Spherical wave expansion of a 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 (7.26) for the Green's function of the scalar Helmholtz equation. Let us take the limit ${\bfm r}'\rightarrow \infty$ of this equation. We can make the substitution $\vert{\bfm r} - {\bfm r}'\vert\simeq r'-{\bfm n}\!\cdot\!{\bfm r}$ on the left-hand-side, where ${\bfm n}$ is a unit vector pointing in the direction of ${\bfm r}'$. On the right-hand side, $r_<= r$ and $r_>=r'$. Furthermore, we can use the asymptotic form (7.16) for $h_l^{(1)}(kr)$. Thus, we obtain

$$\frac{{\rm e}^{\,{\rm i}\,kr'}}{4\pi\,r'} \,{\rm e}^{-{\rm i... ...kr) \,Y_{lm}^\ast(\theta', \varphi')\, Y_{lm}(\theta,\varphi).$$ (1312)

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

$${\rm e}^{\,{\rm i}\,{\bfm k}\cdot{\bfm r}}=4\pi \sum_{l=0}^\... ...}^{+l} Y_{lm}^\ast(\theta,\varphi)\, Y_{lm}(\theta',\varphi'),$$ (1313)

where ${\bfm k}$ is the wave vector with the spherical coordinates $k$, $\theta'$, $\varphi'$. The well known addition theorem for the spherical harmonics states that

$$P_l(\cos\gamma) = \frac{4\pi}{2l+1} \sum_{m=-1}^{+l} Y_{lm}^\ast (\theta,\varphi) \,Y_{lm}(\theta',\varphi'),$$ (1314)

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

$$\cos\gamma =\cos\theta\,\cos\theta' +\sin\theta\,\sin\theta'\,\cos (\varphi-\varphi').$$ (1315)

It follows from Eqs. (7.113) and (7.114) that

$${\rm e}^{\,{\rm i}\,{\bfm k}\cdot{\bfm r}}= \sum_{l=0}^\infty {\rm i}^l\,(2l+1)\,j_l(kr) \,P_l(\cos\gamma),$$ (1316)

or

$${\rm e}^{\,{\rm i}\,{\bfm k}\cdot{\bfm r}}=\sum_{l=0}^\infty{\rm i}^l \sqrt{4\pi\,(2l+1)}\,j_l(kr)\,Y_{l,0}(\gamma),$$ (1317)

since

$$Y_{l,0}(\theta) = \sqrt{\frac{2l+1}{4\pi} }\, P_l(\cos\theta).$$ (1318)

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

$${\bfm E}({\bfm r}) = (\hat{\bfm x} \pm {\rm i}\,\hat{\bfm y})\, {\rm e}^{\,{\rm i}\,kz},$$ (1319)

$$c{\bfm B}({\bfm r}) = \hat{\bfm z}\wedge{\bfm E} = \mp\,{\rm i} \,{\bfm E}.$$ (1320)

Since the plane wave is finite everywhere (including the origin), its multipole expansion (7.54) can only involve the well behaved radial eigenfunctions $j_l(kr)$. Thus,

$${\bfm E} = \sum_{l,m}\left[ a_\pm(l,m)\,j_l(kr)\, {\bfm X}_{lm} +\frac{\rm i}{k}\,b_\pm(l,m)\,\nabla\wedge j_l(kr) {\bfm X}_{lm} \right],$$ (1321)

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

$$[0.5ex]$$ (1322)

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 (7.53) of the vector spherical harmonics:

$$\int [f_l(r){\bfm X}_{l'm'}]^\ast \cdot [g_l(r) {\bfm X}_{lm}]\,d{\mit\Omega} = f_l^\ast \,g_l \,\delta_{ll'}\,\delta_{mm'},$$ (1323)

$$\int [f_l(r) {\bfm X}_{l'm'}]^\ast \cdot [\nabla\wedge g_l(r) {\bfm X}_{lm}]\,d{\mit\Omega} = 0.$$ (1324)

The first of these follows directly from Eq. (7.53a). The second follows from Eqs. (7.31), (7.53b), (7.59), and the identity

$$\nabla = \frac{\bfm r}{r} \frac{\partial}{\partial r} - \frac{\rm i}{r^2} \,{\bfm r}\wedge{\bfm L}.$$ (1325)

The coefficients $a_\pm(l,m)$ and $b_\pm(l,m)$ are obtained by taking the scalar product of Eqs. (7.120) with ${\bfm X}_{lm}^\ast$ and integrating over all solid angle, making use of the orthogonality relations (7.121). This yields

$$a_\pm(l,m)\,j_l(kr) = \int {\bfm X}_{lm}^\ast\!\cdot\!{\bfm E}\,d {\mit \Omega},$$ (1326)

$$b_\pm(l,m)\,j_l(kr) = \int {\bfm X}_{lm}^\ast\!\cdot\! c{\bfm B}\,d {\mit \Omega}.$$ (1327)

Substitution of Eqs. (7.52) and (7.120a) into Eq. (7.123a) gives

$$a_\pm(l,m)\,j_l(kr) = \int \frac{(L_\mp\,Y_{lm})^\ast}{\sqrt{l(l+1)}} \,{\rm e}^{\,{\rm i}\,kz}\,d{\mit \Omega},$$ (1328)

where the operators $L_\pm$ are defined in Eqs. (7.30). Making use of Eqs. (7.33), the above expression reduces to

$$a_\pm(l,m)\,j_l(kr) = \frac{\sqrt{(l\pm m)(l\mp m+1)}}{\sqrt... ...nt Y_{l,m\mp 1}^\ast\,{\rm e}^{\,{\rm i}\,kz}\,d{\mit \Omega}.$$ (1329)

If the expansion (7.117) is substituted for ${\rm e}^{\,{\rm i}\, kz}$, and use is made of the orthogonality properties of the spherical harmonics, then we obtain the result

$$a_\pm(l,m) = {\rm i}^l \,\sqrt{4\pi\,(2l+1)}\,\delta_{m,\pm 1}.$$ (1330)

It is clear from Eqs. (7.119b) and (7.123b) that

$$b_\pm(l,m) = \mp \,{\rm i}\,a_\pm(l,m).$$ (1331)

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

$${\bfm E}({\bfm r}) = \sum_{l=1}^\infty {\rm i}^l\sqrt{4\pi(2l+1)}\left[j_l(kr) {\bfm X}_{l,\pm 1}\pm\frac{1}{k}\,\nabla\wedge j_l(kr){\bfm X}_{l,\pm 1} \right],$$ (1332)

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

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