This chapter introduces the multipolar fields as solutions of
Maxwell’s equations for problems with spherical symmetry. These fields
will be expressed in a basis of the so-called multipolar modes. I will
provide an overview of two different approaches that lead to a
multipolar expansion of the field in different coordinate systems.
In general, we will consider electromagnetic fields in linear,
isotropic, and homogeneous media. Such fields are fully described by an
electric and magnetic field, \(\mathbf{E}\) and \(\mathbf{H}\), that for the aforementioned
conditions need to satisfy the vectorial Helmholtz equation, obtained by
the Maxwell equations: \[\label{eq:helmholtz}
\nabla^2\mathbf{E}+k^2\mathbf{E}=0 \qquad \qquad
\nabla^2\mathbf{H}+k^2\mathbf{H}=0\] where \(k=2\pi/\lambda\) is the wave number. This
section will demonstrate how the above two equations can be solved by
expanding \(\mathbf{E}\) and \(\mathbf{H}\) in multipolar modes,
constructed by first solving the scalar wave equation, \[\label{eq:helmholtz_scalar}
\nabla^2\psi+k^2\psi=0,\] in spherical coordinates by
separation of variables, \(\psi(r,\theta,\varphi)=\zeta(r)Y(\theta,\varphi).\)
Inserting this solution in Eq. [eq:helmholtz_scalar] will
yield independent ordinary differential equations for the radial and
angular part. The radial equation has the spherical Bessel functions,
\(j_j(r)\) and \(y_j(r)\) (or any linear combination
thereof) as solutions, while the angular equation will be satisfied by
the spherical harmonics (See App. [app:multipoles]).
Using the multipolar basis to expand \(\mathbf{E}\) and \(\mathbf{H}\), we may describe any electromagnetic field with such components. We will later see that this is a smart choice when describing the scattering process of spherical particles.
One approach to deriving the multipoles is to construct them from the spherical unit vectors, \(\mathbf{\hat{r}}, \boldsymbol{\hat{\theta}}\) and \(\boldsymbol{\hat{\varphi}}\), and the solution of the scalar wave equation, \(\psi\). This approach is taken in (Craig F.Bohren 2004), which leads to the so-called magnetic mode (or transverse electric mode), \(\mathbf{M}=\nabla \times (\mathbf{r}\psi)\), and an electric mode (transverse magnetic) as \(\mathbf{N}=\nabla \times \mathbf{M}/k\). Here, \(\mathbf{r}=r\hat{\mathbf{r}}\) is the radial vector. Using the solutions of Eq. [eq:helmholtz_scalar], the multipolar modes have the following expressions in the spherical basis and coordinates:
\[\begin{aligned} \mathbf{M}_{jm_z} &= \frac{m_z}{\sin \theta} P_j^{m_z}(\cos\theta) \zeta_j(kr) e^{im_z \phi} \, \boldsymbol{\hat{\theta}} - \frac{\d P_j^{m_z}(\cos\theta)}{\d \theta} \frac{\zeta_j(kr) e^{im_z \phi}}{\zeta_j(kr)} \,\boldsymbol{\hat{\varphi}} \\ \mathbf{N}_{jm_z} &= j(j + 1) P_j^{m_z}(\cos\theta) \frac{\zeta_j(kr)}{kr} e^{im_z \phi} \, \mathbf{\hat{r}} + \frac{1}{kr} \frac{\d \left[ kr \zeta_j(kr) \right]}{\d (kr)} \frac{\d P_j^{m_z}(\cos\theta)}{\d \theta} e^{im_z \phi} \, \boldsymbol{\hat{\theta}} \\ &\quad i e^{im_z \phi} \frac{m_z}{\sin \theta} P_j^{m_z}(\cos\theta) \frac{1}{kr} \frac{\d \left[ kr \zeta_j(kr) \right]}{\d (kr)} \, \boldsymbol{\hat{\varphi}} \end{aligned}\] \(j\in\mathbb{N}_0\ \text{and}\ m_z\in [-j,j]\) are indices appearing in the differential equations.
Any incident EM field can then be expanded in this basis by proving that the modes are orthogonal for all \(j, m_z\) and ensuring that the sum converges and is normalised by appropriate choice of coefficients, \(g^{(m)}_{jm_z}\) & \(g^{(e)}_{jm_z}\)(Craig F.Bohren 2004): \[\label{eq:E_i_expansion} \mathbf{E}_i = \sum_{j=0}^\infty \sum_{m_z=-j}^j g_{jm_z}^{(m)}\mathbf{M}_{jm_z} +g_{jm_z}^{(e)}\mathbf{N}_{jm_z}.\] As an example to be compared with the other approach, I will present the plane-wave expansion in this basis, which in the Cartesian basis has the simple expression for a plane wave propagating along the \(z\) axis and polarised along the \(x\) axis: \[\label{eq:PW_cart} \mathbf{E}_i = E_0e^{ikr\cos\theta}\mathrm{\mathbf{e}}_x\] It can be shown that all coefficients except \(g^{(m)}_{j,1}\) & \(g^{(e)}_{j,1}\) vanish. Following the notation in (Craig F.Bohren 2004), each mode can also be separated in an even and odd component, e.g. \(\mathbf{M}_{ejm_z}\) and \(\mathbf{M}_{ojm_z}\). This yields the simplified expansion of the plane wave \[\label{eq:PW_sph} \mathbf{E}_i=E_0\sum_{j=1}^\infty i^j \frac{2j+1}{j(j+1)}(\mathbf{M}_{oj1}-i\mathbf{N}_{ej1})\]
Another way of defining the multipoles is provided by M.E. Rose in
(Rose 1957). This
section follows his notation. In this approach, the multipoles are
demanded to be eigenfunctions of the z-component of the angular momentum
operator, \(\mathbf{J}=\mathbf{L}+\mathbf{S}\) and its
square, \(\mathbf{J}^2\). The
eigenfunctions of the orbital angular momentum operators are the
spherical harmonics, \(Y_l^m\), \[\label{eq.sphharm}
L_zY_l^{m_z}=m_zY_l^{m_z} \qquad
\mathbf{L}^2Y_l^{m_z}=l(l+1)Y_l^{m_z},\] and the spin
eigenfunctions are the normalized spherical basis vectors: \[\label{eq:S_ev}
\boldsymbol{\xi}_1=-\frac{1}{\sqrt{2}}(\mathbf{\hat{x}}+i\mathbf{\hat{y}})\qquad
\boldsymbol{\xi}_0=\mathbf{\hat{z}}\qquad
\boldsymbol{\xi}_{-1}=\frac{1}{\sqrt{2}}(\mathbf{\hat{x}}-i\mathbf{\hat{y}}),\]
that satisfy the eigenvalue equation \(\mathbf{S}\boldsymbol{\xi}_{\mu}=\mu\mathbf{S}\quad
\mu=1,0,-1\). These polarisation vectors correspond to left,
\(z\) and right circularly polarised
light respectively. Finding eigenfunctions of \(J_z\) requires finding common
eigenfunctions of \(L_z\) and \(S_z\), which is done with the
Clebsch-Gordan coefficients(See App. [app:multipoles]): \[\mathbf{T}_{jlm_z}=\sum_{\mu}
C(l1j;m_z-\mu,\mu)Y_{l}^{m_z-\mu}(\theta,\varphi)\boldsymbol{\xi}_{\mu}\]
These are called vector spherical harmonics (VSHs). The sum over \(\mu\) gives the three polarisation
components, adding up to one value of \(m_z\). From the VSHs, the electric and
magnetic multipoles can be derived by imposing rules for addition of
angular momenta and parity considerations(Tischler, Zambrana-Puyalto, and Molina-Terriza
2012), that simplify the VSHs to those with \(l=j, j+1\) or \(j-1\). Then, leaving only the \(l\) index, the multipoles are written as
the product of the spatial and angular components, \[\begin{aligned}
\text{Magnetic:}\qquad&\mathbf{A}^{(m)}_{lm_z}=C_l^{(m)}\zeta_l(kr)\mathbf{T}_{llm_z}(\theta,\varphi)\\
\text{Electric:}\qquad&\mathbf{A}^{(e)}_{lm_z}=C_{l+1}^{(e)}\zeta_{l+1}(kr)\mathbf{T}_{l,l+1,m_z}(\theta,\varphi)+
C_{l-1}^{(e)}\zeta_{l-1}(kr)\mathbf{T}_{l,l-1,m_z}(\theta,\varphi),
\end{aligned}\]
where the coefficients represent normalisation (or gauge)
constants.
In this basis with these VSHs, one can also expand a circularly
polarised plane wave propagating in z as \[\label{eq:rosePW}
\mathbf{E}_{pw}=ik\mathbf{A}_{pw}=ik\sqrt{2\pi}\sum_{l=1}^{\infty}i^l\sqrt{2l+1}
( \mathbf{A}_{lp}^{(m)}+ip\mathbf{A}_{lp}^{(e)}),\] As in Eq.
[eq:PW_sph], this expansion also only
sums over the index \(j\), keeping a
fixed value of \(m\) (here \(p\), representing the left or right
circular polarisation). The electric and magnetic multipole are, as in
Eq. [eq:PW_sph], also separated by a phase of
\(i\).
As mentioned before, any linear combination of the spherical Bessel functions of the first and second kind (also referred to as the spherical Bessel, \(j_j(kr)\), and Neumann, \(y_j(kr)\), functions) satisfy the radial part of the wave equation. However, for \(r\to0\), only \(j_j(kr)\) is regular, while \(y_j(kr)\) goes to \(-\infty\). At \(r\to\infty\), both functions are oscillating and thus physical. This means that in the expression for the multipoles, \(\zeta_j(kr)=j_j(kr)\) if its domain spans all space. This is for example not the case when working with multipoles in a scattering problem. Here, the spherical Bessel function will be used inside the scatterer, while the Hankel function, \(h_j(kr)=j_j(kr)+iy_j(kr)\), is used outside.