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.

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.

To do.