Abstract

Vortex beams carrying orbital angular momentum (OAM) have been recently investigated intensely in optical communication systems, as using OAM mode multiplexing simultaneously with other conventional multiplexing techniques is the key to further expand data capacity. This article demonstrates a wavelength- and OAM-tunable vortex laser at 1.6 µm in an Er:YAG system. For the first time to the best of our knowledge, a reflective volume Bragg grating (VBG) was theoretically and experimentally proved to be an effective OAM-preserving wavelength selector inside the laser cavity. A z-shaped laser cavity employing a VBG as a folding mirror was constructed for the direct generation of vortex beams, and we finally obtained wavelength-tunable beams of five OAM states (0, ± ħ, and ± 2ħ) with a narrow bandwidth less than 0.04 nm. This laser supplies a new way for optical communication by combining the spatial degree of freedom for multiplexing information channels with the conventionally used wavelength domains in packable and robust resonant cavity.

© 2017 Optical Society of America

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2018 (1)

J. Liu, M. Wu, B. Huang, P. Tang, C. Zhao, D. Shen, D. Fan, and S. K. Turitsyn, “Widely Wavelength-Tunable Mid-Infrared Fluoride Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0900507 (2018).
[Crossref]

2017 (2)

2016 (1)

2015 (1)

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Twisted light communication through turbulent air across Vienna,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14197–14201 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (4)

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(6140), 1545–1548 (2013).
[Crossref] [PubMed]

E. Cavalli, L. Esposito, J. Hostaša, and M. Pedroni, “Synthesis and optical spectroscopy of transparent YAG ceramics activated with Er3+,” J. Eur. Ceram. Soc. 33(8), 1425–1434 (2013).
[Crossref]

R. Wang, C. Q. Gao, L. N. Zhu, M. W. Gao, Y. Zheng, Q. Ye, and Z. Q. Wu, “Continuous-wave and Q-switched operation of a resonantly pumped U-shaped Er:YAG laser at 1617 and 1645 nm,” Laser Phys. Lett. 10(2), 025802 (2013).
[Crossref]

I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, “STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA,” Nat. Methods 10(9), 910–916 (2013).
[Crossref] [PubMed]

2012 (2)

J. Wang, J. Y. 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. Zhao, Z. Wang, H. Yu, S. Zhang, H. Zhang, X. Xu, J. Xu, X. Xu, and J. Wang, “Direct generation of optical vortex pulses,” Appl. Phys. Lett. 101(3), 031113 (2012).
[Crossref]

2011 (2)

N. C. Shie, W. F. Hsieh, and J. T. Shy, “Single frequency 1070 nm Nd:GdVO4 laser using a volume Bragg grating,” Opt. Express 19(22), 21109–21115 (2011).
[Crossref] [PubMed]

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106(10), 100407 (2011).
[Crossref] [PubMed]

2010 (1)

2008 (1)

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic super dense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

2007 (2)

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

2005 (3)

S. D. Setzler, M. P. Francis, Y. E. Young, J. R. Konves, and E. P. Chicklis, “Resonantly pumped eyesafe erbium lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 645–657 (2005).
[Crossref]

R. C. Stoneman, R. Hartman, A. I. Malm, and P. Gatt, “Coherent laser radar using eyesafe YAG laser transmitters,” Proc. SPIE 5791, 167–174 (2005).
[Crossref]

A. Ishaaya, N. Davidson, and A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13(13), 4952–4962 (2005).
[Crossref] [PubMed]

2004 (1)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165–166 (2004).
[Crossref] [PubMed]

2001 (1)

Y. Chen and Y. Lan, “Dynamics of the Laguerre Gaussian TEM0,1* mode in a solid-state laser,” Phys. Rev. A 63(6), 063807 (2001).
[Crossref]

1999 (2)

J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999).
[Crossref]

G. J. Wagner, T. J. Carrig, R. H. Page, K. I. Schaffers, J. O. Ndap, X. Ma, and A. Burger, “Continuous-wave broadly tunable Cr2+:ZnSe laser,” Opt. Lett. 24(1), 19–21 (1999).
[Crossref] [PubMed]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1982 (1)

Ahmed, N.

J. Wang, J. Y. 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]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Barnett, S. M.

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106(10), 100407 (2011).
[Crossref] [PubMed]

Barreiro, J. T.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic super dense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

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(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Broekmans, O. D.

I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, “STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA,” Nat. Methods 10(9), 910–916 (2013).
[Crossref] [PubMed]

Burger, A.

Carrig, T. J.

Cavalli, E.

E. Cavalli, L. Esposito, J. Hostaša, and M. Pedroni, “Synthesis and optical spectroscopy of transparent YAG ceramics activated with Er3+,” J. Eur. Ceram. Soc. 33(8), 1425–1434 (2013).
[Crossref]

Cesar, J.

Chen, Y.

Y. Chen and Y. Lan, “Dynamics of the Laguerre Gaussian TEM0,1* mode in a solid-state laser,” Phys. Rev. A 63(6), 063807 (2001).
[Crossref]

Chicklis, E. P.

S. D. Setzler, M. P. Francis, Y. E. Young, J. R. Konves, and E. P. Chicklis, “Resonantly pumped eyesafe erbium lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 645–657 (2005).
[Crossref]

Chipouline, A.

Courtial, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165–166 (2004).
[Crossref] [PubMed]

Davidson, N.

Dennis, M. R.

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106(10), 100407 (2011).
[Crossref] [PubMed]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165–166 (2004).
[Crossref] [PubMed]

Dolinar, S.

J. Wang, J. Y. 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]

Esposito, L.

E. Cavalli, L. Esposito, J. Hostaša, and M. Pedroni, “Synthesis and optical spectroscopy of transparent YAG ceramics activated with Er3+,” J. Eur. Ceram. Soc. 33(8), 1425–1434 (2013).
[Crossref]

Fan, D.

J. Liu, M. Wu, B. Huang, P. Tang, C. Zhao, D. Shen, D. Fan, and S. K. Turitsyn, “Widely Wavelength-Tunable Mid-Infrared Fluoride Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0900507 (2018).
[Crossref]

X. Zhang, D. Shen, H. Huang, J. Liu, X. Liu, J. Zhang, J. Zhang, D. Tang, and D. Fan, “Widely tunable, narrow bandwidth polycrystalline ceramic Er:YAG laser with a volume Bragg grating,” Opt. Express 22(6), 7154–7159 (2014).
[Crossref] [PubMed]

Farge, G.

I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, “STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA,” Nat. Methods 10(9), 910–916 (2013).
[Crossref] [PubMed]

Fazal, I. M.

J. Wang, J. Y. 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]

Fickler, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Twisted light communication through turbulent air across Vienna,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14197–14201 (2015).
[Crossref] [PubMed]

Fink, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Twisted light communication through turbulent air across Vienna,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14197–14201 (2015).
[Crossref] [PubMed]

Francis, M. P.

S. D. Setzler, M. P. Francis, Y. E. Young, J. R. Konves, and E. P. Chicklis, “Resonantly pumped eyesafe erbium lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 645–657 (2005).
[Crossref]

Franke-Arnold, S.

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106(10), 100407 (2011).
[Crossref] [PubMed]

Friesem, A.

Fujii, M.

Gao, C. Q.

R. Wang, C. Q. Gao, L. N. Zhu, M. W. Gao, Y. Zheng, Q. Ye, and Z. Q. Wu, “Continuous-wave and Q-switched operation of a resonantly pumped U-shaped Er:YAG laser at 1617 and 1645 nm,” Laser Phys. Lett. 10(2), 025802 (2013).
[Crossref]

Gao, M. W.

R. Wang, C. Q. Gao, L. N. Zhu, M. W. Gao, Y. Zheng, Q. Ye, and Z. Q. Wu, “Continuous-wave and Q-switched operation of a resonantly pumped U-shaped Er:YAG laser at 1617 and 1645 nm,” Laser Phys. Lett. 10(2), 025802 (2013).
[Crossref]

Gatt, P.

R. C. Stoneman, R. Hartman, A. I. Malm, and P. Gatt, “Coherent laser radar using eyesafe YAG laser transmitters,” Proc. SPIE 5791, 167–174 (2005).
[Crossref]

Gurbatov, S. O.

Haidar, M. T.

Handsteiner, J.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Twisted light communication through turbulent air across Vienna,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14197–14201 (2015).
[Crossref] [PubMed]

Hartman, R.

R. C. Stoneman, R. Hartman, A. I. Malm, and P. Gatt, “Coherent laser radar using eyesafe YAG laser transmitters,” Proc. SPIE 5791, 167–174 (2005).
[Crossref]

Hell, S. W.

I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, “STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA,” Nat. Methods 10(9), 910–916 (2013).
[Crossref] [PubMed]

Heller, I.

I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, “STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA,” Nat. Methods 10(9), 910–916 (2013).
[Crossref] [PubMed]

Hostaša, J.

E. Cavalli, L. Esposito, J. Hostaša, and M. Pedroni, “Synthesis and optical spectroscopy of transparent YAG ceramics activated with Er3+,” J. Eur. Ceram. Soc. 33(8), 1425–1434 (2013).
[Crossref]

Hsieh, W. F.

Huang, B.

J. Liu, M. Wu, B. Huang, P. Tang, C. Zhao, D. Shen, D. Fan, and S. K. Turitsyn, “Widely Wavelength-Tunable Mid-Infrared Fluoride Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0900507 (2018).
[Crossref]

Huang, H.

X. Zhang, D. Shen, H. Huang, J. Liu, X. Liu, J. Zhang, J. Zhang, D. Tang, and D. Fan, “Widely tunable, narrow bandwidth polycrystalline ceramic Er:YAG laser with a volume Bragg grating,” Opt. Express 22(6), 7154–7159 (2014).
[Crossref] [PubMed]

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(6140), 1545–1548 (2013).
[Crossref] [PubMed]

J. Wang, J. Y. 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]

Ishaaya, A.

Ishizuki, H.

Ito, A.

Jack, B.

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106(10), 100407 (2011).
[Crossref] [PubMed]

Konves, J. R.

S. D. Setzler, M. P. Francis, Y. E. Young, J. R. Konves, and E. P. Chicklis, “Resonantly pumped eyesafe erbium lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 645–657 (2005).
[Crossref]

Kozawa, Y.

Krenn, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Twisted light communication through turbulent air across Vienna,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14197–14201 (2015).
[Crossref] [PubMed]

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(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Küppers, F.

Kwiat, P. G.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic super dense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

Lan, Y.

Y. Chen and Y. Lan, “Dynamics of the Laguerre Gaussian TEM0,1* mode in a solid-state laser,” Phys. Rev. A 63(6), 063807 (2001).
[Crossref]

Leach, J.

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106(10), 100407 (2011).
[Crossref] [PubMed]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165–166 (2004).
[Crossref] [PubMed]

Liu, J.

J. Liu, M. Wu, B. Huang, P. Tang, C. Zhao, D. Shen, D. Fan, and S. K. Turitsyn, “Widely Wavelength-Tunable Mid-Infrared Fluoride Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0900507 (2018).
[Crossref]

X. Zhang, D. Shen, H. Huang, J. Liu, X. Liu, J. Zhang, J. Zhang, D. Tang, and D. Fan, “Widely tunable, narrow bandwidth polycrystalline ceramic Er:YAG laser with a volume Bragg grating,” Opt. Express 22(6), 7154–7159 (2014).
[Crossref] [PubMed]

Liu, Q.

Q. Liu, Y. Zhao, W. Zhou, J. Zhang, L. Wang, W. Yao, and D. Shen, “Control of Vortex Helicity With a Quarter-Wave Plate in an Er:YAG Ceramic Solid State Laser,” IEEE Photonics J. 9(1), 1500408 (2017).
[Crossref]

Y. Zhao, Q. Liu, W. Zhou, and D. Shen, “~1 mJ pulsed vortex laser at 1645 nm with well-defined helicity,” Opt. Express 24(14), 15596–15602 (2016).
[Crossref] [PubMed]

Liu, X.

Lyubopytov, V. S.

Ma, X.

Malekizandi, M.

Malik, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Twisted light communication through turbulent air across Vienna,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14197–14201 (2015).
[Crossref] [PubMed]

Malm, A. I.

R. C. Stoneman, R. Hartman, A. I. Malm, and P. Gatt, “Coherent laser radar using eyesafe YAG laser transmitters,” Proc. SPIE 5791, 167–174 (2005).
[Crossref]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

Menges, C.

I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, “STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA,” Nat. Methods 10(9), 910–916 (2013).
[Crossref] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Ndap, J. O.

Padgett, M. J.

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106(10), 100407 (2011).
[Crossref] [PubMed]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165–166 (2004).
[Crossref] [PubMed]

Page, R. H.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

Paul, S.

Pedroni, M.

E. Cavalli, L. Esposito, J. Hostaša, and M. Pedroni, “Synthesis and optical spectroscopy of transparent YAG ceramics activated with Er3+,” J. Eur. Ceram. Soc. 33(8), 1425–1434 (2013).
[Crossref]

Peterman, E. J. G.

I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, “STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA,” Nat. Methods 10(9), 910–916 (2013).
[Crossref] [PubMed]

Petrov, D. V.

J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999).
[Crossref]

Porfirev, A. P.

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(6140), 1545–1548 (2013).
[Crossref] [PubMed]

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(6140), 1545–1548 (2013).
[Crossref] [PubMed]

J. Wang, J. Y. 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]

Romero, J.

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106(10), 100407 (2011).
[Crossref] [PubMed]

Saikawa, J.

Sato, S.

Schaffers, K. I.

Scheidl, T.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Twisted light communication through turbulent air across Vienna,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14197–14201 (2015).
[Crossref] [PubMed]

Schumann, M. F.

Setzler, S. D.

S. D. Setzler, M. P. Francis, Y. E. Young, J. R. Konves, and E. P. Chicklis, “Resonantly pumped eyesafe erbium lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 645–657 (2005).
[Crossref]

Shen, D.

J. Liu, M. Wu, B. Huang, P. Tang, C. Zhao, D. Shen, D. Fan, and S. K. Turitsyn, “Widely Wavelength-Tunable Mid-Infrared Fluoride Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0900507 (2018).
[Crossref]

Q. Liu, Y. Zhao, W. Zhou, J. Zhang, L. Wang, W. Yao, and D. Shen, “Control of Vortex Helicity With a Quarter-Wave Plate in an Er:YAG Ceramic Solid State Laser,” IEEE Photonics J. 9(1), 1500408 (2017).
[Crossref]

Y. Zhao, Q. Liu, W. Zhou, and D. Shen, “~1 mJ pulsed vortex laser at 1645 nm with well-defined helicity,” Opt. Express 24(14), 15596–15602 (2016).
[Crossref] [PubMed]

X. Zhang, D. Shen, H. Huang, J. Liu, X. Liu, J. Zhang, J. Zhang, D. Tang, and D. Fan, “Widely tunable, narrow bandwidth polycrystalline ceramic Er:YAG laser with a volume Bragg grating,” Opt. Express 22(6), 7154–7159 (2014).
[Crossref] [PubMed]

Shie, N. C.

Shy, J. T.

Sitters, G.

I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, “STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA,” Nat. Methods 10(9), 910–916 (2013).
[Crossref] [PubMed]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Stoneman, R. C.

R. C. Stoneman, R. Hartman, A. I. Malm, and P. Gatt, “Coherent laser radar using eyesafe YAG laser transmitters,” Proc. SPIE 5791, 167–174 (2005).
[Crossref]

Tabosa, J. W. R.

J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999).
[Crossref]

Taira, T.

Tang, D.

Tang, P.

J. Liu, M. Wu, B. Huang, P. Tang, C. Zhao, D. Shen, D. Fan, and S. K. Turitsyn, “Widely Wavelength-Tunable Mid-Infrared Fluoride Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0900507 (2018).
[Crossref]

Thylén, L.

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

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(6140), 1545–1548 (2013).
[Crossref] [PubMed]

J. Wang, J. Y. 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]

Turitsyn, S. K.

J. Liu, M. Wu, B. Huang, P. Tang, C. Zhao, D. Shen, D. Fan, and S. K. Turitsyn, “Widely Wavelength-Tunable Mid-Infrared Fluoride Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0900507 (2018).
[Crossref]

Ursin, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Twisted light communication through turbulent air across Vienna,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14197–14201 (2015).
[Crossref] [PubMed]

Wagner, G. J.

Wang, J.

Y. Zhao, Z. Wang, H. Yu, S. Zhang, H. Zhang, X. Xu, J. Xu, X. Xu, and J. Wang, “Direct generation of optical vortex pulses,” Appl. Phys. Lett. 101(3), 031113 (2012).
[Crossref]

J. Wang, J. Y. 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]

Wang, L.

Q. Liu, Y. Zhao, W. Zhou, J. Zhang, L. Wang, W. Yao, and D. Shen, “Control of Vortex Helicity With a Quarter-Wave Plate in an Er:YAG Ceramic Solid State Laser,” IEEE Photonics J. 9(1), 1500408 (2017).
[Crossref]

Wang, R.

R. Wang, C. Q. Gao, L. N. Zhu, M. W. Gao, Y. Zheng, Q. Ye, and Z. Q. Wu, “Continuous-wave and Q-switched operation of a resonantly pumped U-shaped Er:YAG laser at 1617 and 1645 nm,” Laser Phys. Lett. 10(2), 025802 (2013).
[Crossref]

Wang, Z.

Y. Zhao, Z. Wang, H. Yu, S. Zhang, H. Zhang, X. Xu, J. Xu, X. Xu, and J. Wang, “Direct generation of optical vortex pulses,” Appl. Phys. Lett. 101(3), 031113 (2012).
[Crossref]

Wegener, M.

Wei, T. C.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic super dense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

Wende, W.

I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, “STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA,” Nat. Methods 10(9), 910–916 (2013).
[Crossref] [PubMed]

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(6140), 1545–1548 (2013).
[Crossref] [PubMed]

J. Wang, J. Y. 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]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Wu, M.

J. Liu, M. Wu, B. Huang, P. Tang, C. Zhao, D. Shen, D. Fan, and S. K. Turitsyn, “Widely Wavelength-Tunable Mid-Infrared Fluoride Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0900507 (2018).
[Crossref]

Wu, Z. Q.

R. Wang, C. Q. Gao, L. N. Zhu, M. W. Gao, Y. Zheng, Q. Ye, and Z. Q. Wu, “Continuous-wave and Q-switched operation of a resonantly pumped U-shaped Er:YAG laser at 1617 and 1645 nm,” Laser Phys. Lett. 10(2), 025802 (2013).
[Crossref]

Wuite, G. J. L.

I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, “STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA,” Nat. Methods 10(9), 910–916 (2013).
[Crossref] [PubMed]

Xu, J.

Y. Zhao, Z. Wang, H. Yu, S. Zhang, H. Zhang, X. Xu, J. Xu, X. Xu, and J. Wang, “Direct generation of optical vortex pulses,” Appl. Phys. Lett. 101(3), 031113 (2012).
[Crossref]

Xu, X.

Y. Zhao, Z. Wang, H. Yu, S. Zhang, H. Zhang, X. Xu, J. Xu, X. Xu, and J. Wang, “Direct generation of optical vortex pulses,” Appl. Phys. Lett. 101(3), 031113 (2012).
[Crossref]

Y. Zhao, Z. Wang, H. Yu, S. Zhang, H. Zhang, X. Xu, J. Xu, X. Xu, and J. Wang, “Direct generation of optical vortex pulses,” Appl. Phys. Lett. 101(3), 031113 (2012).
[Crossref]

Yan, Y.

J. Wang, J. Y. 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.

J. Wang, J. Y. 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]

Yao, W.

Q. Liu, Y. Zhao, W. Zhou, J. Zhang, L. Wang, W. Yao, and D. Shen, “Control of Vortex Helicity With a Quarter-Wave Plate in an Er:YAG Ceramic Solid State Laser,” IEEE Photonics J. 9(1), 1500408 (2017).
[Crossref]

Ye, Q.

R. Wang, C. Q. Gao, L. N. Zhu, M. W. Gao, Y. Zheng, Q. Ye, and Z. Q. Wu, “Continuous-wave and Q-switched operation of a resonantly pumped U-shaped Er:YAG laser at 1617 and 1645 nm,” Laser Phys. Lett. 10(2), 025802 (2013).
[Crossref]

Yevick, D.

Young, Y. E.

S. D. Setzler, M. P. Francis, Y. E. Young, J. R. Konves, and E. P. Chicklis, “Resonantly pumped eyesafe erbium lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 645–657 (2005).
[Crossref]

Yu, H.

Y. Zhao, Z. Wang, H. Yu, S. Zhang, H. Zhang, X. Xu, J. Xu, X. Xu, and J. Wang, “Direct generation of optical vortex pulses,” Appl. Phys. Lett. 101(3), 031113 (2012).
[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(6140), 1545–1548 (2013).
[Crossref] [PubMed]

J. Wang, J. Y. 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]

Zeilinger, A.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Twisted light communication through turbulent air across Vienna,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14197–14201 (2015).
[Crossref] [PubMed]

Zhang, H.

Y. Zhao, Z. Wang, H. Yu, S. Zhang, H. Zhang, X. Xu, J. Xu, X. Xu, and J. Wang, “Direct generation of optical vortex pulses,” Appl. Phys. Lett. 101(3), 031113 (2012).
[Crossref]

Zhang, J.

Zhang, S.

Y. Zhao, Z. Wang, H. Yu, S. Zhang, H. Zhang, X. Xu, J. Xu, X. Xu, and J. Wang, “Direct generation of optical vortex pulses,” Appl. Phys. Lett. 101(3), 031113 (2012).
[Crossref]

Zhang, X.

Zhao, C.

J. Liu, M. Wu, B. Huang, P. Tang, C. Zhao, D. Shen, D. Fan, and S. K. Turitsyn, “Widely Wavelength-Tunable Mid-Infrared Fluoride Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0900507 (2018).
[Crossref]

Zhao, Y.

Q. Liu, Y. Zhao, W. Zhou, J. Zhang, L. Wang, W. Yao, and D. Shen, “Control of Vortex Helicity With a Quarter-Wave Plate in an Er:YAG Ceramic Solid State Laser,” IEEE Photonics J. 9(1), 1500408 (2017).
[Crossref]

Y. Zhao, Q. Liu, W. Zhou, and D. Shen, “~1 mJ pulsed vortex laser at 1645 nm with well-defined helicity,” Opt. Express 24(14), 15596–15602 (2016).
[Crossref] [PubMed]

Y. Zhao, Z. Wang, H. Yu, S. Zhang, H. Zhang, X. Xu, J. Xu, X. Xu, and J. Wang, “Direct generation of optical vortex pulses,” Appl. Phys. Lett. 101(3), 031113 (2012).
[Crossref]

Zheng, Y.

R. Wang, C. Q. Gao, L. N. Zhu, M. W. Gao, Y. Zheng, Q. Ye, and Z. Q. Wu, “Continuous-wave and Q-switched operation of a resonantly pumped U-shaped Er:YAG laser at 1617 and 1645 nm,” Laser Phys. Lett. 10(2), 025802 (2013).
[Crossref]

Zhou, W.

Q. Liu, Y. Zhao, W. Zhou, J. Zhang, L. Wang, W. Yao, and D. Shen, “Control of Vortex Helicity With a Quarter-Wave Plate in an Er:YAG Ceramic Solid State Laser,” IEEE Photonics J. 9(1), 1500408 (2017).
[Crossref]

Y. Zhao, Q. Liu, W. Zhou, and D. Shen, “~1 mJ pulsed vortex laser at 1645 nm with well-defined helicity,” Opt. Express 24(14), 15596–15602 (2016).
[Crossref] [PubMed]

Zhu, L. N.

R. Wang, C. Q. Gao, L. N. Zhu, M. W. Gao, Y. Zheng, Q. Ye, and Z. Q. Wu, “Continuous-wave and Q-switched operation of a resonantly pumped U-shaped Er:YAG laser at 1617 and 1645 nm,” Laser Phys. Lett. 10(2), 025802 (2013).
[Crossref]

Appl. Phys. Lett. (1)

Y. Zhao, Z. Wang, H. Yu, S. Zhang, H. Zhang, X. Xu, J. Xu, X. Xu, and J. Wang, “Direct generation of optical vortex pulses,” Appl. Phys. Lett. 101(3), 031113 (2012).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

J. Liu, M. Wu, B. Huang, P. Tang, C. Zhao, D. Shen, D. Fan, and S. K. Turitsyn, “Widely Wavelength-Tunable Mid-Infrared Fluoride Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0900507 (2018).
[Crossref]

S. D. Setzler, M. P. Francis, Y. E. Young, J. R. Konves, and E. P. Chicklis, “Resonantly pumped eyesafe erbium lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 645–657 (2005).
[Crossref]

IEEE Photonics J. (1)

Q. Liu, Y. Zhao, W. Zhou, J. Zhang, L. Wang, W. Yao, and D. Shen, “Control of Vortex Helicity With a Quarter-Wave Plate in an Er:YAG Ceramic Solid State Laser,” IEEE Photonics J. 9(1), 1500408 (2017).
[Crossref]

J. Eur. Ceram. Soc. (1)

E. Cavalli, L. Esposito, J. Hostaša, and M. Pedroni, “Synthesis and optical spectroscopy of transparent YAG ceramics activated with Er3+,” J. Eur. Ceram. Soc. 33(8), 1425–1434 (2013).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Laser Phys. Lett. (1)

R. Wang, C. Q. Gao, L. N. Zhu, M. W. Gao, Y. Zheng, Q. Ye, and Z. Q. Wu, “Continuous-wave and Q-switched operation of a resonantly pumped U-shaped Er:YAG laser at 1617 and 1645 nm,” Laser Phys. Lett. 10(2), 025802 (2013).
[Crossref]

Nat. Methods (1)

I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, “STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA,” Nat. Methods 10(9), 910–916 (2013).
[Crossref] [PubMed]

Nat. Photonics (1)

J. Wang, J. Y. 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]

Nat. Phys. (2)

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic super dense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Nature (1)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165–166 (2004).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (2)

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Y. Chen and Y. Lan, “Dynamics of the Laguerre Gaussian TEM0,1* mode in a solid-state laser,” Phys. Rev. A 63(6), 063807 (2001).
[Crossref]

Phys. Rev. Lett. (3)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999).
[Crossref]

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Entangled optical vortex links,” Phys. Rev. Lett. 106(10), 100407 (2011).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Twisted light communication through turbulent air across Vienna,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14197–14201 (2015).
[Crossref] [PubMed]

Proc. SPIE (1)

R. C. Stoneman, R. Hartman, A. I. Malm, and P. Gatt, “Coherent laser radar using eyesafe YAG laser transmitters,” Proc. SPIE 5791, 167–174 (2005).
[Crossref]

Science (1)

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(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Other (2)

A. Jain, “Creation of Optical Vortices Using an Adjustable Spiral Phase Plate and Computer-Generated Holograms,” Physics Individual Project Siemens-Westinghouse Competition October, 1–21(2005).

O. M. Efimov, L. Glebov, and V. Smirnov, “High efficiency volume diffractive elements in photo-thermo-refractive glass,” United States Patent 6673497, 2004.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup of tunable Er:YAG crystal vortex laser.

Fig. 2
Fig. 2

(a) The output intensity distributions of LG0, ± 1 modes and corresponding interference fringes; (b) the wavelength tuning ranges; measured spectra of vortex with (c) OAM of + ħ and (d) OAM of –ħ.

Fig. 3
Fig. 3

The scattered points are the output power of LG02 mode as a function of wavelength under the maximum incident pumping level of 21.2 W. The inset are the corresponding intensity distribution and interference fringes.

Fig. 4
Fig. 4

Output powers of beams with five OAM states at emission peaks.

Equations (5)

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2 E+ ( ω c ) 2 n 2 ( x ¯ )E=0,
E(r,φ,Δz)=exp[iΔz ( 2 + ( ω c ) 2 (n+δn) 2 ) 1 2 ]E(r,φ,0),
( 2 + ( ω c ) 2 (n+δn) 2 ) 1 2 ~ 2 ( 2 + k 2 ) 1/2 +k +k+ ω c δn,
E L (r,φ,z)=ε(r,z)exp(ilφ)exp(ikz)=ε'(r,φ,z)exp(ikz),
ε'(r,φ,Δz)=exp(iΔzk δn n )exp{ iΔz[ 2 ( 2 + k 2 ) 1/2 +k ] }ε'(r,φ,0)+0(Δ z 3 ) =exp(iΔzk δn n )exp{ iΔz[ 2 ( 2 + k 2 ) 1/2 +k ] }ε(r,0)exp(ilφ)+0(Δ z 3 ).

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