Abstract

In multimode fiber transmission systems, mode-dependent loss and gain (collectively referred to as MDL) pose fundamental performance limitations. In the regime of strong mode coupling, the statistics of MDL (expressed in decibels or log power gain units) can be described by the eigenvalue distribution of zero-trace Gaussian unitary ensemble in the small-MDL region that is expected to be of interest for practical long-haul transmission. Information-theoretic channel capacities of mode-division-multiplexed systems in the presence of MDL are studied, including average and outage capacities, with and without channel state information.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. IEEE, 802.3 Standard, Carrier Sense Multiple Access with Collision Detection (CSMA/CD) Access Method and Physical Layer Specifications, 2008.
  2. A. F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Develop. 49(4), 755–775 (2005).
    [CrossRef]
  3. Y. Koike and S. Takahashi, “Plastic optical fibers: technologies and communication links,” in Optical Fiber Telecommunications VB: Systems and Networks, I. P. Kaminow, T. Li and A. E. Willner eds. (Elsevier Academic, 2008).
  4. D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1780 (1972).
  5. R. Olshansky, “Mode-coupling effects in graded-index optical fibers,” Appl. Opt. 14(4), 935–945 (1975).
    [PubMed]
  6. L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Nowell, “An experimental and theoretical study of the offset launch technique for the enhancement of the bandwidth of multimode fiber links,” J. Lightwave Technol. 16(3), 324–331 (1998).
    [CrossRef]
  7. K.-P. Ho and J. M. Kahn, “Statistics of group delays in multimode fiber with strong mode coupling,” to be published in J. Lightwave Technol., http://arxiv.org/abs/1104.4527
  8. M. L. Mehta, Random Matrices, 3rd ed. (Elsevier Academic, 2004).
  9. B. Rosinski, J. W. D. Chi, P. Grosso, and J. Le Bihan, “Multichannel transmission of a multicore fiber coupled with vertical-cavity surface-emitting lasers,” J. Lightwave Technol. 17(5), 807–810 (1999).
    [CrossRef]
  10. B. Zhu, T. F. Taunay, M. F. Yan, J. M. Fini, M. Fishteyn, E. M. Monberg, and F. V. Dimarcello, “Seven-core multicore fiber transmissions for passive optical network,” Opt. Express 18(11), 11117–11122 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-11-11117 .
    [CrossRef] [PubMed]
  11. H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
    [CrossRef] [PubMed]
  12. A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links,” IEEE Commun. Mag. 45(5), 57–63 (2007).
    [CrossRef]
  13. A. R. Shah, R. C. J. Hsu, A. Tarighat, A. H. Sayed, and B. Jalali, “Coherent optical MIMO (COMIMO),” J. Lightwave Technol. 23(8), 2410–2419 (2005).
    [CrossRef]
  14. R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10(3), 1089–7798 (2006).
    [CrossRef]
  15. M. Nazarathy and A. Agmon, “Coherent transmission direct detection MIMO over short-range optical interconnects and passive optical networks,” J. Lightwave Technol. 26(14), 2037–2045 (2008).
    [CrossRef]
  16. H. Bülow, “Coherent multichannel transmission over multimode-fiber and related signal processing,” Proc. of OSA Topical Meeting on Access Networks and In-House Communications, Karlsruhe, Germany, June 21–24, 2010, paper AThB1.
  17. A. Li, A. Al Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” in OFC ’11, paper PDPB8.
  18. R. Ryf, S. Randel, A. H. Gnuack, C. Bolle, R.-J. Essiambre, P. Winzer, D. W. Peckham, A. McCurdy, and R. Lingle, “Space-division multiplexing over 10 km of three-mode fiber using coherent 6×6 MIMO processing,” in OFC ’11, paper PDPB10.
  19. 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, and G. Charlet, “Transmission at 2×100 Gb/s, over two modes of 40 km-long prototype few-mode fiber, using LCOS based mode multiplexer and demultiplexer,” in OFC ’11, paper PDPB9.
  20. Z. Tong, Q. Yang, Y. Ma, and W. Shieh, “21.4 Gbit/s transmission over 200 km multimode fiber using coherent optical OFDM,” Electron. Lett. 44(23), 1373–1374 (2008).
    [CrossRef]
  21. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-2-841 .
    [CrossRef] [PubMed]
  22. J. Shentu, K. Panta, and J. Armstrong, “Effects of phase noise on performance of OFDM systems using an ICI cancellation scheme,” IEEE Trans. Broadcast 49(2), 221–224 (2003).
    [CrossRef]
  23. S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,” IEEE Trans. Commun. 52(11), 1988–1996 (2004).
    [CrossRef]
  24. W. Shieh and K.-P. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16(20), 15718–15727 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-20-15718 .
    [CrossRef] [PubMed]
  25. C. Xie, “WDM coherent PDM-QPSK systems with and without inline optical dispersion compensation,” Opt. Express 17(6), 4815–4823 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-6-4815 .
    [CrossRef] [PubMed]
  26. K.-P. Ho, A. P. T. Lau, and W. Shieh, “Equalization-enhanced phase noise induced timing jitter,” Opt. Lett. 36(4), 585–587 (2011), http://www.opticsinfobase.org/abstract.cfm?URI=ol-36-4-585 .
    [CrossRef] [PubMed]
  27. P. J. Winzer and G. J. Foschini, “Outage calculations for spatially multiplexed fiber links,” in OFC ’11, paper OThO5.
  28. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
    [CrossRef] [PubMed]
  29. H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion” in Optical Fiber Telecommunications IVB: Systems and Impairments, I. Kaminow and T. Li, eds, (Academic Press, 2002).
  30. C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22(19), 1029–1030 (1986).
    [CrossRef]
  31. M. B. Shemirani, W. Mao, R. A. Panicker, and J. M. Kahn, “Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling,” J. Lightwave Technol. 27(10), 1248–1261 (2009).
    [CrossRef]
  32. A. M. Tulino and S. Verdú, Random Matrix Theory and Wireless Communications (Now, 2004).
  33. D. Tse and P. Viswanath, Fundamentals of Wireless Communication (Cambridge Univ. Press, 2005).
  34. X. Zhu and R. D. Murch, “Layered space-frequency equalization in a single-carrier MIMO system for frequency-selective channels,” IEEE Trans. Wirel. Comm. 3(3), 701–708 (2004).
    [CrossRef]
  35. B. Vucetic and J. Yuan, Space-Time Coding (Wiley, 2003).
  36. V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results,” IEEE J. Sel Areas in Commun. 17(3), 451–460 (1999).
    [CrossRef]
  37. R. Bellman, “Limit theorems for non-commutative operations I,” Duke Math. J. 21(3), 491–500 (1954).
    [CrossRef]
  38. H. Furstenberg and H. Kesten, “Products of random matrices,” Ann. Math. Stat. 31(2), 457–469 (1960).
    [CrossRef]
  39. J. E. Cohen and C. M. Newman, “The stability of large random matrices and their products,” Ann. Probab. 12(2), 283–310 (1984).
    [CrossRef]
  40. A. Crisanti, G. Paladin, and A. Vulpiani, Products of Random Matrices in Statistical Physics (Springer, 1993).
  41. M. A. Berger, “Central limit theorem for products of random matrices,” Trans. Am. Math. Soc. 285(2), 777–803 (1984).
    [CrossRef]
  42. A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14(3), 313–315 (2002).
    [CrossRef]
  43. A. Glatarossa and L. Palmieri, “The exact statistics of polarization-dependent loss in fiber-optic links,” IEEE Photon. Technol. Lett. 15(1), 57–59 (2003).
    [CrossRef]
  44. D. Voiculescu, K. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series, vol. 1, (American Mathematical Society, 1992).
  45. A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, vol. 335, (Cambridge Univ. Press, 2006).
  46. P. Lu, L. Chen, and X. Bao, “Statistical distribution of polarization dependent loss in the presence of polarization mode dispersion in single mode fibers,” IEEE Photon. Technol. Lett. 13(5), 451–453 (2001).
    [CrossRef]
  47. D. Voiculescu, “Limit laws for random matrices and free products,” Invent. Math. 104(1), 201–220 (1991).
    [CrossRef]
  48. T. Tao and V. H. Vu, “From the Littlewood-Offord problem to the circular law: Universality of the spectral distribution of random matrices,” Bull. Am. Math. Soc. 46(3), 377–396 (2009).
    [CrossRef]
  49. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 2nd ed. (Oxford, 1992).
  50. T. S. Rapport, Wireless Communications: Principles & Practice (Prentice Hall, 1996).
  51. K.-P. Ho, “Statistical properties of stimulated Raman crosstalk in WDM systems,” J. Lightwave Technol. 18(7), 915–921 (2000).
    [CrossRef]
  52. K.-P. Ho, “Central limits for the products of free random variables,” http://arxiv.org/abs/1101.5220 .
  53. H. Bercovici and V. Pata, “Limit laws for products of free and independent random variables,” Studia Math. 141, 43–52 (2000).
  54. G. Chistyakov and F. Götze, “Limit theorems in free probability theory. II,” Central Eur. J. Math. 6(1), 87–117 (2008).
    [CrossRef]
  55. V. Kargin, “The norm of products of free random variables,” Probab. Theory Relat. Fields 139(3-4), 397–413 (2007).
    [CrossRef]
  56. E. Wigner, “Characteristic vectors of bordered matrices with infinite dimensions,” Ann. Math. 62(3), 548–564 (1955).
    [CrossRef]
  57. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971).
    [CrossRef] [PubMed]
  58. K. Koebele, M. Salsi, G. Charlet, and S. Bigo, “Nonlinear effects in long-haul transmission over bimodal optical fibre,” in ECOC ’10, paper Mo.2.C.6.
  59. D. E. Knuth, The Art of Computer Programming II: Seminumerical Algorithms, 3rd ed. (Addison Wesley, 1998).
  60. H. G. Golub and C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins, 1996).
  61. I. Dumitriu and A. Edelman, “Matrix models for beta ensembles,” J. Math. Phys. 43(11), 5830–5847 (2002).
    [CrossRef]
  62. A. S. Hedayat, N. J. A. Sloane, and J. Stufken, Orthogonal Arrays: Theory and Applications, (Springer, 1999).

2011

2010

2009

2008

2007

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links,” IEEE Commun. Mag. 45(5), 57–63 (2007).
[CrossRef]

V. Kargin, “The norm of products of free random variables,” Probab. Theory Relat. Fields 139(3-4), 397–413 (2007).
[CrossRef]

2006

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10(3), 1089–7798 (2006).
[CrossRef]

2005

A. F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Develop. 49(4), 755–775 (2005).
[CrossRef]

A. R. Shah, R. C. J. Hsu, A. Tarighat, A. H. Sayed, and B. Jalali, “Coherent optical MIMO (COMIMO),” J. Lightwave Technol. 23(8), 2410–2419 (2005).
[CrossRef]

2004

S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,” IEEE Trans. Commun. 52(11), 1988–1996 (2004).
[CrossRef]

X. Zhu and R. D. Murch, “Layered space-frequency equalization in a single-carrier MIMO system for frequency-selective channels,” IEEE Trans. Wirel. Comm. 3(3), 701–708 (2004).
[CrossRef]

2003

J. Shentu, K. Panta, and J. Armstrong, “Effects of phase noise on performance of OFDM systems using an ICI cancellation scheme,” IEEE Trans. Broadcast 49(2), 221–224 (2003).
[CrossRef]

A. Glatarossa and L. Palmieri, “The exact statistics of polarization-dependent loss in fiber-optic links,” IEEE Photon. Technol. Lett. 15(1), 57–59 (2003).
[CrossRef]

2002

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14(3), 313–315 (2002).
[CrossRef]

I. Dumitriu and A. Edelman, “Matrix models for beta ensembles,” J. Math. Phys. 43(11), 5830–5847 (2002).
[CrossRef]

2001

P. Lu, L. Chen, and X. Bao, “Statistical distribution of polarization dependent loss in the presence of polarization mode dispersion in single mode fibers,” IEEE Photon. Technol. Lett. 13(5), 451–453 (2001).
[CrossRef]

2000

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[CrossRef] [PubMed]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
[CrossRef] [PubMed]

H. Bercovici and V. Pata, “Limit laws for products of free and independent random variables,” Studia Math. 141, 43–52 (2000).

K.-P. Ho, “Statistical properties of stimulated Raman crosstalk in WDM systems,” J. Lightwave Technol. 18(7), 915–921 (2000).
[CrossRef]

1999

B. Rosinski, J. W. D. Chi, P. Grosso, and J. Le Bihan, “Multichannel transmission of a multicore fiber coupled with vertical-cavity surface-emitting lasers,” J. Lightwave Technol. 17(5), 807–810 (1999).
[CrossRef]

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results,” IEEE J. Sel Areas in Commun. 17(3), 451–460 (1999).
[CrossRef]

1998

1991

D. Voiculescu, “Limit laws for random matrices and free products,” Invent. Math. 104(1), 201–220 (1991).
[CrossRef]

1986

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22(19), 1029–1030 (1986).
[CrossRef]

1984

J. E. Cohen and C. M. Newman, “The stability of large random matrices and their products,” Ann. Probab. 12(2), 283–310 (1984).
[CrossRef]

M. A. Berger, “Central limit theorem for products of random matrices,” Trans. Am. Math. Soc. 285(2), 777–803 (1984).
[CrossRef]

1975

1972

D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1780 (1972).

1971

1960

H. Furstenberg and H. Kesten, “Products of random matrices,” Ann. Math. Stat. 31(2), 457–469 (1960).
[CrossRef]

1955

E. Wigner, “Characteristic vectors of bordered matrices with infinite dimensions,” Ann. Math. 62(3), 548–564 (1955).
[CrossRef]

1954

R. Bellman, “Limit theorems for non-commutative operations I,” Duke Math. J. 21(3), 491–500 (1954).
[CrossRef]

Agmon, A.

Armstrong, J.

J. Shentu, K. Panta, and J. Armstrong, “Effects of phase noise on performance of OFDM systems using an ICI cancellation scheme,” IEEE Trans. Broadcast 49(2), 221–224 (2003).
[CrossRef]

Bao, H.

Bao, X.

P. Lu, L. Chen, and X. Bao, “Statistical distribution of polarization dependent loss in the presence of polarization mode dispersion in single mode fibers,” IEEE Photon. Technol. Lett. 13(5), 451–453 (2001).
[CrossRef]

Bar-Ness, Y.

S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,” IEEE Trans. Commun. 52(11), 1988–1996 (2004).
[CrossRef]

Bellman, R.

R. Bellman, “Limit theorems for non-commutative operations I,” Duke Math. J. 21(3), 491–500 (1954).
[CrossRef]

Benner, A. F.

A. F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Develop. 49(4), 755–775 (2005).
[CrossRef]

Bercovici, H.

H. Bercovici and V. Pata, “Limit laws for products of free and independent random variables,” Studia Math. 141, 43–52 (2000).

Berger, M. A.

M. A. Berger, “Central limit theorem for products of random matrices,” Trans. Am. Math. Soc. 285(2), 777–803 (1984).
[CrossRef]

Calderbank, A. R.

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results,” IEEE J. Sel Areas in Commun. 17(3), 451–460 (1999).
[CrossRef]

Chen, L.

P. Lu, L. Chen, and X. Bao, “Statistical distribution of polarization dependent loss in the presence of polarization mode dispersion in single mode fibers,” IEEE Photon. Technol. Lett. 13(5), 451–453 (2001).
[CrossRef]

Chi, J. W. D.

Chistyakov, G.

G. Chistyakov and F. Götze, “Limit theorems in free probability theory. II,” Central Eur. J. Math. 6(1), 87–117 (2008).
[CrossRef]

Cohen, J. E.

J. E. Cohen and C. M. Newman, “The stability of large random matrices and their products,” Ann. Probab. 12(2), 283–310 (1984).
[CrossRef]

Cunningham, D. G.

Dimarcello, F. V.

Dumitriu, I.

I. Dumitriu and A. Edelman, “Matrix models for beta ensembles,” J. Math. Phys. 43(11), 5830–5847 (2002).
[CrossRef]

Edelman, A.

I. Dumitriu and A. Edelman, “Matrix models for beta ensembles,” J. Math. Phys. 43(11), 5830–5847 (2002).
[CrossRef]

Fini, J. M.

Fishteyn, M.

Furstenberg, H.

H. Furstenberg and H. Kesten, “Products of random matrices,” Ann. Math. Stat. 31(2), 457–469 (1960).
[CrossRef]

Glatarossa, A.

A. Glatarossa and L. Palmieri, “The exact statistics of polarization-dependent loss in fiber-optic links,” IEEE Photon. Technol. Lett. 15(1), 57–59 (2003).
[CrossRef]

Gloge, D.

D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1780 (1972).

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

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
[CrossRef] [PubMed]

Götze, F.

G. Chistyakov and F. Götze, “Limit theorems in free probability theory. II,” Central Eur. J. Math. 6(1), 87–117 (2008).
[CrossRef]

Grosso, P.

Ho, K.-P.

Hsu, R. C. J.

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links,” IEEE Commun. Mag. 45(5), 57–63 (2007).
[CrossRef]

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10(3), 1089–7798 (2006).
[CrossRef]

A. R. Shah, R. C. J. Hsu, A. Tarighat, A. H. Sayed, and B. Jalali, “Coherent optical MIMO (COMIMO),” J. Lightwave Technol. 23(8), 2410–2419 (2005).
[CrossRef]

Ignatowski, M.

A. F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Develop. 49(4), 755–775 (2005).
[CrossRef]

Jafarkhani, H.

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results,” IEEE J. Sel Areas in Commun. 17(3), 451–460 (1999).
[CrossRef]

Jalali, B.

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links,” IEEE Commun. Mag. 45(5), 57–63 (2007).
[CrossRef]

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10(3), 1089–7798 (2006).
[CrossRef]

A. R. Shah, R. C. J. Hsu, A. Tarighat, A. H. Sayed, and B. Jalali, “Coherent optical MIMO (COMIMO),” J. Lightwave Technol. 23(8), 2410–2419 (2005).
[CrossRef]

Kahn, J. M.

Kargin, V.

V. Kargin, “The norm of products of free random variables,” Probab. Theory Relat. Fields 139(3-4), 397–413 (2007).
[CrossRef]

Kash, J. A.

A. F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Develop. 49(4), 755–775 (2005).
[CrossRef]

Kesten, H.

H. Furstenberg and H. Kesten, “Products of random matrices,” Ann. Math. Stat. 31(2), 457–469 (1960).
[CrossRef]

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
[CrossRef] [PubMed]

Kuchta, D. M.

A. F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Develop. 49(4), 755–775 (2005).
[CrossRef]

Lau, A. P. T.

Le Bihan, J.

Lu, P.

P. Lu, L. Chen, and X. Bao, “Statistical distribution of polarization dependent loss in the presence of polarization mode dispersion in single mode fibers,” IEEE Photon. Technol. Lett. 13(5), 451–453 (2001).
[CrossRef]

Ma, Y.

Z. Tong, Q. Yang, Y. Ma, and W. Shieh, “21.4 Gbit/s transmission over 200 km multimode fiber using coherent optical OFDM,” Electron. Lett. 44(23), 1373–1374 (2008).
[CrossRef]

Mao, W.

Mecozzi, A.

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14(3), 313–315 (2002).
[CrossRef]

Monberg, E. M.

Murch, R. D.

X. Zhu and R. D. Murch, “Layered space-frequency equalization in a single-carrier MIMO system for frequency-selective channels,” IEEE Trans. Wirel. Comm. 3(3), 701–708 (2004).
[CrossRef]

Nazarathy, M.

Newman, C. M.

J. E. Cohen and C. M. Newman, “The stability of large random matrices and their products,” Ann. Probab. 12(2), 283–310 (1984).
[CrossRef]

Nowell, M. C.

Olshansky, R.

Palmieri, L.

A. Glatarossa and L. Palmieri, “The exact statistics of polarization-dependent loss in fiber-optic links,” IEEE Photon. Technol. Lett. 15(1), 57–59 (2003).
[CrossRef]

Panicker, R. A.

Panta, K.

J. Shentu, K. Panta, and J. Armstrong, “Effects of phase noise on performance of OFDM systems using an ICI cancellation scheme,” IEEE Trans. Broadcast 49(2), 221–224 (2003).
[CrossRef]

Pata, V.

H. Bercovici and V. Pata, “Limit laws for products of free and independent random variables,” Studia Math. 141, 43–52 (2000).

Poole, C. D.

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22(19), 1029–1030 (1986).
[CrossRef]

Raddatz, L.

Ritter, M. B.

A. F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Develop. 49(4), 755–775 (2005).
[CrossRef]

Rosinski, B.

Sayed, A. H.

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links,” IEEE Commun. Mag. 45(5), 57–63 (2007).
[CrossRef]

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10(3), 1089–7798 (2006).
[CrossRef]

A. R. Shah, R. C. J. Hsu, A. Tarighat, A. H. Sayed, and B. Jalali, “Coherent optical MIMO (COMIMO),” J. Lightwave Technol. 23(8), 2410–2419 (2005).
[CrossRef]

Shah, A.

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links,” IEEE Commun. Mag. 45(5), 57–63 (2007).
[CrossRef]

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10(3), 1089–7798 (2006).
[CrossRef]

Shah, A. R.

Shemirani, M. B.

Shentu, J.

J. Shentu, K. Panta, and J. Armstrong, “Effects of phase noise on performance of OFDM systems using an ICI cancellation scheme,” IEEE Trans. Broadcast 49(2), 221–224 (2003).
[CrossRef]

Shieh, W.

Shtaif, M.

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14(3), 313–315 (2002).
[CrossRef]

Stuart, H. R.

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[CrossRef] [PubMed]

Tang, Y.

Tao, T.

T. Tao and V. H. Vu, “From the Littlewood-Offord problem to the circular law: Universality of the spectral distribution of random matrices,” Bull. Am. Math. Soc. 46(3), 377–396 (2009).
[CrossRef]

Tarighat, A.

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links,” IEEE Commun. Mag. 45(5), 57–63 (2007).
[CrossRef]

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10(3), 1089–7798 (2006).
[CrossRef]

A. R. Shah, R. C. J. Hsu, A. Tarighat, A. H. Sayed, and B. Jalali, “Coherent optical MIMO (COMIMO),” J. Lightwave Technol. 23(8), 2410–2419 (2005).
[CrossRef]

Tarokh, V.

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results,” IEEE J. Sel Areas in Commun. 17(3), 451–460 (1999).
[CrossRef]

Taunay, T. F.

Tong, Z.

Z. Tong, Q. Yang, Y. Ma, and W. Shieh, “21.4 Gbit/s transmission over 200 km multimode fiber using coherent optical OFDM,” Electron. Lett. 44(23), 1373–1374 (2008).
[CrossRef]

Voiculescu, D.

D. Voiculescu, “Limit laws for random matrices and free products,” Invent. Math. 104(1), 201–220 (1991).
[CrossRef]

Vu, V. H.

T. Tao and V. H. Vu, “From the Littlewood-Offord problem to the circular law: Universality of the spectral distribution of random matrices,” Bull. Am. Math. Soc. 46(3), 377–396 (2009).
[CrossRef]

Wagner, R. E.

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22(19), 1029–1030 (1986).
[CrossRef]

White, I. H.

Wigner, E.

E. Wigner, “Characteristic vectors of bordered matrices with infinite dimensions,” Ann. Math. 62(3), 548–564 (1955).
[CrossRef]

Wu, S.

S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,” IEEE Trans. Commun. 52(11), 1988–1996 (2004).
[CrossRef]

Xie, C.

Yan, M. F.

Yang, Q.

Z. Tong, Q. Yang, Y. Ma, and W. Shieh, “21.4 Gbit/s transmission over 200 km multimode fiber using coherent optical OFDM,” Electron. Lett. 44(23), 1373–1374 (2008).
[CrossRef]

Zhu, B.

Zhu, X.

X. Zhu and R. D. Murch, “Layered space-frequency equalization in a single-carrier MIMO system for frequency-selective channels,” IEEE Trans. Wirel. Comm. 3(3), 701–708 (2004).
[CrossRef]

Ann. Math.

E. Wigner, “Characteristic vectors of bordered matrices with infinite dimensions,” Ann. Math. 62(3), 548–564 (1955).
[CrossRef]

Ann. Math. Stat.

H. Furstenberg and H. Kesten, “Products of random matrices,” Ann. Math. Stat. 31(2), 457–469 (1960).
[CrossRef]

Ann. Probab.

J. E. Cohen and C. M. Newman, “The stability of large random matrices and their products,” Ann. Probab. 12(2), 283–310 (1984).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1780 (1972).

Bull. Am. Math. Soc.

T. Tao and V. H. Vu, “From the Littlewood-Offord problem to the circular law: Universality of the spectral distribution of random matrices,” Bull. Am. Math. Soc. 46(3), 377–396 (2009).
[CrossRef]

Central Eur. J. Math.

G. Chistyakov and F. Götze, “Limit theorems in free probability theory. II,” Central Eur. J. Math. 6(1), 87–117 (2008).
[CrossRef]

Duke Math. J.

R. Bellman, “Limit theorems for non-commutative operations I,” Duke Math. J. 21(3), 491–500 (1954).
[CrossRef]

Electron. Lett.

Z. Tong, Q. Yang, Y. Ma, and W. Shieh, “21.4 Gbit/s transmission over 200 km multimode fiber using coherent optical OFDM,” Electron. Lett. 44(23), 1373–1374 (2008).
[CrossRef]

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22(19), 1029–1030 (1986).
[CrossRef]

IBM J. Res. Develop.

A. F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B. Ritter, “Exploitation of optical interconnects in future server architectures,” IBM J. Res. Develop. 49(4), 755–775 (2005).
[CrossRef]

IEEE Commun. Lett.

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10(3), 1089–7798 (2006).
[CrossRef]

IEEE Commun. Mag.

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links,” IEEE Commun. Mag. 45(5), 57–63 (2007).
[CrossRef]

IEEE J. Sel Areas in Commun.

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results,” IEEE J. Sel Areas in Commun. 17(3), 451–460 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14(3), 313–315 (2002).
[CrossRef]

A. Glatarossa and L. Palmieri, “The exact statistics of polarization-dependent loss in fiber-optic links,” IEEE Photon. Technol. Lett. 15(1), 57–59 (2003).
[CrossRef]

P. Lu, L. Chen, and X. Bao, “Statistical distribution of polarization dependent loss in the presence of polarization mode dispersion in single mode fibers,” IEEE Photon. Technol. Lett. 13(5), 451–453 (2001).
[CrossRef]

IEEE Trans. Broadcast

J. Shentu, K. Panta, and J. Armstrong, “Effects of phase noise on performance of OFDM systems using an ICI cancellation scheme,” IEEE Trans. Broadcast 49(2), 221–224 (2003).
[CrossRef]

IEEE Trans. Commun.

S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,” IEEE Trans. Commun. 52(11), 1988–1996 (2004).
[CrossRef]

IEEE Trans. Wirel. Comm.

X. Zhu and R. D. Murch, “Layered space-frequency equalization in a single-carrier MIMO system for frequency-selective channels,” IEEE Trans. Wirel. Comm. 3(3), 701–708 (2004).
[CrossRef]

Invent. Math.

D. Voiculescu, “Limit laws for random matrices and free products,” Invent. Math. 104(1), 201–220 (1991).
[CrossRef]

J. Lightwave Technol.

J. Math. Phys.

I. Dumitriu and A. Edelman, “Matrix models for beta ensembles,” J. Math. Phys. 43(11), 5830–5847 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Probab. Theory Relat. Fields

V. Kargin, “The norm of products of free random variables,” Probab. Theory Relat. Fields 139(3-4), 397–413 (2007).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
[CrossRef] [PubMed]

Science

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[CrossRef] [PubMed]

Studia Math.

H. Bercovici and V. Pata, “Limit laws for products of free and independent random variables,” Studia Math. 141, 43–52 (2000).

Trans. Am. Math. Soc.

M. A. Berger, “Central limit theorem for products of random matrices,” Trans. Am. Math. Soc. 285(2), 777–803 (1984).
[CrossRef]

Other

A. Crisanti, G. Paladin, and A. Vulpiani, Products of Random Matrices in Statistical Physics (Springer, 1993).

D. Voiculescu, K. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series, vol. 1, (American Mathematical Society, 1992).

A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, vol. 335, (Cambridge Univ. Press, 2006).

H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion” in Optical Fiber Telecommunications IVB: Systems and Impairments, I. Kaminow and T. Li, eds, (Academic Press, 2002).

P. J. Winzer and G. J. Foschini, “Outage calculations for spatially multiplexed fiber links,” in OFC ’11, paper OThO5.

G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 2nd ed. (Oxford, 1992).

T. S. Rapport, Wireless Communications: Principles & Practice (Prentice Hall, 1996).

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

A. M. Tulino and S. Verdú, Random Matrix Theory and Wireless Communications (Now, 2004).

D. Tse and P. Viswanath, Fundamentals of Wireless Communication (Cambridge Univ. Press, 2005).

B. Vucetic and J. Yuan, Space-Time Coding (Wiley, 2003).

H. Bülow, “Coherent multichannel transmission over multimode-fiber and related signal processing,” Proc. of OSA Topical Meeting on Access Networks and In-House Communications, Karlsruhe, Germany, June 21–24, 2010, paper AThB1.

A. Li, A. Al Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” in OFC ’11, paper PDPB8.

R. Ryf, S. Randel, A. H. Gnuack, C. Bolle, R.-J. Essiambre, P. Winzer, D. W. Peckham, A. McCurdy, and R. Lingle, “Space-division multiplexing over 10 km of three-mode fiber using coherent 6×6 MIMO processing,” in OFC ’11, paper PDPB10.

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, and G. Charlet, “Transmission at 2×100 Gb/s, over two modes of 40 km-long prototype few-mode fiber, using LCOS based mode multiplexer and demultiplexer,” in OFC ’11, paper PDPB9.

Y. Koike and S. Takahashi, “Plastic optical fibers: technologies and communication links,” in Optical Fiber Telecommunications VB: Systems and Networks, I. P. Kaminow, T. Li and A. E. Willner eds. (Elsevier Academic, 2008).

K.-P. Ho and J. M. Kahn, “Statistics of group delays in multimode fiber with strong mode coupling,” to be published in J. Lightwave Technol., http://arxiv.org/abs/1104.4527

M. L. Mehta, Random Matrices, 3rd ed. (Elsevier Academic, 2004).

A. S. Hedayat, N. J. A. Sloane, and J. Stufken, Orthogonal Arrays: Theory and Applications, (Springer, 1999).

K.-P. Ho, “Central limits for the products of free random variables,” http://arxiv.org/abs/1101.5220 .

K. Koebele, M. Salsi, G. Charlet, and S. Bigo, “Nonlinear effects in long-haul transmission over bimodal optical fibre,” in ECOC ’10, paper Mo.2.C.6.

D. E. Knuth, The Art of Computer Programming II: Seminumerical Algorithms, 3rd ed. (Addison Wesley, 1998).

H. G. Golub and C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins, 1996).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

MDL in a two-mode fiber. (a) Comparing the simulated distribution of the overall MDL with the two-sided Maxwellian distribution given in Table 1. The x-axis for each curve is normalized by the simulated STD of the overall MDL σmdl. (b) Comparing the simulated STD of overall MDL with approximations (1) and (8). The simulated mean MDL difference is also shown for comparison.

Fig. 2
Fig. 2

MDL in MMF with D = 64 modes. (a) Comparing the simulated distribution of the overall MDL with the semicircle distribution given in Table 1. The x-axis for each curve is normalized by the simulated STD of the overall MDL σmdl. (b) Comparing the simulated STD of overall MDL with approximation (1). The simulated mean maximum MDL difference is also shown for comparison.

Fig. 3
Fig. 3

MDL in MMF with D = 4 modes. (a) Comparing the simulated distribution of the overall MDL with the four-peak distribution given in Table 1. The x-axis for each curve is normalized by the simulated STD of the overall MDL σmdl. (b) Comparing the simulated STD of overall MDL with approximation (1). The simulated mean maximum MDL difference is also shown for comparison.

Fig. 4
Fig. 4

MDL in MMF with D = 8 modes. (a) Comparing the simulated distribution of the overall MDL with the eight-peak distribution given in Table 1. (b) Comparing the simulated STD of overall MDL with approximation (1). The simulated mean maximum MDL difference is also shown for comparison.

Fig. 5
Fig. 5

Channel capacity of a MMF without MDL as a function of SNR ρt for various numbers of modes D.

Fig. 6
Fig. 6

Average channel capacity of a MMF with MDL and without CSI as a function of ξ for fibers with various numbers of modes D. Equal power is allocated to each mode, and SNR is ρ t = 10 dB. Theoretical results are shown as curves and simulated results are shown as circles.

Fig. 7
Fig. 7

Average channel capacity of a MMF with MDL and with CSI as a function of ξ for fibers with various numbers of modes D. Transmit power is allocated optimally to each mode, and SNR is ρ t = 10 dB. Theoretical results are shown as curves and simulated results are shown as circles.

Fig. 8
Fig. 8

Outage capacity of a MMF with MDL as a function of ξ for fibers with various numbers of modes D. The outage probability is 10−3, and SNR is ρ t = 10 dB. Theoretical results are shown as curves and simulated results are shown as open symbols. Blue curves/blue circles: with CSI; green curves/green squares: without CSI.

Fig. 9
Fig. 9

Spatial non-whiteness of the output noise, quantified by the STD of the spatial spectral distribution as a function of 1 / K , where K is the number of noise sources, for D = 8 modes. The markers are simulation results, and the solid lines are fitted numerically. The upper two data sets are for ξ = 10 dB and the lower two are for ξ = 5 dB. Red and blue data sets represent the maximum and mean of the STD over 100 random realizations at each value of K.

Tables (1)

Tables Icon

Table 1 Probability Distributions of Normalized MDL Having Unit Variance.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

σ mdl = ξ 1 + 1 12 ξ 2 .
M ( k ) ( ω ) = V ( k ) Λ ( k ) ( ω ) U ( k ) ,                 k = 1 , , K ,
Λ ( k ) ( ω ) = diag [ e 1 2 g 1 ( k ) j ω τ 1 ( k ) , e 1 2 g 2 ( k ) j ω τ 2 ( k ) , , e 1 2 g D ( k ) j ω τ D ( k )     ] ,
g ( k ) = ( g 1 ( k ) , g 2 ( k ) , ... , g D ( k ) )
M ( t ) = M ( K ) M ( 2 ) M ( 1 ) .
M ( t ) = V ( t ) Λ ( t ) U ( t ) * ,
ξ 2 = σ g ( 1 ) 2 + σ g ( 2 ) 2 + + σ g ( K ) 2 ,
σ mdl = 3 2 exp ( 2 9 ξ 2 ) 1 ,
C = D log 2 ( 1 + ρ t D ) .
C = D + log 2 [ 1 + χ D exp ( σ mdl x ) ] p D ( x ) d x ,
χ = ρ t + exp ( σ mdl x ) p D ( x ) d x .
χ = ρ t σ mdl 2 I 1 ( σ mdl ) ,
i = 1 D [ μ e g i ( t ) ] + = χ ,
C = E { i = 1 D log 2 ( 1 + [ μ e g i ( t ) 1 ] + ) } ,
Pr { i = 1 D log 2 ( 1 + [ μ e g i ( t ) 1 ] + ) C out } = p out ,
σ mdl = ξ 1 + 1 κ ξ 2 ,
V ( K ) diag [ σ K , 1 2 , σ K , 2 2 , , σ K , D 2 ] V ( K ) * .
E { V ( K ) diag [ σ K , 1 2 , σ K , 2 2 , , σ K , D 2 ] V ( K ) * } = σ K , 1 2 + σ K , 2 2 + + σ K , D 2 D I ,

Metrics