Abstract

In order to control the dispersion and the dispersion slope of index-guiding photonic crystal fibers (PCFs), a new controlling technique of chromatic dispersion in PCF is reported. Moreover, our technique is applied to design PCF with both ultra-low dispersion and ultra-flattened dispersion in a wide wavelength range. A full-vector finite element method with anisotropic perfectly matched layers is used to analyze the dispersion properties and the confinement losses in a PCF with a finite number of air holes. It is shown from numerical results that it is possible to design a fourring PCF with flattened dispersion of 0±0.5 ps/(km·nm) from a wavelength of 1.19 µm to 1.69 µm and a five-ring PCF with flattened dispersion of 0 ±0.4 ps/(km·nm) from a wavelength 1.23 µm to 1.72 µm.

© 2003 Optical Society of America

1. Introduction

Photonic crystal fibers (PCFs) [1, 2] consisting of a central defect region surrounded by multiple air holes that run along the fiber length are attracting much attention in recent years because of unique properties which are not realized in conventional optical fibers. PCFs are divided into two different kinds of fibers. The first one, index-guiding PCF, guides light by total internal reflection between a solid core and a cladding region with multiple air-holes [3, 4]. On the other hand, the second one uses a perfectly periodic structure exhibiting a photonic band-gap (PBG) effect at the operating wavelength to guide light in a low index core-region [5, 6].

Index-guiding PCFs, also called holey fibers or microstructured optical fibers, possess the especially attractive property of great controllability in chromatic dispersion by varying the hole diameter and hole-to-hole spacing. Control 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 [7, 8], an ultra-flattened chromatic dispersion [911], and a large positive dispersion with a negative slope in the 1.55 µm wavelength range [12], have been reported. However, in conventional PCFs, the chromatic dispersion is controlled by using air-holes with same diameter in a cladding region. Using a conventional design technique, it is difficult to control the dispersion slope in wide wavelength range.

In this paper, in order to control dispersion and dispersion slope, a new controlling technique of chromatic dispersion in PCF is reported. Moreover, our technique is applied to design PCF with both ultra-low dispersion and ultra-flattened dispersion in a wide wavelength range. A full-vector finite element method (FEM) is used to analyze the dispersion properties. Here, anisotropic perfectly matched layers (PMLs) [13] are incorporated into a full-vector FEM with curvilinear hybrid edge/nodal elements [14] to evaluate confinement losses in a PCF with a finite number of air holes. It is shown from numerical results that it is possible to design a four-ring PCF with flattened dispersion of 0±0.5 ps/(km·nm) from a wavelength of 1.19 µm to 1.69 µm and a five-ring PCF with flattened dispersion of 0±0.4 ps/(km·nm) from a wavelength of 1.23 µm to 1.72 µm.

2. Analysis method

We consider a PCF surrounded by PML regions 1 to 8 with thicknesses ti (i=1 to 4), the cross section of which is shown in Fig. 1, where x and y are the transverse directions, z is the propagation direction, PML regions 1, 2 and 3, 4 are faced with the x and y directions, respectively, regions 5 to 8 correspond to the four corners, and Wx and Wy are the computational window sizes along the x and y directions, respectively.

Using an anisotropic PML [13], from Maxwell’s equations the following vectorial wave equation is derived:

×([s]1×E)k02n2[s]E=0

with

[s]=[sysx000sxsy000sxsy]

where k 0=2π/λ the free-space wavenumber, λ is the wavelength, E denotes the electric field, and n is the refractive index. The PML parameters sx and sy are given in Table 1, where the values of si (i=1 to 4) are complex as

si=1jαi(ρti)2

where ρ is the distance from the beginning of PML. Attenuation of the field E in PML regions can be controlled by choosing the values of αi appropriately.

 

Fig. 1. Transverse cross section of photonic crystal fiber surrounded by PMLs.

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When applying a full-vector FEM to PCFs, a curvilinear hybrid edge/nodal element [14] is very useful for avoiding spurious solutions and for accurately modeling curved boundaries of air holes [15]. Dividing the fiber cross section into a number of the curvilinear hybrid elements, from Eq. (1) we can obtain the following eigenvalue equations:

[K]{E}=k02neff2[M]{E}

where [K] and [M] are the finite element matrices, {E} is the discretized electric field vector consisting of the edge and nodal variables, and neff is the effective index. Utilizing sparse nature of [K] and [M], Eq. (4) is solved with the multifrontal method [16] which reorganizes the overall factorization of a sparse matrix into a sequence of partial factorizations of dense smaller matrices.

The chromatic dispersion D of a PCF is easily calculated from the neff values versus the wavelength using

D=λcd2Re[neff]dλ2

where c is the velocity of light in a vacuum and Re stands for the real part. The material dispersion given by Sellmeier’s formula is directly included in the calculation. The confinement loss is also an important parameter to design a PCF with a finite number of air holes. The modes of single-material PCFs are inherently leaky because the core index is the same as the index of the outer cladding region without air holes. The confinement loss is deduced from the value of neff as

confinementloss=8.686Im[k0neff]

in dB/m, where Im stands for the imaginary part.

In order to check the accuracy of the full-vector FEM, we consider an index-guiding PCF with a single ring of six equally spaced air holes with hole diameter of 5 µm, hole-to-hole spacing of 6.75 µm, the background refractive index of 1.45, and the operating wavelength λ=1.45 µm [17]. The computed effective index of the fundamental mode neff=1.4453952+j3.19×10-8. This result is in good agreement with that of the full-vector multipole method [17].

Tables Icon

Table 1. PML parameters.

3. Design principle

In conventional PCFs, the cladding structure is usually formed by air holes with the same diameter arrayed in a regular triangular lattice. The chromatic dispersion profile can be easily engineered by varying the hole diameter and hole-to-hole spacing. However, using a PCF with all of the same air-hole diameter in the cladding region, it is difficult to control the dispersion slope in wide 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 propose an index-guiding PCF as shown in Fig. 2 to control both the dispersion and dispersion slope in wide wavelength range, where Λ is the hole-to-hole spacing and di (i=1 ~ n) is the hole diameter of ith air-hole ring. In the short wavelength range, the guided mode is well confined into the core region and the dispersion property is affected by the inner air-hole rings, while in the long wavelength range, the effective core area is increased and the dispersion property is affected not only inner rings but also outer rings, particularly when the hole-to-hole spacing is small. By optimizing the each air-hole diameter di (i=1 ~ n) and hole-to-hole spacing, both the dispersion and dispersion slope can be controlled in wide wavelength range.

In order to show how the chromatic dispersion is affected by varying the hole diameter di of each air-hole ring, we consider four index-guiding PCFs with four rings of air holes as shown in Fig. 3, where the hole-to-hole spacing Λ is 2.0 µm and the each air-hole diameter di (i=1 ~ 4) is d 1=d 2=d 3=d 4=0.5 µm (Fig. 3(a)), d 1=0.5 µm, d 2=d 3=d 4=0.6 µm (Fig. 3(b)), d 1=0.5 µm, d 2=0.6 µm, d 3=d 4=0.7 µm (Fig. 3(c)), d 1=0.5 µm, d 2=0.6 µm, d 3=0.7 µm, d 4=1.8 µm (Fig. 3(d)). The hole diameter d 4 in Fig. 3(d) is very larger than the others to show a distinct difference between the dispersion profile of PCF in Fig. 3(c) and Fig. 3(d). Figure 4 shows the chromatic dispersion of these four PCFs. This result indicates that the dispersion profile is affected by varying the each air-hole diameter di and optimization of di enables index-guiding PCFs to control their dispersion properties.

 

Fig. 2. Cross section of proposed photonic crystal fiber. Λ is the hole-to-hole spacing and di (i=1 ~ n) is the hole diameter of ith air-hole ring.

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Fig. 3. Index-guiding PCFs with four rings of air holes. The air holes are shown as colored circles. The hole-to-hole spacing Λ=2.0 µm and the each air-hole diameter is (a) d 1=d 2=d 3=d 4=0.5 µm, (b) d 1=0.5 µm, d 2=d 3=d 4=0.6 µm, (c) d 1=0.5 µm, d 2=0.6 µm, d 3=d 4=0.7 µm, (d) d 1=0.5 µm, d 2=0.6 µm, d 3=0.7 µm, d 4=1.8 µm.

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Fig. 4. Chromatic dispersion curves as a function of wavelength for PCFs with four rings of air holes in Fig. 3. The hole-to-hole spacing Λ=2.0 µm and the each air-hole diameter is (a) d 1=d 2=d 3=d 4=0.5 µm, (b) d 1=0.5 µm, d 2=d 3=d 4=0.6 µm, (c) d 1=0.5 µm, d 2=0.6 µm, d 3=d 4=0.7 µm, (d) d 1=0.5 µm, d 2=0.6 µm, d 3=0.7 µm, d 4=1.8 µm.

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4. Ultra-low and ultra-flattened dispersion

PCFs with ultra-flattened dispersion have been investigated numerically by Ferrando et al. [8, 9] and demonstrated experimentally by W.H. Reeves et al. [10]. The cladding structure of conventional ultra-flattened dispersion PCFs is formed by air holes with the same diameter d arrayed in a regular triangular lattice. Setting the hole-to-hole spacing Λ and the air-hole diameter d as Λ ≈ 2.6 µm and d/Λ ≈ 0.24, respectively, it is possible to realize a nearly zero ultra-flattened dispersion PCF in the telecommunication window. However, since the value of d/Λ is small, more than twenty rings of arrays of air holes are required in the cladding region to reduce the confinement loss to a level of 0.1 dB/km. On the other hand, using the design principle described in the previous section, it is possible to design a PCF with both ultraflattened dispersion and low confinement loss.

Figure 5 shows two examples of nearly zero ultra-flattened dispersion PCFs with four air-hole rings and five air-hole rings designed by using our design principle, where Λ=1.56 µm, d 1/Λ=0.32, d 2/Λ=0.45, d 3/Λ=0.67, d 4/Λ=0.95 in Fig. 5(a) and Λ=1.58 µm, d 1/Λ=0.31, d 2/Λ=0.45, d 3/Λ=0.55, d 4/Λ=0.63, d 5/Λ=0.95 in Fig. 5(b). These PCFs have small hole-to-hole spacing (Λ ≈ 1.6 µm). In order to control the dispersion and the dispersion slope in the wide wavelength range, a small hole-to-hole spacing is an important parameter. Moreover, it is possible to decrease effective refractive index in the cladding region along the radius by increasing the air-hole diameters along the radius and to realize flattened dispersion by optimizing each air-hole diameter. Increasing the air-hole diameters along the radius is also very useful for designing low loss index-guiding PCFs with small number of air-hole rings. The last row of air holes with very large diameter in Fig. 5 is profitable for not only controlling the dispersion but also reducing the confinement loss.

Figure 6 shows the chromatic dispersion, the confinement loss, and the effective mode area as a function of wavelength for the four-ring PCF in Fig. 5(a). The effective mode area Aeff is calculated with

Aeff=(E2dxdy)2E4dxdy.

The wavelength range for which the PCF dispersion remains between -0.5 and +0.5 ps/(km·nm) is from 1.19 µm to 1.69 µm. The confinement loss is less than 1 dB/km in the wavelength range shorter than 1.6 µm. The effective mode area is 8.55 µm2 at 1.55 µm wavelength. Figure 7 shows the chromatic dispersion, the confinement loss, and the effective mode area as a function of wavelength for the five-ring PCF in Fig. 5(b). The wavelength range for which the PCF dispersion remains between -0.4 and +0.4 ps/(km·nm) is from 1.23 µm to 1.72 µm. The confinement loss is less than 0.1 dB/km in the wavelength range shorter than 1.72 µm. The effective mode area is 8.95 µm2 at 1.55 µm wavelength. Of course, using more air-hole rings and optimizing Λ and di, it is possible to design the flattened dispersion PCFs in wider wavelength range. These PCFs with nearly zero ultra-flattened dispersion have a relatively small effective more area and would be useful for supercontinuum generation, soliton pulse transmission, and so on. Our controlling technique of chromatic dispersion in PCFs is applied to design PCF with nearly zero ultra-flattened dispersion in this paper, however, this design principle can be also applied to control dispersion slope of a low nonlinear PCF with large mode area and design a dispersion compensating PCF.

 

Fig. 5. Ultra-flattened dispersion PCFs with (a) four air-hole rings and (b) five air-hole rings. The hole-to-hole spacing and the air-hole diameters are (a) Λ=1.56 µm, d 1/Λ=0.32, d 2/Λ=0.45, d 3/Λ=0.67, d 4/Λ=0.95 and (b) Λ=1.58 µm, d 1/Λ=0.31, d 2/Λ=0.45, d 3/Λ=0.55, d 4/Λ=0.63, d 5/Λ=0.95.

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Fig. 6. (a) Chromatic dispersion curve, (b) confinement loss, and (c) effective mode area as a function of wavelength for ultra-flattened dispersion PCF with four air-hole rings in Fig. 5(a).

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Fig. 7. (a) Chromatic dispersion curve, (b) confinement loss, and (c) effective mode area as a function of wavelength for ultra-flattened dispersion PCF with five air-hole rings in Fig. 5(b).

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5. Conclusion

A new controlling technique of chromatic dispersion in index-guiding PCFs was proposed and applied to design PCFs with both ultra-low dispersion and ultra-flattened dispersion in a wide wavelength range. It was shown from numerical results that it is possible to design a four-ring PCF with flattened dispersion of 0±0.5 ps/(km·nm) from a wavelength of 1.19 µm to 1.69 µm and a five-ring PCF with flattened dispersion of 0±0.4 ps/(km·nm) from a wavelength of 1.23 µm to 1.72 µm. Fabrication of these PCFs is now under consideration. The knowledge reported in this paper will have a great impact on many engineering applications such as dispersion compensation, wide-band supercontinuum generation, ultra-short soliton pulse transmission, and wavelength-division multiplexing transmission using PCFs. The applications of our design principle to PCFs with other unique dispersion properties will be reported soon.

References and links

1. J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330, (1999). [CrossRef]  

2. T.A. Birks, J.C. Knight, B.J. Mangan, and P.St.J. Russell, “Photonic crystal fibers: An endless variety,” IEICE Trans. Electron. E84-C, 585–592, (2001).

3. J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549, (1996). [CrossRef]   [PubMed]  

4. T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963, (1997). [CrossRef]   [PubMed]  

5. J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fiber,” Science 282, 1476–1478, (1998). [CrossRef]   [PubMed]  

6. R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Singlemode photonic band gap guidance of light in air,” Science 285, 1537–1539, (1999). [CrossRef]   [PubMed]  

7. 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]  

8. J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.St.J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809, (2000). [CrossRef]  

9. A. Ferrando, E. Silvestre, J.J. Miret, and P. Andrés, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792, (2000). [CrossRef]  

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

11. 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]  

12. T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber - A practical derivative of photonic crystal fiber,” Proc. Mater. Res. Soc. Spring Meeting L4.2. (2002).

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

14. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743, (2000). [CrossRef]  

15. M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315, (2001). [CrossRef]  

16. J.W.H. Liu, “The multifrontal method for sparse matrix solutions: theory and practice,” SIAM Rev. 34, 82–109,(1992). [CrossRef]  

17. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330, (2002). [CrossRef]  

References

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  1. J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330, (1999).
    [Crossref]
  2. T.A. Birks, J.C. Knight, B.J. Mangan, and P.St.J. Russell, “Photonic crystal fibers: An endless variety,” IEICE Trans. Electron. E84-C, 585–592, (2001).
  3. J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549, (1996).
    [Crossref] [PubMed]
  4. T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963, (1997).
    [Crossref] [PubMed]
  5. J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fiber,” Science 282, 1476–1478, (1998).
    [Crossref] [PubMed]
  6. R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Singlemode photonic band gap guidance of light in air,” Science 285, 1537–1539, (1999).
    [Crossref] [PubMed]
  7. 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]
  8. J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.St.J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809, (2000).
    [Crossref]
  9. A. Ferrando, E. Silvestre, J.J. Miret, and P. Andrés, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792, (2000).
    [Crossref]
  10. A. Ferrando, E. Silvestre, P. Andrés, J.J. Miret, and M.V. Andrés, “Desinging the properties of dispersionflattened photonic crystal fibers,” Opt. Express 9, 687–697, (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687
    [Crossref] [PubMed]
  11. 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]
  12. T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber - A practical derivative of photonic crystal fiber,” Proc. Mater. Res. Soc. Spring Meeting L4.2. (2002).
  13. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. 38, 927–933, (2002).
    [Crossref]
  14. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743, (2000).
    [Crossref]
  15. M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315, (2001).
    [Crossref]
  16. J.W.H. Liu, “The multifrontal method for sparse matrix solutions: theory and practice,” SIAM Rev. 34, 82–109,(1992).
    [Crossref]
  17. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330, (2002).
    [Crossref]

2002 (3)

2001 (3)

M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315, (2001).
[Crossref]

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

T.A. Birks, J.C. Knight, B.J. Mangan, and P.St.J. Russell, “Photonic crystal fibers: An endless variety,” IEICE Trans. Electron. E84-C, 585–592, (2001).

2000 (3)

1999 (3)

R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Singlemode photonic band gap guidance of light in air,” Science 285, 1537–1539, (1999).
[Crossref] [PubMed]

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]

J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330, (1999).
[Crossref]

1998 (1)

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fiber,” Science 282, 1476–1478, (1998).
[Crossref] [PubMed]

1997 (1)

1996 (1)

1992 (1)

J.W.H. Liu, “The multifrontal method for sparse matrix solutions: theory and practice,” SIAM Rev. 34, 82–109,(1992).
[Crossref]

Allan, D.C.

R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Singlemode photonic band gap guidance of light in air,” Science 285, 1537–1539, (1999).
[Crossref] [PubMed]

Andrés, M.V.

Andrés, P.

Arriaga, J.

J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.St.J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809, (2000).
[Crossref]

Atkin, D.M.

Barkou, S.E.

J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330, (1999).
[Crossref]

Birks, T.A.

T.A. Birks, J.C. Knight, B.J. Mangan, and P.St.J. Russell, “Photonic crystal fibers: An endless variety,” IEICE Trans. Electron. E84-C, 585–592, (2001).

J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.St.J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809, (2000).
[Crossref]

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]

R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Singlemode photonic band gap guidance of light in air,” Science 285, 1537–1539, (1999).
[Crossref] [PubMed]

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fiber,” Science 282, 1476–1478, (1998).
[Crossref] [PubMed]

T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963, (1997).
[Crossref] [PubMed]

J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549, (1996).
[Crossref] [PubMed]

Bjarklev, A.

J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330, (1999).
[Crossref]

Botten, L.C.

Broeng, J.

J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330, (1999).
[Crossref]

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fiber,” Science 282, 1476–1478, (1998).
[Crossref] [PubMed]

Cregan, R.F.

R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Singlemode photonic band gap guidance of light in air,” Science 285, 1537–1539, (1999).
[Crossref] [PubMed]

de Sterke, C. Martijn

Ferrando, A.

Gander, M.J.

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]

Hasegawa, T.

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber - A practical derivative of photonic crystal fiber,” Proc. Mater. Res. Soc. Spring Meeting L4.2. (2002).

Jones, J.D.C.

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]

Knight, J.C.

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]

T.A. Birks, J.C. Knight, B.J. Mangan, and P.St.J. Russell, “Photonic crystal fibers: An endless variety,” IEICE Trans. Electron. E84-C, 585–592, (2001).

J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.St.J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809, (2000).
[Crossref]

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]

R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Singlemode photonic band gap guidance of light in air,” Science 285, 1537–1539, (1999).
[Crossref] [PubMed]

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fiber,” Science 282, 1476–1478, (1998).
[Crossref] [PubMed]

T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963, (1997).
[Crossref] [PubMed]

J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549, (1996).
[Crossref] [PubMed]

Koshiba, M.

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

M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315, (2001).
[Crossref]

M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743, (2000).
[Crossref]

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber - A practical derivative of photonic crystal fiber,” Proc. Mater. Res. Soc. Spring Meeting L4.2. (2002).

Kuhlmey, B.T.

Liu, J.W.H.

J.W.H. Liu, “The multifrontal method for sparse matrix solutions: theory and practice,” SIAM Rev. 34, 82–109,(1992).
[Crossref]

Mangan, B.J.

T.A. Birks, J.C. Knight, B.J. Mangan, and P.St.J. Russell, “Photonic crystal fibers: An endless variety,” IEICE Trans. Electron. E84-C, 585–592, (2001).

R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Singlemode photonic band gap guidance of light in air,” Science 285, 1537–1539, (1999).
[Crossref] [PubMed]

Maystre, D.

McBride, R.

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]

McPhedran, R.C.

Miret, J.J.

Mogilevstev, D.

J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330, (1999).
[Crossref]

Mogilevtsev, D.

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]

Nishimura, M.

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber - A practical derivative of photonic crystal fiber,” Proc. Mater. Res. Soc. Spring Meeting L4.2. (2002).

Onishi, M.

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber - A practical derivative of photonic crystal fiber,” Proc. Mater. Res. Soc. Spring Meeting L4.2. (2002).

Ortigosa-Blanch, A.

J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.St.J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809, (2000).
[Crossref]

Reeves, W.H.

Renversez, G.

Roberts, P.J.

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]

R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Singlemode photonic band gap guidance of light in air,” Science 285, 1537–1539, (1999).
[Crossref] [PubMed]

Russell, P.St.J.

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]

T.A. Birks, J.C. Knight, B.J. Mangan, and P.St.J. Russell, “Photonic crystal fibers: An endless variety,” IEICE Trans. Electron. E84-C, 585–592, (2001).

J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.St.J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809, (2000).
[Crossref]

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]

R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Singlemode photonic band gap guidance of light in air,” Science 285, 1537–1539, (1999).
[Crossref] [PubMed]

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fiber,” Science 282, 1476–1478, (1998).
[Crossref] [PubMed]

T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963, (1997).
[Crossref] [PubMed]

J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549, (1996).
[Crossref] [PubMed]

Saitoh, K.

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

M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315, (2001).
[Crossref]

Sasaoka, E.

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber - A practical derivative of photonic crystal fiber,” Proc. Mater. Res. Soc. Spring Meeting L4.2. (2002).

Silvestre, E.

Tsuji, Y.

M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743, (2000).
[Crossref]

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber - A practical derivative of photonic crystal fiber,” Proc. Mater. Res. Soc. Spring Meeting L4.2. (2002).

Wadsworth, W.J.

J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.St.J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809, (2000).
[Crossref]

White, T.P.

Electron. Lett. (1)

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]

IEEE J. Quantum Electron. (1)

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

IEEE Photon. Technol. Lett. (2)

M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315, (2001).
[Crossref]

J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P.St.J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809, (2000).
[Crossref]

IEICE Trans. Electron. (1)

T.A. Birks, J.C. Knight, B.J. Mangan, and P.St.J. Russell, “Photonic crystal fibers: An endless variety,” IEICE Trans. Electron. E84-C, 585–592, (2001).

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Opt. Express (2)

Opt. Fiber Technol. (1)

J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330, (1999).
[Crossref]

Opt. Lett. (3)

Science (2)

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fiber,” Science 282, 1476–1478, (1998).
[Crossref] [PubMed]

R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Singlemode photonic band gap guidance of light in air,” Science 285, 1537–1539, (1999).
[Crossref] [PubMed]

SIAM Rev. (1)

J.W.H. Liu, “The multifrontal method for sparse matrix solutions: theory and practice,” SIAM Rev. 34, 82–109,(1992).
[Crossref]

Other (1)

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber - A practical derivative of photonic crystal fiber,” Proc. Mater. Res. Soc. Spring Meeting L4.2. (2002).

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Figures (7)

Fig. 1.
Fig. 1.

Transverse cross section of photonic crystal fiber surrounded by PMLs.

Fig. 2.
Fig. 2.

Cross section of proposed photonic crystal fiber. Λ is the hole-to-hole spacing and di (i=1 ~ n) is the hole diameter of ith air-hole ring.

Fig. 3.
Fig. 3.

Index-guiding PCFs with four rings of air holes. The air holes are shown as colored circles. The hole-to-hole spacing Λ=2.0 µm and the each air-hole diameter is (a) d 1=d 2=d 3=d 4=0.5 µm, (b) d 1=0.5 µm, d 2=d 3=d 4=0.6 µm, (c) d 1=0.5 µm, d 2=0.6 µm, d 3=d 4=0.7 µm, (d) d 1=0.5 µm, d 2=0.6 µm, d 3=0.7 µm, d 4=1.8 µm.

Fig. 4.
Fig. 4.

Chromatic dispersion curves as a function of wavelength for PCFs with four rings of air holes in Fig. 3. The hole-to-hole spacing Λ=2.0 µm and the each air-hole diameter is (a) d 1=d 2=d 3=d 4=0.5 µm, (b) d 1=0.5 µm, d 2=d 3=d 4=0.6 µm, (c) d 1=0.5 µm, d 2=0.6 µm, d 3=d 4=0.7 µm, (d) d 1=0.5 µm, d 2=0.6 µm, d 3=0.7 µm, d 4=1.8 µm.

Fig. 5.
Fig. 5.

Ultra-flattened dispersion PCFs with (a) four air-hole rings and (b) five air-hole rings. The hole-to-hole spacing and the air-hole diameters are (a) Λ=1.56 µm, d 1/Λ=0.32, d 2/Λ=0.45, d 3/Λ=0.67, d 4/Λ=0.95 and (b) Λ=1.58 µm, d 1/Λ=0.31, d 2/Λ=0.45, d 3/Λ=0.55, d 4/Λ=0.63, d 5/Λ=0.95.

Fig. 6.
Fig. 6.

(a) Chromatic dispersion curve, (b) confinement loss, and (c) effective mode area as a function of wavelength for ultra-flattened dispersion PCF with four air-hole rings in Fig. 5(a).

Fig. 7.
Fig. 7.

(a) Chromatic dispersion curve, (b) confinement loss, and (c) effective mode area as a function of wavelength for ultra-flattened dispersion PCF with five air-hole rings in Fig. 5(b).

Tables (1)

Tables Icon

Table 1. PML parameters.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

× ( [ s ] 1 × E ) k 0 2 n 2 [ s ] E = 0
[ s ] = [ s y s x 0 0 0 s x s y 0 0 0 s x s y ]
s i = 1 j α i ( ρ t i ) 2
[ K ] { E } = k 0 2 n eff 2 [ M ] { E }
D = λ c d 2 Re [ n eff ] d λ 2
confinement loss = 8.686 Im [ k 0 n eff ]
A eff = ( E 2 dx dy ) 2 E 4 dx dy .

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