Abstract

We present an iterative design method for the coupling and the mode conversion of arbitrary modes to focused surface plasmons using a large array of aperiodically randomly located slits in a thin metal film. As the distance between the slits is small and the number of slits is large, significant mutual coupling occurs between the slits which makes an accurate computation of the field scattered by the slits difficult. We use an accurate modal source radiator model to efficiently compute the fields in a significantly shorter time compared with three-dimensional (3D) full-field rigorous simulations, so that iterative optimization is efficiently achieved. Since our model accounts for mutual coupling between the slits, the scattering by the slits of both the source wave and the focused surface plasmon can be incorporated in the optimization scheme. We apply this method to the design of various types of couplers for arbitrary fiber modes and a mode demultiplexer that focuses three orthogonal fiber modes to three different foci. Finally, we validate our design results using fully vectorial 3D finite-difference time-domain (FDTD) simulations.

© 2014 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  27. D. A. B. Miller, “Reconfigurable add-drop multiplexer for spatial modes,” Opt. Express 21, 20220–20229 (2013).
    [Crossref] [PubMed]
  28. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
    [Crossref]
  29. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
    [Crossref] [PubMed]
  30. J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
    [Crossref] [PubMed]
  31. F. Afshinmanesh, J. S. White, W. Cai, and M. L. Brongersma, “Measurement of the polarization state of light using an integrated plasmonic polarimeter,” Nanophotonics 1, 125–129 (2012).
    [Crossref]
  32. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
    [Crossref] [PubMed]
  33. C. Zhao and J. Zhang, “Binary plasmonics: launching surface plasmon polaritons to a desired pattern,” Opt. Lett. 34, 2417–2419 (2009).
    [Crossref] [PubMed]
  34. T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
    [Crossref] [PubMed]
  35. L. Li, T. Li, S. Wang, S. Zhu, and X. Zhang, “Broad band focusing and demultiplexing of in-plane propagating surface plasmons,” Nano Lett. 11, 4357–4361 (2011).
    [Crossref] [PubMed]
  36. S.-H. Chang, S. K. Gray, and G. C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Express 13, 3150–3165 (2005).
    [Crossref] [PubMed]
  37. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
    [Crossref] [PubMed]
  38. H. Liu and P. Lalanne, “Comprehensive microscopic model of the extraordinary optical transmission,” J. Opt. Soc. Am. A 27, 2542–2550 (2010).
    [Crossref]
  39. X. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B 84, 245120 (2011).
    [Crossref]
  40. Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006).
    [Crossref] [PubMed]
  41. E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000).
    [Crossref]
  42. M. Paulus, P. G. Balmaz, and O. J. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797 (2000).
    [Crossref]
  43. M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A 18, 854–861 (2001).
    [Crossref]
  44. T. Tanemura, P. Wahl, S. Fan, and D. A. B. Miller, “Modal source radiator model for arbitrary two-dimensional arrays of subwavelength apertures on metal films,” IEEE J. Sel. Top. Quant. 19, 4601110 (2013).
    [Crossref]
  45. P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-Source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromag. Res. 138, 467–478 (2013).
  46. S. Kim, H. Kim, Y. Lim, and B. Lee, “Off-axis directional beaming of optical field diffracted by a single sub-wavelength metal slit with asymmetric dielectric surface gratings,” Appl. Phys. Lett. 90, 051113 (2007).
    [Crossref]
  47. B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quant. Electron. 34, 47–87 (2010).
    [Crossref]
  48. J. S. White, G. Veronis, Z. Yu, E. S. Barnard, A. Chandran, S. Fan, and M. L. Brongersma, “Extraordinary optical absorption through subwavelength slits,” Opt. Lett. 34, 686–688 (2009).
    [Crossref] [PubMed]
  49. F. López-Tejeira, S. G. Rodrigo, L. Martíin-Moreno, F. J. Garcíia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
    [Crossref]
  50. D. A. B. Miller, “How complicated must an optical component be?” J. Opt. Soc. Am. A 30, 238–251 (2013).
    [Crossref]

2013 (10)

T. Sakamoto, T. Mori, T. Yamamoto, N. Hanzawa, S. Tomita, F. Yamamoto, K. Saitoh, and M. Koshiba, “Mode-division multiplexing transmission system With DMD-Independent low complexity MIMO processing,” J. Light-wave Technol. 31, 2192–2199 (2013).
[Crossref]

A. M. Bratkovsky, J. B. Khurgin, E. Ponizovskaya, W. V. Sorin, and M. R. T. Tan, “Mode division multiplexed (MDM) waveguide link scheme with cascaded Y-junctions,” Opt. Commun. 309, 85–89 (2013).
[Crossref]

D. A. B. Miller, “Self-aligning universal beam coupler,” Opt. Express 21, 6360–6370 (2013).
[Crossref] [PubMed]

D. A. B. Miller, “Self-configuring universal linear optical component,” Photonics Research 1, 1–15 (2013).
[Crossref]

D. Miller, “Establishing optimal wave communication channels automatically,” J. Lightwave Technol.  31, 3987–3994 (2013).
[Crossref]

D. A. B. Miller, “Reconfigurable add-drop multiplexer for spatial modes,” Opt. Express 21, 20220–20229 (2013).
[Crossref] [PubMed]

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
[Crossref] [PubMed]

T. Tanemura, P. Wahl, S. Fan, and D. A. B. Miller, “Modal source radiator model for arbitrary two-dimensional arrays of subwavelength apertures on metal films,” IEEE J. Sel. Top. Quant. 19, 4601110 (2013).
[Crossref]

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-Source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromag. Res. 138, 467–478 (2013).

D. A. B. Miller, “How complicated must an optical component be?” J. Opt. Soc. Am. A 30, 238–251 (2013).
[Crossref]

2012 (12)

F. Afshinmanesh, J. S. White, W. Cai, and M. L. Brongersma, “Measurement of the polarization state of light using an integrated plasmonic polarimeter,” Nanophotonics 1, 125–129 (2012).
[Crossref]

H. Bulow, “Optical-mode demultiplexing by optical MIMO filtering of spatial samples,” IEEE Photon. Tech. L. 24, 1045–1047 (2012).
[Crossref]

V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20, 28388–28397 (2012).
[Crossref] [PubMed]

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, and P. Sillard, “Mode-division multiplexing of 2 × 100 Gb/s channels using an LCOS-based spatial modulator,” J. Lightwave Technol. 30, 618–623 (2012).
[Crossref]

J. Carpenter and T. D. Wilkinson, “Characterization of multimode fiber by selective mode excitation,” J. Light-wave Technol. 30, 1386–1392 (2012).
[Crossref]

A. E. Willner, J. Wang, and H. Huang, “A different angle on light communications,” Science 337, 655–656 (2012).
[Crossref] [PubMed]

C. P. Tsekrekos and D. Syvridis, “All-fiber broadband mode converter for future wavelength and mode division multiplexing systems,” IEEE Photon. Tech. L. 24, 1638–1641 (2012).
[Crossref]

D. S. Ly-Gagnon, K. C. Balram, J. S. White, P. Wahl, M. L. Brongersma, and D. A. B. Miller, “Routing and photodetection in subwavelength plasmonic slot waveguides,” Nanophotonics 1, 9–16 (2012).
[Crossref]

D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express 20, 23985–23993 (2012).
[Crossref] [PubMed]

N. Riesen, J. D. Love, and J. W. Arkwright, “Few-mode elliptical-core fiber data transmission,” IEEE Photon. Tech. L., IEEE 24, 344–346 (2012).
[Crossref]

J. D. Love and N. Riesen, “Mode-selective couplers for few-mode optical fiber networks,” Opt. Lett. 37, 3990–3992 (2012).
[Crossref] [PubMed]

N. Riesen and J. D. Love, “Weakly-guiding mode-selective fiber couplers,” IEEE J. Quantum Electron. 48, 941–945 (2012).
[Crossref]

2011 (4)

G. Stepniak, L. Maksymiuk, and J. Siuzdak, “Binary-phase spatial light filters for mode-selective excitation of multimode fibers,” J. Lightwave Technol. 29, 1980–1987 (2011).
[Crossref]

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
[Crossref] [PubMed]

L. Li, T. Li, S. Wang, S. Zhu, and X. Zhang, “Broad band focusing and demultiplexing of in-plane propagating surface plasmons,” Nano Lett. 11, 4357–4361 (2011).
[Crossref] [PubMed]

X. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B 84, 245120 (2011).
[Crossref]

2010 (3)

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quant. Electron. 34, 47–87 (2010).
[Crossref]

H. Liu and P. Lalanne, “Comprehensive microscopic model of the extraordinary optical transmission,” J. Opt. Soc. Am. A 27, 2542–2550 (2010).
[Crossref]

2009 (3)

2008 (1)

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[Crossref] [PubMed]

2007 (2)

F. López-Tejeira, S. G. Rodrigo, L. Martíin-Moreno, F. J. Garcíia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[Crossref]

S. Kim, H. Kim, Y. Lim, and B. Lee, “Off-axis directional beaming of optical field diffracted by a single sub-wavelength metal slit with asymmetric dielectric surface gratings,” Appl. Phys. Lett. 90, 051113 (2007).
[Crossref]

2006 (1)

Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006).
[Crossref] [PubMed]

2005 (5)

S.-H. Chang, S. K. Gray, and G. C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Express 13, 3150–3165 (2005).
[Crossref] [PubMed]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[Crossref] [PubMed]

Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator,” Opt. Lett. 30, 141–143 (2005).
[Crossref] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Sub-wavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

2002 (2)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[Crossref] [PubMed]

2001 (1)

2000 (2)

E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000).
[Crossref]

M. Paulus, P. G. Balmaz, and O. J. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797 (2000).
[Crossref]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Afshinmanesh, F.

F. Afshinmanesh, J. S. White, W. Cai, and M. L. Brongersma, “Measurement of the polarization state of light using an integrated plasmonic polarimeter,” Nanophotonics 1, 125–129 (2012).
[Crossref]

Akimov, Y.

Al-Hashimi, H.

H. Bulow, H. Al-Hashimi, and B. Schmauss, “Spatial mode multiplexers and MIMO processing,” in Opto-Electronics and Communications Conference (OECC), 2012 17th, (IEEE, 2012), pp. 562–563.
[Crossref]

Antoniou, N.

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
[Crossref] [PubMed]

Arkwright, J. W.

N. Riesen, J. D. Love, and J. W. Arkwright, “Few-mode elliptical-core fiber data transmission,” IEEE Photon. Tech. L., IEEE 24, 344–346 (2012).
[Crossref]

Balmaz, P. G.

M. Paulus, P. G. Balmaz, and O. J. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797 (2000).
[Crossref]

Balram, K. C.

D. S. Ly-Gagnon, K. C. Balram, J. S. White, P. Wahl, M. L. Brongersma, and D. A. B. Miller, “Routing and photodetection in subwavelength plasmonic slot waveguides,” Nanophotonics 1, 9–16 (2012).
[Crossref]

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
[Crossref] [PubMed]

Barnard, E. S.

Barnett, S. M.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref] [PubMed]

Beijersbergen, M. W.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

Berkhout, G. C. G.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

Bigo, S.

Boutin, A.

Bozhevolnyi, S. I.

F. López-Tejeira, S. G. Rodrigo, L. Martíin-Moreno, F. J. Garcíia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[Crossref]

Bratkovsky, A. M.

A. M. Bratkovsky, J. B. Khurgin, E. Ponizovskaya, W. V. Sorin, and M. R. T. Tan, “Mode division multiplexed (MDM) waveguide link scheme with cascaded Y-junctions,” Opt. Commun. 309, 85–89 (2013).
[Crossref]

Brindel, P.

Brongersma, M. L.

D. S. Ly-Gagnon, K. C. Balram, J. S. White, P. Wahl, M. L. Brongersma, and D. A. B. Miller, “Routing and photodetection in subwavelength plasmonic slot waveguides,” Nanophotonics 1, 9–16 (2012).
[Crossref]

F. Afshinmanesh, J. S. White, W. Cai, and M. L. Brongersma, “Measurement of the polarization state of light using an integrated plasmonic polarimeter,” Nanophotonics 1, 125–129 (2012).
[Crossref]

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H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
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D. S. Ly-Gagnon, K. C. Balram, J. S. White, P. Wahl, M. L. Brongersma, and D. A. B. Miller, “Routing and photodetection in subwavelength plasmonic slot waveguides,” Nanophotonics 1, 9–16 (2012).
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D. S. Ly-Gagnon, K. C. Balram, J. S. White, P. Wahl, M. L. Brongersma, and D. A. B. Miller, “Routing and photodetection in subwavelength plasmonic slot waveguides,” Nanophotonics 1, 9–16 (2012).
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J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
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Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
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N. Riesen, J. D. Love, and J. W. Arkwright, “Few-mode elliptical-core fiber data transmission,” IEEE Photon. Tech. L., IEEE 24, 344–346 (2012).
[Crossref]

N. Riesen and J. D. Love, “Weakly-guiding mode-selective fiber couplers,” IEEE J. Quantum Electron. 48, 941–945 (2012).
[Crossref]

J. D. Love and N. Riesen, “Mode-selective couplers for few-mode optical fiber networks,” Opt. Lett. 37, 3990–3992 (2012).
[Crossref] [PubMed]

Rodrigo, S. G.

F. López-Tejeira, S. G. Rodrigo, L. Martíin-Moreno, F. J. Garcíia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[Crossref]

Ruan, Z.

Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006).
[Crossref] [PubMed]

Saitoh, K.

T. Sakamoto, T. Mori, T. Yamamoto, N. Hanzawa, S. Tomita, F. Yamamoto, K. Saitoh, and M. Koshiba, “Mode-division multiplexing transmission system With DMD-Independent low complexity MIMO processing,” J. Light-wave Technol. 31, 2192–2199 (2013).
[Crossref]

Sakamoto, T.

T. Sakamoto, T. Mori, T. Yamamoto, N. Hanzawa, S. Tomita, F. Yamamoto, K. Saitoh, and M. Koshiba, “Mode-division multiplexing transmission system With DMD-Independent low complexity MIMO processing,” J. Light-wave Technol. 31, 2192–2199 (2013).
[Crossref]

Salsi, M.

Schatz, G. C.

Schmauss, B.

H. Bulow, H. Al-Hashimi, and B. Schmauss, “Spatial mode multiplexers and MIMO processing,” in Opto-Electronics and Communications Conference (OECC), 2012 17th, (IEEE, 2012), pp. 562–563.
[Crossref]

Sillard, P.

Siuzdak, J.

Sorin, W. V.

A. M. Bratkovsky, J. B. Khurgin, E. Ponizovskaya, W. V. Sorin, and M. R. T. Tan, “Mode division multiplexed (MDM) waveguide link scheme with cascaded Y-junctions,” Opt. Commun. 309, 85–89 (2013).
[Crossref]

Sperti, D.

Srituravanich, W.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Steele, J. M.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Stepniak, G.

Sun, C.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Syvridis, D.

C. P. Tsekrekos and D. Syvridis, “All-fiber broadband mode converter for future wavelength and mode division multiplexing systems,” IEEE Photon. Tech. L. 24, 1638–1641 (2012).
[Crossref]

Tan, M. R. T.

A. M. Bratkovsky, J. B. Khurgin, E. Ponizovskaya, W. V. Sorin, and M. R. T. Tan, “Mode division multiplexed (MDM) waveguide link scheme with cascaded Y-junctions,” Opt. Commun. 309, 85–89 (2013).
[Crossref]

Tanemura, T.

T. Tanemura, P. Wahl, S. Fan, and D. A. B. Miller, “Modal source radiator model for arbitrary two-dimensional arrays of subwavelength apertures on metal films,” IEEE J. Sel. Top. Quant. 19, 4601110 (2013).
[Crossref]

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
[Crossref] [PubMed]

Thienpont, H.

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-Source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromag. Res. 138, 467–478 (2013).

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Thomsen, B. C.

J. A. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Optical vortex based mode division multiplexing over graded-index multimode fibre,” (Optical Society of America, 2013), OSA Technical Digest (online),OTh4G.3+.

Tomita, S.

T. Sakamoto, T. Mori, T. Yamamoto, N. Hanzawa, S. Tomita, F. Yamamoto, K. Saitoh, and M. Koshiba, “Mode-division multiplexing transmission system With DMD-Independent low complexity MIMO processing,” J. Light-wave Technol. 31, 2192–2199 (2013).
[Crossref]

Tran, P.

Tsekrekos, C. P.

C. P. Tsekrekos and D. Syvridis, “All-fiber broadband mode converter for future wavelength and mode division multiplexing systems,” IEEE Photon. Tech. L. 24, 1638–1641 (2012).
[Crossref]

Van Erps, J.

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-Source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromag. Res. 138, 467–478 (2013).

Verluise, F.

Vermeulen, N.

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-Source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromag. Res. 138, 467–478 (2013).

Veronis, G.

Vlasko-Vlasov, V. K.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Sub-wavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Wagner, K. H.

K. H. Wagner, “Mode group demultiplexing and modal dispersion compensation using spatial-spectral holography,” in IEEE Photonics Society Summer Topical Meetings, Space Division Multiplexing for Optical Communications,(2013), pp. 89–90.

Wahl, P.

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-Source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromag. Res. 138, 467–478 (2013).

T. Tanemura, P. Wahl, S. Fan, and D. A. B. Miller, “Modal source radiator model for arbitrary two-dimensional arrays of subwavelength apertures on metal films,” IEEE J. Sel. Top. Quant. 19, 4601110 (2013).
[Crossref]

D. S. Ly-Gagnon, K. C. Balram, J. S. White, P. Wahl, M. L. Brongersma, and D. A. B. Miller, “Routing and photodetection in subwavelength plasmonic slot waveguides,” Nanophotonics 1, 9–16 (2012).
[Crossref]

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
[Crossref] [PubMed]

Wang, J.

A. E. Willner, J. Wang, and H. Huang, “A different angle on light communications,” Science 337, 655–656 (2012).
[Crossref] [PubMed]

Wang, Q.

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
[Crossref] [PubMed]

Wang, S.

L. Li, T. Li, S. Wang, S. Zhu, and X. Zhang, “Broad band focusing and demultiplexing of in-plane propagating surface plasmons,” Nano Lett. 11, 4357–4361 (2011).
[Crossref] [PubMed]

Weeber, J. C.

F. López-Tejeira, S. G. Rodrigo, L. Martíin-Moreno, F. J. Garcíia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[Crossref]

Welp, U.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Sub-wavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

White, J. S.

D. S. Ly-Gagnon, K. C. Balram, J. S. White, P. Wahl, M. L. Brongersma, and D. A. B. Miller, “Routing and photodetection in subwavelength plasmonic slot waveguides,” Nanophotonics 1, 9–16 (2012).
[Crossref]

F. Afshinmanesh, J. S. White, W. Cai, and M. L. Brongersma, “Measurement of the polarization state of light using an integrated plasmonic polarimeter,” Nanophotonics 1, 125–129 (2012).
[Crossref]

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
[Crossref] [PubMed]

J. S. White, G. Veronis, Z. Yu, E. S. Barnard, A. Chandran, S. Fan, and M. L. Brongersma, “Extraordinary optical absorption through subwavelength slits,” Opt. Lett. 34, 686–688 (2009).
[Crossref] [PubMed]

Wilkinson, T. D.

J. Carpenter and T. D. Wilkinson, “Characterization of multimode fiber by selective mode excitation,” J. Light-wave Technol. 30, 1386–1392 (2012).
[Crossref]

J. A. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Optical vortex based mode division multiplexing over graded-index multimode fibre,” (Optical Society of America, 2013), OSA Technical Digest (online),OTh4G.3+.

Willner, A. E.

A. E. Willner, J. Wang, and H. Huang, “A different angle on light communications,” Science 337, 655–656 (2012).
[Crossref] [PubMed]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Yamamoto, F.

T. Sakamoto, T. Mori, T. Yamamoto, N. Hanzawa, S. Tomita, F. Yamamoto, K. Saitoh, and M. Koshiba, “Mode-division multiplexing transmission system With DMD-Independent low complexity MIMO processing,” J. Light-wave Technol. 31, 2192–2199 (2013).
[Crossref]

Yamamoto, T.

T. Sakamoto, T. Mori, T. Yamamoto, N. Hanzawa, S. Tomita, F. Yamamoto, K. Saitoh, and M. Koshiba, “Mode-division multiplexing transmission system With DMD-Independent low complexity MIMO processing,” J. Light-wave Technol. 31, 2192–2199 (2013).
[Crossref]

Yin, L.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Sub-wavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Yu, Z.

Yuan, G.

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
[Crossref] [PubMed]

Yuan, X.-C.

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
[Crossref] [PubMed]

Zhang, J.

Zhang, S.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[Crossref] [PubMed]

Zhang, X.

L. Li, T. Li, S. Wang, S. Zhu, and X. Zhang, “Broad band focusing and demultiplexing of in-plane propagating surface plasmons,” Nano Lett. 11, 4357–4361 (2011).
[Crossref] [PubMed]

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Zhao, C.

Zhu, S.

L. Li, T. Li, S. Wang, S. Zhu, and X. Zhang, “Broad band focusing and demultiplexing of in-plane propagating surface plasmons,” Nano Lett. 11, 4357–4361 (2011).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

S. Kim, H. Kim, Y. Lim, and B. Lee, “Off-axis directional beaming of optical field diffracted by a single sub-wavelength metal slit with asymmetric dielectric surface gratings,” Appl. Phys. Lett. 90, 051113 (2007).
[Crossref]

IEEE J. Quantum Electron. (1)

N. Riesen and J. D. Love, “Weakly-guiding mode-selective fiber couplers,” IEEE J. Quantum Electron. 48, 941–945 (2012).
[Crossref]

IEEE J. Sel. Top. Quant. (1)

T. Tanemura, P. Wahl, S. Fan, and D. A. B. Miller, “Modal source radiator model for arbitrary two-dimensional arrays of subwavelength apertures on metal films,” IEEE J. Sel. Top. Quant. 19, 4601110 (2013).
[Crossref]

IEEE Photon. Tech. L. (2)

C. P. Tsekrekos and D. Syvridis, “All-fiber broadband mode converter for future wavelength and mode division multiplexing systems,” IEEE Photon. Tech. L. 24, 1638–1641 (2012).
[Crossref]

H. Bulow, “Optical-mode demultiplexing by optical MIMO filtering of spatial samples,” IEEE Photon. Tech. L. 24, 1045–1047 (2012).
[Crossref]

IEEE Photon. Tech. L., IEEE (1)

N. Riesen, J. D. Love, and J. W. Arkwright, “Few-mode elliptical-core fiber data transmission,” IEEE Photon. Tech. L., IEEE 24, 344–346 (2012).
[Crossref]

J. Light-wave Technol. (2)

T. Sakamoto, T. Mori, T. Yamamoto, N. Hanzawa, S. Tomita, F. Yamamoto, K. Saitoh, and M. Koshiba, “Mode-division multiplexing transmission system With DMD-Independent low complexity MIMO processing,” J. Light-wave Technol. 31, 2192–2199 (2013).
[Crossref]

J. Carpenter and T. D. Wilkinson, “Characterization of multimode fiber by selective mode excitation,” J. Light-wave Technol. 30, 1386–1392 (2012).
[Crossref]

J. Lightwave Technol (1)

D. Miller, “Establishing optimal wave communication channels automatically,” J. Lightwave Technol.  31, 3987–3994 (2013).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. A (3)

Nano Lett. (4)

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Sub-wavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011).
[Crossref] [PubMed]

L. Li, T. Li, S. Wang, S. Zhu, and X. Zhang, “Broad band focusing and demultiplexing of in-plane propagating surface plasmons,” Nano Lett. 11, 4357–4361 (2011).
[Crossref] [PubMed]

Nanophotonics (2)

F. Afshinmanesh, J. S. White, W. Cai, and M. L. Brongersma, “Measurement of the polarization state of light using an integrated plasmonic polarimeter,” Nanophotonics 1, 125–129 (2012).
[Crossref]

D. S. Ly-Gagnon, K. C. Balram, J. S. White, P. Wahl, M. L. Brongersma, and D. A. B. Miller, “Routing and photodetection in subwavelength plasmonic slot waveguides,” Nanophotonics 1, 9–16 (2012).
[Crossref]

Nat. Phys. (1)

F. López-Tejeira, S. G. Rodrigo, L. Martíin-Moreno, F. J. Garcíia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[Crossref]

Nature (2)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[Crossref] [PubMed]

Opt. Commun. (1)

A. M. Bratkovsky, J. B. Khurgin, E. Ponizovskaya, W. V. Sorin, and M. R. T. Tan, “Mode division multiplexed (MDM) waveguide link scheme with cascaded Y-junctions,” Opt. Commun. 309, 85–89 (2013).
[Crossref]

Opt. Express (6)

Opt. Lett. (4)

Photonics Research (1)

D. A. B. Miller, “Self-configuring universal linear optical component,” Photonics Research 1, 1–15 (2013).
[Crossref]

Phys. Rev. B (2)

X. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B 84, 245120 (2011).
[Crossref]

E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000).
[Crossref]

Phys. Rev. E (1)

M. Paulus, P. G. Balmaz, and O. J. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797 (2000).
[Crossref]

Phys. Rev. Lett. (4)

Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006).
[Crossref] [PubMed]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[Crossref] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref] [PubMed]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

Prog. Electromag. Res. (1)

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-Source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromag. Res. 138, 467–478 (2013).

Prog. Quant. Electron. (1)

B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quant. Electron. 34, 47–87 (2010).
[Crossref]

Science (3)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[Crossref] [PubMed]

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
[Crossref] [PubMed]

A. E. Willner, J. Wang, and H. Huang, “A different angle on light communications,” Science 337, 655–656 (2012).
[Crossref] [PubMed]

Other (3)

H. Bulow, H. Al-Hashimi, and B. Schmauss, “Spatial mode multiplexers and MIMO processing,” in Opto-Electronics and Communications Conference (OECC), 2012 17th, (IEEE, 2012), pp. 562–563.
[Crossref]

K. H. Wagner, “Mode group demultiplexing and modal dispersion compensation using spatial-spectral holography,” in IEEE Photonics Society Summer Topical Meetings, Space Division Multiplexing for Optical Communications,(2013), pp. 89–90.

J. A. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Optical vortex based mode division multiplexing over graded-index multimode fibre,” (Optical Society of America, 2013), OSA Technical Digest (online),OTh4G.3+.

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Figures (11)

Fig. 1
Fig. 1 Illustration of the plasmonic mode coupler/demultiplexer that consists of a thin metal layer grown on top of an oxide in which slits are etched. Well optimized slit locations can focus a particular mode to a single location or can demultiplex several orthogonal modes to different focal points.
Fig. 2
Fig. 2 The field inside the slits is decomposed into TE01 and TE02 eigenmodes of the slits.
Fig. 3
Fig. 3 Illustration of the different steps in one iteration of the design method.
Fig. 4
Fig. 4 Illustration of the design method. We assume that the surface plasmon propagates with phase velocity kspp. For slit Sk, the dashed blue circles represent the locations where ΔΦq,k = arg(Ψq,k) and the full black circles represent the locations where ΔΦq,k = 2. Also, δ1 = ΔΦq,k/kspp and δ2 = (2π − ΔΦq,k)/kspp. The design method has the goal to place the slit in a location where ΔΦq,k ≈ 2, like the area inside the red circle.
Fig. 5
Fig. 5 Design of a slit array that focuses an L31 fiber mode (a) at r 1 F = ( 0 , 0 ) μm. In (b) the |Ez|2 field just below the metal layer is depicted and the white lines show the optimized positions of the slits. The cross-section of |Ez|2 through the focus and along the yellow dashed line is depicted in (c). All the fields are normalized to the average field intensity in the fiber core.
Fig. 6
Fig. 6 Design of a slit array that focuses an L31 fiber mode at r 1 F = ( 12 , 0 ) μm. In (a) the |Ez|2 field just below the metal layer is depicted and the white lines show the optimized positions of the slits. The cross-section of |Ez|2 through the focus and along the yellow dashed line is depicted in (b). All the fields are normalized to the average field intensity in the fiber core.
Fig. 7
Fig. 7 | E z ( r 1 F ) | 2 at each iteration step for r 1 F = ( 0 , 0 ) μm (a) and for r 1 F = ( 12 , 0 ) μm (b). One clearly observes the improvement in field intensity obtained at iteration 30 when we remove the slits for which Pk < 0. All the fields are normalized to the average field intensity in the fiber core.
Fig. 8
Fig. 8 Ez field intensity just below the metal layer. The white lines show the optimized positions of the slits for demultiplexing the L01, L21 and L22 fiber modes. The fields are normalized to the average field intensity in the fiber core.
Fig. 9
Fig. 9 (a) Field intensity | E z ( r q F ) | 2 at each iteration step when demultiplexing the L01, L21 and L22 fiber modes and (b) Cross-section at x = 7 μm of the |Ez|2 field intensity for the optimized structure shown in Figs. 8(d)–8(f) along the dashed yellow lines. The fields are normalized to the average field intensity in the fiber core.
Fig. 10
Fig. 10 Cross section of the |Ez|2 field intensity for the optimized structure shown in Fig. 8 along the full line at x = 0 μm for comparison with a full-field FDTD simulation. The FDTD simulation is run on a domain containing all the slits enclosed by the dotted box in Fig. 8. The fields are normalized to the average field intensity in the fiber core.
Fig. 11
Fig. 11 Ez field intensity just below the metal layer. The white lines show the optimized positions of the slits for demultiplexing the L01, L21 and L22 fiber modes focusing to r 1 F = ( 7 , 3 ) μm, r 2 F = ( 7 , 0 ) μm, r 3 F = ( 7 , 3 ) μm The fields are normalized to the average field intensity in the fiber core.

Equations (17)

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

| E = α ( A α e i q α z | e α + + B α e i q α z | e α ) | H = α ( A α e i q α z | h α + + B α e i q α z | h α )
| H t I = α ( A α e i q α z 1 B α e i q α z 1 ) | h t α
| H t I = | H 0 I + α ( A α e i q α z 1 G H I | e α + + B α e i q α z 1 G H I | e α )
| E z I = | E z 0 I + α ( A α e i q α z 1 G E I | e α + + B α e i q α z 1 G E I | e α )
Ψ q , k = | E z 0 I ( r q F ) + α ( in slit k ) ( A α e i q α z 1 G E I | e α + ( r q F ) + B α e i q α z 1 G E I | e α ( r q F ) )
Δ n Φ q , k = arg ( Ψ q , k ) ϕ q ref + 2 n π
ϕ q ref = arg ( exp ( i 2 π ( q 1 ) / n f ) )
C F k ( r ) = q = 1 n f ( min n { | R k q | r q F r | Δ n Φ q , k k spp | } w q ) G k n ( r ) G k ( r )
G k ( r ) = exp ( | r r k | 2 / σ g 2 )
G k n ( r ) = 1 + l k n s exp ( | r r l | 2 / σ n n 2 )
w q new = w q old ( max ( | E z ( r q F ) | 2 ) | E z ( r q F ) | ) v
P k = q = 1 n f [ ( Ψ q , k ) ( ϕ q ref ) + ( Ψ q , k ) ( ϕ q ref ) ] < 0
r k new = ( 1 w ) r k + w r k min
P | | ( x , y ) = z = z 2 ( P ( x , y , z ) × z ) d z
| P | | ( x , y ) | = τ | E z ( x , y ) | 2
η = y 1 y 2 τ | E z ( x F , y ) | 2 d y P z source d x d y
N D = 2 M c ( M I + M O M C 1 2 )

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