1. Introduction

In recent years, the capacity carried by a single fiber has been rapidly approaching its limits [1]. Space-division multiplexing (SDM) can overcome this “capacity crunch” by using the spatial modes of a multi-mode fiber (MMF), or the multiple cores in a multi-core fiber as an additional and independent degree of freedom. Using multiple spatial modes can either increase the link’s spectral efficiency through additional channels occupying the same wavelength, or increase photon-efficiency of the link through coding.

Figure 1(a) illustrates a multiple-input multiple-output (MIMO) based SDM system where N independent information channels are launched onto an orthogonal combination of M spatial modes. Such a system consists of a spatial multiplexer to couple to the fiber modes, a multi-mode fiber (MMF), a mode-demultiplexer, an array of coherent receivers (Coh. Rx), and electrical MIMO processing. Maximal capacity in systems with N = M can occur only when the mode-dependent losses (MDL) are negligible [2]. There are a variety of spatial multiplexers with their respective strengths and weaknesses. Some spatial multiplexers directly excite the spatial modes (i.e., a mode-multiplexer) and others excite an orthogonal combination of modes. Directly exciting the MMF modes often requires large bulk-optic setups to produce the mode profiles using phase masks or spatial-light modulators and beam-combiners to overlap the pro-files onto the MMF [3–5]. Due to passive beam-combining, the coupling losses (CPL) increase proportionally to N. However, since mode scrambling is inevitable in MMFs [2, 3, 6–8], it is not always necessary to excite individual MMF modes, especially for long-haul transmission. It could be beneficial to excite an orthogonal combination of modes such that all channels experience similar modal dependencies. This is expected to reduce the outage probability [6,7,9]. Additionally, electronic MIMO processing can faithfully recover the launched information provided all K demultiplexed signals are coherently detected [7]. Photonic integrated circuits can shrink the coupler’s size, however these devices still suffer from relatively large losses due to the low efficiency of the grating couplers [10, 11]. With a large number of high-speed coherent receivers readily available, coherent MIMO demonstrations over MMF supporting 10 or more spatial modes are expected in the near future. Thus, spatial multiplexers supporting large N with negligible MDL and CPL are highly desirable.

 

Fig. 1 (a) Coherent SDM with electronic MIMO processing (N<=M<=K). (b) A vanishing core photonic lantern to function as the spatial multiplexer and spatial demultiplexer. (c) Approximate geometry of the array pattern during the taper.

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A “photonic-lantern” spatial multiplexer [Fig. 1(b)] adiabatically merges N single-mode fibers (SMF) into a single multi-mode core (or an array of strongly coupled cores) that supports N modes [12, 13]. During the adiabatic transition the light in the N isolated cores evolve into the N modes of the MMF without any power loss. If no power is lost, the signal in each of the N cores must couple to an orthogonal combination of MMF modes (i.e., a unitary transform). Photonic lanterns [12] are used in astronomy to increase the system input aperture while maintaining single-mode performance by collecting starlight with the multi-mode end and distributing it into high-resolution single-mode optics such as fiber-Bragg gratings [14]. Lanterns with over 120 ports have been demonstrated [15,16] and a study was performed on a three-port lantern for SDM [9]. Astronomical applications require low CPL with very high mode counts. MDL caused by some of the modes lost during the adiabatic transition are typically irrelevant and not analyzed. For example, in a 60 port lantern, if one launched mode (or combination of modes) radiates out of the lantern all the information placed onto that mode is lost even though this is a negligible decrease in throughput (e.g., CPL is 59/60 or 0.07 dB). However in SDM, this MDL reduces the system’s capacity and increases the probability of outage [2] therefore it is often preferable to reduce the MDL at the expense of increased CPL especially for a small number of spatial modes. A good example are spatial multiplexers based on phase masks which use passive beam combining to equalize the power launched into each mode at the expense of a factor N in power splitting loss.

In this work, we show how to optimize the photonic lantern for use in SDM systems that interface N isolated cores with a step-index MMF with N Linear Polarized (LP) modes [17]. The lantern can be used as both the spatial multiplexer and spatial demultiplexer. Optimizing the geometrical arrangement of the cores through the taper minimizes the MDL and CPL. Using the scalar beam propagation method (BPM) and modal analysis [18], we construct the lantern’s transfer matrix and quantify the CPL and MDL for several different core arrangements with up to 51 cores.

2. Conditions for spatial multiplexing without information loss

Figure 1(a) shows that a SDM systems transmits N information channels onto an orthogonal mixture of the M modes of MMFs, demultiplexes K modes at the receivers, and recovers the N information channels with MIMO processing. According to the second law of thermodynamics (the brightness theorem), the number of orthogonal modes in a system cannot decrease. At a minimum, N <= M <= K has to be fulfilled, otherwise there is possibility of information loss [2]. Let us consider the simple example of coupling into and out of the fundamental mode of a two-mode MMF using single-mode fibers (N = 1, M = 2, K = 1). All the light couples into the MMF on the fundamental mode (N < M). However, during propagation there is the potential for the information to scramble randomly between the fundamental and higher order MMF modes. In the extreme case, the information transfers completely to the higher-order mode and does not couple to the output. On average, the information has equal probability of being on either mode and thus only half of the information is recovered. However, if both modes of the MMF are coupled out onto two SMFs (K=2) then the information is spread across two outputs and not lost. Recovering the single-channel information requires additional receivers and MIMO processing. In a realistic network that incorporates amplifiers, routers, and cross-connects, it is important to unify the number of orthogonal modes and never let it increase (i.e., N = M = K). For a complete analysis of information throughput and the outage probability in a MIMO system with N <= M and K <= M refer to [2].

3. Photonic lanterns for space division multiplexing

In this section we describe the lantern structure and optimal core arrangements for minimizing MDL and CPL.

3.1. Photonic lantern structure

The photonic lantern structure is illustrated in Fig. 1(c). The geometry consists of individual cores with index n₂ with diameter a surrounded by a cladding with refractive index n₁. The cores are then placed inside a glass capillary (i.e., second cladding) with diameter D which has a slightly lower index n₀. During the taper, the light is initially guided in the single moded cores. As the core diameter shrinks the light is then guided in the cladding material which becomes the new MMF core. In the following figures, the mode index is the effective refractive index of the guided mode. Each core is placed along a ring with number of cores defined as R_n where n is the ring number (larger numbers further from center). For example, the three core lantern in Fig. 1(c) has R₁ = 3. The angle of the taper is defined with respect to the capillary diameter vs. lateral position. Larger angles correspond to shorter tapers which become diabatic. We define NA₁ as the numerical aperture between the core and the cladding material, and NA₂ as the numerical aperture between the cladding and the capillary (using the step-index approximation).

3.2. Optimum core arrangements for a lantern interfacing to a step-index MMF

The starting point for low CPL and MDL lantern design is an uncoupled core geometry that best approximates the MMF modes or samples the modes [19, 20] [see Fig. 2(a)]. In the literature, photonic lantern core geometries consist of hexagonal lattices or square lattices with the number of cores approximately equal to the number of final waveguide modes [12, 14, 21]. However, the arrangements which minimize MDL are unique for each waveguide geometry and have not been previously investigated.

 

Fig. 2 (a) 15 lowest order step index fiber spatial modes. (b) Coupled waveguide arrays whose supermodes closely match the fiber modes. (c) 15 lowest order modes of near optimal 15 core arrangement. (d) 15 lowest order modes of an incorrect 15 core arrangement.

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Intuitively, when the cores are isolated or weakly coupled, their supermodes (or superposition of supermodes) should resemble the final waveguide modes. The arrangement that best approximates the LP_nm modes in a step-index MMF (n is the max azimuthal mode number, and m is the radial mode number) are m concentric rings. The number of spots in each ring is 2p+1 where p is the largest n for each radial-number, m. For example, Fig. 2(a) shows the 15 lowest order LP modes. Each box represents the modes with identical cut-off frequency. Figure 2(b) shows the core patterns that best approximate an MMF supporting 3, 6, 8, 10, 12, and 15 spatial modes. Figure 2(c) shows the matching supermodes of the 15 core array which approximate a MMF supporting 15 spatial modes. In an adiabatic lantern, light launched in a particular super-mode mode will evolve into the corresponding MMF mode. Using these core geometries allow for a photonic lantern couplers with negligible MDL and CPL.

In comparison, Fig. 2(d) shows the supermodes of a 15 core array with an incorrect core arrangement that consists of two rings with 9 and 6 cores. The supermode set does not include a LP₀₃-like mode. Rather, it supports a mode that is unguided in a 15 mode MMF and any energy launched onto that supermode will radiate out of the lantern during the taper. Likewise, if the LP₀₃ is launched from the MMF, it will be guided within the cladding rather than within the cores.

3.3. Fabrication

These lanterns can be fabricated in two different ways: by single mode fiber bundling of different cladding fibers to achieve the right fiber packing geometry [12], or through ultrafast pulsed laser inscription in bulk glasses which gives unrestricted geometrical parameters as demonstrated in [13, 21]. For lanterns with less than three rings we expect that the fiber packing method should produce lanterns with the lowest MDL and CPL since the lantern can be directly spliced to SMF and the MMF.

4. Simulations

In this section we compare different photonic lantern core geometries using modal analysis, BPM, and transfer matrices; those that can have 100% throughput (i.e., CPL = 0 dB and MDL = 0 dB) and those that have non zero MDL and CPL. For all simulations, n₀=1.435, n₁=1.44, n₂=1.445 and NA₁=NA₂=0.12.

4.1. Quantification of coupling

In a coherent MIMO system the MDL sets the upper limit on capacity by reducing the number of available modes and the CPL affects the maximum span length (e.g., 1 dB excess loss means the spans length is reduced by 5 km). We quantify the MDL and CPL of the lantern by analyzing its transfer matrix, H(ω), from each isolated core mode (or the supermodes of the coupled array) to each MMF guided mode. Using the beam-propagation method (BPM), we construct the transfer matrix from the i − th isolated core to the MMF by propagating a guided mode from each isolated core (or core supermodes) to the lantern output [see Fig. 1], L_i(x, y). The overlap of L_i(x, y) with each MMF modes, F_j(x, y), forms the coupling matrix with elements $h_{ij}=\int F_j\left(x,y\right)L_i^{*}\left(x,y\right)dxdy$. Likewise, the coupling matrix in the reverse direction (from the MMF to the isolated cores) could be computed by propagating from each MMF mode to each single core, or by taking the conjugate transpose of H(ω).

H(ω) can be calculated at each frequency and completely describes how the fiber modes couple to the isolated core modes. Singular value decomposition (SVD), H = UΣV^*, is used to find the MDL and CPL where Σ is a diagonal matrix containing the singular values, λ_j, and U, V are unitary matrices of dimensions N × N and M × M, respectively. The singular vectors, v_j, the columns of V, are always orthogonal and represent the required linear combination of L_i(x, y) that couple to an orthogonal combination of MMF modes (F_j) with throughput of ${\lambda }_j^2$. The field at the lantern output corresponding to each λ_j is A_j(x, y) = ∑_i L_iV_ji. Using the A_j’s as a set of orthogonal channels, the the worst case MDL of the coupler is max(λ²)/min(λ²) and the CPL is N/∑_i λ². The A_j’s corresponding to λ_j = 0 span the null space of H(ω) and will not couple between the MMF and isolated cores.

4.2. Three core lantern examples

A three core lantern interfaces to a MMF supporting the LP₀₁ and the double degenerate LP₁₁ modes along with both polarizations. The obvious core arrangement consists of cores placed in a triangle pattern and is shown in Fig. 3(a). To highlight the importance of the core arrangement, Fig. 3(b) shows a lantern where the cores are placed along a line. The supermodes of the triangle lattice match the LP modes, and as the geometry shrinks those supermodes evolve into the MMF modes. The blue lines indicate the cladding modes of the lantern (i.e., those originally guided outside the cores in the cladding) which stay well separated from the core modes (red lines). When the cores are spaced far apart, the propagation constants of the three supermodes approach the same value indicating completely isolated cores. Also, the propagation constants of the core modes remain well separated and do not cross, indicating that if the taper is adiabatic then the light will evolve directly from the MMF modes into the supermodes. Note, although this analysis uses the core supermodes, each isolated core mode is equal to a superposition of the supermodes. For example, the right most core mode is the superposition of the LP₀₁ mode and the first LP₁₁ mode. Finally, since no modes are lost, this lantern can have 100% throughput without MDL.

 

Fig. 3 Comparison of two three-core lanterns using modal analysis. (a) Triangle array which supports the same modes as the fiber, (b) linear array which supports the LP₀₁, LP_11a, and a TEM₂₀ mode. Red lines indicate the three guided MMF modes, blue lines indicate higher-order cladding modes which are not guided in the MMF, and green lines indicate modes that are guided in the array but not in the MMF. NA₁ = NA₂ = 0.12.

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Figure 3(b) shows the same analysis for the linear pattern. The array supports the LP₀₁ mode, only one of the LP₁₁ modes, and a Hermite-Gaussian TEM₂₀ mode. The TEM₂₀ mode does not exist in the MMF and the second LP₁₁ mode is not guided by the linear array. Therefore, during the transition from the MMF to the isolated cores, the second LP₁₁ mode is lost. Likewise, going the opposite direction from the isolated cores into the MMF any energy on TEM₂₀ mode is lost. A third of the energy of each isolated core mode contains is on the TEM₂₀ mode and this lantern has throughput of 2/3 (CPL=1.76 dB) and infinite MDL.

Next, we perform a BPM simulation from the core supermodes into the MMF and from the isolated cores into the MMF and compute H(ω) for both cases. The taper angle is 0.012 rad which results in an adiabatic transition. Figure 4(a,b) show the BPM results and Fig. 4(c,d) show the intensity of the coupling matrices. Through the transition, the radiation pattern looks like the desired mode and evolves directly into individual MMF modes. Thus, the coupling matrix is diagonal and since almost no power is lost during the transition and all input modes reach the output with equal power this lantern has 0 dB CPL and 0 dB MDL (rounded to within 0.1 dB numerical error). Note, since the supermodes evolve directly into the MMF modes, the lantern can act as a 1-1 mode multiplexer by placing an appropriate optical mapping network before the lantern input [22].

 

Fig. 4 BPM simulations from for the three core lantern with 0.012 rad taper angle (a) the input supermodes and the (b) input core modes to the outputs. (c,d) Intensity coupling matrix from the inputs to the MMF modes. Simulation length is 10 mm, core diameter to outer diameter ratio is 1/8, the taper begins with a 70 μm outer diameter and ends with a 14 μm outer diameter.

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A more common scenario is to launch light into the isolated cores and the corresponding BPM simulation is shown in Fig. 4(b). Here, the input radiation pattern can be considered a superposition of the supermodes and mode beating is expected as the supermode propagation constants separate. During the taper, the light propagates back and forth between the cores and eventually evolves into a superposition of the three MMF modes. The coupling matrix is not diagonal and scrambled, however it is free from CPL and MDL. The additional scrambling can reduce large modal dependencies during fiber transmission [6, 7].

Figure 5 shows the supermode evolution in a diabatic (non adiabatic) transition. If the taper occurs too fast, then light will couple to higher order cladding modes that are not guided in the MMF and will possibly radiate outside the lantern. Figure 5(a) shows the BPM simulation illustrating that the light couples to the higher order cladding modes. Also, the spatial pattern at the output of the lantern resembles the LP modes with some higher order radiative modes. The coupling matrix [Fig. 5(b)] is still diagonal, but the diagonal elements have less intensity and are nonuniform. Correspondingly, the CPL is 1.5 dB and the MDL is 0.5 dB.

 

Fig. 5 BPM simulations for the three core lantern with a 0.12 rad taper angle. (a) the input supermodes to the outputs. (b) Intensity coupling matrix from the inputs to the MMF modes. Simulation length is 1 mm, core diameter to outer diameter ratio is 1/8, the taper begins with a 70 μm outer diameter and ends with a 14 μm outer diameter.

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4.3. Wavelength dependence

Since the lanterns use an adiabatic transition to obtain lossless coupling, they are expected to have some wavelength dependence. Across a specified wavelength range, if the M isolated cores are single moded, and the MMF supports M modes than the MDL and CPL can be 0 dB. Rather than affecting the MDL and CPL which reduce capacity, the wavelength dependence results in a unitary rotation of the coupling matrix. The longer the adiabatic taper, the faster the matrix rotates with wavelength. Across the telecommunications C-band, the three core lantern should have negligable MDL or CPL since the MMF supports three modes and the isolated cores are single moded across the whole wavelength range.

4.4. Lossless twelve core lantern arrangements

Figure 6 shows modal and beam-propagation method (BPM) analyses of two 12 core arrangements on two rings that adiabatically taper into a 12 mode MMF: lantern A has an optimum pattern and lantern B has a incorrect square lattice. Lantern A has two rings with three and nine cores. Lantern B has four and eight cores on each ring approximates a rectangular grid. Figure 6(a) shows the propagation constants of the guided modes (supermodes) as the isolated cores evolve into a single core. For lantern A, the 12 originally degenerate supermodes become the 12 MMF modes. Note that each core mode can be represented by a superposition of the 12 degenerate supermodes. Throughout the taper, these supermodes remain well separated from the cladding modes which become cut-off (radiative). An adiabatic taper can ensure that this spatial multiplexer will have zero MDL and CPL.

 

Fig. 6 Comparison of two 12 core lanterns. (a) Propagation constants vs. core separation. BPM simulation showing supermode radiating out of lantern B. (c) CPL and MDL calculated using BPM assuming lantern is cut at different lengths.

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Contrarily in lantern B, one of the supermodes resembles the LP₂₂ mode which is a radiation mode of the MMF. Additionally, one of the cladding modes (guided by the cladding rather than the cores) resembles the LP_41b mode which is guided in the final 12-mode MMF. Figure 6(b) shows the evolution of the LP₂₂ like supermode evolving into the radiative LP₂₂ MMF mode. Also, when coupling in the reverse direction (from the MMF into the SMFs) the originally guided LP_41b MMF mode evolves into a cladding mode of the waveguide array. Since one mode is lost, lantern B’s MDL is infinite and its best CPL is 0.37 dB (11/12).

Figure 6(c) shows the CPL and MDL for the two lanterns cut to different lengths assuming that the fiber modes can be imaged (resized) onto the lantern output. For lantern A, as the cores merge, the MDL and CPL approaches zero. Also, the MDL and CPL fall below 0.5 dB at an effective diameter of 37 μm suggesting that the lantern is a good spatial multiplexer even if it does not fully merge into a single multi-mode core. Lantern B has infinite MDL because its LP₂₂ like supermode evolves into a cut-off fiber mode. As the cores merge, the CPL asymptotically approaches 0.37 dB (i.e., 11/12).

4.5. Core arrangements for large lanterns

Figure 7 shows modal analysis for a 51 core lantern tapering into a MMF supporting 51 spatial modes (102 with polarization). Lantern C has the correct core arrangement as described above, and Lantern D has a slightly modified core arrangement where the 3rd and 4th ring from the center have an even number of cores. The modal analysis indicates that Lantern C can be loss-less since none of the core modes cross into the higher-order cladding modes during the taper and that Lantern D loses one mode during the transition. Considering all modes, lantern D has a CPL of 0.086 dB and infinite MDL (one mode is lost). Therefore, these core arrangements are also important for lanterns coupling to highly-multimode fibers. However, due to the large number of modes, the effects of an improper arrangement may go unnoticed.

 

Fig. 7 Comparison of two 51 core lanterns. Propagation constants vs. cladding diameter. Lantern C has correct core arrangements and cladding and supermodes do not mix. Lantern D has an incorrect core arrangement and the one supermode mixes with one cladding mode.

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5. Conclusions

We have shown through simulations that photonic lantern spatial multiplexers with the proper core arrangements can couple into and out of MMFs supporting a large number of spatial modes (51) and simultaneously achieve low MDL and low CPL. The lanterns with core arrangements whose supermodes approximate the LP modes will only have negligible MDL and CPL. Otherwise, lanterns with incorrect core arrangement with supermodes that do not exist in the MMF will lose at least one mode during the transition. Additionally, if the supermode patterns are launched, they will directly evolve into the corresponding MMF mode suggesting a technique to build a 1-1 mode multiplexer. Alternatively, launching the isolated core modes distributes the information across the MMF modes possibly mitigating the effect of MDL occurring during transmission. These lanterns with optimal core arrangements will be important for large mode count SDM systems.