## Abstract

Confinement loss is comprehensively evaluated for TE, TM, and hybrid modes of Bragg fibers using a multilayer division method newly developed. We show the loss dependence on the core radius, wavelength, cladding index contrast, and the number of cladding pairs. The confinement loss is reduced in proportion to (*a*/*b*)^{2N} and (*n*
^{2}
_{b}
*b*/*n*
^{2}
_{a}
*a*)^{2N} for the TE and other three modes, respectively, with respect to cladding pairs *N* under the quarter-wave stack condition, with cladding high *n _{a}* and low indices

*n*and their corresponding thicknesses

_{b}*a*and

*b*. For sufficiently large core radius, the confinement loss decreases in inverse proportion to the third and first powers of core radius for the TE and other three modes, respectively. Low loss modes are the TE

_{01}, TE

_{02}, HE

_{13}, and TE

_{03}modes in order of increasing confinement loss.

©2008 Optical Society of America

## 1. Introduction

Conventional optical fibers confine optical wave to the core owing to the total reflection between core and cladding. Although their fields penetrate into the cladding due to the evanescent wave, they exponentially decay in the cladding. On the other hand, photonic crystal fibers confine their wave owing to the Bragg diffraction originating from the periodicity in the cladding. The cladding plays an essential role as a guiding mechanism in photonic crystal fibers. A finite periodic cladding causes a decrease in optical confinement, resulting in a loss increase that is called the confinement loss [1]. The confinement loss is a key factor for the appropriate design of photonic crystal fibers.

A Bragg fiber, a kind of photonic crystal fibers, consists of air core surrounded by the periodic cladding with alternating high and low refractive indices. The confinement loss in the Bragg fiber has been treated by the transfer matrix method [2], Chew’s method [3], [4], and asymptotic analysis [5], although the term of ”radiation loss” was used in [2]. In previous papers, the confinement loss has been analyzed for the TE_{01} and some modes. The purpose of the present paper is to offer a comprehensive study on the confinement loss of various modes of the Bragg fiber for the fiber designer.

Since the confinement loss closely relates to the imaginary part of the propagation constant, we must simultaneously calculate its real and imaginary parts of the fiber structure with finite periodic cladding. Refractive index distribution of the Bragg fiber consists of a staircase function. To calculate the propagation constant of such structures, one has used not only the analysis methods described above but also other methods, such as the supercell [6], biorthonormal basis [7], finite-difference [8], and multilayer division methods [9], [10]. The present paper develops the multilayer division method so as to apply to high-index-contrast fibers. Many numerical results will be provided to investigate various properties of the confinement loss of the Bragg fiber, such as the loss dependence on the core radius, wavelength, cladding index contrast, and the number of cladding layer pairs.

This paper proceeds as follows: Section 2 describes the multilayer division method that is used to calculate the confinement loss here, and section 3 briefly describes the Bragg fiber. Section 4 shows many numerical results on confinement loss of various modes to elucidate its properties. In section 5, we present numerical results of the TE_{01} mode that is the lowest mode among all the modes. Finally in section 6, the present numerical results are shown to well agree with those calculated by other methods.

## 2. Multilayer Division Method

#### 2.1. Preparation of Derivation

We assume that a fiber studied has a cylindrical symmetry and its radial index distribution is represented by a staircase function. A cylindrical coordinate system (*r*,*θ*, *z*) is used, with *z* being the propagation direction of light. A cross sectional view of the fiber is shown in Fig. 1. The refractive index of the *i*th layer and its outer radius are represented by *n _{i}* and

*r*, respectively.

_{i}Electromagnetic field components are assumed to have a spatiotemporal factor of *U _{tz}*=exp[i(

*ωt*–

*βz*)], with

*ω*and

*β*being the angular frequency and propagation constant, respectively. The

*β*may be complex. Axial electromagnetic field components in the ith layer are set to be

where *κ _{i}* is the lateral propagation constant of the

*i*th layer and is defined by

Here, *A _{i}*-

*D*denote the amplitude coefficients of the

_{i}*i*th layer,

*H*

^{(1)}

_{ν}=

*J*+i

_{ν}*N*and

_{ν}*H*

^{(2)}

_{ν}=

*J*-i

_{ν}*N*indicate the Hankel functions of the first and second kinds of order

_{ν}*ν*, respectively,

*k*

_{0}=2

*π*/

*λ*

_{0}is the wavenumber of vacuum, and

*λ*

_{0}is the vacuum wavelength. The

*ν*denotes the azimuthal mode number, and

*θ*

_{in}is the initial phase. Result for

*ν*=0 reduces to TE and TM modes,

*θ*

_{in}=0 corresponds to the TM mode, and

*θ*

_{in}=

*π*/2 corresponds to the TE mode.

Lateral field components are represented using the axial field components. Tangential components, *H _{z}*,

*E*,

_{θ}*E*and

_{z}*H*, to be continuous in each layer interface, are selected as fundamental components. Electromagnetic fields are represented in a matrix form:

_{θ}with *r _{i}*−1≤

*r*≤

*r*and

_{i}Here, *D _{i}*(

*r*) is referred to as the representation matrix of the

*i*th layer. Its elements are expressed as

$${d}_{21}=-\frac{{d}_{43}}{{Y}_{i}^{2}}=-\frac{\omega {\mu}_{0}}{{\kappa}_{i}}{H}_{v}^{\left(2\right)\prime}\left({\kappa}_{i}r\right),{d}_{22}=-\frac{{d}_{44}}{{Y}_{i}^{2}}=-\frac{\omega {\mu}_{0}}{{\kappa}_{i}}{H}_{v}^{\left(1\right)\prime}\left({\kappa}_{i}r\right),$$

$${d}_{23}=-{d}_{41}=-\frac{v\beta}{{\kappa}_{i}^{2}r}{H}_{v}^{\left(2\right)}\left({\kappa}_{i}r\right),{d}_{24}=-{d}_{42}=-\frac{v\beta}{{\kappa}_{i}^{2}r}{H}_{v}^{\left(1\right)}\left({\kappa}_{i}r\right).$$

The prime indicates differentiation with respect to the argument. In addition, *Y _{i}*≡

*n*(

_{i}*ε*

_{0}/

*μ*

_{0})

^{1/2}denotes the characteristic admittance in a medium having the refractive index

*n*,

_{i}*ε*

_{0}denotes the dielectric permittivity of vacuum, and

*μ*

_{0}denotes the magnetic permeability of vacuum. In Eq. (3) we omitted angular dependencies of sin(

*νθ*+

*θ*

_{in}) for the

*H*and

_{z}*E*, and cos(

_{θ}*νθ*+

*θ*

_{in}) for the

*E*and

_{z}*H*. For

_{θ}*ν*=0 in Eq. (3), the representation matrix is block-diagonalized into 2×2 matrices, and the left-upper and the right-lower parts reduce to results for TE and TM modes.

Use of Eq. (3) formally yields amplitude coefficients

For two radial points, *r*
_{i1} and *r*
_{i2}, arbitrarily selected within the ith layer, the relationship of field components between them becomes

by substituting Eq. (6) into (3). Here,

Elements of the displacement matrix *Q _{i}*(

*r*

_{i2},

*r*

_{i1}) are written as

$${q}_{12}^{i}=-{Y}_{i}^{2}{q}_{34}^{i}=\frac{\pi {\kappa}_{i}^{2}{r}_{i1}}{4\mathrm{i}\omega {\mu}_{0}}\left[{H}_{v}^{\left(2\right)}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(1\right)}\left({\kappa}_{i}{r}_{i1}\right)-{H}_{v}^{\left(1\right)}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(2\right)}\left({\kappa}_{i}{r}_{i1}\right)\right],$$

$${q}_{13}^{i}={Y}_{i}^{2}{q}_{31}^{i}=\frac{\pi v\beta}{4\mathrm{i}\omega {\mu}_{0}}\left[{H}_{v}^{\left(2\right)}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(1\right)}\left({\kappa}_{i}{r}_{i1}\right)-{H}_{v}^{\left(1\right)}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(2\right)}\left({\kappa}_{i}{r}_{i1}\right)\right],$$

$${q}_{21}^{i}=-\frac{{q}_{43}^{i}}{{Y}_{i}^{2}}=\frac{\mathrm{i}\pi {r}_{i1}}{4\mathrm{\omega}{\epsilon}_{0}{n}_{i}^{2}}\left\{{\left({k}_{0}{n}_{i}\right)}^{2}\right)\left[{H}_{v}^{\left(2\right)\prime}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(1\right)\prime}\left({\kappa}_{i}{r}_{i1}\right)-{H}_{v}^{\left(1\right)\prime}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(2\right)\prime}\left({\kappa}_{i}{r}_{i1}\right)\right]$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\frac{{\left(v\beta \right)}^{2}}{{\kappa}_{i}^{2}{r}_{i1}{r}_{i2}}\left[{H}_{v}^{\left(2\right)}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(1\right)}\left({\kappa}_{i}{r}_{i1}\right)-{H}_{v}^{\left(1\right)}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(2\right)}\left({\kappa}_{i}{r}_{i1}\right)\right]\},$$

$${q}_{22}^{i}={q}_{44}^{i}=\frac{\mathrm{i}\pi {\kappa}_{i}{r}_{i1}}{4}\left[{H}_{v}^{\left(2\right)\prime}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(1\right)}\left({\kappa}_{i}{r}_{i1}\right)-{H}_{v}^{\left(1\right)\prime}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(2\right)}\left({\kappa}_{i}{r}_{i1}\right)\right],$$

$${q}_{23}^{i}=-{q}_{41}^{i}=\frac{\mathrm{i}\pi v\beta}{4{\kappa}_{i}}\{\left[{H}_{v}^{\left(2\right)\prime}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(1\right)}\left({\kappa}_{i}{r}_{i1}\right)-{H}_{v}^{\left(1\right)\prime}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(2\right)}\left({\kappa}_{i}{r}_{i1}\right)\right]$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\frac{{r}_{i1}}{{r}_{i2}}\left[{H}_{v}^{\left(2\right)}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(1\right)\prime}\left({\kappa}_{i}{r}_{i1}\right)-{H}_{v}^{\left(1\right)}\left({\kappa}_{i}{r}_{i2}\right){H}_{v}^{\left(2\right)\prime}\left({\kappa}_{i}{r}_{i1}\right)\right]\},$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}{q}_{24}^{i}=\frac{1}{{Y}_{i}^{2}}{q}_{42}^{i}=\frac{1}{{Y}_{i}^{2}}\frac{{r}_{i1}}{{r}_{i2}}{q}_{13}^{i}.$$

In deriving Eq. (9), we used Lommel’s formula concerning Hankel functions.

#### 2.2. Eigenvalue Equation and Amplitude Coefficients

Electromagnetic fields at the outer radius, *r*=*r _{i}*, of the

*i*th layer are related to those at

*r*=

*r*

_{1}of the first layer as

where

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}=\left(\begin{array}{cccc}{f}_{11}^{i}& {f}_{12}^{i}& {f}_{13}^{i}& {f}_{14}^{i}\\ {f}_{21}^{i}& {f}_{22}^{i}& {f}_{23}^{i}& {f}_{24}^{i}\\ {f}_{31}^{i}& {f}_{32}^{i}& {f}_{33}^{i}& {f}_{34}^{i}\\ {f}_{41}^{i}& {f}_{42}^{i}& {f}_{43}^{i}& {f}_{44}^{i}\end{array}\right).$$

From the requirement that fields must be finite in the origin of the first layer, we obtain amplitude coefficients *B*
_{1}=*A*
_{1} and *D*
_{1}=*C*
_{1} for the first layer. In the external layer, where we set *i*=*N*+1 and *r*=*r _{N}* in Eq. (3), there must exist only the outward-traveling wave, this resulting in a condition,

*B*

_{N+1}=

*D*

_{N+1}=0. In this case, the magnitude of fields is finite even for

*r*≥

*r*. In representing fields we use

_{N}*A*

_{1},

*C*

_{1},

*A*

_{N+1}, and

*C*

_{N+1}as independent variables of amplitude coefficients. Fields at

*r*=

*r*

_{1}and

*r*=

*r*can be expressed using Eqs. (3) and (5). Substituting Eq. (11) and expressions for fields into Eq. (10) and rearranging it yields

_{N}where

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}{s}_{12}={s}_{34}=-{H}_{v}^{\left(2\right)}\left({\kappa}_{N+1}{r}_{N}\right),$$

$${s}_{i3}=2{f}_{i3}^{N}{J}_{v}\left({\kappa}_{1}{r}_{1}\right)+2{f}_{i4}^{N}\frac{\omega {\epsilon}_{0}{n}_{1}^{2}}{{\kappa}_{1}}{J}_{v}^{\prime}\left({\kappa}_{1}{r}_{1}\right)-2{f}_{i2}^{N}\frac{v\beta}{{\kappa}_{1}^{2}{r}_{1}}{J}_{v}\left({\kappa}_{1}{r}_{1}\right)\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\left(i=1-4\right),$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}{s}_{22}=-\frac{{s}_{44}}{{Y}_{N+1}^{2}}=\frac{\omega {\mu}_{0}}{{\kappa}_{N+1}}{H}_{v}^{\left(2\right)\prime}\left({\kappa}_{N+1}{r}_{N}\right),$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}{s}_{24}=-{s}_{42}=\frac{v\beta}{{\kappa}_{N+1}^{2}{r}_{N}}{H}_{v}^{\left(2\right)}\left({\kappa}_{N+1}{r}_{N}\right)=-\frac{v\beta}{{\kappa}_{N+1}^{2}{r}_{N}}{s}_{12}.$$

An eigenvalue equation for hybrid modes can be obtained by an expression where the determinant of Eq. (12) vanishes. Eigenvalue equations for the TE and TM modes are obtained by setting the determinant of the upper-left and lower-right 2×2 matrices zero in Eq. (12).

One obtains amplitude coefficients, *A*
_{N+1}, *C*
_{1}, and *C*
_{N+1}, for hybrid modes in terms of the amplitude coefficient *A*
_{1} through solving Eq. (12), as shown in Appendix A. Amplitude coefficients in the *i*th layer can be expressed as

by inserting Eq. (10) into (6). Consequently, all the amplitude coefficients can be represented as a function of only the amplitude coefficient *A*
_{1} in the first layer. We have amplitude coefficients for the TE and TM modes by setting *C*
_{1}=*C*
_{N+1}=0 and *A*
_{1}=*A*
_{N+1}=0, respectively, in Eq. (14).

The eigenvalue equation derived from Eq. (12) includes solutions for the HE and EH modes simultaneously. It is possible for us to discriminate between HE and EH modes using a parameter *P* in a manner similar to that in a conventional step-index fiber [11]. The *P* is obtained by

using Eq. (A.2) if it is evaluated by value of the first layer. We will describe the *P* of the Bragg fiber with finite cladding pairs in the second paragraph of section 4.

#### 2.3. Miscellaneous properties and some features of this method

If the propagation constant satisfies 0≤Re(*β*/*k*
_{0})≤*n*
_{1} in the first layer, then its fields are represented by the Bessel function *J _{ν}*(

*κ*

_{1}

*r*). In that case, the real part of

*κ*

_{1}is positive. On the contrary, provided that Re(

*β*/

*k*

_{0})≥

*n*

_{1}in the first layer, fields are expressed by the modified Bessel function

*I*(

_{ν}*κ*

_{1}

*r*) of first kind using a relation of

*J*(i

_{ν}*z*)=exp(i

*νπ*/2)

*I*(

_{ν}*z*).

The present multilayer division method has some features: (i) It can analyze electromagnetic properties of radially inhomogeneous fiber. (ii) The propagation constant and fields can be calculated by giving only the relationship between the refractive index and the inner radial coordinate for each layer. Accordingly, we do not need such an absorbing layer or a PML as in the FDTD method even for finite cladding. (iii) It can calculate both real and imaginary parts of propagation constant simultaneously. (iv) It can analyze a fiber structure having the complex index of refraction, if all the parameters are treated in terms of complex numbers. (v) The mode discrimination can be made between the HE and EH modes using Eq.(15). (vi) It can treat fibers with high index contrast unlike Ref. [10] because the present treatment does not restrict to a magnitude of refractive index.

## 3. Brief description of Bragg fiber

The Bragg fiber has a cylindrically symmetric microstructure, and it has a hollow core surrounded by the periodic cladding (see Fig. 2 (a) and (b)). The core index is *n _{c}* and its radius is

*r*. The cladding has a finite number,

_{c}*N*, of layer pairs that consist of high

*n*and low indices

_{a}*n*(

_{b}*n*>

_{a}*n*>

_{b}*n*). Their corresponding layer thicknesses are

_{c}*a*and

*b*, and the cladding period is Λ=

*a*+

*b*. Outside the periodic cladding we add the external layer whose index of refraction is

*n*

_{ex}. A case of

*n*

_{ex}=1.0 corresponds to a situation that the fiber is located in the air. The core index is assumed to be

*n*=1.0 throughout this paper.

_{c}The index distribution of Bragg fiber is expressed by a staircase function. If its cladding extends to infinity, its eigenvalue equation can be simplified using an asymptotic expansion method [12], [13]. However, if its periodic cladding is finite, then we must resort to trouble-some numerical means to investigate its electromagnetic properties. In the present paper, the multilayer division method described above is used to calculate the confinement loss of the Bragg fiber with finite cladding pairs. The propagation constant *β* is treated as complex number, and it is obtained by numerically solving the eigenvalue equation. The imaginary part of *β* is directly related to the confinement loss, where we require Im(*β*)<0 in the present formulation. The accuracy of the present method depends on the accuracy in numerical calculation of the propagation constant.

## 4. Numerical results on confinement loss for various modes

In this section we will present the confinement loss of dependence on the number of cladding layer pairs, core radius, and wavelength for the TE, TM, and hybrid modes. Although the present paper is targeted for Bragg fibers with finite cladding pairs, most fiber parameters treated here are prescribed so as to meet the quarter-wave stack (QWS) condition in the case of infinite cladding pairs.

The mode discrimination is made between HE and EH modes as follows: The parameter *P* is calculated to be complex number for the Bragg fiber with finite cladding pairs using Eq. (15). It is confirmed that if we use the sign of real part of the *P*, then the mode is discriminated in the same way as that of the Bragg fiber with infinite cladding pairs [13].

#### 4.1. Dependence on the number of cladding pairs

In the following most examples, fiber parameters are as follows: wavelength, *λ*
_{0}=1.0*µ*m; core radius, *r _{c}*=2.0

*µ*m; cladding high index,

*n*=2.5; cladding low index,

_{a}*n*=1.5; external layer index,

_{b}*n*

_{ex}=1.5. Cladding layer thicknesses,

*a*and

*b*, are determined according to [13]

which satisfies the QWS condition for the Bragg fiber with infinite periodic cladding. The *b* is obtained by an expression where *a* and *n _{a}* are replaced by

*b*and

*n*, respectively, in Eq. (16). The

_{b}*U*

_{QWS}is a constant peculiar to the mode and relates to the zeros of Bessel function.

Figure 3 is a semi-logarithmic plot for the confinement loss of various modes of Bragg fiber with above parameters as a function of the number, *N*, of cladding pairs. As for TE and TM modes, we plotted all the modes that appear at *λ*
_{0}=1.0 *µ*m. The confinement loss rapidly decreases with increasing the cladding pairs, and it is in nearly linear change except for extremely small *N*. For sufficiently large *N*, the loss of all the TE modes is lower than that of all the TM modes. For the same *N*, the loss is low for lower-order modes in the TE_{0μ} modes, while the loss is low for higher-order modes in the TM_{0μ} modes. On the other hand, for extremely small *N*, the loss of the TM mode is low for lower-order modes. In the TM_{01} mode whose loss is relatively high, the loss shows the highest value near *N*=2. Roughly speaking, the HE and EH modes exhibit losses which are situated between the TE and TM modes. Losses were shown to a level as extremely low as unrealizable to elucidate properties of the confinement loss.

Let us consider the slope in Fig. 3 here. In the Bragg fiber with infinite periodic cladding, amplitude coefficients in the *m*th cladding a layer vary according to exp[-i*K*
^{S}
_{1} (*m*-1)Λ] [13], where *K*
^{S}
_{1} denotes the Bloch wavenumber, and the S is used to distinguish between the TE and TM modes. Since the Bragg fiber has a good optical confinement factor, we can assume that this relation holds even for finite cladding pairs. Then, the confinement loss is proportional to exp(−i2*NK*
^{S}
_{1}Λ).

If the QWS condition is satisfied in the Bragg fiber, then the confinement loss *L* can be expressed as

by employing Eqs. (A3) and (A4) in Ref. [13]. Hybrid modes have field components common to TE and TM modes. The confinement loss of hybrid modes is expected to have the same *N* dependence as that of the TM mode because the TM mode is greatly lossier than the TE mode. Hence, the slope *S* in Fig. 3 becomes

Using Eq. (16) and its relating expression for *b* and applying an approximation, *r _{c}*/

*λ*

_{0}≫1, to them, we have

*S*

_{TE}⋍log[(

*n*

^{2}

_{b}-

*n*

^{2}

_{c})/(

*n*

^{2}

_{a}-

*n*

^{2}

_{c})] and

*S*

_{non-TE}⋍log{[

*n*

^{4}

_{b}(

*n*

^{2}

_{a}-

*n*

^{2}

_{c})]/[

*n*

^{4}

_{a}(

*n*

^{2}

_{b}-

*n*

^{2}

_{c})]}. These approximate expressions for the slope include only the cladding indices, and they agree with results obtained previously [5].

Some results of slope and their relative errors are listed in Table 1. Numerical data are evaluated by an average value of losses between *N*=10 and 20. Results derived from Eq. (18) are in excellent agreement with those calculated from numerical data, whereas errors of the approximate slope increase in the order of increasing mode order in each mode group.

#### 4.2. Dependence on the core radius

The core radius dependence of the confinement loss is logarithmically shown in Fig. 4 for the TE, TM, and hybrid modes at *λ*
_{0}=1.0 *µ*m. The number of cladding pairs is fixed at *N*=10. The confinement loss is roughly inclined to decrease with increasing the core radius except for small core radius, and goes toward infinity at the guiding limit. The TE_{0μ} modes exhibit lower loss than other modes in relatively large core radius. As the core radius increases in TM_{0μ} and EH_{νμ} modes, the confinement loss converges to a certain value peculiar to the individual mode groups regardless of mode numbers, *ν* and *μ*. On the contrary, the loss is dependent on mode numbers even for large core radius in TE and HE modes. At the guiding limit where the real part of propagation constant *β* vanishes, the core radius for finite cladding pairs is larger than that for infinite cladding pairs. This is because the guided mode becomes leaky in the case of the finite cladding pairs.

Figures 5(a) and 5(b) indicate the core radius dependence of confinement loss for TE and TM modes, respectively, as a function of cladding high index *n _{a}*. The confinement loss decreases

with increasing the *n _{a}* for a fixed

*r*. In the TE mode, the loss level is by about three orders reduced from

_{c}*n*=2.5 to 3.5, and it is by about two orders reduced from

_{a}*n*=3.5 to 4.5. Even when the cladding high index

_{a}*n*is changed, we can admit a tendency similar to that in Fig. 4. In the TM

_{a}_{0μ}modes, the convergence loss depends only on the

*n*. We see from Figs. 4 to 5(b) that for

_{a}*r*/

_{c}*λ*

_{0}sufficiently larger than unity, the confinement loss is nearly proportional to

*r*

^{−3}

_{c}and

*r*

^{−1}

_{c}for the TE mode and other three mode groups, respectively. These dependencies on the core radius have been pointed out in a loss called radiation loss [2], which is equivalent to the confinement loss, as can be seen from Fig. 15.

The exponents were evaluated from average values between *r _{c}*=20 and 30

*µ*m in Figs. 4 to 5(b). Discrepancies from the above values slightly increase with increasing mode numbers and

*n*. Although relative errors of TE

_{a}_{01}and TM

_{01}modes are less than 0.4 % even for

*n*=4.5, relative errors amount to 2.2 and 8.1 % for TE

_{a}_{03}and TM

_{04}modes, respectively, for

*n*=4.5. For small core radius, one notices a departure from the

_{a}*r*

^{−3}

_{c}relation [5], as expected from Figs.4 to 5(b). Of course, these discrepancies are reduced with increasing the core radius

*r*.

_{c}Dependencies on the core radius similar to the confinement loss can be found in the optical power confinement factor of the Bragg fiber that has infinite cladding pairs [14]. If we express Γ_{QWS} as the optical power confinement factor to the core under the QWS condition, the fractional power within the cladding is represented by Γ_{clad}≡1-Γ_{QWS}. For sufficiently large *r _{c}*/

*λ*

_{0}, the Γ

_{clad}has the same core radius dependence as that of the confinement loss described above. This is due to a fact that optical power is confined to the cladding close to the core in the Bragg fiber and that the same mechanism is related to both confinement loss and optical power confinement factor. The confinement loss of the TM mode does not agree with those of hybrid modes for large

*r*/

_{c}*λ*

_{0}unlike the optical confinement factor.

#### 4.3. Dependence on the wavelength

In calculating the wavelength dependence of confinement loss, we firstly set cladding indices, *n _{a}*,

*n*, and

_{b}*n*, and external layer index

_{c}*n*

_{ex}. Cladding layer thicknesses,

*a*and

*b*, are determined from Eq. (16) so as to satisfy the QWS condition at

*λ*

_{0}=λ

_{QWS}. After the fiber structure was fixed, wavelength was varied.

The wavelength dependence of confinement loss is semi-logarithmically plotted in Fig. 6 for the TE and TM modes. All the modes are plotted that appear at *r _{c}*/

*λ*

_{0}=2.0 under the prescribed condition. We see that for a fixed wavelength, the confinement loss of TE mode is lower than that of the TM mode in most wavelengths shown. In addition, the loss is low for lower-order modes in the TE

_{0μ}mode, whereas the loss is low for higher-order modes in the TM

_{0μ}mode. Difference in the loss among mode number

*μ*is markedly larger in the TE mode than in the TM mode near

*λ*

_{0}=1.0

*µ*m.

Figures 7(a) and 7(b) show the wavelength dependence of confinement loss for HE and EH modes, respectively. We plotted all the modes that appear at *r _{c}*/

*λ*

_{0}=2.0. We notice from Fig. 7(a) that the loss level of the HE

_{νμ}modes is divided into two groups: The HE

_{ν1}modes exhibit relatively high losses in the vicinity of

*λ*

_{0}=1.0

*µ*m, while other HE modes exhibit low losses. This is because the HE

_{ν1}modes possesses lower optical power confinement factor than the HE

_{νμ}modes (

*μ*≥2) [15]. In the EH mode, the loss is low for higher-order modes near the

*λ*

_{QWS}as well as the TM mode. This is because the EH mode reduces to the TM mode under the QWS condition [13]. It is found from Figs. 6 and 7(a) that losses of the HE

_{ν1}modes are nearly the same as those of the TM mode near

*λ*

_{0}=1.0

*µ*m. In the EH mode, difference in the loss resulting from mode numbers,

*ν*and

*μ*, is relatively small. Losses of the EH mode are roughly situated between the TM and HE

_{νμ}(

*μ*≥2) modes at

*λ*

_{0}=1.0

*µ*m, and amount to a loss level of about 10

^{6}dB/km. Figures 6 to 7(b) show that low loss modes are the TE

_{01}, TE

_{02}, HE

_{13}, and TE

_{03}modes in the order of increasing loss near the

*λ*

_{QWS}.

The wavelength dependence of confinement loss is shown in Fig. 8(a) and 8(b) for the TE and TM modes, respectively, as a function of cladding high index *n _{a}*. All the modes are plotted that appear for each

*n*. In both mode groups, losses are lowered with increasing the

_{a}*n*. In the TE mode, the loss level is by about three orders reduced from

_{a}*n*=2.5 to 3.5, and it is by about two orders reduced from

_{a}*n*=3.5 to 4.5. Wavelength region existed becomes wide with increasing (

_{a}*n*-

_{a}*n*) for each mode. TE

_{b}_{03}and TM

_{04}modes do not exist in long wavelengths compared to other modes. This is due to that the two modes reach the guiding limit because of

*β*=0.

Figure 9 depicts the wavelength dependence of confinement loss for the TE_{01} and TM_{01} modes with several core radii. The confinement loss decreases with increasing the core radius *r _{c}* in the neighborhood of

*λ*

_{QWS}. The decrease due to the

*r*is marked in the TE

_{c}_{01}mode. In the TE

_{01}mode, loss value at a particular wavelength is low for large

*r*. In the TM

_{c}_{01}mode, however, the loss level caused by the

*r*is converted near a wavelength of 0.95

_{c}*µ*m.

One can find a tendency about wavelength *λ*
_{min} showing the minimum loss from Figs. 6-9. In TE and TM modes, the minimum-loss wavelength *λ*
_{min} is shorter than *λ*
_{QWS} in low loss modes, whereas *λ*
_{min} is longer than *λ*
_{QWS} in high loss modes. We observe from Figs. 6 and 8(a) that *λ*
_{min}<*λ*
_{QWS} in the TE_{0μ} mode. Figures 6 and 8(b) show that the loss of the TM_{04} mode is lower than that of other TM modes, and that *λ*
_{min}<*λ*
_{QWS} for the TM_{04} mode while *λ*
_{min}>*λ*
_{QWS} for other TM modes. These tendencies also hold in Fig. 9. Since hybrid modes consist of electromagnetic components common to the TE and TM modes, we can not admit a sure tendency about hybrid modes from Figs. 7(a) and 7(b).

The minimum loss is relatively close to a loss at λ_{QWS}, although the minimum-loss wavelength *λ*
_{min} deviates from the *λ*
_{QWS}. The tendency about discrepancy between *λ*
_{QWS} and *λ*
_{min} is also found in a fiber loss which is estimated from the optical power confinement factor [16].

#### 4.4. Dependence on cladding high index

The cladding high index dependence of confinement loss is shown in Fig. 10 for several modes with *n _{b}*=

*n*

_{ex}=1.5. As for TE and TM modes, we show modes that appear at

*r*/

_{c}*λ*

_{0}=2.0 and

*n*=2.5. The confinement loss decreases with increasing the

_{a}*n*. For example, for the TE

_{a}_{01}mode we have loss values of 0.1 and 0.01 dB/km at

*n*=3.61 and 4.02, respectively, in this small core radius.

_{a}## 5. Numerical results of the TE_{01} mode

Several characteristics of the TE_{01} mode will be given in detail in this section because the TE_{01} mode exhibits the lowest confinement loss among all the modes.

#### 5.1. Dependence on the number of cladding pairs

Figure 11 shows the dependence of confinement loss of the TE_{01} mode on the number, *N*, of cladding pairs as a function of core radius *r _{c}* and cladding high index

*n*. As the number of cladding pairs increases, the confinement loss changes in roughly linear dependence even for

_{a}*N*=1. For example, for

*n*=2.5 and

_{a}*r*=2.0

_{c}*µ*m, the confinement loss is by 0.9996×10

^{3}reduced from

*N*=5 to 10, it is by 0.9982×10

^{3}reduced from

*N*=10 to 15, and it is by 0.9976×10

^{3}reduced from

*N*=15 to 20. In addition, for the TE

_{01}mode with

*r*=2.0

_{c}*µ*m and

*N*=10, the loss is by 1.861×10

^{3}reduced from

*n*=2.5 to 3.5, and it is by 2.080×10

_{a}^{2}reduced from

*n*=3.5 to 4.5.

_{a}These results for the same mode can be estimated using Eqs. (16) and (17) with *U*
_{QWS}=3.8317. When parameters are set to be *λ*
_{0}=1.0 *µ*m, *r _{c}*=2.0

*µ*m,

*n*=2.5, and

_{a}*n*=1.5, one obtains (

_{b}*a*/

*b*)

^{2N}=(0.50135)

^{2N}. Losses decreases by 0.9968×10

^{3}every five cladding pairs. Relative errors between numerical and theoretical values are less than 0.3 % even for

*N*=5. When

*N*=10, we have 1.0065×10

^{-6}, 5.4131×10

^{-10}, and 2.6018×10

^{-12}for

*n*=2.5, 3.5, and 4.5, respectively. Loss ratios are 1.859×10

_{a}^{3}and 2.080×10

^{2}for

*n*=2.5 to 3.5 and 3.5 to 4.5, respectively. Excellent agreement between numerical and theoretical values supports the validity of Eq. (17). Dependence on the

_{a}*r*can not be estimated by using Eq. (17) alone because the

_{c}*r*may also be included in other factors.

_{c}#### 5.2. Dependence on wavelength

The wavelength dependence of confinement loss of the TE_{01} mode is illustrated in Fig. 12 as a function of the cladding high index *n _{a}* and core radius

*r*. The confinement loss is reduced with increasing the core radius and cladding high index. The minimum-loss wavelength

_{c}*λ*

_{min}shifts toward a short wavelength as the cladding high index

*n*becomes large. The

_{a}*λ*

_{min}is smaller than the

*λ*

_{QWS}in spite of

*n*and

_{a}*r*in the TE

_{c}_{01}mode. For example, we obtain a loss value of ≈10

^{-3}dB/km at

*λ*

_{0}=1.0

*µ*m for a combination of

*r*=2.0

_{c}*µ*m and

*n*=4.5 or

_{a}*r*=10.0

_{c}*µ*m and

*n*=3.5. Photonic band width is nearly independent of the

_{a}*r*but it increases with increasing the

_{c}*n*.

_{a}#### 5.3. Dependence on cladding indices

Let us consider a possible optimum TE_{01} mode transmission. TM and hybrid modes tend to be cut off more readily than the TE mode [13]. Although a radiation loss of the TE_{01} mode is about five orders smaller than those of TM and hybrid modes, it is only 2.9 times smaller than that of the TE_{02} mode [17]. If only the TE mode is supported in the Bragg fiber, then single mode transmission can be realized for 0.610 (=*j*
_{1,1}/2*πn _{c}*)≤

*r*/

_{c}*λ*

_{0}≤1.117 (=

*j*

_{1,2}/2

*πn*) under the QWS condition, where

_{c}*j*are the

_{ν,μ}*μ*th zeroes of the Bessel function

*J*of order

_{ν}*ν*. We use the normalized core radius

*r*/

_{c}*λ*

_{0}=1.117. Cladding layer thicknesses are set so as to satisfy the QWS condition for each

*n*and

_{a}*n*, and are obtainable from

_{b}*a*=

*λ*

_{0}/

*n*and

_{a}*b*=

*λ*

_{0}/

*n*.

_{b}Figures 13(a) and 13(b) show the wavelength dependence of confinement loss of the TE_{01} mode as a function of cladding high *n _{a}* and low indices

*n*, respectively. The confinement loss decreases as the cladding index contrast (

_{b}*n*–

_{a}*n*) increases. We see from these figures that for a fixed (

_{b}*n*–

_{a}*n*), say 1.0 or 1.5, the smaller the

_{b}*n*is, the lower the confinement loss is. This is because low cladding indices produce wide cladding layer thicknesses for identical core radius and wavelength under the QWS condition, leading to little field penetration into the cladding, hence the low confinement loss. It is also valid for the cladding index contrast dependence that the minimum-loss wavelength

_{b}*λ*

_{min}becomes short for a low loss case.

#### 5.4. Dependence on external layer index

The confinement loss is expected to have a strong dependence on the external layer index *n*
_{ex} because the *n*
_{ex} severely affects the cladding field distribution. The wavelength dependence of confinement loss is shown in Fig. 14 for three *n*
_{ex} values. Loss values increase with decreasing the nex near the central region of photonic band, although they do not little change near edges of photonic band.

## 6. Comparison with other methods

We show results calculated by the multilayer division method in Figs. 15 and 16 to compare with those calculated by the transfer matrix [2] and Chew’s methods [4], respectively. Parameters used are the same as those in their citations. Although Ref. [2] makes use of the term of radiation loss, excellent agreement can be seen between the present and transfer matrix methods, indicating that their ”radiation loss” corresponds to the confinement loss. We can also find an excellent agreement between the present and Chew’s methods. The multilayer division method has a high accuracy and some features described in subsection 2.3.

## 7. Conclusion

The confinement loss has comprehensively been studied for TE, TM, and hybrid modes of Bragg fibers using the multilayer division method newly developed. The confinement loss decreases with increasing the core radius, cladding index contrast, and the number, *N*, of cladding layer pairs. The confinement loss is reduced in proportion to (*a*/*b*)^{2N} and (*n*
^{2}
_{b}
*b*/*n*
^{2}
_{a}
*a*)^{2N} for the TE and other three modes, respectively, with respect to *N*, with cladding high *n _{a}* and low indices

*n*, and their corresponding thicknesses

_{b}*a*and

*b*under the QWS condition. The confinement loss decreases in inverse proportion to the third and first powers of core radius for the TE and other three modes, respectively, for sufficiently large core radius. The confinement loss of TM

_{0μ}and EH

_{νμ}modes converges to a certain value peculiar to the individual mode group regardless of mode numbers,

*ν*and

*μ*. The HE

_{ν1}modes exhibit higher confinement loss than the HE

_{νμ}modes (

*μ*≥2). Low loss modes are the TE

_{01}, TE

_{02}, HE

_{13}, and TE

_{03}modes in the order of increasing loss. In particular, detailed numerical data were presented for the TE

_{01}mode.

The wavelength *λ*
_{min} showing the minimum loss tends to slightly deviate from *λ*
_{QWS} at which the fiber structure is designed so as to satisfy the QWS condition. In spite of the deviation in wavelength, the minimum loss value is close to a loss at *λ*
_{QWS}. The present results are in excellent agreement with those calculated by the transfer matrix and Chew’s methods.

## Appendix A: Amplitude Coefficients in Eq. (14)

Amplitude coefficients in Eq. (14) can be obtained by solving Eq. (12) as follows:

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