Harmonic Content & Multipoles magnets

The magnetic field can be expressed on specific basis functions. We have identified two famillies of such basis:

cylindrical,
spherical.

Decomposing a field on a basis allows to express the field with only the coefficient in the basis. Moreover, the knowledge of few fields measurements - actually the order of precision we want to achieve on the basis function - provide us the full knwoledge of the field.

To achieve dimensionless unit for the coefficient of the basis function, the field is conveniently scaled with a reference field and radius that have to be specified to fully understand the decomposition.

1. Cylindrical harmonics

Considering a two dimensional multipole fields, one can show we have - writting $\mathbf{B} = \left(B_x, B_y, B_z\right)$ with $B_z$ constant - the relation:

$B_y + i B_x = \sum\limits_{n=0}^{\infty} C_n \left(x + i y\right)^{n-1}$

in the vacuum.

It can be very convenient to write this in the polar coordinates :

$B_{\theta} + i B_r = \sum\limits_{n=1}^{\infty} C_n r^{n-1} e^{i n \theta}$

At least, if the field is measured at $P$ equally-spaced points around a circle of radius $R_{meas}$ , we can use the Fast Fourrier Transform to evaluate the $C_n$ parameters.

We have to provide the various $C_n$ coefficient (real and imaginary parts) at various altitude to fully present the field.

The method is decribed in Determination of magnetic multipoles using a hall probe and Maxwell’s equations for magnets.

A pure multipole magnet of order

$n$ has only

$C_n \neq 0$ (

$C_2 \neq 0$ for a dipole,

$C_3$ for a sextupole and so on).

2. Spherical harmonics

To be done.