## Abstract

We propose a new structure of highly nonlinear dispersion-flattened (HNDF) photonic crystal fiber (PCF) with nonlinear coefficient as large as 30 W^{-1}km^{-1} at 1.55 μm designed by varying the diameters of the air-hole rings along the fiber radius. This innovative HNDF-PCF has a unique effective-index profile that can offer not only a large nonlinear coefficient but also flat dispersion slope and low leakage losses. It is shown through numerical results that the novel microstructured optical fiber with small normal group-velocity dispersion and nearly zero dispersion slope offers the possibility of efficient supercontinuum generation in the telecommunication window using a few ps pulses.

©2004 Optical Society of America

## 1. Introduction

Index-guiding photonic crystal fibers (PCFs) [1], also called holey fibers or microstructured optical fibers, are usually formed by a central solid defect region surrounded by multiple air holes with the same diameter arrayed in a regular triangular lattice. The chromatic dispersion profile can be easily controlled by varying the hole diameter and the hole-to-hole spacing. Controllability of chromatic dispersion in PCFs is a very important problem for practical applications to optical communication systems, dispersion compensation, and nonlinear optics. So far, various PCFs with remarkable dispersion properties as, for example, zero dispersion wavelengths shifted to the visible and near-infrared wavelengths [2] and an ultra-flattened chromatic dispersion [3,4] have been reported. However, using a PCF with all of the same air-hole diameter in the cladding region, it is difficult to control both the dispersion and the dispersion slope in a wide wavelength range and to design a highly nonlinear dispersion-flattened PCF (HNDF-PCF) in 1.55-μm wavelength range. In index-guiding PCFs, since the periodicity in the cladding region is not essential to confine the guiding light into the high-index core region, we have reported that various effective refractive-index profiles can be obtained by varying the hole diameter of each air-hole ring along the radius [5]. In this paper, using this design principle, we propose a new structure of HNDF-PCF with nonlinear coefficient as large as 30 W^{-1}km^{-1} at 1.55 μm. It is shown through numerical results that the proposed HNDF-PCF with small normal group-velocity dispersion and nearly zero dispersion slope is suitable for efficient supercontinuum (SC) generation in the telecommunication window.

## 2. Highly nonlinear dispersion-flattened PCF

Figure 1 shows two examples of PCFs and their effective refractive-index profiles, where *d _{i}*(

*i*= 1 to

*n*) is the hole diameter of

*i*th air-hole ring and the air-hole diameters are

*d*

_{1}>

*d*

_{2}= … =

*d*in Fig. 1(a) and

_{n}*d*

_{2}<

*d*

_{1}=

*d*

_{3}= …; =

*d*in Fig. 1(b). The effective refractive index in the cladding region increases with decreasing the air-hole diameter and the effective refractive index in the cladding region decreases with increasing the air-hole diameter. By optimizing the air-hole diameters

_{n}*d*and the hole-to-hole spacing, both the dispersion and the dispersion slope can be controlled in a wide wavelength range.

_{i}Using a regular triangular PCF with a small core and a large air-hole diameter, high nonlinearity and tight mode confinement could be achieved and various highly nonlinear PCFs have been reported [6, 7]. However, the PCFs with a small core and a large air-hole diameter tend to have zero-dispersion wavelength at shorter wavelength ranges than telecommunication window and also it is difficult to design a HNDF-PCF in 1.55-μm wavelength range. Recently, a nonlinear dispersion-flattened PCF with elliptical core has been designed and fabricated by T. Yamamoto, *et al*. [8], though, its nonlinear coefficient is 19 W^{-1}km^{-1} at 1.55 μm. On the other hand, varying each air-hole diameter along the radius, it is possible to design a PCF with very low and flat dispersion, low confinement loss, and nonlinear coefficient as large as ~30 W^{-1}km^{-1}. Figure 2 shows an example of HNDF-PCF with ten air-hole rings, where the hole-to-hole spacing Λ = 0.89 μm, *d*
_{1} = 0.41 Λ, *d*
_{2} = 0.85Λ, *d*
^{3} = 0.92Λ, *d*
_{4} = 0.53Λ, *d*
_{5} = … = *d*
_{10} = 0.60Λ. It is possible to decrease the effective refractive index in the cladding region by increasing the air-hole diameters along the fiber radius. It is also possible to realize flattened dispersion by using the effective refractive-index profile shown in Fig. 2(b) [9] with small hole-to-hole spacing. The proposed structure is deduced numerically to best meet the performance criteria such as flat and zero dispersion, low confinement loss, high nonlinear coefficient, and single-mode operation. Figure 3 shows the chromatic dispersion, the confinement loss, and the effective mode area *A _{eff}* as a function of wavelength for the PCF in Fig. 2(a) calculated by using full vectorial modal solver based on finite element method [10], where the material dispersion given by Sellmeier’s formula is directly included in the calculations. The wavelength range for which the PCF dispersion remains between -0.5 and 0.0 ps/(km·nm) is from 1.46 μm to 1.64 μm and the dispersion is -0.12 ps/(km·nm) at 1.55 μm wavelength. The confinement loss of the fundamental mode for this PCF is less than 0.1 dB/km for wavelengths shorter than 1.6 μm. In short wavelength ranges, this PCF supports the second order mode, however, the confinement loss of the second order mode is larger than 500 dB/m for wavelengths over 1.4 μm and the effective refractive indices of the second order mode are quite different from those of the fundamental mode, so this PCF effectively operates as a single mode fiber in the telecommunication window. The effective mode area is 2.8 μm

^{2}at 1.55 μm wavelength and the nonlinear coefficient is about 34 W

^{-1}km

^{-1}. This PCF with small normal group-velocity dispersion, nearly zero dispersion slope, and small effective mode area would be useful for broadband SC generation, soliton pulse transmission, nonlinear optical loop mirror, wavelength conversion [11], and so on.

## 3. Supercontinuum generation in HNDF-PCF

SC generation in the HNDF-PCF proposed here is numerically investigated. SC generation in the 1.55-μm band has important applications, such as, a spectral-slicing pulse source used in wavelength division multiplexing optical transmission systems [12], an optical wavelength convertor [11], and an all-optical signal processing. Especially, dispersion-flattened fibers with normal group-velocity dispersion are widely used to generate SC because of the flatness of the SC spectrum and the good quality of the spectrum-sliced channels [13]. The HNDF-PCF with normal group-velocity dispersion would be efficient for SC generation. Our numerical model uses the nonlinear Schrödinger equation (NLSE) [14]:

where *A* is the complex amplitude of the optical field, α is the loss constant of the fiber, β_{i}(*i* = 1 to 4) are the *i*th order of the Taylor series expansion of the propagation constant around the carrier frequency, γ is the nonlinear coefficient, λ_{c} is the center wavelength, and *T _{R}* is the Raman scattering parameter, respectively. This NLSE is solved using split-step Fourier method.

We consider the propagation of the sech^{2} waveform with the full width at half maximum (FWHM), *T _{FWHM}*, of 2.5 ps through the HNDF-PCF in Fig. 2(a). The fiber parameters used in the calculation are given as γ = 34.5 W

^{-1}km

^{-1},

*T*= 3.0 fs, β

_{R}_{2}= 0.15 ps

^{2}/km, β

_{3}= 0.0 ps

^{3}/km, and β

_{4}= 2 × 10

^{-4}ps

^{4}/km at the center wavelength λ

_{c}= 1.558 μm. Figure 4 shows the evolution of the waveforms and their spectral contents, obtained by split-step Fourier method, where the fiber length is 320 m and the peak power of the incident pulse is 6.0 W. Relatively flat SC spectrum with the 10-dB bandwidth of 60 nm is achieved despite the short fiber length of 320 m.

## Conclusions

A new controlling technique of chromatic dispersion in index-guiding PCFs was applied to design a highly nonlinear dispersion-flattened PCF. It was shown through numerical results that it is possible to design a ten-ring PCF with low and flat dispersion, low confinement loss, and large nonlinear coefficient γ > 30 W^{-1}km^{-1} around 1.55-μm wavelength and that this PCF can be used for efficient SC generation. In order to fabricate this PCF, silica capillaries with five different sized air-hole diameters are required. Fabrication of this PCF is now under consideration.

## References and links

**1. **P. St. J. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

**2. **M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevtsev, T.A. Birks, J.C. Knight, and P.St.J. Russell, “Experimantal measurement of group velocity dispersion in photonic crystal fibre,” Electron. Lett. **35**, 63–64 (1999). [CrossRef]

**3. **A. Ferrando, E. Silvestre, P. Andrés, J.J. Miret, and M.V. Andrés, “Desinging the properties of dispersion-flattened photonic crystal fibers,” Opt. Express **9**, 687–697 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687. [CrossRef] [PubMed]

**4. **W.H. Reeves, J.C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express **10**, 609–613 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-14-609. [CrossRef] [PubMed]

**5. **K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion,” Opt. Express **11**, 843–852 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843. [CrossRef] [PubMed]

**6. **A. Bjarklev, J. Broeng, and A.S. Bjarklev, *Photonic Crystal Fibres*, (Kluwer Academic Publishers, 2003). [CrossRef]

**7. **V. Finazzi, T.M. Monro, and D.J. Richardson, “Small-core holey fibers: nonlinearity and confinement loss trade-offs,” J. Opt. Soc. Am. B **20**, 1427–1436 (2003). [CrossRef]

**8. **T. Yamamoto, H. Kubota, S. Kawanishi, M. Tanaka, and S. Yamaguchi, “Supercontinuum generation at 1.55 μm in a dispersion-flattened polarization-maintaining photonic crystal fiber,” Opt. Express **11**, 1537–1540 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1537. [CrossRef] [PubMed]

**9. **W. Lieber, M. Loch, H. Etzkorn, W.E. Heinlein, K.-F. Klein, H.U. Bonewitz, and A. Mühlich, “Three-step index strictly single-mode, only F-doped silica fibers for broad-band low dispersion,” J. Lightwave Technol. **LT-4**, 715–719 (1986). [CrossRef]

**10. **K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. **38**, 927–933 (2002). [CrossRef]

**11. **T. Okuno, M. Hirano, T. Kato, M. Shigematsu, and M. Onishi, “Highly nonlinear and perfectly dispersion-flattened fibers for efficient optical signal processing applications,” Electron. Lett. **39**, 972–974 (2003). [CrossRef]

**12. **T. Morioka, K. Okamoto, M. Ishii, and M. Saruwatari, “Low-noise, pulsewidth tunable picosecond to femtosecond pulse generation by spectral filtering of wideband supercontinuum with variable bandwidth arrayed-waveguide grating filters,” Electron. Lett. **32**, 836–837 (1996). [CrossRef]

**13. **Y. Takushima, F. Futami, and K. Kikuchi, “Generation of over 140-nm-wide super-continuum from a normal dispersion fiber by using a mode-locked semiconductor laser source,” IEEE Photon. Technol. Lett. **10**, 1560–1562 (1998). [CrossRef]

**14. **G. Agrawal, *Nonlinear Fiber Optics*, Academic Press (San Diego, CA), 2dn Edition (1995).