Abstract

We demonstrate the realization of plasmonic analog of electromagnetically induced transparency (EIT) in a system composing of two stub resonators side-coupled to metal-dielectric-metal (MDM) waveguide. Based on the coupled mode theory (CMT) and Fabry-Perot (FP) model, respectively, the formation and evolution mechanisms of plasmon-induced transparency by direct and indirect couplings are exactly analyzed. For the direct coupling between the two stub resonators, the FWHM and group index of transparent window to the inter-space are more sensitive than to the width of one cut, and the high group index of up to 60 can be achieved. For the indirect coupling, the formation of transparency window is determined by the resonance detuning, but the evolution of transparency is mainly attributed to the change of coupling distance. The consistence between the analytical solution and finite-difference time-domain (FDTD) simulations verifies the feasibility of the plasmon-induced transparency system. It is also interesting to notice that the scheme is easy to be fabricated and may pave the way to highly integrated optical circuits.

© 2013 OSA

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2012

2011

2010

L. Yang, C. G. Min, and G. Veronis, “Guided subwavelength slow-light mode supported by a plasmonic waveguide system,” Opt. Lett.35(24), 4184–4186 (2010).
[CrossRef] [PubMed]

R. D. Kekatpure, E. S. Barnard, W. Cai, and M. L. Brongersma, “Phase-coupled plasmon-induced transparency,” Phys. Rev. Lett.104(24), 243902 (2010).
[CrossRef] [PubMed]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[CrossRef]

2009

X. Yang, M. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett.102(17), 173902 (2009).
[CrossRef] [PubMed]

2008

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today61(5), 44–50 (2008).
[CrossRef]

T. Baba, “Slow light in photonic crystal,” Nat. Photonics2(8), 465–473 (2008).
[CrossRef] [PubMed]

2007

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett.98(21), 213904 (2007).
[CrossRef] [PubMed]

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A75(6), 063833 (2007).
[CrossRef]

2006

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett.96(12), 123901 (2006).
[CrossRef] [PubMed]

2005

Y. Zhang, K. Hayasaka, and K. Kasai, “Conditional transfer of quantum correlation in the intensity of twin beams,” Phys. Rev. A71(6), 062341 (2005).
[CrossRef]

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett.94(7), 073903 (2005).
[CrossRef] [PubMed]

1999

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature397(6720), 594–598 (1999).
[CrossRef]

S. Harris and L. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett.82(23), 4611–4614 (1999).
[CrossRef]

1991

K. J. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66(20), 2593–2596 (1991).
[CrossRef] [PubMed]

H. A. Haus and W. P. Huang, “Coupled-mode theory,” Proc. IEEE79(10), 1505–1518 (1991).
[CrossRef]

1969

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev.182(2), 539–554 (1969).
[CrossRef]

Baba, T.

T. Baba, “Slow light in photonic crystal,” Nat. Photonics2(8), 465–473 (2008).
[CrossRef] [PubMed]

Barnard, E. S.

R. D. Kekatpure, E. S. Barnard, W. Cai, and M. L. Brongersma, “Phase-coupled plasmon-induced transparency,” Phys. Rev. Lett.104(24), 243902 (2010).
[CrossRef] [PubMed]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature397(6720), 594–598 (1999).
[CrossRef]

Bogaerts, W.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett.94(7), 073903 (2005).
[CrossRef] [PubMed]

Boller, K. J.

K. J. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66(20), 2593–2596 (1991).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

Z. H. Han and S. I. Bozhevolnyi, “Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices,” Opt. Express19(4), 3251–3257 (2011).
[CrossRef] [PubMed]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[CrossRef]

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today61(5), 44–50 (2008).
[CrossRef]

Brongersma, M. L.

R. D. Kekatpure, E. S. Barnard, W. Cai, and M. L. Brongersma, “Phase-coupled plasmon-induced transparency,” Phys. Rev. Lett.104(24), 243902 (2010).
[CrossRef] [PubMed]

Cai, W.

R. D. Kekatpure, E. S. Barnard, W. Cai, and M. L. Brongersma, “Phase-coupled plasmon-induced transparency,” Phys. Rev. Lett.104(24), 243902 (2010).
[CrossRef] [PubMed]

Chen, Y. L.

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A75(6), 063833 (2007).
[CrossRef]

Darmawan, S.

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature397(6720), 594–598 (1999).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today61(5), 44–50 (2008).
[CrossRef]

Economou, E. N.

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev.182(2), 539–554 (1969).
[CrossRef]

Engelen, R. J.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett.94(7), 073903 (2005).
[CrossRef] [PubMed]

Fan, S.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett.96(12), 123901 (2006).
[CrossRef] [PubMed]

Genet, C.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today61(5), 44–50 (2008).
[CrossRef]

Gersen, H.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett.94(7), 073903 (2005).
[CrossRef] [PubMed]

Gong, Y. K.

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[CrossRef]

Guo, G. C.

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A75(6), 063833 (2007).
[CrossRef]

Guo, Y. H.

Guo, Z.

Han, Z. H.

Harris, S.

S. Harris and L. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett.82(23), 4611–4614 (1999).
[CrossRef]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature397(6720), 594–598 (1999).
[CrossRef]

K. J. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66(20), 2593–2596 (1991).
[CrossRef] [PubMed]

Hau, L.

S. Harris and L. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett.82(23), 4611–4614 (1999).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature397(6720), 594–598 (1999).
[CrossRef]

Haus, H. A.

H. A. Haus and W. P. Huang, “Coupled-mode theory,” Proc. IEEE79(10), 1505–1518 (1991).
[CrossRef]

Hayasaka, K.

Y. Zhang, K. Hayasaka, and K. Kasai, “Conditional transfer of quantum correlation in the intensity of twin beams,” Phys. Rev. A71(6), 062341 (2005).
[CrossRef]

Huang, W. P.

H. A. Haus and W. P. Huang, “Coupled-mode theory,” Proc. IEEE79(10), 1505–1518 (1991).
[CrossRef]

Huang, Y.

Y. Huang, C. Min, and G. Veronis, “Subwavelength slow-light waveguides based on a plasmonic analogue of electromagnetically induced transparency,” Appl. Phys. Lett.99(14), 143117 (2011).
[CrossRef]

Imamolu, A.

K. J. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66(20), 2593–2596 (1991).
[CrossRef] [PubMed]

Jiang, W.

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A75(6), 063833 (2007).
[CrossRef]

Karle, T. J.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett.94(7), 073903 (2005).
[CrossRef] [PubMed]

Kasai, K.

Y. Zhang, K. Hayasaka, and K. Kasai, “Conditional transfer of quantum correlation in the intensity of twin beams,” Phys. Rev. A71(6), 062341 (2005).
[CrossRef]

Kekatpure, R. D.

R. D. Kekatpure, E. S. Barnard, W. Cai, and M. L. Brongersma, “Phase-coupled plasmon-induced transparency,” Phys. Rev. Lett.104(24), 243902 (2010).
[CrossRef] [PubMed]

Kobayashi, N.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett.98(21), 213904 (2007).
[CrossRef] [PubMed]

Koo, S.

Korterik, J. P.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett.94(7), 073903 (2005).
[CrossRef] [PubMed]

Krauss, T. F.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett.94(7), 073903 (2005).
[CrossRef] [PubMed]

Kuipers, L.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett.94(7), 073903 (2005).
[CrossRef] [PubMed]

Kwong, D. L.

X. Yang, M. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett.102(17), 173902 (2009).
[CrossRef] [PubMed]

Lee, K. H.

Lipson, M.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett.96(12), 123901 (2006).
[CrossRef] [PubMed]

Liu, X. M.

Lu, H.

Luo, B.

Luo, X. G.

Mao, D.

H. Lu, X. M. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A85(5), 053803 (2012).
[CrossRef]

H. Lu, X. M. Liu, D. Mao, Y. K. Gong, and G. X. Wang, “Induced transparency in nanoscale plasmonic resonator systems,” Opt. Lett.36(16), 3233–3235 (2011).
[CrossRef] [PubMed]

Mei, T.

Min, C.

Y. Huang, C. Min, and G. Veronis, “Subwavelength slow-light waveguides based on a plasmonic analogue of electromagnetically induced transparency,” Appl. Phys. Lett.99(14), 143117 (2011).
[CrossRef]

Min, C. G.

Pan, W.

Park, N.

Piao, X. J.

Povinelli, M. L.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett.96(12), 123901 (2006).
[CrossRef] [PubMed]

Sandhu, S.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett.96(12), 123901 (2006).
[CrossRef] [PubMed]

Shakya, J.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett.96(12), 123901 (2006).
[CrossRef] [PubMed]

Tobing, L. Y. M.

Tomita, M.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett.98(21), 213904 (2007).
[CrossRef] [PubMed]

Totsuka, K.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett.98(21), 213904 (2007).
[CrossRef] [PubMed]

van Hulst, N. F.

H. Gersen, T. J. Karle, R. J. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett.94(7), 073903 (2005).
[CrossRef] [PubMed]

Veronis, G.

Y. Huang, C. Min, and G. Veronis, “Subwavelength slow-light waveguides based on a plasmonic analogue of electromagnetically induced transparency,” Appl. Phys. Lett.99(14), 143117 (2011).
[CrossRef]

L. Yang, C. G. Min, and G. Veronis, “Guided subwavelength slow-light mode supported by a plasmonic waveguide system,” Opt. Lett.35(24), 4184–4186 (2010).
[CrossRef] [PubMed]

Wang, G. X.

Wen, K. H.

Wong, C. W.

X. Yang, M. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett.102(17), 173902 (2009).
[CrossRef] [PubMed]

Xiao, Y. F.

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A75(6), 063833 (2007).
[CrossRef]

Xu, Q.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett.96(12), 123901 (2006).
[CrossRef] [PubMed]

Yan, L. S.

Yang, L.

Yang, X.

X. Yang, M. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett.102(17), 173902 (2009).
[CrossRef] [PubMed]

Yu, M.

X. Yang, M. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett.102(17), 173902 (2009).
[CrossRef] [PubMed]

Yu, S.

Zhang, D. H.

Zhang, Y.

Y. Zhang, S. Darmawan, L. Y. M. Tobing, T. Mei, and D. H. Zhang, “Coupled resonator-induced transparency in ring-bus-ring Mach-Zehnder interferometer,” J. Opt. Soc. Am. B28(1), 28–36 (2011).
[CrossRef]

Y. Zhang, K. Hayasaka, and K. Kasai, “Conditional transfer of quantum correlation in the intensity of twin beams,” Phys. Rev. A71(6), 062341 (2005).
[CrossRef]

Zou, X. B.

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A75(6), 063833 (2007).
[CrossRef]

Appl. Phys. Lett.

Y. Huang, C. Min, and G. Veronis, “Subwavelength slow-light waveguides based on a plasmonic analogue of electromagnetically induced transparency,” Appl. Phys. Lett.99(14), 143117 (2011).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Photonics

T. Baba, “Slow light in photonic crystal,” Nat. Photonics2(8), 465–473 (2008).
[CrossRef] [PubMed]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[CrossRef]

Nature

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature397(6720), 594–598 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev.182(2), 539–554 (1969).
[CrossRef]

Phys. Rev. A

H. Lu, X. M. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A85(5), 053803 (2012).
[CrossRef]

Y. Zhang, K. Hayasaka, and K. Kasai, “Conditional transfer of quantum correlation in the intensity of twin beams,” Phys. Rev. A71(6), 062341 (2005).
[CrossRef]

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A75(6), 063833 (2007).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(a) Blue and green curves represent the real and imaginary parts of the effective index neff of SPPs mode in MDM waveguide with width h = 100 nm, respectively. (b) Transmission spectra for MDM waveguide coupled to one stub, with w equaling to 100 nm and L being 500 nm. In both figures, insets display the corresponding schematic figures.

Fig. 2
Fig. 2

(a) Schematic of MDM waveguide coupled to two stub resonators: w, the width of the waveguide; w1 and w2, the widths of the stub resonators; L1 and L2, the stub depths; d, the inter-space between the two stub resonators. (b) Illustration of direct coupling between two resonators denoted by R1 and R2. k12 and k21 are coupling coefficients.

Fig. 3
Fig. 3

(a) Transmission spectra of MDM waveguide coupled to two stub resonators at different inter-spaces d. (b) The FWHM and group index of the transparent window versus d. (c-e) Magnetic field distributions at three wavelengths marked by triangles corresponding to d = 100nm, 50 nm, and 25 nm, respectively. The other parameters are changeless, namely, w = w1 = w2 = 100 nm, L1 = L2 = 500 nm.

Fig. 4
Fig. 4

(a) Transmission spectra of the structure shown in Fig. 2(a) for different w2, and the other parameters are set as follows: w = w1 = 100 nm, d = 50 nm, and L1 = L2 = 500 nm. (b) The FWHM and group index of the transparent window versus w2. (c) Resonant frequencies (ω+ and ω-) of the system calculated using the CMT (solid curves) and FDTD method (dots).

Fig. 5
Fig. 5

(a) Transmission spectra of MDM waveguide coupled to two stub resonators at different inter-spaces d with w = w1 = w2 = 100 nm, L1 = 500 nm, L2 = 480 nm. The curve with circles is transmission spectra for the one stub system (shown in Fig. 1(b)) with w = 100 nm, L = 480 nm, and the curve with triangles corresponds to the one stub system with w = 100 nm, L = 500 nm. (b) Transmission spectra for different inter-spaces d at λ = 527 nm. (c) Evolution of the transmission spectra versus δ and λ. (d) The top view of Fig. 5(c).

Equations (6)

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T= (ω- ω 0 ) 2 + ( 1 τ 0 ) 2 (ω- ω 0 ) 2 + ( 1 τ 0 + 1 τ e ) 2 .
d a 1 dt =i ω 1 a 1 - 1 τ 0 a 1 + k 12 a 2 ,
d a 2 dt =i ω 2 a 2 - 1 τ 0 a 2 + k 21 a 1 .
ω= ω ± = ω 1 + ω 2 2 ± ( ω 1 - ω 2 2 ) 2 + | k 12 | 2 ω 1 + ω 2 2 ± Ω 0 .
T(ω)= | t 1 (ω) t 2 (ω) 1 r 1 (ω) r 2 (ω) e i2β(ω)δ | 2 ,
T(ω)= ( | t 1 (ω) t 2 (ω) | 1| r 1 (ω) r 2 (ω) | ) 2 1 1+4 ( | r 1 (ω) r 2 (ω) | 1| r 1 (ω) r 2 (ω) | ) 2 sin 2 θ ,

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