Abstract

The reflection of a TM-polarized light beam from a Kretschmann configuration with a saturable gain medium is investigated theoretically. Here, the dielectric constant of the gain medium is described by a classical Lorentzian oscillator model. When surface plasmon polaritons are effectively excited in this structure, it is demonstrated that the curves of enhanced total reflection (ETR) show different shaped hysteresis loops associated with optical bistability owing to gain saturation effect. The effects of the angle of incidence, the thickness of metal film, and the value of small-signal gain on bistable ETR are discussed in detail in a homogeneously broadened (HB) gain medium at line center. Analogous results can also be obtained in an inhomogeneously broadened (inHB) gain medium, while the two switch thresholds and the width of optical bistability hysteresis in an inHB gain medium are significantly different from those in a HB gain medium.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

2016 (2)

T. U. Tumkur, G. Zhu, and M. A. Noginov, “Strong coupling of surface plasmon polaritons and ensembles of dye molecules,” Opt. Express 24(4), 3921–3928 (2016).
[Crossref] [PubMed]

C. Tzschaschel, M. Sudzius, A. Mischok, H. Fröb, and K. Leo, “Net gain in small mode volume organic microcavities,” Appl. Phys. Lett. 108(2), 023304 (2016).
[Crossref]

2015 (3)

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Scientific Reports 5, 12271 (2015).
[Crossref] [PubMed]

P. Törmä and W. L. Barnes, “Strong coupling between surface plasmon polaritons and emitters: a review,” Rep. Prog. Phys. 78(1), 013901 (2015).
[Crossref]

Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. 40(4), 601–604 (2015).
[Crossref] [PubMed]

2014 (1)

C. Argyropoulos, C. Ciracì, and D. R. Smith, “Enhanced optical bistability with film-coupled plasmonic nanocubes,” Appl. Phys. Lett. 104(6), 063108 (2014).
[Crossref]

2013 (2)

D. Ahn and S. L. Chuang, “High optical gain of I–VII semiconductor quantum wells for efficient light-emitting devices,” Appl. Phys. Lett. 102(12), 121114 (2013).
[Crossref]

P. T. Thanh, D. Tanaka, R. Fujimura, Y. Takanishi, and K. Kajikawa, “Low-Power All-Optical Bistable Device of Twisted-Nematic Liquid Crystal Based on Surface Plasmons in a Metal-Insulator-Metal Structure,” Appl. Phys. Express 6(1), 011701 (2013).
[Crossref]

2012 (6)

P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. 6(1), 16–24 (2012).
[Crossref]

N. C. Lindquist, P. Nagpal, K. M. McPeak, D. J. Norris, and Sang-Hyun Oh, “Engineering metallic nanostructures for plasmonics and nanophotonics,” Rep. Prog. Phys. 75(3), 036501 (2012).
[Crossref] [PubMed]

S. Hayashi and T. Okamoto, “Plasmonics: visit the past to know the future,” J. Phys. D: Appl. Phys. 45(43), 433001 (2012).
[Crossref]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nature Photon. 6(11), 737–748 (2012).
[Crossref]

A. V. Malyshev, “Condition for resonant optical bistability,” Phys. Rev. A 86(6), 065804 (2012).
[Crossref]

J. Zhang and L. Zhang, “Nanostructures for surface plasmons,” Adv. Opt. Photon. 4(2), 157–321 (2012).
[Crossref]

2011 (3)

F. Valmorra, M. Bröll, S. Schwaiger, N. Welzel, D. Heitmann, and S. Mendach, “Strong coupling between surface plasmon polariton and laser dye rhodamine 800,” Appl. Phys. Lett. 99(5), 051110 (2011).
[Crossref]

Y. Chen, J. Li, M. Ren, B. Wang, J. Fu, S. Liu, and Z. Li, “Direct observation of amplified spontaneous emission of surface plasmon polaritons at metal/dielectric interfaces,” Appl. Phys. Lett. 98(26), 261912 (2011).
[Crossref]

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-Plasmonic Modulation via Stimulated Emission of Copropagating Surface Plasmon Polaritons on a Substrate with Gain,” Nano Lett. 11(6), 2231–2235 (2011).
[Crossref] [PubMed]

2010 (4)

2009 (3)

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1(3), 484–588 (2009).
[Crossref]

T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi Splitting and Strong-Coupling Dynamics for Surface-Plasmon Polaritons and Rhodamine 6G Molecules,” Phys. Rev. Lett. 103(5), 053602 (2009).
[Crossref] [PubMed]

J. Chen, P. Wang, X. Wang, Y. Lu, R. Zheng, H. Ming, and Q. Zhan, “Optical bistability enhanced by highly localized bulk plasmon polariton modes in subwavelength metal-nonlinear dielectric multilayer structure,” Appl. Phys. Lett. 94(8), 081117 (2009).
[Crossref]

2008 (3)

2007 (1)

2006 (2)

E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

G. A. Wurtz, R. Pollard, and A.V. Zayats, “Optical Bistability in Nonlinear Surface-Plasmon Polaritonic Crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
[Crossref] [PubMed]

2005 (2)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3–4), 131–314 (2005).
[Crossref]

J. Seidel, S. Grafström, and L. Eng, “Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution,” Phys. Rev. Lett. 94(17), 177401 (2005).
[Crossref] [PubMed]

2004 (3)

J. Bellessa, C. Bonnand, J. C. Plenet, and J. Mugnier, “Strong Coupling between Surface Plasmons and Excitons in an Organic Semiconductor,” Phys. Rev. Lett. 93(3), 036404 (2004).
[Crossref] [PubMed]

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004)
[Crossref]

M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004).
[Crossref] [PubMed]

1994 (1)

R. Reinisch and G. Vitrant, “Optical Bistability,” Prog. Quant. Electr. 18, 1–38 (1994).
[Crossref]

1993 (2)

1986 (2)

1985 (1)

P. Martinot, A. Koster, and S. Laval, “Experimental Observation of Optical Bistability by Excitation of a Surface Plasmon Wave,” IEEE J. Quantum Electron. 21(8), 1140–1143 (1985).
[Crossref]

1981 (1)

1979 (1)

1971 (1)

E. Kretschmann, “The Determination of the Optical Constant of Metals by Excitation of Surface Plasmons,” Z. Physik 241, 313–324 (1971).
[Crossref]

Adegoke, J. A.

Agarwal, G. S.

Ahn, D.

D. Ahn and S. L. Chuang, “High optical gain of I–VII semiconductor quantum wells for efficient light-emitting devices,” Appl. Phys. Lett. 102(12), 121114 (2013).
[Crossref]

Aitchison, J. S.

Alam, M. Z.

Argyropoulos, C.

C. Argyropoulos, C. Ciracì, and D. R. Smith, “Enhanced optical bistability with film-coupled plasmonic nanocubes,” Appl. Phys. Lett. 104(6), 063108 (2014).
[Crossref]

Bahoura, M.

Barnes, W. L.

P. Törmä and W. L. Barnes, “Strong coupling between surface plasmon polaritons and emitters: a review,” Rep. Prog. Phys. 78(1), 013901 (2015).
[Crossref]

Bellessa, J.

J. Bellessa, C. Bonnand, J. C. Plenet, and J. Mugnier, “Strong Coupling between Surface Plasmons and Excitons in an Organic Semiconductor,” Phys. Rev. Lett. 93(3), 036404 (2004).
[Crossref] [PubMed]

Berini, P.

P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. 6(1), 16–24 (2012).
[Crossref]

I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon. 4(6), 382–387 (2010).
[Crossref]

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1(3), 484–588 (2009).
[Crossref]

Bolger, P. M.

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-Plasmonic Modulation via Stimulated Emission of Copropagating Surface Plasmon Polaritons on a Substrate with Gain,” Nano Lett. 11(6), 2231–2235 (2011).
[Crossref] [PubMed]

P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, “Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length,” Opt. Lett. 35(8), 1197–1199 (2010).
[Crossref] [PubMed]

Bonnand, C.

J. Bellessa, C. Bonnand, J. C. Plenet, and J. Mugnier, “Strong Coupling between Surface Plasmons and Excitons in an Organic Semiconductor,” Phys. Rev. Lett. 93(3), 036404 (2004).
[Crossref] [PubMed]

Boriskina, S. V.

Bröll, M.

F. Valmorra, M. Bröll, S. Schwaiger, N. Welzel, D. Heitmann, and S. Mendach, “Strong coupling between surface plasmon polariton and laser dye rhodamine 800,” Appl. Phys. Lett. 99(5), 051110 (2011).
[Crossref]

Chen, G.

Chen, J.

J. Chen, P. Wang, X. Wang, Y. Lu, R. Zheng, H. Ming, and Q. Zhan, “Optical bistability enhanced by highly localized bulk plasmon polariton modes in subwavelength metal-nonlinear dielectric multilayer structure,” Appl. Phys. Lett. 94(8), 081117 (2009).
[Crossref]

Chen, X.

Chen, Y.

Y. Chen, J. Li, M. Ren, B. Wang, J. Fu, S. Liu, and Z. Li, “Direct observation of amplified spontaneous emission of surface plasmon polaritons at metal/dielectric interfaces,” Appl. Phys. Lett. 98(26), 261912 (2011).
[Crossref]

Chuang, S. L.

D. Ahn and S. L. Chuang, “High optical gain of I–VII semiconductor quantum wells for efficient light-emitting devices,” Appl. Phys. Lett. 102(12), 121114 (2013).
[Crossref]

Ciracì, C.

C. Argyropoulos, C. Ciracì, and D. R. Smith, “Enhanced optical bistability with film-coupled plasmonic nanocubes,” Appl. Phys. Lett. 104(6), 063108 (2014).
[Crossref]

Cooper, T. A.

Dai, X.

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Scientific Reports 5, 12271 (2015).
[Crossref] [PubMed]

De Leon, I.

P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. 6(1), 16–24 (2012).
[Crossref]

I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon. 4(6), 382–387 (2010).
[Crossref]

Deck, R. T.

Dickson, W.

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-Plasmonic Modulation via Stimulated Emission of Copropagating Surface Plasmon Polaritons on a Substrate with Gain,” Nano Lett. 11(6), 2231–2235 (2011).
[Crossref] [PubMed]

P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, “Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length,” Opt. Lett. 35(8), 1197–1199 (2010).
[Crossref] [PubMed]

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010), Chap. 3–4.
[Crossref]

Eng, L.

J. Seidel, S. Grafström, and L. Eng, “Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution,” Phys. Rev. Lett. 94(17), 177401 (2005).
[Crossref] [PubMed]

Fainman, Y.

Feng, C.

Fröb, H.

C. Tzschaschel, M. Sudzius, A. Mischok, H. Fröb, and K. Leo, “Net gain in small mode volume organic microcavities,” Appl. Phys. Lett. 108(2), 023304 (2016).
[Crossref]

Fu, J.

Y. Chen, J. Li, M. Ren, B. Wang, J. Fu, S. Liu, and Z. Li, “Direct observation of amplified spontaneous emission of surface plasmon polaritons at metal/dielectric interfaces,” Appl. Phys. Lett. 98(26), 261912 (2011).
[Crossref]

Fu, X.

Fujimura, R.

P. T. Thanh, D. Tanaka, R. Fujimura, Y. Takanishi, and K. Kajikawa, “Low-Power All-Optical Bistable Device of Twisted-Nematic Liquid Crystal Based on Surface Plasmons in a Metal-Insulator-Metal Structure,” Appl. Phys. Express 6(1), 011701 (2013).
[Crossref]

Grafström, S.

J. Seidel, S. Grafström, and L. Eng, “Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution,” Phys. Rev. Lett. 94(17), 177401 (2005).
[Crossref] [PubMed]

Gupta, S. D.

Hakala, T. K.

T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi Splitting and Strong-Coupling Dynamics for Surface-Plasmon Polaritons and Rhodamine 6G Molecules,” Phys. Rev. Lett. 103(5), 053602 (2009).
[Crossref] [PubMed]

Hayashi, S.

S. Hayashi and T. Okamoto, “Plasmonics: visit the past to know the future,” J. Phys. D: Appl. Phys. 45(43), 433001 (2012).
[Crossref]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (2nd Ed., Cambridge, 2012), Chap. 12.
[Crossref]

Heitmann, D.

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C. Tzschaschel, M. Sudzius, A. Mischok, H. Fröb, and K. Leo, “Net gain in small mode volume organic microcavities,” Appl. Phys. Lett. 108(2), 023304 (2016).
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F. Valmorra, M. Bröll, S. Schwaiger, N. Welzel, D. Heitmann, and S. Mendach, “Strong coupling between surface plasmon polariton and laser dye rhodamine 800,” Appl. Phys. Lett. 99(5), 051110 (2011).
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Y. Chen, J. Li, M. Ren, B. Wang, J. Fu, S. Liu, and Z. Li, “Direct observation of amplified spontaneous emission of surface plasmon polaritons at metal/dielectric interfaces,” Appl. Phys. Lett. 98(26), 261912 (2011).
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F. Valmorra, M. Bröll, S. Schwaiger, N. Welzel, D. Heitmann, and S. Mendach, “Strong coupling between surface plasmon polariton and laser dye rhodamine 800,” Appl. Phys. Lett. 99(5), 051110 (2011).
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G. A. Wurtz, R. Pollard, and A.V. Zayats, “Optical Bistability in Nonlinear Surface-Plasmon Polaritonic Crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
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Adv. Opt. Photon. (3)

Appl. Opt. (1)

Appl. Phys. Express (1)

P. T. Thanh, D. Tanaka, R. Fujimura, Y. Takanishi, and K. Kajikawa, “Low-Power All-Optical Bistable Device of Twisted-Nematic Liquid Crystal Based on Surface Plasmons in a Metal-Insulator-Metal Structure,” Appl. Phys. Express 6(1), 011701 (2013).
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Appl. Phys. Lett. (7)

F. Valmorra, M. Bröll, S. Schwaiger, N. Welzel, D. Heitmann, and S. Mendach, “Strong coupling between surface plasmon polariton and laser dye rhodamine 800,” Appl. Phys. Lett. 99(5), 051110 (2011).
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J. Chen, P. Wang, X. Wang, Y. Lu, R. Zheng, H. Ming, and Q. Zhan, “Optical bistability enhanced by highly localized bulk plasmon polariton modes in subwavelength metal-nonlinear dielectric multilayer structure,” Appl. Phys. Lett. 94(8), 081117 (2009).
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IEEE J. Quantum Electron. (1)

P. Martinot, A. Koster, and S. Laval, “Experimental Observation of Optical Bistability by Excitation of a Surface Plasmon Wave,” IEEE J. Quantum Electron. 21(8), 1140–1143 (1985).
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J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

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

S. Hayashi and T. Okamoto, “Plasmonics: visit the past to know the future,” J. Phys. D: Appl. Phys. 45(43), 433001 (2012).
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Nano Lett. (1)

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-Plasmonic Modulation via Stimulated Emission of Copropagating Surface Plasmon Polaritons on a Substrate with Gain,” Nano Lett. 11(6), 2231–2235 (2011).
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Opt. Express (5)

Opt. Lett. (6)

Phys. Rep. (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3–4), 131–314 (2005).
[Crossref]

Phys. Rev. A (1)

A. V. Malyshev, “Condition for resonant optical bistability,” Phys. Rev. A 86(6), 065804 (2012).
[Crossref]

Phys. Rev. Lett. (5)

G. A. Wurtz, R. Pollard, and A.V. Zayats, “Optical Bistability in Nonlinear Surface-Plasmon Polaritonic Crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
[Crossref] [PubMed]

J. Bellessa, C. Bonnand, J. C. Plenet, and J. Mugnier, “Strong Coupling between Surface Plasmons and Excitons in an Organic Semiconductor,” Phys. Rev. Lett. 93(3), 036404 (2004).
[Crossref] [PubMed]

J. Seidel, S. Grafström, and L. Eng, “Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution,” Phys. Rev. Lett. 94(17), 177401 (2005).
[Crossref] [PubMed]

T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi Splitting and Strong-Coupling Dynamics for Surface-Plasmon Polaritons and Rhodamine 6G Molecules,” Phys. Rev. Lett. 103(5), 053602 (2009).
[Crossref] [PubMed]

M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, “Stimulated Emission of Surface Plasmon Polaritons,” Phys. Rev. Lett. 101(22), 226806 (2008).
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Prog. Quant. Electr. (1)

R. Reinisch and G. Vitrant, “Optical Bistability,” Prog. Quant. Electr. 18, 1–38 (1994).
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Rep. Prog. Phys. (2)

N. C. Lindquist, P. Nagpal, K. M. McPeak, D. J. Norris, and Sang-Hyun Oh, “Engineering metallic nanostructures for plasmonics and nanophotonics,” Rep. Prog. Phys. 75(3), 036501 (2012).
[Crossref] [PubMed]

P. Törmä and W. L. Barnes, “Strong coupling between surface plasmon polaritons and emitters: a review,” Rep. Prog. Phys. 78(1), 013901 (2015).
[Crossref]

Science (1)

E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

Scientific Reports (1)

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Scientific Reports 5, 12271 (2015).
[Crossref] [PubMed]

Z. Physik (1)

E. Kretschmann, “The Determination of the Optical Constant of Metals by Excitation of Surface Plasmons,” Z. Physik 241, 313–324 (1971).
[Crossref]

Other (6)

M. A. Parker, Physics of Optoelectronics (CRC, 2005), Chap. 7.
[Crossref]

K. F. Renk, Basics of Laser Physics: For Students of Science and Engineering (Springer, 2012), Chap. 4.
[Crossref]

P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010), Chap. 3–4.
[Crossref]

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).
[Crossref]

L. Novotny and B. Hecht, Principles of Nano-Optics (2nd Ed., Cambridge, 2012), Chap. 12.
[Crossref]

V. M. Shalaev and S. Kawata, Nanophotoncis with surface plasmons (Elsevier, 2007).

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Figures (6)

Fig. 1
Fig. 1 Reflectivity R as a function of the incident angle θ for different thicknesses of metal film. Here i is the imaginary part of d, θc is the critical angle for total internal reflection, and θSPP is the surface plasmon resonance angle. The inset is the Kretschmann configuration for SPPs excitation. For all data sets [17]: 0 = 3.18, m = −15.584 + 0.424i, d = 2.25, and the wavelength of the incident TM-polarized light beam λ = 594nm.
Fig. 2
Fig. 2 Reflectivity R as a function of small-signal gain (associated with i by Eq. (11)) near the surface plasmon resonance angle θSPP; (a) dm = 39nm, (b) dm = 60nm, and (c) dm = 70nm. The color bar on the right represents log10(R). Other parameters are mentioned in the text.
Fig. 3
Fig. 3 Reflectivity curve variation with the incident angle θ at a fixed thickness of metal film dm = 60nm when i = 0. Assume that i = −0.020 and Is = 1MW/cm2. The inset (a) depicts the curves of ETR changes as the variations of the input intensity I in the case of θ < θSPP, while the inset (b) corresponds to θ > θSPP. Other parameters are mentioned in the text.
Fig. 4
Fig. 4 ETR as a function of the input intensity I for different thicknesses of metal film when θ = 65.46° and i = −0.020. Other parameters are mentioned in the text.
Fig. 5
Fig. 5 ETR as a function of the input intensity I for different values of small-signal gain coefficient when dm = 60nm and θ = 65.46°. Other parameters are mentioned in the text.
Fig. 6
Fig. 6 ETR as a function of the input intensity I in a HB (blue-dashed curve) and an inHB (red-solid curve) gain material when dm = 56nm, i = −0.020, and θ = 65.40°. Assume that the saturation intensity in an inHB gain medium is Is = 1MW/cm2. Other parameters are mentioned in the text.

Equations (22)

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k SPP = ω c m d m + d ,
ω c 0 sin ( θ SPP ) = Re ( k SPP ) ,
R = | r 0 m + r md exp ( 2 i k m z d m ) 1 + r 0 m r md exp ( 2 i k m z d m ) | 2 ,
T = | t 0 m t md exp ( i k m z d m ) 1 + r 0 m r md exp ( 2 i k m z d m ) | 2 .
k i z = ± ω c i 0 sin 2 θ , i = 0 , m , d .
P = Ne 2 Vm 1 ( ω 0 2 ω 2 ) + i γ ω E ,
P = ε 0 χ E ,
χ ( ω ) = Ne 2 Vm ε 0 1 ( ω 0 2 ω 2 ) + i γ ω .
χ r = Re χ ( ω ) N e 2 V m ε 0 ω 0 2 ( ω 0 ω ) 4 ( ω 0 ω ) 2 + γ 2 ,
χ i = Im χ ( ω ) N e 2 V m ε 0 ω 0 γ 4 ( ω 0 ω ) 2 + γ 2 .
d = 1 + χ b + χ ( ω ) .
r = 1 + χ b + χ r , i = χ i .
g = ω c i r .
g 0 = Δ N σ ( v , v 0 ) ,
Δ r Δ is = Δ χ r Δ χ i = 2 ( ω 0 ω ) γ .
g L ( v ) = g 0 1 + I d f ( v ) I s f ( v 0 ) .
g L = g 0 1 + I d / I s .
d = r + i i 1 + I d / I s .
I d ¯ 1 2 I md = 1 4 1 Re ( d ) ε 0 c | H md | 2 .
I = 1 2 1 ( 0 ) ε 0 c | H md | 2 T .
g G = g 0 1 + I d / I s .
d = r + i i 1 + I d / I s .

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