Spherical Wave Expansion of Vector Plane Wave

In discussing the scattering or absorption of electromagnetic radiation by localized systems, it is useful to be able to express a plane electromagnetic wave as a superposition of spherical waves.

Consider, first of all, the expansion of a scalar plane wave as a set of scalar spherical waves. This expansion is conveniently obtained from the expansion (1517) for the Green's function of the scalar Helmholtz equation. Let us take the limit $\vert{\bf r}'\vert\rightarrow\infty$ of this equation. We can make the substitution $\vert{\bf r} - {\bf r}'\vert\simeq r'-{\bf n}\cdot{\bf r}$ on the left-hand-side, where ${\bf n}$ is a unit vector pointing in the direction of ${\bf r}'$ . On the right-hand side, $r_<= r$ and $r_>=r'$ . Furthermore, we can use the asymptotic form (1432) for $h_l^{(1)}(k\,r)$ . Thus, we obtain

$$\frac{{\rm e}^{\,{\rm i}\,k\,r'}}{4\pi\,r'} \,{\rm e}^{-{\rm i}\,... ...^{l+1} j_l(k\,r) \,Y_{lm}^{\,\ast}(\theta', \varphi')\, Y_{lm}(\theta,\varphi).$$ (1566)

Canceling the factor ${\rm e}^{\,{\rm i} \,k\,r'}/r'$ on either side of the above equation, and taking the complex conjugate, we get the following expansion for a scalar plane wave,

$${\rm e}^{\,{\rm i}\,{\bf k}\cdot{\bf r}}=4\pi \sum_{l=0,\infty} {... ...k\,r) \sum_{m=-l,+l}Y_{lm}^{\,\ast}(\theta,\varphi)\, Y_{lm}(\theta',\varphi'),$$ (1567)

where ${\bf k}$ is a wavevector with the spherical coordinates $k$ , $\theta'$ , $\varphi'$ . The well-known addition theorem for the spherical harmonics states that

$$P_l(\cos\gamma) = \frac{4\pi}{2\,l+1} \sum_{m=-l,+l}Y_{lm}^{\,\ast} (\theta,\varphi) \,Y_{lm}(\theta',\varphi'),$$ (1568)

where $\gamma$ is the angle subtended between the vectors ${\bf r}$ and ${\bf r'}$ . Consequently,

$$\cos\gamma = \cos\theta\,\cos\theta'+\sin\theta\,\sin\theta'\,\cos(\varphi-\varphi').$$ (1569)

It follows from Equations (1569) and (1570) that

$${\rm e}^{\,{\rm i}\,{\bf k}\cdot{\bf r}}= \sum_{l=0,\infty} {\rm i}^l\,(2\,l+1)\,j_l(k\,r) \,P_l(\cos\gamma),$$ (1570)

or

$${\rm e}^{\,{\rm i}\,{\bf k}\cdot{\bf r}}=\sum_{l=0,\infty}{\rm i}^l \sqrt{4\pi\,(2\,l+1)}\,j_l(k\,r)\,Y_{l0}(\gamma),$$ (1571)

because

$$Y_{l0}(\theta) = \sqrt{\frac{2\,l+1}{4\pi} }\, P_l(\cos\theta).$$ (1572)

Let us now make an equivalent expansion for a circularly polarized plane wave incident along the $z$ -axis:

$${\bf E}({\bf r}) = ({\bf e}_x \pm {\rm i}\,{\bf e}_y)\, {\rm e}^{\,{\rm i}\,k\,z},$$ (1573)

$$c\,{\bf B}({\bf r}) = {\bf e}_z\times{\bf E} = \mp\,{\rm i} \,{\bf E}.$$ (1574)

Because the plane wave is finite everywhere (including the origin), its multipole expansion (1479)-(1480) can only involve the well-behaved radial eigenfunctions $j_l(k\,r)$ . Thus,

$${\bf E} = \sum_{l,m}\left[ a_\pm(l,m)\,j_l(k\,r)\, {\bf X}_{lm} +\frac{\rm i}{k}\,b_\pm(l,m)\,\nabla\times j_l(k\,r) {\bf X}_{lm} \right],$$ (1575)

$$c\,{\bf B} =\sum_{l,m} \left[ \frac{-{\rm i}}{k}\,a_\pm(l,m)\,\nabla\times j_l(k\,r)\,{\bf X}_{lm} +b_\pm(l,m)\,j_l(k\,r)\,{\bf X}_{lm}\right].$$ (1576)

To determine the coefficients $a_\pm(l,m)$ and $b_\pm(l,m)$ , we make use of a slight generalization of the standard orthogonality properties (1477)-(1478) of the vector spherical harmonics:

$$\oint [f_l(r)\,{\bf X}_{l'm'}]^{\,\ast} \cdot [g_l(r)\, {\bf X}_{lm}]\,d{\mit\Omega} = f_l^{\,\ast} \,g_l \,\delta_{ll'}\,\delta_{mm'},$$ (1577)

$$\oint [f_l(r) \,{\bf X}_{l'm'}]^{\,\ast} \cdot [\nabla\times g_l(r)\, {\bf X}_{lm}]\,d{\mit\Omega} =0.$$ (1578)

The first of these follows directly from Equation (1477). The second follows from Equations (1442), (1478), (1486), and the identity

$$\nabla \equiv \frac{\bf r}{r}\, \frac{\partial}{\partial r} - \frac{\rm i}{r^{\,2}} \,{\bf r}\times{\bf L}.$$ (1579)

The coefficients $a_\pm(l,m)$ and $b_\pm(l,m)$ are obtained by taking the scalar product of Equations (1577)-(1578) with ${\bf X}_{lm}^{\,\ast}$ and integrating over all solid angle, making use of the orthogonality relations (1579)-(1580). This yields

$$a_\pm(l,m)\,j_l(k\,r) = \oint {\bf X}_{lm}^{\,\ast}\cdot{\bf E}\,d {\mit \Omega},$$ (1580)

$$b_\pm(l,m)\,j_l(k\,r) = \oint {\bf X}_{lm}^{\,\ast}\cdot\c{\bf B}\,d {\mit \Omega}.$$ (1581)

Substitution of Equations (1476) and (1577) into Equation (1582) gives

$$a_\pm(l,m)\,j_l(k\,r) = \oint \frac{(L_\mp\,Y_{lm})^{\,\ast}}{\sqrt{l\,(l+1)}} \,{\rm e}^{\,{\rm i}\,k\,z}\,d{\mit \Omega},$$ (1582)

where the operators $L_\pm$ are defined in Equations (1439)-(1440). Making use of Equations (1444)-(1446), the above expression reduces to

$$a_\pm(l,m)\,j_l(k\,r) = \frac{\sqrt{(l\pm m)\,(l\mp m+1)}}{\sqrt{l\,(l+1)}} \oint Y_{l,m\mp 1}^{\,\ast}\,{\rm e}^{\,{\rm i}\,k\,z}\,d{\mit \Omega}.$$ (1583)

If the expansion (1573) is substituted for ${\rm e}^{\,{\rm i}\, k\,z}$ , and use is made of the orthogonality properties of the spherical harmonics, then we obtain the result

$$a_\pm(l,m) = {\rm i}^{\,l} \,\sqrt{4\pi\,(2\,l+1)}\,\delta_{m,\pm 1}.$$ (1584)

It is clear from Equations (1576) and (1583) that

$$b_\pm(l,m) = \mp \,{\rm i}\,a_\pm(l,m).$$ (1585)

Thus, the general expansion of a circularly polarized plane wave takes the form

$${\bf E}({\bf r}) = \sum_{l=1,\infty} {\rm i}^{\,l}\sqrt{4\pi\,(2\,l+1)}\left[j_l(k... ... X}_{l,\pm 1}\pm\frac{1}{k}\,\nabla\times j_l(k\,r)\,{\bf X}_{l,\pm 1} \right],$$ (1586)

$${\bf B}({\bf r}) = \sum_{l=1,\infty} {\rm i}^{\,l}\sqrt{4\pi\,(2\,l+1)}\left[\frac... ... j_l(k\,r)\,{\bf X}_{l,\pm 1}\mp {\rm i}\,j_l(k\,r)\, {\bf X}_{l,\pm 1}\right].$$ (1587)

The expansion for a linearly polarized plane wave is easily obtained by taking the appropriate linear combination of the above two expansions.