Vector spherical harmonics explained

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, we define three VSH:

Y _{\ell} = Y _{\ell} \hat{ r },\Psi _{\ell} = r\nabla Y _{\ell} , \Phi _{\ell} = r x \nabla Y _{\ell} ,

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

$$\backslash mathbf\; =\; \backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\; \backslash left(E^r\_(r)\; \backslash mathbf\_\; +\; E^\_(r)\; \backslash mathbf\_\; +\; E^\_(r)\; \backslash mathbf\_\backslash right).$$

The labels on the components reflect that

E ^{r} _{\ell m} is the radial component of the vector field, while E ^{( 1 )} _{\ell m} and E ^{( 2 )} _{\ell m} are transverse components (with respect to the radius vector r ).

Main properties

Symmetry

Like the scalar spherical harmonics, the VSH satisfy

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 :

They are also orthogonal in Hilbert space:

An additional result at a single point

r (not reported in Barrera et al, 1985) is, for all \ell,m,\ell',m' ,

Vector multipole moments

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

The gradient of a scalar field

Given the multipole expansion of a scalar field

$$\backslash phi\; =\; \backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\; \backslash phi\_(r)\; Y\_(\backslash theta,\backslash phi),$$

we can express its gradient in terms of the VSH as

$$\backslash nabla\backslash phi\; =\; \backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\backslash left(\backslash frac\; \backslash mathbf\_+\backslash frac\backslash mathbf\_\backslash right).$$

Divergence

For any multipole field we have

By superposition we obtain the divergence of any vector field:

$$\backslash nabla\backslash cdot\backslash mathbf\; =\; \backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\; \backslash left(\backslash frac+\backslash fracE^r\_-\backslash fracE^\_\backslash right)Y\_.$$

We see that the component on is always solenoidal.

Curl

For any multipole field we have

By superposition we obtain the curl of any vector field:

$$\backslash nabla\backslash times\backslash mathbf\; =\; \backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\backslash left(-\backslash fracE^\_\backslash mathbf\_-\backslash left(\backslash frac+\backslash fracE^\_\backslash right)\backslash mathbf\_+\backslash left(-\backslash fracE^r\_+\backslash frac+\backslash fracE^\_\backslash right)\backslash mathbf\_\backslash right).$$

Laplacian

\Delta = \nabla ⋅ \nabla separates as follows:

$$\backslash Delta\backslash left(f(r)\backslash mathbf\_\backslash right)\; =\; \backslash left(\backslash frac\; \backslash frac\; r^2\; \backslash frac\; \backslash right)\backslash mathbf\_+\; f(r)\backslash Delta\; \backslash mathbf\_,$$where

Z _{\ell} = Y _{\ell} , \Psi _{\ell} , \Phi _{\ell} and

Also note that this action becomes symmetric, i.e. the off-diagonal coefficients are equal to $\backslash frac\backslash sqrt$, for properly normalized VSH.

Examples

First vector spherical harmonics

Expressions for negative values of 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

\omega and complex amplitude

$$\backslash hat=\; J(r)\backslash mathbf\_,$$

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

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

$$\backslash nabla\backslash cdot\backslash hat=0,$$

while Faraday's law decouples as

$$\backslash nabla\backslash times\backslash hat=-i\backslash omega\backslash hat\backslash quad\backslash Rightarrow\backslash quad\backslash begin\; \backslash dfracE\; =\; i\backslash omega\; B^r,\; \backslash \backslash \backslash dfrac\; +\backslash dfrac=\; i\backslash omega\; B^.\backslash end$$

Gauss' law for the magnetic field implies

$$\backslash nabla\backslash cdot\backslash hat\; =\; 0\backslash quad\backslash Rightarrow\; \backslash quad\backslash frac+\backslash fracB^r\; -\; \backslash fracB^=0,$$

and Ampère–Maxwell's equation gives

$$\backslash nabla\backslash times\backslash hat\; =\; \backslash mu\_0\; \backslash hat\; +\; i\backslash mu\_0\backslash varepsilon\_0\backslash omega\; \backslash hat\; \backslash quad\backslash Rightarrow\backslash quad\; -\backslash frac+\backslash frac+\backslash frac\; =\; \backslash mu\_0J+i\backslash omega\backslash mu\_0\backslash varepsilon\_0E.$$

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.^{[6] [7]}

In this case, vector spherical harmonics are generated by scalar functions, which are solutions of scalar Helmholtz equation with the wavevector

k . $$\backslash begin\; \backslash \backslash \backslash end$$here P ^{m} _{n} (\cos \theta) 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 _{m n ^{e} _{o}} = \nabla \psi _{m n ^{e} _{o}}

magnetic harmonics :

M _{m n ^{e} _{o}} = \nabla x \left( r \psi _{m n ^{e} _{o}} \right)

electric harmonics :

N _{m n ^{e} _{o}} = (\nabla x M _{m n ^{e} _{o}})/({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

\rho = kr . In the component form vector spherical harmonics are written as:$$\backslash begin\; \backslash \backslash \; z\_(\backslash rho)\; \backslash mathbf\_\backslash end$$$$\backslash begin\; z\_(\backslash rho)\; \backslash mathbf\_\; \backslash \backslash \; \backslash end$$

$$\backslash begin\; \backslash \backslash \; \backslash frac\; \backslash frac\backslash left[\backslash rho\; z\_\{n\}(\backslash rho)\backslash right]\; \backslash mathbf\_\; \backslash \backslash \; \backslash frac\; \backslash frac\backslash left[\backslash rho\; z\_\{n\}(\backslash rho)\backslash right]\; \backslash mathbf\_\backslash end$$

$$\backslash begin\; \backslash mathbf\_\; (k,\; \backslash mathbf)\; =\backslash qquad\; \backslash frac\; \backslash sin\; (m\; \backslash varphi)\; n(n+1)\; P\_^(\backslash cos\; (\backslash theta))\; \backslash mathbf\_\; \backslash \backslash \; +\backslash sin\; (m\; \backslash varphi)\; \backslash frac\; \backslash frac\; \backslash frac\backslash left[\backslash rho\; z\_\{n\}(\backslash rho)\backslash right]\; \backslash mathbf\_\; \backslash \backslash \; +\; \backslash frac\; \backslash frac\backslash left[\backslash rho\; z\_\{n\}(\backslash rho)\backslash right]\; \backslash mathbf\_\; \backslash end$$There is no radial part for magnetic harmonics. For electric harmonics, the radial part decreases faster than angular, and for big

\rho 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 \rho electric and magnetic harmonics vectors are equal in value and perpendicular to each other.

Longitudinal harmonics:

Orthogonality

The solutions of the Helmholtz vector equation obey the following orthogonality relations:

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^{[8] [9] [10]} $$\backslash hat(\backslash alpha,\; \backslash beta,\; \backslash gamma)\; \backslash mathbf\_^(\backslash theta,\; \backslash varphi)=\; \backslash sum\_^J\; [D^\{(J)\}\_\{MM\text{'}\}(\backslash alpha,\; \backslash beta,\; \backslash gamma)]^*\; \backslash mathbf\_^(\backslash theta,\; \backslash varphi),$$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.$$\backslash hat\backslash mathbf\_(\backslash theta,\; \backslash varphi)=\; (-1)^J\; \backslash mathbf\_(\backslash theta,\; \backslash varphi),$$and magnetic ones have the opposite parity:$$\backslash hat\backslash mathbf\_(\backslash theta,\; \backslash varphi)=\; (-1)^\; \backslash mathbf\_(\backslash theta,\; \backslash varphi),$$

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.,

with the boundary conditions

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

$$\backslash mathbf\_0\; =\; U\_0\backslash left(\backslash cos\backslash theta\backslash ,\; \backslash hat\; -\; \backslash sin\backslash theta\; \backslash ,\backslash hat\backslash right)\; =\; U\_0\; \backslash left(\backslash mathbf\_\; +\; \backslash mathbf\_\backslash right).$$

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

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

Integral relations

Here the following definitions are used:

$$\backslash mathbf\_\backslash left(\backslash frac\backslash right)\; =\; \backslash nabla\; \backslash times\backslash left(\backslash mathbf\; Y\_\backslash left(\backslash frac\backslash right)\backslash right)$$

$$\backslash mathbf\_\backslash left(\backslash frac\backslash right)\; =\; i\; \backslash frac\; \backslash times\; \backslash mathbf\_\backslash left(\backslash frac\backslash right)$$In case, when instead of

z _{n} are spherical Bessel functions, with help of plane wave expansion one can obtain the following integral relations:^[11]

$$\backslash mathbf\; \_^(k,\; \backslash mathbf\; r)\; =\; \backslash frac\; \backslash int\; \backslash mathbf\; Z\_\backslash left(\backslash frac\backslash right)\; e^\; d\backslash Omega\_k$$

$$\backslash mathbf\; \_^(k,\; \backslash mathbf\; r)\; =\backslash frac\; \backslash int\; \backslash mathbf\; X\_\backslash left(\backslash frac\backslash right)\; e^\; d\backslash Omega\_k$$

In case, when

z _{n} are spherical Hankel functions, one should use the different formulae.^[12] For vector spherical harmonics the following relations are obtained:

$$\backslash mathbf\_^(k,\; \backslash mathbf)\; =\; \backslash frac\; \backslash iint\_^\; d\; k\_\; \backslash frac\; \backslash mathbf\_\backslash left(\backslash frac\backslash right)$$

$$\backslash mathbf\_^(k,\; \backslash mathbf)\; =\; \backslash frac\; \backslash iint\_^\; d\; k\_\; \backslash frac\; \backslash mathbf\_\backslash left(\backslash frac\backslash right)$$where $k\_\; =\; \backslash sqrt$, index

( 3 ) means, that spherical Hankel functions are used.

See also

• Spherical harmonics
• Spinor spherical harmonics
• Spin-weighted spherical harmonics
• Electromagnetic radiation
• Spherical basis

References

1. Barrera . R G . Estevez . G A . Giraldo . J . Vector spherical harmonics and their application to magnetostatics . European Journal of Physics . IOP Publishing . 6 . 4 . 1985-10-01 . 0143-0807 . 10.1088/0143-0807/6/4/014 . 287–294. 1985EJPh....6..287B . 10.1.1.718.2001 . 250894245 .
2. Carrascal . B . Estevez . G A . Lee . Peilian . Lorenzo . V . Vector spherical harmonics and their application to classical electrodynamics . European Journal of Physics . IOP Publishing . 12 . 4 . 1991-07-01 . 0143-0807 . 10.1088/0143-0807/12/4/007 . 184–191. 1991EJPh...12..184C . 250886412 .
3. Hill . E. L. . The Theory of Vector Spherical Harmonics . American Journal of Physics . American Association of Physics Teachers (AAPT) . 22 . 4 . 1954 . 0002-9505 . 10.1119/1.1933682 . 211–214. 1954AmJPh..22..211H . 124182424 . https://web.archive.org/web/20200412143916/http://pdfs.semanticscholar.org/e950/1438ddec10ff6baef4614cf3b66f4e27c44b.pdf . dead . 2020-04-12 .
4. Weinberg . Erick J. . Monopole vector spherical harmonics . Physical Review D . American Physical Society (APS) . 49 . 2 . 1994-01-15 . 0556-2821 . 10.1103/physrevd.49.1086 . 1086–1092. 10017069 . hep-th/9308054. 1994PhRvD..49.1086W . 6429605 .
5. P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, New York: McGraw-Hill, 1898-1901 (1953)
6. Bohren, Craig F. and Donald R. Huffman, Absorption and scattering of light by small particles, New York : Wiley, 1998, 530 p.,, (second edition)
7. Book: Stratton, J. A. . Electromagnetic Theory. registration . New York. McGraw-Hill. 1941.
8. D. A. Varhalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum [in Russian], Nauka, Leningrad (1975)
9. Huayong. Zhang. Yiping. Han. 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. 2008. 10.1364/JOSAB.25.000255. 2008JOSAB..25..255Z .
10. Seymour. Stein. Addition theorems for spherical wave functions. Quarterly of Applied Mathematics. 19. 1. 15–24. 1961. 10.1090/qam/120407.
11. Web site: Spherical harmonic lattice sums for gratings. B.. Stout. Popov. E. Gratings: theory and numeric applications. Institut Fresnel, Universite d'Aix-Marseille 6 . 2012.
12. R. C.. Wittmann. 10.1109/8.7220. Spherical wave operators and the translation formulas. IEEE Transactions on Antennas and Propagation. 36. 8. 1078–1087. 1988. 1988ITAP...36.1078W .

External links

• Vector Spherical Harmonics at Eric Weisstein's Mathworld