Fig. 1. Properties of the high-Q modes in the single rectangular NW. (a), (b) Real part and Q-factor of the eigenvalue of modes TE(3,5) and TE(5,3) (type I) as functions of size ratio R. (c), (d) Q-factor and a/λ as functions of m for high-Q mode TE(m,m+2) at the critical ratio. (e), (f) Real part and Q-factor of the eigenvalue of modes TE(3,4) and TE(5,2) versus the size ratio R. (g), (h) Q-factor and a/λ, as functions of m for high-Q mode TE(m,m+1) at the critical ratio.

Fig. 2. Total energy density and scattering efficiency for the rectangular NW with different size ratios. (a), (b) Logarithm total energy density and scattering efficiency mapping versus R and ka. Two modes TE(3,5) and TE(5,3) are labeled as black and red circles. (c), (d) Logarithm of total energy density and scattering efficiency mapping versus R and ka. Two modes TE(3,4) and TE(5,2) are labeled as black and red circles.

Fig. 3. Multipole analysis of the high-Q modes. (a) Multipolar contribution on the scattering cross section of the square NW under the excitation oblique incidence plane wave (θ=15deg). (b), (c) Multipole analysis and Fourier transformation on the eigenfields of two modes TE(3,5) and TE(5,3). (d), E(k0) obtained from a Fourier transformation of eigenfields for two modes. (e) Multipolar contribution on scattering cross section of the rectangular NW with R=0.855 excited by the obliquely incident plane wave (θ=15deg). (f), (g) Multipole analysis and Fourier transformation on the eigenfields of two modes TE(3,4) and TE(5,2). (h) E(k0) obtained from Fourier transformation of eigenfields for two modes. (i) Decomposition of TE(3,5) for the rectangular NW into TE61 and TE22 of eigenmodes for the circular NW. (j) Decomposition of TE(5,3) for the rectangular NW into TE42 and TE03 of eigenmodes for the circular NW.

Fig. 4. High-Q mode for the TM case. (a), (b) Real part and Q-factor of the eigenvalue for modes TM(2,3) and TM(4,1) as functions of the size ratio R. (c), (d) Real part and Q-factor of the eigenvalue for modes TM(3,4) and TM(5,2) versus the size ratio R.

Fig. 5. High-Q mode for the single 3D nanoparticle. (a), (b) Real part and Q-factor of the eigenvalue for the magnetic eigenmode M(1,2,3) and M(1,4,1) in the single cuboid as functions of the size ratio R=c/a while a=b. (c), (d) Real part and Q-factor of the eigenvalue for the magnetic eigenmode M(1,2,4) and M(1,4,2) as functions of the size ratio R=c/a of a cuboid. (e), (f) Real part and Q-factor of the eigenvalue for the electric eigenmode E(1,2,3) and E(1,4,1) as functions of the size ratio R=c/a of the cuboid. (g), (h) Real part and Q-factor of the eigenvalue for the electric eigenmode E(1,2,4) and E(1,4,2) as functions of the size ratio R=c/a of a cuboid.

Fig. 6. (a) Multipole analysis on the eigenfields for two modes of the cuboid, M(1,4,1) and M(1,2,3). (b) E(k0) distribution for two modes.

Fig. 7. Schematic drawing of measurement system.

Fig. 8. Experimental verification of high-Q mode in the single silicon NW on quartz. (a) Schematic drawing of the single Si NW on a quartz substrate under normal incidence for TE polarization. (b) Experimentally measured scattering spectrum for the single NW with a=970nm and b=825nm. Top inset is the eigenfield E distribution of mode TE(3,4) while the bottom inset is an SEM image of fabricated Si NW (the bar in the inset is 500 nm). (c) Simulation and measured scattering spectra for the single NW with different widths while the thickness of the NW is fixed as 825 nm. High-Q modes are indicated with arrows. (d) Extracted Q-factor and (e) Nreal for pair modes TE(3,4) and TE(5,2) as a function of the size ratio.