Linear array N hard spheres

Scattered field

The scattering amplitude from the $p$th sphere is expressed as : $$\begin{equation} f_p^{(s)}(\bb r_p) = \sum_{l=0}^{\infty}c_{p;l}\hl(r_p)Y_l^0(\theta_p) \end{equation}$$

where $\hl$ is the spherical Hankel function of the first kind, $Y_l^0$ are the spherical harmonics with azimuthal order 0.

The coefficients $c_{p;l}$ are found imposing $\big(\sum_{p} f_p^{(s)}+f^{(i)}\big)\big|_{r_p=a_p}=0$ and using the orthogonality of the spherical harmonics. This results in solving the system : $$\begin{eqnarray} c_{p;l} = &&u_l(ka_p)\Bigg[ e^{jkd_p}b_{l}\sqrt{\frac{4\pi}{2l+1}}\\ && + \sum_{q=p+1}^{N}\sum_{n=0}^{\infty}a_{l0;n 0}^{out-in}(kd_{pq},\pi) c_{q;n} + \sum_{q=1}^{p-1}\sum_{n=0}^{\infty}a_{l0;n 0}^{out-in}(kd_{pq},0 ) c_{q;n} \Bigg] \end{eqnarray}$$

where $u_l(ka_p)$ are the single sphere continuity coefficients, $b_l$ are the incident plane wave spherical expansion coefficients and $a_{l0;n0}^{out-in}(kd_{pq})$ are the out-to-in translational coefficients : $$\begin{eqnarray} u_l(ka_p) &=& -\frac{j_l(ka_p)}{\hl(ka_p)}\\ b_l &=& j^l(2l+1)\\ a_{l0;n 0}^{out-in}(kd_{pq},\theta_{pq}) &=& 4\pi\sum_{q=|l-n|}^{l+n}(-j)^{l-n-q} \hl(kd_{pq})Y_l^0(\theta_{pq})\cc G(l,n,q,0,0,0) \end{eqnarray}$$ where $j_l$ is the spherical Bessel function of the first kind and $\cc G$ are the Gaunt coefficients.

Far field scattering

In the far field, $\hl(kr_p)\approx (-j)^{l+1}\frac{e^{jkrp}}{kr_p}$, $r_p=r-d_p\cos\theta$, $\theta_p=\theta$ so the scattering amplitude from the $pth$ sphere can be written $\frac{e^{jkr}}{kr} f_p(\theta)$ where :

$$\begin{equation} f_p(\theta) = \sum_{l=0}^{\infty} (-j)^{l+1}c_ {p;l}Y_l^0(\theta) \end{equation}$$

The total scattering amplitude is the sum of the contribution from individual spheres is :
$$\begin{eqnarray} f(\theta) &=& \sum_{p=1}^{N} f_p(\theta)e^{-jkd_p\cos\theta} \\ &=& \sum_{l=1}^{\infty}(-j)^{l+1}Y_l^0(\theta) \sum_{p=1}^{N} c_{pl}e^{-jkd_p\cos\theta} \\ \end{eqnarray}$$

The normalized differential scattering cross section is therefore : $$\begin{equation} \frac{\sigma(\theta)}{a_p^2} = \frac{|f(\theta)|^2}{\left(ka_p\right)^2} \end{equation}$$

The integrated scattering cross section over the unit sphere can be written using the orthonormality of $Y_l^0$ : $$\begin{eqnarray} \sigma &=&\int_{\cc S} \Big(\sum_p f_p(\theta)e^{-jkd_p\cos\theta}\Big)\Big(\sum_q f_q^{*}(\theta)e^{jkd_q\cos\theta}\Big) d\Omega\\ &=&\sum_{p=1}^{N}\sum_{l=0}^{\infty}c_{pl} \sum_{q=1}^{N}\sum_{n=0}^{\infty}c_{qn}^{*} \int_{\cc S} e^{-jkd_{pq}\cos\theta}Y_l^0(\theta) Y_n^{0*}(\theta)d\Omega\\ &=&\sum_{p=1}^{N}\sigma_p^{(coupled)} + \sum_{l=0}^{\infty}\sum_{n=0}^{\infty} \sum_{p=1}^{N}\sum_{q>p}^{N} 2\Re(c_{pl}c_{qn}^{*})\cc I_{lnpq} \end{eqnarray}$$ where $\sigma_p^{(coupled)}$ is the scattering expression for the scattering of individual sphere. Note however that since $C_{pl}$ are different for each sphere than of the unperturbed case $\sigma_p^{(coupled)}\neq \sigma_p^{(0)}$.

Single hard sphere scattering

a) $\sigma(ka)$	b) $f(\theta;ka)$	c) $abs(h_l(ka))$

d) $c_{pl}(ka=0.5-5)$	(e) $c_{pl}(ka=5-20)$	(f) $c_{pl}(ka=30-40)$

a) Integrated and b) differential cross sections for a single sphere with normalized radius $ka$. c) The magnitude of the Hankel functions up to order $l=50$ and $ka=50$. The modal coefficients $c_{pl}(ka)$ for d) $ka=0.5-5$, e) $ka=5-20$, f) $ka=30-40$.

2 hard spheres scattering

Case for 2 identical spheres of normalized radius $ka$ and inter-distance $kd$.

--	ka=0.5	ka=1.0	ka=2.0	ka=4.0
$C_{0l}(kd)$
$C_{1l}(kd)$

Scattering coefficients of sphere 0 and sphere 1 for the coupled(solid lines) and uncoupled(dashed lines) systems with varying inter-sphere distance $kd$. Different graphs for $ka=0.5,1,2,4$. The different colours on each graph correspond the coefficient of a given mode order.

$\sigma(kd;ka)$	$f(\theta;ka=2.0)$	$f(\theta;ka=5.0)$	$f(\theta;ka=10)$

Scattering amplitudes and total scattering cross section with inter-sphere distance for various normalized radius.

--	$i$	$s$	$t$
p0
p1
p0+p1

Incident, scattered and total wave function for the uncoupled $0th$ sphere, uncoupled $1st$ sphere and coupled hard sphere system. Normalized radius $ka=1$ and inter-distance $kd=\pi$.

Approximate solution

Scattering amplitude

Assuming that the scattering amplitude at the $pth$ sphere from the $qth$ sphere is a plane wave with unknown scattering amplitude $c_q$, the scattering function of the $pth$ sphere is : $$\begin{eqnarray} g_p^{(s)}(\bb r_p) &=&g_{p}^{(s,0)}(r_p,\theta_p)\Big(1 + \sum_{q=1}^{p-1} c_q e^{-jkd_q} \Big) \\ &&+g_{p}^{(s,0)}(r_p,\pi-\theta_p)\sum_{q=p+1}^{N}c_q e^{+jkd_q} \end{eqnarray}$$

where $g_{p}^{(s,0)}$ is the far field scattering function due to an incident plane wave of the $pth$ sphere located at $d_p$ : $$\begin{eqnarray} g_{p}^{(s,0)}(r_p,\theta_p) &=& \frac{e^{jkr_p}}{kr_p}f_p(\theta_p) \\ f_p(\theta_p) &=& \sum_{l=0}^{\infty} (-j)^{l+1} c_{pl}^{(0)}Y_l^0(\theta_p)\\ c_{pl}^{(0)} &=& \sqrt{\frac{4\pi}{2l+1}}b_l u_l(ka_p)e^{jkd_p} \end{eqnarray}$$

System

The coefficients are found solving the system : $$\begin{eqnarray} \left(N-1\right)c_p = \sum_{q\neq p}e^{-jkd_{pq}} &&\Bigg[g_p(d_{pq},\theta_{pq})\\ +&& g_p(d_{pq}, \theta_{pq})\sum_{r=1}^{p-1}c_re^{-jkd_r} \\ +&& g_p(d_{pq},\pi-\theta_{pq})\sum_{r=p+1}^{N}c_re^{jkd_r} \Bigg] \end{eqnarray}$$ where $\theta_{pq}=0$ for $q>p$ and $\theta_{pq}=\pi$ for $q<p$.

For the case of 2 identical spheres with $d_1=0$ and $d_2=d$, using $f_1(\theta)=f(\theta)$, $f_2(\theta)=f(\theta)e^{jkd}$, the system gives : $$\begin{eqnarray} c_1 &=& \frac{f(0 )}{kd} + \frac{e^{jkd}f(\pi)}{kd}c_2\\ c_2 &=& \frac{e^{jkd}f(\pi)}{kd} + \frac{e^{jkd}f(\pi)}{kd}c_1\\ \end{eqnarray}$$

Far field scattering

$$\begin{eqnarray} f(\theta) &=& \sum_{p=1}^{N} f_p(\theta)e^{-jkd_p\cos\theta} \\ f_p(\theta) &=& \sum_{l=0}^{\infty} (-j)^{l+1}c_{pl}^{(0)}\Big( Y_l^0(\theta) + \sum_{q=1}^{p-1} c_qe^{-jkd_q}Y_l^0(\theta) + \sum_{q=p+1}^{N} c_qe^{ jkd_q}Y_l^0(\pi-\theta) \Big)\\ &=&\sum_{l=0}^{\infty} (-j)^{l+1}c_{pl}Y_l^0(\theta)\\ c_{pl} &=&c_{p;l}^{(0)}\Big( 1+\sum_{q=1}^{p-1}c_qe^{-jkd_q} + \sum_{q=p+1}^{N}(-1)^l c_qe^{jkd_q} \Big) \end{eqnarray}$$

where the parity of the spherical harmonics $Y_l^0(\pi-\theta)=(-1)^lY_l^0(\theta)$ has been used.

$f(ka=1.0;kd=20)$	$f(ka=5.0;kd=40)$	$c0l(ka=2;kd)$	$c1l(ka=2;kd)$

apx:Spherical coordinates

Plane wave expansion at normal incidence

The scalar plane wave travelling along $z$ is expanded upon spherical waves using : $$\begin{equation} e^{jkz} = e^{jkr\cos\theta} =\sum_{l=0}^{\infty} j^{l}(2l+1) j_l(kr)P_l(\cos\theta) \end{equation}$$ where $P_l$ are the Legendre polynomials.

Translational addition theorem coefficients

The spherical wave function $\psi_{l,0}^{out}=\hl Y_l^0(\theta_p)$ can be translated from $(r_p,\theta_p,\phi_p)$ to $(r,\theta,\phi)$ where $\bb r=\bb r_p+\bb t$ using : $$\begin{eqnarray} \psi_{l,0}^{out}(\bb r_p) &=& \sum_{n=0}^{\infty} a_{l,0;n,0}^{out-in}(\bb t)\psi_{n,0}^{in}(\bb r)\\ a_{l,0;n,0}^{out-in}(\bb t) &=& 4\pi\sum_{q=|l-n|}^{l+n} \left(-j\right)^{l-n-q}\psi^{out}_{q,0}(\bb t)\cc G(l,n,q,0,0,0) \end{eqnarray}$$

Spherical Hankel functions of the first kind in $(r_p,\theta_p,\phi_p)$ and in reference $(r,\theta,\phi)$ using the translational addition theorem with spherical Bessel functions expansion.

$\psi_{0}$	$\psi_{1}$	$\psi_{2}$	$\psi_{3}$