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Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides

Maziar P. Nezhad, Kevin Tetz, and Yeshaiahu Fainman

Open Access Open Access

Abstract

The propagation of surface plasmon polaritons on metallic waveguides adjacent to a gain medium is considered. It is shown that the presence of the gain medium can compensate for the absorption losses in the metal. The conditions for existence of a surface plasmon polariton and its lossless propagation and wavefront behavior are derived analytically for a single infinite metal-gain boundary. In addition, the cases of thin slab and stripe geometries are also investigated using finite element simulations. The effect of a finite gain layer and its distance from the SPP waveguide is also investigated. The calculated gain requirements suggest that lossless gain-assisted surface plasmon polariton propagation can be achieved in practice for infrared wavelengths.

©2004 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. Schematic illustration of various SPP propagation regimes as a function of ε″1.

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Fig. 2.
Fig. 2. Plots of: (a) Im(kx ), (b) propagation length, (c) wavefront tilt angle and (d) Im( k z 1 ) versus gain. The points corresponding to lossless propagation, zero wavefront tilt and bound surface wave limit, respectively, are also shown.

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Fig. 3.
Fig. 3. FEA simulations of total electric field for SPPs propagating on a silver interface embedded in an InGaAsP-based gain medium: (a) Symmetric mode in slab configuration without gain, kx =14.06+i0.0197 µm-1. (b) Symmetric mode in slab configuration with gain, kx =14.06 µm-1. (c) Symmetric mode in stripe configuration without gain, kx =13.76+i0.0094 µm -1. (d) Symmetric mode in stripe configuration with gain, kx =13.76 µm-1.

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Fig. 4.
Fig. 4. (a) Metallic stripe waveguide of Fig. 3 in proximity to a gain layer with finite thickness. (b) FEA generated results showing variation of gain required for lossless propagation as the gap d increases. Each curve corresponds to a different value of gain layer thickness h.

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Equations (12)

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{ E j = ( E x , 0 , E z j ) exp ( i ( k x x + k z j z − ω t ) ) H j = ( 0 , H y , 0 ) exp ( i ( k x x + k z j z − ω t ) ) , j = 1 , 2
{ ε 1 k z 1 = ε 2 k z 2 k z 2 + k z i 2 = ε i k 0 2 E x = k z i ω ε i H y , E z i = − k x ω ε i H y i = 1 , 2
{ k x 2 = k 0 2 ε 1 ε 2 ε 1 + ε 2 ( a ) k z i 2 = k 0 2 ε i 2 ε 1 + ε 2 ( b )
{ k x 2 = k 0 2 ( ε 1 ′ + i ε 1 ″ ) ( ε 2 ′ + i ε 2 ″ ) ( ε 1 ′ + ε 2 ′ ) + i ( ε 1 ″ + ε 2 ″ ) ( a ) k z i 2 = k 0 2 ( ε i ′ + i ε i ″ ) 2 ( ε 1 ′ + ε 2 ′ ) + i ( ε 1 ″ + ε 2 ″ ) ( b )
k z 1 ≃ k 0 ∣ ε 1 ′ + ε 2 ′ ∣ ( ε 1 ′ + ε 2 ′ ) 2 + ( ε 1 ″ + ε 2 ″ ) 2 ( − ε 1 ″ + i ε 1 ′ ) ( 1 − i ( ε 1 ″ + ε 2 ″ ) 2 ( ε 1 ′ + ε 2 ′ ) )
( ε 1 ″ ) 2 + ε 2 ″ ε 1 ″ + 2 ε 1 ′ ( ε 1 ′ + ε 2 ′ ) < 0
− ε 2 ″ − ( ε 2 ″ ) 2 − 8 ε 1 ′ ( ε 1 ′ + ε 2 ′ ) 2 < ε 1 ″ < − ε 2 ″ + ( ε 2 ″ ) 2 − 8 ε 1 ′ ( ε 1 ′ + ε 2 ′ ) 2
k x 2 = k 0 2 ( ε 1 ′ + ε 2 ′ ) 2 + ( ε 1 ″ + ε 2 ″ ) 2 [ ε 1 ′ ( ( ε 2 ′ ) 2 + ∣ ε 1 ∣ 2 ε 1 ′ ε 2 ′ + ( ε 2 ″ ) 2 ) + i ε 2 ″ ( ( ε 1 ″ ) 2 + ∣ ε 2 ∣ 2 ε 2 ″ ε 1 ″ + ( ε 1 ′ ) 2 ) ]
ε 1 ″ = ∣ ε 2 ∣ 2 2 ε 2 ″ ( − 1 ± 1 − 4 ( ε 1 ′ ε 2 ″ ) 2 ∣ ε 2 ∣ 4 ) ≃ { − ∣ ε 2 ∣ 2 ε 2 ″ + ( ε 1 ′ ) 2 ε 2 ″ ∣ ε 2 ∣ 2 ( a ) − ( ε 1 ′ ) 2 ε 2 ″ ∣ ε 2 ∣ 2 ( b )
γ 0 = 2 π λ 0 ε 2 ″ ( ε 1 ′ ) 3 2 ( ε 2 ′ ) 2 + ( ε 2 ″ ) 2
P = 1 2 Re ( E 1 × H * ) = ∣ H y ∣ 2 2 ω [ Re ( k x ε 1 ) x ̂ + Re ( k z 1 ε 1 ) z ̂ ]
Re ( k z 1 ε 1 ) = Re ( k 0 ε 1 ′ + ε 2 ′ + i ( ε 1 ″ + ε 2 ″ ) ) ≃ k 0 ( ε 1 ″ + ε 2 ″ ) 2 ( ∣ ε 1 ′ + ε 2 ′ ∣ ) 3 2
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