Abstract

In this paper, we propose to generate/detect orbital angular momentum (OAM) states based on the multimode interference (MMI) effect in a piece of ring core fiber. A comprehensive theory for the MMI process inside ring core fibers is presented. The MMI process inside ring core fibers will convert one input image into multiple equally spaced duplicated output images. After phase adjustment by a fixed phase shifter array, these output images will stimulate OAM states in the ring core fiber or in free space. Henceforth, a novel OAM multiplexer/de-multiplexer (MUX/DEMUX) can be realized by a piece of ring core fiber and a fixed phase shifters array.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  4. 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(25), 257901 (2002).
    [Crossref] [PubMed]
  5. M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]

2015 (1)

2014 (3)

2013 (1)

J. Zhou and P. Gallion, “A Novel Mode Multiplexer/De-Multiplexer for Multi-core Fibers,” IEEE Photon. Technol. Lett. 25(13), 1214–1217 (2013).
[Crossref]

2012 (3)

2011 (1)

2010 (1)

2008 (1)

X. Yang, Y. Yang, Q. Cai, Y. Zhao, A. Fang, W. Hu, and A. Xu, “Primary Experiments on 2-D and 1-D Fiber-type Optical Phased Array,” Proc. SPIE, 7136, Optical Transmission, Switching, and Subsystems VI, 71363J (2008).
[Crossref]

2005 (1)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

2002 (1)

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(25), 257901 (2002).
[Crossref] [PubMed]

1998 (1)

1996 (1)

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical wave-front laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

Ahmed, N.

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(25), 257901 (2002).
[Crossref] [PubMed]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical wave-front laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

Besse, P. A.

Birnbaum, K.

Cai, Q.

X. Yang, Y. Yang, Q. Cai, Y. Zhao, A. Fang, W. Hu, and A. Xu, “Primary Experiments on 2-D and 1-D Fiber-type Optical Phased Array,” Proc. SPIE, 7136, Optical Transmission, Switching, and Subsystems VI, 71363J (2008).
[Crossref]

Cai, X.

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

Cappuzzo, M.

Chen, L.

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

Chitgarha, M. R.

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical wave-front laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

Courtial, J.

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(25), 257901 (2002).
[Crossref] [PubMed]

Dolinar, S.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Y. Yan, J. Wang, L. Zhang, J. Y. Yang, I. M. Fazal, N. Ahmed, B. Shamee, A. E. Willner, K. Birnbaum, and S. Dolinar, “Fiber coupler for generating orbital angular momentum modes,” Opt. Lett. 36(21), 4269–4271 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-21-4269 .
[Crossref] [PubMed]

Dolinar, S. J.

Earnshaw, M.

Eckner, J.

Fang, A.

X. Yang, Y. Yang, Q. Cai, Y. Zhao, A. Fang, W. Hu, and A. Xu, “Primary Experiments on 2-D and 1-D Fiber-type Optical Phased Array,” Proc. SPIE, 7136, Optical Transmission, Switching, and Subsystems VI, 71363J (2008).
[Crossref]

Fazal, I. M.

Ferrari, C.

Fontaine, N. K.

Franke-Arnold, S.

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(25), 257901 (2002).
[Crossref] [PubMed]

Gallion, P.

J. Zhou and P. Gallion, “A Novel Mode Multiplexer/De-Multiplexer for Multi-core Fibers,” IEEE Photon. Technol. Lett. 25(13), 1214–1217 (2013).
[Crossref]

J. Zhou and P. Gallion, “Mode De-Multiplexers for Rectangular Multimode Waveguides,” IEEE Photon. Technol. Lett.In press.

Gamper, E.

Golowich, S.

Golowich, S. E.

Gregg, P.

Guan, B.

Hess, R.

Hu, W.

X. Yang, Y. Yang, Q. Cai, Y. Zhao, A. Fang, W. Hu, and A. Xu, “Primary Experiments on 2-D and 1-D Fiber-type Optical Phased Array,” Proc. SPIE, 7136, Optical Transmission, Switching, and Subsystems VI, 71363J (2008).
[Crossref]

Huang, H.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Y. Yan, Y. Yue, H. Huang, J. Y. Yang, M. R. Chitgarha, N. Ahmed, M. Tur, S. J. Dolinar, and A. E. Willner, “Efficient generation and multiplexing of optical orbital angular momentum modes in a ring fiber by using multiple coherent inputs,” Opt. Lett. 37(17), 3645–3647 (2012), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-37-17-3645 .
[PubMed]

Keller, B.

Klemens, F.

Koshiba, M.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical wave-front laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

Kristensen, P.

Leach, J.

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(25), 257901 (2002).
[Crossref] [PubMed]

Leuthold, J.

Lopez-Garcia, M.

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

Melchior, H.

O’Brien, J. L.

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

Padgett, M. J.

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(25), 257901 (2002).
[Crossref] [PubMed]

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

Phillips, D. B.

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

Qin, C.

Ramachandran, S.

Ren, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Saitoh, F.

Saitoh, K.

Scott, R. P.

Shamee, B.

Sorel, M.

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

Strain, M. J.

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

Su, T.

Thompson, M. G.

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

Tur, M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Y. Yan, Y. Yue, H. Huang, J. Y. Yang, M. R. Chitgarha, N. Ahmed, M. Tur, S. J. Dolinar, and A. E. Willner, “Efficient generation and multiplexing of optical orbital angular momentum modes in a ring fiber by using multiple coherent inputs,” Opt. Lett. 37(17), 3645–3647 (2012), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-37-17-3645 .
[PubMed]

Wang, J.

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Y. Yan, L. Zhang, J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, A. E. Willner, and S. J. Dolinar, “Fiber structure to convert a Gaussian beam to higher-order optical orbital angular momentum modes,” Opt. Lett. 37(16), 3294–3296 (2012), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-37-16-3294 .
[Crossref] [PubMed]

Y. Yan, J. Wang, L. Zhang, J. Y. Yang, I. M. Fazal, N. Ahmed, B. Shamee, A. E. Willner, K. Birnbaum, and S. Dolinar, “Fiber coupler for generating orbital angular momentum modes,” Opt. Lett. 36(21), 4269–4271 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-21-4269 .
[Crossref] [PubMed]

Willner, A. E.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical wave-front laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

Xu, A.

X. Yang, Y. Yang, Q. Cai, Y. Zhao, A. Fang, W. Hu, and A. Xu, “Primary Experiments on 2-D and 1-D Fiber-type Optical Phased Array,” Proc. SPIE, 7136, Optical Transmission, Switching, and Subsystems VI, 71363J (2008).
[Crossref]

Yan, Y.

Yang, J.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, J. Y.

Yang, X.

X. Yang, Y. Yang, Q. Cai, Y. Zhao, A. Fang, W. Hu, and A. Xu, “Primary Experiments on 2-D and 1-D Fiber-type Optical Phased Array,” Proc. SPIE, 7136, Optical Transmission, Switching, and Subsystems VI, 71363J (2008).
[Crossref]

Yang, Y.

X. Yang, Y. Yang, Q. Cai, Y. Zhao, A. Fang, W. Hu, and A. Xu, “Primary Experiments on 2-D and 1-D Fiber-type Optical Phased Array,” Proc. SPIE, 7136, Optical Transmission, Switching, and Subsystems VI, 71363J (2008).
[Crossref]

Yoo, S. J. B.

Yu, S.

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

Yue, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Y. Yan, Y. Yue, H. Huang, J. Y. Yang, M. R. Chitgarha, N. Ahmed, M. Tur, S. J. Dolinar, and A. E. Willner, “Efficient generation and multiplexing of optical orbital angular momentum modes in a ring fiber by using multiple coherent inputs,” Opt. Lett. 37(17), 3645–3647 (2012), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-37-17-3645 .
[PubMed]

Zhang, L.

Zhao, Y.

X. Yang, Y. Yang, Q. Cai, Y. Zhao, A. Fang, W. Hu, and A. Xu, “Primary Experiments on 2-D and 1-D Fiber-type Optical Phased Array,” Proc. SPIE, 7136, Optical Transmission, Switching, and Subsystems VI, 71363J (2008).
[Crossref]

Zhou, J.

J. Zhou and P. Gallion, “A Novel Mode Multiplexer/De-Multiplexer for Multi-core Fibers,” IEEE Photon. Technol. Lett. 25(13), 1214–1217 (2013).
[Crossref]

J. Zhou and P. Gallion, “Mode De-Multiplexers for Rectangular Multimode Waveguides,” IEEE Photon. Technol. Lett.In press.

Zhu, J.

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

IEEE Photon. Technol. Lett. (1)

J. Zhou and P. Gallion, “A Novel Mode Multiplexer/De-Multiplexer for Multi-core Fibers,” IEEE Photon. Technol. Lett. 25(13), 1214–1217 (2013).
[Crossref]

J. Lightwave Technol. (1)

Nat. Commun. (1)

M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5, 4856 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Opt. Commun. (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical wave-front laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. Lett. (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(25), 257901 (2002).
[Crossref] [PubMed]

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

Proc. SPIE, 7136, Optical Transmission, Switching, and Subsystems (1)

X. Yang, Y. Yang, Q. Cai, Y. Zhao, A. Fang, W. Hu, and A. Xu, “Primary Experiments on 2-D and 1-D Fiber-type Optical Phased Array,” Proc. SPIE, 7136, Optical Transmission, Switching, and Subsystems VI, 71363J (2008).
[Crossref]

Other (3)

B. Guan, R. P. Scott, N. K. Fontaine, T. Su, C. Ferrari, M. Cappuzzo, F. Klemens, B. Keller, M. Earnshaw, and S. J. B. Yoo, “Integrated optical orbital angular momentum multiplexing device using 3-D waveguides and a silica PLC” in CLEO:2013, OSA Technical Digest (online) (Optical Society of America, 2013), CTu1L.4.

N. Bozinovic, P. Kristensen, and S. Ramachandran, “Long-range fiber-transmission of photons with orbital angular momentum” in Proceedings of the Conference on Lasers and Electro-Optics paper CTuB1, CLEO 2011 (Optical Society of America, 2011).
[Crossref]

J. Zhou and P. Gallion, “Mode De-Multiplexers for Rectangular Multimode Waveguides,” IEEE Photon. Technol. Lett.In press.

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Figures (7)

Fig. 1
Fig. 1 (a) Cross section of the device with input/output port labeling (b) the conceptual illustration of the device as a mode MUX/DEMUX.
Fig. 2
Fig. 2 (a, c) the amplitude and phase distribution when the input signal is at the 0th port. (b, d) the amplitude and phase distribution when the input signal is at the 1st port.
Fig. 3
Fig. 3 OAM modes generated inside the ring core fiber (a, c) amplitude and phase of OAM state 0 generated by injecting the signal into the 0th input port of the OAM MUX/DEMUX. (b, d) amplitude and phase of OAM state −1 generated by injecting the signal into the 1st input port of the OAM MUX/DEMUX.
Fig. 4
Fig. 4 OAM modes generated inside the ring core fiber (a, c) amplitude and phase of OAM state −2 generated by injecting the signal into the 2nd input port of the OAM MUX/DEMUX. (b, d) amplitude and phase of OAM state −3 generated by injecting the signal into the 3rd input port of the OAM MUX/DEMUX.
Fig. 5
Fig. 5 OAM modes generated in free space (a, c) amplitude and phase of OAM state 0 generated by injecting the signal into the 0th input port of the OAM MUX/DEMUX. (b, d) amplitude and phase of OAM state −1 generated by injecting the signal into the 1st input port of the OAM MUX/DEMUX.
Fig. 6
Fig. 6 OAM state detection using the device (a-d) OAM states 0, −1, −2, and −3 (m = 0, 1, 2, and 3) are spatially sampled and passed onto the device, and figures show the field patterns at the output of the device.
Fig. 7
Fig. 7 OAM state detection using the device (a-d) OAM states 4, 3, 2, and 1 (m = 4, 5, 6, and 7) are spatially sampled and passed onto the device, and the figures show the field patterns at the output of the device.

Tables (2)

Tables Icon

Table 1, Phase distributions of the duplicated images before and after phase adjustment when the input signal is injected into the 0th input port (m = 0)

Tables Icon

Table 2, Phase distributions of the duplicated images before and after phase adjustment when the input signal is injected into the 1st input port (m = 1)

Equations (22)

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OAM l,p = R p ( r )exp( jlφ )exp( j β l,p z )
OAM l = R 1 ( r )exp( jlφ )exp( j β l z )
R( r )= R 1 ( r )
1 r r ( r R( r ) r ) l 2 r 2 R( r )+( k 2 β l 2 )R( r )=0
R ( r )+ 1 a R ( r )+( k 2 β l 2 l 2 a 2 )R( r )=0
R ( r )+ 1 a R ( r ) R( r ) = k r 2
β l = k 2 ( l 2 a 2 + k r 2 ) k( 1 l 2 a 2 + k r 2 2 k 2 )
E( r,φ,z )= l R( r ) c l exp( jlφ )exp( l 2 z 2k a 2 )
L N = 2k a 2 π N
E( r,φ, L N )=R( r ) l c l exp( jlφ )exp( j l 2 π N )
R( r ) m=0 N1 a m δ( φ 2mπ N )
c l = 1 2π m=0 N1 a m exp( j 2mlπ N )
E( r,φ, L N )=R( r ) 1 2π n exp( jnNφ ) k=0 N1 m=0 N1 a m exp( j 2mkπ N )exp( j k 2 π N )exp( jkφ )
N 2π n exp( jnNφ ) = n=0 N1 δ( φ 2πn N ) φ[ 0,2π ) k=0 N1 exp( j k 2 π N ) = N exp( j π 4 )
E( r,φ, L N )=R( r ) exp( j π 4 ) N n=0 N1 ( m=0 N1 a m exp( j m 2 π N )exp( j 2mnπ N ) exp( j n 2 π N ) )δ( φ 2πn N )
E( r,φ,0 ) =R( r )( f( φ ) m=0 N1 a m δ( φ 2mπ N ) ) =R( r ) m=0 N1 a m f( φ 2mπ N )
E( r,φ, L N )=R( r ) n=0 N1 b n f( φ 2πn N ) b n = exp( j π 4 ) N m=0 N1 a m exp( j m 2 π N ) exp( j 2mnπ N )exp( j n 2 π N )
b=Ta
T mn = exp( j π 4 ) N exp( j m 2 π N )exp( j 2mnπ N )exp( j n 2 π N ) m=0,,N1 n=0,,N1
exp( j n 2 π N )
b= T a T =DT D=( exp( j 0 2 π N ) exp( j ( N2 ) 2 π N ) exp( j ( N1 ) 2 π N ) )
T mn = exp( j π 4 ) N exp( j 2mnπ N )exp( j m 2 π N )

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