Chapter 2 Principle
2.1 Optical properties of particles
2.1-1 Mie theory
Light scattering by an induced dipole moment is caused by an incident electromagnetic wave. Electromagnetic field of an arbitrary particle (E, H) located at the polar coordination system can be expressed by the combination of incident electromagnetic field (Ein, Hin) and
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scattering electromagnetic field (Esca, Hsca).
sca
in E
E
E ΗHinHs c a (Eq. 2-1)
The Poynting vector S = E × H specifies the magnitude and direction of the rate of the transfer of electromagnetic energy at all points of spaces. Once we have obtained the electromagnetic fields inside the particle and scattered by the paricle, we can determine the Poynting vector at any point. However, we are usually interested only in the points outside the particle. The time-averaged Poynting vector S at any point in the medium surrounding the particle can be written as the sum of three terms [20]:
SSinSscaSext (Eq. 2-2)
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If the orientation of a plane surface with area A is specified by a unit normal vector N, the net rate W at which electromagnetic energy crosses the boundary of a closed surface A which enclose a volume V is
dA rate at which energy is absorbed by the particle. Because of Eq. 2-4, Wa may be written as the sum of three terms [20]: Figure 2.1 Spherical polar coordinate system centered on a spherical particle of radius a.
θ
X
Y
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Figure 2.2 Light scattering, absorption, and extinction by a single particle, whose cross
sections are given as Csca,Ｃabs, andＣext respectively.
Win vanishes identically for a non-absorbing medium; Wsca is the rate at which energy scattered across the surface A. Therefore, Wext is just the sum of the energy absorption rate and the energy scattering rate which corresponds extinction by particle:
abs sca
ext W W
W
(Eq. 2-7)
The ratio of Wext to Ii is a quantity with dimensions of area called cross section:
i ext
ext I
C W
(Eq. 2-8) where Ii is the incident irradiation.
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Similarly to Eq. 2-7, the extinction cross section Cext can be written as the sum of the absorption cross section Cabs and the scattering cross section Csca:
sca
Based on physics, we know that Wext and Wa are independent of the polarization state of the incident light. Therefore, we may take the incident light to be x-polarized [20]:
n n n n
where ω is angular frequency, μ is permeability and ρ = kr. The angle dependent functions πn , τn and ψn are where Pn1 (cosθ) is associated Legendre function and Jn (ρ) is the first term of Bessel function.The corresponding scattered field is
n n n n n n
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And the scattering coefficients of the field inside the particle are[20]
where hn(l)(ρ) is Hankel function, jn (x) is spherical Bessel function and μl, μ are permittivities of the particle and medium.
The size parameter and the refractive index are expressed as
λ
where Nl and N are refractive indices of particle and medium, respectively; a is the radius of particle, λ is the wavelength of incident light.
Together with Eq. 2-6, Eq. 2-10 and Eq. 2-12, we derive expressions for the cross
sections of a sphere more exactly:
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Then the scattering cross section is obtained [20]:
2 2
Similarly, the extinction cross section is:
Expanding the spherical Bessel functions in the scattering coefficients leads an and bn to power series. Retaining the scattering coefficients in a sufficient number of expansions, the four coefficients can be obtained as[20]
)
Here, we have taken permittivity of the sphere to be equal to that of the surrounding medium.
If |m| x << 1, then |b1| << |a1|; with this assumption the amplitude scattering matrix elements are down to terms of order x3.
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If the incident light is unpolarized with irradiance Ii, the scattered irradiance Is calculated
by scattering matrix is
Thus, if the quantity |(m2-1) /(m2+2)| is weakly dependent on wavelength, the irradiance scattered by a sphere small compared with the wavelength or, indeed, any sufficiently small particle regardless of its shape, is proportional to 1/λ4.
“When light is scattered by particles which are very small compared with any of the
wavelengths, the ratio of the amplitudes of the vibrations of the scattered and incident light varies inversely as the square of the wavelength and the intensity of the lights themselves as the inverse fourth power.” Lord Rayleigh showed the simple dimensional analysis in 1871.
However, if the incident light is unpolarized, the scattered light will be partially
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The extinction and scattering cross sections are
2.1-3Refractive index of metallic and organic nanoparticles
The optical property of metallic nanoparticle is strongly dependent on particle size, shape, electron density and the local enviroment, the refracive index of surrounding medium. The application of refractive index now attracts to much attetion because other parameters such as density and temperature can be obtained by detecting refractive index [21].
We consider Au nanoparticles in 100 nm are dispersed on the glass substrate and embedded in non-absorbing medium. The relationship between the shift of surface plasmon band and the refractive index was given by
refractive index induced by the adsorbate; d is the effective adsorbate layer thickness; and ld is the characteristic EM-field-decay length (approximated as an exponential decay)[22].
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Herman suggested the scattering efficiency based on Mie theory could be written [23]:
where R(0°) = |(m-1)/(m+1)|2 is a reflectance of a plane surface at normal incidence and m is the complex relative refractive index of nanoparticle [24].Fig. 2.3 shows the calculated scattering spectra of gold nanoparticle with various refractive index of surrounding medium from 1.33 to 1.53. Based on Mie theory, the calculation result displays that the surface plasmon band red-shifts and the scattering efficiency enhances as the refractive index increases.
400 600 800
Figure 2.3 Calculated spectra of 100 nm Au nanoparticles with various refractive index. With
the increase of refractive index, the plasmon band shifts to longer wavelength, the intensity increases, and the FWHM of the plasmon band broadens.
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