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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 17 — Aug. 15, 2011
  • pp: 16601–16611
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Multimode fiber amplifier with tunable modal gain using a reconfigurable multimode pump

Neng Bai, Ezra Ip, Ting Wang, and Guifang Li  »View Author Affiliations


Optics Express, Vol. 19, Issue 17, pp. 16601-16611 (2011)
http://dx.doi.org/10.1364/OE.19.016601


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Abstract

We propose a method for controlling modal gain in a multimode Erbium-doped fiber amplifier (MM-EDFA) by tuning the mode content of a multimode pump. By adjusting the powers and orientation of input pump modes, modal dependent gain can be tuned over a large dynamic range. Performance impacts due to excitation of undesired pump modes, mode coupling and macro-bending loss within the erbium-doped fiber are also investigated. The MM-EDFA may potentially be a key element for long haul mode-division multiplexed transmission.

© 2011 OSA

1. Introduction

Over the past years, advances in optical coherent detection and signal processing have led to tremendous growth in the spectral efficiency achieved in optical fiber. Recently, 100-Tb/s transmission at a spectral efficiency of 11 b/s/Hz was reported in a single-mode fiber [1

1. D. Qian, M.-F. Huang, E. Ip, Y.-K. Huang, Y. Shao, J. Hu, and T. Wang, “101-Tb/s (370×294-Gb/s) PDM-128QAM-OFDM transmission over 3×55-km SSMF using pilot-based phase noise mitigation,” in Proc. OFC (Los Angeles, CA, USA 2011). Paper PDPB5.

]. Owing to the nonlinear refractive index of silica, it is impossible to continue increasing spectral efficiency indefinitely by merely increasing the launched power. One method to reduce fiber nonlinearity is to increase the effective area of the propagating mode, thus reducing the optical intensity and the resulting nonlinear effects [2

2. H. T. Hattori and A. Safaai-Jazi, “Fiber designs with significantly reduced nonlinearity for very long distance transmission,” Appl. Opt. 37(15), 3190–3197 (1998). [CrossRef] [PubMed]

]. However, mode effective area is limited by bending loss and by the requirement of the waveguide to be single-mode. It is possible to transmit data in the fundamental mode of a “few-mode fiber” (FMF). Provided mode coupling is low, the signal will remain single-mode during propagation. The larger effective area of the fundamental mode in FMF in comparison with that achievable in SMF can further reduce nonlinearity [3

3. F. Yaman, N. Bai, Y.-K. Huang, M.-F. Huang, B. Zhu, T. Wang, and G. Li, “10 x 112Gb/s PDM-QPSK transmission over 5032 km in few-mode fibers,” Opt. Express 18(20), 21342–21349 (2010). [CrossRef] [PubMed]

]. However, a nonlinear capacity limit will always exist. Even if the transmission medium was linear, Shannon’s capacity C=BWlog2(1+SNR) b/s per channel shows that the capacity scales only logarithmically with signal-to-noise ratio. Ultra-high spectral efficiency is therefore very power inefficient. To achieve cost-effective scaling in system capacity, new paradigms in optical transmission are required.

One promising solution is space-division multiplexing, where data is transmitted over parallel channels. Indeed, transmission over parallel orthogonal channels has been well established in wireless systems, where the achievable capacity using multiple-input multiple-output (MIMO) antennas increases with the number of independent “eigenchannels,” which under the assumption of rich multipath, scales as the minimum of the number of antennae deployed at the transmitter and receiver [4

4. G. J. Foschini, “Layered space-time architecture for wireless communications in a fading environment when using multielement antennas,” Bell Labs Tech. J. 1(2), 41–59 (1996). [CrossRef]

].

In optical fiber transmission, two space-division multiplexing (SDM) schemes have been proposed. These are (i) multicore fibers (MCF), where a single strand of glass fiber contains a number of independent single- (or multi-) mode cores each capable of communicating optical signals [5

5. J. Sakaguchi, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, T. Hayashi, T. Taru, T. Kobayashi, and M. Watanabe, “109-Tb/s (7×97×172-Gb/s SDM/WDM/PDM) QPSK transmission through 16.8-km homogeneous multi-core fiber,” Proc. OFC 2011, Paper PDPB6, Los Angeles, CA, USA (2011).

,6

6. B. Zhu, T. G. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, F. V. Dimarcello, K. Abedin, P. W. Wisk, D. W. Peckham, and P. Dziedzic, “Space-, wavelength-, polarization-division multiplexed transmission of 56-Tb/s over a 76.8-km seven-core fiber,” Proc. OFC 2011, Paper PDPB7, Los Angeles, CA, USA (2011).

]; and (ii) multimode fibers (MMF), where a single strand of fiber has one core with sufficiently large cross-section area to support a number of independent guiding modes [7

7. A. Li, A. A. Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” Proc. OFC 2011, Paper PDPB8, Los Angeles, CA, USA (2011).

9

9. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, R.-J. Essiambre, and P. J. Winzer, “Space-division multiplexing over 10 km of three-mode fiber using coherent 6×6 MIMO processing,” Proc. OFC 2011, Paper PDPB10, Los Angeles, CA, USA (2011).

]. SDM transmission experiments have been reported for both types of fibers. Owing to the lack of available inline amplifiers, all MCF and MMF experiments to date have been single-span, with transmission distances up to 76.8 km [6

6. B. Zhu, T. G. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, F. V. Dimarcello, K. Abedin, P. W. Wisk, D. W. Peckham, and P. Dziedzic, “Space-, wavelength-, polarization-division multiplexed transmission of 56-Tb/s over a 76.8-km seven-core fiber,” Proc. OFC 2011, Paper PDPB7, Los Angeles, CA, USA (2011).

] for MCF and 40km [8

8. M. Salsi, C. Koebele, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Transmission at 2×100-Gb/s over two modes of 40km-long prototype few-mode fiber, using LCOS based mode multiplexer and demultiplexer,” Proc. OFC 2011, Paper PDPB9, Los Angeles, CA, USA (2011).

] for MMF.

To enable mode-division multiplexing (MDM) in MMF over long-haul distances, inline erbium-doped fiber amplifiers (EDFA) based on MMF are required [10

10. P. M. Krummrich and K. Petermann, “Evaluation of potential optical amplifier concepts for coherent mode multiplexing,” Proc. OFC 2011, Paper OMH5, Los Angeles, CA, USA (2011).

]. The theory of multimode EDFAs (MM-EDFA) has been studied in [11

11. E. Desurvire, Erbium-doped Fiber Amplifiers-Principles and Applications, (John Wiley & Son Inc. 1994), Chap. 1.

]. Applications for MM-EDFAs have included high-powered lasers and free-space communications, where the multimode optical waveguide is essentially used in a “single-mode” manner, thus mode-dependent gain (MDG) is not critical [11

11. E. Desurvire, Erbium-doped Fiber Amplifiers-Principles and Applications, (John Wiley & Son Inc. 1994), Chap. 1.

,12

12. C. D. Stacey and J. M. Jenkins, “Demonstration of fundamental mode propagation in highly multimode fibre for high power EDFAs,” Conference on Lasers and Electro-Optics Europe (CLEO 2005), Munich, Germany, June 17, p. 558.

]. In MDM transmission however, careful control over MDG is necessary to overcome mode-dependent loss (MDL) in the transmission fiber, and to ensure all signal modes are launched with optimal power maximizing the total system capacity.

Mode dependent gain (MDG) is mainly determined by three factors: (a) the concentration profile of the active dopant ions, (b) the transverse intensity profile of the pump, and (c) the transverse intensity profile of the signal. In general, a signal mode whose profile is better matched to the pump intensity profile will experience higher gain. Hence, by controlling the mode content of the pump, it is possible to control MDG. The organization of this paper is as follows. In Section 2, we review the theory of MM-EDFAs. In Section 3, we provide simulation results for a step-index “two-mode fiber,” demonstrating the feasibility of MDG control by tuning the mode content of the pump. We also explore the dependence of MDG on the excitation of unwanted modes and mode coupling within the EDF.

2. Theory

A schematic of a MM-EDFA is shown in Fig. 1
Fig. 1 Schematic diagram of an MM-EDFA.
. To generate the desired pump intensity profile, we split the pump source into N paths, and use mode converters to transform the spatial mode of the pump source into the N spatial modes of the MMF. The variable attenuators enable N-degree control over the mode content of the pump, and thus the MDG of the device. The pump modes are spatially combined with the signal, which are injected into the erbium-doped MMF. In the paper, we assume the erbium-doped MMF has the profile shown in Fig. 2
Fig. 2 Multimode Erbium-doped fiber amplifier.
, where the core has radius rc, and a region of the core for which r<arc is doped with Erbium atoms at a concentration of N0(r,φ).

The operation of a multimode fiber amplifier is described by coupled differential equations involving: (i) evolution of the intensities of the various signal and pump modes along the amplifying medium, and (ii) population inversion along the amplifying medium [11

11. E. Desurvire, Erbium-doped Fiber Amplifiers-Principles and Applications, (John Wiley & Son Inc. 1994), Chap. 1.

]. In contrast to a single-mode EDFA, the transverse intensity distributions have to be taken into account in a multimode EDFA. We assume both signal and pump are co-propagating. Let Γs,i(r,φ) and Γp,j(r,φ) be the normalized intensity patterns of the i-th signal mode and j-th pump mode of the EDF, respectively; and let Ps,i and Pp,j be their respective power. We further assume the erbium-doped fiber (EDF) can be modeled as a quasi-three-level system at 980 nm pumping, and let N1(r,φ,z)and N2(r,φ,z) with N1(r,φ,z)+N2(r,φ,z)=N0(r,φ) be the population densities of Erbium atoms in the lower and upper levels at position (r,φ,z). Loss is assumed negligible in the EDF. It can be shown that the intensity evolution equations for signal and amplified spontaneous emission (ASE) in the i-th signal mode at the wavelength λs are given by:
dPs,idz=Ps,i02π0ardrdφ   Γs,i(r,φ)[N2(r,φ,z)σes,iN1(r,φ,z)σas,i]k=1msds,ik[Ps,iPs,k]
(1)
dPASE,idz=PASE,i02π0ardrdφ   Γs,i(r,φ)[N2(r,φ,z)σes,iN1(r,φ,z)σas,i]             +02π0ardrdϕ2σes,ihνsΔνN2(r,φ)Γs,i(r,φ)
(2)
where σas,i and σes,i are the absorption and emission cross-section areas at the i-th signal mode,Δν is the equivalent amplifying bandwidth, and the ds,ik’s are coupling coefficients between signal modes [13

13. A. Galvanauskas, “Mode-scalable fiber-based chirped pulse amplification systems,” IEEE J. Sel. Top. Quantum Electron. 7(4), 504–517 (2001). [CrossRef]

]. In the signal propagation Eq. (1), the first term on the right hand side denote net amplification due to stimulated emission, and the second term denote power coupling between the signal modes. In the ASE propagation Eq. (2), the second term on the right hand side represents spontaneous emission of the excited Erbium ions; the coefficient of ‘2’ preceding this term corresponds to two degenerate polarizations modes. The intensity evolution equation for the power in the j-th pump mode at wavelength λp is:
dPp,jdz=Pp,j02π0ardrdφ   Γp,j(r,φ)N1(r,φ,z)σap,jk=1mpdp,jk[Pp,jPp,k]
(3)
Finally, the population density equations are:
N1(r,φ,z)=1τ+i=1ms[Ps,i+PASE,i]σes,iΓs,i(r,φ)hνs1τ+i=1ms[Ps,i+PASE,i](σes,i+σas,i)Γs,i(r,φ)hνs+j=1mpPp,jσap,jΓp,j(r,φ)hνpN0(r,φ)
(4)
N2(r,φ,z)=i=1ms[Ps,i+PASE,i]σas,iΓs,i(r,φ)hνs+j=1mpPp,jσap,jΓp,j(r,φ)hνp1τ+i=1ms[Ps,i+PASE,i](σes,i+σas,i)Γs,i(r,φ)hνs+j=1mpPp,jσap,jΓp,j(r,φ)hνpN0(r,φ)
(5)
where νs and νp are the signal and pump optical frequencies. Other symbols are listed in Table 3

Table 3. List of Variables Used in the Coupled Eqs. (1)(5)

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. Equations (1)(5) can be solved by using the standard fourth-order Runge-Kutta method given initial conditions for pump and signal power [14

14. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]

]. Gains and noise figures for all signal modes may similarly be calculated.

3. Simulation

We consider a MM-EDFA where the EDF has step-index refractive profile. The parameters of the MM-EDFA are shown in Table 1

Table 1. Parameters of a MM-EDFA

table-icon
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. We assume the doped region is the same size as the core (i.e., a = rc). As the normalized frequency at λs = 1.53 μm lies between 2.405 <Vs< 3.832, the EDF supports two degenerate mode groups at this wavelength. We assume a weakly guiding MMF where the modes are well approximated by linearly polarized (LP) modes [15

15. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971). [CrossRef] [PubMed]

]. For the remainder of this paper, we use the notation LPij , s and LPxy , p to denote the LPij mode at λs and LPxy mode at λp, respectively. Figure 3
Fig. 3 (a) Intensity profile of pump and signal modes, (b) normalized intensity profiles viewed along x-axis.
shows the intensity profiles of the modes, and their intensities viewed along the x-axis. We note that for m>0, the LPmn,s and LPmn,p have two spatially degenerate modes. In one of these modes, referred to as the “even” mode, intensity is maximized along the x-axis at (φ = 0); in the other mode, referred to as the “odd mode”, intensity is minimized along the x-axis. All spatial modes, degenerate and non-degenerate, come with two degenerate polarization modes.

3.1 Modal Gain Control for Non-Degenerate Signal Modes

We first consider MDG using a single-mode pump. For the spatially degenerate LP11, p and LP21, p modes, we assume equal power in the even (LP11 e , p or LP21 e , p) and odd modes (LP11 o , p or LP21 o , p) so that the resulting intensity (power) patterns (e.g., LP11,p=12(LP11e,p+LP11o,p)) have no azimuthal dependence (Fig. 3(a)), and hence no MDG between spatially degenerate signal modes. Figure 4
Fig. 4 Modal gain of signal at 1530 nm assuming 0.05 mW power in each degenerate modes of LP01, s and LP11, s, when 980-nm pump is entirely confined in (a) LP01, p, (b) LP11, p and (c) LP21, p.
shows the gain experienced by each signal mode group when pumping in the LP01,p, LP11,p and LP21,p modes. It is assumed that the input signal to the EDF has equal power (0.05 mW) in each of its six (two LP01,s and four LP11,s) spatial and polarization degenerate modes, or 0.3 mW in total. Since the intensity profile of LP01,p is better matched to LP01,s than LP11,s, Fig. 4(a) shows higher gain for LP01,s. Conversely, pumping in LP21,p results in higher gain for LP11,s.

3.2 Modal Gain Control for Spatially Degenerate Signal Modes

3.3 Impact on Performance Due to Inexact Excitation and Mode Coupling

Mode coupling during propagation inside the EDF will have similar impact as excitation of unwanted modes, since both result in power transferred into undesired modes. The strength of mode coupling is inversely proportional to the difference in effective refractive index between the modes (Δneff). Hence, we focus on coupling between LP21, p and LP02, p, as the effective refractive index difference Δneff ,( 21p−02p ) between this pair of modes is the smallest among all the mode pairs in our FMF-based EDF. Assuming the power coupling coefficient between LP21, p and LP02, p, which was defined as dp,2102 in Eq. (3), is constant throughout the EDF, Fig. 10
Fig. 10 Modal gain vs. mode coupling strength from LP21, p to LP01, p (dp,2102).
shows MDG as a function of dp,2102. The value of coupling coefficient refers to [13

13. A. Galvanauskas, “Mode-scalable fiber-based chirped pulse amplification systems,” IEEE J. Sel. Top. Quantum Electron. 7(4), 504–517 (2001). [CrossRef]

]. As expected, ΔG11s–01s decreases with increasing strength of mode coupling. This favors using a shorter length EDF, and a refractive index profile that maximizes effective refractive index difference between the modes to reduce mode coupling.

3.4 Impact on Performance Due to Macro-Bending Loss

In the results thus far, the loss of EDF is assumed to be negligible. In practice, however, the EDF has to be spooled to create a module, which may introduce macro-bending loss. As the higher order LP11, s and LP21, p modes are less confined, they are more likely to couple into cladding modes when the fiber is bent, resulting in higher macro-bending loss than the fundamental modes LP01, s and LP01, p. This must either be taken into account by increasing the power of the higher order pump, or the bending radius has to be large enough to render bending loss negligible. The theoretical macro-bending loss can be calculated using Marcuse’s curvature loss formula [17

17. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). [CrossRef]

]. Figure 11
Fig. 11 Macro-bending loss vs. bending radius.
shows the macro-bending losses for the various signal and pump modes as functions of bend radius.

4. Conclusion

Appendix

References and links

1.

D. Qian, M.-F. Huang, E. Ip, Y.-K. Huang, Y. Shao, J. Hu, and T. Wang, “101-Tb/s (370×294-Gb/s) PDM-128QAM-OFDM transmission over 3×55-km SSMF using pilot-based phase noise mitigation,” in Proc. OFC (Los Angeles, CA, USA 2011). Paper PDPB5.

2.

H. T. Hattori and A. Safaai-Jazi, “Fiber designs with significantly reduced nonlinearity for very long distance transmission,” Appl. Opt. 37(15), 3190–3197 (1998). [CrossRef] [PubMed]

3.

F. Yaman, N. Bai, Y.-K. Huang, M.-F. Huang, B. Zhu, T. Wang, and G. Li, “10 x 112Gb/s PDM-QPSK transmission over 5032 km in few-mode fibers,” Opt. Express 18(20), 21342–21349 (2010). [CrossRef] [PubMed]

4.

G. J. Foschini, “Layered space-time architecture for wireless communications in a fading environment when using multielement antennas,” Bell Labs Tech. J. 1(2), 41–59 (1996). [CrossRef]

5.

J. Sakaguchi, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, T. Hayashi, T. Taru, T. Kobayashi, and M. Watanabe, “109-Tb/s (7×97×172-Gb/s SDM/WDM/PDM) QPSK transmission through 16.8-km homogeneous multi-core fiber,” Proc. OFC 2011, Paper PDPB6, Los Angeles, CA, USA (2011).

6.

B. Zhu, T. G. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, F. V. Dimarcello, K. Abedin, P. W. Wisk, D. W. Peckham, and P. Dziedzic, “Space-, wavelength-, polarization-division multiplexed transmission of 56-Tb/s over a 76.8-km seven-core fiber,” Proc. OFC 2011, Paper PDPB7, Los Angeles, CA, USA (2011).

7.

A. Li, A. A. Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” Proc. OFC 2011, Paper PDPB8, Los Angeles, CA, USA (2011).

8.

M. Salsi, C. Koebele, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Transmission at 2×100-Gb/s over two modes of 40km-long prototype few-mode fiber, using LCOS based mode multiplexer and demultiplexer,” Proc. OFC 2011, Paper PDPB9, Los Angeles, CA, USA (2011).

9.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, R.-J. Essiambre, and P. J. Winzer, “Space-division multiplexing over 10 km of three-mode fiber using coherent 6×6 MIMO processing,” Proc. OFC 2011, Paper PDPB10, Los Angeles, CA, USA (2011).

10.

P. M. Krummrich and K. Petermann, “Evaluation of potential optical amplifier concepts for coherent mode multiplexing,” Proc. OFC 2011, Paper OMH5, Los Angeles, CA, USA (2011).

11.

E. Desurvire, Erbium-doped Fiber Amplifiers-Principles and Applications, (John Wiley & Son Inc. 1994), Chap. 1.

12.

C. D. Stacey and J. M. Jenkins, “Demonstration of fundamental mode propagation in highly multimode fibre for high power EDFAs,” Conference on Lasers and Electro-Optics Europe (CLEO 2005), Munich, Germany, June 17, p. 558.

13.

A. Galvanauskas, “Mode-scalable fiber-based chirped pulse amplification systems,” IEEE J. Sel. Top. Quantum Electron. 7(4), 504–517 (2001). [CrossRef]

14.

M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]

15.

D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971). [CrossRef] [PubMed]

16.

C. D. Poole and S.-C. Wang, “Bend-induced loss for the higher-order spatial mode in a dual-mode fiber,” Opt. Lett. 18(20), 1712–1714 (1993). [CrossRef] [PubMed]

17.

D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). [CrossRef]

OCIS Codes
(060.2330) Fiber optics and optical communications : Fiber optics communications
(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems

ToC Category:
Optical Amplifier

History
Original Manuscript: May 16, 2011
Revised Manuscript: June 27, 2011
Manuscript Accepted: July 7, 2011
Published: August 15, 2011

Virtual Issues
Space Multiplexed Optical Transmission (2011) Optics Express

Citation
Neng Bai, Ezra Ip, Ting Wang, and Guifang Li, "Multimode fiber amplifier with tunable modal gain using a reconfigurable multimode pump," Opt. Express 19, 16601-16611 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16601


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References

  1. D. Qian, M.-F. Huang, E. Ip, Y.-K. Huang, Y. Shao, J. Hu, and T. Wang, “101-Tb/s (370×294-Gb/s) PDM-128QAM-OFDM transmission over 3×55-km SSMF using pilot-based phase noise mitigation,” in Proc. OFC (Los Angeles, CA, USA 2011). Paper PDPB5.
  2. H. T. Hattori and A. Safaai-Jazi, “Fiber designs with significantly reduced nonlinearity for very long distance transmission,” Appl. Opt. 37(15), 3190–3197 (1998). [CrossRef] [PubMed]
  3. F. Yaman, N. Bai, Y.-K. Huang, M.-F. Huang, B. Zhu, T. Wang, and G. Li, “10 x 112Gb/s PDM-QPSK transmission over 5032 km in few-mode fibers,” Opt. Express 18(20), 21342–21349 (2010). [CrossRef] [PubMed]
  4. G. J. Foschini, “Layered space-time architecture for wireless communications in a fading environment when using multielement antennas,” Bell Labs Tech. J. 1(2), 41–59 (1996). [CrossRef]
  5. J. Sakaguchi, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, T. Hayashi, T. Taru, T. Kobayashi, and M. Watanabe, “109-Tb/s (7×97×172-Gb/s SDM/WDM/PDM) QPSK transmission through 16.8-km homogeneous multi-core fiber,” Proc. OFC 2011, Paper PDPB6, Los Angeles, CA, USA (2011).
  6. B. Zhu, T. G. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, F. V. Dimarcello, K. Abedin, P. W. Wisk, D. W. Peckham, and P. Dziedzic, “Space-, wavelength-, polarization-division multiplexed transmission of 56-Tb/s over a 76.8-km seven-core fiber,” Proc. OFC 2011, Paper PDPB7, Los Angeles, CA, USA (2011).
  7. A. Li, A. A. Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” Proc. OFC 2011, Paper PDPB8, Los Angeles, CA, USA (2011).
  8. M. Salsi, C. Koebele, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Transmission at 2×100-Gb/s over two modes of 40km-long prototype few-mode fiber, using LCOS based mode multiplexer and demultiplexer,” Proc. OFC 2011, Paper PDPB9, Los Angeles, CA, USA (2011).
  9. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, R.-J. Essiambre, and P. J. Winzer, “Space-division multiplexing over 10 km of three-mode fiber using coherent 6×6 MIMO processing,” Proc. OFC 2011, Paper PDPB10, Los Angeles, CA, USA (2011).
  10. P. M. Krummrich and K. Petermann, “Evaluation of potential optical amplifier concepts for coherent mode multiplexing,” Proc. OFC 2011, Paper OMH5, Los Angeles, CA, USA (2011).
  11. E. Desurvire, Erbium-doped Fiber Amplifiers-Principles and Applications, (John Wiley & Son Inc. 1994), Chap. 1.
  12. C. D. Stacey and J. M. Jenkins, “Demonstration of fundamental mode propagation in highly multimode fibre for high power EDFAs,” Conference on Lasers and Electro-Optics Europe (CLEO 2005), Munich, Germany, June 17, p. 558.
  13. A. Galvanauskas, “Mode-scalable fiber-based chirped pulse amplification systems,” IEEE J. Sel. Top. Quantum Electron. 7(4), 504–517 (2001). [CrossRef]
  14. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]
  15. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971). [CrossRef] [PubMed]
  16. C. D. Poole and S.-C. Wang, “Bend-induced loss for the higher-order spatial mode in a dual-mode fiber,” Opt. Lett. 18(20), 1712–1714 (1993). [CrossRef] [PubMed]
  17. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). [CrossRef]

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