Abstract

The dynamics of modes and their states of polarizations in multimode fibers as a function of time, space, and wavelength are experimentally and theoretically investigated. The results reveal that the states of polarizations are displaced in Poincaré sphere representation when varying the angular orientations of the polarization at the incident light. Such displacements, which complicate the interpretation of the results, are overcome by resorting to modified Poincaré sphere representation. With such modification it should be possible to predict the output modes and their state of polarization when the input mode and state of polarization are known.

© 2012 Optical Society of America

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  1. P. Pepeljugoski, M. J. Hackert, J. S. Abbott, S. E. Swanson, S. E. Golowich, A. J. Ritger, P. Kolesar, Y. C. Chen, and P. Pleunis, “Development of system specification for laser-optimized 50 μm multimode fiber for multigigabit short-wavelength LANs,” J. Lightwave Technol. 21, 1256–1275 (2003).
    [CrossRef]
  2. C. Lethien, C. Loyez, and J. P. Vilcot, “Potentials of radio over multimode fiber systems for the in-buildings coverage of mobile and wireless LAN applications,” J. Lightwave Technol. 17, 2793–2795 (2005).
  3. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
    [CrossRef]
  4. B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992).
    [CrossRef]
  5. C. D. Poole and R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
    [CrossRef]
  6. W. Shieh and H. Kogelnik, “Dynamic eigenstates of polarization,” IEEE Photon. Technol. Lett. 13, 40–42 (2001).
    [CrossRef]
  7. C. D. Poole, J. H. Winters, and J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 16, 372–374 (1991).
    [CrossRef]
  8. D. Andrescianci, F. Curti, F. Matera, and B. Daino, “Measurement of the group-delay difference between the principal state of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. 12, 844–846 (1987).
    [CrossRef]
  9. C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal state in a 147 km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190(1988).
    [CrossRef]
  10. G. P. Agrawal, Fiber-Optic Communication System, 3rd ed. (Wiley, 2002).
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    [CrossRef]
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    [CrossRef]
  13. G. S. Agarwal, “SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum,” J. Opt. Soc. Am. A 16, 2914–2916 (1999).
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  14. S. Fan and J. M. Kahn, “Principal modes in multimode waveguides,” Opt. Lett. 30, 135–137 (2005).
    [CrossRef]
  15. 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, 1248–1261 (2009).
    [CrossRef]
  16. M. Fridman, M. Nixon, M. Dubinskii, A. A. Friesem, and N. Davidson, “Principal modes in fiber amplifiers,” Opt. Lett. 36, 388–390 (2011).
    [CrossRef]
  17. M. Fridman, M. Nixon, E. Grinvald, N. Davidson, and A. A. Friesem, “Real-time measurement of unique space-variant polarizations,” Opt. Express 18, 10805–10812(2010).
    [CrossRef]
  18. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Express 34, 2525–2527 (2009).
  19. M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008).
    [CrossRef]
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    [CrossRef]
  22. H. Suchowski, Y. Silberberg, and D. B. Uskov, “Pythagorean coupling: Complete population transfer in a four-state system,” Phys. Rev. A 84, 013414 (2011).
  23. P. Du Val, “Homographies, quaternions and rotations,” in Oxford Mathematical Monographs (Clarendon, 1964).

2011 (2)

H. Suchowski, Y. Silberberg, and D. B. Uskov, “Pythagorean coupling: Complete population transfer in a four-state system,” Phys. Rev. A 84, 013414 (2011).

M. Fridman, M. Nixon, M. Dubinskii, A. A. Friesem, and N. Davidson, “Principal modes in fiber amplifiers,” Opt. Lett. 36, 388–390 (2011).
[CrossRef]

2010 (1)

2009 (2)

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, 1248–1261 (2009).
[CrossRef]

S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Express 34, 2525–2527 (2009).

2008 (2)

M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008).
[CrossRef]

D. M. Shyroki, “Exact equivalent straight waveguide model for bent and twisted waveguides,” IEEE Trans. Microwave Theory Tech. 56, 414–419 (2008).
[CrossRef]

2007 (1)

2005 (2)

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

C. Lethien, C. Loyez, and J. P. Vilcot, “Potentials of radio over multimode fiber systems for the in-buildings coverage of mobile and wireless LAN applications,” J. Lightwave Technol. 17, 2793–2795 (2005).

2003 (1)

2001 (1)

W. Shieh and H. Kogelnik, “Dynamic eigenstates of polarization,” IEEE Photon. Technol. Lett. 13, 40–42 (2001).
[CrossRef]

2000 (1)

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

1999 (2)

1992 (1)

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992).
[CrossRef]

1991 (1)

1988 (1)

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal state in a 147 km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190(1988).
[CrossRef]

1987 (1)

1986 (1)

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

1979 (1)

Abbott, J. S.

Agarwal, G. S.

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication System, 3rd ed. (Wiley, 2002).

Andrescianci, D.

Bergano, N. S.

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal state in a 147 km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190(1988).
[CrossRef]

Chen, Y. C.

Courtial, J.

Curti, F.

Daino, B.

Davidson, N.

Du Val, P.

P. Du Val, “Homographies, quaternions and rotations,” in Oxford Mathematical Monographs (Clarendon, 1964).

Dubinskii, M.

Fan, S.

Fridman, M.

Friesem, A. A.

Golowich, S. E.

Gordon, J. P.

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

Grinvald, E.

Hackert, M. J.

Heffner, B. L.

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992).
[CrossRef]

Kahn, J. M.

Kogelnik, H.

W. Shieh and H. Kogelnik, “Dynamic eigenstates of polarization,” IEEE Photon. Technol. Lett. 13, 40–42 (2001).
[CrossRef]

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

Kolesar, P.

Kristensen, P.

S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Express 34, 2525–2527 (2009).

Lethien, C.

C. Lethien, C. Loyez, and J. P. Vilcot, “Potentials of radio over multimode fiber systems for the in-buildings coverage of mobile and wireless LAN applications,” J. Lightwave Technol. 17, 2793–2795 (2005).

Loyez, C.

C. Lethien, C. Loyez, and J. P. Vilcot, “Potentials of radio over multimode fiber systems for the in-buildings coverage of mobile and wireless LAN applications,” J. Lightwave Technol. 17, 2793–2795 (2005).

Machavariani, G.

M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008).
[CrossRef]

Mao, W.

Matera, F.

Nagel, J. A.

Nixon, M.

Padgett, M. J.

Panicker, R. A.

Pepeljugoski, P.

Pleunis, P.

Poole, C. D.

C. D. Poole, J. H. Winters, and J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 16, 372–374 (1991).
[CrossRef]

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal state in a 147 km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190(1988).
[CrossRef]

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

Ramachandran, S.

S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Express 34, 2525–2527 (2009).

Ritger, A. J.

Schulte, H. J.

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal state in a 147 km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190(1988).
[CrossRef]

Shemirani, M. B.

Shieh, W.

W. Shieh and H. Kogelnik, “Dynamic eigenstates of polarization,” IEEE Photon. Technol. Lett. 13, 40–42 (2001).
[CrossRef]

Shyroki, D. M.

D. M. Shyroki, “Exact equivalent straight waveguide model for bent and twisted waveguides,” IEEE Trans. Microwave Theory Tech. 56, 414–419 (2008).
[CrossRef]

Silberberg, Y.

H. Suchowski, Y. Silberberg, and D. B. Uskov, “Pythagorean coupling: Complete population transfer in a four-state system,” Phys. Rev. A 84, 013414 (2011).

Simon, A.

Suchowski, H.

H. Suchowski, Y. Silberberg, and D. B. Uskov, “Pythagorean coupling: Complete population transfer in a four-state system,” Phys. Rev. A 84, 013414 (2011).

Swanson, S. E.

Ulrich, R.

Uskov, D. B.

H. Suchowski, Y. Silberberg, and D. B. Uskov, “Pythagorean coupling: Complete population transfer in a four-state system,” Phys. Rev. A 84, 013414 (2011).

Vilcot, J. P.

C. Lethien, C. Loyez, and J. P. Vilcot, “Potentials of radio over multimode fiber systems for the in-buildings coverage of mobile and wireless LAN applications,” J. Lightwave Technol. 17, 2793–2795 (2005).

Wagner, R. E.

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal state in a 147 km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190(1988).
[CrossRef]

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

Wielandy, S.

Winters, J. H.

Yan, M. F.

S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Express 34, 2525–2527 (2009).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008).
[CrossRef]

Electron. Lett. (1)

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

IEEE Photon. Technol. Lett. (2)

W. Shieh and H. Kogelnik, “Dynamic eigenstates of polarization,” IEEE Photon. Technol. Lett. 13, 40–42 (2001).
[CrossRef]

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

D. M. Shyroki, “Exact equivalent straight waveguide model for bent and twisted waveguides,” IEEE Trans. Microwave Theory Tech. 56, 414–419 (2008).
[CrossRef]

J. Lightwave Technol. (4)

C. Lethien, C. Loyez, and J. P. Vilcot, “Potentials of radio over multimode fiber systems for the in-buildings coverage of mobile and wireless LAN applications,” J. Lightwave Technol. 17, 2793–2795 (2005).

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal state in a 147 km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190(1988).
[CrossRef]

P. Pepeljugoski, M. J. Hackert, J. S. Abbott, S. E. Swanson, S. E. Golowich, A. J. Ritger, P. Kolesar, Y. C. Chen, and P. Pleunis, “Development of system specification for laser-optimized 50 μm multimode fiber for multigigabit short-wavelength LANs,” J. Lightwave Technol. 21, 1256–1275 (2003).
[CrossRef]

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, 1248–1261 (2009).
[CrossRef]

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

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. A (1)

H. Suchowski, Y. Silberberg, and D. B. Uskov, “Pythagorean coupling: Complete population transfer in a four-state system,” Phys. Rev. A 84, 013414 (2011).

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

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

Other (2)

G. P. Agrawal, Fiber-Optic Communication System, 3rd ed. (Wiley, 2002).

P. Du Val, “Homographies, quaternions and rotations,” in Oxford Mathematical Monographs (Clarendon, 1964).

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Figures (6)

Fig. 1.
Fig. 1.

Experimental configuration for measuring the output polarization from a multimode fiber as a function of the input wavelength for different input states of polarization.

Fig. 2.
Fig. 2.

Intensity distributions together with the orientations of the polarization direction for the orthogonal sets of linearly polarized modes LP11. The white arrows in the center of the individual intensity distributions denote the polarization directions. The black arrows and the terms u2 and u3 denote the coupling between the different modes.

Fig. 3.
Fig. 3.

Poincaré sphere representation of the SOP at the output as a function of the wavelength for different angular orientations of the linear polarization at the input. (a) Experimental results and (b) calculated results. The input wavelength ranged from 1062 nm to 1066 nm for each angular orientation.

Fig. 4.
Fig. 4.

Projection of the output polarization on the S1 axes as a function of the input wavelength for two orientations of the input polarization; asterisks (blue) 30° orientation, close to the PM; circles (red) 90° orientation, far from the PM.

Fig. 5.
Fig. 5.

Calculated modified Poincaré sphere representation of the SOP at the output as a function of the wavelength for different angular orientations of linearly polarized light at the input. (a) Using the modified Stokes parameters Sa and (b) using the modified Stokes parameters Sb.

Fig. 6.
Fig. 6.

Average polarization when varying the input wavelength for different input polarizations on (a) the original Poincare sphere and on (b) the modified Poincare sphere.

Equations (11)

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dEoutdω=HEout.
H(2)(ω)=[h1(ω)h2(ω)h2*(ω)h1*(ω)],
H(ω)=[h1(ω)h2(ω)h3(ω)0h2*(ω)h1*(ω)0h3(ω)h3*(ω)0h1(ω)h2(ω)0h3*(ω)h2*(ω)h1*(ω)].
H=(ib1σz+ia2σy+ib2σx)I+I(ib3σx+ia3σy),
E~in,out=[E1E2E3E4]in,out,
E~out(ω)=ΩL(ω)E~inΩR(ω),
ΩL(ω)=[cos(|τb|ω)u~3|τb|sin(|τb|ω)u~3*|τb|sin(|τb|ω)cos(|τb|ω)]
ΩR(ω)=[cos(|τa|ω)+u~1|τa|sin(|τa|ω)u~2*|τa|sin(|τa|ω)u~2|τa|sin(|τa|ω)cos(|τa|ω)u~1|τa|sin(|τa|ω)]
Sa(1)=|E1|2|E2|2+|E3|2|E4|2Sa(2)=2|E1||E2|cos(φ12)+2|E3||E4|cos(φ34)Sa(3)=2|E1||E2|sin(φ12)+2|E3||E4|sin(φ34)
Sb(1)=|E1|2+|E2|2|E3|2|E4|2Sb(2)=2|E1||E3|cos(φ13)+2|E2||E4|cos(φ24)Sb(3)=2|E1||E3|sin(φ13)+2|E2||E4|sin(φ24),
S.a=τa×SaS.b=τb×Sb.

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