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 TE₀₁ 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 TE₀₁ 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 ith layer and its outer radius are represented by n_i and r_i, respectively.

 

Fig. 1. Cross sectional view of the cylindrical symmetric fiber. n_i, refractive index of the ith layer; r_i, outer radius of the ith layer; r _i1 and r _i2, arbitrary radial coordinates within the ith layer; A_i-D_i, amplitude coefficients of the ith layer; D_i(r), representation matrix, Q_i(r _i2, r _i1); displacement matrix between r=r _i1 and r _i2.

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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 ith layer and is defined by

Here, A_i-D_i denote the amplitude coefficients of the ith layer, H ⁽¹⁾ _ν=J_ν+iN_νand H ⁽²⁾ _ν=J_ν-iN_ν indicate the Hankel functions of the first and second kinds of order ν, respectively, k ₀=2π/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_z and H_θ, to be continuous in each layer interface, are selected as fundamental components. Electromagnetic fields are represented in a matrix form:

with r_i−1≤r≤r_i and

Here, D_i(r) is referred to as the representation matrix of the ith layer. Its elements are expressed as

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_z and E_θ, and cos(νθ+θ _in) for the E_z and H_θ. For 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

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 ith layer are related to those at r=r ₁ of the first layer as

where

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_N. In representing fields we use 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_N can be expressed using Eqs. (3) and (5). Substituting Eq. (11) and expressions for fields into Eq. (10) and rearranging it yields

where

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 ith 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.

 

Fig. 2. (Color online) Schematics of a Bragg fiber. r_c, core radius; n_c, refractive index of core; n_a and n_b, indices of layers with thickness a and b, respectively; Λ=a+b, period in the cladding; nex, external layer index.

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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_ν(iz)=exp(iνπ/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_c. The cladding has a finite number, N, of layer pairs that consist of high n_a and low indices n_b (n_a>n_b>n_c). Their corresponding layer thicknesses are 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_c=1.0 throughout this paper.

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=1.0µm; core radius, r_c=2.0µm; cladding high index, n_a=2.5; cladding low index, n_b=1.5; external layer index, 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_b, respectively, in Eq. (16). The 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 µm. 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 TM₀₁ 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 mth cladding a layer vary according to exp[-iK ^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(−i2NK ^S ₁Λ).

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

 

Fig. 3. (Color online) Cladding pair dependence of confinement loss for TE, TM, and hybrid modes. λ ₀=1.0 µm, r_c=2.0 µm, n_a=2.5, and n_b=n _ex=1.5. Cladding thicknesses, a and b, are determined from Eq. (16) for each combination of n_a, n_b, and UQWS.

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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 µm. 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

Table 1. Numerical Results on the Slope in Fig. 3

View Table

 

Fig. 4. (Color online) Core radius dependence of confinement loss for TE, TM, and hybrid modes. λ ₀=1.0 µm, n_a=2.5, n_b=n _ex=1.5, and N=10. Cladding thicknesses, a and b, are determined from Eq. (16) for each combination of r_c, n_a, and n_b.

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with increasing the n_a for a fixed r_c. In the TE mode, the loss level is by about three orders reduced from n_a=2.5 to 3.5, and it is by about two orders reduced from n_a=3.5 to 4.5. Even when the cladding high index n_a is changed, we can admit a tendency similar to that in Fig. 4. In the TM_0μ modes, the convergence loss depends only on the n_a. We see from Figs. 4 to 5(b) that for 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 µm in Figs. 4 to 5(b). Discrepancies from the above values slightly increase with increasing mode numbers and n_a. Although relative errors of TE₀₁ and TM₀₁ modes are less than 0.4 % even for n_a=4.5, relative errors amount to 2.2 and 8.1 % for TE₀₃ and TM₀₄ modes, respectively, for n_a=4.5. For small core radius, one notices a departure from the 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.

 

Fig. 5. (Color online) Core radius dependence of confinement loss for TE and TM modes as a function of cladding high index. λ ₀=1.0 µm, n_b=n _ex=1.5, and N=10. Cladding thicknesses, a and b, are determined from Eq. (16) for each combination of r_c, n_a, and n_b.

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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_b, and n_c, and external layer index 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 µm.

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 µm, 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 µm. 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 µm, and amount to a loss level of about 10⁶ dB/km. Figures 6 to 7(b) show that low loss modes are the TE₀₁, TE₀₂, HE₁₃, and TE₀₃ modes in the order of increasing loss near the λ _QWS.

 

Fig. 6. (Color online) Wavelength dependence of confinement loss for TE and TM modes. λ _QWS=1.0 µm, n_a=2.5, n_b=n _ex=1.5, and N=10. All the modes are shown that exist at r_c/λ ₀=2.0.

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Fig. 7. (Color online) Wavelength dependence of confinement loss for hybrid modes. Parameters are the same as those in Fig. 6.

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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_a. In both mode groups, losses are lowered with increasing the n_a. In the TE mode, the loss level is by about three orders reduced from n_a=2.5 to 3.5, and it is by about two orders reduced from n_a=3.5 to 4.5. Wavelength region existed becomes wide with increasing (n_a-n_b) for each mode. TE₀₃ and TM₀₄ 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 TE₀₁ and TM₀₁ 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_c is marked in the TE₀₁ mode. In the TE₀₁ mode, loss value at a particular wavelength is low for large r_c. In the TM₀₁ mode, however, the loss level caused by the r_c is converted near a wavelength of 0.95 µm.

 

Fig. 8. (Color online) Wavelength dependence of confinement loss for TE and TM modes as a function of cladding high index. λ _QWS=1.0 µm, r_c=2.0 µm, n_b=n _ex=1.5, and N=10. All the modes are shown that exist at λ ₀=1.0 µm for each n_a.

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Fig. 9. (Color online) Wavelength dependence of confinement loss for TE₀₁ and TM₀₁ modes as a function of core radius. λ _QWS=1.0 µm, n_a=2.5, n_b=n _ex=1.5, and N=10.

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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 TM₀₄ mode is lower than that of other TM modes, and that λ _min<λ _QWS for the TM₀₄ 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].

 

Fig. 10. (Color online) Cladding high index dependence of confinement loss for several modes. λ ₀=1.0 µm, r_c=2.0 µm, n_b=n_ex=1.5, and N=10. All the modes are shown that exist at r_c/λ0=2.0 and n_a=2.5.

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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_a=2.5. The confinement loss decreases with increasing the n_a. For example, for the TE₀₁ mode we have loss values of 0.1 and 0.01 dB/km at n_a=3.61 and 4.02, respectively, in this small core radius.

5. Numerical results of the TE₀₁ mode

Several characteristics of the TE₀₁ mode will be given in detail in this section because the TE₀₁ 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 TE₀₁ mode on the number, N, of cladding pairs as a function of core radius r_c and cladding high index n_a. As the number of cladding pairs increases, the confinement loss changes in roughly linear dependence even for N=1. For example, for n_a=2.5 and r_c=2.0 µ m, the confinement loss is by 0.9996×10³ reduced from N=5 to 10, it is by 0.9982×10³ reduced from N=10 to 15, and it is by 0.9976×10³ reduced from N=15 to 20. In addition, for the TE₀₁ mode with r_c=2.0 µ m and N=10, the loss is by 1.861×10³ reduced from n_a=2.5 to 3.5, and it is by 2.080×10² reduced from n_a=3.5 to 4.5.

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 µm, r_c=2.0 µm, n_a=2.5, and n_b=1.5, one obtains (a/b)^2N=(0.50135)^2N. Losses decreases by 0.9968×10³ 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_a=2.5, 3.5, and 4.5, respectively. Loss ratios are 1.859×10³ and 2.080×10² for n_a=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 r_c can not be estimated by using Eq. (17) alone because the r_c may also be included in other factors.

 

Fig. 11. (Color online) Cladding pairs dependence of confinement loss for the TE₀₁ mode as a function of core radius and cladding high index. λ ₀=1.0 µm, and n_b=n _ex=1.5. Cladding thicknesses, a and b, are determined from Eq. (16) for each combination of r_c and n_a. Dotted, solid, and dotted-dashed curves indicate n_a=2.5, 3.5, and 4.5, respectively.

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Fig. 12. (Color online) Wavelength dependence of confinement loss of the TE₀₁ mode. λ _QWS=1.0 µm, n_b=n _ex=1.5, and N=10. Cladding thicknesses, a and b, are determined from Eq. (16) at λ ₀=1.0 µm for each combination of r_c, n_a, and n_b.

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5.2. Dependence on wavelength

The wavelength dependence of confinement loss of the TE₀₁ mode is illustrated in Fig. 12 as a function of the cladding high index n_a and core radius r_c. The confinement loss is reduced with increasing the core radius and cladding high index. The minimum-loss wavelength λ _min shifts toward a short wavelength as the cladding high index n_a becomes large. The λ _min is smaller than the λ _QWS in spite of n_a and r_c in the TE₀₁ mode. For example, we obtain a loss value of ≈10^-3 dB/km at λ ₀=1.0 µm for a combination of r_c=2.0 µm and n_a=4.5 or r_c=10.0 µm and n_a=3.5. Photonic band width is nearly independent of the r_c but it increases with increasing the n_a.

 

Fig. 13. (Color online) Wavelength dependence of confinement loss of the TE₀₁ mode. λ _QWS=1.0 µm, r_c=1.117 µm, and N=10. (a) n_b=n _ex=1.5. (b) n_a=3.5 and n _ex=1.5.

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5.3. Dependence on cladding indices

Let us consider a possible optimum TE₀₁ mode transmission. TM and hybrid modes tend to be cut off more readily than the TE mode [13]. Although a radiation loss of the TE₀₁ mode is about five orders smaller than those of TM and hybrid modes, it is only 2.9 times smaller than that of the TE₀₂ mode [17]. If only the TE mode is supported in the Bragg fiber, then single mode transmission can be realized for 0.610 (=j _1,1/2πn_c)≤r_c/λ ₀≤1.117 (=j _1,2/2πn_c) under the QWS condition, where j_ν,μ are the μth zeroes of the Bessel function J_ν of order 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_a and n_b, and are obtainable from a=λ ₀/n_a and b=λ ₀/n_b.

Figures 13(a) and 13(b) show the wavelength dependence of confinement loss of the TE₀₁ mode as a function of cladding high n_a and low indices n_b, respectively. The confinement loss decreases as the cladding index contrast (n_a–n_b) increases. We see from these figures that for a fixed (n_a–n_b), say 1.0 or 1.5, the smaller the n_b 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 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.

 

Fig. 14. (Color online)Wavelength dependence of confinement loss of the TE₀₁ mode with three values of external layer index n _ex. λ _QWS=1.0 µm, r_c=2.0 µm, n_a=2.5, n_b=1.5, and N=10.

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Fig. 15. (Color online) Comparison in confinement loss between the present and transfer matrix methods. n_a=4.6, n_b=1.6, n _ex=n_b, Λ=0.434 µm, r_c=30Λ, a=0.22Λ, b=0.78Λ, and N=8. Solid curves indicate results obtained by the present method, and dotted curves indicate results obtained by the transfer matrix method.

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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_b, and their corresponding thicknesses 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 TE₀₁, TE₀₂, HE₁₃, and TE₀₃ modes in the order of increasing loss. In particular, detailed numerical data were presented for the TE₀₁ mode.

 

Fig. 16. (Color online) Comparison in confinement loss between the present and Chew’s methods. n_a=1.49, n_b=1.17, n _ex=n_a, a=0.2133 µm, b=0.346 µm, and N=16. Losses are shown for three core radii. Solid curves indicate results calculated by the present method, and dotted curves indicate results calculated by Chew’s method.

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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.