## Abstract

A rigorous full-vector finite element method is effectively applied to evaluating the effective area *A*_{eff}
and the mode field diameter (MFD) of holey fibers (HFs) with finite cross sections. The effective modal spot size (a half of MFD), *w*_{eff}
, is defined with the help of the second
moment of the optical intensity distribution. The influence of hole diameter, hole pitch, operating wavelength, and number of rings of air holes on *A*_{eff}
and *w*_{eff}
is investigated in detail. As a result, it is shown that *A*_{eff}
and *w*_{eff}
are almost independent of the number of hole rings and that the relation *A*_{eff}
=π${w}_{\mathit{\text{eff}}}^{2}$, which is frequently utilized in the conventional optical fibers, does not always hold, especially in smaller air-filling fraction and/or longer wavelength regions. In addition, we find that for HFs with large air holes operating at longer wavelengths, the mode profiles of the two linearly polarized fundamental modes are significantly different from each other, even though they are degenerate. Using the values of *A*_{eff}
and *w*_{eff}
obtained here, the beam divergence and the nonlinear phase shift are calculated and are compared with the earlier experimental results.

©2003 Optical Society of America

## 1. Introduction

Photonic crystal fibers, also called holey fibers (HFs), have been one of the most interesting developments in recent fiber optics [1],[2].

The effective area *A*_{eff}
is a quantity of great importance in fiber optics. It was originally introduced as a measure of nonlinearities; a small effective area would be useful for enhancing nonlinear effects. Often a large effective area is desired, and then high powers can be transmitted without introducing any unwanted nonlinear effects in the fiber. Although several authors have reported theoretical studies of effective area for HFs [3],[4], the influence of the extent of the confining air hole region on effective area has not been investigated. More recently, Monro *et al.* have reported a pioneering work regarding the effective area of HFs with a few rings of air holes and have claimed that *A*_{eff}
is effectively independent of the number of hole rings [5]. However, in [5] the number of rings of air holes has been limited 2 to 4 and the operating wavelength has been fixed to 1550 nm.

The mode field diameter (MFD) is another important parameter in the context of beam divergence, splice loss, bending loss, source-to-fiber coupling efficiency, and so on. Although the MFD can be related to the effective area through the relation *A*_{eff}
=π${w}_{\mathit{\text{eff}}}^{2}$ with *w*_{eff}
being the effective modal spot size (a half of MFD), which is frequently utilized in the conventional optical fibers, this relation is valid only for close-to-Gaussian modes in ordinary axially symmetrical fibers and its applicability to HFs has not been fully investigated so far.

In this paper, using a rigorous full-vector finite element method [6]–[8], *A*_{eff}
and MFD of HFs with finite cross sections are evaluated. In order to treat axially nonsymmetrical fibers such as HFs with air holes arranged in a triangular lattice in the cladding region, the second moment of the optical intensity distribution is introduced as the definition of *w*_{eff}
. The influence of hole diameter, hole pitch, operating wavelength, and number of hole rings on *A*_{eff}
and *w*_{eff}
is investigated in detail, including confinement losses arising from the leaky nature of HFs with finite cross sections. As a result, it is shown that in contrast with the confinement losses which is strongly dependent of the number of hole rings [8]–[10], *A*_{eff}
and *w*_{eff}
are almost independent of it and that the relation *A*_{eff}
=π${w}_{\mathit{\text{eff}}}^{2}$ does not always hold, especially in smaller air-filling fraction and/or longer wavelength regions. In addition, we find that for HFs with large air holes operating at longer wavelengths, the mode profiles of the horizontally polarized and vertically polarized fundamental modes are significantly different from each other, even though these two linearly polarized modes are degenerate in strict sense. Using the values of *A*_{eff}
and *w*_{eff}
obtained here, the beam divergence and the nonlinear phase shift are calculated and are compared with the earlier experimental results [11]–[13].

## 2. Analysis method

We consider a HF with finite cross section as shown in Fig. 1, where *x* and *y* are the transverse directions, *z* is the propagation direction, *d* is the hole diameter, and Λ is the hole pitch. Anisotropic perfectly matched layers (PMLs) [7],[8] are incorporated as absorbing boundary conditions to evaluate confinement losses. From Maxwell’s equations the following vector wave equation is derived [8]:

where * E* is the electric field vector,

*n*is the refractive index, [

*s*] is the PML matrix, and [

*s*]

^{-1}is an inverse matrix of [

^{s}].

When applying a full-vector FEM to HFs, a curvilinear hybrid edge/nodal element [6] as shown in Fig. 2 is very useful for avoiding spurious solutions and for accurately modeling curved boundaries of circular air holes. For the axial field, *E*_{z}
, a nodal element with six variables, *E*_{z}
_{1} to *E*_{z}
_{6}, is employed, while for the transverse fields, *E*_{x}
and *E*_{y}
, an edge element with eight variables, *E*_{t}
_{1} to *E*_{t}
_{8}, is employed, resulting in significantly fast convergence of solutions [6].

Dividing the fiber cross section into curvilinear hybrid edge/nodal elements, we expand the transverse and axial fields in each element as

where β is the propagation constant, {*E*_{t}
}
_{e}
and {*E*_{z}
}
_{e}
are, respectively, the edge and the nodal variables for each element *e*, {*U*} and {*V*} are the shape function vectors for edge elements, {*N*} is the shape function vector for nodal elements, and *T* denotes a transpose.

Applying the standard finite element technique to Eq. (1), we can obtain the following eigenvalue equation:

with

where {*E*} is the global electric field vector and the finite element matrices [*K*] and [*M*] are given in [8]. Utilizing sparse nature of [*K*] and [*M*], Eq. (3) is solved with the multifrontal method [14].

The area of the fundamental mode is clearly related to the effective area of the fiber core, *A*_{eff}
, for which we use the following definition [15]:

where * E_{t}* is the transverse electric field vector and

*S*denotes the whole fiber cross section. Using the eigenvector {

*E*} corresponding to the eigenvalue β

^{2}obtained from Eq. (3) and noting the relation of Eq. (2),

*A*

_{eff}is calculated as

where Σ
_{e}
extends over all different elements and the optical intensity distribution for each element, *I*_{e}
, is given by

In HFs the two linearly polarized fundamental modes exist. One is polarized in the horizontal (*x*) direction and the other in the vertical (*y*) direction, and these two fundamental modes are degenerate [17],[18]. As is well-known, the mode profile of the fundamental mode in step-index fibers can be well approximated with a Gaussian shape, and the MFD is usually calculated using the Petermann’s definition [16]. However, the Petermann’s definition cannot be applied to the MFD of HFs because the mode profile in such fibers is considerably deformed by the existence of air holes arranged in a triangular lattice and may be different from that in ordinary axially symmetrical fibers. So, we adapt the second moment of the optical intensity distribution as the definition of modal spot size (a half of MFD) for HFs which is very useful for axially nonsymmetrical fibers. According to [19], the modal spot sizes in the *x* and *y* directions, *w*_{x}
and *w*_{y}
, are given by

where *x*_{c}
and *y*_{c}
are, respectively, the *x* and *y* coordinates of the center of the field distribution, which are calculated as

In the curvilinear elements with triangular shape the Cartesian coordinates, *x* and *y*, are, in general, approximated with quadratic polynomials using the local coordinates, *L*
_{1}, *L*
_{2}, and *L*
_{3}, which are also called the area coordinates for the rectilinear elements with triangular shape [6]. Noting that all the shape function vectors, {*U*}, {*V*}, and {*N*}, are also given as the functions of *L*
_{1}, *L*
_{2}, and *L*
_{3}, Eqs. (6) and (8) to Eq. (11) can be easily and accurately evaluated by using the well-established numerical integration formulas (see Eq. (16) in [6]).

As will be demonstrated in Section 4, the modal spot sizes in the *x* and *y* directions, *w*_{x}
and *w*_{y}
, are, in general, different from each other, and so we define the following effective modal spot size *w*_{eff}
:

With this definition, *w*_{eff}
remains the 1/*e*
^{2} intensity radius (namely, the 1/*e* field radius) for a Gaussian profile. A similar definition has been introduced to the measurement of the far-field intensity profile with a Gaussian beam approximation [11].

## 3. Effective area and confinement loss

HFs are usually made from pure silica, and so the guided modes are inherently leaky because the core index is the same as the index of the outer cladding region without air holes. Figure 3 shows the normalized effective area and the normalized confinement loss of the fundamental mode in HFs with finite cross sections, taking the ratio of hole diameter to pitch, *d*/Λ, as a parameter, where *N* is the number of rings of air holes, the background silica index *n* is assumed to be 1.45, and the normalized confinement loss in dB, *L*_{c}
Λ, is defined as

Here Im stands for the imaginary part. For example, assuming the value of *L*_{c}
Λ to be 10^{-8} dB, the confinement loss *L*_{c}
becomes 10 dB/km for Λ=1 µm and 1 dB/km for Λ=10 µm.

As expected, increasing the air-hole size, the mode becomes more confined, and thus, the effective area and the confinement loss are both reduced. Also, increasing the number of rings of air holes, the confinement loss is significantly reduced. On the other hand, the effective area is almost independent of the number of hole rings, as claimed by Monro *et al.* [5]. This seems to be due to the fact that even if *A*_{eff}
/Λ^{2} ≅ π, the spread of effective area is in the order of Λ corresponding to the position of the first ring. In Fig. 3(f), the results of a rigorous full-vector multipole method (MM) [5] are also plotted. Our results agree well with those of MM, showing the reliability of a full-vector FEM with PMLs.

Figure 3 is a general “map” of the effective area which can be used for designing various HFs, such as ultra-low nonlinearity HFs and high nonlinearity HFs with desired confinement loss properties. If, for example, the operating wavelength is given, the effective area and the confinement loss can to a large extent be tailored via the choice of hole pitch, hole size, and number of hole rings.

## 4. Effective area and mode field diameter

As described in the previous section, the effective area is almost independent of the number of hole rings, and here, we consider a HF with ten rings.

Figure 4 shows the relationship between effective area *A*_{eff}
and mode field diameter (MFD) as a function of wavelength for different hole sizes, where the MFD is expressed in the form of π${w}_{\mathit{\text{eff}}}^{2}$ with *w*_{eff}
(a half of MFD) being the effective modal spot size given in Eq. (12). Roughly speaking, the results for *A*_{eff}
and π${w}_{\mathit{\text{eff}}}^{2}$ are similar, but in a strict sense, the relation *A*_{eff}
=π${w}_{\mathit{\text{eff}}}^{2}$, which is frequently adopted to the conventional optical fibers, does not hold, especially in smaller air-filling fraction and/or larger wavelength regions. This suggests that the mode field in these cases cannot be approximated by a simple circularly symmetric Gaussian shape. In fact, Fig. 5 shows the intensity profiles of the fundamental modes in a HF with air-hole size of *d*/Λ=0.3 at λ/Λ=0.2, where the intensity counters are spaced by 1 dB. The left and right profiles shows the principal field components of the horizontally polarized and vertically polarized modes, respectively, which are degenerate [17],[18]. These intensity profiles are very similar. As is well-known, these linearly polarized modes have six-fold rotational symmetry. We can see the deviation from the typical circular shape, resulting in a failure to rely on the relation *A*_{eff}
=π${w}_{\mathit{\text{eff}}}^{2}$ to express MFD from *A*_{eff}
.

Figure 6 shows the relationship among three kinds of modal spot sizes for the horizontally polarized mode, *w*_{eff}
, *w*_{x}
, and *w*_{y}
, as a function of wavelength. For the vertically polarized mode, the values of *w*_{x}
and *w*_{y}
are interchanged. The results of the effective modal spot size *w*_{eff}
are the same as those in Fig. 4. The modal spot sizes in the *x* and *y* directions, *w*_{x}
and *w*_{y}
, are, respectively, obtained from Eqs. (8) and (9). It is worth noting that for HFs with large air holes operating at longer wavelengths, these values are significantly different from each other. This suggests that the mode field in these cases may be considerably deviated from that in ordinary axially symmetrical fibers. Figure 7 shows the intensity profiles of the horizontally polarized and vertically polarized modes in a HF with *d*/Λ=0.9 at λ/Λ=1.2, where the intensity contours are spaced by 1 dB. Surprisingly, in contrast to those in Fig. 5, these intensity profiles are entirely different from each other, even though these two linearly polarized modes are degenerate. They have no longer six-fold rotational symmetry. Such curious mode shapes have not been reported so far, to our knowledge. Traditional methods, which rely on Gaussian optics to estimate the splice loss, source-to-fiber coupling efficiency, and so on, would fail in these cases, and therefore, more rigorous numerical methods may be required.

## 5. Beam divergence and nonlinear phase shift

In this section, using the values of effective area and MFD obtained here, the beam divergence and the nonlinear phase shift of HFs are calculated and are compared with the earlier experimental results [11]–[13].

Figure 8 shows the beam divergences of HFs with hole pitch Λ=2.39 µm and hole diameter *d*/Λ=0.26 and with Λ=7.2 µm and *d*/Λ=0.53, where the beam divergence θ is given by

with *w*_{eff}
being the effective modal spot size in (12) and λ being the operating wavelength. In Fig. 8, assuming the relation *A*_{eff}
=π${w}_{\mathit{\text{eff}}}^{2}$, the beam divergences obtained from
$\theta ={\mathrm{tan}}^{-1}\left(\frac{\lambda}{\sqrt{\pi {A}_{\mathit{eff}}}}\right)$
, are also plotted. For a relatively large air-hole case, *d*/Λ=0.53 and λ/Λ < 0.14, the relation *A*_{eff}
=π${w}_{\mathit{\text{eff}}}^{2}$ is approximately satisfied (see Fig. 4), and so the beam divergence obtained from *A*_{eff}
is very similar to that obtained from *w*_{eff}
. For a considerably small air-hole case, *d*/Λ=0.26 and λ/Λ < 0.42, on the other hand, the relation *A*_{eff}
=π${w}_{\mathit{\text{eff}}}^{2}$ does not hold (see again Fig. 4). Irrespective of air hole sizes, the beam divergences obtained from *w*_{eff}
, in other word, MFD, are in good quantitative agreement with the experimental ones [11],[12]. From Fig. 8, we can say that the beam divergence is not well quantified in terms of the concept of effective area.

Then, we consider a highly nonlinear HF as shown in Fig. 9 [13]. In order to estimate *A*_{eff}
and MFD of such an actual fiber, the real-model simulation [20] is indispensable. In the ideal structure, the two linearly polarized fundamental modes are degenerate [17],[18], and so *A*_{eff}
and MFD are independent of the polarization states. In the actual fiber structure, on the other hand, they depend on the polarization states. Figure 10 shows the values of *A*_{eff}
and π${w}_{\mathit{\text{eff}}}^{2}$ of the two fundamental modes called slow-axis and fast-axis modes as a function of wavelength. The real part of propagation constant of the slow-axis mode is slightly larger than that of the fast-axis mode. In Fig. 10 the modal birefringence is also plotted. This fiber exhibits very small effective area and large birefringence of the order of 10^{-3}. The calculated *A*_{eff}
and the modal birefringence are in good agreement with the experimental ones [13]. In this fiber the relation *A*_{eff}
=π${w}_{\mathit{\text{eff}}}^{2}$ is approximately satisfied because the ratio of hole diameter to pitch is relatively large. Figure 11 shows the normalized phase shift given by

where *n*
_{2} is the nonlinear-index coefficient for pure silica, *P* is the signal power, and the operating wavelength is taken as 1.553 µm. The calculated ϕ_{SPM} for the slow-axis mode agree well with the experimental one [13].

## 6. Conclusion

Using a rigorous full-vector finite element method, the effective area *A*_{eff}
and the mode field diameter (MFD) of holey fibers (HFs) with finite cross sections were evaluated. In order to treat axially nonsymmetrical fibers such HFs with air holes arranged in a triangular lattice in the cladding region, the second moment of the optical intensity distribution was introduced as the definition of the effective modal spot size *w*_{eff}
(a half of MFD). We have shown that *A*_{eff}
and *w*_{eff}
are almost independent of the number of rings of air holes and that the relation *A*_{eff}
=π${w}_{\mathit{\text{eff}}}^{2}$ does not always hold, especially in smaller air filling fraction and/or longer wavelengths. In addition, we have found that for HFs with large air holes operating at longer wavelengths, the mode profiles of the two linearly polarized fundamental modes are significantly different from each other. The values of *A*_{eff}
and MFD obtained here were effectively applied to evaluating the beam divergence and the nonlinear phase shift of HFs. The applicability of *A*_{eff}
and MFD to the splice loss, bending loss, and source-to-fiber coupling efficiency are now under consideration.

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