Abstract

For a multimode optical fiber, the number of modes (Nm) can be calculated by analytic theory when the fiber is straight, twist-free, and strain-free. In practice, however, the fiber is subject to distortions that modify its mode characteristics. In this Letter, we present an experimental method to interrogate the mode properties of a multimode optical fiber. We experimentally measured the transmission matrix of a multimode optical fiber and performed singular value decomposition. We proved, both theoretically and experimentally, that the rank of the transmission matrix is equal to Nm. We expect that the suggested method will contribute to the fields of the biomedical optics and optical communications where optical fiber is widely used.

© 2012 Optical Society of America

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References

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  1. M. Saffman and D. Z. Anderson, Opt. Lett. 16, 300 (1991).
    [CrossRef]
  2. H. R. Stuart, Science 289, 281 (2000).
    [CrossRef]
  3. A. M. Tai and A. A. Friesem, Opt. Lett. 8, 57 (1983).
    [CrossRef]
  4. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).
  5. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
    [CrossRef]
  6. I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008).
    [CrossRef]
  7. Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012).
    [CrossRef]
  8. O. N. Dorokhov, Solid State Commun. 51, 381 (1984).
    [CrossRef]
  9. M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
    [CrossRef]
  10. S. Bianchi and R. Di Leonardo, Lab Chip 12, 635 (2012).
    [CrossRef]
  11. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, Opt. Express 20, 10583 (2012).
    [CrossRef]
  12. Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. 107, 023902 (2011).
    [CrossRef]

2012 (4)

Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012).
[CrossRef]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

S. Bianchi and R. Di Leonardo, Lab Chip 12, 635 (2012).
[CrossRef]

I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, Opt. Express 20, 10583 (2012).
[CrossRef]

2011 (1)

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef]

2010 (1)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

2008 (1)

I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008).
[CrossRef]

2000 (1)

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

1991 (1)

1984 (1)

O. N. Dorokhov, Solid State Commun. 51, 381 (1984).
[CrossRef]

1983 (1)

Anderson, D. Z.

Bianchi, S.

S. Bianchi and R. Di Leonardo, Lab Chip 12, 635 (2012).
[CrossRef]

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Choi, W.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef]

Choi, Y.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef]

Dasari, R. R.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef]

Di Leonardo, R.

S. Bianchi and R. Di Leonardo, Lab Chip 12, 635 (2012).
[CrossRef]

Dorokhov, O. N.

O. N. Dorokhov, Solid State Commun. 51, 381 (1984).
[CrossRef]

Fang-Yen, C.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef]

Farahi, S.

Feld, M. S.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef]

Fink, M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Friesem, A. A.

Genack, A. Z.

Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012).
[CrossRef]

Gigan, S.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Kang, P.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef]

Kim, J.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Kim, M.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Lee, K. J.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef]

Lerosey, G.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Moser, C.

Mosk, A. P.

I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008).
[CrossRef]

Papadopoulos, I. N.

Park, Q. H.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

Psaltis, D.

Saffman, M.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).

Shi, Z.

Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012).
[CrossRef]

Stuart, H. R.

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

Tai, A. M.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).

Vellekoop, I. M.

I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008).
[CrossRef]

Yang, T. D.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef]

Yoon, C.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Lab Chip (1)

S. Bianchi and R. Di Leonardo, Lab Chip 12, 635 (2012).
[CrossRef]

Nat. Photonics (1)

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. H. Park, and W. Choi, Nat. Photonics 6, 581 (2012).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (4)

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. 101, 120601 (2008).
[CrossRef]

Z. Shi and A. Z. Genack, Phys. Rev. Lett. 108, 043901 (2012).
[CrossRef]

Science (1)

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

Solid State Commun. (1)

O. N. Dorokhov, Solid State Commun. 51, 381 (1984).
[CrossRef]

Other (1)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).

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Figures (3)

Fig. 1.
Fig. 1.

Experimental setup. BS1 and BS2, beam splitters; L1, L2, and L3, lenses with focal lengths of 50, 500, 400 mm, respectively. GM, two-axis galvanometer mirror;CL, condenser lens; OL, objective lens; TL, tube lens; MMF, multimode optical fiber; P1 (ξ,η) and P2 (x,y) are the input and output planes of the fiber, respectively. (kξ,kη) is the wave vector of the incident beam at P1 plane.

Fig. 2.
Fig. 2.

Transmission matrix of a multimode optical fiber. (a) Theoretically expected output intensity maps at P2 plane for the incident plane waves with wave vector (kξ,kη) at the input plane. The fiber has 50 μm core diameter, 0.22 NA and 1-meter length. (b) Constructed transmission matrix from (a). (c) Experimentally measured output intensity maps for the fiber with the same specifications as the one considered in (a). (d) Constructed transmission matrix from (c). Scale bar indicates 20 μm.

Fig. 3.
Fig. 3.

Singular value distributions of the transmission matrices and their ranks. (a) Square of the singular values sorted in descending order of the mode index. Dashed red and solid blue curves were obtained from the SVD analysis of the transmission matrices in Figs. 2(b) and 2(d), respectively. (b) Rank of the transmission matrix as a function of the V parameter. The fibers used in the experiment were HPSC10, HPSC25, M14L01, GIF625, and M15L01 from Thorlabs, Inc. Square dots: Experimentally measured ranks. Triangular dots: Exact number of modes for ideal fibers for a single polarization. Red curve: Approximate Nm at large V for a single polarization. The points indicated by a black arrow are for the GRIN fiber.

Equations (4)

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

Eout(x,y)=kξ,kηt(x,y;kξ,kη)Ein(kξ,kη).
E(r,ϕ,z=0)=l,malm(kξ,kη)ulm(r,ϕ).
E(r,ϕ,z=z0)=l,malm(kξ,kη)ulm(r,ϕ)exp(iβlmz0),
t=UτV,

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