## 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 [9–11], 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 *t _{i}* (

*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

*W*and

_{x}*W*are the computational window sizes along the

_{y}*x*and

*y*directions, respectively.

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

with

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

*s*and

_{x}*s*are given in Table 1, where the values of

_{y}*s*(

_{i}*i*=1 to 4) are complex as

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 α

_{i}appropriately.

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:

where [*K*] and [*M*] are the finite element matrices, {*E*} is the discretized electric field vector consisting of the edge and nodal variables, and *n _{eff}* 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 *n _{eff}* values versus the wavelength using

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 *n _{eff}* as

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 *n _{eff}*=1.4453952+

*j*3.19×10

^{-8}. This result is in good agreement with that of the full-vector multipole method [17].

## 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 *d _{i}* (

*i*=1 ~

*n*) is the hole diameter of

*i*th 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

*d*(

_{i}*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 *d _{i}* 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

*d*(

_{i}*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

*d*and optimization of

_{i}*d*enables index-guiding PCFs to control their dispersion properties.

_{i}## 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 *A _{eff}* is calculated with

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 µm^{2} 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 µm^{2} at 1.55 µm wavelength. Of course, using more air-hole rings and optimizing Λ and *d _{i}*, 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.

## 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.

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