Abstract

We present a comparative study of four numerical methods to detect the mode content of a laser beam from, at most, two intensity images. The techniques are compared regarding temporal effort, stability, and accuracy, using the example of three multimode optical fibers that differ in the number of supported modes.

© 2013 Optical Society of America

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  1. M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
    [CrossRef]
  2. 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, 1545–1548 (2013).
    [CrossRef]
  3. D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M2 measurement for fiber beams based on modal analysis,” Appl. Opt. 51, 987–993 (2012).
    [CrossRef]
  4. C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
    [CrossRef]
  5. F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36, 689–691 (2011).
    [CrossRef]
  6. C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express 21, 3170–3181 (2013).
    [CrossRef]
  7. D. Flamm, K.-C. Hou, P. Gelszinnis, C. Schulze, S. Schröter, and M. Duparré, “Modal characterization of fiber-to-fiber coupling processes,” Opt. Lett. 38, 2128–2130 (2013).
    [CrossRef]
  8. F. Stutzki, F. Jansen, C. Jauregui, J. Limpert, and A. Tünnermann, “Non-hexagonal large-pitch fibers for enhanced mode discrimination,” Opt. Express 19, 12081–12086 (2011).
    [CrossRef]
  9. S. Ramachandran, J. Fini, M. Mermelstein, J. Nicholson, S. Ghalmi, and M. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
    [CrossRef]
  10. N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
    [CrossRef]
  11. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16, 7233–7243 (2008).
    [CrossRef]
  12. D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (c2) imaging of fiber and waveguide modes,” Opt. Express 19, 13008–13019 (2011).
    [CrossRef]
  13. J. M. O. Daniel, J. S. P. Chan, J. W. Kim, J. K. Sahu, M. Ibsen, and W. A. Clarkson, “Novel technique for mode selection in a multimode fiber laser,” Opt. Express 19, 12434–12439 (2011).
    [CrossRef]
  14. F. Stutzki, C. Jauregui, C. Voigtlaender, J. U. Thomas, S. Nolte, J. Limpert, and A. Tuennermann, “Real-time monitoring of the modal content of monolithic large-mode-area fiber lasers,” in Optical Fiber Communication Conference (Optical Society of America, 2010).
  15. V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).
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    [CrossRef]
  17. C. Schulze, A. Dudley, D. Flamm, M. Duparr, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
    [CrossRef]
  18. D. M. Nguyen, S. Blin, T. N. Nguyen, S. D. Le, L. Provino, M. Thual, and T. Chartier, “Modal decomposition technique for multimode fibers,” Appl. Opt. 51, 450–456 (2012).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36, 4572–4574 (2011).
    [CrossRef]
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    [CrossRef]
  24. M. Skorobogatiy, C. Anastassiou, S. Johnson, O. Weisberg, T. Engeness, S. Jacobs, R. Ahmad, and Y. Fink, “Quantitative characterization of higher-order mode converters in weakly multimoded fibers,” Opt. Express 11, 2838–2847 (2003).
    [CrossRef]
  25. H. Lü, P. Zhou, X. Wang, and Z. Jiang, “Fast and accurate modal decomposition of multimode fiber based on stochastic parallel gradient descent algorithm,” Appl. Opt. 52, 2905–2908 (2013).
    [CrossRef]
  26. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1996).
  27. J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. 42, 59–66 (1988).
  28. C. Borgentun, J. Bengtsson, and A. Larsson, “Full characterization of a high-power semiconductor disk laser beam with simultaneous capture of optimally sized focus and farfield,” Appl. Opt. 50, 1640–1649 (2011).
    [CrossRef]
  29. A. Cutolo, A. Esposito, T. Isernia, R. Pierri, and L. Zeni, “Characterization of the transverse modes in a laser beam: analysis and application to a q-switched Nd:YAG laser,” Appl. Opt. 31, 2722–2733 (1992).
    [CrossRef]
  30. T. Isernia, G. Leone, and R. Pierri, “Phaseless near field techniques: uniqueness conditions and attainment of the solution,” J. Electromagn. Wave Appl. 8, 889–908 (1994).
  31. S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express 15, 15402–15409 (2007).
    [CrossRef]
  32. ISO, “ISO 11146-1:2005 Test methods for laser beam widths, divergence angles and beam propagation ratios part 1: Stigmatic and simple astigmatic beams,” (2005).

2013

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

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, 1545–1548 (2013).
[CrossRef]

C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express 21, 3170–3181 (2013).
[CrossRef]

D. Flamm, K.-C. Hou, P. Gelszinnis, C. Schulze, S. Schröter, and M. Duparré, “Modal characterization of fiber-to-fiber coupling processes,” Opt. Lett. 38, 2128–2130 (2013).
[CrossRef]

C. Schulze, A. Dudley, D. Flamm, M. Duparr, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

H. Lü, P. Zhou, X. Wang, and Z. Jiang, “Fast and accurate modal decomposition of multimode fiber based on stochastic parallel gradient descent algorithm,” Appl. Opt. 52, 2905–2908 (2013).
[CrossRef]

2012

2011

2009

2008

S. Ramachandran, J. Fini, M. Mermelstein, J. Nicholson, S. Ghalmi, and M. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[CrossRef]

N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
[CrossRef]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16, 7233–7243 (2008).
[CrossRef]

2007

2005

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[CrossRef]

2003

1995

1994

T. Isernia, G. Leone, and R. Pierri, “Phaseless near field techniques: uniqueness conditions and attainment of the solution,” J. Electromagn. Wave Appl. 8, 889–908 (1994).

1992

1988

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. 42, 59–66 (1988).

1982

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

Abouraddy, A. F.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[CrossRef]

Ahmad, R.

Anastassiou, C.

Andermahr, N.

N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
[CrossRef]

Barankov, R. A.

Bartelt, H.

Bengtsson, J.

Blin, S.

Borgentun, C.

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, 1545–1548 (2013).
[CrossRef]

Brüning, R.

Chan, J. S. P.

Chartier, T.

Clarkson, W. A.

Courtial, J.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

Cutolo, A.

Daniel, J. M. O.

Dudley, A.

C. Schulze, A. Dudley, D. Flamm, M. Duparr, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

Duparr, M.

C. Schulze, A. Dudley, D. Flamm, M. Duparr, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

Duparré, M.

Eidam, T.

Engeness, T.

Esposito, A.

Fallnich, C.

N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
[CrossRef]

Fienup, J. R.

Fini, J.

S. Ramachandran, J. Fini, M. Mermelstein, J. Nicholson, S. Ghalmi, and M. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[CrossRef]

Fink, Y.

Flamm, D.

Forbes, A.

C. Schulze, A. Dudley, D. Flamm, M. Duparr, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
[CrossRef]

Gaida, C.

Gelszinnis, P.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

Ghalmi, S.

S. Ramachandran, J. Fini, M. Mermelstein, J. Nicholson, S. Ghalmi, and M. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[CrossRef]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16, 7233–7243 (2008).
[CrossRef]

Golub, M.

V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).

Hartung, A.

Hou, K.-C.

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, 1545–1548 (2013).
[CrossRef]

Ibsen, M.

Isernia, T.

Izzo, I.

Jacobs, S.

Jansen, F.

Jauregui, C.

Jiang, Z.

Joannopoulos, J. D.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[CrossRef]

Johnson, S.

Kaiser, T.

Kim, J. W.

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, 1545–1548 (2013).
[CrossRef]

Larsson, A.

Lavery, M. P. J.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

Le, S. D.

Leone, G.

T. Isernia, G. Leone, and R. Pierri, “Phaseless near field techniques: uniqueness conditions and attainment of the solution,” J. Electromagn. Wave Appl. 8, 889–908 (1994).

Limpert, J.

Lorenz, A.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1996).

Lü, H.

Mermelstein, M.

S. Ramachandran, J. Fini, M. Mermelstein, J. Nicholson, S. Ghalmi, and M. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[CrossRef]

Naidoo, D.

Nguyen, D. M.

Nguyen, T. N.

Nicewander, W. A.

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. 42, 59–66 (1988).

Nicholson, J.

S. Ramachandran, J. Fini, M. Mermelstein, J. Nicholson, S. Ghalmi, and M. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[CrossRef]

Nicholson, J. W.

Nolte, S.

F. Stutzki, C. Jauregui, C. Voigtlaender, J. U. Thomas, S. Nolte, J. Limpert, and A. Tuennermann, “Real-time monitoring of the modal content of monolithic large-mode-area fiber lasers,” in Optical Fiber Communication Conference (Optical Society of America, 2010).

Otto, H.-J.

Padgett, M. J.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

Pierri, R.

Provino, L.

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, 1545–1548 (2013).
[CrossRef]

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (c2) imaging of fiber and waveguide modes,” Opt. Express 19, 13008–13019 (2011).
[CrossRef]

S. Ramachandran, J. Fini, M. Mermelstein, J. Nicholson, S. Ghalmi, and M. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[CrossRef]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16, 7233–7243 (2008).
[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, 1545–1548 (2013).
[CrossRef]

Robertson, D. J.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

Rodgers, J. L.

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. 42, 59–66 (1988).

Sahu, J. K.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

Schimpf, D. N.

Schmidt, O. A.

Schröter, S.

Schulze, C.

Shapira, O.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[CrossRef]

Skorobogatiy, M.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1996).

Soifer, V. A.

V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).

Sponselli, A.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

Steinhoff, N. K.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

Steinmetz, A.

Stutzki, F.

Theeg, T.

N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
[CrossRef]

Thomas, J. U.

F. Stutzki, C. Jauregui, C. Voigtlaender, J. U. Thomas, S. Nolte, J. Limpert, and A. Tuennermann, “Real-time monitoring of the modal content of monolithic large-mode-area fiber lasers,” in Optical Fiber Communication Conference (Optical Society of America, 2010).

Thual, M.

Tuennermann, A.

F. Stutzki, C. Jauregui, C. Voigtlaender, J. U. Thomas, S. Nolte, J. Limpert, and A. Tuennermann, “Real-time monitoring of the modal content of monolithic large-mode-area fiber lasers,” in Optical Fiber Communication Conference (Optical Society of America, 2010).

Tünnermann, A.

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, 1545–1548 (2013).
[CrossRef]

Tyler, G. A.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

Voigtlaender, C.

F. Stutzki, C. Jauregui, C. Voigtlaender, J. U. Thomas, S. Nolte, J. Limpert, and A. Tuennermann, “Real-time monitoring of the modal content of monolithic large-mode-area fiber lasers,” in Optical Fiber Communication Conference (Optical Society of America, 2010).

Wang, X.

Weisberg, O.

Wielandy, 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, 1545–1548 (2013).
[CrossRef]

Wilner, A.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

Yablon, A. D.

Yan, M.

S. Ramachandran, J. Fini, M. Mermelstein, J. Nicholson, S. Ghalmi, and M. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[CrossRef]

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, 1545–1548 (2013).
[CrossRef]

Zeni, L.

Zhou, P.

Am. Stat.

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. 42, 59–66 (1988).

Appl. Opt.

Appl. Phys. B

N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
[CrossRef]

J. Electromagn. Wave Appl.

T. Isernia, G. Leone, and R. Pierri, “Phaseless near field techniques: uniqueness conditions and attainment of the solution,” J. Electromagn. Wave Appl. 8, 889–908 (1994).

Laser Photon. Rev.

S. Ramachandran, J. Fini, M. Mermelstein, J. Nicholson, S. Ghalmi, and M. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[CrossRef]

New J. Phys.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. Wilner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[CrossRef]

C. Schulze, A. Dudley, D. Flamm, M. Duparr, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

Opt. Express

T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17, 9347–9356 (2009).
[CrossRef]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16, 7233–7243 (2008).
[CrossRef]

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (c2) imaging of fiber and waveguide modes,” Opt. Express 19, 13008–13019 (2011).
[CrossRef]

J. M. O. Daniel, J. S. P. Chan, J. W. Kim, J. K. Sahu, M. Ibsen, and W. A. Clarkson, “Novel technique for mode selection in a multimode fiber laser,” Opt. Express 19, 12434–12439 (2011).
[CrossRef]

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
[CrossRef]

F. Stutzki, F. Jansen, C. Jauregui, J. Limpert, and A. Tünnermann, “Non-hexagonal large-pitch fibers for enhanced mode discrimination,” Opt. Express 19, 12081–12086 (2011).
[CrossRef]

C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express 21, 3170–3181 (2013).
[CrossRef]

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Supplementary Material (2)

» Media 1: AVI (3469 KB)     
» Media 2: AVI (1766 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

Scheme of the iterative GS algorithm. Am measured near-field amplitude; Ar reconstructed near-field amplitude; A˜m measured far-field amplitude; A˜r reconstructed far-field amplitude; Pr reconstructed phase.

Fig. 2.
Fig. 2.

Scheme of the experimental setup. MO1, MO2, microscope objectives; MS, micro mechanical shifter; P, polarizer; IL, imaging lens; BS, beam splitter; A, aperture; FL, Fourier-lens.

Fig. 3.
Fig. 3.

Example fiber A. (a)–(e) Near-field intensities; (f)–(j) far-field intensities; (k)–(n) near-field phases from right to left: measured, Residual method, Corr NF+FF method, Corr NF method, GS method; (o) relative power spectrum; (p) modal phases. The minimization process of the Corr NF+FF method can be viewed in Media 1.

Fig. 4.
Fig. 4.

Decomposition time in seconds for the different methods depending on the number of modes.

Fig. 5.
Fig. 5.

Ambiguity of the method Corr NF caused by the sign of the phase: the same near-field intensity (a) and (d) yields flipped far-field intensities (c) and (f) when the sign of the phase (b) and (e) is changed.

Fig. 6.
Fig. 6.

Ambiguity with the method Corr NF caused by modal interference: the same reconstructed near-field intensities (a)–(d) correspond to different modal spectra (m); different far-field intensities (e)–(h), which are obtained from numerical propagation starting from the recontructed near-field. (i)–(l) Depict the corresponding phase distributions.

Fig. 7.
Fig. 7.

Influence of OA position on the modal power spectrum, the insets depicts the reconstructed intensity distributions for the shifted and unshifted case.

Tables (5)

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Table 1. Fiber Specifications of SMF and MMF LMA08, LMA25, and LMA30a

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Table 2. Summary of Reconstruction Results

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Table 3. Summary of Reconstruction Results in Terms of the Variance of Modal Weights and Modal Phases

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Table 4. Statistic for the Reconstruction Results for Random Initial Values

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Table 5. Change of Modal Power Spectrum in Percent per Pixel (pixel size 8.8 μm)

Equations (11)

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U(r)=ncnΨn(r).
Ψm|Ψn=ΨmΨn*d2r=δmn,
cn=ϱneiΔφn=Ψn|U,
ΔN,F=[ImN,F(r)IrN,F(r)]2d2rmin,
Δ=1N2ijklcicjck*cl*Λijkl+1Nijcicj*Γij+P.
Λijkl=Ψi(r)Ψj(r)Ψk*(r)Ψl*(r)d2r
Γij=Im(r)Ψi(r)Ψj*(r)d2r
P=[Im(r)]2d2r,
C=|ΔIm(r)ΔIr(r)d2rΔIm(r)2d2rΔIr(r)2d2r|max,
Δ=2CNFCFFmin,
Δ=1CNFmin.

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