Abstract

We describe the effects of nonlinearity on propagation of surface plasmon polaritons (SPPs) at an interface between a metal and an amplifying medium of the externally pumped two-level atoms. Using Maxwell equations we derive the nonlinear dispersion law and demonstrate that the nonlinear saturation of the linear gain leads to formation of stationary SPP modes with the intensities independent from the propagation distance. Transition to the regime of stationary propagation is similar to the threshold crossover in lasers and leads to narrowing of the SPP spectrum.

© 2009 Optical Society of America

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  1. D. J. Bergman and M. I. Stockman, Phys. Rev. Lett. 90, 027402 (2003).
    [CrossRef] [PubMed]
  2. N. I. Zheludev, V. Prosvirnin, N. Papasimakis, and V. A. Fedotov, Nat. Photonics 2, 351 (2008).
    [CrossRef]
  3. M. P. Nezhad, K. Tetz, and Y. Fainman, Opt. Express 12, 4072 (2004).
    [CrossRef] [PubMed]
  4. I. De Leon and P. Berini, Phys. Rev. B 78, 161401(R) (2008).
    [CrossRef]
  5. J. Seidel, S. Grafstrom, and L. Eng, Phys. Rev. Lett. 94, 177401 (2005).
    [CrossRef] [PubMed]
  6. G. Winter, S. Wedge, and W. Barnes, New J. Phys. 8, 125 (2006).
    [CrossRef]
  7. M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, Phys. Rev. Lett. 101, 226806 (2008).
    [CrossRef] [PubMed]
  8. M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nano Lett. 8, 3998 (2008).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2008 (4)

M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, Phys. Rev. Lett. 101, 226806 (2008).
[CrossRef] [PubMed]

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nano Lett. 8, 3998 (2008).
[CrossRef] [PubMed]

I. De Leon and P. Berini, Phys. Rev. B 78, 161401(R) (2008).
[CrossRef]

N. I. Zheludev, V. Prosvirnin, N. Papasimakis, and V. A. Fedotov, Nat. Photonics 2, 351 (2008).
[CrossRef]

2006 (1)

G. Winter, S. Wedge, and W. Barnes, New J. Phys. 8, 125 (2006).
[CrossRef]

2005 (1)

J. Seidel, S. Grafstrom, and L. Eng, Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (2)

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

D. J. Bergman and M. I. Stockman, Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

1987 (1)

Ambati, M.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nano Lett. 8, 3998 (2008).
[CrossRef] [PubMed]

Barnes, W.

G. Winter, S. Wedge, and W. Barnes, New J. Phys. 8, 125 (2006).
[CrossRef]

Bartal, G.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nano Lett. 8, 3998 (2008).
[CrossRef] [PubMed]

Bergman, D. J.

D. J. Bergman and M. I. Stockman, Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Berini, P.

I. De Leon and P. Berini, Phys. Rev. B 78, 161401(R) (2008).
[CrossRef]

Boardman, A. D.

Boyd, R. W.

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

De Leon, I.

I. De Leon and P. Berini, Phys. Rev. B 78, 161401(R) (2008).
[CrossRef]

Eng, L.

J. Seidel, S. Grafstrom, and L. Eng, Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

Fainman, Y.

Fedotov, V. A.

N. I. Zheludev, V. Prosvirnin, N. Papasimakis, and V. A. Fedotov, Nat. Photonics 2, 351 (2008).
[CrossRef]

Genov, D. A.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nano Lett. 8, 3998 (2008).
[CrossRef] [PubMed]

Grafstrom, S.

J. Seidel, S. Grafstrom, and L. Eng, Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

Mayy, M.

M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, Phys. Rev. Lett. 101, 226806 (2008).
[CrossRef] [PubMed]

Mihalache, D.

Nam, S. H.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nano Lett. 8, 3998 (2008).
[CrossRef] [PubMed]

Nezhad, M. P.

Noginov, M. A.

M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, Phys. Rev. Lett. 101, 226806 (2008).
[CrossRef] [PubMed]

Noginova, N.

M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, Phys. Rev. Lett. 101, 226806 (2008).
[CrossRef] [PubMed]

Papasimakis, N.

N. I. Zheludev, V. Prosvirnin, N. Papasimakis, and V. A. Fedotov, Nat. Photonics 2, 351 (2008).
[CrossRef]

Podolskiy, V. A.

M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, Phys. Rev. Lett. 101, 226806 (2008).
[CrossRef] [PubMed]

Prosvirnin, V.

N. I. Zheludev, V. Prosvirnin, N. Papasimakis, and V. A. Fedotov, Nat. Photonics 2, 351 (2008).
[CrossRef]

Ritzo, B. A.

M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, Phys. Rev. Lett. 101, 226806 (2008).
[CrossRef] [PubMed]

Seaton, C. T.

Seidel, J.

J. Seidel, S. Grafstrom, and L. Eng, Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

Stegeman, G. I.

Stockman, M. I.

D. J. Bergman and M. I. Stockman, Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Tetz, K.

Twardowski, T.

Ulin-Avila, E.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nano Lett. 8, 3998 (2008).
[CrossRef] [PubMed]

Wedge, S.

G. Winter, S. Wedge, and W. Barnes, New J. Phys. 8, 125 (2006).
[CrossRef]

Winter, G.

G. Winter, S. Wedge, and W. Barnes, New J. Phys. 8, 125 (2006).
[CrossRef]

Wright, E. M.

Zanoni, R.

Zhang, X.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nano Lett. 8, 3998 (2008).
[CrossRef] [PubMed]

Zheludev, N. I.

N. I. Zheludev, V. Prosvirnin, N. Papasimakis, and V. A. Fedotov, Nat. Photonics 2, 351 (2008).
[CrossRef]

Zhu, G.

M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, Phys. Rev. Lett. 101, 226806 (2008).
[CrossRef] [PubMed]

Nano Lett. (1)

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nano Lett. 8, 3998 (2008).
[CrossRef] [PubMed]

Nat. Photonics (1)

N. I. Zheludev, V. Prosvirnin, N. Papasimakis, and V. A. Fedotov, Nat. Photonics 2, 351 (2008).
[CrossRef]

New J. Phys. (1)

G. Winter, S. Wedge, and W. Barnes, New J. Phys. 8, 125 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (1)

I. De Leon and P. Berini, Phys. Rev. B 78, 161401(R) (2008).
[CrossRef]

Phys. Rev. Lett. (3)

J. Seidel, S. Grafstrom, and L. Eng, Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, Phys. Rev. Lett. 101, 226806 (2008).
[CrossRef] [PubMed]

D. J. Bergman and M. I. Stockman, Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Other (1)

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

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Figures (3)

Fig. 1
Fig. 1

(a) Normalized SPP intensity as a function of detuning δ. Line 1 is the spectrum of stationary (saturated) SPP, I s ( δ ) : α = 1.1 α min . Line 2 is the spectrum of the amplified linear SPP, I l ( δ ) , below the threshold (propagation distance 60 μm, α = 0.8 α min ). (b) FWHM of the SPP spectra versus gain. Line 1 is for the stationary saturated SPPs, I s . Lines 2 (propagation distance 60 μm) and 3 (distance 100 μm) are for the nonstationary linear SPPs, I l .

Fig. 2
Fig. 2

Dependence of the intensity of the stationary SPPs above the threshold on the gain parameter α α min . Line 1 corresponds to δ = 0 , line 2 corresponds to δ = 0.4 , and line 3 corresponds to δ = 0.8 .

Fig. 3
Fig. 3

Dependence of the (a) real and (b) imaginary parts of the SPP propagation constants on the gain parameter for δ = 0 . Solid lines correspond to the stationary nonlinear SPPs ( β s ) , and the dashed lines correspond to the linear SPPs ( β l ) .

Equations (11)

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z z E x z x E z = D x ,
z x E x x x E z = D z .
β 2 E x + i β x E z = [ ε d + γ d ( | E x | 2 + | E z | 2 ) ] E x ,
i β x E x x x E z = [ ε d + γ d ( | E x | 2 + | E z | 2 ) ] E z .
β ε m E z 0 = i q m [ ε d + γ d ( | E x 0 | 2 + | E z 0 | 2 ) ] E x 0 ,
E x = A e q d x { 1 + w x γ d | A | 2 e 2 x Re q d + O ( | γ d | 2 ) } ,
E z = q d i β A e q d x { 1 + w y γ d | A | 2 e 2 x Re q d + O ( | γ d | 2 ) } .
ε m q d + ε d q m = γ d | A | 2 F + O ( | γ d | 2 ) .
F ( | q d | 2 β 2 + 1 ) q d ε m + q m ε d + 2 Re ( q d ) ( β 2 ε m ε d q m q d ) 4 Re ( q d ) ( Re ( q d ) + q d ) .
β = β l + β n l | A | 2 , β n l β l γ d 2 ε d 2 q d ( | q d | 2 + β l 2 ) Re ( q d ) + q d ,
I s = ( α α 0 ) Im α β l ( Im β n l ) ,

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