Abstract

A tunable circularly polarized distributed feedback (DFB) laser based on reflection grating configuration was realized in a solution of 4-(dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4H-pyran (DCM) doped methanol. Chiral photonic DFB structures with tunable photonic bandgap were generated by controlling the state of polarization of the two interfering pump beams. The dual-peak lasing emission spectrum indicates that the periodic chiral DFB grating is affected by a combination of modulations of gain and refractive index.

© 2008 Optical Society of America

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  1. H. Kogelnik and C. V. Shank, "Simulated emission in a periodic structure," Appl. Phys. Lett. 18, 152-154 (1971).
    [CrossRef]
  2. D. Gindre, A. Vesperini, J.-M. Nunzi, H. Leblond and K. D. Dorkenoo, "Refractive-index saturation-mediated multiple line emission in polymer thin-film distributed feedback lasers," Opt. Lett. 31, 1657-1659 (2006).
    [CrossRef] [PubMed]
  3. J. Wang, C. Ye, F. Chen, L. Shi and D. Lo, "Wavelength tunable two-photon-pumped distributed feedback zirconia waveguide lasers," J. Opt. A: Pure Appl. Opt. 7, 261-264 (2005).
    [CrossRef]
  4. F. Chen, J. Wang, C. Ye, W. H. Ni, J. Chan, Y. Yang and D. Lo, "Near infrared distributed feedback lasers based on LDS dye-doped zirconia organically modified silicate channel waveguides," Opt. Express 13, 1643-1650 (2005).
    [CrossRef] [PubMed]
  5. Y. J. Liu, X. W. Sun, P. Shum, H. P. Li, J. Mi, W. Ji and X. H. Zhang, "Low-threshold and narrow-linewidth lasing from dye-doped holographic polymer-dispersed liquid crystal transmission gratings," Appl. Phys. Lett. 88, 061107 (2006).
    [CrossRef]
  6. D. E. Lucchetta, L. Criante, O. Francescangeli and F. Simoni, "Light amplification by dye-doped holographic polymer dispersed liquid crystals," Appl. Phys. Lett. 84, 4893-4895 (2004).
    [CrossRef]
  7. .S.-T. Wu and A. Y.-G. Fuh, "Lasing in photonic crystal based on dye-doped holographic polymer-dispersed liquid crystal reflection gratings," Conference Lasers and Electro-Optics CLEO, 3, 2218-2220 (2005)
  8. F. Chen, D. Gindre and J.-M. Nunzi, "First-order distributed feedback dye laser effect in reflection pumping geometry," Opt. Lett. 32, 805-807 (2007).
    [CrossRef] [PubMed]
  9. D. Lo, C. Ye, and J. Wang, "Distributed feedback laser action by polarization modulation," Appl. Phys. B 76, 649-653 (2003).
    [CrossRef]
  10. C. Ye, J. Wang, L. Shi and D. Lo, "Polarization and threshold energy variation of distributed feedback lasing of oxazine dye in zirconia waveguides and in solutions," Appl. Phys. B 78, 189 (2004).
    [CrossRef]
  11. V. I. Kopp, Z. Q. Zhang and A. Z. Genack, "Lasing in chiral photonic structures," Quantum. Electron. 27, 369-416 (2003).
    [CrossRef]
  12. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  13. V. I. Kopp, B. Fan, H. K. M. Vithana and A. Z. Genack, "Low-threshold lasing at a edge of a photonic stop band in cholesteric liquid crystals," Opt. Lett. 23, 1707-1709 (1998).
    [CrossRef]
  14. Y. Matsuhisa, Y. H. Huang, Y. Zhou, S.-T. Wu, R. Ozaki, Y. Takao, A. Fujii and M. Ozaki, "Low-threshold and high efficiency lasing upon band-edge excitation in a cholesteric liquid crystal," Appl. Phys. Lett. 90, 091114 (2007).
    [CrossRef]
  15. H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327-2335 (1972).
    [CrossRef]
  16. T. Z. Huang and K. H. Wagner, "Coupled Mode Analysis of Polarization Volume Hologram," IEEE J. Quantum Electron. 31, 372-390 (1995).
    [CrossRef]
  17. T. Z. Huang and K. H. Wagner, "Diffraction analysis of photoanisotropic holography: an anisotropic saturation model," J. Opt. Soc. Am. B 13, 282-299 (1996).
    [CrossRef]

2007 (2)

Y. Matsuhisa, Y. H. Huang, Y. Zhou, S.-T. Wu, R. Ozaki, Y. Takao, A. Fujii and M. Ozaki, "Low-threshold and high efficiency lasing upon band-edge excitation in a cholesteric liquid crystal," Appl. Phys. Lett. 90, 091114 (2007).
[CrossRef]

F. Chen, D. Gindre and J.-M. Nunzi, "First-order distributed feedback dye laser effect in reflection pumping geometry," Opt. Lett. 32, 805-807 (2007).
[CrossRef] [PubMed]

2006 (2)

D. Gindre, A. Vesperini, J.-M. Nunzi, H. Leblond and K. D. Dorkenoo, "Refractive-index saturation-mediated multiple line emission in polymer thin-film distributed feedback lasers," Opt. Lett. 31, 1657-1659 (2006).
[CrossRef] [PubMed]

Y. J. Liu, X. W. Sun, P. Shum, H. P. Li, J. Mi, W. Ji and X. H. Zhang, "Low-threshold and narrow-linewidth lasing from dye-doped holographic polymer-dispersed liquid crystal transmission gratings," Appl. Phys. Lett. 88, 061107 (2006).
[CrossRef]

2005 (2)

F. Chen, J. Wang, C. Ye, W. H. Ni, J. Chan, Y. Yang and D. Lo, "Near infrared distributed feedback lasers based on LDS dye-doped zirconia organically modified silicate channel waveguides," Opt. Express 13, 1643-1650 (2005).
[CrossRef] [PubMed]

J. Wang, C. Ye, F. Chen, L. Shi and D. Lo, "Wavelength tunable two-photon-pumped distributed feedback zirconia waveguide lasers," J. Opt. A: Pure Appl. Opt. 7, 261-264 (2005).
[CrossRef]

2004 (2)

C. Ye, J. Wang, L. Shi and D. Lo, "Polarization and threshold energy variation of distributed feedback lasing of oxazine dye in zirconia waveguides and in solutions," Appl. Phys. B 78, 189 (2004).
[CrossRef]

D. E. Lucchetta, L. Criante, O. Francescangeli and F. Simoni, "Light amplification by dye-doped holographic polymer dispersed liquid crystals," Appl. Phys. Lett. 84, 4893-4895 (2004).
[CrossRef]

2003 (2)

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

D. Lo, C. Ye, and J. Wang, "Distributed feedback laser action by polarization modulation," Appl. Phys. B 76, 649-653 (2003).
[CrossRef]

1998 (1)

1996 (1)

1995 (1)

T. Z. Huang and K. H. Wagner, "Coupled Mode Analysis of Polarization Volume Hologram," IEEE J. Quantum Electron. 31, 372-390 (1995).
[CrossRef]

1987 (1)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

1972 (1)

H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

1971 (1)

H. Kogelnik and C. V. Shank, "Simulated emission in a periodic structure," Appl. Phys. Lett. 18, 152-154 (1971).
[CrossRef]

Chan, J.

Chen, F.

Criante, L.

D. E. Lucchetta, L. Criante, O. Francescangeli and F. Simoni, "Light amplification by dye-doped holographic polymer dispersed liquid crystals," Appl. Phys. Lett. 84, 4893-4895 (2004).
[CrossRef]

Dorkenoo, K. D.

Fan, B.

Francescangeli, O.

D. E. Lucchetta, L. Criante, O. Francescangeli and F. Simoni, "Light amplification by dye-doped holographic polymer dispersed liquid crystals," Appl. Phys. Lett. 84, 4893-4895 (2004).
[CrossRef]

Fujii, A.

Y. Matsuhisa, Y. H. Huang, Y. Zhou, S.-T. Wu, R. Ozaki, Y. Takao, A. Fujii and M. Ozaki, "Low-threshold and high efficiency lasing upon band-edge excitation in a cholesteric liquid crystal," Appl. Phys. Lett. 90, 091114 (2007).
[CrossRef]

Genack, A. Z.

Gindre, D.

Huang, T. Z.

T. Z. Huang and K. H. Wagner, "Diffraction analysis of photoanisotropic holography: an anisotropic saturation model," J. Opt. Soc. Am. B 13, 282-299 (1996).
[CrossRef]

T. Z. Huang and K. H. Wagner, "Coupled Mode Analysis of Polarization Volume Hologram," IEEE J. Quantum Electron. 31, 372-390 (1995).
[CrossRef]

Huang, Y. H.

Y. Matsuhisa, Y. H. Huang, Y. Zhou, S.-T. Wu, R. Ozaki, Y. Takao, A. Fujii and M. Ozaki, "Low-threshold and high efficiency lasing upon band-edge excitation in a cholesteric liquid crystal," Appl. Phys. Lett. 90, 091114 (2007).
[CrossRef]

Ji, W.

Y. J. Liu, X. W. Sun, P. Shum, H. P. Li, J. Mi, W. Ji and X. H. Zhang, "Low-threshold and narrow-linewidth lasing from dye-doped holographic polymer-dispersed liquid crystal transmission gratings," Appl. Phys. Lett. 88, 061107 (2006).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

H. Kogelnik and C. V. Shank, "Simulated emission in a periodic structure," Appl. Phys. Lett. 18, 152-154 (1971).
[CrossRef]

Kopp, V. I.

Leblond, H.

Li, H. P.

Y. J. Liu, X. W. Sun, P. Shum, H. P. Li, J. Mi, W. Ji and X. H. Zhang, "Low-threshold and narrow-linewidth lasing from dye-doped holographic polymer-dispersed liquid crystal transmission gratings," Appl. Phys. Lett. 88, 061107 (2006).
[CrossRef]

Liu, Y. J.

Y. J. Liu, X. W. Sun, P. Shum, H. P. Li, J. Mi, W. Ji and X. H. Zhang, "Low-threshold and narrow-linewidth lasing from dye-doped holographic polymer-dispersed liquid crystal transmission gratings," Appl. Phys. Lett. 88, 061107 (2006).
[CrossRef]

Lo, D.

J. Wang, C. Ye, F. Chen, L. Shi and D. Lo, "Wavelength tunable two-photon-pumped distributed feedback zirconia waveguide lasers," J. Opt. A: Pure Appl. Opt. 7, 261-264 (2005).
[CrossRef]

F. Chen, J. Wang, C. Ye, W. H. Ni, J. Chan, Y. Yang and D. Lo, "Near infrared distributed feedback lasers based on LDS dye-doped zirconia organically modified silicate channel waveguides," Opt. Express 13, 1643-1650 (2005).
[CrossRef] [PubMed]

C. Ye, J. Wang, L. Shi and D. Lo, "Polarization and threshold energy variation of distributed feedback lasing of oxazine dye in zirconia waveguides and in solutions," Appl. Phys. B 78, 189 (2004).
[CrossRef]

D. Lo, C. Ye, and J. Wang, "Distributed feedback laser action by polarization modulation," Appl. Phys. B 76, 649-653 (2003).
[CrossRef]

Lucchetta, D. E.

D. E. Lucchetta, L. Criante, O. Francescangeli and F. Simoni, "Light amplification by dye-doped holographic polymer dispersed liquid crystals," Appl. Phys. Lett. 84, 4893-4895 (2004).
[CrossRef]

Matsuhisa, Y.

Y. Matsuhisa, Y. H. Huang, Y. Zhou, S.-T. Wu, R. Ozaki, Y. Takao, A. Fujii and M. Ozaki, "Low-threshold and high efficiency lasing upon band-edge excitation in a cholesteric liquid crystal," Appl. Phys. Lett. 90, 091114 (2007).
[CrossRef]

Mi, J.

Y. J. Liu, X. W. Sun, P. Shum, H. P. Li, J. Mi, W. Ji and X. H. Zhang, "Low-threshold and narrow-linewidth lasing from dye-doped holographic polymer-dispersed liquid crystal transmission gratings," Appl. Phys. Lett. 88, 061107 (2006).
[CrossRef]

Ni, W. H.

Nunzi, J.-M.

Ozaki, M.

Y. Matsuhisa, Y. H. Huang, Y. Zhou, S.-T. Wu, R. Ozaki, Y. Takao, A. Fujii and M. Ozaki, "Low-threshold and high efficiency lasing upon band-edge excitation in a cholesteric liquid crystal," Appl. Phys. Lett. 90, 091114 (2007).
[CrossRef]

Ozaki, R.

Y. Matsuhisa, Y. H. Huang, Y. Zhou, S.-T. Wu, R. Ozaki, Y. Takao, A. Fujii and M. Ozaki, "Low-threshold and high efficiency lasing upon band-edge excitation in a cholesteric liquid crystal," Appl. Phys. Lett. 90, 091114 (2007).
[CrossRef]

Shank, C. V.

H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

H. Kogelnik and C. V. Shank, "Simulated emission in a periodic structure," Appl. Phys. Lett. 18, 152-154 (1971).
[CrossRef]

Shi, L.

J. Wang, C. Ye, F. Chen, L. Shi and D. Lo, "Wavelength tunable two-photon-pumped distributed feedback zirconia waveguide lasers," J. Opt. A: Pure Appl. Opt. 7, 261-264 (2005).
[CrossRef]

C. Ye, J. Wang, L. Shi and D. Lo, "Polarization and threshold energy variation of distributed feedback lasing of oxazine dye in zirconia waveguides and in solutions," Appl. Phys. B 78, 189 (2004).
[CrossRef]

Shum, P.

Y. J. Liu, X. W. Sun, P. Shum, H. P. Li, J. Mi, W. Ji and X. H. Zhang, "Low-threshold and narrow-linewidth lasing from dye-doped holographic polymer-dispersed liquid crystal transmission gratings," Appl. Phys. Lett. 88, 061107 (2006).
[CrossRef]

Simoni, F.

D. E. Lucchetta, L. Criante, O. Francescangeli and F. Simoni, "Light amplification by dye-doped holographic polymer dispersed liquid crystals," Appl. Phys. Lett. 84, 4893-4895 (2004).
[CrossRef]

Sun, X. W.

Y. J. Liu, X. W. Sun, P. Shum, H. P. Li, J. Mi, W. Ji and X. H. Zhang, "Low-threshold and narrow-linewidth lasing from dye-doped holographic polymer-dispersed liquid crystal transmission gratings," Appl. Phys. Lett. 88, 061107 (2006).
[CrossRef]

Takao, Y.

Y. Matsuhisa, Y. H. Huang, Y. Zhou, S.-T. Wu, R. Ozaki, Y. Takao, A. Fujii and M. Ozaki, "Low-threshold and high efficiency lasing upon band-edge excitation in a cholesteric liquid crystal," Appl. Phys. Lett. 90, 091114 (2007).
[CrossRef]

Vesperini, A.

Vithana, H. K. M.

Wagner, K. H.

T. Z. Huang and K. H. Wagner, "Diffraction analysis of photoanisotropic holography: an anisotropic saturation model," J. Opt. Soc. Am. B 13, 282-299 (1996).
[CrossRef]

T. Z. Huang and K. H. Wagner, "Coupled Mode Analysis of Polarization Volume Hologram," IEEE J. Quantum Electron. 31, 372-390 (1995).
[CrossRef]

Wang, J.

J. Wang, C. Ye, F. Chen, L. Shi and D. Lo, "Wavelength tunable two-photon-pumped distributed feedback zirconia waveguide lasers," J. Opt. A: Pure Appl. Opt. 7, 261-264 (2005).
[CrossRef]

F. Chen, J. Wang, C. Ye, W. H. Ni, J. Chan, Y. Yang and D. Lo, "Near infrared distributed feedback lasers based on LDS dye-doped zirconia organically modified silicate channel waveguides," Opt. Express 13, 1643-1650 (2005).
[CrossRef] [PubMed]

C. Ye, J. Wang, L. Shi and D. Lo, "Polarization and threshold energy variation of distributed feedback lasing of oxazine dye in zirconia waveguides and in solutions," Appl. Phys. B 78, 189 (2004).
[CrossRef]

D. Lo, C. Ye, and J. Wang, "Distributed feedback laser action by polarization modulation," Appl. Phys. B 76, 649-653 (2003).
[CrossRef]

Wu, S.-T.

Y. Matsuhisa, Y. H. Huang, Y. Zhou, S.-T. Wu, R. Ozaki, Y. Takao, A. Fujii and M. Ozaki, "Low-threshold and high efficiency lasing upon band-edge excitation in a cholesteric liquid crystal," Appl. Phys. Lett. 90, 091114 (2007).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yang, Y.

Ye, C.

F. Chen, J. Wang, C. Ye, W. H. Ni, J. Chan, Y. Yang and D. Lo, "Near infrared distributed feedback lasers based on LDS dye-doped zirconia organically modified silicate channel waveguides," Opt. Express 13, 1643-1650 (2005).
[CrossRef] [PubMed]

J. Wang, C. Ye, F. Chen, L. Shi and D. Lo, "Wavelength tunable two-photon-pumped distributed feedback zirconia waveguide lasers," J. Opt. A: Pure Appl. Opt. 7, 261-264 (2005).
[CrossRef]

C. Ye, J. Wang, L. Shi and D. Lo, "Polarization and threshold energy variation of distributed feedback lasing of oxazine dye in zirconia waveguides and in solutions," Appl. Phys. B 78, 189 (2004).
[CrossRef]

D. Lo, C. Ye, and J. Wang, "Distributed feedback laser action by polarization modulation," Appl. Phys. B 76, 649-653 (2003).
[CrossRef]

Zhang, X. H.

Y. J. Liu, X. W. Sun, P. Shum, H. P. Li, J. Mi, W. Ji and X. H. Zhang, "Low-threshold and narrow-linewidth lasing from dye-doped holographic polymer-dispersed liquid crystal transmission gratings," Appl. Phys. Lett. 88, 061107 (2006).
[CrossRef]

Zhang, Z. Q.

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

Zhou, Y.

Y. Matsuhisa, Y. H. Huang, Y. Zhou, S.-T. Wu, R. Ozaki, Y. Takao, A. Fujii and M. Ozaki, "Low-threshold and high efficiency lasing upon band-edge excitation in a cholesteric liquid crystal," Appl. Phys. Lett. 90, 091114 (2007).
[CrossRef]

Appl. Phys. B (2)

D. Lo, C. Ye, and J. Wang, "Distributed feedback laser action by polarization modulation," Appl. Phys. B 76, 649-653 (2003).
[CrossRef]

C. Ye, J. Wang, L. Shi and D. Lo, "Polarization and threshold energy variation of distributed feedback lasing of oxazine dye in zirconia waveguides and in solutions," Appl. Phys. B 78, 189 (2004).
[CrossRef]

Appl. Phys. Lett. (4)

Y. Matsuhisa, Y. H. Huang, Y. Zhou, S.-T. Wu, R. Ozaki, Y. Takao, A. Fujii and M. Ozaki, "Low-threshold and high efficiency lasing upon band-edge excitation in a cholesteric liquid crystal," Appl. Phys. Lett. 90, 091114 (2007).
[CrossRef]

H. Kogelnik and C. V. Shank, "Simulated emission in a periodic structure," Appl. Phys. Lett. 18, 152-154 (1971).
[CrossRef]

Y. J. Liu, X. W. Sun, P. Shum, H. P. Li, J. Mi, W. Ji and X. H. Zhang, "Low-threshold and narrow-linewidth lasing from dye-doped holographic polymer-dispersed liquid crystal transmission gratings," Appl. Phys. Lett. 88, 061107 (2006).
[CrossRef]

D. E. Lucchetta, L. Criante, O. Francescangeli and F. Simoni, "Light amplification by dye-doped holographic polymer dispersed liquid crystals," Appl. Phys. Lett. 84, 4893-4895 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Z. Huang and K. H. Wagner, "Coupled Mode Analysis of Polarization Volume Hologram," IEEE J. Quantum Electron. 31, 372-390 (1995).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

J. Wang, C. Ye, F. Chen, L. Shi and D. Lo, "Wavelength tunable two-photon-pumped distributed feedback zirconia waveguide lasers," J. Opt. A: Pure Appl. Opt. 7, 261-264 (2005).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Quantum. Electron. (1)

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

Other (1)

.S.-T. Wu and A. Y.-G. Fuh, "Lasing in photonic crystal based on dye-doped holographic polymer-dispersed liquid crystal reflection gratings," Conference Lasers and Electro-Optics CLEO, 3, 2218-2220 (2005)

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Figures (6)

Fig. 1.
Fig. 1.

Experimental set-up for reflection grating excitation. Inset shows a zoom of the cell.

Fig. 2.
Fig. 2.

Output wavelengths as a function of the internal intersection angle θin in LCP:LCP pumping geometry. The solid curve is the prediction based on the first order Bragg condition. Right inset shows the corresponding tuning spectra. The left inset shows the emission intensity versus the pump energy.

Fig. 3.
Fig. 3.

Polar plots of the DFB lasing output intensity vs. the azimuthal angle of the polarizer without inserting the quarter-wave plate (a); with inserting the quarter-wave plate, measured from the left side of the dye cell (b) and the right side of the dye cell (c).

Fig. 4.
Fig. 4.

Wavelength tuning as a function of the internal intersection angle θin in RCP:RCP pumping geometry. The solid curve is the theoretical fit based on the first-order Bragg condition. Right inset shows the corresponding tuning spectra. The left inset shows the emission intensity versus the pump energy.

Fig. 5.
Fig. 5.

Polar plots of the DFB lasing output intensity vs. the azimuthal angle of the polarizer without inserting the quarter-wave plate (a); with inserting the quarter-wave plate, measured from the left side of the dye cell (b) and the right side of the dye cell (c).

Fig. 6.
Fig. 6.

(a). Scheme of the reflection pumping geometry in the right-handed Cartesian coordinate system; (b). One period of the polarized interference field pattern is projected onto the x-y plane for the two types of reflection pumping schemes (the upper row is for LCP:LCP pumping and the lower one for RCP:RCP pumping).

Equations (4)

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

Λ = λ pump 2 n × sin ( θ in 2 ) ,
λ L = λ pump M × sin ( θ in 2 )
E 1 = 1 2 [ x ̂ i E + y ̂ E sin ( θ in 2 ) + z ̂ E cos ( θ in 2 ) ] e i k 1 · r E 2 = 1 2 [ x ̂ i E y ̂ E sin ( θ in 2 ) + z ̂ E cos ( θ in 2 ) ] e i k 2 · r E LCP : LCP = 1 2 E [ x ̂ i ( 1 + e ) + y ̂ ( e 1 ) sin ( θ in 2 ) + z ̂ ( e + 1 ) cos ( θ in 2 ) ] }
E 1 = 1 2 [ x ̂ i E + y ̂ E sin ( θ in 2 ) + z ̂ E cos ( θ in 2 ) ] e i k 1 · r E 2 = 1 2 [ x ̂ i E y ̂ E sin ( θ in 2 ) + z ̂ E cos ( θ in 2 ) ] e i k 2 · r E RCP : RCP = 1 2 E [ x ̂ i ( 1 + e ) + y ̂ ( e 1 ) sin ( θ in 2 ) + z ̂ ( e + 1 ) cos ( θ in 2 ) ] }

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