Vector spherical harmonics

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

Definition

Several conventions have been used to define the VSH.^[1]^[2]^[3]^[4]^[5] We follow that of Barrera et al.. Given a scalar spherical harmonic $Y ℓm (θ, φ)$ , we define three VSH:

$𝐘_{ℓ m} = Y_{ℓ m} \hat{𝐫},$
$𝜳_{ℓ m} = r \nabla Y_{ℓ m},$
$𝜱_{ℓ m} = 𝐫 \times \nabla Y_{ℓ m},$

with $\hat{𝐫}$ being the unit vector along the radial direction in spherical coordinates and $𝐫$ the vector along the radial direction with the same norm as the radius, i.e., $𝐫 = r \hat{𝐫}$ . The radial factors are included to guarantee that the dimensions of the VSH are the same as those of the ordinary spherical harmonics and that the VSH do not depend on the radial spherical coordinate.

The interest of these new vector fields is to separate the radial dependence from the angular one when using spherical coordinates, so that a vector field admits a multipole expansion

$𝐄 = \sum_{ℓ = 0}^{\infty} \sum_{m = - ℓ}^{ℓ} (E_{ℓ m}^{r} (r) 𝐘_{ℓ m} + E_{ℓ m}^{(1)} (r) 𝜳_{ℓ m} + E_{ℓ m}^{(2)} (r) 𝜱_{ℓ m}) .$

The labels on the components reflect that $E_{ℓ m}^{r}$ is the radial component of the vector field, while $E_{ℓ m}^{(1)}$ and $E_{ℓ m}^{(2)}$ are transverse components (with respect to the radius vector $𝐫$ ).

In physics

In physics, the vector spherical harmonics $𝐘_{j, ℓ, s}^{m_{j}}$ are defined as spin $s = 1$ eigenfunctions of the angular momentum operators $J^{2}, J_{z}, L^{2}$ , and $S^{2}$ , where $𝐉 = 𝐋 + 𝐒$ is the total angular momentum.^[6] They are written as $𝐘_{j, ℓ, 1}^{m_{j}} (𝐤) = \sum_{m_{ℓ} = - ℓ}^{+ ℓ} \sum_{m_{s} = - 1}^{+ 1} ⟨ j m_{j} | ℓ 1 m_{ℓ} m_{s} ⟩ Y_{ℓ}^{m_{ℓ}} (𝐤) {\hat{𝐞}}_{m_{s}},$ which are linear combinations of the scalar spherical harmonics $Y_{ℓ}^{m_{ℓ}}$ with the vector angular momentum basis ${\hat{𝐞}}_{\pm 1} = \mp \frac{\hat{𝐱} \pm i \hat{𝐲}}{\sqrt{2}}, {\hat{𝐞}}_{0} = \hat{𝐳} .$ using the Clebsch-Gordan coefficients $⟨ j m_{j} | ℓ 1 m_{ℓ} m_{s} ⟩$ .

Because vector bosons such as the photon are spin-one, the vector spherical harmonics are commonly used in physics to describe vector and pseudovector interactions, such as electromagnetic transitions, in atomic and nuclear systems. They are a special ( $s = 1$ ) case of the spin spherical harmonics.

To derive these relations, one begins with the plane-wave expansion for plane waves with vector polarization.

Main properties

Symmetry

Like the scalar spherical harmonics, the VSH satisfy

$\begin{matrix} 𝐘_{ℓ, - m} & = (- 1)^{m} 𝐘_{ℓ m}^{*}, \\ 𝜳_{ℓ, - m} & = (- 1)^{m} 𝜳_{ℓ m}^{*}, \\ 𝜱_{ℓ, - m} & = (- 1)^{m} 𝜱_{ℓ m}^{*}, \end{matrix}$

which cuts the number of independent functions roughly in half. The star indicates complex conjugation.

Orthogonality

The VSH are orthogonal in the usual three-dimensional way at each point $𝐫$ :

$\begin{matrix} 𝐘_{ℓ m} (𝐫) \cdot 𝜳_{ℓ m} (𝐫) & = 0, \\ 𝐘_{ℓ m} (𝐫) \cdot 𝜱_{ℓ m} (𝐫) & = 0, \\ 𝜳_{ℓ m} (𝐫) \cdot 𝜱_{ℓ m} (𝐫) & = 0. \end{matrix}$

They are also orthogonal in Hilbert space:

$\begin{matrix} \int 𝐘_{ℓ m} \cdot 𝐘_{ℓ^{'} m^{'}}^{*} d Ω & = δ_{ℓ ℓ^{'}} δ_{m m^{'}}, \\ \int 𝜳_{ℓ m} \cdot 𝜳_{ℓ^{'} m^{'}}^{*} d Ω & = ℓ (ℓ + 1) δ_{ℓ ℓ^{'}} δ_{m m^{'}}, \\ \int 𝜱_{ℓ m} \cdot 𝜱_{ℓ^{'} m^{'}}^{*} d Ω & = ℓ (ℓ + 1) δ_{ℓ ℓ^{'}} δ_{m m^{'}}, \\ \int 𝐘_{ℓ m} \cdot 𝜳_{ℓ^{'} m^{'}}^{*} d Ω & = 0, \\ \int 𝐘_{ℓ m} \cdot 𝜱_{ℓ^{'} m^{'}}^{*} d Ω & = 0, \\ \int 𝜳_{ℓ m} \cdot 𝜱_{ℓ^{'} m^{'}}^{*} d Ω & = 0. \end{matrix}$

An additional result at a single point $𝐫$ (not reported in Barrera et al., 1985) is, for all $ℓ, m, ℓ^{'}, m^{'}$ ,

$\begin{matrix} 𝐘_{ℓ m} (𝐫) \cdot 𝜳_{ℓ^{'} m^{'}} (𝐫) & = 0, \\ 𝐘_{ℓ m} (𝐫) \cdot 𝜱_{ℓ^{'} m^{'}} (𝐫) & = 0. \end{matrix}$

Vector multipole moments

The orthogonality relations allow one to compute the spherical multipole moments of a vector field as

$\begin{matrix} E_{ℓ m}^{r} & = \int 𝐄 \cdot 𝐘_{ℓ m}^{*} d Ω, \\ E_{ℓ m}^{(1)} & = \frac{1}{ℓ (ℓ + 1)} \int 𝐄 \cdot 𝜳_{ℓ m}^{*} d Ω, \\ E_{ℓ m}^{(2)} & = \frac{1}{ℓ (ℓ + 1)} \int 𝐄 \cdot 𝜱_{ℓ m}^{*} d Ω . \end{matrix}$

The gradient of a scalar field

Given the multipole expansion of a scalar field

$ϕ = \sum_{ℓ = 0}^{\infty} \sum_{m = - ℓ}^{ℓ} ϕ_{ℓ m} (r) Y_{ℓ m} (θ, ϕ),$

we can express its gradient in terms of the VSH as

$\nabla ϕ = \sum_{ℓ = 0}^{\infty} \sum_{m = - ℓ}^{ℓ} (\frac{d ϕ_{ℓ m}}{d r} 𝐘_{ℓ m} + \frac{ϕ_{ℓ m}}{r} 𝜳_{ℓ m}) .$

Divergence

For any multipole field we have

$\begin{matrix} \nabla \cdot (f (r) 𝐘_{ℓ m}) & = (\frac{d f}{d r} + \frac{2}{r} f) Y_{ℓ m}, \\ \nabla \cdot (f (r) 𝜳_{ℓ m}) & = - \frac{ℓ (ℓ + 1)}{r} f Y_{ℓ m}, \\ \nabla \cdot (f (r) 𝜱_{ℓ m}) & = 0. \end{matrix}$

By superposition we obtain the divergence of any vector field:

$\nabla \cdot 𝐄 = \sum_{ℓ = 0}^{\infty} \sum_{m = - ℓ}^{ℓ} (\frac{d E_{ℓ m}^{r}}{d r} + \frac{2}{r} E_{ℓ m}^{r} - \frac{ℓ (ℓ + 1)}{r} E_{ℓ m}^{(1)}) Y_{ℓ m} .$

We see that the component on $Φ ℓm$ is always solenoidal.

Curl

For any multipole field we have

$\begin{matrix} \nabla \times (f (r) 𝐘_{ℓ m}) & = - \frac{1}{r} f 𝜱_{ℓ m}, \\ \nabla \times (f (r) 𝜳_{ℓ m}) & = (\frac{d f}{d r} + \frac{1}{r} f) 𝜱_{ℓ m}, \\ \nabla \times (f (r) 𝜱_{ℓ m}) & = - \frac{ℓ (ℓ + 1)}{r} f 𝐘_{ℓ m} - (\frac{d f}{d r} + \frac{1}{r} f) 𝜳_{ℓ m} . \end{matrix}$

By superposition we obtain the curl of any vector field:

$\nabla \times 𝐄 = \sum_{ℓ = 0}^{\infty} \sum_{m = - ℓ}^{ℓ} (- \frac{ℓ (ℓ + 1)}{r} E_{ℓ m}^{(2)} 𝐘_{ℓ m} - (\frac{d E_{ℓ m}^{(2)}}{d r} + \frac{1}{r} E_{ℓ m}^{(2)}) 𝜳_{ℓ m} + (- \frac{1}{r} E_{ℓ m}^{r} + \frac{d E_{ℓ m}^{(1)}}{d r} + \frac{1}{r} E_{ℓ m}^{(1)}) 𝜱_{ℓ m}) .$

Laplacian

The action of the Laplace operator $Δ = \nabla \cdot \nabla$ separates as follows:

$Δ (f (r) 𝐙_{ℓ m}) = (\frac{1}{r^{2}} \frac{\partial}{\partial r} r^{2} \frac{\partial f}{\partial r}) 𝐙_{ℓ m} + f (r) Δ 𝐙_{ℓ m},$ where $𝐙_{ℓ m} = 𝐘_{ℓ m}, 𝜳_{ℓ m}, 𝜱_{ℓ m}$ and

$\begin{matrix} Δ 𝐘_{ℓ m} & = - \frac{1}{r^{2}} (2 + ℓ (ℓ + 1)) 𝐘_{ℓ m} + \frac{2}{r^{2}} 𝜳_{ℓ m}, \\ Δ 𝜳_{ℓ m} & = \frac{2 ℓ (ℓ + 1)}{r^{2}} 𝐘_{ℓ m} - \frac{1}{r^{2}} ℓ (ℓ + 1) 𝜳_{ℓ m}, \\ Δ 𝜱_{ℓ m} & = - \frac{1}{r^{2}} ℓ (ℓ + 1) 𝜱_{ℓ m} . \end{matrix}$

Also note that this action becomes symmetric, i.e. the off-diagonal coefficients are equal to $\frac{2}{r^{2}} \sqrt{ℓ (ℓ + 1)}$ , for properly normalized VSH.

Examples

𝜳_{1 m}

𝜳_{2 m}

𝜳_{3 m}

𝜱_{1 m}

𝜱_{2 m}

𝜱_{3 m}

Visualizations of the real parts of

ℓ = 1, 2, 3

VSHs. Click to expand.

First vector spherical harmonics

$ℓ = 0$ . $\begin{matrix} 𝐘_{00} & = \sqrt{\frac{1}{4 π}} \hat{𝐫}, \\ 𝜳_{00} & = 𝟎, \\ 𝜱_{00} & = 𝟎 . \end{matrix}$
$ℓ = 1$ . $\begin{matrix} 𝐘_{10} & = \sqrt{\frac{3}{4 π}} \cos θ \hat{𝐫}, \\ 𝐘_{11} & = - \sqrt{\frac{3}{8 π}} e^{i φ} \sin θ \hat{𝐫}, \end{matrix}$ $\begin{matrix} 𝜳_{10} & = - \sqrt{\frac{3}{4 π}} \sin θ \hat{𝜽}, \\ 𝜳_{11} & = - \sqrt{\frac{3}{8 π}} e^{i φ} (\cos θ \hat{𝜽} + i \hat{𝝋}), \end{matrix}$ $\begin{matrix} 𝜱_{10} & = - \sqrt{\frac{3}{4 π}} \sin θ \hat{𝝋}, \\ 𝜱_{11} & = \sqrt{\frac{3}{8 π}} e^{i φ} (i \hat{𝜽} - \cos θ \hat{𝝋}) . \end{matrix}$
$ℓ = 2$ . $\begin{matrix} 𝐘_{20} & = \frac{1}{4} \sqrt{\frac{5}{π}} (3 \cos^{2} θ - 1) \hat{𝐫}, \\ 𝐘_{21} & = - \sqrt{\frac{15}{8 π}} \sin θ \cos θ e^{i φ} \hat{𝐫}, \\ 𝐘_{22} & = \frac{1}{4} \sqrt{\frac{15}{2 π}} \sin^{2} θ e^{2 i φ} \hat{𝐫} . \end{matrix}$ $\begin{matrix} 𝜳_{20} & = - \frac{3}{2} \sqrt{\frac{5}{π}} \sin θ \cos θ \hat{𝜽}, \\ 𝜳_{21} & = - \sqrt{\frac{15}{8 π}} e^{i φ} (\cos 2 θ \hat{𝜽} + i \cos θ \hat{𝝋}), \\ 𝜳_{22} & = \sqrt{\frac{15}{8 π}} \sin θ e^{2 i φ} (\cos θ \hat{𝜽} + i \hat{𝝋}) . \end{matrix}$ $\begin{matrix} 𝜱_{20} & = - \frac{3}{2} \sqrt{\frac{5}{π}} \sin θ \cos θ \hat{𝝋}, \\ 𝜱_{21} & = \sqrt{\frac{15}{8 π}} e^{i φ} (i \cos θ \hat{𝜽} - \cos 2 θ \hat{𝝋}), \\ 𝜱_{22} & = \sqrt{\frac{15}{8 π}} \sin θ e^{2 i φ} (- i \hat{𝜽} + \cos θ \hat{𝝋}) . \end{matrix}$

Expressions for negative values of $m$ are obtained by applying the symmetry relations.

Applications

Electrodynamics

The VSH are especially useful in the study of multipole radiation fields. For instance, a magnetic multipole is due to an oscillating current with angular frequency $ω$ and complex amplitude

$\hat{𝐉} = J (r) 𝜱_{ℓ m},$

and the corresponding electric and magnetic fields, can be written as

$\begin{matrix} \hat{𝐄} & = E (r) 𝜱_{ℓ m}, \\ \hat{𝐁} & = B^{r} (r) 𝐘_{ℓ m} + B^{(1)} (r) 𝜳_{ℓ m} . \end{matrix}$

Substituting into Maxwell equations, Gauss's law is automatically satisfied

$\nabla \cdot \hat{𝐄} = 0,$

while Faraday's law decouples as

$\nabla \times \hat{𝐄} = - i ω \hat{𝐁} \Rightarrow {\begin{matrix} \frac{ℓ (ℓ + 1)}{r} E = i ω B^{r}, \\ \frac{d E}{d r} + \frac{E}{r} = i ω B^{(1)} . \end{matrix}$

Gauss' law for the magnetic field implies

$\nabla \cdot \hat{𝐁} = 0 \Rightarrow \frac{d B^{r}}{d r} + \frac{2}{r} B^{r} - \frac{ℓ (ℓ + 1)}{r} B^{(1)} = 0,$

and Ampère–Maxwell's equation gives

$\nabla \times \hat{𝐁} = μ_{0} \hat{𝐉} + i μ_{0} ε_{0} ω \hat{𝐄} \Rightarrow - \frac{B^{r}}{r} + \frac{d B^{(1)}}{d r} + \frac{B^{(1)}}{r} = μ_{0} J + i ω μ_{0} ε_{0} E .$

In this way, the partial differential equations have been transformed into a set of ordinary differential equations.

Alternative definition

In many applications, vector spherical harmonics are defined as fundamental set of the solutions of vector Helmholtz equation in spherical coordinates.^[7]^[8]

In this case, vector spherical harmonics are generated by scalar functions, which are solutions of scalar Helmholtz equation with the wavevector $𝐤$ . $\begin{matrix} ψ_{e m n} = \cos m φ P_{n}^{m} (\cos ϑ) z_{n} (k r) \\ ψ_{o m n} = \sin m φ P_{n}^{m} (\cos ϑ) z_{n} (k r) \end{matrix}$ here $P_{n}^{m} (\cos θ)$ are the associated Legendre polynomials, and $z_{n} (k r)$ are any of the spherical Bessel functions.

Vector spherical harmonics are defined as:

longitudinal harmonics: $𝐋_{^{e}_{o} m n} = 𝜵 ψ_{^{e}_{o} m n}$
magnetic harmonics: $𝐌_{_{o}^{e} m n} = \nabla \times (𝐫 ψ_{_{o}^{e} m n})$
electric harmonics: $𝐍_{_{o}^{e} m n} = \frac{\nabla \times 𝐌_{_{o}^{e} m n}}{k}$

Here we use harmonics real-valued angular part, where $m \geq 0$ , but complex functions can be introduced in the same way.

Let us introduce the notation $ρ = k r$ . In the component form vector spherical harmonics are written as: $\begin{matrix} 𝐌_{e m n} (k, 𝐫) = \frac{- m}{\sin (θ)} \sin (m φ) P_{n}^{m} (\cos (θ)) z_{n} (ρ) 𝐞_{θ} \\ - \cos (m φ) \frac{d P_{n}^{m} (\cos (θ))}{d θ} z_{n} (ρ) 𝐞_{φ} \end{matrix}$ $\begin{matrix} 𝐌_{o m n} (k, 𝐫) = \frac{m}{\sin (θ)} \cos (m φ) P_{n}^{m} (\cos (θ)) z_{n} (ρ) 𝐞_{θ} \\ - \sin (m φ) \frac{d P_{n}^{m} (\cos (θ))}{d θ} z_{n} (ρ) 𝐞_{φ} \end{matrix}$

$\begin{matrix} 𝐍_{e m n} (k, 𝐫) = \frac{z_{n} (ρ)}{ρ} \cos (m φ) n (n + 1) P_{n}^{m} (\cos (θ)) 𝐞_{𝐫} \\ + \cos (m φ) \frac{d P_{n}^{m} (\cos (θ))}{d θ} \frac{1}{ρ} \frac{d}{d ρ} [ρ z_{n} (ρ)] 𝐞_{θ} \\ - m \sin (m φ) \frac{P_{n}^{m} (\cos (θ))}{\sin (θ)} \frac{1}{ρ} \frac{d}{d ρ} [ρ z_{n} (ρ)] 𝐞_{φ} \end{matrix}$

$\begin{matrix} 𝐍_{o m n} (k, 𝐫) = \frac{z_{n} (ρ)}{ρ} \sin (m φ) n (n + 1) P_{n}^{m} (\cos (θ)) 𝐞_{𝐫} \\ + \sin (m φ) \frac{d P_{n}^{m} (\cos (θ))}{d θ} \frac{1}{ρ} \frac{d}{d ρ} [ρ z_{n} (ρ)] 𝐞_{θ} \\ + m \cos (m φ) \frac{P_{n}^{m} (\cos (θ))}{\sin (θ)} \frac{1}{ρ} \frac{d}{d ρ} [ρ z_{n} (ρ)] 𝐞_{φ} \end{matrix}$ There is no radial part for magnetic harmonics. For electric harmonics, the radial part decreases faster than angular, and for big $ρ$ can be neglected. We can also see that for electric and magnetic harmonics angular parts are the same up to permutation of the polar and azimuthal unit vectors, so for big $ρ$ electric and magnetic harmonics vectors are equal in value and perpendicular to each other.

Longitudinal harmonics: $\begin{matrix} 𝐋_{^{e}_{o} m n} (k, 𝐫) = & \frac{\partial}{\partial r} z_{n} (k r) P_{n}^{m} (\cos θ)^{\cos}_{\sin} m φ 𝐞_{r} \\ + & \frac{1}{r} z_{n} (k r) \frac{\partial}{\partial θ} P_{n}^{m} (\cos θ)^{\cos}_{\sin} m φ 𝐞_{θ} \\ \mp & \frac{m}{r \sin θ} z_{n} (k r) P_{n}^{m} (\cos θ)^{\sin}_{\cos} m φ 𝐞_{φ} \end{matrix}$

Orthogonality

The solutions of the Helmholtz vector equation obey the following orthogonality relations:^[8] $\begin{matrix} \int_{0}^{2 π} \int_{0}^{π} 𝐋_{_{o}^{e} m n} \cdot 𝐋_{_{o}^{e} m n} \sin ϑ d ϑ d φ & = (1 + δ_{m, 0}) \frac{2 π}{(2 n + 1)^{2}} \frac{(n + m)!}{(n - m)!} k^{2} {n {[z_{n - 1} (k r)]}^{2} + (n + 1) {[z_{n + 1} (k r)]}^{2}} \\ \int_{0}^{2 π} \int_{0}^{π} 𝐌_{_{o}^{e} m n} \cdot 𝐌_{_{o}^{e} m n} \sin ϑ d ϑ d φ & = (1 + δ_{m, 0}) \frac{2 π}{2 n + 1} \frac{(n + m)!}{(n - m)!} n (n + 1) {[z_{n} (k r)]}^{2} \\ \int_{0}^{2 π} \int_{0}^{π} 𝐍_{_{o}^{e} m n} \cdot 𝐍_{_{o}^{e} m n} \sin ϑ d ϑ d φ & = (1 + δ_{m, 0}) \frac{2 π}{(2 n + 1)^{2}} \frac{(n + m)!}{(n - m)!} n (n + 1) {(n + 1) {[z_{n - 1} (k r)]}^{2} + n {[z_{n + 1} (k r)]}^{2}} \\ \int_{0}^{π} \int_{0}^{2 π} 𝐋_{_{o}^{e} m n} \cdot 𝐍_{_{o}^{e} m n} \sin ϑ d ϑ d φ & = (1 + δ_{m, 0}) \frac{2 π}{(2 n + 1)^{2}} \frac{(n + m)!}{(n - m)!} n (n + 1) k {{[z_{n - 1} (k r)]}^{2} - {[z_{n + 1} (k r)]}^{2}} \end{matrix}$

All other integrals over the angles between different functions or functions with different indices are equal to zero.

Rotation and inversion

Under rotation, vector spherical harmonics are transformed through each other in the same way as the corresponding scalar spherical functions, which are generating for a specific type of vector harmonics. For example, if the generating functions are the usual spherical harmonics, then the vector harmonics will also be transformed through the Wigner D-matrices^[9]^[10]^[11] $\hat{D} (α, β, γ) 𝐘_{J M}^{(s)} (θ, φ) = \sum_{M^{'} = - J}^{J} [D_{M M^{'}}^{(J)} (α, β, γ)]^{*} 𝐘_{J M^{'}}^{(s)} (θ, φ),$ The behavior under rotations is the same for electrical, magnetic and longitudinal harmonics.

Under inversion, electric and longitudinal spherical harmonics behave in the same way as scalar spherical functions, i.e. $\hat{I} 𝐍_{J M} (θ, φ) = (- 1)^{J} 𝐍_{J M} (θ, φ),$ and magnetic ones have the opposite parity: $\hat{I} 𝐌_{J M} (θ, φ) = (- 1)^{J + 1} 𝐌_{J M} (θ, φ),$

Fluid dynamics

In the calculation of the Stokes' law for the drag that a viscous fluid exerts on a small spherical particle, the velocity distribution obeys Navier–Stokes equations neglecting inertia, i.e.,

$\begin{matrix} 0 & = \nabla \cdot 𝐯, \\ 𝟎 & = - \nabla p + η \nabla^{2} 𝐯, \end{matrix}$

with the boundary conditions

$𝐯 = {\begin{matrix} 𝟎 & r = a, \\ - 𝐔_{0} & r \to \infty . \end{matrix}$

where U is the relative velocity of the particle to the fluid far from the particle. In spherical coordinates this velocity at infinity can be written as

$𝐔_{0} = U_{0} (\cos θ \hat{𝐫} - \sin θ \hat{𝜽}) = U_{0} (𝐘_{10} + 𝜳_{10}) .$

The last expression suggests an expansion in spherical harmonics for the liquid velocity and the pressure

$\begin{matrix} p & = p (r) Y_{10}, \\ 𝐯 & = v^{r} (r) 𝐘_{10} + v^{(1)} (r) 𝜳_{10} . \end{matrix}$

Substitution in the Navier–Stokes equations produces a set of ordinary differential equations for the coefficients.

Integral relations

Here the following definitions are used:

$\begin{matrix} Y_{e m n} & = \cos m φ P_{n}^{m} (\cos θ) \\ Y_{o m n} & = \sin m φ P_{n}^{m} (\cos θ) \end{matrix}$

$𝐗_{^{e}_{o} m n} (\frac{𝐤}{k}) = \nabla \times (𝐤 Y_{^{o}_{e} m n} (\frac{𝐤}{k}))$

$𝐙_{^{o}_{e} m n} (\frac{𝐤}{k}) = i \frac{𝐤}{k} \times 𝐗_{^{e}_{o} m n} (\frac{𝐤}{k})$ In case, when instead of $z_{n}$ are spherical Bessel functions, with help of plane wave expansion one can obtain the following integral relations:^[12]

$𝐍_{p m n}^{(1)} (k, 𝐫) = \frac{i^{- n}}{4 π} \int 𝐙_{p m n} (\frac{𝐤}{k}) e^{i 𝐤 \cdot 𝐫} d Ω_{k}$

$𝐌_{p m n}^{(1)} (k, 𝐫) = \frac{i^{- n}}{4 π} \int 𝐗_{p m n} (\frac{𝐤}{k}) e^{i 𝐤 \cdot 𝐫} d Ω_{k}$

In case, when $z_{n}$ are spherical Hankel functions, one should use the different formulae.^[13]^[12] For vector spherical harmonics the following relations are obtained:

$𝐌_{p m n}^{(3)} (k, 𝐫) = \frac{i^{- n}}{2 π k} \iint_{- \infty}^{\infty} d k_{‖} \frac{e^{i (k_{x} x + k_{y} y \pm k_{z} z)}}{k_{z}} 𝐗_{p m n} (\frac{𝐤}{k})$

$𝐍_{p m n}^{(3)} (k, 𝐫) = \frac{i^{- n}}{2 π k} \iint_{- \infty}^{\infty} d k_{‖} \frac{e^{i (k_{x} x + k_{y} y \pm k_{z} z)}}{k_{z}} 𝐙_{p m n} (\frac{𝐤}{k})$ where $k_{z} = \sqrt{k^{2} - k_{x}^{2} - k_{y}^{2}}$ , index $(3)$ means, that spherical Hankel functions are used.

References

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↑ Carrascal, B; Estevez, G A; Lee, Peilian; Lorenzo, V (1991-07-01). "Vector spherical harmonics and their application to classical electrodynamics". European Journal of Physics. 12 (4). IOP Publishing: 184–191. Bibcode:1991EJPh...12..184C. doi:10.1088/0143-0807/12/4/007. ISSN 0143-0807. S2CID 250886412.
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↑ Weinberg, Erick J. (1994-01-15). "Monopole vector spherical harmonics". Physical Review D. 49 (2). American Physical Society (APS): 1086–1092. arXiv:hep-th/9308054. Bibcode:1994PhRvD..49.1086W. doi:10.1103/physrevd.49.1086. ISSN 0556-2821. PMID 10017069. S2CID 6429605.
↑ P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, New York: McGraw-Hill, 1898-1901 (1953)
↑ Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2020). "Complement B-XIX: Angular momentum of radiation". Quantum mechanics. Vol. III (2nd ed.). Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. p. 2053. ISBN 978-3-527-34553-3.
↑ Bohren, Craig F. and Donald R. Huffman, Absorption and scattering of light by small particles, New York : Wiley, 1998, 530 p., ISBN 0-471-29340-7, ISBN 978-0-471-29340-8 (second edition)
1 2 Stratton, J. A. (1941). Electromagnetic Theory. New York: McGraw-Hill.
↑ D. A. Varhalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum [in Russian], Nauka, Leningrad (1975)
↑ Zhang, Huayong; Han, Yiping (2008). "Addition theorem for the spherical vector wave functions and its application to the beam shape coefficients". J. Opt. Soc. Am. B. 25 (2): 255–260. Bibcode:2008JOSAB..25..255Z. doi:10.1364/JOSAB.25.000255.
↑ Stein, Seymour (1961). "Addition theorems for spherical wave functions". Quarterly of Applied Mathematics. 19 (1): 15–24. doi:10.1090/qam/120407.
1 2 Stout, B. (2012). Popov, E (ed.). "Spherical harmonic lattice sums for gratings" (PDF). Institut Fresnel, Universite d'Aix-Marseille 6. Gratings: theory and numeric applications.
↑ Wittmann, R. C. (1988). "Spherical wave operators and the translation formulas". IEEE Transactions on Antennas and Propagation. 36 (8): 1078–1087. Bibcode:1988ITAP...36.1078W. doi:10.1109/8.7220.

External links

Vector Spherical Harmonics at Eric Weisstein's Mathworld

[1] Barrera, R G; Estevez, G A; Giraldo, J (1985-10-01). "Vector spherical harmonics and their application to magnetostatics". European Journal of Physics. 6 (4). IOP Publishing: 287–294. Bibcode:1985EJPh....6..287B. doi:10.1088/0143-0807/6/4/014. ISSN 0143-0807. S2CID 250894245.

[2] Carrascal, B; Estevez, G A; Lee, Peilian; Lorenzo, V (1991-07-01). "Vector spherical harmonics and their application to classical electrodynamics". European Journal of Physics. 12 (4). IOP Publishing: 184–191. Bibcode:1991EJPh...12..184C. doi:10.1088/0143-0807/12/4/007. ISSN 0143-0807. S2CID 250886412.

[3] Hill, E. L. (1954). "The Theory of Vector Spherical Harmonics" (PDF). American Journal of Physics. 22 (4). American Association of Physics Teachers (AAPT): 211–214. Bibcode:1954AmJPh..22..211H. doi:10.1119/1.1933682. ISSN 0002-9505. S2CID 124182424. Archived from the original (PDF) on 2020-04-12.

[4] Weinberg, Erick J. (1994-01-15). "Monopole vector spherical harmonics". Physical Review D. 49 (2). American Physical Society (APS): 1086–1092. arXiv:hep-th/9308054. Bibcode:1994PhRvD..49.1086W. doi:10.1103/physrevd.49.1086. ISSN 0556-2821. PMID 10017069. S2CID 6429605.

[5] P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, New York: McGraw-Hill, 1898-1901 (1953)

[6] Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2020). "Complement B-XIX: Angular momentum of radiation". Quantum mechanics. Vol. III (2nd ed.). Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. p. 2053. ISBN 978-3-527-34553-3.

[BohrenHuffman-7] Bohren, Craig F. and Donald R. Huffman, Absorption and scattering of light by small particles, New York : Wiley, 1998, 530 p., ISBN 0-471-29340-7, ISBN 978-0-471-29340-8 (second edition)

[stratton-8] 1 2 Stratton, J. A. (1941). Electromagnetic Theory. New York: McGraw-Hill.

[9] D. A. Varhalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum [in Russian], Nauka, Leningrad (1975)

[10] Zhang, Huayong; Han, Yiping (2008). "Addition theorem for the spherical vector wave functions and its application to the beam shape coefficients". J. Opt. Soc. Am. B. 25 (2): 255–260. Bibcode:2008JOSAB..25..255Z. doi:10.1364/JOSAB.25.000255.

[11] Stein, Seymour (1961). "Addition theorems for spherical wave functions". Quarterly of Applied Mathematics. 19 (1): 15–24. doi:10.1090/qam/120407.

[stout-12] 1 2 Stout, B. (2012). Popov, E (ed.). "Spherical harmonic lattice sums for gratings" (PDF). Institut Fresnel, Universite d'Aix-Marseille 6. Gratings: theory and numeric applications.

[13] Wittmann, R. C. (1988). "Spherical wave operators and the translation formulas". IEEE Transactions on Antennas and Propagation. 36 (8): 1078–1087. Bibcode:1988ITAP...36.1078W. doi:10.1109/8.7220.

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Definition

In physics

Main properties

Symmetry

Orthogonality

Vector multipole moments

The gradient of a scalar field

Divergence

Curl

Laplacian

Examples

First vector spherical harmonics

Applications

Electrodynamics

Alternative definition

Orthogonality

Rotation and inversion

Fluid dynamics

Integral relations

See also

References

External links