Dependence of mode characteristics on the central defect in elliptical hole photonic crystal fibers

Wang Zhi; Ren Guobin; Lou Shuqin; Jian Shuisheng

doi:10.1364/OE.11.001966

1. Introduction

Intensive research has been generated by the development of photonic crystal fibers (PCFs), which are silica optical fibers that consist of a central defect region in a regular lattice of air holes. According to the mechanism used to guide light, PCFs can be divided into two general classes [1]: a total internal reflection (TIR) PCF and a photonic bandgap (PBG) PCF. These fibers exhibit some unusual properties including endlessly single mode [2,3], very large or very small mode area [4–6], zero dispersion wavelength shifting into the visible spectrum [7–10], and guidance in air [11–13]. Several authors have considered high birefringence PCFs, in which the birefringence is introduced either by a local change in location or size of a few holes [14,15] or by use of elliptical holes [16–18].

Together with the technological advancement in the fabrication [19] of PCFs, powerful theoretical tools have been developed to model the guidance properties. Some of these are the effective index method [2,10,20], the plane wave expansion method [21–23], the localized basis function method [24–28], the finite-element method [29,30], and more recently the beam propagation method [31], and the multipole method [32,33]. When higher-order modes or polarization properties are considered, a full vector approach is crucial to assess the true behavior of the electromagnetic wave in a complex inhomogeneous structure such as a PCF.

The polarization and dispersive properties of elliptical hole PCFs (EHPCFs) have been investigated with the plane wave method [18] in detail. The results show that a number of important quantities, including birefringence and walk-off parameters, exhibit frequency dependence quite unlike conventional birefringent systems. It was found that the presence of a central defect, a smaller hole in the core region, gives rise to larger birefringence than in the case of a missing hole [16]. To our knowledge, there is nothing in the literature that describes how central defects affect mode properties.

Here we develop the supercell overlap method, a full vector model based on our previous work [27], for modeling the triangular lattice EHPCF. Regardless of how the light is guided, by TIR or PBG, we investigated the propagation characteristics of PCFs by recasting Maxwell’s equations into an eigenvalue system based on a novel accurate description of the periodic dielectric profile. Then, for the first time to our knowledge, the dependence of birefringence on the central defect in microstructure optical fibers is discussed in detail.

2. Full Vector Model

2.1 Eigenequation

We believe that the PCF is lossless and uniform in the propagation (z) direction, so our main task is to investigate the transverse mode field distribution, which satisfies a pair of coupled wave equations [25,34]. When the transverse mode field is expanded as the decomposition of an orthogonal set of Hermite-Gaussian basis functions, the eigenvalue equation, shown as Eq. (1), will be obtained by use of the orthonormality of the Hermite-Gaussian basis functions [35].

L [\begin{matrix} e_{x} \\ e_{y} \end{matrix}] \equiv [\begin{matrix} I_{abcd}^{(1)} + k^{2} I_{abcd}^{(2)} + I_{abcd}^{(3) x} & I_{abcd}^{(4) x} \\ I_{abcd}^{(4) y} & I_{abcd}^{(1)} + k^{2} I_{abcd}^{(2)} + I_{abcd}^{(3) y} \end{matrix}] [\begin{matrix} e_{x} \\ e_{y} \end{matrix}] = β_{j}^{2} [\begin{matrix} e_{x} \\ e_{y} \end{matrix}],

where β_j is the propagation constant that corresponds to the transverse mode (e_x , e_y ), I ⁽¹⁾, I ⁽²⁾, I ⁽³⁾, and I ⁽⁴⁾ are overlap integrals of the modal functions, which are defined as

I_{abcd}^{(1)} = \iint_{- \infty}^{+ \infty} ψ_{a} (x) ψ_{b} (y) \nabla_{t}^{2} [ψ_{c} (x) ψ_{d} (y)] dx dy,

I_{abcd}^{(2)} = \iint_{- \infty}^{+ \infty} ε ψ_{a} (x) ψ_{b} (y) ψ_{c} (x) ψ_{d} (y) dx dy,

I_{abcd}^{(3) x} = \iint_{- \infty}^{+ \infty} ψ_{a} (x) ψ_{b} (y) \frac{\partial}{\partial x} [ψ_{c} (x) ψ_{d} (y) \frac{\partial \ln ε}{\partial x}] dx dy,

I_{abcd}^{(3) y} = \iint_{- \infty}^{+ \infty} ψ_{a} (x) ψ_{b} (y) \frac{\partial}{\partial y} [ψ_{c} (x) ψ_{d} (y) \frac{\partial \ln ε}{\partial y}] dx dy,

I_{abcd}^{(4) x} = \iint_{- \infty}^{+ \infty} ψ_{a} (x) ψ_{b} (y) \frac{\partial}{\partial x} [ψ_{c} (x) ψ_{d} (y) \frac{\partial \ln ε}{\partial y}] dx dy,

I_{abcd}^{(4) y} = \iint_{- \infty}^{+ \infty} ψ_{a} (x) ψ_{b} (y) \frac{\partial}{\partial y} [ψ_{c} (x) ψ_{d} (y) \frac{\partial \ln ε}{\partial x}] dx dy,

where ε is the dielectric constant distribution ε(x,y), ψ_i (s) (I=a,b,c,d, s=x,y) are elements of the orthogonal set of Hermite-Gaussian basis functions, and $ψ_{i} (s) = \frac{2^{\frac{- i}{2}} π^{\frac{- 1}{4}}}{\sqrt{i! ω}} \exp (- \frac{s^{2}}{2 ω^{2}}) H_{i} (\frac{s}{ω}),$ where H_i (s/ω) is the ith-order Hermite polynomial and ω is the characteristic width of the basis set.

2.2 Dielectric Constant Profile

We use the supercell overlap method [27,35,36] to describe the transverse dielectric structure of PCF. In this method, we introduce two two-dimensional (2-D) virtually perfect photonic crystals (PC1 and PC2) that can be combined to reconstruct the dielectric structure of a supercell lattice PCF. The dielectric structure of the PCF can be expressed as the sum of the decomposition of both PC1 and PC2. Figure 1 shows an example to illustrate the way in which the transverse dielectric structure is constructed, and Table 1 lists the parameters of both virtual PCs.

Fig. 1. Schematic of the way in which the transverse dielectric structure is constructed. PC1 is a perfect triangular lattice constructed by the blue elliptical holes, the dielectric constant is ε_air in the hole and ε_si outside the hole. PC2 is another perfect triangular lattice constructed by the coaxial elliptical rings with which the inner major axis is d_c and the outer major axis is d, the dielectric constant is 0 in d_c or outside d and ε_si -ε_air between d_c and d.

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Table 1. Parameters of Virtual PCs for Triangular Lattice PCFs

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Decomposing the dielectric structure of both PC1 and PC2 by use of cosine functions, the dielectric constant of the supercell lattice PCF can be expressed as

ε (x, y) = ε |_{PC 1} + ε |_{PC 2}, \ln ε = (\ln ε) |_{PC 1} + (\ln ε) |_{PC 2},

where

ε |_{PC 1} = \sum_{a, b = 0}^{P_{1} - 1} P_{1 ab} \cos \frac{2 π a x}{l_{1 x}} \cos \frac{2 π b y}{l_{1 y}},

ε ∣_{PC 2} = \sum_{a, b = 0}^{P_{2} - 1} P_{2 ab} \cos \frac{2 π a x}{l_{2 x}} \cos \frac{2 π b y}{l_{2 y}},

(\ln ε) ∣_{PC 1} = \sum_{a, b = 0}^{P_{1} - 1} P_{1 ab}^{\ln} \cos \frac{2 π a x}{l_{1 x}} \cos \frac{2 π b y}{l_{1 y}},

(\ln ε) ∣_{PC 2} = \sum_{a, b = 0}^{P_{2} - 1} P_{2 ab}^{\ln} \cos \frac{2 π a x}{l_{2 x}} \cos \frac{2 π b y}{l_{2 y}},

and where P ₁ and P ₂ are the number of decomposition terms of PC1 and PC2, respectively; and P _1ab, P _2ab, $P_{1 ab}^{ln}$ , and $P_{2 ab}^{ln}$ are the coefficients that can be obtained through the Fourier transform. P _1ab and $P_{1 ab}^{ln}$ can be obtained as described in Refs. [27, 36]; the approach to evaluate P _2ab and $P_{2 ab}^{ln}$ is simply described as follows.

For the EHPCF, the central defect elliptical hole size is d_c along the y axis, elliptical ratio η is the same as the cladding holes. There are two fractional parameters, f_c and f ₂, of PC2 that are defined as follows:

f_{c} = \frac{π d_{c}^{2}}{(2 η l_{2 x} l_{2 y})}, f_{2} = \frac{π d^{2}}{(2 η l_{2 x} l_{2 y})},

where η is defined as η=d/e, and d and e represent the length of the elliptical holes along the y and x axes, respectively, as shown in Fig. 1.

When the dielectric structure of PC2 is expressed in the form of a Fourier transform as

ε (x, y) ∣_{PC 2} = \sum_{m, n = - (P_{2} - 1)}^{P_{2} - 1} F_{2} (K_{mn}) \cos (k_{1} x) \cos (k_{2} y),

the Fourier transform coefficient will be

F_{2} (K_{mn}) = 2 (n_{si}^{2} - n_{air}^{2}) [f . \frac{J_{1} (∣ K_{mn} ∣ \frac{d}{2})}{∣ K_{mn} ∣ \frac{d}{2}} - f_{c} . \frac{J_{1} (∣ K_{mn} ∣ \frac{d_{c}}{2})}{∣ K_{mn} ∣ \frac{d_{c}}{2}}], K_{mn} \neq 0,

F_{2} (0) = (f - f_{c}) (n_{si}^{2} - n_{air}^{2}),

where J ₁ is the first-order Bessel function, k ₁ and k ₂ are the real and imaginary parts of K_mn , i.e., K_{m n} =k ₁+ik ₂ (i is the imaginary unit). K_mn depends on the reciprocal lattice vector [21,22] of PCFs and can be expressed as

K_{mn} = (m + n) k_{x} - i (m - n) k_{y}, k_{x} = \frac{2 π}{\frac{l_{2 x}}{η}}, k_{y} = \frac{2 π}{l_{2 y}} .

P _2ab can be obtained when F ₁ is replaced with F ₂ in Eq. (9) in Ref. 28. $P_{2 ab}^{ln}$ can be analytically obtained in the same way as P _2ab, but only when the dielectric constant parameters are replaced with their logarithm as shown in Table 1.

Figure 2 shows the simulation results of the triangular lattice EHPCF, with a structure parameter hole pitch of D=2.3 µm, a cladding elliptical hole size along the y axis of d=0.8D, a central defect elliptical hole size along the y axis of d_c =0.4D, and an elliptical ratio of both the cladding hole and the central defect η=2. To show the dielectric structure clearly, the supercell parameter N is set at 4 in Fig. 2, whereas N=10 in the following simulations. Figures 2(a), 2(b), and 2(c) are the periodic dielectric profiles of the reconstructed PCF, the virtual PC1, and the virtual PC2, respectively. The unit cell in Fig. 2(c) is a coaxial elliptical ring with which the inner major axis is d_c and the outer major axis is d, the dielectric constant is 0 in d_c or outside d, and ε_si -ε_air between d_c and d.

Fig. 2. Simulation results of the dielectric constant profiles of the PCF and both virtually perfect PCs with parameters of D=2.3 µm, d=0.8D, d_c =0.4D, η=2, and N=4.

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2.3 Overlap Integrals

Substituting decomposition Eq. (4) into Eq. (2), the overlap integrals can be calculated by use of the orthonormality of the Hermite-Gaussian basis functions and the standard integration by parts technique combined with some of the definite integrals available in collections [37] along with some identities [38]. The fact that all these integrals can be performed analytically is such a major advantage that the numerical precision and efficiency can be significantly improved, especially when a significant number of terms are needed in the expansions. The expressions for overlap integrals I ⁽¹⁾, I ⁽²⁾, and I ⁽³⁾ can be found in Refs. [27, 36]; I ^(4)x and I ^(4)y are shown here:

I_{abcd}^{(4) x} = - \sum_{f, g = 0}^{P_{1} - 1} P_{1 fg}^{\ln} I_{fac}^{(42) x} I_{gbd}^{(41) y} - \sum_{f, g = 0}^{P_{2} - 1} P_{2 fg}^{\ln} {IN}_{fac}^{(42) x} {IN}_{gbd}^{(41) y},

I_{abcd}^{(4) y} = - \sum_{f, g = 0}^{P_{1} - 1} P_{1 fg}^{\ln} I_{fac}^{(41) x} I_{gbd}^{(42) y} - \sum_{f, g = 0}^{P_{2} - 1} P_{2 fg}^{\ln} {IN}_{fac}^{(41) x} {IN}_{gbd}^{(42) y},

where

I_{i_{1} i_{2} i_{3}}^{(41) s} = \int_{- \infty}^{+ \infty} \frac{\partial \cos (\frac{2 π i_{1} s}{l_{1 s}})}{\partial s} ψ_{i_{2}} (s) ψ_{i_{3}} (s) d s,

I_{i_{1} i_{2} i_{3}}^{(42) s} = \int_{- \infty}^{+ \infty} \frac{\partial ⌊ ψ_{i_{3}} (s) \cos (\frac{2 π i_{1} s}{l_{1 s}}) ⌋}{\partial s} ψ_{i_{2}} (s) d s .

When l _1s is replaced with l _2s, IN ^(41)s and IN ^(42)s can be evaluated in the same way.

By solving the eigenequation (1) at a particular wavelength λ, the eigenvalue-eigenvector pair of PCFs at λ can be calculated, and the transverse modal electric field can be obtained by substitution of the eigenvector into its decomposition equation.

3. Mode Symmetry and Classification

From the theory of group representations, we know that the symmetry of a waveguide controls several of the important characteristics of waveguide modes [39,40]. A determination of the symmetry type of a particular waveguide enables one to classify the possible modes in mode classes, to predict the mode degeneracy between mode classes, and to determine the azimuthal symmetry of the modal electromagnetic fields in each mode class. In addition, based on the azimuthal symmetry, one can specify a minimum sector of waveguide cross section together with associated boundary conditions for this sector, which is necessary and sufficient to determine all the modes of that mode class completely.

The modes of a waveguide with C_nν symmetry either exhibit the full waveguide symmetry (C_nν ) and are nondegenerate or occur in degenerate pairs that support this symmetry only in combination [36,39–41]. Because the cross section of the triangular lattice EHPCF studied here possesses C _2ν symmetry, all the modes can be classified into four classes. These classes are nondegenerate and exhibit the full waveguide symmetry individually [39,40].

Fig. 3. Minimum sectors for waveguides with C _2ν symmetry. The waveguide modes are classified into four classes (p=1,2,3,4). The solid lines indicate short-circuit boundary conditions, the dashed lines indicate open-circuit boundary conditions.

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Figure 3 shows the minimum sector for waveguides with C _2ν symmetry. All four nondegenerate mode classes are labeled p=1, 2, 3, 4, in which the difference between modal classes differs from the minimum sector that combines with boundary conditions that apply to the tangential component of electric or magnetic fields at the edge of the minimum sector. These boundary conditions are either short circuit (solid lines of classes p=2, e_z =0, e_r =0) or open circuit (dashed lines of classes p=1, h_z =0, e_φ =0) [35,39,40] or combinations of these (class p=3,4). Based on this analysis, we can classify the modes of PCF by the behavior of the transverse electric field at the boundary of each minimum sector in Fig. 3.

By applying our approach to the triangular lattice EHPCF with structure parameters: D=2.3 µm, d/D=0.8, d_c =D, and η=2, we extract the first 12 eigenvalues β_j/k (modal index) at wavelength λ=D/0.8 from the eigenequation and plot them in Fig. 4, and the electric field vector of the first six modes is shown in Fig. 5. We found that all modes are nondegenerate, with modes 1–6 belonging to mode class p=4, 1, 2, 3, 3, 1, respectively.

Fig. 4. Modal index of the first 12 modes of the triangular lattice EHPCF with the parameters D=2. 3 µm, d=0.8D, d_c =D, η=2, λ=D/0.8.

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Fig. 5. Electric field vector of the first six modes in the PCF with parameters as in Fig. 4.

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There is one important fact, that is, the electric field of modes 1 and 4 looks like the HE₁₁ mode in step-index fibers, which we refer to as HE_11y and HE_11x. The doublet components of the fundamental mode, as we know, are degenerate in conventional standard fibers and in circular hole triangular lattice PCFs [35,39–41]. However, when the air holes are elliptical, the degeneracy splits significantly, one is mode 1, and the other is mode 4. Regardless of whether both modes are confined or leaky, the birefringence will be defined as Δn=n ₁-n ₄ rather than Δn=n ₁-n ₂, which will be discussed in detail in the following text. The reason for this notation is that, when the central defect is large, here d_c =D, the light propagates in the PCF guided by the PBG mechanism, not TIR. In air-silica PBG fibers, it is true that higher-order modes are bounded whereas lower-order modes are leaky [11,18]. Here we contribute to the mode disorder and the birefringence properties regardless of whether the light is TIR guided or PBG guided.

When the central defect is small, such as d_c =0.2D, the modal indices of the first 12 modes at wavelength λ=D/0.8 are demonstrated in Fig. 6, and the electric field vector of the first three modes, which belong to the mode class p=4, 3, and 1, are plotted in Fig. 7. It can be seen that HE_11y and HE_11x are the first two modes. At wavelength λ=D/4, the case is the same and not discussed substantially here. It can be deduced from these numerical results that the light-guide mechanism of TIR is still efficient when the central defect is not too large. It is too difficult to present the critical edge between the TIR and the PBG because of the complicated large computations.

Fig. 6. Modal index of the first 12 modes of the triangular lattice EHPCF with parameters D=2.3 µm, d=0.8D, d_c =0.2D, η=2, λ=D/0.8.

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Fig. 7. Electric field vector of the first three modes in the PCF with parameters as in Fig. 6.

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4. Mode Disorder in Electronic Hole Photonic Crystal Fiber

At a certain wavelength, all the eigenvalue-eigenvector pairs can be labeled in the decreasing order of the propagation constant, whereas the mode with the greatest β is labeled as mode 1. Consequently, all the modes can be arranged in descending order of β and an increasing order of labels. Conventionally, the two lowest-order modes are the HE_11x and HE_11y, which are called the fundamental modes. When the fiber structure is complicated, the order of modes is changed, and all the modes are rearranged in different orders, which is called mode disorder.

What is the physical reason for mode disorder in EHPCFs? The propagation constant, as is well known, is the longitudinal (z axis of fibers) component of the wave vector. The term mode means the transverse electric field pattern, which can be considered as the interference pattern of electromagnetic waves in the cross section. In polar coordinates, a wave vector can be decomposed into three orthogonal vectors along three different basis vectors as k=k _r+k _φ+k _z, where k _z is just the propagation constant β. Standing cylinder waves along the r direction with wave vector k _r and standing waves along the φ direction with wave vector k _φ will interfere to produce the mode pattern. All the modes are labeled with two numbers m and n, which are the numbers of modes of the two standing waves along the φ and r directions, respectively, such as mode HE_mn or mode EH_mn. At wavelength λ, the amplitude of the wave vector is |k|=2π/λ, which can be rewritten in the form of |k|²=|k _r|²+|k _φ|²+|k _z|². One can easily deduce that two or more different (k _r, k _φ) pairs could exist with the same k _z at a fixed wavelength. We can also state that there are two or more modes with the same propagation constants (β) at this wavelength. Modes with the same β at a wavelength will exchange their mode order near that wavelength, at which point the modes are then disordered and rearranged.

4.1. Mode Disorder between the Fundamental and the Higher-Order Modes

In Fig. 5, where the air holes are elliptical and the central hole is large enough, the HE_11x mode is the fourth mode, the degeneracy between the fundamental doublet is significantly split. It can be said that the mode disorder occurs between the fundamental and the higher-order modes.

Fig. 8. Mode order of HE_11y and HE_11x of an EHPCF. The parameters are shown at top right.

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The property of mode disorder depends on many parameters such as the structure of PCFs and the operating wavelength. To demonstrate mode disorder between the fundamental and the higher-order modes clearly, a triangular lattice EHPCF with parameters D=2.3µm, d=0.8D, d_c =1.2D, and η=2 is considered. The mode order of both HE_11y and HE_11x varies with the operating wavelength, which is shown in Fig. 8. It is obvious that the HE_11y mode keeps the lowest mode within a wide wavelength range, and the HE_11x mode keeps the second lowest mode within a narrower wavelength range. As the wavelength decreases, the mode disorder of both doublets increases, i.e., there are some other modes that become the first or the second mode.

4.2. Mode Disorder between the Fundamental Doublets

In Fig. 8, the order of HE_11y is always lower than HE_11x, and mode disorder does not exist between the fundamental doublets. When the size of the cladding air hole varies, some novelty emerges. Figure 9(a) illustrates the mode order of the fundamental doublets in an EHPCF, which has the structure parameters D=2.3µm, d=0.4D, d_c =1.2D, and η=2. Figure 9 also shows that both modes are the lowest two, but the mode orders exchange with each other when the wavelength is nearly 0.8 µm.

For clarity, Δn=n_y -n_x is introduced and demonstrated as the solid blue curve in Fig. 9(b). It was found that the mode order of both doublets exchanges at point A where Δn is zero. When the wavelength is longer than λ _A, the HE_11y mode is the lowest, its mode index is greater than HE_11x, hence the slower axis of the EHPCF is along the y axis. When the wavelength is shorter than λ _A, it is inverse, and the y axis becomes the faster axis of the EHPCF. The fact that Δn increases as λ decreases at a much shorter wavelength needs to be researched, so it is not discussed in this contribution.

At point A, the mode indices of both modes are exactly the same, so they degenerate at wavelength λ _A. How does this happen and what is the mechanism of the degeneracy? Figure 10 provides a qualitative understanding of the answer to the question. Figure 10(a) is an EHPCF with a central defect air hole, and only one ring of cladding air holes is plotted for simplicity. It can be considered as the composition of two structures: one is an EHPCF with the same photonic crystal cladding and without the central air hole, the other is a fiber with a central elliptical air hole and a homogeneous cladding, all of which are illustrated in Figs. 10(b) and 10(c), respectively, and labeled as fibers a, b, and c in the following text.

Fig. 9. (a) Mode orders of HE_11y and HE_11x of an EHPCF with the parameters shown at right. (b) Relationship between Δn and λ of fibers a, b, and c.

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Fig. 10. Schematic of fibers a, b, and c, which is helpful to understand the mode disorder between the fundamental doublets.

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With regard to fiber b, the polarization property of this kind of EHPCF has been investigated by a few researchers [17,18], who all reached the same conclusion that the mode index of HE_11y is greater than that of HE_11x. The birefringence of the fundamental doublets can be analyzed and is shown as the dotted green curve in Fig. 9(b). The birefringence of fiber c is also plotted as the dash-dot red curve in Fig. 9(b).

Δn of fiber b increases monotonically when λ increases. At high frequencies, the vector terms in the wave equation [i.e., I ⁽³⁾ and I ⁽⁴⁾ in the eigenequation Eq. (1)] become negligible, and the birefringence must vanish in the scalar regime at short wavelengths. When λ increases, the effective mode area increases [4,5,27] because the electric field penetrates into the cladding area more easily, and the more elliptical holes will be sampled by the modes. The long-wavelength behavior of the birefringence is due to the cladding region, which acts as a birefringent material.

In contrast, birefringence Δn of fiber c, which can be regarded as an inversion of the conventional standard elliptical core fiber, tends to zero at both high and low frequencies. At high frequencies, the zero birefringence exists for the same reason as in fiber b, whereas at low frequencies the bulk of the mode energy is distributed far from the core-cladding interface, which induces birefringence. The maximum value at intermediate frequencies scales with the relative refractive-index difference between the fiber core and the cladding in the weak guidance limit [18].

In Fig. 9(b), the a curve can be approximately understood as the sum of curve b and curve c, i.e., the birefringence of fiber a can be considered as the sum of both fibers b and c. When the birefringence of fibers b and c are equal and have opposite sign, fiber a will have a zero Δn. As the wavelength decreases, the mode field becomes more confined, so the birefringence property can be approximate to fiber c, which has only nonsymmetrical geometry in the core region. As the wavelength increases, the case is reversed. The mode field penetrates more into the cladding area and fills more elliptical holes that have nonsymmetrical geometry everywhere, so fiber a can be approximate to fiber b at long wavelengths.

5. Birefringence

The dependence of birefringence on the hole area and the elliptical ratio in the PCFs with the central defect d_c =0 has been discussed in detail in several publications [14,15,18]. The central defect is taken into account in this paper. Birefringence Δn, defined as the modal index difference between the HE_11y and the HE_11x modes, i.e., Δn=n_y -n_x , is shown in Fig. 11.

Δn increases monotonically when λ/D increases for a certain PCF (fixed d_c /D), which can also be seen in Fig. 12, which shows the relationship between Δn and D/λ at different d_c /D. The birefringence profile is the same as curve b in Fig. 9(b), which has been discussed above. In the long-wavelength regime, a majority of the mode energy is located beyond the first ring of air holes, and the field attempts to approach the distribution of the fundamental space filling mode (FSM) as closely as possible [18]. It can thus be concluded that the birefringence between the FSMy and the FSMx modes must be the maximum value at a fixed wavelength for a certain PCF.

Fig. 11. Relationship between Δn and central defect size d_c /D at different wavelengths. The structure parameters of the PCF are D=2.3 µm, d=0.8D, and η=2.

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Fig. 12. Relationship between Δn and normalized frequency D/λ in the PCFs with different central defect sizes. The parameters are D=2.3 µm, d=0.8D, and η=2.

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At a fixed d and λ/D, when the central defect diameter d_c /D increases from 0 (without the central air hole), Δn increases. Once the central defect increases to the same size as the cladding holes, the PCF becomes a perfect 2D photonic crystal, hence the fundamental mode is the FSM mode. We thus expect that Δn will reach its peak value at point d_c /D=d/D. It is fortunate that the conclusion is almost validated by the numerical results shown in Fig. 11, where the cladding hole size is d/D=0.8. When d_c /D is greater than 0.8, Δn then decreases, which is not shown in Fig. 12 because the mode disorder is too complicated to obtain an accurate Δn.

Fig. 13. Relationship between Δn and normalized frequency D/λ in PCFs with different elliptical ratios. The parameters are D=2.3 µm and d=0.8D. The major axis of the elliptical hole is along the (a) y axis or (b) x axis. The major axis length is d_c =0.4D.

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Figure 13 shows the relationship between Δn and the normalized frequency D/λ at different elliptical ratios η with structure parameters D=2.3 µm and d=0.8D. In Fig. 13(a) the major axis of the elliptical hole is along the y axis and is maintained at d_c =0.4D. In Fig. 13(b) the major axis is along the x axis and is also maintained at 0.4D, which is the same as in Fig.11 or Fig. 12 in which birefringence Δn increases as D/λ decreases. As η decreases, Δn increases gradually when the major axis is along the y axis, and Δn increases significantly when the major axis is along the x axis. By comparing Figs. 13(a) and 13(b) one will find that the birefringence of the x major axis is greater than that of the y major axis, a fact that can be easily understood from the waveguide structure shown in the Fig. 13 inset, which has been discussed in detail in Refs. 18 and 42 and will not be repeated here.

6. Discussion and Conclusion

The full vector supercell overlap method, based on our previous studies, has been developed to model the triangular lattice EHPCF. Some mode characteristics of the PCF with a central defect air hole were obtained by numerical computations. When the central defect is large enough, the modes are disordered. The birefringence (Δn) dependence on the central defect has been discussed in detail by numerical analysis. Δn increases monotonically when λ/D increases for a certain structure PCF. At a fixed d and λ/D, when the central defect diameter d_c /D increases from 0, Δn increases; once the central defect increases to the same size as the cladding holes, the PCF becomes a perfect 2D photonic crystal, Δn will reach its peak value and then decrease as d_c /D increases continuously. The dependence of the birefringence on the orientation of the elliptical hole and its elliptical ratio has also been discussed.

The accuracy and efficiency of the full vector supercell overlap method have been discussed in a prior publication [27]. In general, the number of expansion terms of the electric field and the dielectric structure affects the accuracy of the numerical calculations monotonically. The supercell parameter N also affects the accuracy, and it should be set to a higher value to obtain more accurate results. At long wavelengths and especially for PCFs with a central defect larger than the cladding holes, the mode field would extend far into the cladding area and is likely to reach the next supercell or light could even possibly be PBG guided. So a large supercell must be considered, i.e., N must have a high value.

Some modes are cut off as the frequency decreases. From the electric field distribution it can be found that all the modes are cut off in Fig. 5 and only the first two modes are bounded in Fig. 7. When we discussed the mode disorder properties, only the electric field distribution was considered. For example, in Fig. 5 the first mode is HE_11y and the fourth mode is HE_11x according to the vector distribution of the transverse electric field, no matter whether it is bounded or leaky or whether it is guided by a PBG or TIR.

We thank Wu Chongqing for beneficial discussions about the physical basis of mode disorder as well as information about dielectric waveguides.

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	Hole Pitch	l_1x or l_2x	l_1y or l_2y	Elliptical Cylinder Size Along y axis	Dielectric Constant
					Background		Elliptical Area
					ε	lnε	ε	lnε
PC1	D	D	√3D	d	$n_{si}^{2}$	ln $n_{si}^{2}$	$n_{air}^{2}$	0
PC2	ND	ND	√3ND	dc	0	0	$n_{si}^{2}$ - $n_{air}^{2}$	ln $n_{si}^{2}$

Dependence of mode characteristics on the central defect in elliptical hole photonic crystal fibers

Abstract

1. Introduction

2. Full Vector Model

2.1 Eigenequation

2.2 Dielectric Constant Profile

2.3 Overlap Integrals

3. Mode Symmetry and Classification

4. Mode Disorder in Electronic Hole Photonic Crystal Fiber

4.1. Mode Disorder between the Fundamental and the Higher-Order Modes

4.2. Mode Disorder between the Fundamental Doublets

5. Birefringence

6. Discussion and Conclusion

References and links

Cited By

Figures (13)

Tables (1)

Equations (21)

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