Abstract

We generalize the concept of principal states of polarization and prove the existence of principal modes in multimode waveguides. Principal modes do not suffer from modal dispersion to first order of frequency variation and form orthogonal bases at both the input and the output ends of the waveguide. We show that principal modes are generally different from eigenmodes, even in uniform waveguides, unlike the special case of a single-mode fiber with uniform birefringence. The difference is most pronounced when different eigenmodes possess similar group velocities and when their field patterns vary as a function of frequency. This work may provide a new basis for analysis and control of dispersion in multimode fiber systems.

© 2005 Optical Society of America

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References

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  1. C. F. Lam, in Optical Fiber Telecommunications IVB: Systems and Impairments, I. Kaminow and T. Li, eds. (Academic, San Diego, Calif., 2002), Chap. 11.
  2. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, New York, 2002).
    [CrossRef]
  3. C. D. Poole and R. E. Wagner, Electron. Lett. 22, 1029 (1986).
    [CrossRef]
  4. J. P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
    [CrossRef]
  5. G. J. Foschini and C. D. Poole, J. Lightwave Technol. 9, 1439 (1991).
    [CrossRef]
  6. E. Alon, V. Stojanovi?, J. M. Kahn, S. Boyd, and M. A. Horowitz, “Equalization of modal dispersion in multimode fiber using spatial light modulators,” to be presented at IEEE Global Telecommunications Conference, Dallas, Texas, November 29–December 3, 2004.
  7. H. R. Stuart, Science 289, 281 (2000).
    [CrossRef] [PubMed]
  8. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  9. M. Skorobogaity, M. Ibanescu, S. G. Johnson, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, J. Opt. Soc. Am. B 19, 2867 (2002).
    [CrossRef]

2002 (1)

2000 (2)

J. P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
[CrossRef]

H. R. Stuart, Science 289, 281 (2000).
[CrossRef] [PubMed]

1991 (1)

G. J. Foschini and C. D. Poole, J. Lightwave Technol. 9, 1439 (1991).
[CrossRef]

1986 (1)

C. D. Poole and R. E. Wagner, Electron. Lett. 22, 1029 (1986).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, New York, 2002).
[CrossRef]

Alon, E.

E. Alon, V. Stojanovi?, J. M. Kahn, S. Boyd, and M. A. Horowitz, “Equalization of modal dispersion in multimode fiber using spatial light modulators,” to be presented at IEEE Global Telecommunications Conference, Dallas, Texas, November 29–December 3, 2004.

Boyd, S.

E. Alon, V. Stojanovi?, J. M. Kahn, S. Boyd, and M. A. Horowitz, “Equalization of modal dispersion in multimode fiber using spatial light modulators,” to be presented at IEEE Global Telecommunications Conference, Dallas, Texas, November 29–December 3, 2004.

Engeness, T. D.

Fink, Y.

Foschini, G. J.

G. J. Foschini and C. D. Poole, J. Lightwave Technol. 9, 1439 (1991).
[CrossRef]

Gordon, J. P.

J. P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
[CrossRef]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Horowitz, M. A.

E. Alon, V. Stojanovi?, J. M. Kahn, S. Boyd, and M. A. Horowitz, “Equalization of modal dispersion in multimode fiber using spatial light modulators,” to be presented at IEEE Global Telecommunications Conference, Dallas, Texas, November 29–December 3, 2004.

Ibanescu, M.

Jacobs, S. A.

Johnson, S. G.

Kahn, J. M.

E. Alon, V. Stojanovi?, J. M. Kahn, S. Boyd, and M. A. Horowitz, “Equalization of modal dispersion in multimode fiber using spatial light modulators,” to be presented at IEEE Global Telecommunications Conference, Dallas, Texas, November 29–December 3, 2004.

Kogelnik, H.

J. P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
[CrossRef]

Lam, C. F.

C. F. Lam, in Optical Fiber Telecommunications IVB: Systems and Impairments, I. Kaminow and T. Li, eds. (Academic, San Diego, Calif., 2002), Chap. 11.

Poole, C. D.

G. J. Foschini and C. D. Poole, J. Lightwave Technol. 9, 1439 (1991).
[CrossRef]

C. D. Poole and R. E. Wagner, Electron. Lett. 22, 1029 (1986).
[CrossRef]

Skorobogaity, M.

Soljacic, M.

Stojanovic, V.

E. Alon, V. Stojanovi?, J. M. Kahn, S. Boyd, and M. A. Horowitz, “Equalization of modal dispersion in multimode fiber using spatial light modulators,” to be presented at IEEE Global Telecommunications Conference, Dallas, Texas, November 29–December 3, 2004.

Stuart, H. R.

H. R. Stuart, Science 289, 281 (2000).
[CrossRef] [PubMed]

Wagner, R. E.

C. D. Poole and R. E. Wagner, Electron. Lett. 22, 1029 (1986).
[CrossRef]

Weisberg, O.

Electron. Lett. (1)

C. D. Poole and R. E. Wagner, Electron. Lett. 22, 1029 (1986).
[CrossRef]

J. Lightwave Technol. (1)

G. J. Foschini and C. D. Poole, J. Lightwave Technol. 9, 1439 (1991).
[CrossRef]

J. Opt. Soc. Am. B (1)

Proc. Natl. Acad. Sci. USA (1)

J. P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
[CrossRef]

Science (1)

H. R. Stuart, Science 289, 281 (2000).
[CrossRef] [PubMed]

Other (4)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

E. Alon, V. Stojanovi?, J. M. Kahn, S. Boyd, and M. A. Horowitz, “Equalization of modal dispersion in multimode fiber using spatial light modulators,” to be presented at IEEE Global Telecommunications Conference, Dallas, Texas, November 29–December 3, 2004.

C. F. Lam, in Optical Fiber Telecommunications IVB: Systems and Impairments, I. Kaminow and T. Li, eds. (Academic, San Diego, Calif., 2002), Chap. 11.

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, New York, 2002).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Schematic of the dispersion relations for a multimode waveguide system in the vicinity of the phase-matching frequency. The system, shown in the inset consists of two single-mode waveguides coupled together. The dashed lines represent the dispersion relations of the individual waveguides. The solid curves represent the dispersion relations of the two eigenmodes of the coupled system. ω0 is the frequency at which phase-matched coupling between the two waveguides occurs.

Fig. 2
Fig. 2

Properties of the two principal modes, represented as solid and dashed curves, as a function of propagation distance z, for the system shown in Fig. 1. κ is the coupling constant between the waveguides. (a) Normalized group delay. The vertical axis corresponds to τ/2κ1/ν1-1/ν2, where τ is the group delay and ν1 and ν2 are the group velocities of the individual waveguides. (b) Fraction of the total optical power that is localized in waveguide 1 at the input. Notice that the principal modes vary as a function of propagation distance, unlike eigenmodes in this uniform system.

Equations (17)

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b|b=Ta|a,
Tω=expϕωUω,
bω|b+b|bω=eϕϕωU+Uω|a
Fω|a=τ|a,
Fω-iU+ωUωω
τ=iϕω-1bbω
Θω|n=βn|n,
mFn=imddωn-mU+ddωUn=zdβndωδmn+i{1-exp[iβn-βmz}×mddωn.
dn|dω=-βnI-Θ-1dβndωI-dΘdωn|.
mFn=zdβndωδmn+i1-expiβn-βmzβn-βm×mdΘdωn1-δmn=zdβndωδmn+z sincβn-βmz]2×expiβn-βmz2mdΘdωn1-δmn,
Θ=β0+δκκβ0-δ,
β012ω-ω0ν1+ω-ω0ν2, δ12ω-ω0ν1-ω-ω0ν2
|1=1211,    |2=121-1.
dΘdω=121ν1-1ν2100-1
F=z21ν1-1ν2sincκzcosκzi sinκz-i sinκz-cosκz.
τ1,2=±z21ν1-1ν2sincκz=±12κ1ν1-1ν2sinκz.
|p1=cosκz2-i sinκz2,    |p2=sinκz2i cosκz2.

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