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We report a trench-assisted heterogeneous multicore fiber optimized in terms of higher-order dispersion and crosstalk for radiofrequency true time delay operation. The analysis of the influence of the core refractive index profile on the dispersion slope and effective index reveals a tradeoff between the behavior of the crosstalk against fiber curvatures and the linearity of the propagation group delay. We investigate the optimization of the multicore fiber in the framework of this tradeoff and present a design that features a group delay relative error below 5% for an optical wavelength range up to 100 nm and a crosstalk level below −80 dB for bending radii larger than 103 mm. The performance of the true time delay line is validated in the context of microwave signal filtering and optical beamforming for phased array antennas. This work opens the way towards the development of compact fiber-integrated solutions that enable the implementation of a variety of distributed signal processing functionalities that will be key in future fiber-wireless communications networks and systems.
© 2016 Optical Society of America
1. Introduction
Next generation Information Technology scenarios, such as 5G communications and the Internet of Things, will require new technologies to address the current limitations to massive capacity and connectivity [1]. This involves a full convergence between the optical fiber and the wireless networks in terms of: (1) radio access distribution between a central office and different remote locations, including Multiple Input Multiple Output (MIMO) antenna connectivity; and (2) broadband microwave photonics (MWP) signal processing [2–5]. We have recently proposed the use of multicore fibers (MCFs) as a compact medium to provide both distribution and processing functionalities “simultaneously”, leading to the concept of “fiber-distributed signal processing” [6–8]. This approach requires heterogeneous MCFs that are dispersion engineered to operate as group-index-variable delay lines offering true time delay line (TTDL) operation for radiofrequency (RF) signals. The TTDL is the basis of many MWP signal processing applications that will be key in future fiber-wireless communications such as reconfigurable signal filtering and squint-free beamsteering in phased array antennas [2,3].
A group-index-variable delay line implies a variation in the propagation velocity of the optical fiber cores or waveguides involved [9]. Sampled TTDL operation requires, in first place, a constant group delay difference between adjacent signal samples. In addition, tunability of this delay difference with the optical wavelength requires a linear spectral behavior of the group delay. Both linear performance and tunability can be achieved by a proper design of the refractive index profile of each core [7,8]. In particular, the minimization of the higher-order dispersion will extend the optical wavelength range in which the MCF fulfils these TTDL requirements with negligible degradation of the differential group delay. A broad optical operation bandwidth will guarantee the viability to provide, at the same time, different microwave signal processing functionalities together with purely radio-over-fiber delivery within a single fiber. In addition, it will assure the compatibility of the MCF-based TTDL with wavelength division multiplexing (WDM) techniques. Figure 1 illustrates the rationale of a fiber-wireless communications scenario where, once the MCF is fabricated and deployed, we can use different optical wavelength bands, for instance, for passive optical network (PON) or Fiber-to-the-home distribution, MIMO antenna connectivity, squint-free radio beamsteering, arbitrary waveform generation and microwave signal filtering.
Fig. 1 Application scenario where a single heterogeneous MCF provides both fiber distribution and different signal processing functionalities over different optical bandwidths.
After the higher-order dispersion, the second source of impairment of the proposed TTDL is the intercore crosstalk and its behavior against fiber curvatures. The crosstalk in heterogeneous MCFs is strongly affected by the curvature radius of possible cable bends since they change the phase-matching condition between cores [10–14]. Since we consider MCF lengths up to 20 km, the design refractive index profile must assure a minimum intercore crosstalk for possible curvature radius above 100 mm affecting the deployed link.
In this paper, we report a trench-assisted heterogeneous MCF optimized in terms of nonlinear spectral group delay and intercore crosstalk for broadband TTDL operation. For the first time to our knowledge, we analyze the dispersion slope dependence on the refractive index profile parameters of each individual core. We then design an optimized MCF where the group delay error due to higher-order dispersion is minimized and, therefore, the optical wavelengthoperation range is increased. Finally, the viability of the designed TTDL is demonstrated in the context of microwave signal filtering and optical beamforming for phased array antennas.
2. Dispersion and crosstalk optimization
2.1 Differential group delay: Spatial and optical wavelength diversities
The design of a heterogeneous MCF to behave as a group-index-variable delay line implies that each core features an independent group delay with a linear dependence on the optical wavelength λ, as shown in Fig. 2(c). We can expand the group delay per unit length, τn(λ), as a third-order Taylor series around an anchor wavelength λ0 as:
where Dn is the chromatic dispersion parameter and Sn the dispersion slope of the core n. For proper operation, we must design the refractive index profile of each core such that, first, Dn increases with the core number in an incremental fashion and, secondly, we ensure a linear behavior of the group delay along the desired wavelength range. As Eq. (1) shows, to reduce the quadratic wavelength-dependent term, we must address a rigorous higher-order dispersion analysis and management. This group-index-variable delay line can work on two different regimes whether we exploit the spatial or the optical wavelength diversities [6,7]. The delay difference between adjacent samples, that is, the basic differential delay, follows a different law depending on this operation regime and, as consequence, we must address each regime individually for a proper dispersion optimization.Fig. 2 (a) Heterogeneous N-core MCF. (b) Refractive index profile of a trench-assisted core. (c) Group delay slopes for the N cores showing spatial and optical wavelength diversities.
In the case of spatial diversity, the differential group delay Δτn,n+1 is given by the propagation difference created between each pair of adjacent cores for a particular optical wavelength λm:
where ΔD = Dn+1 - Dn is the incremental dispersion parameter that is kept constant for every pair of cores. We see from Eq. (2), that the differential group delay is affected by a quadratic term that depends on the difference between the dispersion slopes of the cores involved. We can define the differential group delay relative error induced by this nonlinear variation as the ratio between the second and the first terms of the right-hand side of Eq. (2), that is:which shows that the detrimental nonlinear effect raised by the dispersion slope variability increases with the operation wavelength.When the delay line operates in optical wavelength diversity, the differential group delay experienced between two contiguous wavelengths (λm, λm+1) in a particular core n is given by:
where δλ = λm+1 – λm is the separation between the two adjacent optical sources, λ1 the wavelength of the first optical source and 1 ≤ m ≤ M − 1, being M the total number of optical sources. In this case, the undesired variation on Δτn is characterized by both the linear (δλ) and the quadratic (δλ2) dependence on the dispersion slope in Eq. (4), leading to a differential group delay relative error defined as:As Eq. (5) shows, when we exploit the wavelength diversity, the differential group delay relative error depends on the values of both Sn and Dn of the particular core used. When we use the spatial diversity instead, this error depends on the variabilities between cores ΔS and ΔD, as shown in Eq. (3).
We must note that we have neglected the additional group delay induced by the polarization mode dispersion along each one of the cores [15]. In view of the MCF lengths considered for the true time delay line (few km) and assuming a polarization mode dispersion parameter DP < 0.1 ps/√km, this additional group delay is at least three orders of magnitude lower than the target group delay given by Eq. (1).
2.2 Dispersion and crosstalk evaluation
The MCF-based TTDL not only requires optimization in terms of nonlinear spectral group delay, but also in terms of intercore crosstalk. This implies that, first, both the dispersion parameter and the dispersion slope of each core must fulfil a set of specific rules for TTDL operability [7,8], and, secondly, that we must assure a range of effective indices as to minimize the intercore crosstalk [16]. We must, therefore, select the parameters defining the trench-assisted refractive index profile of each core to satisfy both conditions. The evaluation of the influence of these parameters on the MCF design was carried out by means of a full vector finite-element method implemented using the Photon Design FIMMWAVE software. This tool provides so far the most accurate solution for arbitrarily-shaped waveguides with curved boundaries.
Figure 3 shows the computed dispersion parameter D, dispersion slope S and effective index neff as a function of the core radius a1 for three representative sets of design parameters that are defined in Fig. 2(b). Each colored zone corresponds to a particular group of values for the core-to-cladding relative index difference Δ1, the core-to-trench separation a2 and the trench width w. The yellow zone is characterized by having high core radii (4.5 ≤ a1 ≤ 6 µm) and low core-to-cladding relative index differences (0.25 ≤ Δ1 ≤ 0.31%), while the rest of the parameters are within the range 3 ≤ a2 ≤ 7 µm and 4 ≤ w ≤ 7 µm. The orange areas have moderate core radii (3.4 ≤ a1 ≤ 5 µm) and core-to-cladding relative index differences (0.33 ≤ Δ1 ≤ 0.39%), for a combination of 2 ≤ a2 ≤ 6 µm and 3 ≤ w ≤ 6 µm. Finally, the blue zones have low core radii (2 ≤ a1 ≤ 3.4 µm), high core-to-cladding relative index differences (0.72 ≤ Δ1 ≤ 0.8%), for 2 ≤ a2 and w ≤ 5 µm. In general, we have selected the upper and lower limits of each variable as to satisfy, respectively, the single-mode condition and low bend losses [16,17]. Furthermore, we observe that a value of a2 above its upper limit decreases the value of D below the target range for delay line operability. This target range is chosen in a way that the values of D are high enough as to reduce the influence of the nonlinearities raised by S in the wavelength-diversity mode, [see Eq. (5)]. We must note that quite the opposite is found in other applications, such as long-haul high-capacity digital communications, where the target is to reduce the effect of the chromatic dispersion as much as possible.
Fig. 3 Comparison between the possible (a) dispersion parameter D, (b) dispersion slope S and (c) effective index neff values versus the core radius (a1) for three representative zones (different colors) that are characterized by a particular set of values of the core-to-cladding relative index difference (Δ1), core-to-trench distance (a2) and trench width (w).
Since we consider a 7-core MCF characterized by ΔD = 1 ps/(km·nm), we must assure first a range of dispersion parameters up to D7 – D1 = 6 ps/(km·nm) to implement a 7-sample delay line. From Fig. 3(a) we observe that, as long as the core radius increases, the range of dispersion values decreases, limiting the number of samples. Secondly, from the previous section, we learned that to reach broadband operability we must: (1) reduce the dispersion slope variability Sn+1 – Sn between cores as much as possible for spatial-diversity operation [Eq. (3)]; and (2) assure a dispersion parameter value D as high as possible for the wavelength-diversity regime [Eq. (5)]. From Figs. 3(a) and 3(b), we see that increasing the core radius leads to a lower dispersion slope variability and also to higher values of D. Thus, we conclude that the optimum zone for TTDL operation is the orange one, which allows the required 6 ps/(km·nm) dispersion range with the highest dispersion parameter values and the lowest dispersion slope variability possible.
One of the major challenges in the design of heterogeneous MCFs is the management of the crosstalk dependence on the phase-matching condition between the cores when the fiber is bent [10–14]. To prevent this phase matching, the curvature radius must be larger than the fiber threshold bending radius Rpk, which is defined as the highest radius than can induce phase matching [12–14]. This threshold bending radius is inversely proportional to the difference between the effective refractive indices of two adjacent cores [16]. A 7-core fiber with hexagonal disposition requires at least 3 types of cores with similar effective indices to reduce the threshold bending radius. Bend-insensitive MCFs require a minimum effective index difference between adjacent cores of 0.1% for a 35-µm core pitch [18]. Keeping this in mind, we should choose a design zone that assures a range of effective index differences above 0.2%. Figure 3(c) shows that the areas with low core radii (blue zone) are the optimum ones in terms of intercore crosstalk. However, we must keep in mind that we need to fulfil (to the greatest extent possible) the requirements for both delay line operation and intercore crosstalk. Therefore, we finally choose the orange design area as the candidate to build the proposed dispersion- and crosstalk-engineered MCF.
Once we have identified the range of parameters that enable linear group delay characteristics with the best crosstalk performance possible, we carefully evaluate the influence of the core and trench refractive index profile on the behavior of both the higher-order dispersion and the effective index.
We analyze in first place the behavior of the dispersion slope S. As far as we know, this analysis is the first one carried out for trench-assisted core configurations and provides a useful tool for the design of both homogeneous and heterogeneous MCFs, not only in the context of MWP signal processing, but also for digital communications and sensing applications. Figure 4(a) shows the dependence of the computed dispersion slope S with the core-to-trench distance a2 for a 4-μm trench width w and three significant core radii a1 (3.4, 4.3 and 5.0 μm plotted in different colors) and core-to-cladding relative index difference Δ1 (0.33, 0.36 and 0.39% in different line styles). We observe that an increase in a1 results in a shorter range of variability for S, as we have previously deduced from Fig. 3(b), while an increase in Δ1 has a similar but less significant effect. We find as well that a2 is the parameter that induces the highest variation on the dispersion slope, provided that a1 is small enough. Figure 4(b) illustrates the computed dispersion slope behavior against a2 for different values of w with fixed values of a1 = 4 μm and Δ1 = 0.36%. In this case, wider trenches result in a negligible increment on S for high values of a2, which gradually gains relevance as a2 decreases.
Fig. 4 Dispersion slope dependence on the core-to-trench distance a2 versus (a) different core radii a1 (color lines) and core-to-cladding relative index differences Δ1 (line styles) for a fixed trench width w = 4 μm; and (b) different trench widths for a1 = 4 μm and Δ1 = 0.36%.
The next step in the optimization technique is the management of the crosstalk and its sensitivity against fiber curvatures. This requires a design of the refractive index profile as to maximize the effective index difference between adjacent cores. Figure 5(a) shows the dependence of the effective index neff with the core radius a1 and the core-to-trench relative index difference Δ1. We see that increasing a1 and/or Δ1 leads to higher effective indices. In this case, we can achieve a variation up to 0.15% by varying both a1 and Δ1 (for a fixed a2 = 4 µm and w = 4 µm). Figure 5(b) illustrates the effective index variation as a function of both a2 and w for a1 = 4 µm and Δ1 = 0.36%. We see that the effective index is not affected by a variation in w at all, while it raises as a2 increases. Actually, the separation between the core and the trench affects in a similar trend both the effective index and the dispersion slope in the sense that, as long as this distance is kept in a small range (up to 3 µm), a small change causes the higher variation on the effective index. All in all, only a maximum effective index variation of around 0.02% can be achieved by tuning a2. As a consequence, we conclude that the core region of the refractive index profile (defined by a1 and Δ1) has the biggest influence on the effective index.
Fig. 5 Effective index dependence on (a) the core radius a1 versus the core-to-cladding relative index difference Δ1 for a core-to-trench distance a2 = 4 µm and trench width w = 4 µm; and (b) the trench width w versus the core-to-trench distance a2 for a1 = 4 µm and Δ1 = 0.36%.
2.3 Design optimization in terms of higher-order dispersion
Once evaluated the influence of the trench-assisted refractive index profile on both the dispersion slope and the effective index, we develop a particular higher-order-dispersion- and crosstalk- engineered design that ensures a linear group delay in a broad input wavelength range. The designed fiber is composed of seven trench-assisted cores (35-µm core pitch and 125-µm cladding diameter) placed in a hexagonal disposition. By using again the numerical software Fimmwave, we obtained a range of D values from 14.75 up to 20.75 ps/(km·nm) with an incremental dispersion ΔD = 1 ps/(km·nm). The fiber satisfies as well the common group index condition at λ0 = 1550 nm, as reported in [7,8]. Table 1 gathers all the core design variables as well as the computed dispersion slopes, effective indices and effective areas. The cladding-to-trench relative index was fixed to Δ2 ≈1% in all cores.
Table 1. Core design parameters and properties for the dispersion-slope-optimized design.
The design technique was carried out by distributing three groups of cores on the cross-sectional area of the fiber, where each group comprises cores that feature similar effective indices. The groups are formed as follows: (group 1) a central or inner core identified with the lowest effective index; (group 2) three outer cores placed in non-adjacent positions with intermediate values of the effective index and (group 3) three outer cores placed in alternate positions as well featuring the highest effective indices. The core labelled as 5 was chosen as the inner core, having the lowest effective index. To achieve this effective index reduction, its core radius and core-to-trench distance were reduced, while the trench width was highly increased and the core-to-cladding relative index difference was set to an intermediate value. Group 2 is formed by cores 1, 2 and 3, which feature a low core radius, high core-to-cladding relative index difference and high core-to-trench distance. This leads to a S variability below 0.001 ps/(km·nm2) and an effective index around 1.4534. Cores 4, 6 and 7 constitute the group 3. They are characterized by a higher core radius, lower core-to-cladding relative index difference and lower core-to-trench distance to keep a low S variability along all the cores.
The design of the MCF reveals an important tradeoff between the dispersion slope and the effective index difference between adjacent cores in the sense that maximizing the effective index difference increases the range of variability of the dispersion slope. Figure 6 shows the computed dispersion slope versus the effective index for the set of design parameters that satisfy the target true time delay line requirements, i.e., common group index and dispersion requirements. Each small circle corresponds to the effective index and the dispersion slope obtained for particular values of the design variables. The seven colors are used to distinguish the values of the dispersion parameter D that are linked to specific core numbers. Filled squares represent each of the designed cores, named as C1-C7 respectively for cores 1 to 7. We see here how the magnitude of the dispersion slope is reduced as the effective index decreases, leading to an increment on the dispersion slope variability between the core featuring the lowest effective index (i.e., C5) and the rest. As shown, all the cores of the optimized MCF (i.e., filled squares) have the highest effective index possible and the most similar dispersion slope possible. On the other hand, we observe that the intercore crosstalk improves if the limitation on the maximum dispersion slope variability is less restrictive. Therefore, we propose to develop a second heterogeneous MCF in which we prioritize the effective index optimization over the dispersion slope optimization, so that we can compare both structures in terms of crosstalk and TTDL operation.
Fig. 6 Relationship between the computed dispersion slopes S and effective indices neff for a group index of 1.4755 and dispersion values D ranging from 14.75 up to 20.75 ps/(km·nm), plotted in different colored circles. Filled squares illustrate the S and neff of cores 1-7 for the dispersion-slope-optimized MCF, while filled triangles for the effective-index-optimized MCF.
2.4 Design optimization in terms of intercore crosstalk
We present here a new design that prioritizes the optimization of the effective index over the optimization of the dispersion slope. This MCF shares the same TTDL requirements and characteristics, (such as the range of dispersion parameters D and the common group index) as the previous design reported in subsection 2.3. Table 2 gathers the core design parameters and propagation properties. The main difference between both fibers is that the core with the lowest effective index (core 5) is designed as to minimize the value of the effective index. This way, all the core radius, core-to-cladding relative index difference and core-to-trench distance have been reduced to achieve an effective index close to 1.4525, leading to a maximum dispersion slope variability between adjacent cores at least six times higher. Figure 5 shows the comparison between the effective indices and dispersion slopes, where the cores of this new design are identified with filled triangles and the superscript (2). This figure is very helpful to illustrate the tradeoff between S and neff that actually exists in heterogeneous MCFs. On one hand, this tradeoff limits the maximum effective index difference that we can reach if our goal is to reduce the variability in the higher-order dispersion and, on the other hand, it limits the minimum range of dispersion slopes that we can obtain when our goal is to reduce the crosstalk.
Table 2. Core design parameters and properties for the effective-index-optimized design
From both Table 1 and Table 2, we see, as expected, that the effective area Aeff increases with the core radius. Actually, the effective areas are gathered around two values: 60 µm2 for groups of cores 1 and 2, and 80 µm2 for group 3. Since the fiber-wireless scenarios that we consider for the MCF-based distributed signal processing approach require link lengths usually not greater than 20 km and low levels of optical power, we can discard the existence of nonlinear optical effects, such as the Kerr effect [19].
3. True time delay operation and crosstalk validation
Once we have designed both the higher-order-dispersion-optimized and the crosstalk-optimized fibers, we evaluate their performance in terms of true time delay line operation and intercore crosstalk robustness against curvatures.
3.1 True time delay line operation
The performance evaluation of both delay lines will be focused only on the spatial diversity mode of operation. Since both fibers share identical TTDL properties (such as the dispersion parameter D), they feature a very similar differential group delay relative error [as given by Eq. (5)] when we work in the wavelength-diversity regime. Therefore, the evaluation of the wavelength diversity regime is not included here.
Figure 7(a) shows the computed group delay for each core as a function of the optical wavelength for both fibers. We observe here that the first requirement for TTDL operation is fulfilled, that is, all the cores share a common group index (and thus common group delay) at the anchor wavelength λ0 = 1550 nm [6,7]. To a great extent, the second requirement, i.e. linearly incremental group delay slopes, is also achieved in both fibers. Figures 7(b) and 7(c) illustrate the differential group delay due to the nonlinear terms of Eq. (2) as a function of the wavelength, respectively, for the dispersion-slope-optimized and the effective-index-optimized fibers. As a reference, we also include in dashed lines the group delay relative error calculated from Eq. (3). We see how the maximum relative error due to the dispersion slope variability increases up to 15% within a 50-nm range for the effective-index-optimized design, while it is kept below 2.5% for the dispersion-slope-optimized fiber. In general, we checked that a relative error around 5-10% can be considered as the lower limit from which the target response of a typical MWP application (such as signal filtering or radio beamsteering) is excessively damaged. This implies that the wavelength operation range is wider (larger than 100 nm) in the dispersion-optimized fiber than in effective-index-optimized one (25-35 nm range).
Fig. 7 (a) Computed core group delays versus wavelength for the fiber (dashed lines); Computed differential group delay due to the nonlinear dispersion effect (given by the nonlinear part of Eq. (3)) as a function of the wavelength for (b) the dispersion-slope-optimized fiber and (c) the effective-index-optimized fiber. Dashed lines represent the differential group delay relative error calculated from Eq. (3).
3.2 Crosstalk
One of the major detrimental effects that can degrade the performance of heterogeneous MCFs arises from the crosstalk dependence on the phase-matching condition between adjacent cores when the fiber is bent [12–14]. To overcome this phenomenon, we have optimized our designs by maximizing, as much as possible, the effective index difference Δneff between adjacent cores as to improve the threshold bending radius Rpk [16]. Figures 8(a) and 8(b) show the location of the cores on the cross-sectional area of the fiber for the effective-index-optimized and the dispersion-slope-optimized designs, respectively. As described in subsection 2.5, the selected spatial distribution ensures that each of the three groups of similar effective index cores are placed in nonadjacent positions. Figure 8(c) illustrates the numerical evaluation of the intercore crosstalk dependence on the fiber bending radius for the pair of cores bringing the worst-case scenario. These results have been computed with the numerical software Fimmprop by Photon Design. As shown, the fiber optimized in terms of effective index presents a threshold bending radius close to 69 mm that corresponds to Δneff ≈0.074%. On the other hand, the design optimized in terms of higher-order dispersion presents a threshold bending radius shifted to around 103 mm as the effective index difference is reduced down to 0.050%. In addition, we see that the worst-case crosstalk above the phase-matching region is kept below −80 dB in both designs, which stays in the order of the −70 dB reported for trench-assisted 7-core MCFs [12,20]. When the fibers are bent right at their threshold bending radii, the results show that both designs are very robust against fiber curvature losses as well, reaching a worst-case bend loss below 0.1 dB/km.
Fig. 8 Cross-sectional fiber view for (a) the effective-index-optimized and (b) the dispersion-slope-optimized designs; (c) Computed crosstalk as a function of the bending radius for the effective-index (solid red) and dispersion-slope-optimized designs (dashed blue).
4. Application to Microwave Photonics distributed signal processing
The developed delay lines serve as a compact and energy efficient solution to implement a variety of signal processing functionalities that will be especially demanded in fiber-wireless communications networks and subsystems. As a proof of concept, we evaluate here their performance as distributed signal processing elements when they are applied to two Microwave Photonics functionalities: tunable microwave signal filtering and optical beamforming for phased array antennas [2,3]. We show the importance of properly engineering the higher-order dispersion of the heterogeneous MCF in terms of broadband TTDL operation. We compare the designs described in section 3 to the ideal responses given by the linear parts of Eqs. (2) and (4) when applying, respectively, the space-diversity or the wavelength-diversity regimes. The block diagrams describing how to implement both the filter and the beamforming network from the MCF-based TTDL can be found in [6].
4.1 Microwave signal filtering
A frequency filtering effect over RF signals results from combining and collectively photodetecting (with a single receiver) the delayed signal samples coming from the TTDL output. This incoherent Finite Impulse Response filter is characterized by a transfer function H(f) that is given by [3]:
where an is the weight (amplitude and phase) corresponding to the nth sample and f the RF frequency. The frequency period or Free Spectral Range (FSR) of the filter is given by FSR = 1/∆τ, where ∆τ is the basic differential delay as defined in section 2.We evaluate the performance of a TTDL implemented with a 10-km MCF comparing both space and wavelength modes of operation. Figures 8(a)-8(d) illustrate de computed transfer function of the microwave filter as a function of the RF frequency when the TTDL operates in wavelength diversity. We compare the response of the filters implemented with the designed TTDLs (blue-solid line) to the ideal response (red-dashed line) obtained by setting S to zero in the basic differential delay given by Eq. (4). As commented before, we obtain the same performance in both dispersion- and crosstalk-optimized fibers when we exploit diversity in optical wavelength. Figure 9(a) corresponds to the case when we use an array of M = 5 lasers from λ = 1550 up to 1554 nm with a 1-nm separation. We see a perfect match between the responses from the ideal and the designed delay elements. If we increase the wavelength range up to 1590-1594 nm, we observe a considerable mismatch between both responses, as shown in Fig. 9(b). This is caused by the linear δλ-dependence of S in Eq. (4) (second term in right-hand side). However, this linear displacement does not distort the filter shape, so we can compensate a priori this effect by taking into account this displacement when designing the TTDL. Actually, we can re-tune the response by properly managing the operation wavelengths of the lasers, as shown in Fig. 9(c) where the separation between the input wavelengths is reduced down to 0.89 nm. We can as well compensate that effect by increasing the number of lasers, as shown Fig. 9(d) for M = 10. Note that the optical wavelength shift δλ’ that compensates this displacement is obtained from Eq. (4) as:
Fig. 9 Comparison of the computed wavelength-diversity transfer function H as a function of the RF frequency for both designs (blue solid line) and the ideal filter (red dashed line) for a 10-km fiber: (a)-(b) set of 5 lasers with a 1-nm separation at an initial wavelength λ1 of (a) 1550 nm and (b) 1590 nm; (c)-(d) 0.89-nm wavelength separation and λ1 = 1590 nm for (c) 5 lasers and (d) 10 lasers. Comparison of the computed spatial-diversity H between the dispersion-optimized (blue solid line), the effective-index-optimized (green solid line) and the ideal filters (red dashed line) for a 10-km fiber at an operation wavelength of (e) 1560 nm, (f) 1575 nm, (g) 1600 nm and (h) 1650 nm.
Figures 9(e)-9(h) represent the computed transfer function of the microwave filter working in spatial diversity when we compare: (1) the MCF optimized in terms of higher-order dispersion (blue solid lines); (2) the MCF optimized in terms of effective index (green solid lines) and (3) the ideal response (red dashed lines). The 10-km MCF length results in a Free Spectral Range of (e) 10 GHz for an operation wavelength of λ = 1560 nm, (f) 4 GHz for an operation wavelength of λ = 1575 nm, (g) 2 GHz for λ = 1600 nm and (h) 1 GHz for λ = 1650 nm. As shown, the dispersion-slope-optimized fiber overcomes the limitations induced by the nonlinear spectral group delay even for a wavelength range up to 100 nm, while the filter response for the-effective index-optimized fiber is highly degraded for wavelengths above 1575 nm.
4.2 Optical beamforming for phased array antennas
Optical beamforming networks are implemented using a similar configuration as in microwave signal filtering, with the particularity that each sample is individually photodetected and, then, feeds one of the radiating elements that conformed the phased array antenna [2]. In the case of 1D architectures, the normalized angular far-field pattern of the radiated electric field, or array factor AF(θ), is given by [21]
where θ is the far field angular coordinate, ν is the optical frequency (ν = c/λ0) and dx is the spacing between adjacent radiating elements. From Eq. (8), the direction θ0 of maximum radiated energy can be adjusted by tuning the basic differential delay since Δτ = dx sin (θ0) / c.We evaluate the influence of the nonlinear spectral group delay in both fibers in the case of a phased array antenna characterized by dx = 3 cm, a 5-GHz RF signal and a link length of 10 km. As we have pointed out in the microwave filtering analysis, when we operate in wavelength diversity, the nonlinearities arisen in the basic differential delay [term proportional to δλ in the second term of the right-hand side of Eq. (4)] can be further compensated. Then, for simplicity, our evaluation here focuses only on the use of the spatial diversity. We compare again the dispersion-optimized MCF (blue solid lines), the effective-index-optimized fiber (green solid lines) and the ideal response (red dashed lines). Figure 10(a) shows the computed array factor as a function of the beam pointing angle (in degrees) in both polar coordinates (left) and decibels (right) at an operation wavelength of λm = 1570 nm. We see that, for a 20-nm wavelength range (anchor wavelength λ0 = 1550 nm), the array factor offered by the effective-index-optimized MCF is slightly mismatched from the ideal one, while the one given by the dispersion-slope-optimized fiber matches it perfectly. If the operation wavelength is increased up to 1600 nm, Fig. 10(b) shows that the array factor is highly degraded when we use the effective-index-optimized fiber, but stays practically unaltered when we resort to the dispersion-slope-optimized fiber instead.
Fig. 10 Comparison between the computed array factor (AF) as a function of the beam pointing angle (in degrees) for the dispersion-slope-optimized MCF (blue solid line), the effective-index-optimized MCF (green solid line) and the ideal delay line for a 10-km fiber with a 3-cm separation between antennas for a RF frequency of 5 GHz at an operation wavelength of (a) 1570 nm and (b) 1600 nm. Left: in polar coordinates; Right: in decibels.
5. Conclusions
We have presented a trench-assisted heterogeneous multicore fiber optimized in terms of higher-order dispersion that is designed to operate as a broadband tunable delay line for radiofrequency signals. For the first time to our knowledge, we have analyzed the influence of the core and trench refractive index profiles on the dispersion slope of a multicore fiber. This allows us to optimize the performance of the multicore fiber in terms of the propagation nonlinearities that affect the group delay of each one of the cores. In general, the higher-order dispersion optimization provides a useful tool for the design, not only of homogenous or heterogeneous multicore fibers, but also of singlemode singlecore fibers and plastic optical fibers. The optimization results are applicable to a variety of applications besides Microwave Photonics, such as minimization of the optical pulse broadening in high-speed broadband fiber communications (including spatial division multiplexing transmission), fiber distributed sensing or chromatic dispersion compensation. All these applications will benefit from the extension of the operation optical wavelength range in terms of group delay nonlinearities, especially in WDM schemes with a high number of transmission channels. In the particular scenario of microwave signal processing for fiber-wireless networks, this optimization is essential for the implementation of true time delay lines that can operate in a broad optical wavelength range, giving support to different functionalities in the same single optical fiber. The investigation of both the dispersion slope and the effective index reveals an important tradeoff between minimizing the sensitivity of the intercore crosstalk against fiber curvatures and the nonlinear group delay. We have designed and compared a dispersion-slope-optimized fiber and an effective-index-optimized fiber in order to demonstrate the importance of a proper management of the dispersion slope of the cores. The MCF optimized in terms of higher-order dispersion ensures a delay relative error below 2.5% or 5%, respectively, for an optical wavelength range up to 50 or 100 nm, and a crosstalk level below −80 dB for bending radii larger than 103 mm. We have demonstrated the viability of this optical fiber as linear true time delay line when it is applied to microwave signal filtering and optical beamforming in phased array antennas.
This work opens the way towards the development of distributed signal processing for microwave and millimeter-wave signals in a single optical fiber where a variety of functionalities can be implemented exploiting different optical wavelength bands. Future fiber-wireless access networks will benefit the proposed approach in terms of: (1) compactness as compared to a set of parallel singlecore singlemode fibers, (2) performance stability against mechanical or environmental conditions and (3) operation versatility offered by the simultaneous use of the spatial- and wavelength-diversity domains. In addition, the MCF-based TTDL can be applied to many Information and Communication Technology applications besides fiber-wireless access networks, such as broadband satellite communications, physical, chemical and smart structure sensing, medical imaging, optical coherence tomography, broadband electronic and RF measurement instrumentation, and quantum communications.
Funding
Spanish MINECO (TEC2015-62520-ERC Project); Spanish MINECO (TEC2014-60378-C2-1-R MEMES Project); Spanish MINECO (BES-2015-073359 fellowship); Spanish MINECO Ramón y Cajal Program (RYC-2014-16247).
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