In this paper, a non-orthogonal coupled mode theory is proposed to analyze the super-modes of multi-core fibers (MCFs). The theory is valid in the strong coupling regime and can provide accurate analytical formulas for the super-modes inside MCFs. MCFs with circularly distributed cores are analyzed as an example. Analytical formulas are derived both for the refractive indexes and the eigen vectors of the super-modes. It is rigorously revealed that the eigen vectors for the super-modes of such MCFs are the row vectors of the inverse discrete Fourier transform (IDFT) matrix. Therefore, by pre-coding the signal channels via IDFT, one is able to generate the super-modes for the MCFs with circularly distributed cores.
© 2014 Optical Society of America
Multi-core fibers (MCFs) have recently aroused significant attentions among the industrial/research communities due to its higher information capacity in comparison with its single core counterpart . To transmit signals in MCFs, one can either use individual cores provided that the cores are separated far enough to avoid crosstalk , or use the super-modes among the cores to transmit signals [3,4]. The later approach can enable a denser arrangement of the cores and therefore is able to provide a higher capacity in one fiber in comparison with the former. Apart from the optical communication systems [1–6], MCFs are also useful in lasing [7–9], switching  and fiber endoscope .
There have been quite a few publications studying the properties of the super-modes inside MCFs. Some are based on the numerical simulations [3–7,11–13] while others focus on the analytical solutions [8,10,14,15]. In , the super-modes of a linear fiber core array are derived. It is found that the super-modes vectors obey the sinusoidal function . In [10,14,15], the circularly arranged fiber core array is solved analytically with the super-modes to be row vectors of the inverse discrete Fourier transform (IDFT) matrix [10,14,15]. In comparison with the numerical approach, the analytical approaches give a deeper physical insight into the design and analysis. However, these analytical solutions are usually based on the simplified conventional coupled mode theory (CMT) [8,10,14,15], which only considers adjacent-core coupling and assumes orthogonality between the modes. These two assumptions are not valid in the strong coupling regime, which is usually the case for MCFs designed for super-modes multiplexing as well as other applications. Therefore, the conclusions derived in  and [10,14,15] need to be reexamined carefully when the cores are closely arranged. Apart from the super-modes vectors, the propagation constants are also important, especially during the evaluation of the modal dispersion. Obviously, the conventional CMT will give a deviated estimation for the propagation constants.
Along with the analysis of the super-modes inside the multi-core fibers, methods for super-modes multiplexing have been proposed. F. Saitoh et. al.  proposed a mode index matching method to form a multiplexer via embedded waveguides near the multiple cores of the MCF. Y. Kokubun et. al  proposed to multiplex the signals using arrayed waveguide gratings (AWGs). J. Zhou et. al.  discussed a MMI coupler based MCF super-modes multiplexer (MUX). These proposals are mostly designed for MCFs with linearly aligned cores [6,12,16] and are physically realized via the fiber technology or the planar lightwave circuit (PLC) technology.
In this paper, we propose to analyze MCFs using a rigorous non-orthogonal CMT [17,18], which is valid in the strong coupling regime. As an example, MCFs with circularly arranged cores are analyzed. The analytical formulas for the refractive index and eigen modes of super-modes of the circularly aligned MCFs are derived and they are consistent with the existing theory. The results show the super-modes vectors are strictly the row vectors of the IDFT matrix when the cores are circularly arranged even in the case of strong coupling. Numerical simulations are performed to verify the formulas with excellent agreement.
Based on this conclusion, a digital signal processing (DSP) technique based on the IDFT algorithm is proposed for the signal channel pre-coding. The method does not require any physical components to act as the MCF super-modes MUXs and can greatly ease the super-modes generation. The method can be extended for circularly aligned MCFs within multiple rings  as well as the linearly aligned MCFs, which will be discussed elsewhere.
2. A rigorous non-orthogonal coupled mode theory
The analytical analysis of the super-modes inside MCFs usually relies on the simplified conventional CMT [8,10,14,15]. According to the assumption of simplified conventional CMT, the cross-power term is neglected, so is the coupling between non-adjacent cores. This will result in discrepancies in describing the super-modes [17–20]. Therefore, the conventional CMT is only accurate in the weakly coupling regime [17–19]. To extend the CMT to the strongly coupling regime and get accurate analytical formulas for the super-modes inside MCFs, a theory based on the non-orthogonal CMT  will be derived. The theory takes the cross-power term as well as the coupling between non-adjacent cores into account and is valid in the strong coupling regime as demonstrated below.
The guided super-modes of the MCF should satisfy the scalar Holmholtz equation 13]13].
Since the cores are single mode, φn can be treated as a real function . Multiplying φm on both sides of Eq. (4) and integrating over the x-y plane, Eq. (4) can be recast into the following coupled equation 
The theory can be used to analyze the super-modes of the MCFs and provide analytical expressions as demonstrated below.
3. Analytical formulation of super-modes inside circularly aligned MCFs
Assuming that the MCFs are composed of N equally spaced and circularly aligned single mode fiber cores, we have
Therefore, it is possible to define E1…EN, and S1…SN, such as21]
Hence Eq. (5) can be rewritten as
Therefore, the eigen vectors which describe the amplitudes distribution for super-modes will be the row vectors of Q and the propagation constants for the N super-modes should be calculated as
Since the neighboring coupling/mode overlapping coefficients are equal19]
When N is an even number
When N is an odd number
4. Comparison with the published results
To verify the proposed theory, we compare the derived formulas with the published results. In the case of two-core MCF, according to Eq. (14), the effective indexes of the two super-mode modes can be written as
Equation (18) is in exact agreement with the results in  (Eq. (2).1a-2.1b)), which was derived for the effective indexes of the super-modes resulting from the coupling between two closely aligned identical waveguides based on the non-orthogonal CMT.
4. Numerical simulations and discussions
To demonstrate the formulas and method derived in the above two sections, numerical simulations based on the three dimensional beam propagation method (3DBPM) is used to simulate a closely aligned six core MCF. The cross section of the MCF is demonstrated in Fig. 1.
The cladding index is 1.45 and the index difference for the core with respect to the cladding is 1.2%, the radii of the cores are 2.5μm. The index values and the radii are from  to make the example more sound and realistic. The distances between the centers of the cores are 5μm and therefore they are closely aligned and there are no gaps between the cores. As demonstrated in , such a core arrangement will result in a strong coupling among the cores and the conventional CMT will have a quite significant discrepancy during the evaluation of the effective indexes of the super-modes.
The effective refractive indexes of super-modes calculated by the 3DBPM, by Eq. (14) as well as by the simplified conventional CMT are compared in Table 1.The comparison shows the validity of the theory and the greatly improved accuracy using the non-orthogonal CMT in comparison with the simplified conventional CMT which only considers the neighboring coupling effect.
Afterwards, the super-modes fields obtained from the BPM results and the analytical expressions are compared in Fig. 2 and Fig. 3.Since the BPM simulation usually results in the super-mode fields whose phases are identical along the x-y plane, Eq. (17) is used instead of Eq. (10). The correlation coefficients between the mode pairs in Fig. 2 and Fig. 3, which are defined as the normalized overlap integrals, are calculated as 1, 0.99, 0.98, 0.99, 0.93 and 0.92. Therefore, very good agreement is reached by comparing Fig. 2 with Fig. 3.
To further verify the validity of the formulas, a four-core MCF packed in a circular array is investigated. The radii of the cores are 2.5μm. The MCF has cladding index of 1.45, and the index difference for the core with respect to the cladding is 0.6%, indicating a lower refractive index contrast. There are no gaps between the cores and hence they are very tightly packed. The effective refractive indexes of the three super-modes are calculated via BPM simulations and Eq. (14). They are listed in Table 2 and it can be seen that effective refractive indexes from Eq. (14) match well with the BPM results.The reason for the four-core MCF to possess only three super-modes is that the interaction between the modal fields becomes very strong when the phase difference between the adjacent cores is π. Such a mode (a mode with the phase difference between the adjacent cores to be π) cannot exist in the four-core MCF. This fact is also confirmed by the simulations based on finite element method (FEM). Even in such an extreme case, the formulas do provide an accurate estimation for effective refractive indexes the existing super-modes.
5. Signal pre-coding based on the concept
As analyzed above, the eigen vectors, which describes the super-mode distribution inside the MCFs with circularly arranged cores, are the row vectors of the IDFT matrix. Therefore, we propose to process the data streams, which are expected to be transmitted in the MCF, by the IDFT algorithm. Therefore, at the transmitter side, the signal vector g are reformulated as
These signals are injected into the corresponding cores of the MCFs and they will transmit as the supermodes of the MCF. After propagating in the MCF with the length L, they becomeFig. 4.
The multiplexing techniques are illustrated as follows. The MCF to be simulated is the six-core MCF discussed in the previous section. The signal wavelength is still 1550nm. The input vector is assumed to be [1 0 0 0 0 0]. After IDFT, it becomes the coefficients of the single modes of each core. The input signal field is illustrated in Fig. 5(a) and the output signals after propagating inside the fiber for 1m, 5m and 10m are plotted in Figs. 5(b)–5(d). It can be seen from Fig. 5 that the input and the output signals actually act as the fundamental super-mode of the MCF, and they remain unchanged after propagation. This clearly demonstrates the proposed pre-coding concept.
It should be noted that due to the degeneracy among the super-modes, there might be crosstalk between the degenerated super-modes. This problem, however, can be solved via DSP algorithms like constant modulus algorithms (CMA).
6. De-multiplexing the signals based on IDFT
To de-multiplex the signals, one may fabricate a tapered fiber which can gradually separate the cores embedded inside the MCF. If the distances between the cores maintain equal, as discussed in the previous section, the super-modes vectors will remain as the row vectors of the IDFT matrix. Therefore, the super-modes information will be preserved when the cores are separated. The information on each core can be detected by a photodiode (PD) array when the distances between the cores are large enough. If coherent detection scheme is used, one is able to detect both the amplitudes as well as the phases of the modes on each core of the MCF. After doing IDFT on the detected signal, the original signal can be recovered. The schematic for the de-multiplexing technique is illustrated in Fig. 6.
As a proof of the concept, a tapered fiber with the tapering length of 10000μm is simulated. The taper has the same core and cladding indexes. At the input end, the cores are closely arranged while at the output end, the adjacent cores have the distance of 20μm. The fundamental super-mode of the six-core MCF is assumed to be launched into the taper. The field evolution inside it is demonstrated in the Figs. 7(a)–7(d). It can be seen that the super-mode is separated into the individual modes of the cores and can be detected by the receiver array at the output. The detected output vector is processed by IDFT and the final results can be calculated as [1 0 0 0 0 0].
In summary, a rigorous non-orthogonal CMT has been used to analyze the circularly aligned MCFs. Accurate analytical formulas are derived for the eigen vectors as well as the eigen values of the super-modes in such MCFs. Those formulas are valid in the strong coupling regime and are verified by the numerical simulations. It is revealed that the eigen vectors of the super-modes coincides with the row vectors of the IDFT matrix. Based on this discovery, a multiplexing method is proposed based on the IDFT algorithm. A proof of concept for the method is demonstrated via BPM simulations. It should be noted that the results presented here can be extended to circularly aligned MCFs within multiple rings  as well as the linearly aligned MCFs. These studies will be presented elsewhere.
The author would like to thank the three anonymous reviewers for their valuable suggestions during the review process. This work is partially supported by the National science foundation of China (Grant No. 61201068).
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