Analytical formulation of super-modes inside multi-core fibers with circularly distributed cores

Junhe Zhou

doi:10.1364/OE.22.000673

1. Introduction

Multi-core fibers (MCFs) have wide applications in various fields, such as mode division multiplexing (MDM) systems [1–4], fiber lasers [5, 6], transmission of ultra-short laser pulses [7], passive optical switch and star couplers [8], and fiber endoscope [9]. To implement MCFs in these applications, the properties of the super-modes needs to be studied carefully. For instance, the MCFs based MDM system can use the super-modes of the MCFs to carry information [3–5] instead of the modes inside individual cores, which can significantly improve the spectral efficiency of optical fiber communication systems. Also, in fiber lasers formed by MCFs, the near/far field property and the longitude mode wavelength are highly dependent on the modal distributions and the effective indexes of the super-modes [5, 6].

Recently, there have been quite a few publications focusing on the super-modes of the MCFs or the fiber arrays [3–6, 8 ,10–18]. Some of them used numerical simulations [3–5, 15–18] to demonstrate the modal characteristics of the super-modes inside the MCFs, while others analyze the eigen modes and the propagation constants of the super-modes inside the MCFs/the fiber arrays analytically [6,8,10–14]. Compared with the numerical results, analytical expressions give a deeper insight into the physical nature of the super-modes and present a clearer picture for the MCF design and analysis. Therefore, they are of great interests both to the academic and the industrial societies.

Currently, analytical expressions have been proposed for the propagation constants and the eigen mode vectors for the MCFs with linearly aligned cores [8,10,13] and the MCFs with circularly aligned cores within one ring [8,11,13]. However, to align the cores of the MCFs more efficiently, it is preferred to distribute them in multiple rings, e. g. two [12], three or more rings. The arrangement can increase the number of cores inside one MCF, which will increase the number of the super-modes and therefore increase the transmission capacity of the MCF based MDM systems or increase the output power in MCF based fiber lasers while maintaining high beam quality. Furthermore, at the center of the rings, usually, one more core can be inserted. For MCF based MDM system, the center core can further increase the MCF transmission capacity [4]; for MCF based lasers, the center core can increase the mode area and further increase the laser output power [5]; for MCF based star couplers [8], the center core can increase the routing directivity. For such arrangements, there have been very few analytical discussions. In [12], C. Alexeyev et. al. discussed the analytical formulation of the super-modes for the two-ring fiber arrays with the assumption that the coupling coefficients between the adjacently cores within both rings are identical, but this simplification is only valid when the diameters of the rings are very large. In [13,14], N. Kishi et. al and S. Peleš et. al. demonstrated analytical solutions for the one-ring fiber array with an additional core at the center of the ring, which couples to each of the cores within the ring. In particular, two orthogonal modes in each core were considered in [13]. They were, however, not referring to the multiple-ring case.

In this paper, we analyze the super-modes of the MCFs with cores distributed within multiple rings. The propagation constants as well as the eigen modes distributions are studied with explicit analytical expressions. The results are confirmed by the numerical simulations based on the beam propagation method (BPM).

It should be mentioned that in the analysis below, it assumed that coupling only takes place between adjacent fiber cores and the numbers of the cores within individual rings are the same. Also single mode is assumed for each core of the MCF as is adopted by most of the publications [3–6, 8, 10–12, 14].

2. MCFs with cores in one ring

The simplest case for the MCFs with circularly distributed cores is the one-ring fiber array case, which has been analyzed by Janice Hudgings et. al. [8]. Assuming that all the cores are identical single mode fibers, the field on each fiber should vary as a_kexp(-jβz) [8], where a_k is the amplitude of the field at the k^th core and β the common propagation constant of the single mode cores. As indicated in [4,8], the amplitudes of each core should fulfill the coupled mode equation

\begin{array}{l} \frac{d A}{d z} = - j κ A \\ A = (\begin{matrix} a_{1} \\ ⋮ \\ a_{N} \end{matrix}) \\ κ = (\begin{matrix} 0 & κ & κ \\ κ & 0 & κ \\ ⋱ & ⋱ & ⋱ \\ κ & 0 & κ \\ κ & κ & 0 \end{matrix}) \end{array}

where N is the number of the cores within the ring, κ the coupling coefficients between the adjacent cores [4,8].

Since matrix κ is a symmetric and Hermitian matrix, it can be decomposed into QDQ^H, where Q is an orthogonal matrix or a unitary matrix, D a diagonal matrix whose diagonal terms are the eigen value of κ, and H denotes the Hermitian operation. Therefore, the solution of the above equation can be written as [14]

A (L) = Q \exp (- j D L) Q^{H} A (0)

Obviously, the columns of the matrix Q form the eigen mode vectors of the super-modes and β + d_n serve as the propagation constants of the corresponding super-modes, where d_n is the n^th diagonal element of the matrix D.

It has been revealed in [8] that the propagation constants of the super-modes can be analytically formulated as

β + 2 κ \cos (\frac{2 π (n - 1)}{N})

where n is the order of the super-modes and the element of matrix Q is [8]

Q_{m n} = \sqrt{\frac{1}{N}} \exp (- j (m - 1) (n - 1) \frac{2 π}{N})

Equation (4) clearly demonstrates the eigen vectors of the system. It is, however, in the complex form. When conducting the numerical simulations, the mode amplitudes are usually real values [4]. To transform the complex amplitudes of the super-modes into real ones, we propose to use the following expression for Q

\begin{array}{l} when N is an even number \\ Q_{m n} = {\begin{matrix} \sqrt{\frac{1}{N}} & n = 1 \\ \sqrt{\frac{2}{N}} \cos ((m - 1) (n - 1) \frac{2 π}{N}) & 1 < n \leq \frac{N}{2} \\ \sqrt{\frac{1}{N}} {(- 1)}^{m - 1} & n = 1 + \frac{N}{2} \\ \sqrt{\frac{2}{N}} \sin ((m - 1) (N - n + 1) \frac{2 π}{N}) & \frac{N}{2} + 1 < n \leq N \end{matrix} \\ when N is an odd number \\ Q_{m n} = {\begin{matrix} \sqrt{\frac{1}{N}} & n = 1 \\ \sqrt{\frac{2}{N}} \cos ((m - 1) (n - 1) \frac{2 π}{N}) & 1 < n \leq \frac{N + 1}{2} \\ \sqrt{\frac{2}{N}} \sin ((m - 1) (N - n + 1) \frac{2 π}{N}) & \frac{N + 1}{2} < n \leq N \end{matrix} \end{array}

In this case, we have Q^H = Q^T, where T denotes transpose operation on the matrix. It is useful to compare Eq. (5) with the numerical results in [4], and very precise matching can be observed. It should be noted that Q is not unique, because mode n and mode N + 2-n are degenerated modes and their corresponding eigen mode vectors can be the super-position of the eigen mode vectors indicated by Eq. (5).

It should be further noticed that Q is not related to the coupling coefficient κ, and therefore it is universal for all MCFs with one ring cores.

3. MCFs with cores in two rings

3.1 Mathematical formulation

In this analysis, it is still assumed that mode coupling remains the same within the same ring but the two coupling coefficients of the two rings, i. e. κ₁ and κ₂, are different. Also, there are coupling taking place between adjacent cores when they belong to different rings and the coupling coefficient c is not equal to κ₁ or κ₂.

The fiber cores are arranged as the following Fig. 1.

Fig. 1 The arrangement of the fiber array and the coupling between the cores.

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As demonstrated above, the coupled mode equation should be

\begin{array}{l} \frac{d A}{d z} = - j M A \\ M = (\begin{matrix} κ_{1} & C \\ C & κ_{2} \end{matrix}) \end{array}

where the two matrixes κ₁ and κ₂ represent the coupling between the cores inside the rings while the diagonal matrix C stands for the coupling between the cores within different rings and it is proportional to the identity matrix. Since the eigen vectors of κ₁ and κ₂ are not related to the values of coupling coefficients κ₁ and κ₂, they can be written as

\begin{array}{l} κ_{1} = Q D_{1} Q^{H} \\ κ_{2} = Q D_{2} Q^{H} \end{array}

Hence the total coupling matrix M can be formulated as

\begin{array}{l} M = Q_{T o t a l} (\begin{matrix} D_{1} & C \\ C & D_{2} \end{matrix}) Q_{T o t a l}^{H} \\ Q_{T o t a l} = (\begin{matrix} Q & 0 \\ 0 & Q \end{matrix}) \end{array}

The matrix in the middle of the expression can be decomposed into

(\begin{matrix} D_{1} & C \\ C & D_{2} \end{matrix}) = U N U^{H}

where

N = (\begin{matrix} d_{1}^{1} & c \\ c & d_{2}^{1} \\ ⋱ \\ d_{1}^{N} & c \\ c & d_{2}^{N} \end{matrix})

and matrix U is such a matrix which originates from the row exchange of the identity matrix. The k^th row of the identity matrix is exchanged with the (2k-1)^th row when k is between [1, N] or is exchanged with the 2(k-N)^th row when k is between [N + 1, 2N]

The matrix N is a symmetric matrix and can be further decomposed into

N = V Λ V^{H}

The diagonal matrix Λ has the elements corresponding to the eigen values of M and N, and its (2n-1)^th element and (2n)^th element are the eigen values of the sub-matrix $(\begin{matrix} d_{1}^{n} & c \\ c & d_{2}^{n} \end{matrix})$ . Therefore, they are the two roots of the following equation

λ^{2} - (d_{1}^{n} + d_{2}^{n}) λ + d_{1}^{n} d_{2}^{n} - c^{2} = 0

and can be calculated as

λ = \frac{(d_{1}^{n} + d_{2}^{n}) \pm \sqrt{{(d_{1}^{n} - d_{2}^{n})}^{2} + 4 c^{2}}}{2}

Hence the propagation constants of the two-ring MCFs are

β + (κ_{1} + κ_{2}) \cos (\frac{2 π (n - 1)}{N}) \pm \sqrt{{(κ_{1} - κ_{2})}^{2} \cos {(\frac{2 π (n - 1)}{N})}^{2} + c^{2}}

The two corresponding eigenvectors for the sub-matrix are

\begin{array}{l} \sqrt{\frac{c^{2}}{2 k^{2} + 2 c^{2} \pm 2 k \sqrt{k^{2} + c^{2}}}} (\frac{k \pm \sqrt{k^{2} + c^{2}}}{c}, 1) \\ k = (κ_{1} - κ_{2}) \cos (\frac{2 π (n - 1)}{N}) \end{array}

Therefore, the matrix V will be composed by the block matrixes whose elements are formed by the two orthogonal vectors in Eq. (15). Totally, the eigen vectors of matrix M which describes the modal amplitudes distribution within the cores will be the rows of the matrix Q_totalUV. The (2n-1)^th and the (2n)^th eigen vectors, i. e. the corresponding rows of the matrix Q_totalUV, will be the row vectors of the following N × 2 matrix

(\begin{matrix} v_{n}^{11} q_{n} & v_{n}^{12} q_{n} \\ v_{n}^{21} q_{n} & v_{n}^{22} q_{n} \end{matrix})

where v_ij stands for the elements of the sub-matrix of V, which is formulated by Eq. (15). It can be interpreted as the superposition of the original super-modes of the individual rings.

3.2 Two special cases

When c = 0, the two propagation constants in Eq. (14) become

β + (κ_{1} + κ_{2}) \cos (\frac{2 π (n - 1)}{N}) \pm (κ_{1} - κ_{2}) \cos (\frac{2 π (n - 1)}{N})

which are the same as the results indicated by Eq. (3). This can be understood by the fact that when c = 0, the two rings are decoupled and they should act as two individual rings and therefore the propagation constants should obey Eq. (3).

When κ₁, κ₂ and c are equal, Eq. (14) becomes

β + (2 κ_{1} \cos (\frac{2 π (n - 1)}{N})) \pm κ_{1}

This is in agreement with the results in [10], which assumes equal coupling coefficients between the adjacent cores of the two rings as well as the coupling coefficients between the adjacent cores within different rings.

Therefore, the corresponding eigen modes of the whole MCF are

{(\pm \frac{1}{\sqrt{2}} q^{T}, \frac{1}{\sqrt{2}} q^{T})}^{T}

which is in exact agreement with the results in [10].

3.3 Numerical verifications

There have been many proposals for the design and fabrication of multi-core fibers [15–18]. In order to demonstrate the validity of the formulas proposed above, a two-ring MCF was calculated using three dimensional Beam propagation method (3DBPM). The refractive indexes of the cores and the cladding as well as the radii of the cores are exactly the same as those in [15]. The purpose to choose those parameters is to make the example realistic and sound. The core array is illustrated in Fig. 2. The six cores are arranged within two triangles. The distances between the center of cores within the two rings are d₁ = 6.0622μm and d₂ = 17.3205μm respectively. The distances between the adjacent cores within different rings are d = 6.5μm. The diameters of the cores are equal and the value is a = 5μm [15]. The index of the cladding is 1.45 and the core index difference is 1.2%, which are the same as those in [15]. The wavelength of the optical signal is 1550nm.

Fig. 2 The 6-core-MCF core array arrangement.

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To illustrate the validity of the theory, we calculate the effective indexes of the super-modes using Eq. (14) and the BPM. The effective indexes are obtained by dividing the propagation constants of the modes by the free space wave number k₀. The coupling coefficients between the cores used in Eq. (14) are numerically obtained using the overlap integral of the modal fields and the electric permittivity perturbations according to the coupling mode theory [19].

The comparison of the results is illustrated in Table 1. It is clearly demonstrated that the results obtained by Eq. (14) matches with the BPM predications very precisely. The discrepancy between the results is less than 0.1%.

Table 1. Effective Indexes of the Supermodes Obtained by Both Methods

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After calculating the propagation constants of the super-modes, we also investigate the amplitude distribution on each of the cores for the super-modes.

The BPM results for the mode profiles of the super-modes are plotted in Fig. 3. As demonstrated in the figure, the super-modes of the MCF have such properties that the amplitudes of cores within each ring fulfill the same distribution law, which is in accordance with the analysis from Eq. (16). It should be noted that since mode 2, 3 [Fig. 3(b) and Fig. 3(e)] and mode 5, 6 [Fig. 3(c) and Fig. 3(f)] are degenerated modes to each other, the vectors for these modes are not unique and they can be the superposition of the corresponding degenerated mode vectors derived from Eqs. (15) and (16). To further verify the theory, we evaluate super-modes analytically and plot them in Fig. 4. The single modes for each core were multiplied with the corresponding elements of the super-mode vectors derived from Eqs. (5), (15), and (16). The super-modes vectors for the degenerated modes are not unique and they are chosen carefully to match the BPM results. The results in Fig. 4 precisely coincide with the ones obtained from the BPM simulations in Fig. 3 and clearly indicate the effectiveness of the method proposed in this paper.

Fig. 3 The supermodes of the 6-core-MCF obtained by BPM. (a-f) mode profiles of the supermodes 1-6.

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Fig. 4 The supermodes of the 6-core-MCF obtained by the analytical formulas (a-f) mode profiles of the supermodes 1-6.

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4. MCFs with cores in three rings

If the MCFs cores are distributed in three rings, the coupling matrix M inside the coupled equations becomes

M = (\begin{matrix} κ_{1} & C_{12} & 0 \\ C_{21} & κ_{2} & C_{23} \\ 0 & C_{32} & κ_{3} \end{matrix})

where C_ij denotes the coupling between the i^th ring and the j^th ring and C_ij = C_ji. Similar to the two-ring case, we have

M = (\begin{matrix} Q \\ Q \\ Q \end{matrix}) (\begin{matrix} D_{1} & C_{12} & 0 \\ C_{21} & D_{2} & C_{23} \\ 0 & C_{32} & D_{3} \end{matrix}) (\begin{matrix} Q^{H} \\ Q^{H} \\ Q^{H} \end{matrix})

where Q is an orthogonal matrix and D_i a diagonal matrix. The matrix in the middle of the expression can be decomposed into

(\begin{matrix} D_{1} & C_{12} & 0 \\ C_{21} & D_{2} & C_{23} \\ 0 & C_{32} & D_{3} \end{matrix}) = U N U^{H}

where N is a block matrix

N = (\begin{matrix} d_{1}^{1} & c_{12} \\ c_{21} & d_{2}^{1} & c_{23} \\ c_{32} & d_{3}^{1} \\ ⋱ \\ d_{1}^{N} & c_{12} \\ c_{21} & d_{2}^{N} & c_{23} \\ c_{32} & d_{3}^{N} \end{matrix})

and matrix U is such a matrix which originates from the row exchange of the identity matrix. The k^th row of the identity matrix is exchanged with the (3k-2)^th row when k is between [1, N], with the (3(k-N)-1) ^th row when k is between [N + 1, 2N] or with the 3(k-2N) ^th row when k is between [2N + 1, 3N].

Similar to the two-ring case, the block matrix N has the eigen values and the eigen vectors are related to the sub-matrix of

(\begin{matrix} d_{1}^{n} & c_{12} \\ c_{21} & d_{2}^{n} & c_{23} \\ c_{32} & d_{3}^{n} \end{matrix})

Noting that c_ij = c_ji, the characteristic polynomial for the above matrix can be formulated as

\begin{array}{l} λ^{3} + A λ^{2} + B λ + C = 0 \\ A = - (d_{1}^{n} + d_{2}^{n} + d_{3}^{n}) \\ B = (d_{1}^{n} d_{2}^{n} + d_{1}^{n} d_{3}^{n} + d_{2}^{n} d_{3}^{n} - c_{12}^{2} - c_{23}^{2}) \\ C = - d_{1}^{n} d_{2}^{n} d_{3}^{n} + c_{23}^{2} (d_{1}^{n} + d_{3}^{n}) \end{array}

It is generally known that the three roots of Eq. (25) can be obtained analytically as [20]

λ_{k} = - \frac{1}{3} (b + u_{k} D + \frac{Δ_{0}}{u_{k} D})

where [20]

\begin{array}{l} u_{1} = 1, u_{2} = \frac{- 1 + j \sqrt{3}}{2}, u_{3} = \frac{- 1 - j \sqrt{3}}{2} \\ D = \sqrt[3]{\frac{Δ_{1} + \sqrt{Δ_{1}^{2} - 4 Δ_{0}^{3}}}{2}} \\ Δ_{0} = B^{2} - 3 c \\ Δ_{1} = 2 B^{3} - 9 B C + 27 d \end{array}

After obtaining the eigen values, the eigen vectors of the sub-matrix can also be obtained analytically. Therefore, the propagation constants and the eigen mode vectors can be obtained similarly to the two-ring case. The new eigen vectors are still the super-position of the eigen vectors which originate from the coupling matrixes of the individual rings.

5. MCFs with cores in L rings

5.1 Mathematical formulation

For MCFs with cores distributed in L rings, the coupling matrix becomes

M = (\begin{matrix} κ_{1} & C_{12} \\ C_{21} & κ_{2} & C_{23} \\ ⋱ \\ C_{L, L - 1} & κ_{L} \end{matrix})

where C_ij denotes the coupling matrix between the i^th ring and the j^th ring and C_ij = C_ji. The matrix M can be decomposed as

M = U N U^{H}

where N is a block matrix formed by the sub-matrixes with the form of

(\begin{matrix} d_{1}^{n} & c_{12} \\ c_{21} & d_{2}^{n} & c_{23} \\ ⋱ \\ c_{L, L - 1} & d_{L}^{n} \end{matrix})

where c_ij = c_ji. The above matrix is a Hermitian/symmetrical matrix with real eigen values. It is also a band matrix with the characteristic polynomial of order L. Its eigen values and eigen vectors can be obtained using the fast numerical methods and quite a few publications have focused on this topic [21].

The propagation constants will be β plus the eigen values of Eq. (30) and the super-modes amplitudes distribution vectors will be the super-position of eigen vectors which originate from the coupling matrixes of the individual rings, and the coefficients for the superposition will be determined by the eigen vectors of the sub-matrix in Eq. (30).

5.2 A special case when the radii of the rings are very large

When the radii of the rings are very large, the coupling between the adjacent cores can be considered approximately equal. Therefore, the matrix in Eq. (30) becomes

(\begin{matrix} 2 κ \cos (\frac{2 π (n - 1)}{N}) & κ_{a} \\ κ_{a} & 2 κ \cos (\frac{2 π (n - 1)}{N}) & κ_{a} \\ ⋱ \\ κ_{a} & 2 κ \cos (\frac{2 π (n - 1)}{N}) \end{matrix})

where κ denotes the coupling coefficient inside individual rings and κ_a denotes the coupling coefficient between the adjacent rings. The m^th eigen value of the above equations is

2 κ \cos (\frac{2 π (n - 1)}{N}) + 2 κ_{a} \cos (\frac{π m}{L + 1})

and the corresponding eigen vectors are

\sqrt{\frac{2}{L + 1}} {(\sin (\frac{π m}{L + 1}), \sin (\frac{π 2 m}{L + 1}), \dots \sin (\frac{π l m}{L + 1}))}^{T}

which means modal amplitudes distribution fulfills the sinusoidal function in the radial direction, while angular direction amplitudes distribution obeys Eqs. (4) and (5).

6. MCFs with a core in the middle of the rings

6.1 Mathematical formulation

In order to increase the transmission capacity, an additional core is usually added at the center of the rings. The coupling matrix for the MCFs with a core in the middle of the rings is

M = (\begin{matrix} 0 & b^{T} \\ b & κ_{1} & C_{12} \\ C_{21} & κ_{2} & C_{23} \\ ⋱ \\ C_{L - 1, L} & κ_{L} \end{matrix})

where b = b₀(1,1,….,1)^T stands for the coupling between the center core and the cores inside the first ring.

As indicated by [14], the matrix

M_{1} = (\begin{matrix} 0 & b^{T} \\ b & Q D_{1} Q^{H} \end{matrix})

has N-1 same eigen values as κ₁, while the rest two eigen values can be calculated as [13]

κ_{1} \pm \sqrt{κ_{1}^{2} + N b_{0}^{2}}

The N-1 corresponding eigen vectors remain the same as the ones of matrix κ₁ except an additional 0 element while the rest two eigen vectors can be formulated as [13]

{(\frac{- κ_{1} \pm \sqrt{κ_{1}^{2} + N b_{0}^{2}}}{b_{0}}, 1, 1, \dots, 1)}^{T}

After normalization, they can form the two columns of the orthogonal matrix Q' and we have M₁ = Q'D₁Q'^H.

Hence, the coupling matrix M in Eq. (33) can be decomposed as

M = (\begin{matrix} Q' \\ Q \\ ⋱ \\ Q \\ Q \end{matrix}) (\begin{matrix} D_{1} & C'^{H} \\ C' & D_{2} & C_{23} \\ ⋱ \\ C_{L - 1, L - 2} & D_{L - 1} & C_{L - 1, L} \\ C_{L, L - 1} & D_{L} \end{matrix}) (\begin{matrix} Q'^{H} \\ Q^{H} \\ ⋱ \\ Q^{H} \\ Q^{H} \end{matrix})

where Q is the matrix whose rows are the eigen vectors of the individual rings coupling matrix, Q' is the matrix whose rows are the eigen vectors of coupling matrix of the center core and the cores inside the first ring, i. e. matrix M₁. Matrix C' is

\begin{array}{l} C' = Q^{H} (\begin{matrix} 0 & c_{1, 2} \\ ⋮ & ⋱ \\ 0 & c_{1, 2} \end{matrix}) Q' = (\begin{matrix} c'_{+} & c'_{-} \\ ⋮ & ⋱ \\ 0 & c_{1, 2} \end{matrix}) \\ c'_{\pm} = c_{1, 2} \sqrt{\frac{N}{N + {(\frac{- κ_{1} \pm \sqrt{κ_{1}^{2} + N b_{0}^{2}}}{b_{0}})}^{2}}} \end{array}

and the matrix in the center of Eq. (38) can be decomposed into UNU^H, where the matrix N should be formed by the block matrixes indicated by Eq. (30) for n = 2…N while the first block matrix is

(\begin{matrix} d_{1 -}^{1} & c'_{-} \\ d_{1 +}^{1} & c'_{+} \\ c'_{-} & c'_{+} & d_{2}^{1} \\ ⋱ \\ c_{L - 1, L - 2} & d_{L - 1}^{1} & c_{L - 1, L} \\ c_{L, L - 1} & d_{L}^{1} \end{matrix})

where the first two diagonal elements in Eq. (39) are calculated from Eq. (35). From Eq. (39), it can be concluded that the last (N-1)L eigen values of the matrix N remain unchanged in comparison with the results from Eq. (30) and so do the corresponding eigen vectors except an additional 0. The rest L + 1 eigen values can be found by solving the eigen values of Eq. (39). The corresponding eigen vectors can be obtained respectively according to these eigen values.

6.2 An example

To verify the theory and formulas proposed in this section, we propose to investigate a MCF with two rings and a center core. The MCF is composed of eleven cores, with five in each of the rings and one in the middle of the array. Without loss of generality, the coupling coefficients are assumed to be 7c for the inner ring and 3c for the outer ring. As indicated in Fig. 5, the coupling coefficient between the adjacent cores in different rings is 6c and the coupling coefficient between the center core and the cores within the inner ring is 10c.

Fig. 5 The 11-core-MCF core array arrangement and the assumed coupling coefficients between the cores.

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To verify the assertion that the MCF with a core in the middle of the ring and MCF with no center core share (N-1)L identical eigen values, we also investigate a MCF similar to that in Fig. 4 but with no center core in the middle.

The eigen values are computed directly from the expression of the two coupling matrixes of the MCFs using the numerical approach and they are listed in Table 2. It can be seen clearly that the two matrixes have many common eigen values except three unique eigen values for the 1st matrix, which is in agreement with the conclusion above. The eigen values can also be obtained by Eq. (14), while the three unique eigen values can be calculated by Eq. (39) with exactly the same results as Table 2.

Table 2. Eigen Values of theTwo Coupling Matrix of the MCFs with and without a Center Core

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7. Case of strong coupling among the cores

The analysis in this work has been based on the assumption that the coupling only takes place between adjacent cores, which is a common practice during the analysis of the super-modes of the MCFs [5, 6, 8, 10–14]. In the case of very close core arrangement, coupling may take place between non-adjacent cores. Furthermore, the coupled mode theory is accurate in the case of weak/moderate coupling, but the accuracy degrades when the coupling is strong [19]. Therefore, the accuracy of the theory presented here will be impacted when the cores of the MCFs are very close to each other due to these two reasons. The discrepancies arise from the above factors can be studied by comparing the results from the analytical formulas with the results from the numerical simulations, which take all the coupling effects into account [22] and maintain the accuracy in the case of strong coupling.

To investigate the impact of non-adjacent core coupling and strong coupling, a strongly coupled six-core MCF is studied. The six-core fiber has the same cladding and core indexes as the case in section 3.3. The cores have the diameters of 5μm and the distances between the cores are 5.5μm, i.e. the gaps between the adjacent cores are only 0.5μm. The six cores are arranged in one ring as a diamond lattice. The effective indexes of the super-modes can be estimated by Eq. (3) under the adjacent coupling assumption. Numerical simulations based on the BPM and the results estimated by the theory are compared in Fig. 6. The effective indexes of the 1st order and the 2nd order modes are plotted as a function of wavelength. The figure clearly demonstrates the accuracy of the theory in comparison with the numerical counterparts. When the gap size decreases to 0μm, the inaccuracy rises, as demonstrated in Fig. 7.

Fig. 6 The effective indexes of the super-modes of the six-core MCF. (gap size = 0.5μm).

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Fig. 7 The effective indexes of the super-modes of the six-core MCF. (gap size = 0μm).

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To investigate the impact of the gap size/coupling strength between cores on the accuracy of the formulas, the effective indexes of the super-modes versus the gap size as well as the coupling coefficient are plotted in Fig. 8 and Fig. 9. It should be noted that the signal wavelength is 1550nm and the coupling coefficient has been normalized over the free space wave number in Fig. 9. The figures demonstrate that the discrepancy increases as the gap decreases or the coupling coefficient increases. But even in the case of gap size = 0μm, the proposed formulas can still predict the refractive index of the super-modes, albeit with some discrepancies. Although not presented here, it can be noted that the mode amplitude distribution among cores fulfills Eq. (5) with acceptable discrepancies when the gap size is reduced to 0. Therefore, the formulas derived in this work can still provide some guidance in the case of strong coupling for the effective index and the mode field distribution of the super-modes, albeit with reduced accuracies.

Fig. 8 The effective indexes of the super-modes of the six-core MCF VS Gap size.

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Fig. 9 The effective indexes of the super-modes of the six-core MCF VS Normalized coupling coefficient (the coupling coefficient over the free space wave number).

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To further investigate the impact of coupling between non-adjacent cores, a four-core MCF with cores in a square lattice is investigated. MCF with four cores in a ring has the strongest non-adjacent core coupling in comparison with other structures. The four-core MCF for the simulation has the same core/cladding indexes and the core diameters as those of the six-core MCF. The gap size is first assumed to be 0.5μm. The effective indexes obtained from both methods are listed in the following Table 3 when the signal wavelength is at 1550nm. The inaccuracy increases when the gap size is reduced to 0 as illustrated in Table 4. The discrepancy is comparative with the six-core case.

Table 3. Effective Indexes of the Super-modes of the Four-core MCF (gap size = 0.5μm)

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Table 4. Effective Indexes of the Super-modes of the Four-core MCF (gap size = 0μm)

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7. Conclusion

We have analyzed the MCFs with circularly distributed cores. Analytical expressions have been given for the two-ring, the three-ring as well as the multiple-ring cases. Furthermore, the case of the MCFs with an additional core at the center of the circular fiber array is also investigated. One is going to be beneficiary from the expressions derived in this paper during the design and analysis of MCFs.

Acknowledgments

The author would like to thank the three anonymous reviewers and the associated editor for their valuable suggestions during the review process. This work is partially supported by the Fundamental Research Funds for the Central Universities of China and National science foundation of China (Grant No. 61201068).

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Super-mode number	BPM results	Results from Eq. (14)
1	1.4603	1.4602
2	1.4590	1.4590
3	1.4590	1.4590
4	1.4586	1.4586
5	1.4577	1.4577
6	1.4577	1.4577

Eigen values of the coupling matrix with a center core	Eigen values of the coupling matrix of the two ring MCF
−16.9918c
−14.9072c	−14.9072c
−14.9072c	−14.9072c
−3.0358c	−3.0358c
−3.0358c	−3.0358c
−1.2731c	−1.2731c
−1.2731c	−1.2731c
5.6296c	2.7889c
9.2162c	9.2162c
9.2162c	9.2162c
31.3622c	17.2111c

Mode number	BPM results	Results from Eq. (3)
1	1.4610	1.4609
2	1.4588	1.4587
3	1.4588	1.4587
4	1.4566	1.4565

Mode number	BPM results	Results from Eq. (3)
1	1.4621	1.4619
2	1.4591	1.4587
3	1.4591	1.4587
4	1.4558	1.4555

Super-mode number	BPM results	Results from Eq. (14)
1	1.4603	1.4602
2	1.4590	1.4590
3	1.4590	1.4590
4	1.4586	1.4586
5	1.4577	1.4577
6	1.4577	1.4577

Analytical formulation of super-modes inside multi-core fibers with circularly distributed cores

Abstract

1. Introduction

2. MCFs with cores in one ring

3. MCFs with cores in two rings

3.1 Mathematical formulation

3.2 Two special cases

3.3 Numerical verifications

4. MCFs with cores in three rings

5. MCFs with cores in L rings

5.1 Mathematical formulation

5.2 A special case when the radii of the rings are very large

6. MCFs with a core in the middle of the rings

6.1 Mathematical formulation

6.2 An example

7. Case of strong coupling among the cores

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Tables (4)

Equations (39)

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