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The ultimate number of transmission channels in a fiber for the space division multiplexing (SDM) is shown by designing an air-hole-assisted double-cladding few-mode multi-core fiber. The propagation characteristics such as the dispersion and the mode field diameter are almost equalized for all cores owing to the double cladding structure, and the crosstalk between adjacent cores is extremely suppressed by the heterogeneous arrangement of cores and the air holes surrounding the cores. Optimizing the structure of the air-hole-assisted double-cladding, ultra dense core arrangements, e.g. 129 cores in a core accommodated area with 200 μm diameter, can be realized with low crosstalk of less than −34.3 dB at 100km transmission. In this design, each core supports 3 modes i.e. LP01, LP11a, and LP11b as the transmission channels, so that the number of transmission channels can be 3-hold greater than the number of cores. Therefore, 387 transmission channels can be realized.
© 2014 Optical Society of America
1. Introduction
To break through the capacity crunch of the optical fiber network, space division multiplexing (SDM) has been proposed [1, 2]. Multi-core fibers (MCFs) [3] and few-mode fibers (FMFs) [4] are the prominent candidates for the ultra-high capacity SDM transmission. Recently, ultra high bit rate transmission experiments using MCFs have been demonstrated such as 305Tb/s transmission using a 19-core fiber [5], 1.01Pb/s transmission using a 12-core fiber [6], and 1.05Pb/s transmission using a 14-core fiber [7]. However, further increase of the number of cores is difficult due to the thick diameter of core with a trench structure and the combination of homogenous cores. Generally, the minimum core spacing between identical cores to achieve low crosstalk, e.g. −30dB at 100km transmission, is much greater than that between non-identical cores, which is the primary factor limiting the core packing density. Thus, the incorporation of non-identical cores have been suggested as a method to increase the packing density [8]. However, when employing many types of non-identical cores, the difference of V parameter among non-identical cores will be large, resulting in significant differences in the propagation characteristics among the cores. To overcome this problem, a double cladding structure which create differences in the equivalent index among several types of cores while maintaining nearly identical propagation characteristics was proposed [9].
Subsequently, we proposed a novel heterogeneous MCF with air hole assisted double cladding structure, which can drastically suppress the crosstalk between adjacent cores and can achieve extremely high density of cores [10]. However, since the adjacent cores almost contact with each other owing to the extremely low crosstalk, further increase of the packing density of cores is almost impossible. Although some MCFs with hole assisted structure have been demonstrated [11, 12], identical cores were used and so the core density have not reached its limit. Therefore in this paper, to increase the number of transmission channels in the MCF, we propose a FM heterogeneous MCF with air hole assisted double cladding structure. Firstly we show the design principle of air-hole-assisted double-cladding structure and the design example of dense heterogeneous MCF. Next, the optimization of FM air-hole-assisted double-cladding structure and the ultimate number of transmission channels in a MCF are shown.
2. Dense heterogeneous air-hole-assisted double-cladding single mode MCF
2.1. Principle of double-cladding structure
Let us consider a core surrounded by a double-cladding structure as shown in Fig. 1. The double cladding structure can be utilized for heterogeneous arrangement of several types of cores. The field profile of the LP01 mode is mostly confined in the core and the first cladding region owing to the refractive index contrast between the core and the first cladding, Δ. If the index contrast Δ is fixed the same value for all cores, the propagation characteristics such as the dispersion and the mode field diameter (MFD) can be kept identical even for different types of cores. On the other hand, the equivalent index neq can be made different by varying the index difference between the first cladding and the second cladding, Δnc(= n2 − n3). Using this double-cladding structure with low index contrast (Δ=0.375%), we assumed multiple cores with different propagation constants while maintaining the same index contrast, Δ. The refractive index of the second cladding, n3, was assumed to be 1.44402 (pure SiO2 at λ=1550nm). To maximize the core packing density, the radius of the first cladding a1 must be minimized, while keeping the effect of the second cladding on the equivalent index negligibly small. To clarify the radius of the first cladding a1 which required to minimize the effect of second cladding, the relationship between the equivalent index n′eq and the radius of the first cladding a1 was calculated using the perturbation theory and the equation,
where neq is the changed equivalent index due to the existence of the second cladding and n′eq is the equivalent index determined by the normal step-index profile consisting of the core and the cladding. I is the perturbation term given by where, E(r) represents the unperturbed field profile for the case that the refractive index of the second cladding is equal to that of the first cladding, i.e. n2 = n3. The calculated result is shown in Fig. 2. It can be seen that a1 = 8μm is sufficient to minimize the effect of the first cladding, because the change in the perturbed equivalent index is less than 2% of the equivalent index determined by Δ, even for the case of sufficiently large value of Δnc = 0.003 which is the cutoff value of LP11 mode given in the next section.
2.2. Principle of hole assisted double-cladding structure
In the double cladding structure, the core region is mainly utilized to guide the light and the fundamental mode (LP01 mode) is supported by the index difference between the core and the first cladding. However, since there is also the index difference between the first cladding and the second cladding, the first cladding mode, which is the LP11 mode in the whole structure, is supported and may result in the crosstalk for the fundamental mode guided in the core region. To suppress the excitation of the LP11 mode, air-holes are added to the double-cladding structure. The relation between the equivalent index neq of each mode and the refractive index difference Δnc between the first cladding and the second cladding was numerically calculated using a mode solver (FemSIM by Rsoft). The calculated results are shown in Fig. 3. It can be seen from this figure that the cutoff Δnc of LP11 mode of the double cladding structure without air holes is as small as 3 × 10−4. On the other hand, the cutoff Δnc of air-hole-assisted double-cladding structure is extended to 3 × 10−3 when the diameter of air-hole is 5μm and the center of the air-holes are located on the outer circumference of the first cladding. When the equivalent index is less than 1.44402 (refractive index of SiO2), it was confirmed that the field spreads out into the whole area and is no longer confined in the core region. Therefore, this corresponds to the cutoff of LP11. Thus, the index difference Δnc of non-identical cores can be determined so that the value of Δnc for different cores are distributed within the range of the single-mode condition. Since the cutoff Δnc is 0.003, we assumed the value of Δnc of three types of non-identical cores to be 0.001, 0.002 and 0.003, for the heterogeneous uncoupled triangle arranged multi-core fiber. Now, the increment of Δnc between adjacent cores is defined by δnc in the following sections (see Fig. 5(b)), and in this case δnc = 0.001.
2.3. Evaluation of crosstalk using coupled power theory
The crosstalk (XT) between adjacent cores was evaluated using the coupled power theory (CPT) [13]. In this work, the bending radius R was assumed to be sufficient larger than the critical bending radius i.e. non-phase matching region, and the exponential autocorrelation function for the statistical characteristics of the power coupling coefficients (PCCs) was adopted. Thus, the average PCC [14] between the mth- and nth-core is given by,
where Kmn is the coupling coefficient calculated by the coupled mode theory (see Eq. (10) in Ref. [13]), βmn is the propagation-constant difference and dc is the correlation length. The correlation length is defined for the coupling between the fundamental modes of two adjacent cores. These definitions are the same as that in Ref. [14]. The correlation length of 0.05m was hypothetically adopted in this work which was obtained experimentally in Ref. [13].The calculated results of crosstalk are shown in Fig. 4. In this calculation, we assumed that Δnc of one of the two adjacent cores is zero, i.e. n2 = n3, because the coupling in this case is stronger than the case of Δnc ≥ 0. The crosstalk of the double-cladding structure without air-holes is about 0dB in the case of 100km transmission when the core pitch is 21μm. When the air-holes are located on the outer circumference of the first cladding, the crosstalk for 100km transmission is as low as −40dB or less in the range of 0.0005 < Δnc < 0.003. Here, the core spacing can be greatly reduced to 21 μm, and so the adjacent core regions consisting of the core and the first cladding are in contact with each other. These data show that the crosstalk between adjacent cores can be extremely suppressed by the air holes.
In addition to the crosstalk, another important parameter is the critical bending radius, because the crosstalk increases when the bending radius approaches the critical bending radius. Therefore, the critical bending radius should be as small as possible. The critical bending radius Rc is given by
where Λ is the adjacent core spacing and δneq is the difference of the equivalent index between adjacent cores. In this case, since Λ can be greatly reduced to 21 μm and δneq nearly corresponds to δnc of 0.001, the critical bending radius can be reduced to 30 mm.3. Optimization of few mode air hole assisted double cladding structure
Figure 5(a) shows the schematic view and the definition of structural parameters of air hole assisted double cladding structure. a and a1 are the core and first cladding radii, respectively, and h is the diameter of the air hole. In this work, a and a1 were assumed to be 4.5μm and 8μm, respectively, the same as the case of single mode design. The core pitch is Λ and the gap between circumferences of adjacent first cladding is g. Then, ahole, the distance between the center of the core and the center of the hole, is half of Λ, which means holes are located at the midpoint between adjacent cores. Here, Δ is the refractive index contrast between the core and the first cladding which is kept identical for all cores. To increase the number of modes without deteriorating the core density, Δ was increased. In this design, the index contrast Δ of 0.7% was adopted so that 3 modes, i.e. LP01, LP11a and LP11b modes can be supported. Then, in this paper, since the triangle arrangement of non-identical three types of cores was adopted and numbered as #1, #2, and #3, three different values of Δnc were determined to be Δnc = 0.001 for core #1, 0.002 for core #2 and 0.003 for core #3. Here, δnc represents the index difference of first cladding between the core #i and the core #i + 1 as shown in Fig. 5(b).
First, the theoretical cutoff wavelength of the LP21 for this structure was calculated. Fig. 6 shows the relationship between g and the theoretical cutoff wavelength of LP21 mode against the hole diameter h. Here, the theoretical cutoff wavelength λc was calculated for the core #3, because that of core #3 should be longest among the three types of cores due to the large Δnc. It can be seen that the theoretical cutoff wavelength can be shorter as the overlap area of the air hole to the first cladding become larger. The design condition for the cutoff wavelength λc was assumed to be 1530nm in consideration for the use of C- and L-band.
Next, the crosstalk of LP11 modes between adjacent cores at the 100km transmission was calculated using the PCC given by Eq. (3). Although the correlation length was defined for the calculation of the crosstalk between the fundamental modes in Ref. [13], the correlation length dc was assumed to be 0.05m [13]. In this work, since the difference between the equivalent index of LP11 mode and the refractive index of the first cladding n2 is almost equal to that in the single mode case, we assumed this value. Fig. 7 shows the calculated results. It is seen that the crosstalk is extremely suppressed as the core pitch is reduced, when the hole diameter is constant. Usually, the crosstalk becomes large as the core pitch is reduced. However, in this case, the field is more effectively shielded by the holes as the hole approaches the core with the decrease of core pitch according to the relation ghole = (Λ − 2a1 − h)/2. Therefore, the crosstalk is suppressed with the decrease of the core pitch. In the case of hole diameter of 9μm, the crosstalk is less than −30dB for any core pitch as shown in Fig. 7.
Lastly, the optimum design condition was obtained using the calculated results of the theoretical cutoff wavelength of LP21 and the crosstalk of LP11 modes between adjacent cores as shown in Fig. 8. Here, ghole indicates the gap between the circumference of the hole and the core, and dhole indicates the gap between the circumferences of the adjacent holes as shown Fig. 5. In the sky blue region which is surrounded by the lines of ghole=0 and dhole=0, and the curves of XT=−30dB and λth=1530nm, the air hole assisted double cladding structure can be designed. However, it is not practical that the holes contact each other, i.e. dhole=0μm, and the holes and the core contact with each other. Therefore, the deep blue region which is surrounded by the lines of ghole = 2μm and dhole=2μm and the curves of XT=−30dB and λth=1530nm represents the more practical design region.
4. Number of transmission channels
Using the optimum design parameters obtained above, ultra-high density multi-core fiber can be designed. In this study, the triangle arrangement using three types of non-identical cores was adopted since it can realize the highest core density. Once the core pitch, the core arrangement, the number of non-identical core types and the diameter of core accommodation area are given, the number of cores can be calculated using the lattice algorithm [15]. The algorithm to calculate the maximum core capacity in the triangle lattice arrangement can be briefly summarized as follows.
Let us consider a case #1 that core center is located at the vertex of the triangular lattice as shown in Fig. 9. The distance between the center of fiber and the lattice point Rik is given by Eq. (5). The number of cores existing in the circle with radius R is determined by counting the lattice points satisfying Rik < R. This counting procedure is equivalent to the sum of the unit step function U(R − Rik) with respect to i and k. Thus the summation of the lattice points in the normalized circle R/D is given by Eq. (6), where μ and ν are the maximum ranges of summation with respect to i and k and D is the core spacing between identical cores which is equal to .
The maximum core capacity for cases #2 and #3 where the center of fiber is located at the middle point between adjacent lattice points and the center of triangle, respectively as shown in Fig. 9, can be calculated in the same manner. The maximum core capacity for a given R/D is determined by comparing these numbers using Eq. (7). Using the lattice algorithm, the maximum core capacities for triangle lattice were calculated for three cases where the diameter of effective fiber area 2R were assumed to be 100μm, 150μm and 200μm, as shown in Fig. 10. Here, we sampled three designed parameters from Fig. 8 and the case of single mode MCF as described in Sec. 2. The maximum number of cores was evaluated using Fig. 10 and is summarized in Table 1. Cases A, B, and C correspond to the points A, B, and C in Fig. 8, respectively. In the design of few mode transmission, three channels, i.e. LP01, LP11a and LP11b modes, are supported per core. Thus, the number of transmission channels in a fiber will be 3 times the number of the cores. In the case of these three sampled structures, the number of transmission channels ranges from 129 to 387 when the effective fiber area diameter 2R is assumed to be 200μm. Even in the case of 2R=100μm which is almost equivalent to the conventional single mode fiber, the number of transmission channels can be increased to 57–93. Fig. 11 shows the example of case C for 2R = 200μm, which corresponds to the number of transmission channels of 210.
Table 1. Design examples of ultra-large number of channels.
In addition, the critical bending radius for the cases A, B, and C were calculated to be 24mm, 29mm, and 33mm, respectively, using Eq. (4).
5. Conclusion
The air hole assisted double cladding few-mode multi-core fiber has been proposed to achieve the ultra number of transmission channels in a fiber. Optimizing the air hole assisted double cladding structure from the view point of crosstalk suppression and cutoff wavelength, we have shown that several hundreds of transmission channels can be accommodated in a fiber. We represented the tremendous potential of the air hole assisted structure by optimizing the core structure.
In the view point of fabrication, two fabrication methods, i.e. rod-in-tube method [16] and stack and draw method [11, 17], can be used to realize the proposed hole-assisted double cladding structure. In the case of rod-in-tube method, the air-holes are formed by drilling in the preform rod which has double-cladding structure. In the case of stack and draw method, the preform rod is prepared by stacking rods which have several types of index and diameter and thin capillary tubes in a silica tube.
Acknowledgments
This work was supported by the National Institute of Information and Communications Technology (NICT), Japan, under “R&D of Innovative Optical Fiber and Communication Technology.”
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