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 harmonicYℓm(θ, φ), we define three VSH:
with being the
unit vector along the radial direction in
spherical coordinates and the vector along the radial direction with the same norm as the radius, i.e., . 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
The labels on the components reflect that is the radial component of the vector field, while and are transverse components (with respect to the radius vector ).
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 :
They are also orthogonal in Hilbert space:
An additional result at a single point (not reported in Barrera et al, 1985) is, for all ,
Vector multipole moments
The orthogonality relations allow one to compute the spherical multipole moments of a vector field as
Also note that this action becomes
symmetric, i.e. the off-diagonal coefficients are equal to , for properly
normalized VSH.
Examples
Visualizations of the real parts of VSHs. Click to expand.
First vector spherical harmonics
.
.
.
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 and complex amplitude
and the corresponding electric and magnetic fields, can be written as
Substituting into Maxwell equations, Gauss's law is automatically satisfied
while Faraday's law decouples as
Gauss' law for the magnetic field implies
and Ampère–Maxwell's equation gives
In this way, the partial differential equations have been transformed into a set of ordinary differential equations.
Alternative definition
Angular part of magnetic and electric vector spherical harmonics. Red and green arrows show the direction of the field. Generating scalar functions are also presented, only the first three orders are shown (dipoles, quadrupoles, octupoles).
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 .
here are the
associated Legendre polynomials, and are any of the
spherical Bessel functions.
Vector spherical harmonics are defined as:
longitudinal harmonics
magnetic harmonics
electric harmonics
Here we use harmonics real-valued angular part, where , but complex functions can be introduced in the same way.
Let us introduce the notation . In the component form vector spherical harmonics are written as:
There is no radial part for magnetic harmonics. For electric harmonics, the radial part decreases faster than angular, and for big 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 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:[7]
All other integrals over the angles between different functions or functions with different indices are equal to zero.
Rotation and inversion
Illustration of the transformation of vector spherical harmonics under rotations. One can see that they are transformed in the same way as the corresponding scalar functions.
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]
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.
and magnetic ones have the opposite parity:
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
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.
In case, when are spherical Hankel functions, one should use the different formulae.[12][11] For vector spherical harmonics the following relations are obtained:
where , index means, that spherical Hankel functions are used.
^P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, New York: McGraw-Hill, 1898-1901 (1953)
^Bohren, Craig F. and Donald R. Huffman, Absorption and scattering of light by small particles, New York : Wiley, 1998, 530 p.,
ISBN0-471-29340-7,
ISBN978-0-471-29340-8 (second edition)
^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.
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 harmonicYℓm(θ, φ), we define three VSH:
with being the
unit vector along the radial direction in
spherical coordinates and the vector along the radial direction with the same norm as the radius, i.e., . 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
The labels on the components reflect that is the radial component of the vector field, while and are transverse components (with respect to the radius vector ).
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 :
They are also orthogonal in Hilbert space:
An additional result at a single point (not reported in Barrera et al, 1985) is, for all ,
Vector multipole moments
The orthogonality relations allow one to compute the spherical multipole moments of a vector field as
Also note that this action becomes
symmetric, i.e. the off-diagonal coefficients are equal to , for properly
normalized VSH.
Examples
Visualizations of the real parts of VSHs. Click to expand.
First vector spherical harmonics
.
.
.
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 and complex amplitude
and the corresponding electric and magnetic fields, can be written as
Substituting into Maxwell equations, Gauss's law is automatically satisfied
while Faraday's law decouples as
Gauss' law for the magnetic field implies
and Ampère–Maxwell's equation gives
In this way, the partial differential equations have been transformed into a set of ordinary differential equations.
Alternative definition
Angular part of magnetic and electric vector spherical harmonics. Red and green arrows show the direction of the field. Generating scalar functions are also presented, only the first three orders are shown (dipoles, quadrupoles, octupoles).
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 .
here are the
associated Legendre polynomials, and are any of the
spherical Bessel functions.
Vector spherical harmonics are defined as:
longitudinal harmonics
magnetic harmonics
electric harmonics
Here we use harmonics real-valued angular part, where , but complex functions can be introduced in the same way.
Let us introduce the notation . In the component form vector spherical harmonics are written as:
There is no radial part for magnetic harmonics. For electric harmonics, the radial part decreases faster than angular, and for big 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 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:[7]
All other integrals over the angles between different functions or functions with different indices are equal to zero.
Rotation and inversion
Illustration of the transformation of vector spherical harmonics under rotations. One can see that they are transformed in the same way as the corresponding scalar functions.
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]
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.
and magnetic ones have the opposite parity:
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
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.
In case, when are spherical Hankel functions, one should use the different formulae.[12][11] For vector spherical harmonics the following relations are obtained:
where , index means, that spherical Hankel functions are used.
^P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, New York: McGraw-Hill, 1898-1901 (1953)
^Bohren, Craig F. and Donald R. Huffman, Absorption and scattering of light by small particles, New York : Wiley, 1998, 530 p.,
ISBN0-471-29340-7,
ISBN978-0-471-29340-8 (second edition)
^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.