9. Dispersion relation of metal nanorods and nanotips

Transcription

1 9. Dispersion relation of metal nanorods and nanotips D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, Strong coupling of single emitters to surface plasmons, PR B 76, (2007) M. I. Stockman, Nanofocusing of Optical Energy in Tapered Plasmonic Waveguides, Phys. Rev. Lett. 93, (2004)

2 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, (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)

3 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 Function: For large x, Recurrence Relation:

4 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!

5 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

6 B. Dispersion relation of metal nanotips M. I. Stockman, Nanofocusing of Optical Energy in Tapered Plasmonic Waveguides, Phys. Rev. Lett. 93, (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)

7 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,

8 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 1( r > R, z) = A( z) i K1( k0κdr) ρ+ K0( k0κdr) zˆ 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