Abstract

The modal group delays (GDs) are a key property governing the dispersion of signals propagating in a multimode fiber (MMF). An MMF is in the strong-coupling regime when the total length of the MMF is much greater than the correlation length over which local principal modes can be considered constant. In this regime, the GDs can be described as the eigenvalues of zero-trace Gaussian unitary ensemble, and the probability density function (pdf) of the GDs is the eigenvalue distribution of the ensemble. For fibers with two to seven modes, the marginal pdf of the GDs is derived analytically. For fibers with a large number of modes, this pdf is shown to approach a semicircle distribution. In the strong-coupling regime, the delay spread is proportional to the square root of the number of independent sections, or the square root of the overall fiber length.

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2010 (5)

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, C. Laperle, "100 G and beyond with digital coherent signal processing," IEEE Commun. Mag. 48, 62-69 (2010).

J. Yu, X. Zhou, "Ultra-high-capacity DWDM transmission system for 100 G and beyond," IEEE Commun. Mag. 48, S56-S64 (2010).

P. J. Winzer, "Beyond 100 G Ethernet," IEEE. Commun. Mag. 48, 26-30 (2010).

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, B. Goebel, "Capacity limits of optical fiber networks," J. Lightw. Technol. 28, 662-701 (2010).

L. Erdős, J. Ramírez, B. Schlein, T. Tao, V. H. Vu, H.-T. Yau, "Bulk universality for Wigner hermitian matrices with subexponential decay," Math. Res. Lett. 17, 667-674 (2010).

2009 (3)

T. Tao, V. H. Vu, "From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices," Bull. Amer. Math. Soc. 46, 337-396 (2009).

M. B. Shemirani, J. M. Kahn, "Higher-order modal dispersion in graded-index multimode fiber," J. Lightw. Technol. 27, 5461-5468 (2009).

M. B. Shemirani, W. Mao, R. A. Panicker, J. M. Kahn, "Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling," J. Lightw. Technol. 27, 1248-1261 (2009).

2008 (1)

M. Nazarathy, A. Agmon, "Coherent transmission direct detection MIMO over short-range optical interconnects and passive optical networks," J. Lightw. Technol. 26, 2037-2045 (2008).

2005 (4)

A. R. Shah, R. C. J. Hsu, A. Tarighat, A. H. Sayed, B. Jalali, "Coherent optical MIMO (COMIMO)," J. Lightw. Technol. 23, 2410-2419 (2005).

A. F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, M. B. Ritter, "Exploitation of optical interconnects in future server architectures," IBM J. Res. Dev. 49, 755-775 (2005).

S. Fan, J. M. Kahn, "Principal modes in multi-mode waveguides," Opt. Lett. 30, 135-137 (2005).

X. Shen, J. M. Kahn, M. A. Horowitz, "Compensation for multimode fiber dispersion by adaptive optics," Opt. Lett. 30, 2985-2987 (2005).

2004 (1)

J. M. Kahn, K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Topics Quantum Electron. 10, 259-272 (2004).

2003 (1)

P. Pepeljugoshi, S. E. Golowich, A. J. Ritger, P. Kolesar, A. Risteski, "Modeling and simulation of next-generation multimode fiber links," J. Lightw. Technol. 21, 1242-1255 (2003).

2001 (3)

G. J. Foschini, L. E. Nelson, R. M. Jopson, H. Kogelnik, "Statistics of second-order PMD depolarization," J. Lightw. Technol. 19, 1882-1886 (2001).

P. P. Mitra, J. B. Stark, "Nonlinear limits to the information capacity of optical fibre communications," Nature 411, 1027-1030 (2001).

M. Karlsson, "Probability density functions of the differential group delay in optical fiber communication systems," J. Lightw. Technol. 19, 324-331 (2001).

2000 (3)

J. P. Gordon, H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. 97, 4541-4550 (2000).

H. R. Stuart, "Dispersive multiplexing in multimode optical fiber," Science 289, 281-283 (2000).

G. J. Foschini, L. E. Nelson, R. M. Jopson, H. Kogelnik, "Probability densities of second-order polarization mode dispersion including polarization dependent chromatic fiber dispersion," IEEE Photon. Technol. Lett. 12, 293-295 (2000).

1998 (3)

M. J. Li, D. A. Nolan, "Fiber spin-profile designs for producing fibers with low polarization mode dispersion," Opt. Lett. 23, 1659-1661 (1998).

L. Raddatz, I. H. White, D. G. Cunningham, M. C. Nowell, "An experimental and theoretical study of the offset launch technique for the enhancement of the bandwidth of multimode fiber links," J. Lightw. Technol. 16, 324-331 (1998).

A. F. Garito, J. Wang, R. Gao, "Effects of random perturbations in plastic optical fibers," Science 281, 962-967 (1998).

1997 (1)

D. Gottlieb, C.-W. Shu, "On the Gibbs phenomenon and its resolution," SIAM Rev. 39, 644-668 (1997).

1993 (1)

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, "Elliptical-core dual-mode fiber dispersion compensator," IEEE Photon. Technol. Lett. 5, 194-197 (1993).

1991 (2)

D. Voiculescu, "Limit laws for random matrices and free products," Invent. Math. 104, 201-220 (1991).

G. J. Foschini, C. D. Poole, "Statistical theory of polarization dispersion in single mode fibers," J. Lightw. Technol. 9, 1439-1456 (1991).

1987 (1)

1986 (1)

C. D. Poole, R. E. Wagner, "Phenomenological approach to polarization dispersion in long single-mode fibers," Electron. Lett. 22, 1029-1030 (1986).

1982 (1)

1980 (2)

A. Ghatak, K. Thyagarajan, "Graded index waveguides—A review," Prog. Opt. 18, 1-126 (1980).

K.-I. Kitayama, S. Sikai, N. Uchida, "Impulse response prediction based on experimental mode-coupling coefficient in a 10-km long graded-index fiber," J. Quantum Electron. QE-16, 356-362 (1980).

1979 (1)

E. Hewitt, R. E. Hewitt, "The Gibbs–Wilbraham phenomenon: An episode in Fourier analysis," Arch. History Exact Sci. 21, 129-160 (1979).

1975 (1)

1972 (1)

D. Gloge, "Optical power flow in multimode fiber," Bell Sys. Tech. J. 51, 1767-1780 (1972).

1970 (1)

1965 (1)

J. Ginibre, "Statistical ensembles of complex, quaternion, and real matrices," J. Math. Phys. 6, 440-450 (1965).

1958 (1)

E. Wigner, "On the distribution of the roots of certain symmetric matrices," Ann. Math. 67, 325-328 (1958).

1955 (1)

E. Wigner, "Characteristic vectors of bordered matrices with infinite dimensions," Ann. Math. 62, 548-564 (1955).

Ann. Math. (2)

E. Wigner, "Characteristic vectors of bordered matrices with infinite dimensions," Ann. Math. 62, 548-564 (1955).

E. Wigner, "On the distribution of the roots of certain symmetric matrices," Ann. Math. 67, 325-328 (1958).

Appl. Opt. (3)

Arch. History Exact Sci. (1)

E. Hewitt, R. E. Hewitt, "The Gibbs–Wilbraham phenomenon: An episode in Fourier analysis," Arch. History Exact Sci. 21, 129-160 (1979).

Bell Sys. Tech. J. (1)

D. Gloge, "Optical power flow in multimode fiber," Bell Sys. Tech. J. 51, 1767-1780 (1972).

Bull. Amer. Math. Soc. (1)

T. Tao, V. H. Vu, "From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices," Bull. Amer. Math. Soc. 46, 337-396 (2009).

Electron. Lett. (1)

C. D. Poole, R. E. Wagner, "Phenomenological approach to polarization dispersion in long single-mode fibers," Electron. Lett. 22, 1029-1030 (1986).

IBM J. Res. Dev. (1)

A. F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, M. B. Ritter, "Exploitation of optical interconnects in future server architectures," IBM J. Res. Dev. 49, 755-775 (2005).

IEEE Commun. Mag. (2)

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, C. Laperle, "100 G and beyond with digital coherent signal processing," IEEE Commun. Mag. 48, 62-69 (2010).

J. Yu, X. Zhou, "Ultra-high-capacity DWDM transmission system for 100 G and beyond," IEEE Commun. Mag. 48, S56-S64 (2010).

IEEE J. Sel. Topics Quantum Electron. (1)

J. M. Kahn, K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Topics Quantum Electron. 10, 259-272 (2004).

IEEE Photon. Technol. Lett. (2)

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, "Elliptical-core dual-mode fiber dispersion compensator," IEEE Photon. Technol. Lett. 5, 194-197 (1993).

G. J. Foschini, L. E. Nelson, R. M. Jopson, H. Kogelnik, "Probability densities of second-order polarization mode dispersion including polarization dependent chromatic fiber dispersion," IEEE Photon. Technol. Lett. 12, 293-295 (2000).

IEEE. Commun. Mag. (1)

P. J. Winzer, "Beyond 100 G Ethernet," IEEE. Commun. Mag. 48, 26-30 (2010).

Invent. Math. (1)

D. Voiculescu, "Limit laws for random matrices and free products," Invent. Math. 104, 201-220 (1991).

J. Lightw. Technol. (10)

P. Pepeljugoshi, S. E. Golowich, A. J. Ritger, P. Kolesar, A. Risteski, "Modeling and simulation of next-generation multimode fiber links," J. Lightw. Technol. 21, 1242-1255 (2003).

G. J. Foschini, C. D. Poole, "Statistical theory of polarization dispersion in single mode fibers," J. Lightw. Technol. 9, 1439-1456 (1991).

M. Karlsson, "Probability density functions of the differential group delay in optical fiber communication systems," J. Lightw. Technol. 19, 324-331 (2001).

M. B. Shemirani, J. M. Kahn, "Higher-order modal dispersion in graded-index multimode fiber," J. Lightw. Technol. 27, 5461-5468 (2009).

L. Raddatz, I. H. White, D. G. Cunningham, M. C. Nowell, "An experimental and theoretical study of the offset launch technique for the enhancement of the bandwidth of multimode fiber links," J. Lightw. Technol. 16, 324-331 (1998).

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, B. Goebel, "Capacity limits of optical fiber networks," J. Lightw. Technol. 28, 662-701 (2010).

A. R. Shah, R. C. J. Hsu, A. Tarighat, A. H. Sayed, B. Jalali, "Coherent optical MIMO (COMIMO)," J. Lightw. Technol. 23, 2410-2419 (2005).

M. Nazarathy, A. Agmon, "Coherent transmission direct detection MIMO over short-range optical interconnects and passive optical networks," J. Lightw. Technol. 26, 2037-2045 (2008).

G. J. Foschini, L. E. Nelson, R. M. Jopson, H. Kogelnik, "Statistics of second-order PMD depolarization," J. Lightw. Technol. 19, 1882-1886 (2001).

M. B. Shemirani, W. Mao, R. A. Panicker, J. M. Kahn, "Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling," J. Lightw. Technol. 27, 1248-1261 (2009).

J. Math. Phys. (1)

J. Ginibre, "Statistical ensembles of complex, quaternion, and real matrices," J. Math. Phys. 6, 440-450 (1965).

J. Quantum Electron. (1)

K.-I. Kitayama, S. Sikai, N. Uchida, "Impulse response prediction based on experimental mode-coupling coefficient in a 10-km long graded-index fiber," J. Quantum Electron. QE-16, 356-362 (1980).

Math. Res. Lett. (1)

L. Erdős, J. Ramírez, B. Schlein, T. Tao, V. H. Vu, H.-T. Yau, "Bulk universality for Wigner hermitian matrices with subexponential decay," Math. Res. Lett. 17, 667-674 (2010).

Nature (1)

P. P. Mitra, J. B. Stark, "Nonlinear limits to the information capacity of optical fibre communications," Nature 411, 1027-1030 (2001).

Opt. Lett. (4)

Proc. Natl. Acad. Sci. (1)

J. P. Gordon, H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. 97, 4541-4550 (2000).

Prog. Opt. (1)

A. Ghatak, K. Thyagarajan, "Graded index waveguides—A review," Prog. Opt. 18, 1-126 (1980).

Science (2)

H. R. Stuart, "Dispersive multiplexing in multimode optical fiber," Science 289, 281-283 (2000).

A. F. Garito, J. Wang, R. Gao, "Effects of random perturbations in plastic optical fibers," Science 281, 962-967 (1998).

SIAM Rev. (1)

D. Gottlieb, C.-W. Shu, "On the Gibbs phenomenon and its resolution," SIAM Rev. 39, 644-668 (1997).

Other (15)

G. H. Golub, C. F. Van Loan, Matrix Computations (The Johns Hopkins Univ. Press, 1996).

H. Kogelnik, R. M. Jopson, L. E. Nelson, Optical Fiber Telecommunications IVB: Systems and Impairments (Academic, 2002).

C. D. Poole, J. A. Nagel, Optical Fiber Telecommunications IIIA (Academic, 1997) pp. 114-161.

D. Voiculescu, K. Dykema, A. Nica, Free Random Variables, CRM Monograph Series (American Mathematical Society, 1992).

A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series (Cambridge Univ. Press, 2006).

S. S.-H. Yam, F.-T. An, M. E. Marhic, L. G. Kazovsky, "Polarization sensitivity of 40 Gb/s transmission over short-reach 62.5 $\mu$m multimode fiber," Opt. Fiber Commun. Los AngelesCA (2011) Paper FA-5.

Carrier Sense Multiple Access With Collision Detection (CSMA/CD) Access Method and Physical Layer Specifications IEEE 802.3 Standard (2008).

Y. Koike, S. Takahashi, Optical Fiber Telecommunications VB: Systems and Networks (Academic, 2008).

A. Li, A. Al Amin, X. Chen, W. Shieh, "Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber," Proc. Opt. Fiber Commun. (2011) pp. 1-3.

M. Salsi, C. Koebele, D. Sperti, P. Tran, P. Brindel, H. Margoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Bigot-Astruc, L. Provost, F. Cerou, G. Charlet, "Transmission at 2$\,\times\,$100 Gb/s, over two modes of 40 km-long prototype few-mode fiber, using LCOS based mode multiplexer and demultiplexer," Opt. Fiber Commun. Los AngelesCA (2011) Paper PDPB9.

R. Ryf, S. Randel, A. H. Gnuack, C. Bolle, R.-J. Essiambre, P. Winzer, D. W. Peckham, A. McCurdy, R. Lingle, "Space-division multiplexing over 10 km of three-mode fiber using coherent 6$\,\times\,$6 MIMO processing," Opt. Fiber Commun. Los AngelesCA (2011) Paper PDPB10.

Carrier Sense Multiple Access With Collision Detection (CSMA/CD) Access Method and Physical Layer Specifications Amendment 2: Physical Layer and Management Parameters for 10 Gb/s Operation, Type 10 GBASE-LRM IEEE Standard 802.3aq (2006).

Carrier Sense Multiple Access With Collision Detection (CSMA/CD) Access Method and Physical Layer Specifications Amendment 4: Media Access Control Parameters, Physical Layers and Management Parameters for 40 Gb/s and 100 Gb/s Operation IEEE Standard 802.3ba (2010).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2007).

M. L. Mehta, Random Matrices (Academic, 2004).

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