How the GG calculation works

This page summarizes the mathematics behind gg_fit. The field-expansion table is linear in the GG functions and their s-derivatives, which makes the fit a (weighted) linear least-squares problem.

The field expansion

For each field component \(c \in \{B_x, B_y, B_s\}\),

\[B_c(x,y,z) = \sum_{(n,m)} CS_{c,b}(n,m; x,y)\, b_{(n,m)}(z) + \sum_{(n,m)} CS_{c,a}(n,m; x,y)\, a_{(n,m)}(z) + \sum_{m} CS_{c,bs}(m; x,y)\, bs_{(m)}(z),\]

where each coefficient is a polynomial in the transverse coordinates,

\[CS_{c,f}(n,m; x,y) = \sum (\text{coeff}\cdot g_{\text{ref}}^{\,k}\, x^p\, y^q),\]

and the derivative towers are \(b_{(n,m)} = \mathrm{d}^m b_n/\mathrm{d}z^m\), \(a_{(n,m)} = \mathrm{d}^m a_n/\mathrm{d}z^m\), and \(bs_{(m)} = \mathrm{d}^{m+1} a_0/\mathrm{d}z^{m+1}\).

Plane-by-plane least squares

The unknowns at a base plane \(z_0\) are the function values and their derivatives \(f_{(n,m)}(z_0)\) for \(m = 0 \ldots m_{\max}\). The field on a neighbouring plane at offset \(\mathrm{d}z = z - z_0\) is obtained by Taylor-extrapolating each derivative,

\[f_{(n,m)}(z_0 + \mathrm{d}z) = \sum_{j \ge m} \frac{\mathrm{d}z^{\,j-m}}{(j-m)!}\, f_{(n,j)}(z_0),\]

which makes the model linear in the base-plane unknowns. Each base plane is then solved by weighted linear least squares over all field points lying within n_planes_add planes of it. Adding planes (m_max = 2·n_planes_add) lets the fit resolve higher derivatives and smooths the result; past some point, using more planes makes the polynomial approximation less accurate, so there is an optimum.

The merit function and weights

For a base plane, the merit function minimized is

\[\text{Merit} = \sum \text{weight}\,(B_{\text{table}} - B_{\text{GG}})^2 ,\]

with a per-point weight that factors as \(\text{weight}(x,y,\mathrm{d}z) = w_{\text{core}}(x,y)\, w_{\text{plane}}(\mathrm{d}z)\):

\[w_{\text{core}}(x,y) = \texttt{core\_weight}\, \frac{r_{\max}^2}{r_{\max}^2 + r^2(\texttt{core\_weight}-1)}, \qquad r^2 = x^2 + y^2,\]

\[w_{\text{plane}}(\mathrm{d}z) = 1 + (\texttt{outer\_plane\_weight}-1)\,\frac{|\mathrm{d}z|}{\mathrm{d}z_{\max}} .\]

core_weight = 1 (the default) weights all transverse points equally; a value

1 favours near-axis points, which is usually desirable since a beam spends most of its time near the core. outer_plane_weight = 1 weights all planes equally; a value below 1 (but non-negative) down-weights the outer planes. When n_planes_add = 0 (so \(\mathrm{d}z_{\max} = 0\)), \(w_{\text{plane}}\) is taken to be 1.

Conversion to Bmad’s convention

Bmad’s gen_grad_map uses azimuthal-harmonic gradients \(C_{m,\alpha}(z)\), \(\alpha \in \{\sin, \cos\}\), rather than this project’s midplane-derivative GGs. Equating the two expansions on the midplane gives exact recursions (solved in order of increasing \(m\)):

\[C^{[j]}_{m,s} = \frac{1}{m!}\Big[ b^{[j]}_m - (m-1)! \sum_{n\ge 1,\, m-2n\ge 1} W_n(m,n)\, C^{[j+2n]}_{m-2n,s} \Big],\]

\[C^{[j]}_{m,c} = \frac{1}{m!}\Big[ a^{[j]}_m - (m-1)! \sum_{n\ge 1,\, m-2n\ge 1} W_c(m,n)\, C^{[j+2n]}_{m-2n,c} - [m \text{ even}]\,(m-1)!\,U_s(m)\, b_s^{[m+j-1]} \Big],\]

\[C^{[j]}_{0,c} = b_s^{[j-1]} \quad (j \ge 1),\]

where \(x^{[j]} \equiv \mathrm{d}^j x/\mathrm{d}s^j\) is supplied directly by the fit. The full derivation, including the mixing weights \(W_n\), \(W_c\), and \(U_s\), is in the write_bmad_gg_fit docstring (see the API Reference).

References

• S. Van der Schueren et al., “Magnetic Field Modelling and Symplectic Integration of Magnetic Fields on Curved Reference Frames for Improved Synchrotron Design: First Steps” (copy in the papers/ directory).

• D. Sagan, IPAC’23 — the Venturini–Dragt azimuthal-harmonic field expansion used by Bmad.