Supplementary information
Broadband optical absorption by tunable Mie resonances in silicon nanocone
arrays
Z.Y. Wang1, R. J. Zhang1,*, S. Y. Wang1, 2, 3,*1, M. Lu1, X. Chen4, Y. X. Zheng1, L. Y. Chen1, Z. Ye3,
C. Z. Wang3 and K. M. Ho3
1
Shanghai Engineering Research Center of Ultra-Precision Optical Manufacturing and
Department of Optical Science and Engineering, Fudan University, Shanghai, 200433, China
2
Key Laboratory for Information Science of Electromagnetic Waves (MoE), Shanghai 200433,
China
3
Ames Laboratory, U. S. Department of Energy and Department of Physics and Astronomy, Iowa
State University, Ames, Iowa 50011, USA
4
National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese
Academy of Sciences, Shanghai 200083, China
Mie theory
The Mie theoryS1,S2 was developed for spheres embedded in a nonmagnetic transparent medium
with relative dielectric constant  d . It is assumed that an incident electromagnetic wave propagates
in the z direction and is polarized in the x direction. The extinction cross section could be
represented as:
Cext 
2
k2

 (2n  1) Re(a
n 1
n
 bn )
(1)
and the scattering coefficients for the first Mie resonance mode are:
3
  i F () 3
2
a1  i (kd2 d  d ) 2 d
r ,
3
2d  i F ()
(2)
3
   i F () 3
2
b1  i (kd2 d  d ) 2 d
r ,
3
2 d   i F ()
where F ()  2(sin    cos ) / [(2  1)sin   cos ],   k0 r  i i . And  i and i are
complex permittivity and complex permeability of the material in the sphere, k d is the wave
vector in the embedding medium and r is radius of the sphere. The effective electric and magnetic
polarizabilities have the following form:
*Corresponding author: songyouwang@fudan.edu.cn and rjzhang@fudan.edu.cn
1
M  i
6 r  d
6
a1 ,  E  i 3 b1
3
kd
kd
(3)
The Mie resonance can be seen from the spectra of  E and  M in Supplementary Fig.
S1(a). The Qsca calculated by discrete dipole approximation (DDA) reveals a significant
correlation with the polarizabilities. The resonance modes will be excited at the wavelength where
the imagine part of polarizabilities and the Qsca reach to maxima. The electric field distributions in
Supplementary Fig. S1(b) have shown a good agreement with the Mie theory.
Effective Radius:
If the actual volume of a particle (e.g., a nanocone) is V, then “effective radius” of the particle
is defined as:
Reff  (3V / 4 )1/3
(4)
Average Absorbance
The average absorbance is defined as:
Absavg  
max
min
 ( ) d 
(5)
where  ( ) is the absorbance of the material as a function of the wavelength.
2
Supplementary Figure S1
Supplementary Figure S1: Mie resonance of a Si sphere, the sphere is 156 nm in diameter. (a) The
normalized scattering cross section (black) and the effective electric (E) and magnetic (M)
polarizabilities (red and blue) of Si sphere. (b) Normalized electric field density (|E|/|E0|) at the
electric dipole and magnetic dipole resonant at the wavelengths of 502 nm and 620 nm. The
sphere is irradiated by a plane wave from bottom to the top. The electric and magnetic field and
light propagation direction was labeled accordingly.
3
Supplementary Figure S2
Supplementary Figure S2: In actual simulation, the Si nanocone is approximated by stacked
cylinders of different diameters as shown in the figure.
4
Supplementary Figure S3
Supplementary Figure S3: (a,b) Comparisons of reflectance and absorbance spectra between
experiment and simulations (BD=180 nm, H=450 nm). (c) The unit cell of “case 1” atructure used
in the simulation: the black background represents the Si substrate and white circles are the
nanocones of different size in the unit cell. (d) the simulated reflectance spectra are compared with
experimental result. The incident angle is 8 degree.
5
Supplementary Figure S4
Supplementary Figure S4: The average absorbance of nanocone array with (a) different heights
and (b) different base diameters, respectively.
6
Supplementary Figure 5
Supplementary Figure S5: Simulated absorbance and reflectance (average value of TE and TM
modes) spectra of Si nanocone arrays with P=BD=160 nm, H=400 nm with different light incident
angles.
7
Supplementary Table 1
Supplementary Table 1: The effective radius of nanocone particle with varied height. The base
diameter of these nanocones is 160 nm.
Particle Height (nm)
Effective Radius (nm)
300
78.3
320
80.0
400
86.2
500
92.8
Supplementary Table 2
Supplementary Table 2: Origin of absorption peaks in Fig. 4(e).
Height (nm)
Period (nm)
Mie resonance (nm)
WR anomaly (nm)
400
120
426 (2nd), 502 (1st)
384 (2nd)
400
160
600 (1st)
413 (2nd)
400
200
715 (1st)
392 (3rd), 455 (2nd)
400
240
820 (1st)
412(3rd), 502 (2nd)
Reference
S1. Bohren, C.F. & Huffman, D.R. Absorption and scattering of light by small particles, Bohren,C.
F. & Huffman D. R. (ed.) 83-129. (John Wiley & Sons, Inc., New York, 1998).
S2. Jylhä, L., Kolmakov, I., Maslovski, S. & Tretyakov, S. Modeling of isotropic backward-wave
materials composed of resonant spheres. J. Appl. Phys. 99, 043102 (2006).
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