Abstract

For the first time, we experimentally and theoretically research about the second-order nonlinear signal (SNS) including electromagnetically induced absorbing (EIA) and electromagnetically induced gain (EIG), six wave mixing band gap signal (SWM BGS) resulting from photonic band gap structure in an inverted Y-type four level system with the electromagnetically induced grating. The interplay between the SNS and SWM BGS is illustrated clearly for the first time. When we change the frequency detuning to make the SWM BGS and SNS overlap, the SWM BGS is suppressed and the intensity of SNS is strongest near the resonance point. We can control the intensity of the SWM BGS and EIG caused by the classic effect through changing the power of coupling field. And the changes on the EIA generated by the quantum effect are obtained by changing the power of dressing field. Since the SWM BGS is the enhancement of the four wave mixing band gap signal (FWM BGS), when we set FWM BGS as the input and SNS as the modulation role to control the amplification amplitude for the FWM BGS in our scheme, the adjustable optical amplifier can be obtained.

© 2015 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  13. J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
    [Crossref] [PubMed]
  14. C. Li, H. Zheng, Z. Zhang, X. Yao, Y. Zhang, Y. Zhang, and Y. Zhang, “Electromagnetically induced transparency and fluorescence in blockaded Rydberg atomic system,” J. Chem. Phys. 139(16), 164316 (2013).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  25. M. Gao, Z. Wang, Z. Ullah, H. Chen, D. Zhang, Y. Zhang, and Y. Zhang, “Modulated photonic band gaps generated by high-order wave mixing,” J. Opt. Soc. Am. B 32(1), 179–187 (2015).
    [Crossref]

2015 (1)

2013 (3)

Y. P. Zhang, C. Z. Yuan, Y. Q. Zhang, H. B. Zheng, H. X. Chen, C. B. Li, Z. G. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett. 10(5), 055406 (2013).
[Crossref]

H. Zheng, X. Zhang, C. Li, H. Lan, J. Che, Y. Zhang, and Y. Zhang, “Suppression and enhancement of coexisting super-fluorescence and multi-wave mixing processes in sodium vapor,” J. Chem. Phys. 138(20), 204315 (2013).
[Crossref] [PubMed]

C. Li, H. Zheng, Z. Zhang, X. Yao, Y. Zhang, Y. Zhang, and Y. Zhang, “Electromagnetically induced transparency and fluorescence in blockaded Rydberg atomic system,” J. Chem. Phys. 139(16), 164316 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (4)

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[Crossref] [PubMed]

A. Schilke, C. Zimmermann, P. W. Courteille, and W. Guerin, “Photonic Band Gaps in One-Dimensionally Ordered Cold Atomic Vapors,” Phys. Rev. Lett. 106(22), 223903 (2011).
[Crossref] [PubMed]

L. Yuan, A. A. Lanin, P. K. Jha, A. J. Traverso, A. M. Zheltikov, and M. O. Scully, “Coherent Raman umklappscattering,” Laser Phys. Lett. 8(10), 736–741 (2011).
[Crossref]

H. Zhou, D. Wang, D. Wang, J. Zhang, and S. Zhu, “Efficient reflection via four-wave mixing in a Doppler-free electromagnetically-induced-transparency gas system,” Phys. Rev. A 84(5), 053835 (2011).
[Crossref]

2010 (1)

2009 (1)

C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009).
[Crossref]

2006 (3)

J. Qi and A. M. Lyyra, “Electromagnetically induced transparency and dark fluorescence in a cascade three-level di-atomic lithium system,” Phys. Rev. A 73(4), 043810 (2006).
[Crossref]

Y. V. Rostovtsev, Z. E. Sariyanni, and M. O. Scully, “Electromagnetically induced coherent backscattering,” Phys. Rev. Lett. 97(11), 113001 (2006).
[Crossref] [PubMed]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006).
[Crossref] [PubMed]

2004 (1)

D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

2003 (1)

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426(6967), 638–641 (2003).
[Crossref] [PubMed]

2002 (1)

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
[Crossref] [PubMed]

1999 (2)

J. B. Qi, G. Lazarov, X. J. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: Prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83(2), 288–291 (1999).
[Crossref]

M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999).
[Crossref]

1998 (3)

S. Wielandy and A. Gaeta, “Investigation of electromagnetically induced transparency in the strong probe regime,” Phys. Rev. A 58(3), 2500–2505 (1998).
[Crossref]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quaside generate levels in Rb vapor,” Phys. Rev. A 57(4), 2996–3002 (1998).
[Crossref]

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
[Crossref]

1997 (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

1996 (1)

Akulshin, A. M.

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quaside generate levels in Rb vapor,” Phys. Rev. A 57(4), 2996–3002 (1998).
[Crossref]

Artoni, M.

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006).
[Crossref] [PubMed]

Bajcsy, M.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426(6967), 638–641 (2003).
[Crossref] [PubMed]

Balic, V.

D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

Barreiro, S.

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quaside generate levels in Rb vapor,” Phys. Rev. A 57(4), 2996–3002 (1998).
[Crossref]

Braje, D. A.

D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

Che, J.

H. Zheng, X. Zhang, C. Li, H. Lan, J. Che, Y. Zhang, and Y. Zhang, “Suppression and enhancement of coexisting super-fluorescence and multi-wave mixing processes in sodium vapor,” J. Chem. Phys. 138(20), 204315 (2013).
[Crossref] [PubMed]

Chen, H.

Chen, H. X.

Y. P. Zhang, C. Z. Yuan, Y. Q. Zhang, H. B. Zheng, H. X. Chen, C. B. Li, Z. G. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett. 10(5), 055406 (2013).
[Crossref]

Courteille, P. W.

A. Schilke, C. Zimmermann, P. W. Courteille, and W. Guerin, “Photonic Band Gaps in One-Dimensionally Ordered Cold Atomic Vapors,” Phys. Rev. Lett. 106(22), 223903 (2011).
[Crossref] [PubMed]

Field, R. W.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
[Crossref] [PubMed]

Fry, E.

M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999).
[Crossref]

Gaeta, A.

S. Wielandy and A. Gaeta, “Investigation of electromagnetically induced transparency in the strong probe regime,” Phys. Rev. A 58(3), 2500–2505 (1998).
[Crossref]

Gao, M.

Goda, S.

D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

Guerin, W.

A. Schilke, C. Zimmermann, P. W. Courteille, and W. Guerin, “Photonic Band Gaps in One-Dimensionally Ordered Cold Atomic Vapors,” Phys. Rev. Lett. 106(22), 223903 (2011).
[Crossref] [PubMed]

Harris, S. E.

D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Hollberg, L.

M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999).
[Crossref]

Jha, P. K.

L. Yuan, A. A. Lanin, P. K. Jha, A. J. Traverso, A. M. Zheltikov, and M. O. Scully, “Coherent Raman umklappscattering,” Laser Phys. Lett. 8(10), 736–741 (2011).
[Crossref]

Jia, S.

Kash, M.

M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999).
[Crossref]

Kirova, T.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
[Crossref] [PubMed]

La Rocca, G. C.

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006).
[Crossref] [PubMed]

Lan, H.

H. Zheng, X. Zhang, C. Li, H. Lan, J. Che, Y. Zhang, and Y. Zhang, “Suppression and enhancement of coexisting super-fluorescence and multi-wave mixing processes in sodium vapor,” J. Chem. Phys. 138(20), 204315 (2013).
[Crossref] [PubMed]

Lanin, A. A.

L. Yuan, A. A. Lanin, P. K. Jha, A. J. Traverso, A. M. Zheltikov, and M. O. Scully, “Coherent Raman umklappscattering,” Laser Phys. Lett. 8(10), 736–741 (2011).
[Crossref]

Lazarov, G.

J. B. Qi, G. Lazarov, X. J. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: Prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83(2), 288–291 (1999).
[Crossref]

Lazoudis, A.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
[Crossref] [PubMed]

Lezama, A.

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quaside generate levels in Rb vapor,” Phys. Rev. A 57(4), 2996–3002 (1998).
[Crossref]

Li, C.

C. Li, H. Zheng, Z. Zhang, X. Yao, Y. Zhang, Y. Zhang, and Y. Zhang, “Electromagnetically induced transparency and fluorescence in blockaded Rydberg atomic system,” J. Chem. Phys. 139(16), 164316 (2013).
[Crossref] [PubMed]

H. Zheng, X. Zhang, C. Li, H. Lan, J. Che, Y. Zhang, and Y. Zhang, “Suppression and enhancement of coexisting super-fluorescence and multi-wave mixing processes in sodium vapor,” J. Chem. Phys. 138(20), 204315 (2013).
[Crossref] [PubMed]

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[Crossref] [PubMed]

Y. Zhang, Z. Nie, Z. Wang, C. Li, F. Wen, and M. Xiao, “Evidence of Autler-Townes splitting in high-order nonlinear processes,” Opt. Lett. 35(20), 3420–3422 (2010).
[Crossref] [PubMed]

Li, C. B.

Y. P. Zhang, C. Z. Yuan, Y. Q. Zhang, H. B. Zheng, H. X. Chen, C. B. Li, Z. G. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett. 10(5), 055406 (2013).
[Crossref]

C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009).
[Crossref]

Li, L.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
[Crossref] [PubMed]

J. B. Qi, G. Lazarov, X. J. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: Prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83(2), 288–291 (1999).
[Crossref]

Li, N.

Li, P.

Li, Y.

Li, Y. Q.

Li, Y.-Q.

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
[Crossref]

Ling, H. Y.

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
[Crossref]

Lu, K.

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[Crossref] [PubMed]

Lukin, M.

M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999).
[Crossref]

Lukin, M. D.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426(6967), 638–641 (2003).
[Crossref] [PubMed]

Lyyra, A. M.

J. Qi and A. M. Lyyra, “Electromagnetically induced transparency and dark fluorescence in a cascade three-level di-atomic lithium system,” Phys. Rev. A 73(4), 043810 (2006).
[Crossref]

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
[Crossref] [PubMed]

J. B. Qi, G. Lazarov, X. J. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: Prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83(2), 288–291 (1999).
[Crossref]

Magnes, J.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
[Crossref] [PubMed]

Narducci, L. M.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
[Crossref] [PubMed]

J. B. Qi, G. Lazarov, X. J. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: Prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83(2), 288–291 (1999).
[Crossref]

Nie, Z.

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[Crossref] [PubMed]

Y. Zhang, Z. Nie, Z. Wang, C. Li, F. Wen, and M. Xiao, “Evidence of Autler-Townes splitting in high-order nonlinear processes,” Opt. Lett. 35(20), 3420–3422 (2010).
[Crossref] [PubMed]

Nie, Z. Q.

C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009).
[Crossref]

Qi, J.

J. Qi and A. M. Lyyra, “Electromagnetically induced transparency and dark fluorescence in a cascade three-level di-atomic lithium system,” Phys. Rev. A 73(4), 043810 (2006).
[Crossref]

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
[Crossref] [PubMed]

Qi, J. B.

J. B. Qi, G. Lazarov, X. J. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: Prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83(2), 288–291 (1999).
[Crossref]

Rostovtsev, Y.

M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999).
[Crossref]

Rostovtsev, Y. V.

Y. V. Rostovtsev, Z. E. Sariyanni, and M. O. Scully, “Electromagnetically induced coherent backscattering,” Phys. Rev. Lett. 97(11), 113001 (2006).
[Crossref] [PubMed]

Sariyanni, Z. E.

Y. V. Rostovtsev, Z. E. Sariyanni, and M. O. Scully, “Electromagnetically induced coherent backscattering,” Phys. Rev. Lett. 97(11), 113001 (2006).
[Crossref] [PubMed]

Sautenkov, V.

M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999).
[Crossref]

Schilke, A.

A. Schilke, C. Zimmermann, P. W. Courteille, and W. Guerin, “Photonic Band Gaps in One-Dimensionally Ordered Cold Atomic Vapors,” Phys. Rev. Lett. 106(22), 223903 (2011).
[Crossref] [PubMed]

Scully, M.

M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999).
[Crossref]

Scully, M. O.

L. Yuan, A. A. Lanin, P. K. Jha, A. J. Traverso, A. M. Zheltikov, and M. O. Scully, “Coherent Raman umklappscattering,” Laser Phys. Lett. 8(10), 736–741 (2011).
[Crossref]

Y. V. Rostovtsev, Z. E. Sariyanni, and M. O. Scully, “Electromagnetically induced coherent backscattering,” Phys. Rev. Lett. 97(11), 113001 (2006).
[Crossref] [PubMed]

Song, J. P.

C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009).
[Crossref]

Spano, F. C.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
[Crossref] [PubMed]

J. B. Qi, G. Lazarov, X. J. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: Prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83(2), 288–291 (1999).
[Crossref]

Traverso, A. J.

L. Yuan, A. A. Lanin, P. K. Jha, A. J. Traverso, A. M. Zheltikov, and M. O. Scully, “Coherent Raman umklappscattering,” Laser Phys. Lett. 8(10), 736–741 (2011).
[Crossref]

Ullah, Z.

Wang, D.

H. Zhou, D. Wang, D. Wang, J. Zhang, and S. Zhu, “Efficient reflection via four-wave mixing in a Doppler-free electromagnetically-induced-transparency gas system,” Phys. Rev. A 84(5), 053835 (2011).
[Crossref]

H. Zhou, D. Wang, D. Wang, J. Zhang, and S. Zhu, “Efficient reflection via four-wave mixing in a Doppler-free electromagnetically-induced-transparency gas system,” Phys. Rev. A 84(5), 053835 (2011).
[Crossref]

Wang, X. J.

J. B. Qi, G. Lazarov, X. J. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: Prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83(2), 288–291 (1999).
[Crossref]

Wang, Z.

Wang, Z. G.

Y. P. Zhang, C. Z. Yuan, Y. Q. Zhang, H. B. Zheng, H. X. Chen, C. B. Li, Z. G. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett. 10(5), 055406 (2013).
[Crossref]

Welch, G.

M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999).
[Crossref]

Wen, F.

Wielandy, S.

S. Wielandy and A. Gaeta, “Investigation of electromagnetically induced transparency in the strong probe regime,” Phys. Rev. A 58(3), 2500–2505 (1998).
[Crossref]

Xiao, M.

Y. P. Zhang, C. Z. Yuan, Y. Q. Zhang, H. B. Zheng, H. X. Chen, C. B. Li, Z. G. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett. 10(5), 055406 (2013).
[Crossref]

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[Crossref] [PubMed]

Y. Zhang, Z. Nie, Z. Wang, C. Li, F. Wen, and M. Xiao, “Evidence of Autler-Townes splitting in high-order nonlinear processes,” Opt. Lett. 35(20), 3420–3422 (2010).
[Crossref] [PubMed]

C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009).
[Crossref]

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
[Crossref]

Y. Q. Li and M. Xiao, “Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms,” Opt. Lett. 21(14), 1064–1066 (1996).
[Crossref] [PubMed]

Yao, X.

C. Li, H. Zheng, Z. Zhang, X. Yao, Y. Zhang, Y. Zhang, and Y. Zhang, “Electromagnetically induced transparency and fluorescence in blockaded Rydberg atomic system,” J. Chem. Phys. 139(16), 164316 (2013).
[Crossref] [PubMed]

Yin, G. Y.

D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

Yuan, C. Z.

Y. P. Zhang, C. Z. Yuan, Y. Q. Zhang, H. B. Zheng, H. X. Chen, C. B. Li, Z. G. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett. 10(5), 055406 (2013).
[Crossref]

Yuan, L.

L. Yuan, A. A. Lanin, P. K. Jha, A. J. Traverso, A. M. Zheltikov, and M. O. Scully, “Coherent Raman umklappscattering,” Laser Phys. Lett. 8(10), 736–741 (2011).
[Crossref]

Zhang, D.

Zhang, J.

H. Zhou, D. Wang, D. Wang, J. Zhang, and S. Zhu, “Efficient reflection via four-wave mixing in a Doppler-free electromagnetically-induced-transparency gas system,” Phys. Rev. A 84(5), 053835 (2011).
[Crossref]

Zhang, X.

H. Zheng, X. Zhang, C. Li, H. Lan, J. Che, Y. Zhang, and Y. Zhang, “Suppression and enhancement of coexisting super-fluorescence and multi-wave mixing processes in sodium vapor,” J. Chem. Phys. 138(20), 204315 (2013).
[Crossref] [PubMed]

Zhang, Y.

M. Gao, Z. Wang, Z. Ullah, H. Chen, D. Zhang, Y. Zhang, and Y. Zhang, “Modulated photonic band gaps generated by high-order wave mixing,” J. Opt. Soc. Am. B 32(1), 179–187 (2015).
[Crossref]

M. Gao, Z. Wang, Z. Ullah, H. Chen, D. Zhang, Y. Zhang, and Y. Zhang, “Modulated photonic band gaps generated by high-order wave mixing,” J. Opt. Soc. Am. B 32(1), 179–187 (2015).
[Crossref]

H. Zheng, X. Zhang, C. Li, H. Lan, J. Che, Y. Zhang, and Y. Zhang, “Suppression and enhancement of coexisting super-fluorescence and multi-wave mixing processes in sodium vapor,” J. Chem. Phys. 138(20), 204315 (2013).
[Crossref] [PubMed]

H. Zheng, X. Zhang, C. Li, H. Lan, J. Che, Y. Zhang, and Y. Zhang, “Suppression and enhancement of coexisting super-fluorescence and multi-wave mixing processes in sodium vapor,” J. Chem. Phys. 138(20), 204315 (2013).
[Crossref] [PubMed]

C. Li, H. Zheng, Z. Zhang, X. Yao, Y. Zhang, Y. Zhang, and Y. Zhang, “Electromagnetically induced transparency and fluorescence in blockaded Rydberg atomic system,” J. Chem. Phys. 139(16), 164316 (2013).
[Crossref] [PubMed]

C. Li, H. Zheng, Z. Zhang, X. Yao, Y. Zhang, Y. Zhang, and Y. Zhang, “Electromagnetically induced transparency and fluorescence in blockaded Rydberg atomic system,” J. Chem. Phys. 139(16), 164316 (2013).
[Crossref] [PubMed]

C. Li, H. Zheng, Z. Zhang, X. Yao, Y. Zhang, Y. Zhang, and Y. Zhang, “Electromagnetically induced transparency and fluorescence in blockaded Rydberg atomic system,” J. Chem. Phys. 139(16), 164316 (2013).
[Crossref] [PubMed]

N. Li, Z. Zhao, H. Chen, P. Li, Y. Li, Y. Zhao, G. Zhou, S. Jia, and Y. Zhang, “Observation of dressed odd-order multi-wave mixing in five-level atomic medium,” Opt. Express 20(3), 1912–1929 (2012).
[Crossref] [PubMed]

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[Crossref] [PubMed]

Y. Zhang, Z. Nie, Z. Wang, C. Li, F. Wen, and M. Xiao, “Evidence of Autler-Townes splitting in high-order nonlinear processes,” Opt. Lett. 35(20), 3420–3422 (2010).
[Crossref] [PubMed]

Zhang, Y. P.

Y. P. Zhang, C. Z. Yuan, Y. Q. Zhang, H. B. Zheng, H. X. Chen, C. B. Li, Z. G. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett. 10(5), 055406 (2013).
[Crossref]

C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009).
[Crossref]

Zhang, Y. Q.

Y. P. Zhang, C. Z. Yuan, Y. Q. Zhang, H. B. Zheng, H. X. Chen, C. B. Li, Z. G. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett. 10(5), 055406 (2013).
[Crossref]

Zhang, Z.

C. Li, H. Zheng, Z. Zhang, X. Yao, Y. Zhang, Y. Zhang, and Y. Zhang, “Electromagnetically induced transparency and fluorescence in blockaded Rydberg atomic system,” J. Chem. Phys. 139(16), 164316 (2013).
[Crossref] [PubMed]

Zhao, Y.

Zhao, Z.

Zheltikov, A. M.

L. Yuan, A. A. Lanin, P. K. Jha, A. J. Traverso, A. M. Zheltikov, and M. O. Scully, “Coherent Raman umklappscattering,” Laser Phys. Lett. 8(10), 736–741 (2011).
[Crossref]

Zheng, H.

C. Li, H. Zheng, Z. Zhang, X. Yao, Y. Zhang, Y. Zhang, and Y. Zhang, “Electromagnetically induced transparency and fluorescence in blockaded Rydberg atomic system,” J. Chem. Phys. 139(16), 164316 (2013).
[Crossref] [PubMed]

H. Zheng, X. Zhang, C. Li, H. Lan, J. Che, Y. Zhang, and Y. Zhang, “Suppression and enhancement of coexisting super-fluorescence and multi-wave mixing processes in sodium vapor,” J. Chem. Phys. 138(20), 204315 (2013).
[Crossref] [PubMed]

Zheng, H. B.

Y. P. Zhang, C. Z. Yuan, Y. Q. Zhang, H. B. Zheng, H. X. Chen, C. B. Li, Z. G. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett. 10(5), 055406 (2013).
[Crossref]

C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009).
[Crossref]

Zhou, G.

Zhou, H.

H. Zhou, D. Wang, D. Wang, J. Zhang, and S. Zhu, “Efficient reflection via four-wave mixing in a Doppler-free electromagnetically-induced-transparency gas system,” Phys. Rev. A 84(5), 053835 (2011).
[Crossref]

Zhu, S.

H. Zhou, D. Wang, D. Wang, J. Zhang, and S. Zhu, “Efficient reflection via four-wave mixing in a Doppler-free electromagnetically-induced-transparency gas system,” Phys. Rev. A 84(5), 053835 (2011).
[Crossref]

Zibrov, A.

M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999).
[Crossref]

Zibrov, A. S.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426(6967), 638–641 (2003).
[Crossref] [PubMed]

Zimmermann, C.

A. Schilke, C. Zimmermann, P. W. Courteille, and W. Guerin, “Photonic Band Gaps in One-Dimensionally Ordered Cold Atomic Vapors,” Phys. Rev. Lett. 106(22), 223903 (2011).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009).
[Crossref]

J. Chem. Phys. (2)

H. Zheng, X. Zhang, C. Li, H. Lan, J. Che, Y. Zhang, and Y. Zhang, “Suppression and enhancement of coexisting super-fluorescence and multi-wave mixing processes in sodium vapor,” J. Chem. Phys. 138(20), 204315 (2013).
[Crossref] [PubMed]

C. Li, H. Zheng, Z. Zhang, X. Yao, Y. Zhang, Y. Zhang, and Y. Zhang, “Electromagnetically induced transparency and fluorescence in blockaded Rydberg atomic system,” J. Chem. Phys. 139(16), 164316 (2013).
[Crossref] [PubMed]

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

Laser Phys. Lett. (2)

Y. P. Zhang, C. Z. Yuan, Y. Q. Zhang, H. B. Zheng, H. X. Chen, C. B. Li, Z. G. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett. 10(5), 055406 (2013).
[Crossref]

L. Yuan, A. A. Lanin, P. K. Jha, A. J. Traverso, A. M. Zheltikov, and M. O. Scully, “Coherent Raman umklappscattering,” Laser Phys. Lett. 8(10), 736–741 (2011).
[Crossref]

Nature (1)

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426(6967), 638–641 (2003).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (5)

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
[Crossref]

S. Wielandy and A. Gaeta, “Investigation of electromagnetically induced transparency in the strong probe regime,” Phys. Rev. A 58(3), 2500–2505 (1998).
[Crossref]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quaside generate levels in Rb vapor,” Phys. Rev. A 57(4), 2996–3002 (1998).
[Crossref]

J. Qi and A. M. Lyyra, “Electromagnetically induced transparency and dark fluorescence in a cascade three-level di-atomic lithium system,” Phys. Rev. A 73(4), 043810 (2006).
[Crossref]

H. Zhou, D. Wang, D. Wang, J. Zhang, and S. Zhu, “Efficient reflection via four-wave mixing in a Doppler-free electromagnetically-induced-transparency gas system,” Phys. Rev. A 84(5), 053835 (2011).
[Crossref]

Phys. Rev. Lett. (8)

M. Kash, V. Sautenkov, A. Zibrov, L. Hollberg, G. Welch, M. Lukin, Y. Rostovtsev, E. Fry, and M. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82(26), 5229–5232 (1999).
[Crossref]

J. B. Qi, G. Lazarov, X. J. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: Prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83(2), 288–291 (1999).
[Crossref]

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002).
[Crossref] [PubMed]

D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

Y. V. Rostovtsev, Z. E. Sariyanni, and M. O. Scully, “Electromagnetically induced coherent backscattering,” Phys. Rev. Lett. 97(11), 113001 (2006).
[Crossref] [PubMed]

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[Crossref] [PubMed]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006).
[Crossref] [PubMed]

A. Schilke, C. Zimmermann, P. W. Courteille, and W. Guerin, “Photonic Band Gaps in One-Dimensionally Ordered Cold Atomic Vapors,” Phys. Rev. Lett. 106(22), 223903 (2011).
[Crossref] [PubMed]

Phys. Today (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Other (1)

Y. Zhang, and M. Xiao, Multi-Wave Mixing Processes: From Ultrafast Polarization Beats to Electromagnetically Induced Transparency (HEP & Springer, 2009).

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Figures (4)

Fig. 1
Fig. 1 (a) The setup of our experiment. (b) Inverted-Y type energy system. (c) Schematic of an electromagnetically induced grating formed by two coupling beams E3 and E′3. (d) The single dressed energy level schematic diagrams and the calculated single dressed period energy levels. (e) The double dressed energy level schematic diagrams and the calculated double dressed periodic energy levels.
Fig. 2
Fig. 2 Measured (a) probe transmission signal, (b) reflection signal, and (c) fluorescence versus Δ2 from 200 MHz to 550 MHz when different beams are blocked. (a1)- (c1) E 3 and E 3 blocked, (a2)-(c2) E 3 blocked, (a3)-(c3) E 1 blocked, and (a4)-(c4) no beam blocked. Δ1 = −280 MHz and Δ3 = −450MHz.
Fig. 3
Fig. 3 For Fig. 3(a)-3(c), measured (a1) probe transmission signal, (b1) reflection signal and (c1) fluorescence versus Δ2, when we select five different discrete values of P2 as black (3.5 mW), red(2.4 mW), blue(1.5 mW), green(1.2 mW) and purple(0.7mW) and Δ3 = −110 MHz, Δ1 = −280 MHz. (a2), (b2) and (c2) are the theoretical calculations of (a1), (b1) and (c1), respectively. For Fig. 3(d)-3(f), measured (d1) probe transmission signal, (e1) reflection signal and (f1) fluorescence versus Δ2, when we select five different discrete values of P1 as black (10.05 mW), red(8.62 mW), blue(6.1 mW), green(3.83 mW) and purple(1.86 mW) and Δ3 = −410 MHz, Δ1 = −280 MHz. (d2), (e2) and (f2) are the theoretical calculations of (d1), (d1) and (f1), respectively.
Fig. 4
Fig. 4 For Fig. 4(a)-4(c) measured (a1) probe transmission signal, (b1) reflection signal and (c1) fluorescence versus Δ2, when we select five different discrete values of Δ3 as black (−150MHz), red (−200MHz), blue (−280MHz), green (−350MHz) and purple (−450MHz) and Δ1 = −280 MHz. (a2), (b2) and (c2) are the theoretical calculations of (a1), (b1) and (c1), respectively. For Fig. 4(d)-4(f) measured (d1) probe transmission signal, (e1) reflection signal and (f1) fluorescence versus Δ2, when we select five different discrete values of Δ1 as black (−80MHz), red (−180MHz), blue (−280MHz), green (−300MHz) and purple (−380MHz) and Δ3 = −280 MHz. (d2), (e2) and (f2) are the theoretical calculations of (d1), (d1) and (f1), respectively. (g) the schematic diagram of the adjustable optical amplifier.

Equations (15)

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

ρ 10 ( 1 ) = i G 1 d 10 + | G 30 | 2 / d 30 + | G 2 | 2 / d 20
ρ 13 ( 1 ) = i G 3 d 13 + | G 1 | 2 / d 03 + | G 2 | 2 / d 23 + | G 30 | 2 / Γ 33
ρ 20 ( 2 ) = G 1 G 2 ( d 10 + | G 1 | 2 / Γ 00 + | G 2 | 2 / d 20 + | G 30 | 2 / d 30 ) * d 20
ρ 23 ( 2 ) = G 2 G 3 ( d 13 + | G 1 | 2 / d 03 + | G 2 | 2 / d 23 + | G 30 | 2 / Γ 33 ) * d 23
ρ 10 ( 3 ) = i G 1 G 3 G 3 ( d 10 + | G 30 | 2 / d 30 + | G 2 | 2 / d 20 ) 2 d 30
ρ 10 ( 5 ) = i G 1 G 3 G 3 | G 2 | 2 ( d 10 + | G 30 | 2 / d 30 + | G 2 | 2 / d 20 ) 3 d 30 d 20
χ L = i N μ 2 ε 0 1 d 10 + | G 30 | 2 / d 30 + | G 2 | 2 / d 20 + i N μ 2 ε 0 1 d 13 + | G 1 | 2 / d 03 + | G 30 | 2 / Γ 33 + | G 2 | 2 / d 23
χ N L = i N μ 2 ε 0 [ 1 ( d 10 + | G 30 | 2 / d 30 + | G 2 | 2 / d 20 ) 2 d 30 1 ( d 10 + | G 30 | 2 / d 30 + | G 2 | 2 / d 20 ) 3 d 30 d 20 ] N μ 2 ε 0 1 ( d 10 + | G 1 | 2 / Γ 00 + | G 2 | 2 / d 20 + | G 30 | 2 / d 30 ) * d 20 N μ 2 ε 0 1 ( d 13 + | G 1 | 2 / d 03 + | G 2 | 2 / d 23 + | G 30 | 2 / Γ 33 ) * d 23
E 1 ( x ) / x = α E 1 ( x ) + k e i Δ k x x E r ( x )
E r ( x ) / x = α E r ( x ) + k e i Δ k x x E 1 ( x )
R = | 1 k e λ 2 + d x e λ 2 d x e λ 2 + d x ( λ 1 + + α ) 1 e λ 2 d x ( λ 1 + α ) 1 | 2
T = | e ( λ 1 + + λ 1 ) d x ( λ 1 λ 1 + ) ( λ 1 + α ) e λ 1 d x ( λ 1 + + α ) e λ 1 + d x | 2
ρ 11 D D ( 2 ) = | G 1 | 2 ( d 10 + | G 30 | 2 / d 30 + | G 2 | 2 / d 20 ) Γ 11
ρ 22 D ( 4 ) = | G 1 | 2 | G 2 | 2 Γ 22 d 10 d 21 [ d 20 + | G 1 | 2 / d 21 + | G 2 | 2 / ( d 10 + | G 30 | 2 / d 30 ) ]
ρ 22 ( 4 ) = | G 3 | 2 | G 2 | 2 Γ 22 d 13 d 21 [ d 23 + | G 2 | 2 / ( d 13 + | G 1 | 2 / d 03 ) + | G 30 | 2 / d 21 ]

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