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Dynamic near-field calculations of surface-plasmon polariton pulses resonantly scattered at sub-micron metal defects

José A. Sánchez-Gil and Alexei A. Maradudin

Open Access Open Access

Abstract

We investigate theoretically the near-field dynamics of the scattering of a surface-plasmon polariton (SPP) pulse impinging normally on a rectangular groove on an otherwise planar metal surface. Our formulation is based on solving the reduced Rayleigh equation (derived through the use of an impedance boundary condition) for every component of the spectral decomposition of the incoming SPP pulse. Numerical calculations are carried out of the time dependence of the near-field resonant scattering effects produced at the rectangular groove. The scattering process is tracked through the (time-resolved) repartition of the incoming SPP electromagnetic energy into reflected and transmitted SPP pulses, and into pulsed scattered light. Furthermore, we directly show evidence of the excitation of single resonances, as manifested by the concentration of electric field intensity within the groove, and its subsequent leakage, over the resonance lifetime. The near-field formation of oscillations caused by the interference between two adjacent resonances simultaneously excited is also considered.

©2004 Optical Society of America

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Supplementary Material (4)

Media 1: AVI (304 KB)     
Media 2: AVI (255 KB)     
Media 3: AVI (320 KB)     
Media 4: AVI (244 KB)     

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Figures (9)
Fig. 1.
Fig. 1. Illustration of the scattering geometry. The dashed rectangle shows the configuration of the near-field region scanned in Figs. 3, 4, 7 and 9.

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Fig. 2.
Fig. 2. Spectral dependence of the monochromatic, SPP reflection (green curve) and transmission (red curve) coefficients (R SPP and T SPP, respectively), and total radiated energy S (blue curve) for a rectangular defect of half-width L=789 nm. Solid curves: Groove, h=-157 nm; Dashed curves: Ridge, h=157 nm. λp =2πc/ωp =157 nm (Ag). The spectral amplitudes (normalized to 1) of the SPP pulses considered below are superimposed (black curves).

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Fig. 3.
Fig. 3. (308 KB) Movie of the time evolution of the near (electric) field intensity (log scale) images in an area of 8λ×4λ perpendicular to the Ag surface for a rectangular groove (located at the bottom center, see Fig. 1) of half-width L=785 nm and depth h=-157 nm. The incident SPP pulse parameters are: ω 0/ωp =0.275 (λ0=571 nm) and Δω/ω 0=0.02. Front picture: t=0 (incoming SPP pulse maximum at x 1=0).

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Fig. 4.
Fig. 4. (260 KB) Same as in Fig. 3 but for a rectangular ridge (h=157 nm).

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Fig. 5.
Fig. 5. Pulse amplitudes for ω 0/ωp =0.275 and Δω/ω 0=0.035 at a distance x=400λ 0 scattered from rectangular defects of half-width L=785 nm and height h=±157 nm: Reflected (green curve) and transmitted (red curve) SPP, and radiated (blue curve) at θ=θmax . (a) Groove, θmax =46.6°; (b) Ridge, θmax =62.5°. The freely propagating SPP pulse is also shown (black curve).

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Fig. 6.
Fig. 6. Time dependence of the angular distribution of intensity scattered (for ω 0/ωp =0.275) from rectangular defects of half-width L=785 nm and height h=±157 nm. (a) Groove; (b) Ridge.

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Fig. 7.
Fig. 7. (328 KB) Same as in Fig. 3 but for different incident SPP pulse parameters: ω 0/ωp =0.247 (λ 0=636 nm) and Δω/ω 0=0.035.

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Fig. 8.
Fig. 8. Same as in Fig. 5(a) but for different incident SPP pulse parameters: ω 0/ωp =0.247 and Δω/ω 0=0.035. θmax =69.6°.

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Fig. 9.
Fig. 9. (252 KB) Movie of the time evolution of the near (electric) field intensity (log scale) images in an area of 8λ×4λ perpendicular to the Ag surface for a Gaussian groove (located at the bottom center, see Fig. 1) of 1/e-width a=157 nm and height δ=-785 nm. The incident SPP pulse parameters are: ω 0/ωp =0.239 (λ 0=657 nm) and Δω/ω 0=0.035. Front picture: t=0 (incoming SPP pulse maximum at x 1=0).

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Equations (17)

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H 2 ( i ) ( x 1 , x 3 ; t ) = ∫ − ∞ ∞ d ω F ( ω ) exp [ ik ( ω ) x 1 − β 0 ( ω ) x 3 ] exp ( − i ω t ) ,
F ( ω ) = exp [ − ( ω 2 − ω 0 2 ) ⁄ ( Δ ω ) 2 ] ⁄ ( π Δ ω ) ,
k ( ω ) = ω c ( 1 − 1 ε ( ω ) ) 1 ⁄ 2 , β 0 ( ω ) ≡ ( k ( ω ) 2 − ω 2 c 2 ) 1 ⁄ 2 = ω c [ − ε ( ω ) ] − 1 ⁄ 2 ,
H 2 ( sc ) ( x 1 , x 3 ; t ) = ∫ − ∞ ∞ d ω F ( ω ) e − i ω t ∫ − ∞ ∞ d q 2 π R ( q , ω ) exp [ i ( q x 1 + α 0 ( q , ω ) x 3 ) ] ,
R ( q , ω ) = G 0 ( q , ω ) T ( q , ω ) ,
G 0 ( q , ω ) = i ε ( ω ) ε ( ω ) α 0 ( q , ω ) + i ( ω ⁄ c ) [ − ε ( ω ) ] 1 ⁄ 2 ≡ C ( q , ω ) ( 1 q − k ( ω ) − 1 q + k ( ω ) ) ,
T ( q , ω ) = V ( q ∣ k ( ω ) ) + ∫ − ∞ ∞ d p 2 π V ( q ∣ p ) G 0 ( p , ω ) T ( p , ω ) ,
V ( q ∣ p ) ≡ β 0 ( ω ) s ̂ ( q − p ) , s ̂ ( Q ) = ∫ − ∞ ∞ d x 1 e − i Q x 1 s ( x 1 ) .
H 2 ( x 1 , 0 ; t ) = H 2 ( i ) ( x 1 , 0 ; t ) + H 2 ( r ) ( x 1 , 0 ; t )
= H 2 ( i ) ( x 1 , 0 ; t ) + ∫ − ∞ ∞ d ω F ( ω ) e − i ω t ρ ( ω ) exp [ − ik ( ω ) x 1 − β 0 ( ω ) x 3 ] , x 1 ≪ 0 ;
H 2 ( x 1 , 0 ; t ) = H 2 ( t ) ( x 1 , 0 ; t )
= ∫ − ∞ ∞ d ω F ( ω ) e − i ω t τ ( ω ) exp [ ik ( ω ) x 1 − β 0 ( ω ) x 3 ] , x 1 ≫ 0 ,
ρ ( ω ) = iT ( − k R ( ω ) , ω ) C ( − k R ( ω ) , ω )
τ ( ω ) = 1 + iT ( k R ( ω ) , ω ) C ( k R ( ω ) , ω ) ,
H 2 ( s ) ( r , θ ; t ) = e − i π 4 cos θ 2 π r ∫ − ∞ ∞ d ω F ( ω ) e − i ω t ω c R ( ( ω ⁄ c ) sin θ , ω ) exp ( i ω r c ) ,
I ( θ , t ) ≡ r ∣ H s ∣ 2 .
s ( x 1 , ω ) = 1 − ε ( ω ) d ( ω ) ε ( ω ) h [ Θ ( x 1 + L ) − Θ ( x 1 − L ) ] ,
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