Abstract

We show that quasi-closed subwavelength electromagnetic rectangular resonators can be constructed using stacked dielectric-plasmonic bilayer structures, in an analogy to the dielectric resonators formed by materials with large permittivity commonly seen in microwave engineering. We establish an analytic framework for designing such subwavelength resonators by providing formulas for calculating the resonant frequency and the radiation/material loss Q-factors of the fundamental mode. The provided analytic formulas yield accurate results when compared with the numerical simulations for a broad range of design parameter sets. The proposed stack resonator in principle can be made arbitrarily small and, in the absence of material loss, can have Q-factors larger than 105.

© 2012 Optical Society of America

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Errata

Hongyuan Zhao and Guanghao Zhu, "Analytical design of quasi-closed subwavelength electromagnetic rectangular resonators using stacks of dielectric–plasmonic bilayers: errata," J. Opt. Soc. Am. B 30, 2928-2929 (2013)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-30-11-2928

References

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2012 (3)

2011 (4)

2010 (4)

2009 (4)

2008 (2)

E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express 16, 16529–16537 (2008).
[CrossRef]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials,” Science 321, 930 (2008).
[CrossRef]

2007 (3)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90, 181102 (2007).
[CrossRef]

Y. Gong and J. Vuckovic, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90, 033113 (2007).
[CrossRef]

2006 (4)

H. Li, J. Hao, L. Zhou, Z. Wei, L. Gong, H. Chen, and C. T. Chan, “All-dimensional subwavelength cavities made with metamaterials,” Appl. Phys. Lett. 89, 104101 (2006).
[CrossRef]

T. Jiang, Y. Chen, and Y. Feng, “Subwavelength rectangular cavity partially filled with left-handed material,” Chin. Phys. 15, 1154–1160 (2006).
[CrossRef]

R. Wangberg, J. Elser, E. E. Narimanov, and V. A. Podolskiy, “Nonmagnetic nanocomposites for optical and infrared negative-refractive-index media,” J. Opt. Soc. Am. B 23, 498–505 (2006).
[CrossRef]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

2005 (3)

S. He, Y. Jin, Z. Ruan, and J. Kuang, “On subwavelength and open resonators involving metamaterials of negative refraction index,” New J. Phys. 7, 210 (2005).
[CrossRef]

Y. Li, L. Ran, H. Chen, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental realization of a one-dimensional LHM–RHM resonator,” IEEE Trans. Microwave Theory Tech. 53, 1522–1526 (2005).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

2003 (1)

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

2002 (1)

N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterial with negative permittivity and permeability,” IEEE Antennas Wireless Propagat. Lett. 1, 10–13 (2002).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of negative refraction,” Science 292, 77–79 (2001).
[CrossRef]

2000 (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2000).
[CrossRef]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Ahn, B. H.

Akimov, Y. A.

Albrektsen, O.

Babinec, T.

Bai, P.

Barnard, E. S.

Bartal, G.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. USA 108, 11327–11331 (2011).

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials,” Science 321, 930 (2008).
[CrossRef]

Bergner, N.

Borghs, G.

Bozhevolnyi, S. I.

Brongersma, M. L.

Bruderer, L.

Bulu, I.

Caglayan, H.

Cai, W.

Chan, C. T.

H. Li, J. Hao, L. Zhou, Z. Wei, L. Gong, H. Chen, and C. T. Chan, “All-dimensional subwavelength cavities made with metamaterials,” Appl. Phys. Lett. 89, 104101 (2006).
[CrossRef]

Chandran, A.

Chau, Y. F.

Chen, H.

H. Li, J. Hao, L. Zhou, Z. Wei, L. Gong, H. Chen, and C. T. Chan, “All-dimensional subwavelength cavities made with metamaterials,” Appl. Phys. Lett. 89, 104101 (2006).
[CrossRef]

Y. Li, L. Ran, H. Chen, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental realization of a one-dimensional LHM–RHM resonator,” IEEE Trans. Microwave Theory Tech. 53, 1522–1526 (2005).
[CrossRef]

Chen, J.

Chen, K.

Y. Li, L. Ran, H. Chen, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental realization of a one-dimensional LHM–RHM resonator,” IEEE Trans. Microwave Theory Tech. 53, 1522–1526 (2005).
[CrossRef]

Chen, W. T.

Chen, Y.

T. Jiang, Y. Chen, and Y. Feng, “Subwavelength rectangular cavity partially filled with left-handed material,” Chin. Phys. 15, 1154–1160 (2006).
[CrossRef]

Chettiar, U. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

Choi, M.

Choy, J. T.

Chu, H. S.

Drachev, V. P.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

Elser, J.

Engheta, N.

N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterial with negative permittivity and permeability,” IEEE Antennas Wireless Propagat. Lett. 1, 10–13 (2002).
[CrossRef]

Fedotov, V.

Feng, Y.

T. Jiang, Y. Chen, and Y. Feng, “Subwavelength rectangular cavity partially filled with left-handed material,” Chin. Phys. 15, 1154–1160 (2006).
[CrossRef]

Gelfand, R. M.

Gong, L.

H. Li, J. Hao, L. Zhou, Z. Wei, L. Gong, H. Chen, and C. T. Chan, “All-dimensional subwavelength cavities made with metamaterials,” Appl. Phys. Lett. 89, 104101 (2006).
[CrossRef]

Gong, Q. H.

Gong, Y.

Y. Gong and J. Vuckovic, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90, 033113 (2007).
[CrossRef]

Grzegorczyk, T. M.

Y. Li, L. Ran, H. Chen, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental realization of a one-dimensional LHM–RHM resonator,” IEEE Trans. Microwave Theory Tech. 53, 1522–1526 (2005).
[CrossRef]

Hao, J.

H. Li, J. Hao, L. Zhou, Z. Wei, L. Gong, H. Chen, and C. T. Chan, “All-dimensional subwavelength cavities made with metamaterials,” Appl. Phys. Lett. 89, 104101 (2006).
[CrossRef]

Hausmann, B.

He, S.

S. He, Y. Jin, Z. Ruan, and J. Kuang, “On subwavelength and open resonators involving metamaterials of negative refraction index,” New J. Phys. 7, 210 (2005).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principle of NanoOoptics (Cambridge University, 2006).

Helgert, C.

Ho, Y. Z.

Hosseini, A.

A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90, 181102 (2007).
[CrossRef]

Huang, Y. W.

Huangfu, J.

Y. Li, L. Ran, H. Chen, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental realization of a one-dimensional LHM–RHM resonator,” IEEE Trans. Microwave Theory Tech. 53, 1522–1526 (2005).
[CrossRef]

Jiang, T.

T. Jiang, Y. Chen, and Y. Feng, “Subwavelength rectangular cavity partially filled with left-handed material,” Chin. Phys. 15, 1154–1160 (2006).
[CrossRef]

Jin, Y.

S. He, Y. Jin, Z. Ruan, and J. Kuang, “On subwavelength and open resonators involving metamaterials of negative refraction index,” New J. Phys. 7, 210 (2005).
[CrossRef]

Kang, J. H.

Kildishev, A. V.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

Kim, M. K.

Kley, E. B.

Kong, J. A.

Y. Li, L. Ran, H. Chen, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental realization of a one-dimensional LHM–RHM resonator,” IEEE Trans. Microwave Theory Tech. 53, 1522–1526 (2005).
[CrossRef]

Kottmann, J. P.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2000).
[CrossRef]

Kuang, J.

S. He, Y. Jin, Z. Ruan, and J. Kuang, “On subwavelength and open resonators involving metamaterials of negative refraction index,” New J. Phys. 7, 210 (2005).
[CrossRef]

Kwon, S. H.

Lagae, L.

Lederer, F.

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

Lee, S. H.

Lee, Y. H.

Li, E. P.

Li, H.

H. Li, J. Hao, L. Zhou, Z. Wei, L. Gong, H. Chen, and C. T. Chan, “All-dimensional subwavelength cavities made with metamaterials,” Appl. Phys. Lett. 89, 104101 (2006).
[CrossRef]

Li, Y.

Y. Li, L. Ran, H. Chen, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental realization of a one-dimensional LHM–RHM resonator,” IEEE Trans. Microwave Theory Tech. 53, 1522–1526 (2005).
[CrossRef]

Li, Z.

Litchinitser, N. M.

Liu, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials,” Science 321, 930 (2008).
[CrossRef]

Liu, Z.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials,” Science 321, 930 (2008).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

Loncar, M.

Martin, O. J. F.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2000).
[CrossRef]

Massoud, Y.

A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90, 181102 (2007).
[CrossRef]

Min, B.

Mohseni, H.

Narimanov, E. E.

Neutens, P.

Ni, X.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

No, Y. S.

Novotny, L.

L. Novotny and B. Hecht, Principle of NanoOoptics (Cambridge University, 2006).

Ozbay, E.

Park, H. G.

Park, N.

Pendry, J. B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

Pertsch, T.

Petschulat, J.

Podolskiy, V. A.

Pors, A.

Pozar, D. M.

D. M. Pozar, Microwave Engineering (Wiley, 1998).

Ran, L.

Y. Li, L. Ran, H. Chen, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental realization of a one-dimensional LHM–RHM resonator,” IEEE Trans. Microwave Theory Tech. 53, 1522–1526 (2005).
[CrossRef]

Rho, J.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photon. 6, 450–454 (2012).
[CrossRef]

Rockstuhl, C.

Ruan, Z.

S. He, Y. Jin, Z. Ruan, and J. Kuang, “On subwavelength and open resonators involving metamaterials of negative refraction index,” New J. Phys. 7, 210 (2005).
[CrossRef]

Sarychev, A. K.

Savinov, V.

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of negative refraction,” Science 292, 77–79 (2001).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2000).
[CrossRef]

Schurig, D.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

Shalaev, V. M.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

N. M. Litchinitser and V. M. Shalaev, “Metamaterials: transforming theory into reality,” J. Opt. Soc. Am. B 26, B161–B169 (2009).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

W. Cai, and V. M. Shalaev, Optical Metamaterials (Springer, 2010).

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of negative refraction,” Science 292, 77–79 (2001).
[CrossRef]

Smith, D. R.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of negative refraction,” Science 292, 77–79 (2001).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2000).
[CrossRef]

Stacy, A.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials,” Science 321, 930 (2008).
[CrossRef]

Steinert, M.

Sun, C.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials,” Science 321, 930 (2008).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

Tsai, D. P.

Tünnermann, A.

Van Dorpe, P.

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Vuckovic, J.

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X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photon. 6, 450–454 (2012).
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Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
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Figures (2)

Fig. 1.
Fig. 1.

(a) Electromagnetic resonator based on dielectric-plasmonic bilayer stack and (b) sketch of effective permittivity as a function of frequency.

Fig. 2.
Fig. 2.

(a) Intensity profile of the magnetic field of the TE11 mode, (b) normalized resonant frequency as a function of the normalized resonator width, (c) radiation loss Q-factor as a function of the normalized resonator width, and (d) overall loss Q-factor as a function of the normalized collision rate.

Equations (33)

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εx=(ηε11+(1η)ε21)1,
εy=εz=ηε1+(1η)ε2,
ε2=ε0(1ωp2ω2),
ωcωp=(1+(1η)ε1ηε0)1/2.
(1εy2x2+1εx2y2+μ0ω2)Hz=0,
Hz=H0sin(πxLx)sin(πyLy),
ω=π2μ0(1εyLx2+1εxLy2).
εyεx=Ly2Lx24εyε0ω2ωp2Ly2λp2,
Qrad=π2512·(ε0ε¯xεx2LxLy+ε0ε¯yεy2LyLx+4ω2ωp2LxLyλp2)·ωp4ω4·λp4Lx2Ly2,
Qmat(1+4λp2/Lx2+λp2/Ly2)ωγp.
Im=E·dl,
H=H0sin(πxLx)sin(πyLy)z^,
E=Ex0sin(πxLx)cos(πyLy)x^+Ey0cos(πxLx)sin(πyLy)y^,
Ex0=1iωεxπLyH0,
Ey0=1iωεyπLxH0.
Im=4πEx0Lx+4πEy0Ly.
dWdt=k8ε0μ0Im2.
dWdt=2ε0H02ω(1εxLy2+1εyLx2)2Lx2Ly2.
dWdt=2H02π4ωε0k4Lx2Ly2.
W=We+Wm=(14ε¯|E|2+14μ¯|H|2)dxdy.
W=π2H0216ω2(ε¯xεx2LxLy+ε¯yεy2LyLx+4ω2ωp2LxLyλp2).
dWdt=γW=2H02π4ωε0k4Lx2Ly2=γπ2H0216ω2(ε¯xεx2LxLy+ε¯yεy2LyLx+4ω2ωp2LxLyλp2),
γ=512π2·(ε0ε¯xεx2LxLy+ε0ε¯yεy2LyLx+4ω2ωp2LxLyλp2)1·ω4ωp4·Lx2Ly2λp4·ω.
Qrad=ωγ=π2512·(ε0ε¯xεx2LxLy+ε0ε¯yεy2LyLx+4ω2ωp2LxLyλp2)·ωp4ω4·λp4Lx2Ly2.
ε2=ε0(1ωp2ω2iωγp),
2ωδω=π2μ0(δ[1εy]1Lx2+δ[1εx]1Ly2),
δ[1εx]=2(1η)ε2(1ω02/ω2)2ω02ω2δωiγd/2ω=Γxε2(δωωiγd2ω),
δ[1εy]=2(1η)ε2(ηε1+(1η)ε2(1ω02/ω2))2ω02ω2δωiγd/2ω=Γyε2(δωωiγd2ω),
Γx=2(1η)(1ω02/ω2)2ω02ω2,
Γy=2(1η)(ηε1/ε2+(1η)(1ω02/ω2))2ω02ω2.
2ωδω=π2μ0(Γyε21Lx2Γxε21Ly2)(δωωiγd2ω).
δω=(1+2ω2π2μ0(Γyε21Lx2+Γxε21Ly2))1·iγd2(1+4λp2/Lx2+λp2/Ly2)1·iγd2,
Qmat(1+4λp2/Lx2+λp2/Ly2)ωγp.

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