Abstract

We use high-resolution imaging of Rayleigh scattered light through the side of few-mode optical fibers to measure the local spatial structure of propagating vector fields. We demonstrate the technique by imaging both pure modes and superpositions of modes in the LP01 and LP11 families. Direct imaging not only gives high-resolution beat length measurements, but also records the local propagation dynamics including those due to perturbations. The imaging setup uses polarization discrimination to monitor both the transverse and the longitudinal polarization components simultaneously.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]

2013 (5)

2012 (3)

M. C. Frawley, A. Petcu-Colan, V. G. Truong, and S. Nic Chormaic, “Higher order mode propagation in an optical nanofiber,” Opt. Commun. 285(23), 4648–4654 (2012).
[Crossref]

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref] [PubMed]

E. Vetsch, S. T. Dawkins, R. Mitsch, D. Reitz, P. Schneeweiss, and A. Rauschenbeutel, “Nanofiber-based optical trapping of cold neutral atoms,” IEEE J. Sel. Top. Quantum Electron. 18(6), 1763–1770 (2012).
[Crossref]

2011 (2)

2010 (2)

R. R. J. Maier, W. N. MacPherson, J. S. Barton, S. McCulloch, and B. J. S. Jones, “Distributed sensing using Rayleigh scatter in polarization-maintaining fibres for transverse load sensing,” Meas. Sci. Technol. 21(9), 094019 (2010).
[Crossref]

F. Yaman, N. Bai, B. Zhu, T. Wang, and G. Li, “Long distance transmission in few-mode fibers,” Opt. Express 18(12), 13250–13257 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

G. Sagué, A. Baade, and A. Rauschenbeutel, “Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibers,” New J. Phys. 10(11), 113008 (2008).
[Crossref]

2006 (2)

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

E. Li, X. L. Wang, and C. Zhang, “Fiber-optic temperature sensor based on interference of selective higher-order modes,” Appl. Phys. Lett. 89(9), 091119 (2006).
[Crossref]

2005 (1)

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252(1-3), 12–21 (2005).
[Crossref]

2004 (1)

P. Mazumder, S. L. Logunov, and S. Raghavan, “Analysis of excess scattering in optical fibers,” J. Appl. Phys. 96(8), 4042–4049 (2004).
[Crossref]

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

1989 (1)

R. Calvani, R. Caponi, and F. Cisternino, “Polarization measurements on single-mode fibers,” J. Lightwave Technol. 7(8), 1187–1196 (1989).
[Crossref]

1985 (1)

1981 (1)

M. Nakazawa, T. Horiguchi, M. Tokuda, and N. Uchida, “Polarisation beat length measurement in a single-mode optical fibre by backward Rayleigh scattering,” Electron. Lett. 17(15), 513–515 (1981).
[Crossref]

1976 (1)

W. Eickhoff and O. Krumpholz, “Determination of the ellipticity of monomode glass fibres from measurements of scattered light intensity,” Electron. Lett. 12(16), 405–407 (1976).
[Crossref]

Alton, D. J.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref] [PubMed]

Baade, A.

G. Sagué, A. Baade, and A. Rauschenbeutel, “Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibers,” New J. Phys. 10(11), 113008 (2008).
[Crossref]

Bai, N.

Barton, J. S.

R. R. J. Maier, W. N. MacPherson, J. S. Barton, S. McCulloch, and B. J. S. Jones, “Distributed sensing using Rayleigh scatter in polarization-maintaining fibres for transverse load sensing,” Meas. Sci. Technol. 21(9), 094019 (2010).
[Crossref]

Beadie, G.

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Booth, T.

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Calvani, R.

R. Calvani, R. Caponi, and F. Cisternino, “Polarization measurements on single-mode fibers,” J. Lightwave Technol. 7(8), 1187–1196 (1989).
[Crossref]

Caponi, R.

R. Calvani, R. Caponi, and F. Cisternino, “Polarization measurements on single-mode fibers,” J. Lightwave Technol. 7(8), 1187–1196 (1989).
[Crossref]

Choi, K. S.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref] [PubMed]

Cisternino, F.

R. Calvani, R. Caponi, and F. Cisternino, “Polarization measurements on single-mode fibers,” J. Lightwave Technol. 7(8), 1187–1196 (1989).
[Crossref]

Courjon, D.

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252(1-3), 12–21 (2005).
[Crossref]

Dawkins, S. T.

E. Vetsch, S. T. Dawkins, R. Mitsch, D. Reitz, P. Schneeweiss, and A. Rauschenbeutel, “Nanofiber-based optical trapping of cold neutral atoms,” IEEE J. Sel. Top. Quantum Electron. 18(6), 1763–1770 (2012).
[Crossref]

Desantolo, A.

Digiovanni, D. J.

Dimarcello, F. V.

Ding, D.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref] [PubMed]

Duan, W.

Y. Yang, W. Duan, and M. Ye, “High precision measurement technology for beat length of birefringence optical fiber,” Meas. Sci. Technol. 24(2), 025201 (2013).
[Crossref]

Eickhoff, W.

W. Eickhoff and O. Krumpholz, “Determination of the ellipticity of monomode glass fibres from measurements of scattered light intensity,” Electron. Lett. 12(16), 405–407 (1976).
[Crossref]

Fatemi, F. K.

Fini, J. M.

Frawley, M. C.

M. C. Frawley, A. Petcu-Colan, V. G. Truong, and S. Nic Chormaic, “Higher order mode propagation in an optical nanofiber,” Opt. Commun. 285(23), 4648–4654 (2012).
[Crossref]

Ghalmi, S.

Goban, A.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref] [PubMed]

Grosjean, T.

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252(1-3), 12–21 (2005).
[Crossref]

Headley, C.

Hlubina, P.

Hoffman, J. E.

Hollenbach, U.

Horiguchi, T.

M. Nakazawa, T. Horiguchi, M. Tokuda, and N. Uchida, “Polarisation beat length measurement in a single-mode optical fibre by backward Rayleigh scattering,” Electron. Lett. 17(15), 513–515 (1981).
[Crossref]

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Jones, B. J. S.

R. R. J. Maier, W. N. MacPherson, J. S. Barton, S. McCulloch, and B. J. S. Jones, “Distributed sensing using Rayleigh scatter in polarization-maintaining fibres for transverse load sensing,” Meas. Sci. Technol. 21(9), 094019 (2010).
[Crossref]

Kim, K.

Kimble, H. J.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref] [PubMed]

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Krumpholz, O.

W. Eickhoff and O. Krumpholz, “Determination of the ellipticity of monomode glass fibres from measurements of scattered light intensity,” Electron. Lett. 12(16), 405–407 (1976).
[Crossref]

Lacroûte, C.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref] [PubMed]

Li, E.

E. Li, X. L. Wang, and C. Zhang, “Fiber-optic temperature sensor based on interference of selective higher-order modes,” Appl. Phys. Lett. 89(9), 091119 (2006).
[Crossref]

Li, G.

Liu, X.

Logunov, S. L.

P. Mazumder, S. L. Logunov, and S. Raghavan, “Analysis of excess scattering in optical fibers,” J. Appl. Phys. 96(8), 4042–4049 (2004).
[Crossref]

MacPherson, W. N.

R. R. J. Maier, W. N. MacPherson, J. S. Barton, S. McCulloch, and B. J. S. Jones, “Distributed sensing using Rayleigh scatter in polarization-maintaining fibres for transverse load sensing,” Meas. Sci. Technol. 21(9), 094019 (2010).
[Crossref]

Maier, R. R. J.

R. R. J. Maier, W. N. MacPherson, J. S. Barton, S. McCulloch, and B. J. S. Jones, “Distributed sensing using Rayleigh scatter in polarization-maintaining fibres for transverse load sensing,” Meas. Sci. Technol. 21(9), 094019 (2010).
[Crossref]

Mazumder, P.

P. Mazumder, S. L. Logunov, and S. Raghavan, “Analysis of excess scattering in optical fibers,” J. Appl. Phys. 96(8), 4042–4049 (2004).
[Crossref]

McCulloch, S.

R. R. J. Maier, W. N. MacPherson, J. S. Barton, S. McCulloch, and B. J. S. Jones, “Distributed sensing using Rayleigh scatter in polarization-maintaining fibres for transverse load sensing,” Meas. Sci. Technol. 21(9), 094019 (2010).
[Crossref]

Mielke, M.

Mitsch, R.

E. Vetsch, S. T. Dawkins, R. Mitsch, D. Reitz, P. Schneeweiss, and A. Rauschenbeutel, “Nanofiber-based optical trapping of cold neutral atoms,” IEEE J. Sel. Top. Quantum Electron. 18(6), 1763–1770 (2012).
[Crossref]

Mohr, J.

Monberg, E.

Monberg, E. M.

Nakazawa, M.

M. Nakazawa, T. Horiguchi, M. Tokuda, and N. Uchida, “Polarisation beat length measurement in a single-mode optical fibre by backward Rayleigh scattering,” Electron. Lett. 17(15), 513–515 (1981).
[Crossref]

Napiorkowski, M.

Nic Chormaic, S.

M. C. Frawley, A. Petcu-Colan, V. G. Truong, and S. Nic Chormaic, “Higher order mode propagation in an optical nanofiber,” Opt. Commun. 285(23), 4648–4654 (2012).
[Crossref]

Nicholson, J. W.

Noda, J.

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Orozco, L. A.

Peng, X.

Petcu-Colan, A.

M. C. Frawley, A. Petcu-Colan, V. G. Truong, and S. Nic Chormaic, “Higher order mode propagation in an optical nanofiber,” Opt. Commun. 285(23), 4648–4654 (2012).
[Crossref]

Pototschnig, M.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref] [PubMed]

Raghavan, S.

P. Mazumder, S. L. Logunov, and S. Raghavan, “Analysis of excess scattering in optical fibers,” J. Appl. Phys. 96(8), 4042–4049 (2004).
[Crossref]

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

Rauschenbeutel, A.

E. Vetsch, S. T. Dawkins, R. Mitsch, D. Reitz, P. Schneeweiss, and A. Rauschenbeutel, “Nanofiber-based optical trapping of cold neutral atoms,” IEEE J. Sel. Top. Quantum Electron. 18(6), 1763–1770 (2012).
[Crossref]

G. Sagué, A. Baade, and A. Rauschenbeutel, “Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibers,” New J. Phys. 10(11), 113008 (2008).
[Crossref]

Ravets, S.

Reitz, D.

E. Vetsch, S. T. Dawkins, R. Mitsch, D. Reitz, P. Schneeweiss, and A. Rauschenbeutel, “Nanofiber-based optical trapping of cold neutral atoms,” IEEE J. Sel. Top. Quantum Electron. 18(6), 1763–1770 (2012).
[Crossref]

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Rolston, S. L.

Sabac, A.

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252(1-3), 12–21 (2005).
[Crossref]

Sagué, G.

G. Sagué, A. Baade, and A. Rauschenbeutel, “Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibers,” New J. Phys. 10(11), 113008 (2008).
[Crossref]

Schneeweiss, P.

E. Vetsch, S. T. Dawkins, R. Mitsch, D. Reitz, P. Schneeweiss, and A. Rauschenbeutel, “Nanofiber-based optical trapping of cold neutral atoms,” IEEE J. Sel. Top. Quantum Electron. 18(6), 1763–1770 (2012).
[Crossref]

Sieber, H.

Stern, N. P.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref] [PubMed]

Szczurowski, M.

Takada, K.

Thiele, T.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref] [PubMed]

Tokuda, M.

M. Nakazawa, T. Horiguchi, M. Tokuda, and N. Uchida, “Polarisation beat length measurement in a single-mode optical fibre by backward Rayleigh scattering,” Electron. Lett. 17(15), 513–515 (1981).
[Crossref]

Truong, V. G.

M. C. Frawley, A. Petcu-Colan, V. G. Truong, and S. Nic Chormaic, “Higher order mode propagation in an optical nanofiber,” Opt. Commun. 285(23), 4648–4654 (2012).
[Crossref]

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Uchida, N.

M. Nakazawa, T. Horiguchi, M. Tokuda, and N. Uchida, “Polarisation beat length measurement in a single-mode optical fibre by backward Rayleigh scattering,” Electron. Lett. 17(15), 513–515 (1981).
[Crossref]

Ulrich, R.

Urbanczyk, W.

Vetsch, E.

E. Vetsch, S. T. Dawkins, R. Mitsch, D. Reitz, P. Schneeweiss, and A. Rauschenbeutel, “Nanofiber-based optical trapping of cold neutral atoms,” IEEE J. Sel. Top. Quantum Electron. 18(6), 1763–1770 (2012).
[Crossref]

Wang, T.

Wang, X. L.

E. Li, X. L. Wang, and C. Zhang, “Fiber-optic temperature sensor based on interference of selective higher-order modes,” Appl. Phys. Lett. 89(9), 091119 (2006).
[Crossref]

Westbrook, P. S.

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Windeler, R. S.

Wisk, P.

Yaman, F.

Yan, M. F.

Yang, Y.

Y. Yang, W. Duan, and M. Ye, “High precision measurement technology for beat length of birefringence optical fiber,” Meas. Sci. Technol. 24(2), 025201 (2013).
[Crossref]

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Y. Yang, W. Duan, and M. Ye, “High precision measurement technology for beat length of birefringence optical fiber,” Meas. Sci. Technol. 24(2), 025201 (2013).
[Crossref]

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L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
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Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

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Zhang, C.

E. Li, X. L. Wang, and C. Zhang, “Fiber-optic temperature sensor based on interference of selective higher-order modes,” Appl. Phys. Lett. 89(9), 091119 (2006).
[Crossref]

Zhu, B.

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Appl. Opt. (2)

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E. Li, X. L. Wang, and C. Zhang, “Fiber-optic temperature sensor based on interference of selective higher-order modes,” Appl. Phys. Lett. 89(9), 091119 (2006).
[Crossref]

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[Crossref]

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Phys. Rev. Lett. (2)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

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Science (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1 (a) Vector modes in the LP01 and LP11 families. (b) Example of TM01 in fiber. Only the polarization component orthogonal to the viewing direction is observed, giving a two-lobed profile from each direction. (c) Schematic of setup. CCD camera images fiber-under-test (FUT) over its length. Sample image shown, with TM01 input. Structure of Rayleigh scattering for TM01 is clearly resolved in the core. WP = λ/2 waveplate.
Fig. 2
Fig. 2 (a) Images and line profiles of the core of HI1060 fiber from 0 and 45 degrees for pure modes in the LP01 and LP11 families. Horizontal axis is scaled by the fiber radius. (b) RS images of the TM01 and TE01 modes over the 50 cm fiber length. The images do not change longitudinally, indicating high modal purity.
Fig. 3
Fig. 3 (a) Top: The superposition of HE21 and TM01 modes produces an oscillating two-lobe pattern. RS images are shown over two beat lengths. Bottom: RS image and profile over 50 cm. Scattered power in each image (blue), calculated by summing the transverse pixels of the RS images, is fit to a sinusoid (black) to determine the beat length. Red: RS intensity along the fiber axis relative to the peak RS intensity in the image. (b) Same plots for a superposition of HE21 and TE01. (c) Cutback measurements of the HE21-TM01 superposition at maximum RS power (top) and minimum RS power (bottom).
Fig. 4
Fig. 4 (top) Superpositions of the LP01 and LP11 families are made by translating a pi-phase-plate across a Gaussian beam. (middle) Beam profile entering the fiber. (bottom) Rayleigh scattering image over a 1mm fiber length. (b) Cross sectional profile of the LP01-LP11 superposition at two locations in the image.
Fig. 5
Fig. 5 (a) Simultaneous image (using 50x microscope objective) of the longitudinal and transverse components in the fiber when TM01 is propagating, along the with the line profiles for each. (b) Line profiles for the HE11 (fundamental) mode and the HE21, TE01, and TM01 modes. Profiles have been displaced horizontally from one another for visual comparison.
Fig. 6
Fig. 6 Image of a fiber splice using RS light. The splice location is indicated by the dashed vertical line. Fiber cores are marked by thin white lines and extensions to the sides of the image. Light from the HI1060 core that is not coupled into the 980HP core can be seen diverging into the 980HP cladding. The fibers have 125 micron cladding diameter.

Equations (2)

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E(r,θ,z)= j=1 N c j f j (r,θ)exp( i β j z)
H E 11 x ( r,θ )= F 01 (r) x ^ H E 11 y ( r,θ )= F 01 ( r ) y ^ T M 01 ( r,θ )= F 11 ( r )( cosθ x ^ +sinθ y ^ ) T E 01 ( r,θ )= F 11 ( r )( cosθ y ^ sinθ x ^ ) H E 21 e ( r,θ )= F 11 ( r )( cosθ x ^ sinθ y ^ ) H E 21 o ( r,θ )= F 11 ( r )( cosθ y ^ +sinθ x ^ )

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