Geometries and materials for subwavelength surface plasmon modes Plasmon slot waveguides : Metal-Insulator-Metal (MIM) Metal nanorods and nanotips Metal nanoparticles Metal Dielectric Dielectric Metal Metal R y x Metal-Insulator-Metal (MIM) Metal nanorod Metal nanotip Metal nanoparticle
MIM structures Rashid Zia, Mark D. Selker, Peter B. Catrysse, and Mark L. Brongersma J. Opt. Soc. Am. A/Vol. 21, No. 12/December 2004, 2442-2446. To achieve subwavelength pitches, a metal insulator metal geometry is required with higher confinement factors and smaller spatial extent than conventional insulator metal insulator structures. The resulting trade-off between propagation and confinement for surface plasmons is discussed, and optimization by materials selection is described.
Consider the isotropic wave equation for a generic three-layer plasmonic slab waveguide with metallic and dielectric regions, where z is the propagation direction and thus k z is the conserved quantity. For a guided surface-plasmon mode to exist, If the radiation is unconfined in the y dimension (i.e., k y = 0),
Ultimate confinement of the IMI structure is limited by the decay length into the dielectric cladding. For confinement below the limit of a conventional dielectric waveguide (λ/2n), 2π x (1/k x,dielectric ) < (λ/2n) Note that this condition is met only near the surface plasmon resonance frequency. Confinement of the MIM structure is limited by the decay length into the metallic regions, which can be approximated as follows for metals below the surface-plasmon resonance: k 1/2 2 ω ε metal ω x, metal = > 2 c εmetal + εd c ε dielec
Power confinement factor (Γ) of field-symmetric TM modes - MIM and IMI plasmonic waveguides - (Au air, λ = 1.55 μm) 99.4% 2% If plasmonic waveguides are intended to propagate light in subwavelength modes, MIM geometries with higher confinement factors and shorter spatial extents are much better suited for this purpose.
Plasmon dispersion in MIM waveguides z Metal ε 2 y x d Insulator ε 1 Metal ε 2 E = ( E,0, E ) B= (0, B,0) x z y E = (0, E,0) B= ( B,0, B ) y x z
z y x Metal ε 2 d Insulator ε 1 Metal ε 2
Mode L+ Mode L- Tangential (E x ) Electric Field Profiles (TM modes)
(infinite d) a b : antisymmetric bound Tangential (E x ) Electric Field Profiles s b : Symmetric bound d = 250 nm (infinite d) y z x d Metal ε 2 Insulator ε 1 d = 100 nm Metal ε 2
IMI MIM (infinite d)
D = 250 nm a b s b (infinite d) SP modes conventional waveguide modes In S i O 2 core SP modes
D = 100 nm (infinite d) S b conventional waveguide modes within ΔE ~ 1 ev SP modes
s b SP : D = 50 nm 30 nm 25 nm 12 nm y z x d Metal ε 2 Insulator ε 1 Metal ε 2 The dispersion of the 50-nm-thick sample lies completely to the left of the decoupled SP mode. Low-energy asymptotic behavior follows a light line of n = 1.5. It suggests that polariton modes of MIM more highly sample the imaginary dielectric component. In the low energy limit, the S b SP truly represent a photon trapped on the metal surface.
a b SP : D = 50 nm 30 nm 25 nm 12 nm Purely plasmonic nature of the mode The cutoff frequencies remains essentially unchanged, possibly by the Goos-Hanchen effect. As waveguide dimensions are decreased, energy densities are more highly concentrated at the metal surface. This enhanced field magnifies Goss-Hanchen contributions significantly. In the limit of d << s (skin depth), complete SP dephasing could result.
MIM (Ag/SiO 2 /Ag) TM-polarized propagation distance and skin depth Forbidden band ( SiO 2, D = 250 nm ) 80 μm (infinite d) a b s b 15 μm 20 nm Note that only a slight correlation between propagation distance and skin depth (σ). The metal absorption is not the limiting loss mechanism in MIM structures.
MIM (Ag/SiO 2 /Ag) TM-polarized propagation distance and skin depth ( D = 12 nm, 20 nm, 35 nm, 50 nm, and 100 nm ) a b s b σ ~ 20 nm Approximately constant in the Ag cladding. Thus, MIM can achieve micron-scale propagation with nanometer-scale confinement. Evanescent within 10 nm for all wavelength Local minima corresponding to the transition between quasibound modes and radiation modes Unlike IMI, extinction (prop. distance) is determined not by ohmic loss (metal absorption), but by field interference upon phase shifts induced by the metal.
TE modes in MIM structures (~ 4 ev: ~300 nm)
EM energy density profiles of MIM structures (Ag/SiO2/Ag) d = 250 nm d = 100 nm
2007/5/1 ~ With an optical range resonator based on single mode metal-insulator-metal plasmonic gap waveguides,. a small bridge between the resonator and the input waveguide can be used to tune the resonance frequency. FDTD with the perfectly matched layer boundary conditions
Dispersion relation of metal nanorods and nanotips
A. Dispersion relation of metal nanorods D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, Strong coupling of single emitters to surface plasmons, PR B 76,035420 (2007) For nonmagnetic media, the electric and magnetic fields in frequency space satisfy the wave equation, ε 1 (dielectric) ˆρ ˆ φ ε 2 (metal) R ẑ In cylindrical coordinates, the electric field is given by Plot of Bessel function of the first kind J m (x) The scalar solutions of the wave equations satisfying the necessary boundary conditions take the form, Plot of Bessel function of the second kind ( outside: ρ > R ) ( inside : ρ < R ) Y m (x) J m : Bessel functions of the first kind H m : Hankel functions of the first kind H m (x) = J m (x) + iy m (x)
NOTE : Bessel functions and Hankel functions Bessel's Differential Equation is defined as: The solutions of this equation are called Bessel Functions of order n. Since Bessel's differential equation is a second order ordinary differential equation, two sets of functions, the Bessel function of the first kind and the Bessel function of the second kind (also known as the Weber Function) are needed to form the general solution: Bessel 1 st and 2 nd Functions: Hankel Function: the Hankel function of the first kind and second kind, prominent in the theory of wave propagation, are defined as For large x, For small x, Modified Bessel & Hankel Functions: For large x, Recurrence Relation:
Two independent vector solutions of are given by ˆρ ε 1 (dielectric) ˆφ The curl relations of Maxwell s equations then imply that E and H must take the form ε 2 (metal) R ẑ where a i and b i are constant coefficients. (silver nanorod) Continuity of the tangential field components at ρ = R gives the dispersion relation, m=0 all higher-order modes : purely imaginary exhibit a cutoff as m=1 2 3 the m=0 fundamental plasmon mode exhibits a unique behavior of the m=0 field outside the wire becomes tightly localized on a scale of R around the metal surface, leading to a small effective transverse mode area that scales like confined well below the diffraction limit!
For the special case a TM mode ( H z = 0) with no winding m=0 : fundamental mode. (TM mode with m = 0) : a i = 0 E φ = 0, H z = 0 Continuity of the remaining tangential field components E z and H φ at the boundary requires that ε 1 (dielectric) ε 2 (metal) ˆρ R ˆφ ẑ Setting the determinant of the above matrix equal to zero (det M=0) immediately yields the dispersion relation, In the limit of where I m, K m are modified Bessel functions When (nanoscale-radius wire) The fields themselves are given by
B. Dispersion relation of metal nanotips M. I. Stockman, Nanofocusing of Optical Energy in Tapered Plasmonic Waveguides, Phys. Rev. Lett. 93, 137404 (2004) Note that the TM, fundamental mode ( E φ = 0, H z = 0 ) on a nanorod was given by y x In the eikonal (WKB) approximation (slowly varying in z direction), this field on a nanotip may have the form where r is a two-dimensional (2D) vector in the xy plane and A(z) is a slow-varying preexponential factor. In the limit of (eikonal approximation) where n(z) is the effective surface index of the plasmonic waveguide at a point z, which is determined by the equation The dispersion relation obtained from the boundary conditions is, In the limit of ε 1 (dielectric) ε 2 (metal) ˆρ R ˆφ ẑ k = n( z) k 0 ε d = ε 1 (dielectric) ε m = ε 2 (metal)
Dispersion relation of metal nanotips y ε m x ε d For a thin, nanoscale-radius wire k = nk 0 For, the phase velocity v c/ n( z) 0 p = and the group velocity v c [ d nω dω] The time to reach the point R = 0 (or z = 0) g = / ( )/ 0 The eikonal parameter (also called WKB or adiabatic parameter) is defined as For the applicability of the eikonal (WKB) approximation, it necessary and sufficient that At the nanoscale tip of the wire,
The SPP electric fields are found from the Maxwell equations in eikonal (WKB) approximation in the form: For a nanorod [ Rz ( ) = R; constant] y 2 (ε m ) x 1 (ε d ) For a nanotip [ R = R( z); not fixed] I0( k0κ mr) n E ˆ ˆ 1( r > R, z) = A( z) i K1( k0κdr) ρ+ K0( k0κdr) z e K0( k0κmr) κd n ink0z E ˆ ˆ 2( r < R, z) = A( z) i I1( k0κmr) ρ+ I0( k0κmr) z e κm ink z 0 To determine the preexponential A(z), we use the energy flux conservation in terms of the Pointing vector integrated over the normal (xy) plane, Intensity Energy density
Particle (Localized) surface plasmons ( Plasmons in metal nanostructures, Dissertation, University of Munich by Carsten Sonnichsen, 2001) Lycurgus cup, 4th century (now at the British Museum, London). The colors originates from metal nanoparticles embedded in the glass. At places, where light is transmitted through the glass it appears red, at places where light is scattered near the surface, the scattered light appears greenish. Focusing and guidance of light at nanometer length scales
Quasi-static approximation Rayleigh scattering by a small particle Rayleigh scattering ( Scattering of Electromagnetic Waves: Theories and Applications, Leung Tsang, Jin Au Kong, Kung-Hau Ding, 2000 John Wiley & Sons, Inc. This approach effectively means that a region in space is investigated which is much smaller than the wavelength of light, so the electromagnetic phase is constant throughout the region of interest. For small metal particles with diameters below 40 nm, this proves to be a reasonable simplification. The incident wave to the direction of k i is The far field radiated by a dipole p in the direction k s is ε (surrounding medium, ε m ) ε p (particle, ε ) v 0 (particle volume) p The polarization per unit volume inside the particle is P = P P = ( ε ε ) E ( ε ε ) E = ( ε ε) E int particle medium p 0 int 0 int p int The dipole moment of the particle is where v 0 is the volume of the particle. For a sphere of radius a << λ, The power scattered is : scattering cross section, where From Ohm s law, the power absorbed is
Rayleigh Theory for metal = dipole surface-plasmon resonance of a metal nanoparticle Rayleigh scattering ( Plasmons in metal nanostructures, Dissertation, University of Munich by Carsten Sonnichsen, 2001) r p 4 R ε ε r r = πε εe = αεe 3 p 0 0 0 ε p + 2ε : dipole moment of the particle The polarizability α of the metal sphere is ε (surrounding medium, ε m ) ε (particle, ) p ε The scattering and absorption cross-section are then Scattering and absorption exhibit the plasmon resonance where, Frohlich condition Re εp ( ω) + 2ε = 0 For free particles in vacuum, resonance energies of 3.48 ev for silver (near UV) and 2.6 ev for gold (blue) are calculated. When embedded in polarizable media, the resonance shifts towards lower energies (the red side of the visible spectrum).
Rayleigh Theory : Scattering by elliptical particles Rayleigh scattering a) prolate (cigar-shaped) sheroid (a > b = c), b) oblate (pancake-shaped) sheroid (a = b > c) The polarizability α i of such a spheroidal particle along the axis i is given by L i : a geometrical factor related to the shape of the particle. for a prolate particles
Beyond the quasi-static approximation : Mie scattering Theory Mie scattering For particles of larger diameter (> 100 nm in visible), the phase of the driving field significantly changes over the particle volume. Mie theory valid for larger particles than wavelength from smaller particles than the mean free-path of its oscillating electrons. Mie calculations for particle shapes other than spheres are not readily performed. The spherical symmetry suggests the use of a multipole extension of the fields, here numbered by n. The Rayleigh-type plasmon resonance, discussed in the previous sections, corresponds to the dipole mode n = 1. In the Mie theory, the scattering and extinction efficiencies are calculated by: Frohlich condition n + 1 Re εp( ω) = εembedded n ( Plasmons in metal nanostructures, Dissertation, University of Munich by Carsten Sonnichsen, 2001) For the first (n=1) TM mode of Mie s formulation is
For a 60 nm gold nanosphere embedded in a medium with refractive index n = 1.5. (use of bulk dielectric functions (e.g. Johnson and Christy, 1972)) Mie scattering By the Mie theory for cross-sections By the Mie theory for spherical particle By the Rayleigh theory for ellipsoidal particles. a/b = 1+3.6 (2.25 E res / ev) The red-shift observed for increasing size is partly due to increased damping and to retardation effects. The broadening of the resonance is due to increasing radiation damping for larger nanospheres. Influence of the refractive index of the embedding medium on the resonance position and linewidth of the particle plasmon resonance of a 20 nm gold nanosphere. Calculated using the Mie theory. Resonance energy for a 40 nm gold nanosphere embedded in water (n = 1.33) with increasing thickness d of a layer with refractive index n = 1.5.
Experimental measurement of particle plasmons measurement Scanning near-field microscopy(snom) SNOM images gold nanodisks 633 nm SEM image 550 nm Dark-field microscopy in reflection Total internal reflection microscopy(tirm) Dark-field microscopy in transmission
Plasmon Damping (Plasmon life time) in metal nanoparticles C. Sönnichsen, et. al, Drastic Reduction of Plasmon Damping in Gold Nanorods, PRL, 88, 077402 (2002). Radiative (left) and nonradiative (right) decay The nonradiative decay occurs via excitation of electron-hole pairs either within the conduction band (intraband excitation) or between the d band and the conduction band (interband excitation). Dephasing times T 2, directly relate to the plasmon lifetime, can be deduced from the measured homogeneous linewidths Γ, Γ T 2 = 2 / Γ Quality factor of the resonance Q = E res /Γ Dephasing time decreases with increasing particle diameter, possibly due to increased radiation damping Gold diameters (150 nm) Solid line (Mie calculation) diameters from 100 20 nm 20 nm Diameters (nm)
Interaction between particles an isolated sphere is symmetric, so the polarization direction doesn t matter. LONGITUDINAL: restoring force reduced by coupling to neighbor Resonance shifts to lower frequency TRANSVERSE: restoring force increased by coupling to neighbor Resonance shifts to higher frequency pair of silver nanospheres with 60 nm diameter Focusing and guidance of light at nanometer length scales