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# 1. Derivation of the discrete dipole approximation ## 1.1 Introduction The discrete dipole approximation (DDA), also called the coupled dipole method, is a technique for computing scattering and absorption of electromagnetic radiation by particles of arbitrary geometry. DDA is applicable to targ...

# 1. Derivation of the discrete dipole approximation ## 1.1 Introduction The discrete dipole approximation (DDA), also called the coupled dipole method, is a technique for computing scattering and absorption of electromagnetic radiation by particles of arbitrary geometry. DDA is applicable to targets of any material. The DDA is a volume discretization technique. The scattering object is replaced by an array of $N$ point dipoles. The point dipoles become polarized in response to the local electric field. The local field is the sum of the incident field and the fields due to all the other dipoles in the array. ### 1.1.1 Basic references - Purcell, E.M. and Pennypacker, C.R., Astrophys. J. 186, 705 (1973). - Draine, B.T. and Flatau, P. J. Opt. Soc. Am. A 11, 1491 (1994). ### 1.1.2 Other references - Schatz, G.C., Young, M.A. Chem. Phys. Lett. 229, 316 (1994). - Draine, B.T., оптике и спектроскопии 88, 784 (2000). - Bohren, C.F. and Huffman, D.R., Absorption and scattering of light by small particles (Wiley, New York, 1983). - Draine, B.T., Interstellar Dust (Princeton University Press, Princeton, 2003). ## 1.2 Polarizability In the DDA each point dipole has a polarizability $\alpha_j$, where $j$ is the index of the dipole. The electric dipole moment of the $j$-th dipole is given by $\qquad \mathbf{P}_j = \alpha_j \mathbf{E}_{loc,j}$ where $\mathbf{E}_{loc,j}$ is the local electric field at the location of the $j$-th dipole. The most common choice for the polarizability is the "Clausius-Mossotti" polarizability $\qquad \alpha_j = \alpha_{CM} = \frac{3v_j}{4\pi} \frac{\epsilon_j - 1}{\epsilon_j + 2}$ where $v_j$ is the volume of the $j$-th dipole and $\epsilon_j$ is the dielectric function of the material at the location of the $j$-th dipole. Another choice for the polarizability is the "Loratentz-Lorenz" polarizability $\qquad \alpha_j = \alpha_{LL} = \frac{v_j}{4\pi} (\epsilon_j - 1)$ A third choice for the polarizability is the "radiative-reaction" polarizability $\qquad \alpha_j = \alpha_{RR} = \left[ \frac{1}{\alpha_{LL}} - \frac{2}{3}ik^3 \right]^{-1}$ where $k = \omega/c$ is the wavenumber. Draine and Flatau (1994) recommend using the following polarizability $\qquad \alpha_j = \alpha_{DF} = \left[ \frac{1}{\alpha_{RR}} + \frac{1}{3} \frac{2}{3}ik^3 (\epsilon_j - 1) \right]^{-1} = \left[ \frac{1}{\alpha_{LL}} - \frac{2}{3}ik^3 + \frac{1}{3} (\epsilon_j - 1) (ik)^3 \right]^{-1}$ ## 1.3 Basic equations Let $\mathbf{r}_j$ be the position of the $j$-th dipole. The electric field at $\mathbf{r}_j$ is the sum of the incident field and the fields due to all the other dipoles in the array: $\qquad \mathbf{E}_j = \mathbf{E}_{inc,j} + \sum_{l \neq j}^{N} \mathbf{E}_{l,j}$ where $\mathbf{E}_{inc,j}$ is the incident electric field at $\mathbf{r}_j$ and $\mathbf{E}_{l,j}$ is the electric field at $\mathbf{r}_j$ due to the $l$-th dipole. The electric field at $\mathbf{r}_j$ due to the $l$-th dipole is given by $\qquad \mathbf{E}_{l,j} = A \mathbf{G}_{jl} \mathbf{P}_l$ where $\qquad A = k^2 e^{ik|\mathbf{r}_j - \mathbf{r}_l|}/|\mathbf{r}_j - \mathbf{r}_l|$ and $\qquad \mathbf{G}_{jl} = \left[ \hat{\mathbf{I}} - (\mathbf{\hat{r}}_j - \mathbf{r}_l) (\mathbf{\hat{r}}_j - \mathbf{r}_l) \right] - \left[ 1 - ik|\mathbf{r}_j - \mathbf{r}_l| - k^2 |\mathbf{r}_j - \mathbf{r}_l|^2 \right]$ where $\hat{\mathbf{I}}$ is the unit dyadic. The electric field at $\mathbf{r}_j$ is then given by $\qquad \mathbf{E}_j = \mathbf{E}_{inc,j} + \sum_{l \neq j}^{N} A \mathbf{G}_{jl} \mathbf{P}_l$ Using the relation $\mathbf{P}_j = \alpha_j \mathbf{E}_j$, we obtain $\qquad \mathbf{P}_j = \alpha_j \mathbf{E}_{inc,j} + \alpha_j \sum_{l \neq j}^{N} A \mathbf{G}_{jl} \mathbf{P}_l$ or $\qquad \mathbf{P}_j - \alpha_j \sum_{l \neq j}^{N} A \mathbf{G}_{jl} \mathbf{P}_l = \alpha_j \mathbf{E}_{inc,j}$ For each dipole we have three equations (one for each component of the electric dipole moment). Therefore, for $N$ dipoles we have $3N$ equations. These equations can be written in matrix form as $\qquad \mathbf{X} \mathbf{P} = \mathbf{E}_{inc}$ where $\mathbf{X}$ is a $3N \times 3N$ matrix, $\mathbf{P}$ is a $3N \times 1$ vector containing the electric dipole moments of all the dipoles, and $\mathbf{E}_{inc}$ is a $3N \times 1$ vector containing the incident electric fields at all the dipoles. The matrix $\mathbf{X}$ is given by $\qquad \mathbf{X}_{jj} = \alpha_j^{-1} \hat{\mathbf{I}}$ and, for $l \neq j$, $\qquad \mathbf{X}_{jl} = -A \mathbf{G}_{jl}$ The vector $\mathbf{E}_{inc}$ is given by $\qquad \mathbf{E}_{inc,j} = \mathbf{E}_{inc}(\mathbf{r}_j)$ The vector $\mathbf{P}$ is given by $\qquad \mathbf{P}_j = \mathbf{P}(\mathbf{r}_j)$ Once the $3N$ equations have been solved for the $\mathbf{P}_j$, the absorption and scattering properties of the particle can be determined.