Superposition of Plane-Wave Contributions

The result analogous to Eq. (19) for scattering of light propagating along $\hat{\boldsymbol{z}}$ and polarized in the $\hat{\boldsymbol{y}}$ direction is obtained through a rotation of $\pi/2$ about $\hat{\boldsymbol{z}}$ : $\mathbb{LM}_2(r,\theta,\phi) = \mathbb{R}(-\frac{\pi}{2}) \, \mathbb{LM}_1(r,\theta,\phi-\pi/2)$ . The scattered wave due to a plane wave incident along $\hat{\boldsymbol{s}} = (\theta_s,\phi_s)$ has the same form in the coordinate system, $\vec{r}^\prime = (x^\prime,y^\prime,z^\prime)$ , that is rotated so that $\hat{\boldsymbol{s}}$ is aligned with $\hat{\boldsymbol{z}}^\prime$ . The necessary coordinate transformation can be performed with the Euler rotation tensor

$$\mathbb{E}(\hat{\boldsymbol{s}}) = \begin{pmatrix}(\cos\theta_s-1... ...s \cos\phi_s & -\sin^2\phi_s (\cos\theta_s-1) + 1 & \cos\theta_s \end{pmatrix},$$ (26)

such that $\vec{r}^\prime = \mathbb{E}(\hat{\boldsymbol{s}}) \, \vec{r}$ . The general solution for the scattered wave therefore has the form

$$\vec{A}^{\text{S}}(\vec{r},\hat{\boldsymbol{s}}) = \mathbb{E}^{-1... ...\, \mathbb{E}(\hat{\boldsymbol{s}}) \, \vec{A}^{\text{I}}(\hat{\boldsymbol{s}})$$ (27)

This can be combined with Eq. (17) for the incident field's vector potential and substituted into Eqs. (1) through (4) to obtain the force and torque acting on the sphere.