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Tailoring optical pulling force on gain coated nanoparticles with nonlocal effective medium theory

X. Bian, D. L. Gao, and L. Gao

Open Access Open Access

Abstract

We study the optical scattering force on the coated nanoparticles with gain core and nonlocal plasmonic shell in the long-wavelength limit, and demonstrate negative optical force acting on the nanoparticles near the symmetric and/or antisymmetric surface plasmon resonances. To understand the optical force behavior, we propose nonlocal effective medium theory to derive the equivalent permittivity for the coated nanoparticles with nonlocality. We show that the imaginary part of the equivalent permittivity is negative near the surface resonant wavelength, resulting in the negative optical force. The introduction of nonlocality may shift the resonant wavelength of the optical force, and strengthen the negative optical force. Two examples of Fano-like resonant scattering in such coated nanoparticles are considered, and Fano resonance-induced negative optical force is found too. Our findings could have some potential applications in plasmonics, nano-optical manipulation, and optical selection.

© 2017 Optical Society of America

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Figures (7)
Fig. 1
Fig. 1 Schematic of the coated nanosphere embedded in the host medium with relative permittivity ε h . The coated particle is composed of a gain core with inner radius a and the relative permittivity ε c , and a nonlocal plasmonic shell with outer radius b and ε s ( k , ω ) .

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Fig. 2
Fig. 2 Evolution of the equivalent permittivity as gain is increased with Eq. (13), from (a) to (f). Parameters: ε g 1 = 2.1025 and f = 0.03 . The real and imaginary parts are denoted by the red solid lines and blue dash-dotted lines. The black dash line represents the value of − 2 ε h .

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Fig. 3
Fig. 3 Normalized optical force and equivalent permittivity as a function of the incident wavelength with f = 0.03 (a-c) and f = 0.3 (e-f) in nonlocal (red solid lines) and local (blue dash-dotted lines) cases.

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Fig. 4
Fig. 4 Two plasmonic resonant wavelengths (blue lines), and corresponding normalized resonant optical forces (red lines) with increasing volume fraction f , in nonlocal (solid lines) and local (dash-dotted lines) cases, respectively.

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Fig. 5
Fig. 5 Normalized optical force F / F 0 with respect to incident wavelength and volume fraction f in nonlocal theory. Gray region indicates the parameter space for the pushing force, colored region indicates the pulling force. The circled region is magnified in the inset. The black region shows extremely large negative optical force much stronger than −15.

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Fig. 6
Fig. 6 (a) Dependence of scattering efficiency on incident wavelength and aspect ratio for core-shell spheres under nonlocal frameworks. The yellow and blue lines show resonant and cloaking modes, respectively. (b)-(d) show corresponding scattering efficiency, normalized optical force and the equivalent permittivity with a = 1.2   n m   ( η = 0.12 ) , respectively.

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Fig. 7
Fig. 7 (a) Scattering efficiency, (b) normalized optical force and (c) The real (solid line) and imaginary (dash-dotted line) parts of equivalent permittivity with a weaker damping coefficient above/below the plasma frequency/wavelength as a function of incident wavelength for f = 0.6 in nonlocal theory.

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Equations (15)

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D ( r ) = ∫ ε s ( r − r ' , ω ) E ( r ' ) d 3 r ' .
∇ 2 φ D ( r ) = E 0 [ A δ ( r − a ) + B δ ( r − b ) ] cos θ ,
− k 2 φ D ( k ) = E 0 ∫ [ A δ ( r − a ) + B δ ( r − b ) ] e − i k ⋅ r cos θ d 3 r .
ε s ( k , ω ) = ε b − ω p 2 ω ( ω + i γ ) − β 2 k 2 ,
φ ( k ) = E 0 i 4 π [ A a 2 j 1 ( k a ) + B b 2 j 1 ( k b ) ] e i θ k k 2 ε s ( k , ω )
φ D ( r ) = 1 ( 2 π ) 3 ∫ φ D ( k ) e i k ⋅ r d 3 k = − 1 3 E 0 [ A a 3 r 2 + B r ] cos θ
φ ( r ) = 1 ( 2 π ) 3 ∫ φ ( k ) e i k ⋅ r d 3 k = − E 0 2 π cos θ ∫ [ A a 2 j 1 ( k a ) + B b 2 j 1 ( k b ) ] j 1 ( k r ) ε s ( k , ω ) d k
{ φ c ( r ) = − E 0 C r cos θ , r < a φ s ( r ) = − E 0 2 π cos θ ∫ [ A a 2 j 1 ( k a ) + B b 2 j 1 ( k b ) ] j 1 ( k r ) ε s ( k , ω ) d k , a < r < b φ h ( r ) = − E 0 ( D / r 2 − r ) cos θ , r > b
φ D ( r ) = − 1 3 E 0 [ A a 3 r 2 + B r ] cos θ , a < r < b
A = − 9 ( G a G a b − ε c G a ) ε h G a b ( 1 + 2 ε h / G b ) ( ε c + 2 G a ) + 2 G a ( ε h + ε c − G a b − ε h ε c / G a b ) f B = − 9 ( 2 G a G a b + ε c G a b ) ε h G a b ( 1 + 2 ε h / G b ) ( ε c + 2 G a ) + 2 G a ( ε h + ε c − G a b − ε h ε c / G a b ) f C = − 3 ( G a b + 2 G a ) ε h G a b ( 1 + 2 ε h / G b ) ( ε c + 2 G a ) + 2 G a ( ε h + ε c − G a b − ε h ε c / G a b ) f D = b 3 G a b ( 1 − ε h / G b ) ( ε c + 2 G a ) + G a [ 2 ( ε c − G a b ) + ε h ( ε c / G a b − 1 ) ] f G a b ( 1 + 2 ε h / G b ) ( ε c + 2 G a ) + 2 G a ( ε h + ε c − G a b − ε h ε c / G a b ) f
α = α 0 / ( 1 − i 2 3 k 3 α 0 4 π ε 0 ε h )
〈 F 〉 = 1 2 k E 0 2 I m ( α ) ,
ε e q = G b G a b [ ( G a b + 2 f G a ) ε c + 2 G a G a b ( 1 − f ) ] ( G a b 2 − f G a G b ) ε c + G a G a b ( 2 G a b + f G b ) ,
ε e q = ε s [ ( 1 + 2 f ) ε c + 2 ( 1 − f ) ε s ] ( 1 − f ) ε c + ( 2 + f ) ε s .
〈 F 〉 = 2 π ε 0 ε h k b 3 E 0 2 3 I m ( ε e q ) ε h + 2 ( k b ) 3 [ R e ( ε e q ) − ε h ] 2 / 3 [ R e ( ε e q ) + 2 ε h ] 2 + [ I m ( ε e q ) ] 2 + 4 ( k b ) 3 I m ( ε e q ) ε h
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