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Investigation of the supercell based orthonormal basis function method for different kinds of fibers
Optical Fiber Technology 10 (2004) 296–311 www.elsevier.com/locate/yofte
Investigation of the supercell based orthonormal basis function method for different kinds of fibers Zhi Wang ∗ , Guobin Ren, Shuqin Lou, Weijun Liang Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China Received 8 August 2003; revised 24 November 2003 Available online 27 July 2004
Abstract The supercell based orthonormal basis function method is proposed in this paper, in which the eigenequation about the propagation constant and the modal field is derived from the full vectorial coupled wave equations. It is efficient and accurate because all of the overlap integrals and the decomposition coefficients can be evaluated analytically. This method is efficient to investigate almost all the mode characteristics, such as the vector, degeneracy, coupling, absorption or gain properties, for so many kinds of fibers as ring fibers, Bragg fibers, multi-core fibers, elliptical fibers and photonic crystal fibers (PCF). 2004 Elsevier Inc. All rights reserved. Keywords: Supercell; Orthonormal basis function; Bragg fiber; Multi-core fiber; PCF
1. Introduction Optical fibers have attracted lots interests for about 40 years, and the technology of the optical telecommunications have been developed more and more with the successfulness of kinds of novel fibers and photonic components [1]. For the conventional fibers, the transfer matrix method (TMM) [2,3] of multi-layered dielectric waveguide are successfully used with high accuracy. It is efficient to obtain almost all the transmission characteristics, * Corresponding author.
E-mail address: [email protected] (Z. Wang). 1068-5200/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.yofte.2004.05.002
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such as propagation constant, group velocity dispersion, mode field distribution, birefringence and polarization, mode coupling. There exist some practiced commercial tools for the conventional fibers. It is more complex for the elliptical-core fibers. The wave equations, which are Bessel equations for the circular-core fibers, will become into Mathieu equations after the coordinates transform, and the complicated computations about the Mathieu functions will have to be carried out. There are many approximate approaches to simplify the elliptical-core fiber, but they usually have different results about the birefringence, and are different from the numerical calculations directly from the Mathieu functions [4]. Multi-core fibers had been investigated with some efficient methods in the mid of 1980’s, such as point-matching model [5–7] based on the group theory, and some approximate methods [2] especially for twin-core fibers. With the approximation of weaklyguiding and weakly-coupling, the mode coupling can be explained clearly in the twin-core fibers, but the split of the degeneracy [2] will still be incomprehensive with these approximations. At the beginning of the optical fiber progress, the ring fiber had been investigated by the transfer matrix method [2,3]. There was not more attention paid to it because the index difference between every layer was very small and low-loss propagation necessitates a very large number of periods [8]. The ring fiber became one of the hot subjects again when the great index difference dielectric materials were used at the end of last century [9–13]. The ring fiber has some new names called “Coaxial fiber,” “Omniguide fiber,” or “Bragg fiber,” which can be considered as a one-dimensional photonic crystal fiber because the dielectric constant structure is periodic along the radius direction. There are a few different approaches to calculate the modes of the Bragg fibers, such as the semi-analytic approach based on the transfer matrix method [2,3], the asymptotic matrix method [14,15], the bi-orthonormal-basis method [17,18], and the approach involving a numerical solution of Maxwell’s equations in the frequency domain with the use of the conjugate gradient method within the supercell approximation [18]. Photonic crystal fibers (PCFs), which had been proposed by Russell in 1995 [19], are microstructured fibers which are constructed by periodically distributed air holes in silica background. There are a few models and numerical methods to study the propagation characteristics of the photonic crystal fibers, such as the effective index method (EIM), localized orthogonal basis function method (LFM), plane wave expansion method (PWM), beam propagation method (BPM), scattering matrix method (SMM), finite time domain difference (FDTD), finite element method (FEM) and the multi-pole method [19–25]. Only a few classical numerical methods, such as FDTD, FEM, TMM, etc., are efficient to analyze kinds of fibers. The supercell based orthonormal basis function method is proposed to investigate the propagation properties of different fibers. A square lattice is constructed by the whole transverse profile (dielectric constant distribution) of the fiber which is considered as a supercell, and the periodical dielectric structure of the square lattice is decomposed using periodic functions (cosine). The modal electric field is expanded as the sum of the orthonormal set of Hermite-Gaussian basis functions. The propagation characteristics can be obtained after recasting the wave equations into an eigenvalue system. This method can be used to investigate many different kinds of fibers with considerable efficiency and accuracy.
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Fig. 1. Scheme of the construction of the supercell square lattice with the lattice constant D.
2. The model 2.1. Supercell In Fig. 1, each square area with length D is the transverse profile of the whole fiber which is considered as a supercell, in which the rings are used to show the dielectric constant structures. The square lattice is constructed by the supercell periodically along x and y axis with the lattice constant D. The effect between the adjacent supercells can be ignored when D is large enough. When the dielectric constant structure is x- and y-axial symmetry in the coordinates, which is built for the case of simplicity and convenience, it can be expressed as the sum of cosine functions, ε(r) = ε(x, y) =
P
Pab cos
a,b=0
ln ε(r) = ln ε(x, y) =
P a,b=0
2πax 2πby cos , D D
Pabln cos
2πby 2πax cos , D D
(1a)
(1b)
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where ε(r) is the dielectric structure in the supercell, r = (x, y) is the space position, ln are the expansion coefficients (P + 1) is the number of the expansion items, Pab , Pab which can be analytically evaluated from the Fourier transform of ε(r). In the supercell based method, the periodic dielectric constant structure can be transferred into its Fourier transform, then the wave equation will be able to be solved. For the supercell with length D in Fig. 1, the Fourier transform of the dielectric constant can be written as [23] 1{ 1{ εF (K) = ε(r)e−iK·r ds, ln εF (K) = ln ε(r)e−iK·r ds, (2) A A A
A
where A = D 2 is the area of the supercell, K = (Kx , Ky ), the vector in the reciprocal space of the supercell lattice, is linearly combined by the primitive reciprocal lattice vectors (2π/D, 2π/D). Then, Pab can be expressed as Pab = εF (Ka+P ,b+P ) + εF (Ka+P ,−b+P ) + εF (K−a+P ,b+P ) + εF (K−a+P ,−b+P ), for a = 0 or b = 0, for a = 0 and b = 0,
(3a)
Pab = εF (Ka+P ,b+P ) + εF (Ka+P ,−b+P ), P00 = εF (KP ,P ),
(3b) (3c)
ln can also be analytically evaluated from the where Km,n = (Kx , Ky ) = 2π/D × (m, n). Pab Fourier transform of the logarithm of the dielectric constant distribution ln ε(r).
2.2. Full vectorial coupled eigensystem The waveguide is considered uniform along the longitudinal direction (z-axis) and without absorption loss and confinement loss due to the finite size in the cross section, then the transverse electric field, et (x, y), which can be expressed as the sum of x- and ypolarization components as ˆ et (x, y) = ex (x, y)xˆ + ey (x, y)y,
(4)
will satisfy the full vectorial coupled wave equations [2] 2 ∂ ∂ ln ε ∂ ln ε ∇t − β 2 + k 2 ε ex = − + ey ex , ∂x ∂x ∂y 2 ∂ ln ε ∂ ln ε ∂ ex + ey , ∇t − β 2 + k 2 ε ey = − ∂y ∂x ∂y
(5)
where k = 2π/λ is the vacuum wave number, β is the propagation constant corresponding to the mode field distribution (ex , ey ). In order to obtain the characteristics of the modes in the waveguide, the transverse electric field is expanded using the localized orthonormal basis Hermite-Gaussian functions as follows [24], ex (x, y) =
F −1 a,b=0
x εab ψa (x)ψb (y),
ey (x, y) =
F −1 a,b=0
y
εab ψa (x)ψb (y),
(6)
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where F is the number of the expansion items, ψi (s) (i = a, b, s = x, y) is the normalized Hermite-Gaussian functions [24]. Substituting the decomposition equations of the dielectric constant and the electric field (Eqs. (1) and (6)) into Eq. (5), the eigenequation is now obtained as L
(1) (2) (3)x Iabcd + k 2 Iabcd + Iabcd εx ≡ (4)y εy Iabcd x ε = β2 y , ε
(4)x
Iabcd (3)y (1) (2) Iabcd + k 2 Iabcd + Iabcd
εx εy
(7)
where L is a four-dimensional matrix, I (1) , I (2) , I (3) , and I (4) are the overlapping integrals which have been expressed in our previous works [25,26] and not rewritten here. Through the subscript transform, the matrix L and the eigenvector can be transferred into a [2 × F 2 ] × [2 × F 2 ] 2-D matrix and a vector with 2 × F 2 elements. β 2 is now the eigenvalue of the matrix L, and the corresponding eigenvector of β 2 are the coefficients in Eq. (6). All the overlapping integrals about the localized HermiteGaussian functions can be analytically evaluated, and the decomposition coefficients of the dielectric constant can be analytically expressed with its Fourier transform, so the supercell based orthonormal basis function method is efficient as we expected. All information of the transverse electromagnetic field are included in the eigenequation (7). The vector property is reflected by both x- and y-components; the coupling property is included in the coupled terms I (4)x and I (4)y ; the fiber with complex dielectric structure can be investigated while it is made of the dielectric constant with negative or positive imaginary part (gain or absorption); the degeneracy property can be obtained from the full vectorial coupling wave equation.
3. Implementation in different fibers For different fibers, when a suitable coordinates is built in order to make the dielectric constant structure be both x- and y-axial symmetry, the cosine decomposition can be implemented only with different Fourier transform. The following procedures in different fibers will be completely same after one evaluates the Fourier transform of the dielectric constant distribution, then the modal characteristics and its transmission properties can be obtained. 3.1. Bragg fiber The dielectric structure ε(r) of the Bragg fiber can be expressed in different regions as follows: ε , r ε − ε , r i i−1 < r < ri , b i−1 < r < ri , ε(r) = = εb + i (8) εb , r > rm , 0, r > rm ,
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where ri is the radius of every layer, and r0 is set zero, m is the total number of the layers, εb is the background dielectric constant, and εb = 1 (air) in this article. Equation (2) will be analytically expressed as m 2fi J1 (Kri ) 2fi−1 J1 (Kri−1 ) εF (K) = εb δ(K) + (εi − εb ) − , (9) Kri Kri−1 i=1
is the filling ratio which is defined as the ratio of the areas where K = |K|, fi = between the ith layer and the supercell. The limitation will be used when K = 0 in Eq. (9). The high index-core Bragg fiber [16], which is a cylindrically symmetric microstructured fiber, has a high index-core (silica in this paper) surrounded by a multi-layered cladding made of alternating layers of a higher and a lower refractive-index dielectric. The dielectric constants are alternately ε1 and ε2 , which are selected as silica/air in this article. The radius of the high index-core is R = Λ − a. Fig. 2 is the simulation result of the dielectric constant of the Bragg fiber, with which the structure parameters are ε1 = 1.4572 (silica at 632.8 nm), ε2 = 1.0 (air), Λ = 1.5 µm, a = 0.3 µm, m = 20 (10 bilayers), the supercell period D is 30 µm and the expansion items P is 500. It is obvious that it is accurate and efficient to describe the dielectric constant using the cosine function. From the scaling property of the Maxwell’s equations [21], one can deduce the scaling property of the waveguide dispersion which says, when the length scale is changed by a factor M, the waveguide dispersion and the corresponding wavelength will both scale by the factor 1/M; when we multiply the refractive index everywhere by a factor N , the waveguide dispersion at wavelength λ will be same as that at wavelength Nλ. It can be used to check the efficiency and accuracy of a numerical method. When the dielectric structure (Λ and n/n0 ) varies, the zero-dispersion wavelength (λzero ) and the min-dispersion (λmin ) wavelength are plotted in Fig. 3. It can be seen that all relations are linear and approximately through the origin of the coordinates which coincides with the scaling property as discussed above. πri2 /A
3.2. Multi-core fibers The mode coupling properties of multi-core fibers with symmetric dielectric constant structure [2,5–7] have been studied since the mid of 1980’s. A multi-core profile is shown on the top panel of Fig. 4. All parameters in different core area are distinguished by its subscript i and, dielectric constant εi , core radius Ri , coordinates of the core center ri . All cores are inside an inner cladding in which the radius is Ra and dielectric constant is εa , and the dielectric constant in the out-cladding is εb . When there are m cores in the inner cladding, the Fourier transform will be 2fa J1 (KRa ) 2fi J1 (KRi ) −iK·ri + (εi − εa ) e , (10) KRa KRi m
εF (K) = εb δ(K) + (εa − εb )
i=1
where fa = fi = are the ratios between every core area and the supercell area. The bottom panel of Fig. 4 is the simulation result of a 4-core fiber at wavelength 1.55 µm, and the out-cladding is pure silica, inner cladding is high-doped silica and εa = (1 + 0.25%)2εb , Ra = 10 µm, the radius and dielectric constant of 4 cores πRa2 /D 2 ,
πRi2 /D 2
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Fig. 2. The simulation result of the dielectric constant of the Bragg fiber, with the parameters ε1 = 1.4572 , ε2 = 1.0, Λ = 1.5 µm, a = 0.3 µm, and m = 20. Supercell lattice constant D = 30 µm, and P = 500. The top panel is the contour line and the bottom is the distribution in the y = 0 cross section.
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Fig. 3. The zero-dispersion wavelength and the min-dispersion wavelength vary with the structure parameters, the period Λ (top) and the refractive index (bottom).
are same, and εi = (1 + 0.5%)2 εb , Ri = 3/2 µm, the center positions of every core are expressed in complex as ri = {8, −8, 6i, −6i}, the decomposition parameter P = 1000. Twin-core fiber [2,27] is the most interest in multi-core fibers, and is usually used to equal pumping based on the modes weak-coupling between both cores. The coupling length may reach over 10 m when both cores are separated away enough [28].
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Fig. 4. Dielectric constant profile of multi-core fiber, top panel is the cross section and bottom panel is the simulation of a 4-core fiber with the structure parameters εb = silica at 1.55 µm, εa = (1 + 0.25%)2 εb , Ra = 10 µm, εi = (1 + 0.5%)2 εb , Ri = 3/2 µm, ri = {8, −8, 6i, −6i}.
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(a)
(b) Fig. 5. The electric field quiver distribution (a), the birefringence (b) and the coupling length (c) of a twin-core fiber, with the structure parameters εa = εb = silica at 1.55 µm, ε1 = ε2 = (1 + 0.5%)2 εb , R1 = R2 = 3/2 µm, ri = {3, −3}, P = 1000.
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(c) Fig. 5. (Continued.)
Electric field distribution of the first 4 lowest modes are demonstrated as Fig. 5a, in which the twin-core fiber is constructed as pure silica in out-cladding and inner cladding, ε1 = ε2 = (1 + 0.5%)2εb , R1 = R2 = 3/2 µm, ri = {3, −3}, P = 1000. Conventionally, all the 4 shown modes are approximately considered degenerate fundamental modes [2], and labeled as mode 1 to mode 4 in the descending order of the mode index, or labeled as x+, y+, y−, x− according to the modal symmetry. From the group theory and symmetry analysis [26,29], all modes are non-degenerate because the twincore fiber has C2ν point-group symmetry. Consequently, all modes shown in Fig. 5a are non-degenerate, and the mode index difference between modes x+ and y+ or between modes y− and x− should be regarded as the birefringence ( n) of the fundamental mode. Fig. 5b shows the difference between the propagation constants of both modes x+ and y+, in which the solid line is obtained by the supercell method, and the ‘+’ is obtained from the analytical result (Eq. (19)) in Ref. [27]. From Fig. 5b, in which both results can be considered as accordance, it can be said that the supercell method is validity for multi-core fibers. The mode coupling occurs between both modes with almost same polarization, i.e., between mode x+ and x− or y+ and y−. The mode index difference between modes x+ and x−, which is almost same as that between y+ and y−, is much greater than the fundamental mode birefringence n. The coupling length, defined as LC = 2π/(βx+ − βx− ) ≈ 2π/(βy+ − βy− ) [2], is shown in Fig. 5c. LC is very short (only a few millimeters) due to very strong-coupling because both cores are very close to each other.
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Fig. 6. The simulation dielectric constant structure of conventional step-index fiber (top) and elliptical-core fiber (bottom), with parameters core radius or half length of the major axis is 4.5 µm, = 0.5%, and P = 1000.
3.3. Conventional step-index fiber and elliptical-core fiber A conventional step-index fiber and elliptical-core fiber can be considered as a special case of the ring fibers or multi-core fibers with only one circular or elliptical core. Fig. 6
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Fig. 7. Waveguide dispersion (top) of the conventional step-index fiber and the fundamental mode birefringence (bottom) due to numerical calculation.
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shows the simulation dielectric structure of a step-index fiber and an elliptical-core fiber, with which the core radius (half length of the major axis for the elliptical-core fiber) is 4.5 µm, the relative refractive index difference is 0.5%, for elliptical-core fiber, the ratio of the major axis to the minor axis is η = 3, and P = 1000. The waveguide dispersion of the step-index fiber is plotted in Fig. 7 (top panel), in which the solid line is calculated by the conventional transfer matrix approach [2], and “+” is obtained by the supercell based method discussed above. It can be seen that both results are almost identical in the optical communication band. | n|, the birefringence of the doublet HE11x and HE11y of the fundamental mode is plotted on the bottom panel. HE11x and HE11y must be critically degenerate in the waveguide which has circular symmetry. Hence | n| must result from the numerical computations and should be little enough to be ignored. Less | n| is, more accurate the numerical method is [25]. Over the wavelength range from 0.6 to 2.2 µm, | n| is less than 1.6 × 10−14 with the supercell based method. The accuracy can be improved while increasing P and F , but it will cost more computations and elapse more. The simulation of the elliptical-core fiber can afford its transmission properties and the birefringence which are almost same as conventional [2,4], while they are not shown here for the case of simplicity. 3.4. Photonic crystal fibers Periodical lattice is constructed by supercell which is a representative region including the defects and enough atoms in the plane wave expansion method [23] and bi-orthonormal basis method [17] for investigating the photonic crystal fibers. The supercell based method, described in our previous works [25,26], can also be used to calculate the propagation properties of PCFs, such as dispersion, birefringence, mode field distribution. For the case of simplicity, it will not be discussed again.
4. Conclusion The supercell based orthonormal basis function method is proposed, in which a square lattice is constructed by the whole transverse profile (dielectric constant distribution) of the fiber which is considered as a supercell, and the periodical dielectric structure of the square lattice is decomposed using periodic functions (cosine). The modal electric field is expanded as the sum of the orthonormal set of Hermite-Gaussian basis functions. The propagation characteristics can be obtained after recasting the full vectorial coupled wave equations into an eigenvalue system. It is efficient and accurate because all of the overlap integrals and the decomposition coefficients can be evaluated analytically. This method is efficient to investigate almost all the mode characteristics, such as the vector, degeneracy, coupling, absorption or gain properties, for so many kinds of fibers as ring fibers, Bragg fibers, multi-core fibers, elliptical fibers and photonic crystal fibers (PCF).
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Investigation of the supercell based orthonormal basis function method for different kinds of fibers Zhi Wang ∗ , Guobin Ren, Shuqin Lou, Weijun Liang Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China Received 8 August 2003; revised 24 November 2003 Available online 27 July 2004
Abstract The supercell based orthonormal basis function method is proposed in this paper, in which the eigenequation about the propagation constant and the modal field is derived from the full vectorial coupled wave equations. It is efficient and accurate because all of the overlap integrals and the decomposition coefficients can be evaluated analytically. This method is efficient to investigate almost all the mode characteristics, such as the vector, degeneracy, coupling, absorption or gain properties, for so many kinds of fibers as ring fibers, Bragg fibers, multi-core fibers, elliptical fibers and photonic crystal fibers (PCF). 2004 Elsevier Inc. All rights reserved. Keywords: Supercell; Orthonormal basis function; Bragg fiber; Multi-core fiber; PCF
1. Introduction Optical fibers have attracted lots interests for about 40 years, and the technology of the optical telecommunications have been developed more and more with the successfulness of kinds of novel fibers and photonic components [1]. For the conventional fibers, the transfer matrix method (TMM) [2,3] of multi-layered dielectric waveguide are successfully used with high accuracy. It is efficient to obtain almost all the transmission characteristics, * Corresponding author.
E-mail address: [email protected] (Z. Wang). 1068-5200/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.yofte.2004.05.002
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such as propagation constant, group velocity dispersion, mode field distribution, birefringence and polarization, mode coupling. There exist some practiced commercial tools for the conventional fibers. It is more complex for the elliptical-core fibers. The wave equations, which are Bessel equations for the circular-core fibers, will become into Mathieu equations after the coordinates transform, and the complicated computations about the Mathieu functions will have to be carried out. There are many approximate approaches to simplify the elliptical-core fiber, but they usually have different results about the birefringence, and are different from the numerical calculations directly from the Mathieu functions [4]. Multi-core fibers had been investigated with some efficient methods in the mid of 1980’s, such as point-matching model [5–7] based on the group theory, and some approximate methods [2] especially for twin-core fibers. With the approximation of weaklyguiding and weakly-coupling, the mode coupling can be explained clearly in the twin-core fibers, but the split of the degeneracy [2] will still be incomprehensive with these approximations. At the beginning of the optical fiber progress, the ring fiber had been investigated by the transfer matrix method [2,3]. There was not more attention paid to it because the index difference between every layer was very small and low-loss propagation necessitates a very large number of periods [8]. The ring fiber became one of the hot subjects again when the great index difference dielectric materials were used at the end of last century [9–13]. The ring fiber has some new names called “Coaxial fiber,” “Omniguide fiber,” or “Bragg fiber,” which can be considered as a one-dimensional photonic crystal fiber because the dielectric constant structure is periodic along the radius direction. There are a few different approaches to calculate the modes of the Bragg fibers, such as the semi-analytic approach based on the transfer matrix method [2,3], the asymptotic matrix method [14,15], the bi-orthonormal-basis method [17,18], and the approach involving a numerical solution of Maxwell’s equations in the frequency domain with the use of the conjugate gradient method within the supercell approximation [18]. Photonic crystal fibers (PCFs), which had been proposed by Russell in 1995 [19], are microstructured fibers which are constructed by periodically distributed air holes in silica background. There are a few models and numerical methods to study the propagation characteristics of the photonic crystal fibers, such as the effective index method (EIM), localized orthogonal basis function method (LFM), plane wave expansion method (PWM), beam propagation method (BPM), scattering matrix method (SMM), finite time domain difference (FDTD), finite element method (FEM) and the multi-pole method [19–25]. Only a few classical numerical methods, such as FDTD, FEM, TMM, etc., are efficient to analyze kinds of fibers. The supercell based orthonormal basis function method is proposed to investigate the propagation properties of different fibers. A square lattice is constructed by the whole transverse profile (dielectric constant distribution) of the fiber which is considered as a supercell, and the periodical dielectric structure of the square lattice is decomposed using periodic functions (cosine). The modal electric field is expanded as the sum of the orthonormal set of Hermite-Gaussian basis functions. The propagation characteristics can be obtained after recasting the wave equations into an eigenvalue system. This method can be used to investigate many different kinds of fibers with considerable efficiency and accuracy.
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Fig. 1. Scheme of the construction of the supercell square lattice with the lattice constant D.
2. The model 2.1. Supercell In Fig. 1, each square area with length D is the transverse profile of the whole fiber which is considered as a supercell, in which the rings are used to show the dielectric constant structures. The square lattice is constructed by the supercell periodically along x and y axis with the lattice constant D. The effect between the adjacent supercells can be ignored when D is large enough. When the dielectric constant structure is x- and y-axial symmetry in the coordinates, which is built for the case of simplicity and convenience, it can be expressed as the sum of cosine functions, ε(r) = ε(x, y) =
P
Pab cos
a,b=0
ln ε(r) = ln ε(x, y) =
P a,b=0
2πax 2πby cos , D D
Pabln cos
2πby 2πax cos , D D
(1a)
(1b)
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where ε(r) is the dielectric structure in the supercell, r = (x, y) is the space position, ln are the expansion coefficients (P + 1) is the number of the expansion items, Pab , Pab which can be analytically evaluated from the Fourier transform of ε(r). In the supercell based method, the periodic dielectric constant structure can be transferred into its Fourier transform, then the wave equation will be able to be solved. For the supercell with length D in Fig. 1, the Fourier transform of the dielectric constant can be written as [23] 1{ 1{ εF (K) = ε(r)e−iK·r ds, ln εF (K) = ln ε(r)e−iK·r ds, (2) A A A
A
where A = D 2 is the area of the supercell, K = (Kx , Ky ), the vector in the reciprocal space of the supercell lattice, is linearly combined by the primitive reciprocal lattice vectors (2π/D, 2π/D). Then, Pab can be expressed as Pab = εF (Ka+P ,b+P ) + εF (Ka+P ,−b+P ) + εF (K−a+P ,b+P ) + εF (K−a+P ,−b+P ), for a = 0 or b = 0, for a = 0 and b = 0,
(3a)
Pab = εF (Ka+P ,b+P ) + εF (Ka+P ,−b+P ), P00 = εF (KP ,P ),
(3b) (3c)
ln can also be analytically evaluated from the where Km,n = (Kx , Ky ) = 2π/D × (m, n). Pab Fourier transform of the logarithm of the dielectric constant distribution ln ε(r).
2.2. Full vectorial coupled eigensystem The waveguide is considered uniform along the longitudinal direction (z-axis) and without absorption loss and confinement loss due to the finite size in the cross section, then the transverse electric field, et (x, y), which can be expressed as the sum of x- and ypolarization components as ˆ et (x, y) = ex (x, y)xˆ + ey (x, y)y,
(4)
will satisfy the full vectorial coupled wave equations [2] 2 ∂ ∂ ln ε ∂ ln ε ∇t − β 2 + k 2 ε ex = − + ey ex , ∂x ∂x ∂y 2 ∂ ln ε ∂ ln ε ∂ ex + ey , ∇t − β 2 + k 2 ε ey = − ∂y ∂x ∂y
(5)
where k = 2π/λ is the vacuum wave number, β is the propagation constant corresponding to the mode field distribution (ex , ey ). In order to obtain the characteristics of the modes in the waveguide, the transverse electric field is expanded using the localized orthonormal basis Hermite-Gaussian functions as follows [24], ex (x, y) =
F −1 a,b=0
x εab ψa (x)ψb (y),
ey (x, y) =
F −1 a,b=0
y
εab ψa (x)ψb (y),
(6)
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where F is the number of the expansion items, ψi (s) (i = a, b, s = x, y) is the normalized Hermite-Gaussian functions [24]. Substituting the decomposition equations of the dielectric constant and the electric field (Eqs. (1) and (6)) into Eq. (5), the eigenequation is now obtained as L
(1) (2) (3)x Iabcd + k 2 Iabcd + Iabcd εx ≡ (4)y εy Iabcd x ε = β2 y , ε
(4)x
Iabcd (3)y (1) (2) Iabcd + k 2 Iabcd + Iabcd
εx εy
(7)
where L is a four-dimensional matrix, I (1) , I (2) , I (3) , and I (4) are the overlapping integrals which have been expressed in our previous works [25,26] and not rewritten here. Through the subscript transform, the matrix L and the eigenvector can be transferred into a [2 × F 2 ] × [2 × F 2 ] 2-D matrix and a vector with 2 × F 2 elements. β 2 is now the eigenvalue of the matrix L, and the corresponding eigenvector of β 2 are the coefficients in Eq. (6). All the overlapping integrals about the localized HermiteGaussian functions can be analytically evaluated, and the decomposition coefficients of the dielectric constant can be analytically expressed with its Fourier transform, so the supercell based orthonormal basis function method is efficient as we expected. All information of the transverse electromagnetic field are included in the eigenequation (7). The vector property is reflected by both x- and y-components; the coupling property is included in the coupled terms I (4)x and I (4)y ; the fiber with complex dielectric structure can be investigated while it is made of the dielectric constant with negative or positive imaginary part (gain or absorption); the degeneracy property can be obtained from the full vectorial coupling wave equation.
3. Implementation in different fibers For different fibers, when a suitable coordinates is built in order to make the dielectric constant structure be both x- and y-axial symmetry, the cosine decomposition can be implemented only with different Fourier transform. The following procedures in different fibers will be completely same after one evaluates the Fourier transform of the dielectric constant distribution, then the modal characteristics and its transmission properties can be obtained. 3.1. Bragg fiber The dielectric structure ε(r) of the Bragg fiber can be expressed in different regions as follows: ε , r ε − ε , r i i−1 < r < ri , b i−1 < r < ri , ε(r) = = εb + i (8) εb , r > rm , 0, r > rm ,
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where ri is the radius of every layer, and r0 is set zero, m is the total number of the layers, εb is the background dielectric constant, and εb = 1 (air) in this article. Equation (2) will be analytically expressed as m 2fi J1 (Kri ) 2fi−1 J1 (Kri−1 ) εF (K) = εb δ(K) + (εi − εb ) − , (9) Kri Kri−1 i=1
is the filling ratio which is defined as the ratio of the areas where K = |K|, fi = between the ith layer and the supercell. The limitation will be used when K = 0 in Eq. (9). The high index-core Bragg fiber [16], which is a cylindrically symmetric microstructured fiber, has a high index-core (silica in this paper) surrounded by a multi-layered cladding made of alternating layers of a higher and a lower refractive-index dielectric. The dielectric constants are alternately ε1 and ε2 , which are selected as silica/air in this article. The radius of the high index-core is R = Λ − a. Fig. 2 is the simulation result of the dielectric constant of the Bragg fiber, with which the structure parameters are ε1 = 1.4572 (silica at 632.8 nm), ε2 = 1.0 (air), Λ = 1.5 µm, a = 0.3 µm, m = 20 (10 bilayers), the supercell period D is 30 µm and the expansion items P is 500. It is obvious that it is accurate and efficient to describe the dielectric constant using the cosine function. From the scaling property of the Maxwell’s equations [21], one can deduce the scaling property of the waveguide dispersion which says, when the length scale is changed by a factor M, the waveguide dispersion and the corresponding wavelength will both scale by the factor 1/M; when we multiply the refractive index everywhere by a factor N , the waveguide dispersion at wavelength λ will be same as that at wavelength Nλ. It can be used to check the efficiency and accuracy of a numerical method. When the dielectric structure (Λ and n/n0 ) varies, the zero-dispersion wavelength (λzero ) and the min-dispersion (λmin ) wavelength are plotted in Fig. 3. It can be seen that all relations are linear and approximately through the origin of the coordinates which coincides with the scaling property as discussed above. πri2 /A
3.2. Multi-core fibers The mode coupling properties of multi-core fibers with symmetric dielectric constant structure [2,5–7] have been studied since the mid of 1980’s. A multi-core profile is shown on the top panel of Fig. 4. All parameters in different core area are distinguished by its subscript i and, dielectric constant εi , core radius Ri , coordinates of the core center ri . All cores are inside an inner cladding in which the radius is Ra and dielectric constant is εa , and the dielectric constant in the out-cladding is εb . When there are m cores in the inner cladding, the Fourier transform will be 2fa J1 (KRa ) 2fi J1 (KRi ) −iK·ri + (εi − εa ) e , (10) KRa KRi m
εF (K) = εb δ(K) + (εa − εb )
i=1
where fa = fi = are the ratios between every core area and the supercell area. The bottom panel of Fig. 4 is the simulation result of a 4-core fiber at wavelength 1.55 µm, and the out-cladding is pure silica, inner cladding is high-doped silica and εa = (1 + 0.25%)2εb , Ra = 10 µm, the radius and dielectric constant of 4 cores πRa2 /D 2 ,
πRi2 /D 2
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Fig. 2. The simulation result of the dielectric constant of the Bragg fiber, with the parameters ε1 = 1.4572 , ε2 = 1.0, Λ = 1.5 µm, a = 0.3 µm, and m = 20. Supercell lattice constant D = 30 µm, and P = 500. The top panel is the contour line and the bottom is the distribution in the y = 0 cross section.
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Fig. 3. The zero-dispersion wavelength and the min-dispersion wavelength vary with the structure parameters, the period Λ (top) and the refractive index (bottom).
are same, and εi = (1 + 0.5%)2 εb , Ri = 3/2 µm, the center positions of every core are expressed in complex as ri = {8, −8, 6i, −6i}, the decomposition parameter P = 1000. Twin-core fiber [2,27] is the most interest in multi-core fibers, and is usually used to equal pumping based on the modes weak-coupling between both cores. The coupling length may reach over 10 m when both cores are separated away enough [28].
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Fig. 4. Dielectric constant profile of multi-core fiber, top panel is the cross section and bottom panel is the simulation of a 4-core fiber with the structure parameters εb = silica at 1.55 µm, εa = (1 + 0.25%)2 εb , Ra = 10 µm, εi = (1 + 0.5%)2 εb , Ri = 3/2 µm, ri = {8, −8, 6i, −6i}.
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(a)
(b) Fig. 5. The electric field quiver distribution (a), the birefringence (b) and the coupling length (c) of a twin-core fiber, with the structure parameters εa = εb = silica at 1.55 µm, ε1 = ε2 = (1 + 0.5%)2 εb , R1 = R2 = 3/2 µm, ri = {3, −3}, P = 1000.
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(c) Fig. 5. (Continued.)
Electric field distribution of the first 4 lowest modes are demonstrated as Fig. 5a, in which the twin-core fiber is constructed as pure silica in out-cladding and inner cladding, ε1 = ε2 = (1 + 0.5%)2εb , R1 = R2 = 3/2 µm, ri = {3, −3}, P = 1000. Conventionally, all the 4 shown modes are approximately considered degenerate fundamental modes [2], and labeled as mode 1 to mode 4 in the descending order of the mode index, or labeled as x+, y+, y−, x− according to the modal symmetry. From the group theory and symmetry analysis [26,29], all modes are non-degenerate because the twincore fiber has C2ν point-group symmetry. Consequently, all modes shown in Fig. 5a are non-degenerate, and the mode index difference between modes x+ and y+ or between modes y− and x− should be regarded as the birefringence ( n) of the fundamental mode. Fig. 5b shows the difference between the propagation constants of both modes x+ and y+, in which the solid line is obtained by the supercell method, and the ‘+’ is obtained from the analytical result (Eq. (19)) in Ref. [27]. From Fig. 5b, in which both results can be considered as accordance, it can be said that the supercell method is validity for multi-core fibers. The mode coupling occurs between both modes with almost same polarization, i.e., between mode x+ and x− or y+ and y−. The mode index difference between modes x+ and x−, which is almost same as that between y+ and y−, is much greater than the fundamental mode birefringence n. The coupling length, defined as LC = 2π/(βx+ − βx− ) ≈ 2π/(βy+ − βy− ) [2], is shown in Fig. 5c. LC is very short (only a few millimeters) due to very strong-coupling because both cores are very close to each other.
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Fig. 6. The simulation dielectric constant structure of conventional step-index fiber (top) and elliptical-core fiber (bottom), with parameters core radius or half length of the major axis is 4.5 µm, = 0.5%, and P = 1000.
3.3. Conventional step-index fiber and elliptical-core fiber A conventional step-index fiber and elliptical-core fiber can be considered as a special case of the ring fibers or multi-core fibers with only one circular or elliptical core. Fig. 6
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Fig. 7. Waveguide dispersion (top) of the conventional step-index fiber and the fundamental mode birefringence (bottom) due to numerical calculation.
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shows the simulation dielectric structure of a step-index fiber and an elliptical-core fiber, with which the core radius (half length of the major axis for the elliptical-core fiber) is 4.5 µm, the relative refractive index difference is 0.5%, for elliptical-core fiber, the ratio of the major axis to the minor axis is η = 3, and P = 1000. The waveguide dispersion of the step-index fiber is plotted in Fig. 7 (top panel), in which the solid line is calculated by the conventional transfer matrix approach [2], and “+” is obtained by the supercell based method discussed above. It can be seen that both results are almost identical in the optical communication band. | n|, the birefringence of the doublet HE11x and HE11y of the fundamental mode is plotted on the bottom panel. HE11x and HE11y must be critically degenerate in the waveguide which has circular symmetry. Hence | n| must result from the numerical computations and should be little enough to be ignored. Less | n| is, more accurate the numerical method is [25]. Over the wavelength range from 0.6 to 2.2 µm, | n| is less than 1.6 × 10−14 with the supercell based method. The accuracy can be improved while increasing P and F , but it will cost more computations and elapse more. The simulation of the elliptical-core fiber can afford its transmission properties and the birefringence which are almost same as conventional [2,4], while they are not shown here for the case of simplicity. 3.4. Photonic crystal fibers Periodical lattice is constructed by supercell which is a representative region including the defects and enough atoms in the plane wave expansion method [23] and bi-orthonormal basis method [17] for investigating the photonic crystal fibers. The supercell based method, described in our previous works [25,26], can also be used to calculate the propagation properties of PCFs, such as dispersion, birefringence, mode field distribution. For the case of simplicity, it will not be discussed again.
4. Conclusion The supercell based orthonormal basis function method is proposed, in which a square lattice is constructed by the whole transverse profile (dielectric constant distribution) of the fiber which is considered as a supercell, and the periodical dielectric structure of the square lattice is decomposed using periodic functions (cosine). The modal electric field is expanded as the sum of the orthonormal set of Hermite-Gaussian basis functions. The propagation characteristics can be obtained after recasting the full vectorial coupled wave equations into an eigenvalue system. It is efficient and accurate because all of the overlap integrals and the decomposition coefficients can be evaluated analytically. This method is efficient to investigate almost all the mode characteristics, such as the vector, degeneracy, coupling, absorption or gain properties, for so many kinds of fibers as ring fibers, Bragg fibers, multi-core fibers, elliptical fibers and photonic crystal fibers (PCF).
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