Meshed index profile method for photonic crystal fibers with arbitrary structures

Kwang No Park; Kyung Shik Lee

doi:10.1364/OE.16.013175

1. Introduction

Recently, there has been an increasing interest in photonic crystal fibers (PCFs) which consist of a region with a central defect in a regular lattice of air-holes. The PCFs have various interesting characteristics including endless single-mode operation, ultra-flattened dispersion, a large or ultra-small effective area, large nonlinearity, etc. Because of their unique characteristics, a number of numerical methods have been developed to analyze their modal characteristics, such as the effective index method (EIM)[1–7], multipole method (MPM)[8], finite element method (FEM)[9–12], supercell overlapping method (SOM)[13], localized function method (LFM)[14–16], plane-wave expansion method (PWM)[17, 18] and finite difference method (FDM)[19].

The EIM has the advantage of short simulation time, although it is less accurate than the other numerical methods. The EIM can approximately analyze some optical properties of the PCF, with the exception of some important modal properties, such as the polarization property and the mode field intensity distribution. To increase the accuracy of the EIM, an improved EIM (IEIM) was proposed and demonstrated [20]. The proposed IEIM has greater accuracy than the original EIM, but the IEIM still has a problem similar to the original EIM. The unsolved problem is that the IEIM can never be used to analyze the exact mode field intensity distribution and polarization property. Meanwhile, numerical methods such as FEM, FDM, PWM and MPM allow the modal properties of PCFs to be analyzed more accurately. Therefore, commercially available software is based on the FEM. The FEM based software is very accurate and can analyze the PCF with arbitrary structure. But these FEM based programs are very inefficient especially when analyzing the PCF over a broad wavelength range because recalculation is needed whenever operating wavelength is changed. And some freeware, as the MPB based on the PWM, is used for analyzing the PCF. However these numerical methods for analyzing PCFs require a large amount of computational time and the simulation model of PCF is very complex.

The LFM, which is based on the composition of the localized basis function, has been developed for the analysis of the PCFs with a triangular air-hole lattice. The LFM is based on an eigenvalue matrix obtained by recasting Maxwell’s equations with a representation of the refractive index and the field distribution as sums of localized basis functions, such as the Hermite-Gaussian function. Some numerical methods, such as the SOM[13], supercell lattice method[21] and the orthogonal function method[22] are based on the LFM and can analyze various modal properties of the PCF with great accuracy. But PCFs for which those numerical methods based on the LFM are restricted to the cases of a PCF with a triangular air-hole lattice.

In this paper, a meshed index profile method (MIPM) based on the LFM is proposed in order to analyze various PCFs with arbitrary air-hole lattices. The modal characteristics of the PCF, such as the transverse electric field distribution, the dispersion and the birefringence, are analyzed by using the MIPM. The transverse electric field is decomposed into several orthogonal sets of Hermite-Gaussian basis functions. The transverse index distribution of the PCF can be expressed using the meshed index profile. The orthogonal basis function and the meshed index profile function are substituted into the Maxwell’s equations, and then the eigenvalue matrix is obtained by recasting the Maxwell’s equation. By solving this eigenvalue matrix, we finally analyze the modal property of various PCFs with arbitrary air-hole lattices.

2. Electric field decomposition

The PCF used in this paper is assumed to be uniform in the direction of light propagation. Therefore, the transverse mode field distribution in the PCF is only affected by the transverse PCF structure. The main objective of the MIPM is to investigate the transverse mode field distribution which is composed of two polarized components and their modal characteristics. The transverse mode field consists of both x-polarized and y-polarized components. Therefore, the transverse mode field distribution can be divided into the two polarized components and can be expressed as,

{\vec{E}}_{t} = (x, y) = E_{x} (x, y) \hat{x} + E_{y} (x, y) \hat{y},

where E_x(x,y) and E_y(x,y) are the field distributions of the modes linearly polarized along the x-axis and y-axis, respectively. And the mode field equations as sums of the localized basis functions can be expressed as,

E_{x} (x, y) = \sum_{m = 0}^{S} \sum_{n = 0}^{S} e_{mn}^{x} ψ_{m} (x) ψ_{n} (y),

E_{y} (x, y) = \sum_{m = 0}^{S} \sum_{n = 0}^{S} e_{mn}^{y} ψ_{m} (x) ψ_{n} (y),

where (S+1) is the number of basis functions used to express the transverse electric field distribution. Ψ_m(x) and Ψ_n(x) are the Hermite-Gaussian basis functions used to express the mode field distribution in the LFM and are expressed as,

ψ_{i} (s) = \frac{2^{- \frac{i}{2}} π^{- \frac{1}{4}}}{\sqrt{i! w}} \exp (- \frac{s^{2}}{2 w^{2}}) H_{i} (\frac{s}{w}),

where H_i(s/w) is the i th order Hermite polynomial and w is the characteristics width of the basis function. e^x_mn and e^y_mn are the coefficients to express the mode field distributions and will be calculated by using the MIPM in the next section. This indicates that the method of selecting the basis functions, the characteristic width of the basis function, and the number of basis functions of the expansion can determine the numerical results. We can express more exact mode field distribution by increasing the number of basis functions. However, if the number of terms is increased, considerably more time is required. Inserting E_x(x,y) and E_y(x,y) into the full vector wave equation [22, 23], we obtain a pair of coupled wave equations,

[\frac{\nabla^{2}}{k^{2}} - \frac{β_{x}^{2}}{k^{2}} + n^{2} (x, y)] E_{x} = - \frac{1}{k^{2}} \frac{\partial}{\partial x} [E_{x} \frac{\partial \ln n^{2} (x, y)}{\partial x} + E_{y} \frac{{\partial \ln n}^{2} (x, y)}{\partial y}],

[\frac{\nabla^{2}}{k^{2}} - \frac{β_{y}^{2}}{k^{2}} + n^{2} (x, y)] E_{y} = - \frac{1}{k^{2}} \frac{\partial}{\partial y} [E_{x} \frac{\partial \ln n^{2} (x, y)}{\partial x} + E_{y} \frac{{\partial \ln n}^{2} (x, y)}{\partial y}],

where k=2π/λ is the wave number, β_x and β_y are the propagation constants of the x and y polarized modes, respectively. By solving these coupled wave equations, we can then analyze the modal characteristics of the PCF including the effective index, the mode field distribution and the dispersion. If we express the transverse index profile (n ²(x,y)) in a simple form in the coupled wave equations, the coupled wave equations can be easily solved by using the overlap integrals of the Hermite-Gaussian basis functions and the some integrals with specific identities. A two-dimensional Fourier transform has been used to express the transverse index profile (n ²(x,y)) in the earlier simulation methods such as the SOM [13] and the orthogonal function method [22]. It is efficient and easy to analyze the PCF with a triangular air-hole lattice by using the SOM and orthogonal function method. However, the PCF has more complex structure or non-symmetric air-hole lattice. Therefore, many virtual photonic crystals are needed to analyze the modal property of the PCF. For the reason, it is very difficult and takes huge computation time to analyze the PCF with an arbitrary structure by using the SOM and orthogonal function method. This means that an efficient method needs to be developed in order to express the index profile of the PCF and to analyze the PCF with an arbitrary structure by using the simulation method based on the LFM. In the next section, we demonstrate a MIPM which efficiently express the index profile of a PCF with an arbitrary structure and analyze the modal characteristics of the PCF.

3. Meshed index profile of the PCF and eigenvalue matrix

The transverse index profile of the PCF has recently been determined by using the SOM [13] and the orthogonal function method [22]. The similarity between the SOM and the orthogonal function method is that they use a combination of several virtual photonic crystal structures to determine the index profile of the PCF. In order to describe the index profile of the PCF, two two-dimensional virtual photonic crystals were used in the SOM [13, 22]. Several different two two-dimensional photonic crystal structures were also proposed and demonstrated in reference 22. These methods are simple and efficient in determining the index profile of the PCF, but restricted to the PCF with a triangular air-hole lattices. If the PCF has more complex structures with non-symmetric air-hole lattice, elliptical air-holes or different sized air-holes, many virtual photonic crystals should be used to fully express the index profile of the PCF. The number of virtual photonic crystals used to express the index profile is strongly correlated with the computation time. Therefore, these methods are not efficient for analyzing a PCF with non-symmetric air-hole lattice, elliptical air-holes or different sized air-holes.

In this paper, we propose a more efficient method for determining the index profile of a PCF with an arbitrary air-hole structure. Fig. 1 shows the process for expressing the index profile of the PCF in the MIPM. The transverse structure of the PCF for analyzing the modal characteristics is shown in Fig. 1(a). It is assumed that the PCF has air-holes and a central defect region. Fig. 1(b) shows an example to make an index profile function (n ²(x,y)) from the transverse structure of the PCF. The transverse plane of the PCF is divided into (2N+1)×(2N+1) unit cells and then the total index profile of the PCF is expressed as the sum of unit row matrix functions (n_i(x)) or as the sum of column matrix functions (n_i(y)). Because x_i and y_i are the constants in a respective same row matrix and column matrix, we can define the unit row and column matrix functions as n(x,y_i)=n_i(x) and n(x_i,y)=n_i(y), respectively. Then, the total index profile of the PCF can be expressed as,

n^{2} (x, y) = n^{2} (x, y_{- N}) + n^{2} (x, y_{- (N - 1)}) + \dots + n_{0}^{2} (x, y_{0}) + \dots + n^{2} (x, y_{N - 1}) + n^{2} (x, y_{N})

where 2N+1 is the number of unit row (n ² _i(x)) or column (n ² _i(y)) matrix. In this way, an index profile function (n ²(x,y)) is deduced directly and easily from the index profile of the PCF. Therefore, the index composition method used in the MIPM is efficient and very useful in expressing the index profile of the PCF.

Fig. 1. (a) Meshed transverse dielectric structure of a PCF and (b) the detailed definition of index profile function.

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Next, we can obtain the eigenvalue equation for analyzing the modal characteristics of the PCF by substituting the index profile function (n ²(x,y)) into the full vector wave equations (5) and (6). Then the two eigenvalue equations are obtained and given by,

M_{abcd}^{x} e^{x} = (I_{abcd}^{(1)} + k^{2} I_{abcd}^{(2)} + I_{abcd}^{(3) x}) e^{x} = β_{x}^{2} e^{x},

M_{abcd}^{y} e^{y} = (I_{abcd}^{(1)} + k^{2} I_{abcd}^{(2)} + I_{abcd}^{(3) y}) e^{y} = β_{y}^{2} e^{y},

where k=2π/λ is the wave number, β_x and β_y are the propagation constants of the modal electric fields E_x and E_y, respectively. Here, M^x_abcd and M^y_abcd are (S+1)×(S+1)×(S+1)×(S+1)-order four-dimensional matrices which are used in calculating the modal characteristics of the PCF. The overlap integrals I⁽¹⁾_abcd, I⁽²⁾_abcd, I^(3)x_abcd and I^(3)y_abcd in (8) and (9) are defined as follows:

I_{abcd}^{(1)} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ψ_{a} (x) ψ_{b} (y) \nabla^{2} [ψ_{c} (x) ψ_{d} (y)] d x d y,

I_{abcd}^{(2)} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} n^{2} (x, y) ψ_{a} (x) ψ_{b} (y) ψ_{c} (x) ψ_{d} (y) d x d y,

I_{abcd}^{(3) x} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ψ_{a} (x) ψ_{b} (y) {\frac{\partial}{\partial x} [ψ}_{c} (x) ψ_{d} (y) \frac{\partial \ln n^{2} (x, y)}{\partial x}] d x d y,

I_{abcd}^{(3) y} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ψ_{a} (x) ψ_{b} (y) {\frac{\partial}{\partial y} [ψ}_{c} (x) ψ_{d} (y) \frac{\partial \ln n^{2} (x, y)}{\partial y}] d x d y,

where Ψ_a(x), Ψ_b(y), Ψ_c(x) and Ψ_d(y) are the Hermite-Gaussian basis functions.

Next, the meshed index profile function n ²(x,y) expressed by (7) is substituted into the overlap integrals. The overlap integrals can be calculated with the orthonormality of the Hermite-Gaussian basis functions and are rewritten as [24, 25],

I_{abcd}^{(1)} = - \frac{2 a + 2 b + 1}{w^{2}} δ_{ac} δ_{bd} + \frac{\sqrt{2 b (2 b + 1)}}{2 w^{2}} δ_{ac} δ_{b - 1, d} + \frac{\sqrt{2 a (2 a + 1)}}{2 w^{2}} δ_{a - 1, c} δ_{bd}

I_{abcd}^{(2)} = \sum_{i = - N}^{N} \int_{Δ y} \int_{- \infty}^{\infty} n_{i}^{2} (x) ψ_{a} (x) ψ_{b} (y) ψ_{c} (x) ψ_{d} (y) d x d y,

I_{abcd}^{(3) x} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ψ_{a} (x) ψ_{b} (y) \frac{\partial}{\partial x} [ψ_{c} (x) ψ_{d} (y) \frac{\partial \sum_{i = - N}^{N} \ln n_{i}^{2} (x)}{\partial x}] d x d y

I_{abcd}^{(3) y} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ψ_{a} (x) ψ_{b} (y) \frac{\partial}{\partial y} [ψ_{c} (x) ψ_{d} (y) \frac{\partial \sum_{i = - N}^{N} \ln n_{i}^{2} (y)}{\partial y}] d x d y

Then, we obtain four-dimensional matrices M^x_abcd, M^y_abcd to analyze the modal characteristics of the PCF by substituting Eqs. (14)–(17) into Eq. (8) and (9). The four-dimensional matrices M^x_abcd, M^y_abcd can be transferred into the (S+1)²×(S+1)² two-dimensional matrices[21], so the eigenvalues and eigenvectors of the matrices M^x_abcd, M^y_abcd become a vector with (S+1)² elements. The Propagation constants of the guided modes β_x and β_y are the eigenvalues of M^x_abcd and M^y_abcd, respectively. The eigenvectors corresponding to the eigenvalues β_x and β_y are used as the mode field distribution coefficients (e^x_mn, e^y_mn) in Eqs. (2) and (3), respectively.

4. Modal electric field distribution, dispersion and birefringence

We analyze the modal electric field distribution of the PCF by using the proposed MIPM. Two PCFs used in this simulation have air-hole lattice periods Λ of 2.3µm and 4.6µm, respectively. And d/Λ for the PCF is 0.2, where d is the air-hole diameter of the PCF. The dimension of index profile matrix used in the MIPM is 1201×1201 and the number of basis functions (S+1) is 16.

Fig. 2 x-polarized fundamental mode field intensities (|E_x|²) of PCFs with different air-hole periods Λ and d/Λ=0.2.

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Fig. 3 x-polarized contour plots of the PCFs with the same parameters shown in Fig. 2. The contour plots are spaced by 2dB(a,b) and 1dB(c,d) from -30dB.

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Figure 2 shows the x-polarized fundamental mode field intensities (|E_x|²) at two different wavelengths 0.633µm and 1.55µm. Figure 3 displays the corresponding contour plots shown in Fig. 2. From the contour plots shown in Fig. 3, we establish that the mode field intensity distribution at the short wavelengths has greater localization over the core region. Meanwhile, at longer wavelengths, mode field intensity distribution spreads over the cladding region as predicted.

Figure 4(a) and Fig. 4(b) are the x-polarized mode field intensity distribution and its contour plot of the PCF with an elliptical air-hole lattice, respectively. In this simulation, Λ of 2.3µm and the elliptical ratio of the air-hole of 0.5 were used, where the major width of the elliptical air-hole is 0.4Λ and the minor is 0.2Λ. We expect that the mode field intensity distribution is affected by the shape and location of the air-hole. Figure 4 shows that the mode field intensity distribution of the PCF with the elliptical air-hole lattice is elliptical and this result matches with our expectation.

Fig. 4 x-polarized mode intensity distribution and contour plot of the PCF with an elliptical air-hole lattice. The contour plots are spaced by 2dB from -40dB.

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Fig. 5. Transverse structure of the PM-PCF with Λ=4.4µm, D₁=4.5µm and D₂=2.2µm.

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Fig. 6. x-polarized mode intensity distribution and contour plot of the PM-PCF shown in Fig. 5. The contour plots are spaced by 2dB from -30dB

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Figure 5 shows the transverse structure of the polarization maintaining PCF (PM-PCF) with Λ=4.4µm, large air-hole diameter D₁=4.5µm and small air-hole diameter D₂=2.2µm. In order to calculate the effective index, birefringence and fundamental mode field intensity distribution of this PM-PCF, first, the transverse structure of the PM-PCF is used to derive the meshed index profile. The meshed index profile derived from the transverse structure of the PM-PCF is then substituted into the eigenvalue equation and the modal characteristics of the PM-PCF are obtained by solving this eigenvalue equation. If the PCF has an arbitrary structure, these processes for analyzing the modal characteristics of the PCF are identical. Figure 6(a) and Fig. 6(b) are the x-polarized fundamental mode intensity distribution and the contour plot of the PM-PCF in Fig. 5, respectively. Figure 7 is the birefringence (Δn=|n_x-n_y|) of the PM-PCF shown in Fig. 5. If the PCF has the symmetric air-hole lattice, the two polarized fundamental modes are degenerate and the mode field intensity distributions are nearly identical. Then the modal birefringence becomes very small for the fundamental mode in the PCF with a symmetrically triangular air-hole lattice. But the PM-PCF has two air-holes of which size are different from the other air-holes in the triangular lattice as shown in Fig. 5. In that case the birefringence of the PM-PCF becomes larger than the birefringence of the PCF with a symmetrically triangular air-hole lattice. Figure 7 shows the birefringence of the PM-PCF computed by the MIPM, which increases with wavelength as predicted [26].

Fig. 7. Birefringence of a PM-PCF as a function of wavelength.

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The total dispersion of the PCF is defined as the sum of the material dispersion (D_m) and the waveguide dispersion (D_w). Because the PCF is mainly made of pure silica, D_m is same for all silica PCFs with different structures. It means that the total dispersion of the PCF is dominantly affected by the waveguide dispersion. The waveguide dispersions of the PCFs with different parameters are calculated by the MIPM. And the obtained waveguide dispersion of the PCF should satisfy the condition of scaling transformation property of the Maxwell’s equations which are expressed as [27],

D_{w} (λ, M Λ, f) = \frac{1}{M} D_{w} (\frac{λ}{M}, Λ, f),

where M is the scaling factor and f is air filling fraction of the PCF. The scaling property of the waveguide dispersion expressed in Eq (18) is well known and very useful in designing the waveguide dispersion of PCF.

Figure 8(a) shows the waveguide dispersion for five different air-hole pitches Λ with fixed d/Λ=0.3. Figure 8(b) shows the waveguide dispersion for five different values d/Λ with fixed Λ=2.3µm. Note that the dispersion magnitude, dispersion slope, the zero dispersion wavelength and the dispersion curve of the PCFs vary strongly with parameters Λ and d/Λ. This scaling property is very useful in designing the PCF for various dispersion-related applications.

Fig. 8. Waveguide dispersions of PCFs with different parameters.

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5. Accuracy verification of the MIPM

We demonstrate the MIPM for different PCFs to analyze the modal characteristics, such as the mode field intensity distribution, the effective index, the waveguide dispersion, and the birefringence. It is obvious that the spread-out distribution of the transverse mode at a given wavelength is one of the key factors used to verify the accuracy of the MIPM. Other key elements are the effective index of the guide mode and the modal birefringence Δn=|n_x-n_y| (n_x and n_y are effective indices of HE_11x and HE_11y mode, respectively).

Fig. 9. Rms error of the effective index as a function of wavelength in the range between 0.5µm and 1.8µm.

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To check the accuracy of the MIPM, we compute the rms error defined as the root-mean-square(rms) values of deviation of the effective indices evaluated by the MIPM from those computed by the MPM. Fig. 9 shows the rms errors of the effective indices calculated by using the MIPM of the PCF with Λ=4.6µm and d/Λ=0.2. The rms error decreases as the number of unit row or column matrix, used to express the index profile of the PCF, increases. Note that the rms error is reduced less than 5×10^-6 and saturated by increasing the number of unit matrix above 1200 as shown in the solid boxed region of the Fig. 9. From these results, one can say that the accuracy of the MIPM can be improved by increasing the number of unit matrix above 1200. However, increasing the number of unit matrix and the number of basis functions require additional computation time.

Meanwhile, the fundamental modes of the fibers with sixfold rotational symmetry should be degenerate [28]. This means that the fundamental modes (HE_11x and HE_11y) in the PCF with a triangular air-hole lattice, which means sixfold rotational symmetry, should be degenerate modes. Therefore, the PCF with a triangular air-hole lattice should exhibit very small birefringence.

Fig. 10. Birefringence of a PCF in the wavelength range between 0.5µm and 1.8µm

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The modal birefringences between the two degenerate fundamental modes are computed for the PCF (Λ=2.3µm, d/Λ=0.2) and the PCF (Λ=4.6µm, d/Λ=0.2) by the MIPM and are less than 6×10^-5 and 1.3×10^-5, respectively, over the wavelength range between 0.5µm and 1.8µm as shown in Fig. 10. This very small value of the birefringence <6×10^-5 over the wide wavelength range implies that the MIPM is accurate [28].

6. Discussion and conclusion

We present a meshed index profile method (MIPM) for efficiently analyzing the modal characteristics of the PCFs with arbitrary transverse structures. Meshed index profile was proposed for expressing the transverse structure of the PCF with arbitrary air-hole lattice and applied to the MIPM. By using the MIPM, we analyzed various modal characteristics, such as effective index, mode field distribution, waveguide dispersion and birefringence. Index decomposition of any number of virtual photonic crystal structures are needed in order to analyze the modal property of the PCF in the earlier simulation methods such as the SOM and the orthogonal method. These simulation methods are restricted to the PCF with a symmetric air-hole lattice. However, our proposed MIPM can efficiently express the index profile of a PCF with arbitrary transverse structure, and therefore, effectively analyze the modal characteristics of the PCF. The modal property of the PCF is obtained by solving the full vector wave equation which is substituted into the meshed index profile of the PCF. And the eigenvalues and eigenvectors corresponding to the fundamental and higher order modes can be simultaneously evaluated from the same eigenvalue matrix. Because the continuous localized basis functions are used in the MIPM, the calculated mode field distribution by the MIPM is also continuous function. Strictly speaking, the mode field distribution has discontinuity at the boundary of air-hole, and this discontinuity causes some numerical error. But Fig. 9 shows that the accuracy of the MIPM can be improved by increasing the number of unit matrix.

The PCFs that we analyzed their modal characteristics in this work include the PCFs with symmetric air-hole lattice, elliptical air-hole lattice and PM-PCF with two large air-holes. To verify the validation of the MIPM, we computed the rms error of the effective index of a PCF and found that the accuracy of the MIPM can be improved by increasing the number of unit matrix and saturated above 1200. The birefringence of the PCF with a triangular air-hole lattice was also computed to be less than 6×10^-5, indicating that the MIPM is reasonably accurate. The proposed MIPM has high accuracy like other simulation methods such as FEM, MPM and FDM. The eigenvalue matrix calculated by the MIPM used to analyze the modal property of the PCF is independent of wavelength. Even though the operating wavelength is changed, we can calculate the modal property without reconstructing the eigenvalue matrix. Therefore, MIPM is very efficient for designing the PCF with ultra-flattened dispersion or the ultra-wideband birefringent PCF. In conclusion, we are convinced that the MIPM is an efficient method for analyzing modal characteristics of the PCF with a complex index profile, because an arbitrary index profile of the PCF and its modal characteristics can be effectively obtained.

Acknowledgments

This paper was supported by the 63 Research Fund, from the Sungkyunkwan University, in 2006 and by the Second Stage of Brain Korea 21 Project.

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Meshed index profile method for photonic crystal fibers with arbitrary structures

Abstract

1. Introduction

2. Electric field decomposition

3. Meshed index profile of the PCF and eigenvalue matrix

4. Modal electric field distribution, dispersion and birefringence

5. Accuracy verification of the MIPM

6. Discussion and conclusion

Acknowledgments

References and links

Cited By

Figures (10)

Equations (29)

Optics Express