A non-orthogonal coupled mode theory for super-modes inside multi-core fibers Junhe Zhou* Dept. of Electronics Science and Engineering, Tongji University, Shanghai 200092, China * [email protected]
Abstract: 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 OCIS codes: (060.0060) Fiber optics and optical communications; (060.4005) Microstructured fibers.
References and links 1. 2. 3.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000). T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Ultra-low-crosstalk multi-core fiber feasible to ultra-long-haul transmission,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2011, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC2. C. Xia, N. Bai, R. Amezcua-Correa, E. Antonio-Lopez, A. Schulzgen, M. Richardson, X. Zhou, and G. Li, “Supermodes in strongly-coupled multi-core fibers,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper OTh3K.5. C. Xia, N. Bai, I. Ozdur, X. Zhou, and G. Li, “Supermodes for optical transmission,” Opt. Express 19(17), 16653–16664 (2011). K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, S. Matsuo, K. Saitoh, and M. Koshiba, “A large effective area multi-core fiber with an optimized cladding thickness,” Opt. Express 19(26), B543–B550 (2011). F. Saitoh, K. Saitoh, and M. Koshiba, “A design method of a fiber-based mode multi/demultiplexer for modedivision multiplexing,” Opt. Express 18(5), 4709–4716 (2010). Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010). E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9(4), 125–127 (1984). T. Mansuryan, Ph. Rigaud, G. Bouwmans, V. Kermene, Y. Quiquempois, A. Desfarges-Berthelemot, P. Armand, J. Benoist, and A. Barthélémy, “Spatially dispersive scheme for transmission and synthesis of femtosecond pulses through a multicore fiber,” Opt. Express 20(22), 24769–24777 (2012). J. Hudgings, L. Molter, and M. Dutta, “Design and modeling of passive optical switches and power dividers using non-planar coupled fiber arrays,” IEEE J. Quantum Electron. 36(12), 1438–1444 (2000). K. L. Reichenbach and C. Xu, “Numerical analysis of light propagation in image fibers or coherent fiber bundles,” Opt. Express 15(5), 2151–2165 (2007). Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: Proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009). A. Mafi and J. Moloney, “Shaping modes in multicore photonic crystal fibers,” IEEE Photon. Technol. Lett. 17(2), 348–350 (2005). Y. C. Meng, Q. Z. Guo, W. H. Tan, and Z. M. Huang, “Analytical solutions of coupled-mode equations for multiwaveguide systems, obtained by use of Chebyshev and generalized Chebyshev polynomials,” J. Opt. Soc. Am. A 21(8), 1518–1528 (2004). S. Peleš, J. L. Rogers, and K. Wiesenfeld, “Robust synchronization in fiber laser arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026212 (2006). J. Zhou and P. Gallion, “A novel mode multiplexer/de-multiplexer for multi-core fibers,” IEEE Photon. Technol. Lett. 25(13), 1214–1217 (2013).
#207471 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 21 Apr 2014; accepted 22 Apr 2014; published 28 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010815 | OPTICS EXPRESS 10815
17. B. E. Little and W. P. Huang, “Coupled-mode theory for optical waveguides,” Prog. Electromagn. Res. 10, 217– 270 (1995). 18. W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994). 19. J. Zhou, “Analytical formulation of super-modes inside multi-core fibers with circularly distributed cores,” Opt. Express 22(1), 673–688 (2014). 20. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62(11), 1267–1277 (1972). 21. R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2(3), 155–239 (2005).
1. Introduction 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 [1]. To transmit signals in MCFs, one can either use individual cores provided that the cores are separated far enough to avoid crosstalk [2], 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 [10] and fiber endoscope [11]. 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 [8], the super-modes of a linear fiber core array are derived. It is found that the super-modes vectors obey the sinusoidal function [8]. 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 [8] 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. [6] proposed a mode index matching method to form a multiplexer via embedded waveguides near the multiple cores of the MCF. Y. Kokubun et. al [12] proposed to multiplex the signals using arrayed waveguide gratings (AWGs). J. Zhou et. al. [16] 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 supermodes 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 supermodes generation. The method can be extended for circularly aligned MCFs within multiple rings [19] as well as the linearly aligned MCFs, which will be discussed elsewhere.
#207471 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 21 Apr 2014; accepted 22 Apr 2014; published 28 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010815 | OPTICS EXPRESS 10816
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 supermodes inside MCFs, a theory based on the non-orthogonal CMT [17] 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] ∇ 2ϕ + k 2ϕ = 0
(1)
and they can be expanded by the modes of the individual single mode cores [13] N
ϕ = cnϕ n e j β z
(2)
n =1
where φn stands for the mode of each individual core, cn the slowly varying amplitude of the mode, β the propagation constant, and they satisfy the following equation [13]. ∇ ⊥ 2ϕ n + ( kn 2 − β 2 ) ϕ n = 0
(3)
where kn stands for the propagation constants in the free space multiplied with the local refractive indexes of the each single mode core and the cladding, the symbol ∇ ⊥ 2 stands for the two dimensional Laplace operator. Substituting Eqs. (2) and (3) into Eq. (1) and using the paraxial approximation, the following equation can be derived N j ( k 2 − kn2 ) ∂cn ϕ n = cn ϕn (4) 2β n =1 ∂z n =1 Since the cores are single mode, φn can be treated as a real function [9]. 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 [17] N
S
∂c = jEc ∂z
(5)
where the vector c has its ith elements as ci and the two matrixes E and S have the elements as Emn =
+∞ +∞
(k
−∞ −∞
S mn =
2
− kn2 )
2β
ϕmϕn dxdy
+∞ +∞
(6)
ϕ ϕ dxdy m
n
−∞ −∞
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
#207471 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 21 Apr 2014; accepted 22 Apr 2014; published 28 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010815 | OPTICS EXPRESS 10817
Emn = Em′n′ S mn = Sm′n′
(7)
m − n = m′ − n ′ Therefore, it is possible to define E1…EN, and S1…SN, such as Emn = E m − n +1 S mn = S m − n +1
(8)
The matrixes E and S are circulant matrixes and therefore they can be decomposed as [19,21] E = QGQ H S = QDQ H
(9)
where Q is the normalized IDFT matrix with the elements of Qmn =
2π ( m − 1)( n − 1) exp j N N
1
(10)
and QH is the normalized discrete Fourier transform (DFT) matrix. The two diagonal matrixes G and D have their diagonal elements as [21] N 2π ( n − 1)( m − 1) Gnn = Em exp − j N m =1 N 2π ( n − 1)( m − 1) Dnn = S m exp − j N m =1 Hence Eq. (5) can be rewritten as
∂c = jMc ∂z
(11)
(12)
where M = QD−1GQ H (13) 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
Gnn Dnn Since the neighboring coupling/mode overlapping coefficients are equal
βn = β +
Em = EN − m Sm = S N − m
(14)
(15)
we have
βn = β N −n+2
(16)
which indicates mode n and mode N-n + 2 are degenerated modes. Therefore, matrix Q is not unique. To transform the complex amplitudes of the super-modes into real ones, the following expression for Q was proposed [19]
#207471 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 21 Apr 2014; accepted 22 Apr 2014; published 28 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010815 | OPTICS EXPRESS 10818
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
Qmn
1 N 2 2π = cos ( m − 1)( n − 1) N N 2 2π sin ( m − 1)( N − n + 1) N N
n =1 1< n ≤
N 2
N n = 1+ 2
(17a)
N +1 < n ≤ N 2
n =1 1< n ≤
N +1 2
(17b)
N +1
Abstract: 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 OCIS codes: (060.0060) Fiber optics and optical communications; (060.4005) Microstructured fibers.
References and links 1. 2. 3.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000). T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Ultra-low-crosstalk multi-core fiber feasible to ultra-long-haul transmission,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2011, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC2. C. Xia, N. Bai, R. Amezcua-Correa, E. Antonio-Lopez, A. Schulzgen, M. Richardson, X. Zhou, and G. Li, “Supermodes in strongly-coupled multi-core fibers,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper OTh3K.5. C. Xia, N. Bai, I. Ozdur, X. Zhou, and G. Li, “Supermodes for optical transmission,” Opt. Express 19(17), 16653–16664 (2011). K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, S. Matsuo, K. Saitoh, and M. Koshiba, “A large effective area multi-core fiber with an optimized cladding thickness,” Opt. Express 19(26), B543–B550 (2011). F. Saitoh, K. Saitoh, and M. Koshiba, “A design method of a fiber-based mode multi/demultiplexer for modedivision multiplexing,” Opt. Express 18(5), 4709–4716 (2010). Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010). E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9(4), 125–127 (1984). T. Mansuryan, Ph. Rigaud, G. Bouwmans, V. Kermene, Y. Quiquempois, A. Desfarges-Berthelemot, P. Armand, J. Benoist, and A. Barthélémy, “Spatially dispersive scheme for transmission and synthesis of femtosecond pulses through a multicore fiber,” Opt. Express 20(22), 24769–24777 (2012). J. Hudgings, L. Molter, and M. Dutta, “Design and modeling of passive optical switches and power dividers using non-planar coupled fiber arrays,” IEEE J. Quantum Electron. 36(12), 1438–1444 (2000). K. L. Reichenbach and C. Xu, “Numerical analysis of light propagation in image fibers or coherent fiber bundles,” Opt. Express 15(5), 2151–2165 (2007). Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: Proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009). A. Mafi and J. Moloney, “Shaping modes in multicore photonic crystal fibers,” IEEE Photon. Technol. Lett. 17(2), 348–350 (2005). Y. C. Meng, Q. Z. Guo, W. H. Tan, and Z. M. Huang, “Analytical solutions of coupled-mode equations for multiwaveguide systems, obtained by use of Chebyshev and generalized Chebyshev polynomials,” J. Opt. Soc. Am. A 21(8), 1518–1528 (2004). S. Peleš, J. L. Rogers, and K. Wiesenfeld, “Robust synchronization in fiber laser arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026212 (2006). J. Zhou and P. Gallion, “A novel mode multiplexer/de-multiplexer for multi-core fibers,” IEEE Photon. Technol. Lett. 25(13), 1214–1217 (2013).
#207471 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 21 Apr 2014; accepted 22 Apr 2014; published 28 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010815 | OPTICS EXPRESS 10815
17. B. E. Little and W. P. Huang, “Coupled-mode theory for optical waveguides,” Prog. Electromagn. Res. 10, 217– 270 (1995). 18. W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994). 19. J. Zhou, “Analytical formulation of super-modes inside multi-core fibers with circularly distributed cores,” Opt. Express 22(1), 673–688 (2014). 20. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62(11), 1267–1277 (1972). 21. R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2(3), 155–239 (2005).
1. Introduction 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 [1]. To transmit signals in MCFs, one can either use individual cores provided that the cores are separated far enough to avoid crosstalk [2], 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 [10] and fiber endoscope [11]. 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 [8], the super-modes of a linear fiber core array are derived. It is found that the super-modes vectors obey the sinusoidal function [8]. 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 [8] 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. [6] proposed a mode index matching method to form a multiplexer via embedded waveguides near the multiple cores of the MCF. Y. Kokubun et. al [12] proposed to multiplex the signals using arrayed waveguide gratings (AWGs). J. Zhou et. al. [16] 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 supermodes 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 supermodes generation. The method can be extended for circularly aligned MCFs within multiple rings [19] as well as the linearly aligned MCFs, which will be discussed elsewhere.
#207471 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 21 Apr 2014; accepted 22 Apr 2014; published 28 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010815 | OPTICS EXPRESS 10816
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 supermodes inside MCFs, a theory based on the non-orthogonal CMT [17] 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] ∇ 2ϕ + k 2ϕ = 0
(1)
and they can be expanded by the modes of the individual single mode cores [13] N
ϕ = cnϕ n e j β z
(2)
n =1
where φn stands for the mode of each individual core, cn the slowly varying amplitude of the mode, β the propagation constant, and they satisfy the following equation [13]. ∇ ⊥ 2ϕ n + ( kn 2 − β 2 ) ϕ n = 0
(3)
where kn stands for the propagation constants in the free space multiplied with the local refractive indexes of the each single mode core and the cladding, the symbol ∇ ⊥ 2 stands for the two dimensional Laplace operator. Substituting Eqs. (2) and (3) into Eq. (1) and using the paraxial approximation, the following equation can be derived N j ( k 2 − kn2 ) ∂cn ϕ n = cn ϕn (4) 2β n =1 ∂z n =1 Since the cores are single mode, φn can be treated as a real function [9]. 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 [17] N
S
∂c = jEc ∂z
(5)
where the vector c has its ith elements as ci and the two matrixes E and S have the elements as Emn =
+∞ +∞
(k
−∞ −∞
S mn =
2
− kn2 )
2β
ϕmϕn dxdy
+∞ +∞
(6)
ϕ ϕ dxdy m
n
−∞ −∞
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
#207471 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 21 Apr 2014; accepted 22 Apr 2014; published 28 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010815 | OPTICS EXPRESS 10817
Emn = Em′n′ S mn = Sm′n′
(7)
m − n = m′ − n ′ Therefore, it is possible to define E1…EN, and S1…SN, such as Emn = E m − n +1 S mn = S m − n +1
(8)
The matrixes E and S are circulant matrixes and therefore they can be decomposed as [19,21] E = QGQ H S = QDQ H
(9)
where Q is the normalized IDFT matrix with the elements of Qmn =
2π ( m − 1)( n − 1) exp j N N
1
(10)
and QH is the normalized discrete Fourier transform (DFT) matrix. The two diagonal matrixes G and D have their diagonal elements as [21] N 2π ( n − 1)( m − 1) Gnn = Em exp − j N m =1 N 2π ( n − 1)( m − 1) Dnn = S m exp − j N m =1 Hence Eq. (5) can be rewritten as
∂c = jMc ∂z
(11)
(12)
where M = QD−1GQ H (13) 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
Gnn Dnn Since the neighboring coupling/mode overlapping coefficients are equal
βn = β +
Em = EN − m Sm = S N − m
(14)
(15)
we have
βn = β N −n+2
(16)
which indicates mode n and mode N-n + 2 are degenerated modes. Therefore, matrix Q is not unique. To transform the complex amplitudes of the super-modes into real ones, the following expression for Q was proposed [19]
#207471 - $15.00 USD (C) 2014 OSA
Received 3 Mar 2014; revised 21 Apr 2014; accepted 22 Apr 2014; published 28 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010815 | OPTICS EXPRESS 10818
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
Qmn
1 N 2 2π = cos ( m − 1)( n − 1) N N 2 2π sin ( m − 1)( N − n + 1) N N
n =1 1< n ≤
N 2
N n = 1+ 2
(17a)
N +1 < n ≤ N 2
n =1 1< n ≤
N +1 2
(17b)
N +1
