed in section 4.3, this phenomenon is also helpful to reduce coupling between LP01 and LP11b. The maximum of 1/Δβ(z), O(z) and C(z) for different D is shown in Figs. 11(d)-(e). All of the three curves between LP01 and LP11b almost linearly decrease with the increase of D, which verifies our optimization direction is correct. But due to the same reason mentioned in the first method, the coupling between LP11 and LP21 will increase with the downward movement of cores and lead to the decrease of coupling efficiency.
Fig. 11. (a) 1/Δβ(z), (b) O(z) and (c) C(z) between LP01 and LP11b when D is 5 µm, 15 µm and 25 µm, respectively. Maximum of (d) 1/Δβ(z), (e) O(z) and (f) C(z) for different D with markers corresponding to curves in (a)-(c).
To find proper parameters for this PL, the mode selectivity and coupling efficiency of LP01 and LP11 with different diameter of core 2/3 and D is calculated and shown in Fig. 12. The smallest diameter of core 2/3 is set to be 4.5 µm because the effective index of fundamental mode in core 2/3 must be larger than the effective indexes of higher order modes in core 1. Otherwise, the effective index curves will cross with each other, which is not supposed to happen in the MGS-PL. The calculated result is a good confirmation of our previous inference. With the decrease of diameter of core 2/3 and increase of D, the mode selectivity of LP01 and LP11 can reach a very high level from only about 10 dB to about 50 dB. But the coupling efficiency of LP11 decreases sharply within a certain range of geometric parameters, while the coupling efficiency of LP01 is always close to 1 due to weak coupling between LP01 and higher order modes. It’s worth noting that the mode selectivity of LP11 is not completely negatively correlated with the coupling efficiency of LP11. If the coupling efficiency is too low, the LP11 mode power will decrease sharply and lead to the decrease of mode selectivity. Thus, LP01 and LP11 may not reach maximum mode selectivity at the same parameters. For example, if we restrict the coupling efficiency to more than 95%, the highest mode selectivity is 44.5 dB for LP01 and 54.7 dB for LP11 (5 µm in diameter of core 2/3 and 20 µm in D, green dots in Fig. 12). But if we restrict the coupling efficiency to more than 90%, the highest mode selectivity is 45.7 dB for LP01 while the mode selectivity of LP11 is reduced to 53.8 dB (4.5 µm in diameter of core 2/3 and 15 µm in D, blue dots in Fig. 12).
Fig. 12. Mode selectivity and coupling efficiency of LP01 and LP11 with different diameter of core 2/3 and D.
5. Conclusion
We have proposed a novel full vector numerical simulation method based on FEM and LCMT to analyze the mode transmission characteristics of PL with arbitrary input mode field. We introduced the detailed process of the method and compared it to traditional simulation methods by applying it on an example of three-core MGS-PL. The validity and high efficiency of the method was verified. With the clues from coupled mode equation and coupling coefficient, we optimized the geometric parameters of the PL and improved the mode selectivity for LP01 and LP11 up to 44.5 dB and 54.7 dB, respectively, while the coupling efficiency for both are higher than 95%. The simulation and optimization methods can be applied to PL with more input cores for improving the mode selectivity, coupling efficiency or other characteristics, and provide a guidance for the manufacture of high performance PL.
Funding
National Key Research and Development Program of China (2018YFB1801802); National Natural Science Foundation of China (11674177, 61775107, 61835006); Higher Education Discipline Innovation Project (B16027).
Disclosures
The authors declare no conflicts of interest.
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