1. Introduction

Photonic crystal fibers (PCFs) are optical fibers which guide light in a defect surrounded by a periodic array of air holes running along the entire length of the fiber. These fibers have been shown to possess numerous unusual properties, including highly tunable dispersion and nonlinearity [1,2] and single-mode operation at all wavelengths [3]. Such properties are of fundamental interest in optical physics (soliton formation and propagation, for example) and are also of practical importance in the design of increasingly sophisticated broadband optical telecommunications networks.

Two different guiding mechanisms for PCFs have been identified. The first mechanism uses a defect mode in a two-dimensional photonic band gap; the second is analogous to conventional guiding, and relies on a form of total internal reflection. The former utilizes structure which stops propagation in any transverse direction, is typically narrowband, but, in principle, allows light to propagate in the air core [4,5]. The latter achieves a total internal reflection condition because the effective index of the cladding is lower that the dielectric core. This type of PCF, which we consider in this paper, does not need the strict periodicity of air holes or the high air filling ratio required for the existence of a photonic band gap.

In the analysis of a PCF, the effective cladding index is an important parameter; it not only facilitates our understanding of PCFs but allows the application of conventional theories of fiber propagation and dispersion without necessarily resorting to detailed numerical models. The effective cladding index can be obtained by investigating the propagation of the fundamental cladding mode (which is also called the space filling mode [6]). So far, the computation of the cladding effective index has been performed by using either full-vectorial treatments [7–10], or approximate scalar [3] and vectorial [6] analytical approaches. However, these methods all have some limitations. For example, the approaches which make use of general plane-wave-expansions [7,8] usually cannot take into account the material dispersion because the unknown mode frequency is required to determine the dielectric constant before it is known. The approximate scalar and analytical approaches [3,6] lack general applicability. And the biorthonormal-basis method [9,10] is somewhat complicated. In this paper, we present a simple but efficient vectorial approach, which overcomes the general limitations of the plane-wave-expansion-methods. In section 2 we outline the method. In section 3 we use the method to analyze the modal properties and polarization of the cladding modes, with emphasis on the fundamental cladding mode. Section 4 provides a discussion and conclusion.

2. The Method

We take the PCF to be uniform in the propagation (z) direction, and write the modal magnetic field in the form

where β is the propagation constant of the mode, H _t and H_z and are the transverse and longitudinal components of the modal magnetic field, respectively. Substituting Eq. (1) into the wave equation

we get the equation for H _t :

where k=ω/c=2π/λ is the wavevector and ε=ε(x, y) is the transverse dielectric constant profile. We write H _t as a column vector

${\mathbf{H}}_t=\left(\begin{array}{c}{H}_x\\ H_y\end{array}\right)$

then Eq. (3) is the coupled equation for H_x and H_y .

To solve the coupled equations for H_x and H_y , we expand ε(x,y), lnε (x,y) and H_t (x,y) as

where x _t =x̂ x+ŷ y, and G(l)=l_x b _x +l_y b _y is a vector in the reciprocal space. Here l_x and l_y are any two integers that we denote collectively by l, b _x and b _y are the primitive vector of the reciprocal lattice. When these expansions are substituted into Eq. (3), it becomes the algebraic eigenvalue problem

where $L_{G\prime }^G$ are the matrix coefficients of the operator L in the plane-wave basis.

The elements of the matrix L can be written as

with

Comparing Eq. (3) with the master equation usually used in photonic crystals $\nabla \times \left(\frac{1}{\epsilon }\nabla \times \mathbf{H}\right)={\left(\frac{\omega }{c}\right)}^2\mathbf{H}$, we can see that the former is more suitable to calculate the modal properties of PCFs when the material dispersion must be considered. We here solve for magnetic fields, similarly one can start this method from electric fields. In fact, in our numerical calculations, we always get the same result for β in both cases.

When considering the air holes are circular, we obtain the Fourier coefficients $\widehat{\epsilon }$(G)and $\widehat{\kappa }$(G)in Eq. (4) as

and

where f=πR ²/A_c is the air filling ratio, R is the radius of the air holes, A_c is the area of the primitive cell. ε_a and ε_b are the dielectric constants of air and silica, respectively.

3. Cladding modes

We consider PCFs with claddings of triangular lattice of circular air holes, as illustrated in Fig. 1. Λ is the lattice pitch, R is the radius of the air-holes. The two vectors of the primitive cell are ${\mathbf{a}}_1=(\frac{1}{2}\Lambda ,\frac{\sqrt{3}}{2}\Lambda )$ and ${\mathbf{a}}_2=(\frac{1}{2}\Lambda ,-\frac{\sqrt{3}}{2}\Lambda )$. The area of the primitive cell is $A_c=\frac{\sqrt{3}}{2}{\Lambda }^2$. The filling ratio is thus $f=\frac{2\pi }{\sqrt{3}}{(R⁄\Lambda )}^2$. The primitive vector in the reciprocal space are ${\mathbf{b}}_1=\frac{2\pi }{\Lambda }(1,1⁄\sqrt{3})$ and ${\mathbf{b}}_2=\frac{2\pi }{\Lambda }(1,-1\sqrt{3})$. The dielectric constant of silica εb is calculated using the Sellmeier equation with the parameters for fused silica as given in Ref. 11. The dielectric constant of air ε_a is assumed to be 1.0 at all wavelengths considered.

 

Fig. 1. Schematic of the triangular lattice of air holes in the cladding. The lattice pitch is Λ, and R is the radius of the air holes. Dielectric constants of air and silica are ε_a =1.0 and ε_b (λ), respectively.

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The fundamental cladding mode (space filling mode) is defined to be the mode that exhibits the highest effective index. We use the method described in the above section and calculated the dispersion curves for the several lower order cladding modes, as shown in Fig. 2. The chosen parameters correspond to a previously investigated sample [10]: Λ=2.3 µm, R=0.3 µm. The fundamental cladding mode is a two-fold degenerate mode. Fig. 3 shows the transverse magnetic fields for these two degenerate modes at λ=1.5 µm. We can see that they are essentially linearly polarized, with minute deviation from linear polarization near silica-air interfaces. In Fig. 4 we show the transverse magnetic field intensity of the x-polarized fundamental mode. As expected, the field is strongly concentrated in the high dielectric region (silica).

 

Fig. 2. Dispersions of the some lower cladding modes. The fundamental mode (FCM) is a two-fold degenerate mode. The next lower mode (HCM) includes six nearly degenerate modes. The material dispersion of the silica core is also shown in the figure.

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Fig. 3. Transverse magnetic fields of the two degenerate modes of the fundamental cladding mode.

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Also we calculate the dispersions of the fundamental cladding modes with different air-filling ratios. Since our method is a fully vectorial one, it can be used to analyze the cladding modes at very large air-filling ratio. To see the differences between scalar approximation and rigorous vectorial treatment, we show also in Fig. 5 the results from the scalar version of the presented method by neglecting the term [$Q_{\mathrm{G}\prime }^{\mathrm{G}}$] in Eq. (6). We can see from Fig. 5 that at short wavelength and low air-filling ratio, the scalar method is a good approximation. This is easy to understand: at short wavelengths or with small air holes, the modal fields tend to concentrate more in the high dielectric region (silica), the contribution of the vectorial terms is therefore small compared to the scalar terms [2].

The effective cladding index n_eff,cl is a function of wavelength λ, dielectric constant of silica ε_b , lattice pitch Λ, and air hole radius R. Due to fact that Maxwell’s equations have no fundamental length scale, n_eff,cl is in fact a function of ε_b and normalized parameters λ/Λ and R/Λ. In Fig. 6 we show a surface plot of n_eff,cl . The material dispersion of silica is considered in the calculations.

 

Fig. 4. Transverse magnetic field intensity of the fundamental cladding mode (x-polarized).

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Fig. 5. Dispersion curves calculated using the present vectorial method (solid lines) and the scalar method (dashed lines) for claddings of different air-hole radius R.

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Once the effective cladding indices are determined, one can use them to determine the width of the spectral range over which a PCF is single-mode [3]. The V parameter of the PCF defined as V=kΛ($n_{\mathit{\text{core}}}^2$ -$n_{\mathit{\text{eff}}\mathit{,}\mathit{\text{cl}}}^2$ )^1/2 gives a good estimation of the number of guided modes inside the PCF. Fig. 7 shows how V varies with wavelength for PCFs of different lattice pitches and air-filling ratios. We can see that the V is well below the red dashed line (which corresponds to the single-mode cutoff value for traditional step-index optical fibers) for PCFs with small air holes. It is clearly that PCFs with relatively small air holes can be single mode over a very large spectral range. To appreciate the difference between the vectorial and scalar methods, we also show the results from the scalar approximation in Fig. 7(a). The relative difference between the two approaches reaches about 10% in the results shown. So, in order to accurately predict the single-mode behavior of PCFs (especially near the single-mode cutoff), it is advisable to use the vectorial method.

 

Fig. 6. Effective cladding index as a function of λ/Λ and R/Λ. Λ is chosen to be 2.3 µm and the material dispersion of silica is considered in the calculations.

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Fig. 7. Calculated parameter V vs. wavelength for claddings of different pitches and air-hole radii. The red dashed line indicates the single-mode cutoff value of V (=2.405). (a) Lattice pitch Λ=2.3 µm. Results from the vectorial (scalar) method are shown by blue (black) solid lines. (b) Lattice pitch Λ=5.0 µm. Only the vectorial results are shown.

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In all the numerical calculations, a total of 289 plane waves was used. We estimate that the relative error in the calculated effective cladding index is less than 0.06% at λ=2 µm and R=0.8 µm. More plane waves are needed to keep the accuracy for increasing wavelength λ and air-hole radius R.

4. Discussion and Conclusion

The method presented in this paper has broad applicability beyond the analysis of cladding modes. In fact, under the so-called supercell approximation, our approach is readily used in characterization of PCFs with various cladding and core structures, such as irregular air-hole cladding PCFs and multiple-core PCFs. Such structures can yield either accidental birefringence (due to disorder, perhaps) or may be engineered for very high birefringence (for polarization compensation). The presented method provides a full vectorial and efficient tool in modeling PCFs. It should be emphasized that the localized function method used in Refs. [2,12] incorporates material dispersion easily as we do. However, this method is incapable of calculating the effective cladding index of PCFs, because it makes use of the fact that the fields of guided modes are localized around the defect core.

In summary, we present a rigorous and efficient method to analyze the cladding modes in PCFs. It overcomes the limitation of general plane-wave-expansion based techniques that material dispersions are not easily taken into account. Using this approach, we analyze the cladding modes in PCFs with different air-filling ratios. Comparisons with scalar approximation are also given.