Home | Survey | Topics | Index | References | Dictionary | Contributing | Gallery | Community
| Mie theory: Numerical aspects of integrating particle size-dependent patterns | Prev topic | Next topic |
The particle-size-dependent patterns of Mie theory exhibit complex oscillation structure (see Fig. 1 in Mie theory: Particle size-dependent patterns). This structure ranges from low-frequency oscillations, through ripple, to sharp peaks, termed optical resonances (see Mie theory: Optical resonances and ripple and Fourier analysis of particle size-dependent patterns).
What would be an optimum integration grid for these patterns? This question can be answered with the help of Fourier analysis of the oscillations in a Mie size-dependent pattern. In the case of the M11 element of the scattering matrix that is related to the volume scattering function of a dispersion, such analysis indicates that the dominant maximum particle size-related frequency of the pattern depends linearly on the scattering angle in a wide angular range as expressed by Eq. 1 in Mie theory: Fourier analysis of particle size-dependent patterns.
The frequency spectra of the size-dependent patterns may contain components with frequencies much greater than the dominant frequency discussed above, as such higher frequencies are required to express the sharp optical resonance peaks. Zender et al (2006), who examined the effect of such peaks on the absorption of light by a dispersion, concluded that the effect of these peaks is minimal on wide-range integrals over the wavalength of light, which can be taken here (but not always) to be equivalent to the particle size via the concept of the relative particle size x. However, one should not mistake that conclusion for a suggestion that the use of a sparse integration grid is a reasonable approach. As pointed out by Hill (2003), when an integration grid point happens to coincide with a resonance peak, significant errors in the integrals of the absorption coefficient may result. However, the findings of Zender et al would imply that skipping such integration grid points may be an acceptable trade-off for a wide wavelength or particle-size range. If the resonance peaks are expected to contribute significantly to a size-dependent pattern, as in integration of the absorption coefficient over a narrow (~10 nm, Zender et al., to ~40 nm, Hill) wavelength range, the peak positions and widths can be found, their spectral shapes can be approximated by the Lorentzian functions and integrated analytically (Hill 2003).
An interesting approach was used by Quenzel and Müller (1978) to circumvent the complexity of the particle-size dependent patterns. They used a randomly-chosed increment in the relative particle size, x, with the maximum set to 0.05x, where x is presumably the upper integration limit.
| CITATION: Jonasz M. 2006. Mie theory: Numerical aspects of integrating particle size-dependent patterns (www.tpdsci.com/Tpc/MiePtnSzIntgNum.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 30-Jun-2006 Modified: 25-Jul-2006 Reviewed: PENDING |
| Copyright 2005-2008 MJC Optical Technology. All rights reserved. | Terms of use | Menu |