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We report a novel idea for achieving highly efficient dispersion-compensating Bragg fiber by exploiting a modified quarter-wave stack condition. Our Bragg fiber ...
August 1, 2005 / Vol. 30, No. 15 / OPTICS LETTERS

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Design of dispersion-compensating Bragg fiber with an ultrahigh figure of merit Sonali Dasgupta, Bishnu P. Pal, and M. R. Shenoy Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016, India Received January 24, 2005 We report a novel idea for achieving highly efficient dispersion-compensating Bragg fiber by exploiting a modified quarter-wave stack condition. Our Bragg fiber yielded an average dispersion of ⬇−1800 ps/ 共nm km兲 across the C band for the fundamental TE mode and an ultrahigh figure of merit of ⬇180,000 ps/ 共nm dB兲, which is at least 2 orders of magnitude higher than that of conventional dispersioncompensating fibers. The proposed methodology could be adopted for the design of a dispersion compensator across any desired wavelength range. © 2005 Optical Society of America OCIS codes: 060.2340, 260.2030.

Bragg fibers are one-dimensional photonic bandgap (PBG) fibers, which consist of a low-index core surrounded by a cladding that comprises alternate layers of high- and low-refractive-index materials.1 The radially periodic refractive-index profile of the fiber results in a one-dimensional photonic bandgap. The loss and dispersion characteristics of these fibers can be tailored by judicious choice of the core radius, refractive-index contrast, and thickness of the cladding layers. In recent years there has been considerable interest in obtaining dispersion-compensating fibers (DCFs),2–4 even though the first proposal for a DCF was made as early as in 1980.5 Typically the fundamental TE mode 共TE01兲 of a Bragg fiber, designed according to the “quarter-wave stack condition,”1 exhibits large positive dispersion in its characteristic low-loss wavelength window, which is contrary to the negative dispersion required for a DCF. Nevertheless, Bragg fibers have been designed to generate large negative dispersion. They were based on exploiting the HE11 mode of a Bragg fiber. The dispersion coefficient for the HE11 mode of one such dispersion-compensating Bragg fiber (DCBF) was estimated to be ⬇−20,000 ps/ 共nm km兲.6 However, the hybrid nature of the HE11 mode and the small core radius 共⬍1 ␮m兲 required for achieving such characteristics limited its applicability. An alternative design, which was based on introducing a defect layer into an otherwise periodic multilayer cladding, exploited the TE01 mode of a Bragg fiber to generate a large negative dispersion coefficient.7 Calculations showed that such a Bragg fiber could yield very large negative dispersion, up to ⬇−50,000 ps/ 共nm km兲.7 However, it is conceivable that transmission loss in this design may not be very low in practice, since the functional principle necessitates that the modal field penetrate into the cladding for the required interactions in the defect layer. It may be worthwhile to draw the reader’s attention to other novel PBG waveguides8,9: Fresnel waveguides with the potential to yield dispersioncompensating features have been proposed. In this Letter we propose the use of multiple quarter-wave stacks, for the first time to our knowledge, to design a 0146-9592/05/151917-3/$15.00

DCBF with high figure of merit (FOM) based on the fundamental TE mode. Here we outline our design recipe. The well-known quarter-wave stack condition minimizes the radiation loss of the TE01 mode of a Bragg fiber.1 Functionally, the periodic cladding of a Bragg fiber is analogous to the multilayers of alternate refractive indices in thin-film interference filters, which exhibit high reflectivity if the thickness of the adjacent layers (forming a period) corresponds to a round-trip phase of 2␲. In view of this analogy, we explored the feasibility of imposing the following multiple quarterwave stack condition on the design of applicationspecific Bragg fibers: 2␲ ␭0

n 1l 1 =

2␲

␲ n 2l 2 = m , ␭0 2

共1兲

where m is an odd integer 共m ⬎ 1兲; 共l1 , l2兲 and 共n1 , n2兲 are the thicknesses and the refractive indices, respectively, of the cladding layers, and ␭0 is the central wavelength of the bandgap that corresponds to the cladding layers. Consistent with our expectations, it was indeed found that the fundamental quarter-wave stack condition 共m = 1兲 is not necessarily an essential requirement for good confinement of TE modal fields; tight modal confinement is also achievable if the cladding layers of the Bragg fibers are chosen according to Eq. (1). We exploited this physical feature to tailor the dispersion characteristics of the TE01 mode in a Bragg fiber to obtain a DCBF while we retained its low-loss characteristics. Once the material system (values of n1 and n2) is chosen, the thicknesses of the cladding layers (l1 and l2) are initially chosen according to the multiple quarter-wave stack condition at ␭0, such that m = 3 in Eq. (1). ␭0 is chosen such that it is ⬇20% greater than the operating wavelength ␭op, at which negative dispersion is desired. Under this condition, the modal field becomes sensitive to any small change in the operating wavelength. As a result, the TE01 mode exhibits large negative dispersion, similar to the mechanism responsible for generation of large negative dispersion in conventional DCFs.2 Also, such a choice for the cladding thickness © 2005 Optical Society of America

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OPTICS LETTERS / Vol. 30, No. 15 / August 1, 2005

Fig. 1. Dispersion and radiation loss spectra for the TE01 mode of the proposed DCBF.

results in a reduced sensitivity of the radiation loss to variations in wavelength, thereby yielding a wide wavelength window for low-loss and large negative dispersion. We may note that ␭0 needs to be judiciously chosen so that ␭op does not lie completely outside the photonic bandgap. If it did, the radiation loss of the mode would be very large and would render the design useless as an efficient dispersion compensator. Essentially, there exists a trade-off between low-loss and high negative dispersion. A cladding thickness corresponding to ␭op = ␭0 yields minimal radiation loss at ␭op, as the mode lies well inside the bandgap and the field is tightly confined within the core. However, such a well-confined mode would not exhibit high negative dispersion, which is achievable only through the choice of a wavelength smaller than ␭0, at the cost of increased radiation loss of the mode. We assumed an operating wavelength of 1550 nm and hence ␭0 was chosen to be ⬇1900 nm to satisfy Eq. (1), with m = 3. The fiber was assumed to have an air core of radius 3.5 ␮m, and the refractive indices of the cladding layers were chosen to be 2.8 and 1.5, respectively.10 We used the matrix theory11 to analyze the fiber and calculate the effective index of the propagating mode, and the dispersion and loss. Once the initial dispersion curve was attained by use of the above-mentioned multiple quarter-wave stack condition, we carried out fine tuning by altering the thickness of the high-index layer to achieve the desired dispersion spectrum of the DCF. Figure 1 shows the variation of dispersion coefficient and radiation loss of the TE01 mode of the proposed DCBF. Our DCBF exhibits dispersion and radiation losses of ⬇−1245 ps/ 共nm km兲 and ⬇0.006 dB/ km (with 20 cladding bilayers), respectively, at 1550 nm. These values amount to an effective FOM of ⬇200,000 ps/ 共nm dB兲, which is ⬃2 orders of magnitude larger than that of conventional DCFs. The average dispersion of the DCBF is ⬇−1800 ps/ 共nm km兲 across the C-band, with an estimated average radiation loss of ⬇0.1 dB/ km. We have also investigated the sensitivity of the results given above to the choice of the fiber’s core radius. The core radius was found to have a significant effect on the dispersion slope of the TE01 mode. A

smaller core radius leads to an increase in dispersion slope and also to a shift of the zero-dispersion wavelength to slightly longer wavelengths (Fig. 2). To retain the dispersion characteristics with a larger core (e.g., 10 ␮m), we must increase the separation between ␭0 and ␭op such that ␭0 is greater than 20% of ␭op, which essentially pushes ␭op further toward the edge of the bandgap, thereby yielding large negative dispersion. The multiple quarter-wave stack condition [m = 3 in Eq. (1)] enabled us to achieve low radiation loss and large negative dispersion simultaneously in a Bragg fiber via the TE01 mode without necessitating any defect in the cladding layers. Efficient coupling to the transmission fibers should be possible through appropriate tapering of the designed DCBF to attain good modal overlap. Though the proposed design supports more than one mode in the operating wavelength band, the high differential loss suffered by the higher-order modes with respect to the TE01 mode makes the fiber effectively single moded. For example, the estimated radiation loss of the TE02 mode was ⬇1.39⫻ 108 dB/ km at 1550 nm, compared to 0.006 dB/ km for the TE01 mode, in the proposed DCBF. Therefore the inherent nondegeneracy of the TE01 mode makes the proposed DCF free from issues of polarization mode dispersion, which should be an attractive feature from a system point of view. Moreover, the threshold power for nonlinear effects to occur in the proposed air-core DCBF is expected to be much higher than for the solid-core conventional DCFs, thereby leading to better power-handling capability.12 We may mention that the estimated values of FOM might be a little optimistic, as we have not included material-related losses in the calculations and have assumed no imperfections at the cladding layer interfaces. The material absorption loss suffered by the TE01 mode in air-core Bragg fibers is negligible compared with the radiation loss.13 We also point out that the dispersion curve is highly sensitive to cladding thickness. Nevertheless, assuming that perfection in fabrication technology will be achievable in the near future, DCBFs should be useful as efficient dispersion compensators.

Fig. 2. Effect of core radius on the dispersion spectrum of the designed DCBF.

August 1, 2005 / Vol. 30, No. 15 / OPTICS LETTERS

In conclusion, we have proposed, for the first time to the best of our knowledge, a simple multiple quarter-wave stack condition for the choice of cladding layers of Bragg fibers to tailor their dispersion characteristics and achieve large negative dispersion while maintaining low propagation loss for the fundamental TE mode. We exploited this property to design an efficient DCBF whose average dispersion was ⬇−1800 ps/ 共nm dB兲 across the C-band of an erbiumdoped fiber amplifier. It exhibits a dispersion of ⬇−1245 共ps/ km兲 nm at 1550 nm with an estimated radiation loss of ⬇0.006 dB/ km at the same wavelength, thereby yielding a FOM of ⬇200,000 ps共nm dB兲. Tight confinement of the propagating mode within the core is expected to minimize material-related losses and thereby provide greater flexibility in the choice of cladding materials. We have also studied the dependence of dispersion characteristics on the core radius of the DCBF. Further work is in progress to explore the feasibility of designing a broadband DCBF. S. Dasgupta thanks the Council of Scientific and Industrial Research, Government of India, for the award of a Shyama Prasad Mukherjee fellowship. B. P. Pal’s e-mail address is [email protected]. References 1. P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. 68, 1196 (1978).

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2. B. Jopson and A. Gnauck, IEEE Commun. Mag. 33(6), 96 (1995). 3. L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, T. Magnussen, C. C. Larsen, and H. Daamsgard, Opt. Fiber Technol. 6, 164 (2000). 4. J. L. Auguste, J. M. Blondy, J. Monry, J. Marcou, B. Dussardier, G. Monnom, R. Jindal, K. Thyagarajan, and B. P. Pal, Opt. Fiber Technol. 8, 89 (2002). 5. C. Lin, H. Kogelnik, and L. G. Cohen, Opt. Lett. 5, 476 (1980). 6. G. Ouyang, Y. Xu, and A. Yariv, Opt. Express 10, 899 (2002). 7. T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, Opt. Express 11, 1175 (2003). 8. J. Canning, Opt. Commun. 207, 35 (2002). 9. J. Canning, E. Buckley, and K. Lyytikainen, Opt. Lett. 28, 230 (2003). 10. S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, Science 296, 510 (2002). 11. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, J. Lightwave Technol. 20, 428 (2002). 12. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, Opt. Express 9, 748 (2001). 13. Y. Xu, A. Yariv, J. G. Fleming, and S. Y. Lin, Opt. Express 11, 1039 (2003).