## Abstract

A supercell lattice method, believed to be novel, deduced from the plane-wave expansion method and the localized basis function method, is presented for analyzing photonic crystal fibers (PCFs). The electric field is decomposed by use of Hermite—Gaussian functions, and the dielectric constant of PCFs missing a central air hole is considered as the sum of two virtual different periodic dielectric structures of perfect photonic crystals (PCs). The structures of both virtual PCs are expanded in cosine functions. From the wave equation and the orthonormality of the Hermite—Gaussian functions, the propagation characteristics of the PCFs, such as the mode field distribution, the effective area, and the dispersion property, are obtained. The accuracy of the novel method is demonstrated as we obtain the same results when the dielectric constant is split into two virtual ideal PCs in different ways.

© 2003 Optical Society of America

## 1. Introduction

In recent years there has been an explosion of interest in the new science of photonic crystal fibers (PCFs)—silica optical fibers consisting of a central defect region in a regular lattice of air holes. According to their light-guided mechanisms, PCFs may be divided into two general categories, namely, photonic bandgap (PBG) and total-internal-reflection (TIR) PCFs. A number of numerical methods [1] have been developed to analyze the modal characteristics of PCFs, some of which are the effective index method (EIM), the plane-wave expansion method (PWM), the localized basis function method (LFM), the finite-element method (FEM), and the finite-difference method. The key to these different methods is how to describe accurately the periodic dielectric structure ε and the electric or magnetic field **E** or **H**.

The PWM [2–4] solves the full vector wave equation for the magnetic field and, as the name implies, is based on a plane-wave expansion of the field and the position-dependent dielectric constant, which is equal to the Fourier transform. It allows one to calculate the mode field distribution and the photonic band structure of the PCFs and thereby the possible existence, width, and positioning of any PBGs. The plane-wave expansion of the periodic dielectric constant is simple, and the coefficients of all terms are analytical, but it will cost too much time because it requires a large number of terms for expanding the field and the dielectric constant for certain accuracy.

In the EIM [5–9] model for the TIR PCF, the PCF is considered to be a conventional step-index fiber in which a silica core (the defect) is surrounded by a uniform cladding with a lower effective index, which can be related to the band structure of the perfect photonic crystal in absence of the core. The main task of this model is to calculate the effective index of the PBG cladding associated with the so-called fundamental space-filling mode (FSM). The EIM model may be used for approximate calculation of the propagation properties of the PCF, including the loss and the dispersion properties. It can roughly show some concepts and explain some optical properties of the PCFs, with the exception of a few important propagation properties, such as the birefringence and the mode field distribution.

The LFM model [10–13] has been developed for the modeling of triangular PCFs or holey fibers. This method is based on recasting Maxwell’s equations into an eigenvalue system, using a representation of the refractive index and the field distribution as sums of localized basis functions, such as the Hermite—Gaussian function. Some properties of the PCFs, such as the transverse modal electric field, the mode area, the propagation constant, and thereby the dispersion and the birefringence properties, can be accurately calculated from the eigenvalue equation. However, the means of selecting the basis functions, the characteristic width of the basis set, and the number of terms of the expansion will determine the numerical results. Considerable time will be required if the number of terms is great.

There are many other traditional algorithms for analyzing the electromagnetic field, such as the FEM [14] and the finite-difference time domain (FDTD) [15], which can also be used to investigate the PCFs with very high accuracy. But again the time requirement will be great because of the complicated structure of PCFs. In fact, the LFM model is more practical because it is almost at the same precision level as the FEM or the FDTD [16].

In this paper, a novel supercell lattice method (SLM), based on the combination of PWM and LFM, is developed. The modal electric field, like the LFM model, is expanded as the sum of the orthogonal set of Hermite—Gaussian basis functions; the periodic dielectric structure with a central defect is considered to be a supercell lattice that can be constructed with two different periodic dielectric structures of perfect two-dimensional (2-D) photonic crystals (PCs). The dielectric structure of every perfect 2-D PC, such as the PWM model, can be decomposed by use of periodic functions (cosines). with the function relations and the orthonormality of Hermite—Gaussian functions, the transmission properties of PCFs can be obtained after the wave equation is recast into an eigenvalue system.

## 2. Supercell lattice

Triangular lattice microstructure silica-air TIR PCF is shown in Fig. 1. The practical PCF fiber has a limited number of air rings, and its modal characteristics can be investigated when we construct a supercell lattice using the practical PCF as a supercell, as shown in Fig. 1(a). The structural parameters of the PCF are the hole pitch *D*, the hole diameter *d*, and the supercell lattice period *ND*. To describe the dielectric structure of Fig. 1(a), two 2-D perfect PCs (PC1 and PC2) are introduced, as shown in Fig. 1(b) and Fig. 1(c), with the parameters shown in Table 1. Adding the dielectric structures of both PC1 [Fig. 1(b)] and PC2 [Fig. 1(c)] will form the dielectric structure of the supercell lattice PCF [Fig. 1(a)].

In fact, the split up into PC1 and PC2 could be made in different ways as demonstrated by the two gray lines of gray color Table 1. It is assumed that *x, y, z* and *z*’ are selected as desired; then the following identities must be satisfied in order to reconstruct the missing hole region.

Regardless of the dielectric constant parameters (*x, y, z* and *z*’) of PC1 are selected, the results will not change. For the case of simplicity, they are set as the last two lines in Table 1. Here *x* and *y* of PC1 are selected as ${n}_{\text{si}}^{2}$ and ln ${n}_{\text{si}}^{2}$; consequently the dielectric constant and the logarithm of the background of PC2 are both zero. This choice seems to be incorrect, but we use only the sum of both PC1 and PC2 in this novel method and do not investigate them individually. When the sum of both PC1 and PC2 is fixed, the results will not be affected by the way in which the PCF is split.

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

where

and where (*P*
_{1}+1) and (*P*
_{2}+1) are the number of decomposition terms of PC1 and PC2, respectively, and *P*
_{1ab}, *P*
_{2ab}, ${P}_{1\mathit{\text{ab}}}^{1\mathrm{n}}$, and ${P}_{2\mathit{\text{ab}}}^{1\mathrm{n}}$ present the coefficients that can be obtained through the Fourier transform. To show how to obtain the coefficients, *P*
_{1ab} is illustrated as an example.

The dielectric constant of PC1 can be expressed in the form of a Fourier transform as

where *F*(*K*
_{mn}), the 2-D Fourier transform coefficients, can be written as [4]

$$F\left(0\right)={n}_{\mathit{air}}^{2}+f\left({n}_{\mathit{air}}^{2}-{n}_{\mathit{si}}^{2}\right)$$

where *J*
_{1} is the first-order Bessel function and

*k*
_{x} and *k*
_{y} depend on the reciprocal lattice vector of the triangular lattice [3,4]. *f*, a fraction parameter, is defined as the ratio of the hole volume to the cell volume:

When we transform the subscripts *m* and *n* from [-*P*
_{1}, *P*
_{1}] to [0, 2*P*
_{1}], Eq. (4) will become Eq. (8).

When we compare Eq. (8) with Eq. (3a), *P*
_{1ab} can be analytically expressed as the composition of *F*(*K*
_{mn}) shown as Eq. (9).

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+F\left({K}_{\frac{a+b+2k}{2},\frac{a-b+2k}{2}}\right)+F\left({K}_{\frac{a-b+2k}{2},\frac{a+b+2k}{2}}\right),$$

Other coefficients can also be evaluated analytically in the same way. Figure 2 demonstrates the simulation results along the *y*=0 axis of the dielectric reconstructions for a PCF with the structure parameters *D*=2.3 µm, *d*=0.69 µm, and *P*
_{1}=50, *N*=10, *P*
_{2}=500. Obviously, it can describe the supercell lattice efficiently and accurately. In fact, when *P*
_{2}=*P*
_{1}**N*, the reconstructed dielectric structure will be the most accurate according to the Fourier transform.

## 3. Electric field and eigenvalue system

It is assumed that the PCF is lossless and uniform in the propagation (*z*) direction, so a main task is to investigate the transverse mode field distribution ${\overrightarrow{e}}_{t}$
(*x,y*), which can be divided into two polarization components along the *x* or *y* axis:

Both *e*_{x}
(*x,y*) and *e*_{y}
(*x,y*) satisfy a pair of coupled wave equations [10–12,17]. The modal electric field will be expanded as

where (*F*+1) is the number of terms retained in this expansion, ψ_{i}(*s*) (*i*=*a,b, s*=*x,y*) are elements of the orthogonal set of Hermite—Gaussian basis functions:

where *H*
_{2i}(*s*/ω) is the 2*i*th-order Hermite polynomial and ω is the characteristic width of the basis set. As discussed by Monro *et al*. [11], only even-order Hermite polynomials are used because the observed fundamental mode profiles in these fibers are even.

When the decompositions in Eq. (11) are substituted into the wave equation, the eigenvalue equation will be obtained and shown as Eq. (13) [10,11].

where *M*
^{s}=*I*
^{(1)}+*k*
^{2}
*I*
^{(2)}+*I*
^{(3)s} is an (*F*+1)×(*F*+1)×(*F*+1)×(*F*+1)-order four-dimensional (4D) matrix; β_{s} is the propagation constant of *e*
_{x} or *e*
_{y}; ε^{s} is an (*F*+1)×(*F*+1)-order 2-D matrix; *k*=2π/λ is the wave number; and *I*
^{(1)}, *I*
^{(2)}, and *I*
^{(3)} are overlap integrals of the modal functions, which are defined as [10,11]

$${I}_{\mathit{abcd}}^{\left(2\right)}={\iint}_{-\infty}^{+\infty}\epsilon {\psi}_{a}\left(x\right){\psi}_{b}\left(y\right){\psi}_{c}\left(x\right){\psi}_{d}\left(y\right)dxdy,$$

$${I}_{\mathit{abcd}}^{\left(3\right)x}={\iint}_{-\infty}^{+\infty}{\psi}_{a}\left(x\right){\psi}_{b}\left(y\right)\frac{\partial}{\partial x}\left[{\psi}_{c}\left(x\right){\psi}_{d}\left(y\right)\frac{\partial \phantom{\rule{.2em}{0ex}}\mathrm{ln}\phantom{\rule{.2em}{0ex}}\epsilon}{\partial x}\right]dxdy,$$

$${I}_{\mathit{abcd}}^{\left(3\right)y}={\iint}_{-\infty}^{+\infty}{\psi}_{a}\left(x\right){\psi}_{b}\left(y\right)\frac{\partial}{\partial y}\left[{\psi}_{c}\left(x\right){\psi}_{d}\left(y\right)\frac{\partial \phantom{\rule{.2em}{0ex}}\mathrm{ln}\phantom{\rule{.2em}{0ex}}\epsilon}{\partial y}\right]dxdy.$$

When we substitute the decomposition equation Eq. (3) into Eq. (14), the overlap integrals can be calculated with 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 [18] along with some identities [19]. The fact that all these integrals can be performed analytically is a significant advantage, especially when a great number of terms are needed in the expansions. Then Eq. (14) can be written as

$$+\frac{\sqrt{2\left(b+1\right)\left(2b+1\right)}}{2{\varpi}^{2}}{\delta}_{\mathit{ac}}{\delta}_{b,d-1}+\frac{\sqrt{2\left(a+1\right)\left(2a+1\right)}}{2{\varpi}^{2}}{\delta}_{a,c-1}{\delta}_{\mathit{bd}},$$

where

where *l*
_{x}=*D* and *l*
_{y}=sqrt(3)*D*. Both parts of the overlap integrals *I*
^{(2)}, *I*
^{(3)x}, and *I*
^{(3)y} depend on the decomposition coefficients *P*
_{1ab}, *P*
_{2ab}, ${P}_{1\mathit{\text{ab}}}^{1\mathrm{n}}$, and ${P}_{2\mathit{\text{ab}}}^{1\mathrm{n}}$, and this is the only way that the division of PCF into both virtual PC1 and PC2 is reflected in the eigenvalue equation. When the division satisfies Eq. (1) and Table 1, the overlap integrals *I*
^{(2)}, *I*
^{(3)x}, and *I*
^{(3)y} will stay unchanged no matter how the PCF is split.

Through the subscript transform as in Eq. (17), *M*
^{s} and ε^{s} can be transferred into an (*F*+1)^{2}×(*F*+1)^{2} 2-D matrix and a vector with (*F*+1)^{2} elements, which are still written as *M*
^{s} and ε^{s} for compactness.

β_{s} is now the eigenvalue of the matrix *M*
^{s}, and ε^{s} is the corresponding eigenvector of β_{s}. When we solve Eq. (13) at a particular wavelength λ, the modes and the corresponding β_{s} of the PCF at λ can be calculated. Because *M*
^{s} is an (*F*+1)^{2}×(*F*+1)^{2} matrix, the eigenvalue problem can be solved to produce (*F*+1)^{2} eigenvalues and their corresponding eigenvectors. Only one or a few of the eigenvalue-eigenvector pairs are physically relevant quantities corresponding to the guided modes of the fiber. The remaining eigenvalues are related to radiation modes of the fiber. The guided-mode solutions can be distinguished from the radiation modes by extraction of the eigenvalues that fall within the following range allowed by the structure: *n*
_{eff}<β/*k*<*n*
_{si}, where *n*
_{eff} is the effective index [6,9,20] of the photonic crystal cladding at the wavelength λ. The transverse modal electric field can be obtained by substitution of the eigenvector ε^{s} into the decomposition Eq. (11).

## 4. Results

A triangular-lattice circular-hole PCF with the same structural parameters as in Fig. 2 was investigated in this paper. The modal electric field distribution, the effective modal area, and the dispersion properties are discussed in detail.

#### 4.1 Modal electric field distribution

The simulation results of the *x*-polarized fundamental modal intensity profiles (|*E*_{x}
|^{2}) at 0.633 and 1.55 µm are shown in Figs. 3(a) and 3(b), respectively. Figure 4 demonstrates the corresponding contour lines, and the dielectric constant profile is superimposed. The intensity contours are spaced by 2 dB from -30 dB. Obviously, the wavelength is one of the key factors in determining the localization extent of the transverse mode. At shorter wavelengths, the field is limited in the high-index core, and at longer wavelengths, the field penetrates further into the periodic cladding region.

#### 4.2 Effective modal area A_{eff}

One issue of importance in fiber design is the size of the fundamental mode. The modal area can be tailored to a large extent by choice of the hole pitch *D* and the hole size *d*. Much current research focuses on designing fibers with very large or very small effective modal areas [21–23]; such specially designed fibers can be used to weaken or enhance the nonlinear effects. For conventional fibers, such special designing must have quite a complex refractiveindex profile, but for PCFs, both types of application can be obtained simply by altering of either *D* or *d* during the fabrication process by means of drawing the fiber under different conditions.

*A*
_{eff} is defined as Eq. (18) [17].

When we substitute the decomposition equation Eq. (11) into Eq. (18), and again use the orthogonality of the basis set and the Feldhiem’s identity, *A*
_{eff} can be expressed as

where

Γ(κ_{s}) is gamma function. Using the evaluated eigenvector from the eigenvalue Eq. (13), *A*
_{eff} can be calculated efficiently and accurately for a given PCF at any wavelength.

Figure 5(a) shows the effective modal areas of PCFs with different *d*/*D*, whereas the hole spacing remains constant at *D*=2.3 µm; and Fig. 5(b) shows *A*
_{eff} with different hole spacing *D*, whereas *d*/*D*=0.3. *A*
_{eff} of the PCFs can be several times less than or larger than that of conventional fibers, even extremely less or larger with extreme structure parameters. Hence more attention is paid to the nonlinearity characteristics of PCFs. When *d*/*D* is greater, the mode field will be confined more in the central area of the PCF. When *D*, the distance between the nearest holes, is greater, it is natural that the mode field will extend outside. When the wavelength increases, the same as with conventional fibers, the electric field will penetrate into the outer cladding more easily. This can also be seen from Figs. 3 and 4. In general, when *D* increases, or *d*/*D* decreases, or the wavelength increases, *A*
_{eff} will increase.

We should note that *A*
_{eff} of this paper is twice that shown in the results of Monro *et al*. in Refs. [10] and [11] because of the factor 2 in Eq. (19).

#### 4.3 Dispersion

Considerable attention has been paid to the ability of PCFs to generate novel group-velocity dispersion profiles. The total dispersion coefficient *D*
_{t} is proportional to the second-order derivative of the modal effective index (or the propagation constant) with respect to the wavelength λ.

The total dispersion coefficient *D*
_{t} could be written as the sum of the waveguide dispersion *D*
_{w} and the material dispersion *D*
_{m}, and *D*
_{m} can be calculated by application of the Sellmeyer formula. Because PCF is made of silica, the material dispersion is the same for PCFs with different structural parameters, and the total dispersion coefficient will be dominated by *D*
_{w}. When we take account of the scaling transformation property of Maxwell’s equations, there is a scaling property of the waveguide dispersion expressed as Eq. (21) [24].

Equation (21) tells us that if the scale of PCF is magnified by a factor *M*, the waveguide dispersion coefficient is reduced to 1/*M*, and at the same time the corresponding wavelength shifts to *M*λ. This equation is useful for analyzing the waveguide dispersion of PCFs.

The dispersion profiles of the *x*-polarized mode *e*_{x}
are shown in Fig. 6, where the material dispersion (*D*_{m}
) and the waveguide dispersion (*D*
_{w}) are clearly displayed. The dispersion profile of *e*
_{y} is not shown because *e*
_{x} and *e*
_{y} must be degenerate theoretically [25]. Figure 6(a) shows the waveguide dispersion for different *D*, with fixed *d*/*D*=0.3. The change of *D* produces a shift in the curve and a variation in the amplitude simultaneously. When we evaluate the zero-dispersion and minimum-dispersion wavelengths, the scaling transformation property Eq. (21) can be proved numerically [26]. Figure 6(b) shows the waveguide dispersion for different *d*/*D*, with fixed *D*=2.3 µm. The curve shifts to longer wavelength when *d*/*D* increases. A remarkable feature of these curves is that when *d*/*D* increases, the slope of the curve remains approximately the same in the region where it is uniformly decreasing. This property is helpful in designing PCFs for dispersion-related applications [27].

## 5. Accuracy and efficiency

We used the novel supercell lattice method for a few different lattice PCFs to analyze transmission characteristics, such as the propagation constant, the modal field distribution, the effective modal area, the waveguide dispersion, and the birefringence. Compared with some other calculations from the recent literature, with out method there are fewer computations, and all the elements of the eigenvalue equation are analytical. We are convinced that this novel method is accurate and efficient because of the numerical calculation processes and its results.

The novelty of this method is in how it splits the PCF into two virtual perfect photonic crystals. We performed this method in different ways to split the PCF, and it is shown that regardless of the dielectric constant parameters of both virtual photonic crystals selected, the results will not change when the sum of both PC1 and PC2 is fixed. In fact, the results shown if Fig. 6(a) are obtained by selection of the structural parameters *z*=1/2 and *z*’=0.

On the other hand, as discussed by Steel *et al*. [25], for any fibers with rotational symmetry of order higher than two, a mode that has a preferred direction must be one of a pair of degenerate modes. In particular, for all existing PCFs, the fundamental modes are almost uniformly polarized HE_{11}-like modes, so they must be degenerate. As a result, the observation of birefringence must be a result of asymmetry in the structure or the error produced by the numerical calculations. When an ideal PCF is examined, the modal birefringence Δ*n*=*n*
_{x}-*n*
_{y} can be used to scale the accuracy of the algorithm. The lower the value of |Δ*n*|, the more accurate the numerical method. Figure 7 shows the modal birefringence evaluated from the SLM model. Obviously, the result is accurate, since |Δ*n*| is less than 5×10^{-5} over the wavelength range of 0.9–1.7 µm. The accuracy can be improved while increasing *P*_{1}
, *N*, and *P*_{2}
for a certain PCF, but this will cost more time.

## 6. Discussion and conclusion

A novel supercell lattice method has been developed on the basis of the model of PWM and LFM. The modal electric field is expanded as the sum of the orthogonal set of Hermite—Gaussian basis functions, and the periodical dielectric structure with a central defect is considered to be a supercell lattice that can be constructed with two perfect 2-D photonic crystals with different periods. The dielectric structure of every perfect 2-D PC is decomposed by use of periodic functions (cosine). Considering the function relations and the orthonormality of Hermite—Gaussian functions, the propagation characteristics of PCFs are obtained after recasting of the wave equation into an eigenvalue system. Since all the decomposition coefficients and the overlap integrals are evaluated analytically, the novel SLM model is efficient and accurate.

The supercell lattice method can be used to analyze many other structures of PCFs, such as the elliptical-hole PCF, the square-lattice PCF, and the PBG PCF. For example, when the fraction parameters *f* and *k*
_{x} in Eq. (4) are divided by the ellipticity ratio η=*b*/*a*, where *b* and *a* are the lengths of the major and minor axes of the elliptical holes, the SLM can be used to investigate the elliptical hole PCF [28–30] with a straightforward extension of the approach described in this paper. Figure 8 shows the dielectric structure of an elliptical hole PCF with the parameters *D*=2.3 µm, b/*D*=0.8, η=3, *P*_{1}
=30, *N*=10, *P*_{2}
=300.

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