Abstract

In this paper, a numerical analysis to model the amplification of nonlinear surface plasmon polaritons (SPPs) propagating along a metal active dielectric interface is presented. Nonlinearity of the medium with gain is considered through the full saturation susceptibility parameter of a two-level model. Above the linear threshold of the gain and loss balance, the nonlinear performance of the amplified SPPs is verified, which agrees well with the experimental results reported in the literature. An important feature of the proposed method is to predict an abrupt increase in output intensity due to amplified spontaneous emission.

© 2014 Optical Society of America

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References

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  1. R. Zia, M. Selker, P. Catrysse, and M. L. Brongersma, “Geometries and materials for sub-wavelength surface plasmon modes,” J. Opt. Soc. Am. A 21, 2442–2446 (2004).
    [CrossRef]
  2. P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14, 13030–13042 (2006).
    [CrossRef]
  3. I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 37, 1–6 (2010).
  4. F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. ’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110, 206802 (2013).
    [CrossRef]
  5. R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
    [CrossRef]
  6. 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, 1197–1199 (2010).
    [CrossRef]
  7. M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12, 4072–4079 (2004).
    [CrossRef]
  8. 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, 226806 (2008).
    [CrossRef]
  9. A. Marini, A. V. Gorbach, D. V. Skryabin, and A. V. Zayats, “Amplification of surface plasmon polaritons in the presence of nonlinearity and spectral signatures of threshold crossover,” Opt. Lett. 34, 2864–2866 (2009).
    [CrossRef]
  10. R. W. Boyd, Nonlinear Optics (Academic, 2003).
  11. M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics 4, 457–461 (2010).
    [CrossRef]

2013 (1)

F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. ’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110, 206802 (2013).
[CrossRef]

2010 (3)

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics 4, 457–461 (2010).
[CrossRef]

I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 37, 1–6 (2010).

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, 1197–1199 (2010).
[CrossRef]

2009 (2)

A. Marini, A. V. Gorbach, D. V. Skryabin, and A. V. Zayats, “Amplification of surface plasmon polaritons in the presence of nonlinearity and spectral signatures of threshold crossover,” Opt. Lett. 34, 2864–2866 (2009).
[CrossRef]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

2008 (1)

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, 226806 (2008).
[CrossRef]

2006 (1)

2004 (2)

’t Hooft, G. W.

F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. ’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110, 206802 (2013).
[CrossRef]

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Berini, P.

I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 37, 1–6 (2010).

P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14, 13030–13042 (2006).
[CrossRef]

Bolger, P. M.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Brongersma, M. L.

Catrysse, P.

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Danz, N.

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics 4, 457–461 (2010).
[CrossRef]

de Dood, M. J. A.

F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. ’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110, 206802 (2013).
[CrossRef]

Dickson, W.

Fainman, Y.

Gather, M. C.

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics 4, 457–461 (2010).
[CrossRef]

Geluk, E. J.

F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. ’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110, 206802 (2013).
[CrossRef]

Gladden, C.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Gorbach, A. V.

Hickey, S. G.

Krasavin, A. V.

Leon, I. D.

I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 37, 1–6 (2010).

Leosson, K.

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics 4, 457–461 (2010).
[CrossRef]

Liebscher, L.

Ma, R. M.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Marini, A.

Mayy, M.

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, 226806 (2008).
[CrossRef]

Meerholz, K.

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics 4, 457–461 (2010).
[CrossRef]

Nezhad, M. P.

Noginov, M. A.

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, 226806 (2008).
[CrossRef]

Noginova, N.

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, 226806 (2008).
[CrossRef]

Oulton, R. F.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Podolskiy, V. A.

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, 226806 (2008).
[CrossRef]

Ritzo, B. A.

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, 226806 (2008).
[CrossRef]

Selker, M.

Skryabin, D. V.

Sorger, V. J.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Tetz, K.

van Beijnum, F.

F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. ’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110, 206802 (2013).
[CrossRef]

van Exter, M. P.

F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. ’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110, 206802 (2013).
[CrossRef]

van Veldhoven, P. J.

F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. ’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110, 206802 (2013).
[CrossRef]

Zayats, A. V.

Zentgraf, T.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Zhang, X.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Zhu, G.

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, 226806 (2008).
[CrossRef]

Zia, R.

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

Nat. Photonics (2)

I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 37, 1–6 (2010).

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics 4, 457–461 (2010).
[CrossRef]

Nature (1)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. Lett. (2)

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, 226806 (2008).
[CrossRef]

F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. ’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110, 206802 (2013).
[CrossRef]

Other (1)

R. W. Boyd, Nonlinear Optics (Academic, 2003).

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Figures (3)

Fig. 1.
Fig. 1.

Nonlinear electric field profile (a) for different values of α(δ=0), α=1.1α0 (red solid lines), α=2α0 (blue dashed lines), and α=5α0 (black dotted lines) and (b) for different values of δ(α=2α0), δ=0 (red solid lines), δ=0.4 (blue dashed lines), and δ=0.8 (black dotted lines).

Fig. 2.
Fig. 2.

Dependence of electric field intensity on gain parameter α/α0 for δ=0; numerical analysis (red solid line) and perturbative theory (blue dashed lines).

Fig. 3.
Fig. 3.

Dependency of β on gain parameter α/α0 for δ=0; numerical analysis (red solid lines) and perturbative theory (dashed blue lines).

Equations (23)

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×E⃗=icμ0H⃗,
×H⃗=icϵ0ϵE⃗,
zx2Ezzz2Ex=Dx,
zx2Exxx2Ez=Dz.
Ex=iβqm2xEz,
xxEzqm2Ez=0,
Ex(x)=iβqmEz0eqmx,
Ez(x)=Ez0eqmx,
χ=αδi1+δ2+|E⃗Es|2,
χαδi1+δ2+αiδ(1+δ2)2|E⃗Es|2.
D⃗=[ϵb+αδi1+δ2+(|Ex(x)|2+|Ez(x)|2)|eβz|2]E⃗,
β2Ex+iβxEz=[ϵb+αδi1+δ2+(|Ex(x)|2+|Ez(x)|2)]Ex,
iβxExxxEz=[ϵb+αδi1+δ2+(|Ex(x)|2+|Ez(x)|2)]Ez.
iβϵmqmEz0=[ϵb+αδi1+δ2+(|Ex(x)|2+|Ez(x)|2)]Ex0.
xEz=1iβ[ϵb+αδi1+δ2+(|Ex(x)|2+|Ez(x)|2)β2]Ex,
xEx=[PN+Q1PM]+i[M(PN+Q1PM)+N],
M=2[Re(B)Im(Ex)+Im(B)Re(Ex)]Re(Ex)1+2Im(Ex)[Re(B)Im(Ex)+Im(B)Re(Ex)],
N=βRe(Ez)2Re(Ez*Ex(A+iβ))1[Im(B)Im(Ex)+Re(B)Re(Ex)],×[Re(B)Im(Ex)Im(B)Re(Ex)],
P=Im(B)Re(Ex)+Re(B)Im(Ex)1+Re(B)Re(Ex)Im(B)Im(Ex),
Q=βIm(Ez)+2Re(Ez*Ex(A+iβ))1+Re(B)Re(Ex)Im(B)Im(Ex),×[Im(B)Im(Ex)Re(B)Re(Ex)],
A=1iβ[ϵb+αδi1+δ2+(|Ex(x)|2+|Ez(x)|2)],
B=α(iδ)[ϵb+αδi1+δ2+(|Ex|2+|Ez|2)]×1[1+δ2+(|Ex|2+|Ez|2)]2.
f(Ez0,β)=xixj(|Ex|2+|Ez|2)dx.

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