Abstract

We propose a chiral Fabry-Perot cavity in which only the cavity modes in almost pure spin (circular polarization) states lase in the presence of gain. In absence of imposed nonreciprocal environments and time-reversal symmetry breaking of emitter states to favor the emission of circularly-polarized photons, only the resonance of modes with a specific spin orientation remains in the cavity. We demonstrate a prototype of the cavity using distributed Bragg reflectors and cholesteric liquid crystals. This reciprocal cavity may provide a method to control the angular momentum state of emitters based on stimulated emissions.

© 2012 OSA

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    [CrossRef]
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  3. 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, 8185–8189 (1992).
    [CrossRef] [PubMed]
  4. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  5. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
    [CrossRef]
  6. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Optical helices and spiral interference fringes,” Opt. Lett. 22, 52–54 (1997).
    [CrossRef] [PubMed]
  7. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef] [PubMed]
  8. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
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  11. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
    [CrossRef] [PubMed]
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  13. H. Fujino, S. Koh, S. Iba, T. Fujimoto, and H. Kawaguchi, “Circularly polarized lasing in a (110)-oriented quantum well vertical-cavity surface-emitting laser under optical spin injection,” Appl. Phys. Lett. 94, 131108 (2009).
    [CrossRef]
  14. M. Holub, J. Shin, S. Chakrabarti, and P. Bhattacharya, “Electrically injected spin-polarized vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 091108 (2005).
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  16. M. Holub and P. Bhattacharya, “Spin-polarized light-emitting diodes and lasers,” J. Phys. D: Appl. Phys. 40, R179–R203 (2007).
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    [CrossRef]
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  20. H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics 4, 676–685 (2010).
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  21. V. I. Kopp and A. Z. Genack, “Twist defect in chiral photonic structures,” Phys. Rev. Lett. 89, 033901 (2002).
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  23. J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
    [CrossRef] [PubMed]
  24. V. I. Kopp, Z. Q. Zhang, and A. Z. Genack, “Lasing in chiral photonic structures,” Prog. Quantum Electron. 27, 369–416 (2003).
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    [CrossRef] [PubMed]
  26. Y. Matsuhisa, R. Ozaki, M. Ozaki, and K. Yoshino, “Single-mode lasing in one-dimensional periodic structure containing helical structure as a defect,” Jpn. J. Appl. Phys. Part 2 44, L629–L632 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
  28. B. Park, M. Kim, S. W. Kim, and I. T. Kim, “Circularly polarized unidirectional lasing from a cholesteric liquid crystal layer on a 1-D photonic crystal substrate,” Opt. Express 17, 12323–12331 (2009).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  33. A. L. Shelankov and G. E. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
    [CrossRef]
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  35. P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011).
    [CrossRef]
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    [CrossRef]
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2011 (4)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[CrossRef]

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef] [PubMed]

S. W. Chang, “Full frequency-domain approach to reciprocal microlasers and nanolasers-perspective from Lorentz reciprocity,” Opt. Express 19, 21116–21134 (2011).
[CrossRef] [PubMed]

P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011).
[CrossRef]

2010 (1)

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics 4, 676–685 (2010).
[CrossRef]

2009 (2)

B. Park, M. Kim, S. W. Kim, and I. T. Kim, “Circularly polarized unidirectional lasing from a cholesteric liquid crystal layer on a 1-D photonic crystal substrate,” Opt. Express 17, 12323–12331 (2009).
[CrossRef] [PubMed]

H. Fujino, S. Koh, S. Iba, T. Fujimoto, and H. Kawaguchi, “Circularly polarized lasing in a (110)-oriented quantum well vertical-cavity surface-emitting laser under optical spin injection,” Appl. Phys. Lett. 94, 131108 (2009).
[CrossRef]

2008 (1)

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[CrossRef]

2007 (3)

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, “Electrical spin injection and threshold reduction in a semiconductor laser,” Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

M. Holub and P. Bhattacharya, “Spin-polarized light-emitting diodes and lasers,” J. Phys. D: Appl. Phys. 40, R179–R203 (2007).
[CrossRef]

Y. Matsuhisa, Y. Huang, Y. Zhou, S. T. Wu, Y. Takao, A. Fujii, and M. Ozaki, “Cholesteric liquid crystal laser in a dielectric mirror cavity upon band-edge excitation,” Opt. Express 15, 616–622 (2007).
[CrossRef] [PubMed]

2006 (1)

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

2005 (3)

M. Holub, J. Shin, S. Chakrabarti, and P. Bhattacharya, “Electrically injected spin-polarized vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 091108 (2005).
[CrossRef]

Y. Matsuhisa, R. Ozaki, M. Ozaki, and K. Yoshino, “Single-mode lasing in one-dimensional periodic structure containing helical structure as a defect,” Jpn. J. Appl. Phys. Part 2 44, L629–L632 (2005).
[CrossRef]

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

2003 (2)

V. I. Kopp, Z. Q. Zhang, and A. Z. Genack, “Lasing in chiral photonic structures,” Prog. Quantum Electron. 27, 369–416 (2003).
[CrossRef]

J. Schmidtke, W. Stille, and H. Finkelmann, “Defect mode emission of a dye doped cholesteric polymer network,” Phys. Rev. Lett. 90, 083902 (2003).
[CrossRef] [PubMed]

2002 (2)

V. I. Kopp and A. Z. Genack, “Twist defect in chiral photonic structures,” Phys. Rev. Lett. 89, 033901 (2002).
[CrossRef] [PubMed]

J. Schmidtke, W. Stille, H. Finkelmann, and S. T. Kim, “Laser emission in a dye doped cholesteric polymer network,” Adv. Mater. 14, 746 (2002).
[CrossRef]

2001 (1)

B. Taheri, A. F. Munoz, P. Palffy-Muhoray, and R. Twieg, “Low threshold lasing in cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 358, 73–82 (2001).
[CrossRef]

1998 (2)

V. I. Kopp, B. Fan, H. K. M. Vithana, and A. Z. Genack, “Low-threshold lasing at the edge of a photonic stop band in cholesteric liquid crystals,” Opt. Lett. 23, 1707–1709 (1998).
[CrossRef]

H. Ando, T. Sogawa, and H. Gotoh, “Photon-spin controlled lasing oscillation in surface-emitting lasers,” Appl. Phys. Lett. 73, 566–568 (1998).
[CrossRef]

1997 (1)

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

1992 (4)

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, 8185–8189 (1992).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

S. M. Barnett, B. Huttner, and R. Loudon, “Spontaneous emission in absorbing dielectric media,” Phys. Rev. Lett. 68, 3698–3701 (1992), (The dyadic Green’s function in that work has a sign difference from that used here).
[CrossRef] [PubMed]

A. L. Shelankov and G. E. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
[CrossRef]

1973 (2)

1951 (1)

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta. Cryst. 4, 219–226 (1951).
[CrossRef]

1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[CrossRef]

1896 (1)

H. A. Lorentz, “The theorem of poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Amsterdammer Akademie der Wetenschappen 4, 176 (1896).

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Optical helices and spiral interference fringes,” Opt. Lett. 22, 52–54 (1997).
[CrossRef] [PubMed]

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, 8185–8189 (1992).
[CrossRef] [PubMed]

Ando, H.

H. Ando, T. Sogawa, and H. Gotoh, “Photon-spin controlled lasing oscillation in surface-emitting lasers,” Appl. Phys. Lett. 73, 566–568 (1998).
[CrossRef]

Arakawa, Y.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef] [PubMed]

Barnett, S. M.

S. M. Barnett, B. Huttner, and R. Loudon, “Spontaneous emission in absorbing dielectric media,” Phys. Rev. Lett. 68, 3698–3701 (1992), (The dyadic Green’s function in that work has a sign difference from that used here).
[CrossRef] [PubMed]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

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, 8185–8189 (1992).
[CrossRef] [PubMed]

Berreman, D. W.

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[CrossRef]

Bhattacharya, P.

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, “Electrical spin injection and threshold reduction in a semiconductor laser,” Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

M. Holub and P. Bhattacharya, “Spin-polarized light-emitting diodes and lasers,” J. Phys. D: Appl. Phys. 40, R179–R203 (2007).
[CrossRef]

M. Holub, J. Shin, S. Chakrabarti, and P. Bhattacharya, “Electrically injected spin-polarized vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 091108 (2005).
[CrossRef]

Chakrabarti, S.

M. Holub, J. Shin, S. Chakrabarti, and P. Bhattacharya, “Electrically injected spin-polarized vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 091108 (2005).
[CrossRef]

Chang, S. W.

Chuang, S. L.

S. L. Chuang, Physics of Optoelectronic Devices (Wiley and Sons, 1995), 1st ed.

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Coles, H.

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics 4, 676–685 (2010).
[CrossRef]

Coles, H. J.

P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011).
[CrossRef]

de Vries, H.

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta. Cryst. 4, 219–226 (1951).
[CrossRef]

Dholakia, K.

Dobson, C. A.

P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011).
[CrossRef]

Elachi, C.

Fan, B.

Finkelmann, H.

J. Schmidtke, W. Stille, and H. Finkelmann, “Defect mode emission of a dye doped cholesteric polymer network,” Phys. Rev. Lett. 90, 083902 (2003).
[CrossRef] [PubMed]

J. Schmidtke, W. Stille, H. Finkelmann, and S. T. Kim, “Laser emission in a dye doped cholesteric polymer network,” Adv. Mater. 14, 746 (2002).
[CrossRef]

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[CrossRef]

Friedberg, S. H.

S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra (Prentice Hall, 1989), 2nd ed.

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Fujii, A.

Fujimoto, T.

H. Fujino, S. Koh, S. Iba, T. Fujimoto, and H. Kawaguchi, “Circularly polarized lasing in a (110)-oriented quantum well vertical-cavity surface-emitting laser under optical spin injection,” Appl. Phys. Lett. 94, 131108 (2009).
[CrossRef]

Fujino, H.

H. Fujino, S. Koh, S. Iba, T. Fujimoto, and H. Kawaguchi, “Circularly polarized lasing in a (110)-oriented quantum well vertical-cavity surface-emitting laser under optical spin injection,” Appl. Phys. Lett. 94, 131108 (2009).
[CrossRef]

Gardiner, D. J.

P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011).
[CrossRef]

Genack, A. Z.

V. I. Kopp, Z. Q. Zhang, and A. Z. Genack, “Lasing in chiral photonic structures,” Prog. Quantum Electron. 27, 369–416 (2003).
[CrossRef]

V. I. Kopp and A. Z. Genack, “Twist defect in chiral photonic structures,” Phys. Rev. Lett. 89, 033901 (2002).
[CrossRef] [PubMed]

V. I. Kopp, B. Fan, H. K. M. Vithana, and A. Z. Genack, “Low-threshold lasing at the edge of a photonic stop band in cholesteric liquid crystals,” Opt. Lett. 23, 1707–1709 (1998).
[CrossRef]

Gotoh, H.

H. Ando, T. Sogawa, and H. Gotoh, “Photon-spin controlled lasing oscillation in surface-emitting lasers,” Appl. Phys. Lett. 73, 566–568 (1998).
[CrossRef]

Hands, P. J. W.

P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

Holub, M.

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, “Electrical spin injection and threshold reduction in a semiconductor laser,” Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

M. Holub and P. Bhattacharya, “Spin-polarized light-emitting diodes and lasers,” J. Phys. D: Appl. Phys. 40, R179–R203 (2007).
[CrossRef]

M. Holub, J. Shin, S. Chakrabarti, and P. Bhattacharya, “Electrically injected spin-polarized vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 091108 (2005).
[CrossRef]

Huang, Y.

Huttner, B.

S. M. Barnett, B. Huttner, and R. Loudon, “Spontaneous emission in absorbing dielectric media,” Phys. Rev. Lett. 68, 3698–3701 (1992), (The dyadic Green’s function in that work has a sign difference from that used here).
[CrossRef] [PubMed]

Hwang, J.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Iba, S.

H. Fujino, S. Koh, S. Iba, T. Fujimoto, and H. Kawaguchi, “Circularly polarized lasing in a (110)-oriented quantum well vertical-cavity surface-emitting laser under optical spin injection,” Appl. Phys. Lett. 94, 131108 (2009).
[CrossRef]

Insel, A. J.

S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra (Prentice Hall, 1989), 2nd ed.

Ishikawa, K.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Iwamoto, S.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef] [PubMed]

Kawaguchi, H.

H. Fujino, S. Koh, S. Iba, T. Fujimoto, and H. Kawaguchi, “Circularly polarized lasing in a (110)-oriented quantum well vertical-cavity surface-emitting laser under optical spin injection,” Appl. Phys. Lett. 94, 131108 (2009).
[CrossRef]

Kim, I. T.

Kim, M.

Kim, S. T.

J. Schmidtke, W. Stille, H. Finkelmann, and S. T. Kim, “Laser emission in a dye doped cholesteric polymer network,” Adv. Mater. 14, 746 (2002).
[CrossRef]

Kim, S. W.

Koh, S.

H. Fujino, S. Koh, S. Iba, T. Fujimoto, and H. Kawaguchi, “Circularly polarized lasing in a (110)-oriented quantum well vertical-cavity surface-emitting laser under optical spin injection,” Appl. Phys. Lett. 94, 131108 (2009).
[CrossRef]

Kong, J. A.

J. A. KongElectromagnetic Wave Theory (EMW Publishing, 2008), last ed.

Konishi, K.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef] [PubMed]

Kopp, V. I.

V. I. Kopp, Z. Q. Zhang, and A. Z. Genack, “Lasing in chiral photonic structures,” Prog. Quantum Electron. 27, 369–416 (2003).
[CrossRef]

V. I. Kopp and A. Z. Genack, “Twist defect in chiral photonic structures,” Phys. Rev. Lett. 89, 033901 (2002).
[CrossRef] [PubMed]

V. I. Kopp, B. Fan, H. K. M. Vithana, and A. Z. Genack, “Low-threshold lasing at the edge of a photonic stop band in cholesteric liquid crystals,” Opt. Lett. 23, 1707–1709 (1998).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Kumagai, N.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef] [PubMed]

Kuwata-Gonokami, M.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef] [PubMed]

Lorentz, H. A.

H. A. Lorentz, “The theorem of poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Amsterdammer Akademie der Wetenschappen 4, 176 (1896).

Loudon, R.

S. M. Barnett, B. Huttner, and R. Loudon, “Spontaneous emission in absorbing dielectric media,” Phys. Rev. Lett. 68, 3698–3701 (1992), (The dyadic Green’s function in that work has a sign difference from that used here).
[CrossRef] [PubMed]

Manzo, C.

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

Matsuhisa, Y.

Y. Matsuhisa, Y. Huang, Y. Zhou, S. T. Wu, Y. Takao, A. Fujii, and M. Ozaki, “Cholesteric liquid crystal laser in a dielectric mirror cavity upon band-edge excitation,” Opt. Express 15, 616–622 (2007).
[CrossRef] [PubMed]

Y. Matsuhisa, R. Ozaki, M. Ozaki, and K. Yoshino, “Single-mode lasing in one-dimensional periodic structure containing helical structure as a defect,” Jpn. J. Appl. Phys. Part 2 44, L629–L632 (2005).
[CrossRef]

McDuff, R.

Morris, S.

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics 4, 676–685 (2010).
[CrossRef]

Morris, S. M.

P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011).
[CrossRef]

Munoz, A. F.

B. Taheri, A. F. Munoz, P. Palffy-Muhoray, and R. Twieg, “Low threshold lasing in cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 358, 73–82 (2001).
[CrossRef]

Nishimura, S.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Nomura, M.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef] [PubMed]

Ozaki, M.

Y. Matsuhisa, Y. Huang, Y. Zhou, S. T. Wu, Y. Takao, A. Fujii, and M. Ozaki, “Cholesteric liquid crystal laser in a dielectric mirror cavity upon band-edge excitation,” Opt. Express 15, 616–622 (2007).
[CrossRef] [PubMed]

Y. Matsuhisa, R. Ozaki, M. Ozaki, and K. Yoshino, “Single-mode lasing in one-dimensional periodic structure containing helical structure as a defect,” Jpn. J. Appl. Phys. Part 2 44, L629–L632 (2005).
[CrossRef]

Ozaki, R.

Y. Matsuhisa, R. Ozaki, M. Ozaki, and K. Yoshino, “Single-mode lasing in one-dimensional periodic structure containing helical structure as a defect,” Jpn. J. Appl. Phys. Part 2 44, L629–L632 (2005).
[CrossRef]

Padgett, M.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[CrossRef]

Padgett, M. J.

Palffy-Muhoray, P.

B. Taheri, A. F. Munoz, P. Palffy-Muhoray, and R. Twieg, “Low threshold lasing in cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 358, 73–82 (2001).
[CrossRef]

Paparo, D.

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

Park, B.

B. Park, M. Kim, S. W. Kim, and I. T. Kim, “Circularly polarized unidirectional lasing from a cholesteric liquid crystal layer on a 1-D photonic crystal substrate,” Opt. Express 17, 12323–12331 (2009).
[CrossRef] [PubMed]

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Pikus, G. E.

A. L. Shelankov and G. E. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
[CrossRef]

Qasim, M. M.

P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Saha, D.

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, “Electrical spin injection and threshold reduction in a semiconductor laser,” Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

Schmidtke, J.

J. Schmidtke, W. Stille, and H. Finkelmann, “Defect mode emission of a dye doped cholesteric polymer network,” Phys. Rev. Lett. 90, 083902 (2003).
[CrossRef] [PubMed]

J. Schmidtke, W. Stille, H. Finkelmann, and S. T. Kim, “Laser emission in a dye doped cholesteric polymer network,” Adv. Mater. 14, 746 (2002).
[CrossRef]

Shelankov, A. L.

A. L. Shelankov and G. E. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
[CrossRef]

Shin, J.

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, “Electrical spin injection and threshold reduction in a semiconductor laser,” Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

M. Holub, J. Shin, S. Chakrabarti, and P. Bhattacharya, “Electrically injected spin-polarized vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 091108 (2005).
[CrossRef]

Simpson, N. B.

Smith, C. P.

Sogawa, T.

H. Ando, T. Sogawa, and H. Gotoh, “Photon-spin controlled lasing oscillation in surface-emitting lasers,” Appl. Phys. Lett. 73, 566–568 (1998).
[CrossRef]

Song, M. H.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Spence, L. E.

S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra (Prentice Hall, 1989), 2nd ed.

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, 8185–8189 (1992).
[CrossRef] [PubMed]

Stille, W.

J. Schmidtke, W. Stille, and H. Finkelmann, “Defect mode emission of a dye doped cholesteric polymer network,” Phys. Rev. Lett. 90, 083902 (2003).
[CrossRef] [PubMed]

J. Schmidtke, W. Stille, H. Finkelmann, and S. T. Kim, “Laser emission in a dye doped cholesteric polymer network,” Adv. Mater. 14, 746 (2002).
[CrossRef]

Taheri, B.

B. Taheri, A. F. Munoz, P. Palffy-Muhoray, and R. Twieg, “Low threshold lasing in cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 358, 73–82 (2001).
[CrossRef]

Takanishi, Y.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Takao, Y.

Takezoe, H.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Toyooka, T.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Twieg, R.

B. Taheri, A. F. Munoz, P. Palffy-Muhoray, and R. Twieg, “Low threshold lasing in cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 358, 73–82 (2001).
[CrossRef]

Vithana, H. K. M.

White, A. G.

Wilkinson, T. D.

P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011).
[CrossRef]

Woerdman, J.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[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, 8185–8189 (1992).
[CrossRef] [PubMed]

Wu, J. W.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Wu, S. T.

Yao, A. M.

Yeh, C.

Yoshino, K.

Y. Matsuhisa, R. Ozaki, M. Ozaki, and K. Yoshino, “Single-mode lasing in one-dimensional periodic structure containing helical structure as a defect,” Jpn. J. Appl. Phys. Part 2 44, L629–L632 (2005).
[CrossRef]

Zhang, Z. Q.

V. I. Kopp, Z. Q. Zhang, and A. Z. Genack, “Lasing in chiral photonic structures,” Prog. Quantum Electron. 27, 369–416 (2003).
[CrossRef]

Zhou, Y.

Acta. Cryst. (1)

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta. Cryst. 4, 219–226 (1951).
[CrossRef]

Adv. Mater. (1)

J. Schmidtke, W. Stille, H. Finkelmann, and S. T. Kim, “Laser emission in a dye doped cholesteric polymer network,” Adv. Mater. 14, 746 (2002).
[CrossRef]

Adv. Opt. Photon. (1)

Amsterdammer Akademie der Wetenschappen (1)

H. A. Lorentz, “The theorem of poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Amsterdammer Akademie der Wetenschappen 4, 176 (1896).

Appl. Phys. Lett. (3)

H. Ando, T. Sogawa, and H. Gotoh, “Photon-spin controlled lasing oscillation in surface-emitting lasers,” Appl. Phys. Lett. 73, 566–568 (1998).
[CrossRef]

H. Fujino, S. Koh, S. Iba, T. Fujimoto, and H. Kawaguchi, “Circularly polarized lasing in a (110)-oriented quantum well vertical-cavity surface-emitting laser under optical spin injection,” Appl. Phys. Lett. 94, 131108 (2009).
[CrossRef]

M. Holub, J. Shin, S. Chakrabarti, and P. Bhattacharya, “Electrically injected spin-polarized vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 091108 (2005).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. D: Appl. Phys. (1)

M. Holub and P. Bhattacharya, “Spin-polarized light-emitting diodes and lasers,” J. Phys. D: Appl. Phys. 40, R179–R203 (2007).
[CrossRef]

Jpn. J. Appl. Phys. Part 2 (1)

Y. Matsuhisa, R. Ozaki, M. Ozaki, and K. Yoshino, “Single-mode lasing in one-dimensional periodic structure containing helical structure as a defect,” Jpn. J. Appl. Phys. Part 2 44, L629–L632 (2005).
[CrossRef]

Laser Photon. Rev. (1)

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[CrossRef]

Mol. Cryst. Liq. Cryst. (1)

B. Taheri, A. F. Munoz, P. Palffy-Muhoray, and R. Twieg, “Low threshold lasing in cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 358, 73–82 (2001).
[CrossRef]

Nat. Mater. (1)

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Nat. Photonics (1)

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics 4, 676–685 (2010).
[CrossRef]

Opt. Commun. (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[CrossRef]

Phys. Rev. A (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, 8185–8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (1)

A. L. Shelankov and G. E. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
[CrossRef]

Phys. Rev. Lett. (7)

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef] [PubMed]

S. M. Barnett, B. Huttner, and R. Loudon, “Spontaneous emission in absorbing dielectric media,” Phys. Rev. Lett. 68, 3698–3701 (1992), (The dyadic Green’s function in that work has a sign difference from that used here).
[CrossRef] [PubMed]

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

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, “Electrical spin injection and threshold reduction in a semiconductor laser,” Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

V. I. Kopp and A. Z. Genack, “Twist defect in chiral photonic structures,” Phys. Rev. Lett. 89, 033901 (2002).
[CrossRef] [PubMed]

J. Schmidtke, W. Stille, and H. Finkelmann, “Defect mode emission of a dye doped cholesteric polymer network,” Phys. Rev. Lett. 90, 083902 (2003).
[CrossRef] [PubMed]

Proc. SPIE (1)

P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011).
[CrossRef]

Prog. Quantum Electron. (1)

V. I. Kopp, Z. Q. Zhang, and A. Z. Genack, “Lasing in chiral photonic structures,” Prog. Quantum Electron. 27, 369–416 (2003).
[CrossRef]

Other (3)

S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra (Prentice Hall, 1989), 2nd ed.

J. A. KongElectromagnetic Wave Theory (EMW Publishing, 2008), last ed.

S. L. Chuang, Physics of Optoelectronic Devices (Wiley and Sons, 1995), 1st ed.

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Figures (6)

Fig. 1
Fig. 1

The schematic diagram of the 1D FP cavity. Reflector 1 is isotropic while reflector 2 is chiral with specific characteristics of reflections.

Fig. 2
Fig. 2

The polarization states (a) v1,2 (case I), and (b) u1,2 (case II) when κ ≠ 0. The phase angle of κ is set to arg ( a b ). When κ = 0, the polarization states v1,2 become ê while u1,2 turn into ê+.

Fig. 3
Fig. 3

(a) Reflector 1 is a λ/4 DBR at the left side of the gain medium. (b) Reflector 2 is the cascade of a λ/4 DBR and CLC at the right side of the gain medium.

Fig. 4
Fig. 4

(a) The reflectivity of reflector 1 versus the wavelength. (b) The square magnitudes of matrix elements in r̿2 versus the wavelength. The square magnitudes |r2,CP,+,+|2 and |r2,CP,−,−|2 are equal (upper graph), while |r2,CP,+,−|2 is much larger than |r2,CP,−,+|2 near 850 nm (lower graph).

Fig. 5
Fig. 5

(a)–(d) Spectra of the Stokes parameters 〈Sn(z,ω)〉 (n = 0 – 3). The magnitudes of 〈S1(z,ω)〉 and 〈S2(z,ω)〉 are much smaller than those of 〈S0(z,ω)〉 and 〈S3(z,ω)〉.

Fig. 6
Fig. 6

The energy levels of double two-level systems forming a time-reversal pair. The dipole moment μvc between states c (|1/2,1/2〉) and v (|3/2,3/2〉) is −μê and that μṽc̃ between (|1/2,–1/2〉) and (|3/2,–3/2〉) is μê+. Due to the dominance of photons with a spin quantum number ms,ph = 1, only the STEs from to take place.

Equations (24)

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

Ω d r J s , 1 ( r , ω ) E 2 ( r , ω ) = Ω d r J s , 2 ( r , ω ) E 1 ( r , ω ) ,
R sp , c v R sp , c ˜ v ˜ = 2 μ 0 ω c v 2 h ¯ Im [ μ v c * G ¯ ¯ ee ( r s , r s , ω c v ) μ v c ] 2 μ 0 ω c ˜ v ˜ 2 h ¯ Im [ μ v ˜ c ˜ * G ¯ ¯ ee ( r s , r s , ω c ˜ v ˜ ) μ v ˜ c ˜ ] = 2 μ 0 ω c v 2 h ¯ Im [ μ v c * { G ¯ ¯ ee ( r s , r s , ω c v ) G ¯ ¯ ee T ( r s , r s , ω c ˜ v ˜ ) } μ v c ] = 0 ,
R st , c v R st , c ˜ v ˜ = ε 0 2 h ¯ d r Im [ E v c ( r ) Δ ε ¯ ¯ r , a * ( r , ω c v ) E v c * ( r ) E v ˜ c ˜ ( r ) Δ ε ¯ ¯ r , a * ( r , ω c ˜ v ˜ ) E v ˜ c ˜ * ( r ) ] = 2 ε 0 μ 0 2 ω c v 4 h ¯ d r st , c v , c ˜ v ˜ ( r , r s ) ,
st , c v , c ˜ v ˜ ( r , r s ) = Im [ μ v c * { G ¯ ¯ ee ( r , r s , ω c v ) Δ ε ¯ ¯ r , a ( r , ω c v ) G ¯ ¯ ee ( r , r s , ω c v ) G ¯ ¯ ee T ( r , r s , ω c v ) Δ ε ¯ ¯ r , a * ( r , ω c v ) G ¯ ¯ ee * ( r , r s , ω c v ) } μ v c ] ,
E v c ( v ˜ c ˜ ) ( r ) = 2 μ 0 ω c v ( c ˜ v ˜ ) 2 G ¯ ¯ ee ( r , r s , ω c v ( c ˜ v ˜ ) ) μ v c ( v ˜ c ˜ ) .
st , c v , c ˜ v ˜ ( r , r s ) = 2 Δ ε r , a , I ( r , ω c v ) tr { Im [ G ¯ ¯ ee ( r s , r , ω c v ) G ¯ ¯ ee * ( r , r s , ω c v ) ] Im [ μ v c μ v c ] } ,
r ¯ ¯ 2 = ( r 2 , x x , r 2 , x y r 2 , y x , r 2 , y y ) = ( a , c c , b ) ,
r ¯ ¯ 2 , CP ( r 2 , CP , + + , r 2 , CP , + r 2 , CP , + , r 2 , CP , ) = p ¯ ¯ r ¯ ¯ 2 p ¯ ¯ = ( a + b 2 , a b 2 i c a b 2 + i c , a + b 2 ) ,
p ¯ ¯ = ( 1 / 2 , 1 / 2 i / 2 , i / 2 ) ,
Λ = Λ 1 , 2 = ( a + b ) 2 ± κ ( a b ) κ 2 ( 1 , 2 for  + , ) .
w = v CP 1 , 2 = ( v CP , + 1 , 2 v CP , 1 , 2 ) = 1 D ( a , b , κ ) ( ± κ ( a b ) κ 2 ) ( case I ) ,
w = u CP 1 , 2 = ( u CP , + 1 , 2 u CP , 1 , 2 ) = 1 D ( a , b , κ ) ( ( a b ) κ 2 ± κ ) ( case II ) ,
J s , α * ( z , ω ) J s , α ( z , ω ) = δ α , α δ ( z z ) c , v D c v ( z , ω ) | j sp , v c ( ω ) | 2 2 ,
E ( z , ω ) = z 1 z 2 d z G ¯ ¯ ee ( z , z , ω ) i ω μ 0 J s ( z , ω ) ,
G ¯ ¯ ee ( z , z , ω ) = U ( z z ) G ¯ ¯ ee < ( z , z , ω ) + U ( z z ) G ¯ ¯ ee > ( z , z , ω ) ,
G ¯ ¯ ee < ( z , z , ω ) = η e i k h 2 i ω μ 0 [ e i k ( z z 1 ) r ¯ ¯ 1 + e i k ( z z 1 ) I ¯ ¯ 2 ] [ r ¯ ¯ 2 r ¯ ¯ 1 e 2 i k h I ¯ ¯ 2 ] 1 × [ e i k ( z z 2 ) r ¯ ¯ 2 + e i k ( z z 2 ) I ¯ ¯ 2 ] ,
G ¯ ¯ ee > ( z , z , ω ) = η e i k h 2 i ω μ 0 [ e i k ( z z 2 ) r ¯ ¯ 2 + e i k ( z z 2 ) I ¯ ¯ 2 ] [ r ¯ ¯ 1 r ¯ ¯ 2 e 2 i k h I ¯ ¯ 2 ] 1 × [ e i k ( z z 1 ) r ¯ ¯ 1 + e i k ( z z 1 ) I ¯ ¯ 2 ] ,
S n ( z , ω ) = E * ( z , ω ) M ¯ ¯ n E ( z , ω ) ,
𝒤 n ( z , ω ) z 1 z 2 d z tr [ G ¯ ¯ ee ( z , z , ω ) G ¯ ¯ ee ( z , z , ω ) M ¯ ¯ n ] .
st , c v , c ˜ v ˜ ( z , z ) Re [ tr { G ¯ ¯ ee ( z , z , ω ) G ¯ ¯ ee ( z , z , ω ) M ¯ ¯ 3 } ] | ω = ω c v .
G ¯ ¯ ee ( z , z , ω ) v n = g n ( z , z , ω ) Γ Λ n e 2 i k h 1 v n ( n = 1 , 2 ) ,
g n ( z , z , ω ) = { η e i k h 2 i ω μ 0 [ e i k ( z z 1 ) Γ + e i k ( z z 1 ) ] [ e i k ( z z 2 ) Λ n + e i k ( z z 2 ) ] z ( z 1 , z ) , η e i k h 2 i ω μ 0 [ e i k ( z z 1 ) Γ + e i k ( z z 1 ) ] [ e i k ( z z 2 ) Λ n + e i k ( z z 2 ) ] z ( z , z 2 ) ,
ε ¯ ¯ r ( ω ) = ε r , bgd I ¯ ¯ 3 + N e h ¯ ε 0 1 ω c v ω i γ c v [ μ v c μ v c ( f v f c ) + μ v ˜ c ˜ μ v ˜ c ˜ ( f v ˜ f c ˜ ) ] ,
ε ¯ ¯ r ( ω ) = ε r , bgd I ¯ ¯ 3 + N e h ¯ ε 0 μ 2 ω c v ω i γ c v { [ ( f v f c ) + ( f v ˜ f c ˜ ) ] 2 ( I ¯ ¯ 2 , 0 2 × 1 0 1 × 2 , 0 ) + [ ( f v f c ) ( f v ˜ f c ˜ ) ] 2 ( σ ¯ ¯ 2 , 0 2 × 1 0 1 × 2 , 0 ) } ,

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