# 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.

## 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.

Remark

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).

To do.