Dispersion tailoring in Bragg fibers Sonali Dasgupta, Bishnu P. Pal, * and M. R. Shenoy Department of Physics, Indian Institute of Technology Delhi Hauz Khas, New Delhi-110016, INDIA. We report a study on the dispersion tailoring of 1-D photonic-bandgap Bragg fibers and propose two designs to demonstrate their utility as efficient dispersion compensators and as a metro fiber. The dispersion compensator design is based on exploiting the TE01 mode of a defect-free air-core Bragg fiber to achieve an average negative dispersion of ~ 1817 ps/km.nm and an average radiation loss less than 0.36 dB/km across the C-band. The dispersion-flattened Bragg fiber, designed for metro applications, has an average dispersion of 9.9 ps/km.nm across the same wavelength range.
1. INTRODUCTION In a conventional optical fiber, light is guided by total internal reflection due to the refractive index contrast between a finite sized core and surrounding cladding of lower refractive index. A Bragg fiber (as shown in Fig. 1), on the other hand, consists of a core surrounded by a series of periodic layers of alternate high and low refractive index materials (each of which has a refractive index higher than that of the core). The thickness of the cladding layers is chosen suitably so as to induce Bragg reflections. The periodic layers constituting the cladding result in a photonic bandgap (PBG). The PBG, analogous to the Low index electronic bandgap for electrons in a solid, does not allow light of certain frequencies to propagate along the direction of periodicity. Thus, in a Bragg fiber, light within a certain frequency range is confined within the Periodic core, along the fiber, through the mechanism of photonic bandgap guidance, in contrast to total internal reflection in conventional fibers. The multilayer cladding of the Bragg fiber acts like a highly reflecting mirror due to in-phase multiple reflections that add up from each Fig. 1. Cross sectional view of a boundary between the high and the low refractive index layers. The Bragg fiber precise nature of the bandgap structure depends on the quantitative values of the refractive index contrast and periodicity of the cladding layers. Light confinement through photonic bandgap guidance allows the possibility of having an air core. Also, large refractive index contrast in cladding layers allows a large fraction of the guided light to be confined within the air core; thereby enabling low-loss optical transmission and reduced sensitivity to optical nonlinearity. Bragg fibers are characterized by several independent physical parameters such as core size, refractive indices and thickness of the cladding layers, and hence offer a wide choice of parametric avenues to tailor their propagation characteristics. As is true with any fiber, loss and dispersion are the two most important propagation characteristics of a Bragg fiber. Owing to their unique guidance mechanism, Bragg fibers can exhibit dispersion characteristics that are otherwise nearly impossible to achieve in conventional silica fibers. Reported designs of Bragg fibers have indicated the potential feasibility of achieving zero dispersion at wavelength ~ 1 µm, multiple zero-dispersion, and high negative dispersion ~ 20,000 ps/km.nm [1-3]. In this paper, we propose two designs of Bragg fibers: one to achieve a highly efficient dispersion compensator for longhaul networks, and the second one to achieve non-zero dispersion-shifted fiber that is suitable for application as a metro fiber.
2. BASIC FIBER DESIGN 2.1 Multiple quarter-wave stack condition for modal confinement The radiation loss of a mode propagating through a Bragg fiber with infinite number of cladding layers is zero. However, in practice, any fabricated fiber would have only a finite number of cladding layers and since, the modes of a Bragg fiber are leaky in nature, the propagating mode would suffer a finite radiation loss. The loss can be minimized by appropriately choosing the cladding layer parameters. For a given number of cladding layers, the following quarterwave stack condition (Eq. 1) minimizes the radiation loss of the TE modes supported by a Bragg fiber [4]: *
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2π
λ0
n1 l1 =
2π
λ0
n2 l2 = π
(1)
2
where n1, n2 and l1, l2 are the refractive indices and thickness of odd and even numbered layers, respectively. In the case of multilayer dielectric mirrors and interference filters, it is well known that high reflectance can be achieved if the layer 2π 2π thicknesses are so chosen that the condition: n1l1 = n l = nπ , where n is an odd integer is satisfied [5]. Since 2 λ λ 22 0
0
functionally the cladding of Bragg fibers is analogous to thin film filters (or multi-layer mirrors), we anticipated a similar reflectance characteristic in Bragg fibers, and we investigated the effect of variation of the thickness of cladding layers on the propagation loss. It turns out that the quarter-wave stack condition (n = 1) is not an essential requirement for good confinement of the modal fields; tight modal confinement of the TE modes is also feasible in Bragg fibers, whose cladding layers satisfy the following multiple quarter-wave stack condition (Eq. 2) as opposed to precise quarter-wave stack condition (cf. Eq. 1) 2π
λ0
n1 l1 =
2π
λ0
n2 l2 = nπ , 2
(2)
where n is an odd integer (n > 1). However, the modal confinement is minimal when the cladding bi-layers add up to a 2π 2π round-trip phase of odd integral multiples of π, which corresponds to: n1l1 = n 2 l 2 = nπ , where n is an integer.
λ0
λ0
We exploited these physical features with regards to loss characteristics, to design a highly efficient dispersion compensating Bragg fiber (Sec. 3). 2.2 Analytical technique used In case of conventional fibers, the effective index of a guided mode can be obtained by employing the boundary condition of transverse field decaying exponentially to zero, outside the cladding-air interface. However, the same condition cannot be applied for modes of a Bragg fiber as they support quasi-modes, which are essentially leaky in nature and are oscillatory even in the cladding layers. Hence, to obtain the effective indices of these quasi-modes, the net outward flux through the fiber has to be minimized (optimization procedure) [4]. The optimization procedure for the Bragg fiber is relatively complex and time-consuming, more so if the fiber supports multiple modes. However, the semiasymptotic matrix approach simplifies the problem to a great extent [6]. According to the semi-asymptotic matrix theory, the fields in the core of the fiber and within a certain finite number of cladding layers are assumed to be Bessel functions, which are the exact solution of Maxwell’s equations in cylindrical co-ordinates. In the remaining (outer) cladding layers, asymptotic form of Bessel functions is used to approximate the fields. In the asymptotic limit, the Bessel functions resemble plane waves, whose amplitude decrease by a factor of r along the direction of propagation. Hence, by applying Bloch theorem to the outer cladding layers and using the continuity conditions for the electric and magnetic fields at each interface, the eigen value equation for determining the propagation constant of a guided mode can be obtained, which subsequently is used in the radiation loss calculations. The radiation loss coefficient, αTE/TM, is defined as the ratio of the radial power flux to the longitudinal component of the flux through the core
α TE / TM =
PrTE / TM PzTE / TM dz
(3)
We have used the above-mentioned semi-asymptotic matrix theory to calculate the effective indices of the TE01 modes of the proposed Bragg fibers. Since the TE01 modes were found to be well confined within the initial few cladding layers, we have neglected the material-related absorption losses and restricted ourselves to only the radiation loss for calculating propagation loss [7].
3. RESULTS 3.1 Dispersion compensating Bragg fiber Dispersion compensation forms an integral functional task in the design of a long-haul optical communication link. Out of various options that are available to fulfill this task, the most popular choice has been a dispersion compensating fiber (DCF). A typical DCF possesses a high negative dispersion so that a short length of the DCF could be used to cancel out
the positive dispersion that the signal acquires after propagating through much longer lengths of the transmission fiber within the EDFA wavelength band. Both loss and dispersion together, determine the performance efficiency of a DCF, which is measured in terms of an integral parameter known as figure of merit (FOM). FOM is defined through the ratio D , where D is the dispersion coefficient and α is the propagation loss constant of the DCF. Our proposed Bragg
α
fiber-based DCF design yields an average FOM exceeding 100,000 ps/nm.dB across the C-band. In contrast, the FOM of conventional DCFs typically range from 200 − 400 ps/nm.dB. Other reported designs for achieving high negative dispersion in Bragg fibers are based on either exploiting higher order hybrid modes (azimuthal modal index greater than zero) [3], or by incorporating a defect layer in the multilayer cladding [2]. In contrast, our design of the Bragg fiberbased DCF utilizes the circularly symmetric fundamental TE mode of a Bragg fiber that is devoid of any intentional structural defect, to achieve large negative dispersion coefficient. The TE mode also offers the advantage of attaining potentially low splice loss with the transmission fiber due to its good modal overlap with the LP01 mode of a conventional SMF, e.g. G.652, fiber. We have exploited the multiple quarter-wave stack condition [cf. Eq. (2)] with n = 3, to design a DCF with FOM ≈ 69,000 ps/nm.dB at 1550 nm and suitable for operation in the entire Cband. The dispersion coefficient and radiation loss of the TE01 mode of the proposed Bragg fiber, as a function of wavelength have been plotted in Figs. 2 and 3, respectively. The utilization of the multiple quarter-wave stack condition ensures low transmission loss for the TE01 mode in such a Bragg fiber while radically altering the dispersion characteristics as compared to the corresponding Bragg fiber with cladding layers satisfying the precise quarter-wave stack condition. The 15 cladding bi-layers of the proposed fiber have a refractive index contrast of 3.0, while the air-core radius is 4.0 µm. Though the fiber supports multiple modes, the large differential loss of the higher order modes with respect to the TE01 mode makes it effectively single-moded, similar to that reported in [8]. As explained in Sec. 2.2, the semiasymptotic matrix theory was used to obtain the propagation constant of the TE01 mode of the Bragg fiber. The dispersion coefficient of the TE01 mode is ~ − 1216 ps/km.nm, and it has an estimated radiation loss of ~ 0.02 dB/km, at 1550 nm. The average dispersion and FOM of the proposed DCF are ≈ − 1817 ps/km.nm and 190,000 ps/nm.dB, respectively, across the C-band of the EDFA. Due to the large magnitude of waveguide dispersion, we have neglected material dispersion in our estimations of D. We may also mention that the estimated values of the FOM may be a little optimistic, as we have not included the material-related losses in the calculations. However, as has been reported in [7], the material absorption loss suffered by the TE01 mode in air-core Bragg fibers is negligible as compared to the radiation loss, due to small penetration of modal field into the cladding.
Fig. 2. Dispersion spectrum of TE01 mode of proposed DCF
Fig. 3. Radiation loss spectrum of TE01 mode of proposed DCF
3.2 Non-zero dispersion shifted Bragg fiber for metro applications Metropolitan networks bridge the gap between local/access networks and long distance telecommunication networks. As a rule-of-thumb, metro networks are required to be suitably designed to address features like low installation cost, high degree of scalability, and dynamism that is capable of accommodating unpredicted traffic growth besides having the flexibility to add/drop individual signals at any central office in the network, and interoperability to support protocols such as SONET/SDH and IP. The low cost requirement implies that metro-specific fiber designs are aimed at minimizing use of components like amplifiers, dispersion compensators, gain flattening filters etc. Reported designs for metro fibers are based on their operation as negative dispersion fibers [9, 10] or positive dispersion fibers across the C-
band [11], so as to achieve a span length of ~ 100 km, without the need for a dispersion compensating device. However these networks still need an amplifier after approximately every 80 km. In this paper, we propose the potential use of a Bragg fiber as a metro fiber. Our designed Bragg fiber is potentially suitable for metro applications to achieve a dispersion limited span length of ~ 100 km @ 10 Gbits/s, without the need for any amplifier. The elimination of amplifiers should significantly reduce the overall system cost while drastically reducing complexities in network design. Our designed Bragg metro-fiber is a dispersion-flattened positive dispersion fiber, with ultra-low loss across the C-band. It has a core (air) radius of 10 µm and refractive index contrast of 2.0 in the cladding corresponding to the quarter-wave stack. The fiber exhibits an average dispersion of 9.9 ps/km.nm (cf. Fig. 4), and an average dispersion slope of 0.23 ps/km.nm2 across the C-band. The average radiation loss of the TE01 mode is ~ 0.04 dB/km across the same wavelength range. Again, the high differential loss of the higher order modes with respect to the TE01 mode makes the fiber essentially single-moded [8], while the non-degeneracy of the TE01 mode eliminates any difficult issues related to polarization mode dispersion.
Fig. 4. Dispersion spectrum of TE01 mode of proposed metro Bragg fiber Table I. Variation of radiation loss of the proposed metro Bragg fiber with wavelength Wavelength (nm)
1530
1540
1550
1560
1570
1580
1590
1600
1610
Radiation Loss (dB/km)
1.05E-01
4.03E-02
1.69E-02
7.65E-03
3.70E-03
1.89E-03
1.02E-03
5.74E-04
3.37E-04
4. CONCLUSION We have shown for the first time, to the best of our knowledge, that the multiple quarter-wave stack condition for the cladding layers can be used to tailor the dispersion characteristics of Bragg fibers to achieve high negative dispersion while maintaining low propagation loss. This feature was exploited to design an efficient dispersion compensating Bragg fiber, whose average waveguide dispersion was ~ −1817 ps/km.nm, across the C-band of the EDFA. It exhibits a dispersion of ~ − 1216 ps/km.nm at 1550 nm and has an estimated radiation loss of ~ 0.02 dB/km, at the same wavelength, thereby yielding an FOM ~ 69,000 ps/nm.dB at 1550 nm. The estimated FOM of the proposed DCF is at least two orders of magnitude higher than that of conventional DCFs. Tight confinement of the propagating mode within the core minimizes material-related losses and provides greater flexibility in choice of cladding materials. We have also shown the potential use of Bragg fibers as metro-specific fiber through suitable tailoring of its dispersion and loss characteristics. We have proposed a Bragg fiber design with an average dispersion slope of 0.23 ps/km.nm2 and an average radiation loss of 0.04 dB/km, across the C-band. This should allow a span length of ~ 100 km, devoid of any dispersion-compensator and amplifier, thereby significantly reducing overall installation cost and operational complexity of the network.
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