Quadrupole Magnetic Field Optimisation

Numerical optimisation of a quadrupole magnet modelled as four infinite straight current-carrying wires. The wire angular positions are optimised so that the resulting field best approximates an ideal linear quadrupole field inside the beam aperture.

Features

• Biot-Savart Solver: Vectorized 2D magnetic field calculation for infinite straight wires.
• Least-Squares Gradient Estimation: Analytical closed-form estimation of the field gradient ($G$) across a polar sampling grid, offering high numerical stability over single-point forward differences.
• Pure NumPy Nelder-Mead: A custom, dependency-free implementation of the simplex optimization algorithm.
• Multipole Spectrum Analysis: Fast Fourier Transform (FFT) analysis on a circular probe path ($N_\phi = 2048$) to decompose normal ($b_n$) and skew ($a_n$) multipole coefficients.
• Scaling Law Verification: Numerical verification of the theoretical scaling law $\frac{|b_n(r)|}{|b_2(r)|} \propto \left(\frac{r}{r_0}\right)^{n-2}$ for allowed higher-order harmonics ($n = 6, 10, 14$).

Physics background

A quadrupole magnet focuses a charged-particle beam in one transverse plane while defocusing it in the other. The ideal field satisfies:

Bx = G·y, By = G·x

where G [T/m] is the field gradient. This study models the magnet as four wires on a circle of radius r₀ and asks: what wire positions minimise the deviation from this ideal field?

Key results:

• Fourfold rotational symmetry + alternating currents → only multipole orders n = 2, 6, 10, 14, … are non-zero; all others vanish by symmetry, not by optimisation.
• The leading perturbation beyond the quadrupole (n = 2) is the 12-pole (n = 6), scaling as (r/r₀)⁴.
• Manufacturing tolerance: wire angles must be held to within Δφ < 0.1°.

Repository structure

File Name	Description
quadrupole_optimize.py	Main script — run this
quadrupole_paper_en.pdf	Compiled PDF (13 pages)
fig1_field_geometry.png	Wire positions + field lines + beam region
fig2_multipole_spectrum.png	Allowed vs. forbidden harmonics (bar chart)
fig3_scaling_law.png	|bₙ|/|b₂| ∝ (r/r₀)ⁿ⁻² verification
fig4_convergence_sensitivity.png	Nelder-Mead history + angular tolerance

Running the code

Requirements: Python 3.9+, NumPy, Matplotlib — no other dependencies.

pip install numpy matplotlib
python quadrupole_optimize.py

The script prints the optimised wire angles, field gradient, final cost, multipole amplitudes, and scaling law slopes to stdout, then saves all four figures to the current directory.

What the code does

Step	Function	Description
1	biot_savart_wire	2-D Biot-Savart field of one infinite wire
2	superposed_field	Sum over four wires with alternating currents
3	make_polar_grid	Polar sampling grid inside the beam aperture
4	least_squares_gradient	Analytic best-fit gradient G (closed form)
5	relative_residual_cost	Dimensionless figure of merit C ∈ [0, 1]
6	nelder_mead	Derivative-free simplex optimiser (pure NumPy)
7	multipole_spectrum	FFT-based multipole decomposition

All "magic numbers" are declared as named constants at the top of the file (e.g. WIRE_CIRCLE_RADIUS, NM_CONVERGENCE_TOL). Results are fully reproducible: RANDOM_SEED = 42.

Key parameters

Constant	Value	Meaning
WIRE_CURRENT	1000 A	Current in each wire
WIRE_CIRCLE_RADIUS	50 mm	Radius of wire placement circle
BEAM_RADIUS	10 mm	Beam aperture radius
APERTURE_TO_WIRE_RATIO	0.20	R_beam / r₀
RANDOM_SEED	42	Fixed for reproducibility

References

1. H. Wiedemann, Particle Accelerator Physics, 4th ed., Springer, 2015.
2. S. Y. Lee, Accelerator Physics, 2nd ed., World Scientific, 2004.
3. E. D. Courant, M. S. Livingston, H. S. Snyder, Phys. Rev. 88, 1190 (1952).
4. D. J. Griffiths, Introduction to Electrodynamics, 4th ed., Cambridge, 2017.
5. J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
6. J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999.