Abstract

We present a novel method to validate the relative amount of power carried by high order modes in a multimode fiber using a Spatial and Spectral (S2) imaging technique. The method can be utilized to calibrate the S2 set-up and uses Fresnel reflections from a thin glass plate to compare theoretical values with experimental results. We have found that, in the most general case, spectral leakage and sampling errors can lead S2 to underestimate the multipath interference (MPI) of high order modes by several decibels, thus significantly impairing the result of the measurement. On the other hand, by applying suitable corrections as described in this work, we demonstrate that the S2 produces MPI estimates that are accurate to within 1dB or better.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
  3. O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete Modal Decomposition for Optical Waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
    [Crossref] [PubMed]
  4. D. Gray, Z. Li, F. Poletti, R. Slavík, N. Wheeler, N. Baddela, M. Petrovich, A. Obeysekara, and D. Richardson, “Complementary Analysis of Modal Content and Properties in a 19-cell Hollow Core Photonic Band Gap Fiber using Time-of-Flight and S2 Techniques,” in European Conference and Exhibition on Optical Communication, OSA Technical Digest (online) (Optical Society of America, 2012), paper Mo.2.F.1.
    [Crossref]
  5. J. Y. Lee, T.-J. Ahn, S. Moon, Y. C. Youk, Y. M. Jung, K. Oh, and D. Y. Kim, “Fourier-domain low-coherence interferometry for differential mode delay analysis of an optical fiber,” Opt. Lett. 31(16), 2396–2398 (2006).
    [Crossref] [PubMed]
  6. T. Ahn, S. Moon, S. Kim, K. Oh, D. Kim, J. Kobelke, K. Schuster, and J. Kirchhof, “Optical Frequency-Domain Modal Dispersion Measurement in Multimode Fibers Using Intermodal Interferometer,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OWI17.
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    [PubMed]
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    [Crossref] [PubMed]
  9. Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry,” Appl. Phys. B 96(2–3), 345–353 (2009).
    [Crossref]
  10. M. Paurisse, L. Lévèque, M. Hanna, F. Druon, and P. Georges, “Complete measurement of fiber modal content by wavefront analysis,” Opt. Express 20(4), 4074–4084 (2012).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  12. S. Blin, D. M. Nguyen, T. N. Nguyen, L. Provino, M. Thual, and T. Chartier, “Simple Modal Analysis Method for Multi-Mode Fibers,” in Proceedings of European Conference on Optical Communications,(2009), paper P1.16.
  13. H. J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tunnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
    [Crossref]
  14. B. Sévigny, G. Le Cocq, C. Carrero, C. Valentin, P. Sillard, G. Bouwmans, L. Bigot, and Y. Quiquempois, “Advanced S2 Imaging: Application of Multivariate Statistical Analysis to Spatially and Spectrally Resolved Datasets,” J. Lightwave Technol. 32(23), 4004–4010 (2014).
    [Crossref]
  15. 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(4), 450–456 (2012).
    [Crossref] [PubMed]
  16. D. R. Gray, S. R. Sandoghchi, N. V. Wheeler, G. T. Jasion, J. P. Wooler, M. N. Petrovich, F. Poletti, and D. J. Richardson, “Towards Real-Time Mode Content Characterization of Multimode Fibers,” in Proceedings of European Conference on Optical Communications, (Cannes, 2014), paper Th.1.4.3.
    [Crossref]
  17. L. Grüner-Nielsen and J. W. Nicholson, “Stable mode converter for conversion between LP01 and LP11 using a thermally induced long period grating,” in Proceedings of IEEE Summer Topical Meeting, (2012), pp. 214–215, paper WC1.2.
  18. E. Hecht and A. Zajac, Optics (Addison-Wesley Publishing Company, 1973)
  19. S. W. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing (California Technical Publishing, 1997)
  20. A. J. Jerri, “The Shannon sampling theorem—Its various extensions and applications: A tutorial review,” Proc. IEEE 65(11), 1565–1596 (1977).
    [Crossref]
  21. M. Unser, A. Aldroubi, and M. Eden, “Polynomial spline signal approximations: Filter design and asymptotic equivalence with Shannon’s sampling theorem,” IEEE Trans. Inf. Theory 38(1), 95–103 (1992).
    [Crossref]
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  23. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
    [Crossref]
  24. J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the Modal Content of Large-Mode-Area Fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009).
    [Crossref]
  25. Crystran Sapphire Windows, “Sapphire Windows,” (Crystran, 2014) http://www.crystran.co.uk/windows/sapphire-windows
  26. N. V. Wheeler, M. N. Petrovich, R. Slavik, N. K. Baddela, E. R. Numkam Fokoua, J. R. Hayes, D. Gray, F. Poletti, and D. Richardson, “Wide-bandwidth, low-loss, 19-cell hollow core photonic band gap fiber and its potential for low latency data transmission,” in Optical Fiber Communication Conference, (Optical Society of America, 2012), paper PDP5A.2.
    [Crossref]

2014 (3)

2013 (1)

H. J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tunnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

2012 (2)

2009 (2)

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry,” Appl. Phys. B 96(2–3), 345–353 (2009).
[Crossref]

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the Modal Content of Large-Mode-Area Fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009).
[Crossref]

2008 (2)

2006 (1)

2005 (1)

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete Modal Decomposition for Optical Waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref] [PubMed]

1992 (2)

Y. Painchaud, M. A. Duguay, and F. Ouellette, “Interferometric time measurements of intermodal dispersion in optical fibers by using a CCD photodetector array,” Opt. Lett. 17(20), 1423–1425 (1992).
[Crossref] [PubMed]

M. Unser, A. Aldroubi, and M. Eden, “Polynomial spline signal approximations: Filter design and asymptotic equivalence with Shannon’s sampling theorem,” IEEE Trans. Inf. Theory 38(1), 95–103 (1992).
[Crossref]

1978 (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

1977 (1)

A. J. Jerri, “The Shannon sampling theorem—Its various extensions and applications: A tutorial review,” Proc. IEEE 65(11), 1565–1596 (1977).
[Crossref]

Abdolvand, A.

Abouraddy, A. F.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete Modal Decomposition for Optical Waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref] [PubMed]

Ahn, T.-J.

Alam, S.

Y. Jung, Q. Kang, J. K. Sahu, B. Corbett, R. Winfield, F. Poletti, S. Alam, and D. J. Richardson, “Reconfigurable modal gain control of a few-mode EDFA supporting six spatial modes,” IEEE Photon. Technol. Lett. 26(11), 1100–1103 (2014).
[Crossref]

Aldroubi, A.

M. Unser, A. Aldroubi, and M. Eden, “Polynomial spline signal approximations: Filter design and asymptotic equivalence with Shannon’s sampling theorem,” IEEE Trans. Inf. Theory 38(1), 95–103 (1992).
[Crossref]

Bigot, L.

Blin, S.

Bouwmans, G.

Carrero, C.

Chartier, T.

Chen, J. S. Y.

Corbett, B.

Y. Jung, Q. Kang, J. K. Sahu, B. Corbett, R. Winfield, F. Poletti, S. Alam, and D. J. Richardson, “Reconfigurable modal gain control of a few-mode EDFA supporting six spatial modes,” IEEE Photon. Technol. Lett. 26(11), 1100–1103 (2014).
[Crossref]

Demas, J.

Druon, F.

Duguay, M. A.

Eden, M.

M. Unser, A. Aldroubi, and M. Eden, “Polynomial spline signal approximations: Filter design and asymptotic equivalence with Shannon’s sampling theorem,” IEEE Trans. Inf. Theory 38(1), 95–103 (1992).
[Crossref]

Euser, T. G.

Fini, J. M.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the Modal Content of Large-Mode-Area Fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009).
[Crossref]

Fink, Y.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete Modal Decomposition for Optical Waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref] [PubMed]

Georges, P.

Ghalmi, S.

Hanna, M.

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

Jansen, F.

H. J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tunnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Jauregui, C.

H. J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tunnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Jerri, A. J.

A. J. Jerri, “The Shannon sampling theorem—Its various extensions and applications: A tutorial review,” Proc. IEEE 65(11), 1565–1596 (1977).
[Crossref]

Joannopoulos, J. D.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete Modal Decomposition for Optical Waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref] [PubMed]

Jung, Y.

Y. Jung, Q. Kang, J. K. Sahu, B. Corbett, R. Winfield, F. Poletti, S. Alam, and D. J. Richardson, “Reconfigurable modal gain control of a few-mode EDFA supporting six spatial modes,” IEEE Photon. Technol. Lett. 26(11), 1100–1103 (2014).
[Crossref]

Jung, Y. M.

Kaminski, C. F.

Kang, Q.

Y. Jung, Q. Kang, J. K. Sahu, B. Corbett, R. Winfield, F. Poletti, S. Alam, and D. J. Richardson, “Reconfigurable modal gain control of a few-mode EDFA supporting six spatial modes,” IEEE Photon. Technol. Lett. 26(11), 1100–1103 (2014).
[Crossref]

Kim, D. Y.

Le, S. D.

Le Cocq, G.

Lee, J. Y.

Leuchs, G.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry,” Appl. Phys. B 96(2–3), 345–353 (2009).
[Crossref]

Lévèque, L.

Limpert, J.

H. J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tunnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Ma, Y. Z.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry,” Appl. Phys. B 96(2–3), 345–353 (2009).
[Crossref]

Mermelstein, M. D.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the Modal Content of Large-Mode-Area Fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009).
[Crossref]

Moon, S.

Nguyen, D. M.

Nguyen, T. N.

Nicholson, J. W.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the Modal Content of Large-Mode-Area Fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (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(10), 7233–7243 (2008).
[Crossref] [PubMed]

Nold, J.

Oh, K.

Onishchukov, G.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry,” Appl. Phys. B 96(2–3), 345–353 (2009).
[Crossref]

Otto, H. J.

H. J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tunnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Ouellette, F.

Painchaud, Y.

Paurisse, M.

Peschel, U.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry,” Appl. Phys. B 96(2–3), 345–353 (2009).
[Crossref]

Poletti, F.

Y. Jung, Q. Kang, J. K. Sahu, B. Corbett, R. Winfield, F. Poletti, S. Alam, and D. J. Richardson, “Reconfigurable modal gain control of a few-mode EDFA supporting six spatial modes,” IEEE Photon. Technol. Lett. 26(11), 1100–1103 (2014).
[Crossref]

Provino, L.

Quiquempois, Y.

Ramachandran, S.

Richardson, D. J.

Y. Jung, Q. Kang, J. K. Sahu, B. Corbett, R. Winfield, F. Poletti, S. Alam, and D. J. Richardson, “Reconfigurable modal gain control of a few-mode EDFA supporting six spatial modes,” IEEE Photon. Technol. Lett. 26(11), 1100–1103 (2014).
[Crossref]

Russell, P. S.

Sahu, J. K.

Y. Jung, Q. Kang, J. K. Sahu, B. Corbett, R. Winfield, F. Poletti, S. Alam, and D. J. Richardson, “Reconfigurable modal gain control of a few-mode EDFA supporting six spatial modes,” IEEE Photon. Technol. Lett. 26(11), 1100–1103 (2014).
[Crossref]

Scharrer, M.

Schmauss, B.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry,” Appl. Phys. B 96(2–3), 345–353 (2009).
[Crossref]

Sévigny, B.

Shapira, O.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete Modal Decomposition for Optical Waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref] [PubMed]

Sillard, P.

Stutzki, F.

H. J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tunnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Sych, Y.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry,” Appl. Phys. B 96(2–3), 345–353 (2009).
[Crossref]

Thual, M.

Tunnermann, A.

H. J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tunnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

Unser, M.

M. Unser, A. Aldroubi, and M. Eden, “Polynomial spline signal approximations: Filter design and asymptotic equivalence with Shannon’s sampling theorem,” IEEE Trans. Inf. Theory 38(1), 95–103 (1992).
[Crossref]

Valentin, C.

Whyte, G.

Winfield, R.

Y. Jung, Q. Kang, J. K. Sahu, B. Corbett, R. Winfield, F. Poletti, S. Alam, and D. J. Richardson, “Reconfigurable modal gain control of a few-mode EDFA supporting six spatial modes,” IEEE Photon. Technol. Lett. 26(11), 1100–1103 (2014).
[Crossref]

Yablon, A. D.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the Modal Content of Large-Mode-Area Fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (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(10), 7233–7243 (2008).
[Crossref] [PubMed]

Youk, Y. C.

Appl. Opt. (1)

Appl. Phys. B (1)

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry,” Appl. Phys. B 96(2–3), 345–353 (2009).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the Modal Content of Large-Mode-Area Fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Y. Jung, Q. Kang, J. K. Sahu, B. Corbett, R. Winfield, F. Poletti, S. Alam, and D. J. Richardson, “Reconfigurable modal gain control of a few-mode EDFA supporting six spatial modes,” IEEE Photon. Technol. Lett. 26(11), 1100–1103 (2014).
[Crossref]

IEEE Trans. Inf. Theory (1)

M. Unser, A. Aldroubi, and M. Eden, “Polynomial spline signal approximations: Filter design and asymptotic equivalence with Shannon’s sampling theorem,” IEEE Trans. Inf. Theory 38(1), 95–103 (1992).
[Crossref]

J. Lightwave Technol. (2)

H. J. Otto, F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tunnermann, “Improved modal reconstruction for spatially and spectrally resolved imaging,” J. Lightwave Technol. 31(8), 1295–1299 (2013).
[Crossref]

B. Sévigny, G. Le Cocq, C. Carrero, C. Valentin, P. Sillard, G. Bouwmans, L. Bigot, and Y. Quiquempois, “Advanced S2 Imaging: Application of Multivariate Statistical Analysis to Spatially and Spectrally Resolved Datasets,” J. Lightwave Technol. 32(23), 4004–4010 (2014).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete Modal Decomposition for Optical Waveguides,” Phys. Rev. Lett. 94(14), 143902 (2005).
[Crossref] [PubMed]

Proc. IEEE (2)

A. J. Jerri, “The Shannon sampling theorem—Its various extensions and applications: A tutorial review,” Proc. IEEE 65(11), 1565–1596 (1977).
[Crossref]

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

Other (10)

Crystran Sapphire Windows, “Sapphire Windows,” (Crystran, 2014) http://www.crystran.co.uk/windows/sapphire-windows

N. V. Wheeler, M. N. Petrovich, R. Slavik, N. K. Baddela, E. R. Numkam Fokoua, J. R. Hayes, D. Gray, F. Poletti, and D. Richardson, “Wide-bandwidth, low-loss, 19-cell hollow core photonic band gap fiber and its potential for low latency data transmission,” in Optical Fiber Communication Conference, (Optical Society of America, 2012), paper PDP5A.2.
[Crossref]

National Instruments white paper, “The Fundamentals of FFT-Based Signal Analysis and Measurement in LabVIEW and LabWindows/CVI,” (National instruments, 2009) http://www.ni.com/white-paper/4278/en/

D. Gray, Z. Li, F. Poletti, R. Slavík, N. Wheeler, N. Baddela, M. Petrovich, A. Obeysekara, and D. Richardson, “Complementary Analysis of Modal Content and Properties in a 19-cell Hollow Core Photonic Band Gap Fiber using Time-of-Flight and S2 Techniques,” in European Conference and Exhibition on Optical Communication, OSA Technical Digest (online) (Optical Society of America, 2012), paper Mo.2.F.1.
[Crossref]

T. Ahn, S. Moon, S. Kim, K. Oh, D. Kim, J. Kobelke, K. Schuster, and J. Kirchhof, “Optical Frequency-Domain Modal Dispersion Measurement in Multimode Fibers Using Intermodal Interferometer,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OWI17.

S. Blin, D. M. Nguyen, T. N. Nguyen, L. Provino, M. Thual, and T. Chartier, “Simple Modal Analysis Method for Multi-Mode Fibers,” in Proceedings of European Conference on Optical Communications,(2009), paper P1.16.

D. R. Gray, S. R. Sandoghchi, N. V. Wheeler, G. T. Jasion, J. P. Wooler, M. N. Petrovich, F. Poletti, and D. J. Richardson, “Towards Real-Time Mode Content Characterization of Multimode Fibers,” in Proceedings of European Conference on Optical Communications, (Cannes, 2014), paper Th.1.4.3.
[Crossref]

L. Grüner-Nielsen and J. W. Nicholson, “Stable mode converter for conversion between LP01 and LP11 using a thermally induced long period grating,” in Proceedings of IEEE Summer Topical Meeting, (2012), pp. 214–215, paper WC1.2.

E. Hecht and A. Zajac, Optics (Addison-Wesley Publishing Company, 1973)

S. W. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing (California Technical Publishing, 1997)

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Figures (13)

Fig. 1
Fig. 1 Diagram of a tunable laser source (TLS) & CCD based S2 imaging system.
Fig. 2
Fig. 2 (a) schematic of the double reflection though an optical element (b) diagram of the propagation of a light beam through a t = 4mm sapphire window tilted at θ1 = 20° to the direction of light propagation.
Fig. 3
Fig. 3 Ideal theoretical Fresnel based MPI values for a “virtual” HOM as a function of angle of incidence (a) parallel (b) perpendicular incident polarization
Fig. 4
Fig. 4 Schematic of the modified S2 setup to measure the MPI value from the double reflection of a sapphire window.
Fig. 5
Fig. 5 DGD graph of the double reflection “virtual” HOM generated by a 3mm thick sapphire (a). Comparison between theoretical and experimental MPI values for the “virtual” HOM, as a function of angle of incidence, (b) parallel (c) perpendicular incident polarization
Fig. 6
Fig. 6 An illustration of spectral leakage; (a) A pure sine wave and top hat function, (b) processed with an FFT to produce a delta and sinc function respectively. (c) Multiplication of the sine and top hat function produces a band-limited function. (d) The FFT is a sinc function centered at delta function Fourier position (FP). (e) The band-limited function is picket fenced producing (f) the FFT of the sampled sine wave. Spectral leakage, (f) black arrows, occurs when the sampled sin wave does not contain an integer number of periods.
Fig. 7
Fig. 7 (a) Analysis of spectral leakage and sampling errors vs. DGD for a given set of experimental conditions (1550-1557 nm bandwidth, 10pm resolution), showing the sampling error increasing with DGD and the spectral leakage (SL) error oscillating between 0 and ~3.9dB. (b) Strategies to minimize errors: SL is reduced to ~0.3 dB by estimating the MPI as sum of 3 points at either side of the peak value; the sampling error can be reduced to <1 dB by reducing the measurement bandwidth (to ≤3 nm in this particular instance).
Fig. 8
Fig. 8 Plot showing how the MPI is underestimated due to sampling error. The latter is plotted as a function of the DGD normalized to the Nyquist frequency (abscissa) and to a normalized bandwidth, which allows “universal” correction factors to be obtained.
Fig. 9
Fig. 9 Dependence of the various MPI error components, including sampling error and spectral leakage, on the absolute value of the MPI, showing virtually no dependence as long as there is a dominant FM relative to a single HOM [11].
Fig. 10
Fig. 10 (a) DGD graph of the measured sapphire window double reflection modal interference peak summed MPI value. Comparison between theoretical and experimental MPI values for a “virtual” HOM generated by the sapphire window, as a function of angle of incidence, (b) parallel (c) perpendicular incident polarization.
Fig. 11
Fig. 11 Residual MPI error (both polarizations) after spectral leakage and sampling error corrections.
Fig. 12
Fig. 12 S2 of a 19c HC-PBGF with a selection of HOMs measured to produce MPI and DGD values with intensity and phase imaged
Fig. 13
Fig. 13 Spectral leakage and sampling errors corresponding to the selection of peaks measured in Fig. 12.

Equations (6)

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R ( n, θ dielectric,air )= ( ncos( θ dielectic )cos( θ air ) ncos( θ dielectic )+cos( θ air ) ) 2 ,
R ( n, θ dielectric,air )= ( cos( θ dielectric )ncos( θ air ) cos( θ dieelectric )+ncos( θ air ) ) 2 ,
T , ( n, θ dielectric,air )=1 R , ( n, θ dielectric,air )
DGD(n( λ ), θ air )= 2tn( λ )2t sin 2 ( θ air ) ccos( arcsin( sin( θ air )/n( λ ) ) ) ,
MPI , ( n( λ ), θ air ) =10 log 10 ( R , 2 ( n( λ ), θ air ) )
I( x,y,ω )= A FM ( x,y,ω ) 2 + A HOM ( x,y,ω ) 2 +2 A FM ( x,y,ω ) A HOM ( x,y,ω )exp( ( β FM ( ω ) β HOM ( ω ) )L+Δφ ),

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