Abstract

Multimode fibers (MMFs) are widely used for short fiber links. However, the data rates through MMFs is limited owing to modal dispersion. The so-called “principal modes” (PMs) permit transmission and multiplexing through the MMFs without modal dispersion for small modulation bandwidths. For larger modulation bandwidths, however, they lose their dispersion-free nature. In this paper, we model the impact of modulation bandwidth and mode coupling strength on the performance of PMs. We develop a simulator that characterizes the dispersion and cross-talk of the PMs of few-mode and large-core graded-index MMFs with mode-dependent losses (MDL). Simulations reveal that for fibers without MDL, for modulation frequencies beyond 10 GHz diminishes the PMs’ frequency response by more than 1 dB for 100 m in large-core MMF links and 10 km few-mode fiber links. With MDL, simulations reveal that for modulation bandwidths beyond 2 GHz diminishes the frequency response by 3 dB for a 1 km few-mode fiber and by more than 4 dB for a 1 km large-core multimode fiber. While multiplexing using PMs in large-core MMFs with MDL, we find that for modulation bandwidths beyond 3 GHz, the cross-talk is 20 dB in 1 km large-core MMF links, thereby limiting system performance.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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  24. J. C. Jacob, R. K. Mishra, and K. Appaiah, “Quantization and feedback of principal modes for dispersion mitigation and multiplexing in multimode fibers,” IEEE Trans. on Comm. 64(12), 5149–5161 (2016).
    [Crossref]

2017 (2)

J. Carpenter, B. J. Eggleton, and J. Schroder, “Comparison of principal modes and spatial eigenmodes in multimode optical fibre,” Laser Photon. Rev. 11(1), 1600259 (2017).
[Crossref]

W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rotter, and H. Cao, “Principal modes in multimode fibers: exploring the crossover from weak to strong mode coupling,” Opt. Express 25(3), 2709–2724 (2017).
[Crossref]

2016 (2)

J. C. Jacob, R. K. Mishra, and K. Appaiah, “Quantization and feedback of principal modes for dispersion mitigation and multiplexing in multimode fibers,” IEEE Trans. on Comm. 64(12), 5149–5161 (2016).
[Crossref]

Wen Xiong, Philipp Ambichl, Yaron Bromberg, Brandon Redding, Stefan Rotter, and Hui Cao, “Spatiotemporal control of light transmission through a multimode fiber with strong mode coupling,” Phys. Rev. Let. 117(5), 053901 (2016)
[Crossref]

2015 (2)

J. Carpenter, B. J. Eggleton, and J. Schroder, “Observation of Eisenbud-Wigner-Smith states as principal modes in multimode fibre,” Nat. Phot. 9(11), 751–757 (2015).
[Crossref]

M. T. Hassan and E. H. Doha, “Recursive differentiation method: application to the analysis of beams on two parameter foundations,” J. Theor. Appl. Mech. 53, 15–26 (2015).

2012 (3)

B. Franz and H. Bulow, “Experimental evaluation of principal mode groups as high-speed transmission channels in spatial multiplex systems,” IEEE Photon. Technol. Lett. 24(16), 1363–1365 (2012).
[Crossref]

A. A. Juarez, C. A. Bunge, S. Warm, and K. Petermann, “Perspectives of principal mode transmission in mode-division-multiplex operation,” Opt. Express 20(13), 13810–13824 (2012).
[Crossref] [PubMed]

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, and P. J. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightw. Technol. 30(4), 521–531 (2012).
[Crossref]

2011 (2)

2009 (1)

M. Shemirani, W. Mao, R. Panicker, and J. Kahn, “Principal Modes in Graded-Index Multimode Fiber in Presence of Spatial and Polarization-Mode Coupling,” J. Lightw. Technol. 27(10), 1248–1261 (2009).
[Crossref]

2007 (1)

M. Greenberg, M. Nazarathy, and M. Orenstein, “Data parallelization by optical MIMO transmission over multimode fiber with intermodal coupling,” J. Lightw. Technol. 25(6), 1503–1514 (2007).
[Crossref]

2005 (2)

M. B. Shemirani and J. M. Kahn, “Higher-order modal dispersion in graded-index multimode fiber,” J. Lightw. Technol. 27(23), 5461–5468 (2005).
[Crossref]

S. Fan and J. M. Kahn, “Principal modes in multimode waveguides,” Opt. Lett. 30(2), 135–137 (2005).
[Crossref] [PubMed]

1999 (1)

A. Eyal, W. Marshall, M. Tur, and A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35(19), 1658–1659 (1999).
[Crossref]

1986 (1)

C. Poole and R. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibres,” Electron. Lett. 22(19), 1029–1030(1986).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems(Wiley, 1992).

Ambichl, P.

Ambichl, Philipp

Wen Xiong, Philipp Ambichl, Yaron Bromberg, Brandon Redding, Stefan Rotter, and Hui Cao, “Spatiotemporal control of light transmission through a multimode fiber with strong mode coupling,” Phys. Rev. Let. 117(5), 053901 (2016)
[Crossref]

Appaiah, K.

J. C. Jacob, R. K. Mishra, and K. Appaiah, “Quantization and feedback of principal modes for dispersion mitigation and multiplexing in multimode fibers,” IEEE Trans. on Comm. 64(12), 5149–5161 (2016).
[Crossref]

Bolle, C.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, and P. J. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightw. Technol. 30(4), 521–531 (2012).
[Crossref]

Bolle, C. A.

Bromberg, Y.

Bromberg, Yaron

Wen Xiong, Philipp Ambichl, Yaron Bromberg, Brandon Redding, Stefan Rotter, and Hui Cao, “Spatiotemporal control of light transmission through a multimode fiber with strong mode coupling,” Phys. Rev. Let. 117(5), 053901 (2016)
[Crossref]

Bulow, H.

B. Franz and H. Bulow, “Experimental evaluation of principal mode groups as high-speed transmission channels in spatial multiplex systems,” IEEE Photon. Technol. Lett. 24(16), 1363–1365 (2012).
[Crossref]

Bunge, C. A.

Burden, L.

J. D. F. Richard and L. Burden, Numerical Analysis, (Cengage Learning, 2000).

Burrows, E. C.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, and P. J. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightw. Technol. 30(4), 521–531 (2012).
[Crossref]

Cao, H.

Cao, Hui

Wen Xiong, Philipp Ambichl, Yaron Bromberg, Brandon Redding, Stefan Rotter, and Hui Cao, “Spatiotemporal control of light transmission through a multimode fiber with strong mode coupling,” Phys. Rev. Let. 117(5), 053901 (2016)
[Crossref]

Carpenter, J.

J. Carpenter, B. J. Eggleton, and J. Schroder, “Comparison of principal modes and spatial eigenmodes in multimode optical fibre,” Laser Photon. Rev. 11(1), 1600259 (2017).
[Crossref]

J. Carpenter, B. J. Eggleton, and J. Schroder, “Observation of Eisenbud-Wigner-Smith states as principal modes in multimode fibre,” Nat. Phot. 9(11), 751–757 (2015).
[Crossref]

J. Carpenter, B. J. Eggleton, and J. Schroder, “First demonstration of principal modes in a multimode fibre,”in “Optical Communication (ECOC) 2014 European Conference on”, 1–3 (2014).

Doha, E. H.

M. T. Hassan and E. H. Doha, “Recursive differentiation method: application to the analysis of beams on two parameter foundations,” J. Theor. Appl. Mech. 53, 15–26 (2015).

Eggleton, B. J.

J. Carpenter, B. J. Eggleton, and J. Schroder, “Comparison of principal modes and spatial eigenmodes in multimode optical fibre,” Laser Photon. Rev. 11(1), 1600259 (2017).
[Crossref]

J. Carpenter, B. J. Eggleton, and J. Schroder, “Observation of Eisenbud-Wigner-Smith states as principal modes in multimode fibre,” Nat. Phot. 9(11), 751–757 (2015).
[Crossref]

J. Carpenter, B. J. Eggleton, and J. Schroder, “First demonstration of principal modes in a multimode fibre,”in “Optical Communication (ECOC) 2014 European Conference on”, 1–3 (2014).

Esmaeelpour, M.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, and P. J. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightw. Technol. 30(4), 521–531 (2012).
[Crossref]

Essiambre, R.-J.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, and P. J. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightw. Technol. 30(4), 521–531 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R.-J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle, “6 × 56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6 × 6 MIMO equalization,” Opt. Express 19(17), 16697–19707 (2011).
[Crossref] [PubMed]

Eyal, A.

A. Eyal, W. Marshall, M. Tur, and A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35(19), 1658–1659 (1999).
[Crossref]

Fan, S.

Franz, B.

B. Franz and H. Bulow, “Experimental evaluation of principal mode groups as high-speed transmission channels in spatial multiplex systems,” IEEE Photon. Technol. Lett. 24(16), 1363–1365 (2012).
[Crossref]

Gnauck, A. H.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, and P. J. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightw. Technol. 30(4), 521–531 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R.-J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle, “6 × 56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6 × 6 MIMO equalization,” Opt. Express 19(17), 16697–19707 (2011).
[Crossref] [PubMed]

Greenberg, M.

M. Greenberg, M. Nazarathy, and M. Orenstein, “Data parallelization by optical MIMO transmission over multimode fiber with intermodal coupling,” J. Lightw. Technol. 25(6), 1503–1514 (2007).
[Crossref]

Hassan, M. T.

M. T. Hassan and E. H. Doha, “Recursive differentiation method: application to the analysis of beams on two parameter foundations,” J. Theor. Appl. Mech. 53, 15–26 (2015).

Ho, K.-P.

Jacob, J. C.

J. C. Jacob, R. K. Mishra, and K. Appaiah, “Quantization and feedback of principal modes for dispersion mitigation and multiplexing in multimode fibers,” IEEE Trans. on Comm. 64(12), 5149–5161 (2016).
[Crossref]

Juarez, A. A.

Kahn, J.

M. Shemirani, W. Mao, R. Panicker, and J. Kahn, “Principal Modes in Graded-Index Multimode Fiber in Presence of Spatial and Polarization-Mode Coupling,” J. Lightw. Technol. 27(10), 1248–1261 (2009).
[Crossref]

Kahn, J. M.

Lingle, R.

Mao, W.

M. Shemirani, W. Mao, R. Panicker, and J. Kahn, “Principal Modes in Graded-Index Multimode Fiber in Presence of Spatial and Polarization-Mode Coupling,” J. Lightw. Technol. 27(10), 1248–1261 (2009).
[Crossref]

Marcuse, D.

D. Marcuse, Theory of Dielectric Waveguides (Academic PressNew York, 1974).

Marshall, W.

A. Eyal, W. Marshall, M. Tur, and A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35(19), 1658–1659 (1999).
[Crossref]

McCurdy, A.

Mishra, R. K.

J. C. Jacob, R. K. Mishra, and K. Appaiah, “Quantization and feedback of principal modes for dispersion mitigation and multiplexing in multimode fibers,” IEEE Trans. on Comm. 64(12), 5149–5161 (2016).
[Crossref]

Mumtaz, S.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, and P. J. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightw. Technol. 30(4), 521–531 (2012).
[Crossref]

Nazarathy, M.

M. Greenberg, M. Nazarathy, and M. Orenstein, “Data parallelization by optical MIMO transmission over multimode fiber with intermodal coupling,” J. Lightw. Technol. 25(6), 1503–1514 (2007).
[Crossref]

Orenstein, M.

M. Greenberg, M. Nazarathy, and M. Orenstein, “Data parallelization by optical MIMO transmission over multimode fiber with intermodal coupling,” J. Lightw. Technol. 25(6), 1503–1514 (2007).
[Crossref]

Panicker, R.

M. Shemirani, W. Mao, R. Panicker, and J. Kahn, “Principal Modes in Graded-Index Multimode Fiber in Presence of Spatial and Polarization-Mode Coupling,” J. Lightw. Technol. 27(10), 1248–1261 (2009).
[Crossref]

Peckham, D. W.

Petermann, K.

Poole, C.

C. Poole and R. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibres,” Electron. Lett. 22(19), 1029–1030(1986).
[Crossref]

Randel, S.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, and P. J. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightw. Technol. 30(4), 521–531 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R.-J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle, “6 × 56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6 × 6 MIMO equalization,” Opt. Express 19(17), 16697–19707 (2011).
[Crossref] [PubMed]

Redding, B.

Redding, Brandon

Wen Xiong, Philipp Ambichl, Yaron Bromberg, Brandon Redding, Stefan Rotter, and Hui Cao, “Spatiotemporal control of light transmission through a multimode fiber with strong mode coupling,” Phys. Rev. Let. 117(5), 053901 (2016)
[Crossref]

Richard, J. D. F.

J. D. F. Richard and L. Burden, Numerical Analysis, (Cengage Learning, 2000).

Rotter, S.

Rotter, Stefan

Wen Xiong, Philipp Ambichl, Yaron Bromberg, Brandon Redding, Stefan Rotter, and Hui Cao, “Spatiotemporal control of light transmission through a multimode fiber with strong mode coupling,” Phys. Rev. Let. 117(5), 053901 (2016)
[Crossref]

Ryf, R.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, and P. J. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightw. Technol. 30(4), 521–531 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R.-J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle, “6 × 56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6 × 6 MIMO equalization,” Opt. Express 19(17), 16697–19707 (2011).
[Crossref] [PubMed]

Schroder, J.

J. Carpenter, B. J. Eggleton, and J. Schroder, “Comparison of principal modes and spatial eigenmodes in multimode optical fibre,” Laser Photon. Rev. 11(1), 1600259 (2017).
[Crossref]

J. Carpenter, B. J. Eggleton, and J. Schroder, “Observation of Eisenbud-Wigner-Smith states as principal modes in multimode fibre,” Nat. Phot. 9(11), 751–757 (2015).
[Crossref]

J. Carpenter, B. J. Eggleton, and J. Schroder, “First demonstration of principal modes in a multimode fibre,”in “Optical Communication (ECOC) 2014 European Conference on”, 1–3 (2014).

Shemirani, M.

M. Shemirani, W. Mao, R. Panicker, and J. Kahn, “Principal Modes in Graded-Index Multimode Fiber in Presence of Spatial and Polarization-Mode Coupling,” J. Lightw. Technol. 27(10), 1248–1261 (2009).
[Crossref]

Shemirani, M. B.

M. B. Shemirani and J. M. Kahn, “Higher-order modal dispersion in graded-index multimode fiber,” J. Lightw. Technol. 27(23), 5461–5468 (2005).
[Crossref]

Sierra, A.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, and P. J. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightw. Technol. 30(4), 521–531 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R.-J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle, “6 × 56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6 × 6 MIMO equalization,” Opt. Express 19(17), 16697–19707 (2011).
[Crossref] [PubMed]

Tsekrekos, C.

C. Tsekrekos, “Mode group diversity multiplexing in multimode fiber transmission systems,” Ph.D. dissertation, Technische UniversiteitEindhoven, (2008).

Tur, M.

A. Eyal, W. Marshall, M. Tur, and A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35(19), 1658–1659 (1999).
[Crossref]

Wagner, R.

C. Poole and R. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibres,” Electron. Lett. 22(19), 1029–1030(1986).
[Crossref]

Warm, S.

Winzer, P. J.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, and P. J. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightw. Technol. 30(4), 521–531 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R.-J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle, “6 × 56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6 × 6 MIMO equalization,” Opt. Express 19(17), 16697–19707 (2011).
[Crossref] [PubMed]

Xiong, W.

W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rotter, and H. Cao, “Principal modes in multimode fibers: exploring the crossover from weak to strong mode coupling,” Opt. Express 25(3), 2709–2724 (2017).
[Crossref]

W. Xiong, “Experimental Realization of Principal Modes in a Multimode Fiber with Strong Mode Mixing,” in “Frontiers in Optics 2015 OSA Technical Digest (online) (Optical Society of America, 2015), paper FW4F.3.

Xiong, Wen

Wen Xiong, Philipp Ambichl, Yaron Bromberg, Brandon Redding, Stefan Rotter, and Hui Cao, “Spatiotemporal control of light transmission through a multimode fiber with strong mode coupling,” Phys. Rev. Let. 117(5), 053901 (2016)
[Crossref]

Yariv, A.

A. Eyal, W. Marshall, M. Tur, and A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35(19), 1658–1659 (1999).
[Crossref]

Electron. Lett. (2)

A. Eyal, W. Marshall, M. Tur, and A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35(19), 1658–1659 (1999).
[Crossref]

C. Poole and R. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibres,” Electron. Lett. 22(19), 1029–1030(1986).
[Crossref]

IEEE Photon. Technol. Lett. (1)

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Figures (14)

Fig. 1
Fig. 1 Input column matrix a projected on graded index fiber using spatial light modulator (SLM) and get output as a column matrix b.
Fig. 2
Fig. 2 Schematic of the multi-section model of the fiber.
Fig. 3
Fig. 3 The fiber can be considered to be a cascade of a first order system, consisting only of F(1), and a higher order system, as described in Eq. (10). The input x(t) is projected on to the complete system. If the input to the first order system is in a principal mode, it experiences no distortion.
Fig. 4
Fig. 4 Frequency response of few-mode fiber with different σκ.
Fig. 5
Fig. 5 Frequency response of large-core multimode fiber with different σκ.
Fig. 6
Fig. 6 Cross-talk of few-mode fiber with different σκ.
Fig. 7
Fig. 7 Cross-talk of multimode fiber with different σκ.
Fig. 8
Fig. 8 Frequency response of few-mode lossy fiber with different σκ.
Fig. 9
Fig. 9 Frequency response of multimode lossy fiber with different σκ.
Fig. 10
Fig. 10 Frequency response of different core diameter and numerical aperture (NA) of length 1 km.
Fig. 11
Fig. 11 Cross-talk of few-mode fiber with different σκ.
Fig. 12
Fig. 12 Cross-talk of lossy multimode fiber with different σκ.
Fig. 13
Fig. 13 Frequency responce of different PM of length 1 km.
Fig. 14
Fig. 14 Comparison of bit error rate of different length (no additional dispersion and cross-talk compensation).

Tables (3)

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Table 1 Differences between fiber without MDL and with MDL.

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Table 2 Criterion of few mode and large-core MMFs.

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Table 3 Modulation bandwidth at which fiber response is degraded by 1 dB with σk = 7 m−1.

Equations (19)

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b = U ( ω ) a
U ( ω ) = U N ( ω ) U ( N 1 ) ( ω ) U ( i ) ( ω ) U ( 1 ) ( ω )
C ( k ) = R ( k ) M l ( k )
M l ( k ) = [ B M × M 0 0 B M × M ] 2 M × 2 M
b ( m , n ) = f m ( x , y ) f n ( x , y ) d x d y
f n ( x , y ) = f n ( x cos Φ + y sin Φ + a x , x sin Φ + y cos Φ + a y )
U ( ω ) = U ( S ) ( ω ) C ( S 1 ) U ( S 1 ) ( ω ) C ( k ) ( ω ) U ( k ) ( ω ) U ( k 1 ) ( ω ) C ( 1 ) U ( 1 ) ( ω )
U ( ω ) = exp ( j Θ ( ω ) )
U ( ω 0 + Ω ) = exp ( j n = 0 Ω n n ! d n Θ ( ω 0 + Ω ) d Ω n | Ω = 0 ) = U ( ω 0 ) U ^ ( Ω ) = U 0 exp ( j Ω F ( 1 ) ) exp ( 1 2 Ω F ( 2 ) ) = U 0 exp ( k = 1 ( j Ω ) k F ( k ) k ! )
U ( ω 0 + Ω ) = U ( ω 0 ) U ^ ( Ω ) = U ( ω 0 ) exp ( j Ω F ( 1 ) )
F ( 1 ) = j U ( ω 0 ) H U ( ω 0 + Ω ) Ω | Ω = 0
U ( i ) Ω = U ( i ) ( ω 0 + Ω + h ) U ( i ) ( ω 0 + Ω ) h
U 2 = U ( 2 ) U ( 1 ) , U 3 = U ( 3 ) U 2 , U n + 1 = U ( n + 1 ) = U ( n + 1 ) U n ,
U n + 1 Ω = U ( n + 1 ) U n Ω + U ( n + 1 ) Ω U n
P = [ p 1 p 2 p k p M ] , Q = [ q 1 q 2 q k q M ]
q k = U ( ω 0 ) p k
H MDL ( Ω ) = [ h MDL , 1 ( Ω ) , h MDL , 2 ( Ω ) , h MDL , M ( Ω ) ] T = Q inv U ( ω 0 + Ω ) P = Q inv [ U 0 exp ( j Ω F ( 1 ) ) exp ( 1 2 Ω 2 F ( 2 ) ) ] P
H MDL ( Ω ) = [ h MDL , 1 ( Ω ) , h MDL , 2 ( Ω ) , h MDL , M ( Ω ) ] T = Q inv U ( ω 0 + Ω ) P = Q inv [ U 0 exp ( j Ω F ( 1 ) ) exp ( 1 2 Ω 2 F ( 2 ) ) ] P
cross-talk for PM i = 10 log 10 ( k = 1 , k i M | h k | 2 | h i | 2 ) ( dB )

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