Analytical formulation of super-modes inside multi-core fibers with circularly distributed cores Junhe Zhou* Dept. of electronics science and engineering, Tongji University, Shanghai 200092, China * [email protected]

Abstract: In this paper, super-modes inside multi-core fibers with circularly distributed cores are analyzed in detail. Cores are arranged within one ring, two rings, and multiple rings. Also, MCFs with a center core embedded inside the rings are discussed. In these analyses, analytical formulas are derived for the propagation constants as well as the modal distribution vectors of the super-modes. ©2014 Optical Society of America OCIS codes: (060.0060) Fiber optics and optical communications; (060.4005) Microstructured fibers.

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Received 7 Oct 2013; revised 26 Nov 2013; accepted 16 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000673 | OPTICS EXPRESS 673

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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].

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Received 7 Oct 2013; revised 26 Nov 2013; accepted 16 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000673 | OPTICS EXPRESS 674

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 akexp(-jβz) [8], where ak is the amplitude of the field at the kth 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 dA = − jκA dz  a1    A=   a   N

(1)

κ 0 κ   κ 0 κ   κ =      κ 0 κ  κ κ 0   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 QDQH, 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 ( − jDL ) Q H A ( 0 ) (2) Obviously, the columns of the matrix Q form the eigen mode vectors of the super-modes and β + dn serve as the propagation constants of the corresponding super-modes, where dn is the nth 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

 2π ( n − 1)   N  

β + 2κ cos 

(3)

where n is the order of the super-modes and the element of matrix Q is [8] 1 2π   (4) exp  − j ( m − 1)( n − 1)  N 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 Qmn =

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Received 7 Oct 2013; revised 26 Nov 2013; accepted 16 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000673 | OPTICS EXPRESS 675

when N is an even number  1  N   2 2π   cos  ( m − 1)( n − 1)   N N    Qmn =  1 m −1  ( −1)  N   2 2π    N sin  ( m − 1)( N − n + 1) N     when N is an odd number

n =1 1< n ≤

N 2

n = 1+

N 2

N +1 < n ≤ N 2

(5)

 1 n =1  N   2 2π  N +1  Qmn =  cos  ( m − 1)( n − 1) 1< n ≤  N N 2     2 2π  N + 1  