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Simulation Library
Downloadable Python starting points
The repository includes small, readable scripts that produce numerical output and plots:
- Array factor and beam steering
- Free-space link budget
- Lorentz-force particle tracking with a Boris pusher
- Driven dissipative two-level optical Bloch equations
- Ideal Halbach-cylinder field scaling
- Python examples README
Model-selection map
| Problem | First model | When a more complete model is needed |
|---|---|---|
| Plane-wave propagation | analytic phasor | layered/inhomogeneous medium |
| Transmission line | telegrapher equations | full-wave launch/discontinuity |
| Antenna array | array factor | mutual coupling / installed platform |
| Waveguide | modal solution | bends, irises, dielectric loading |
| Complex 3-D RF structure | FEM / FDTD / MoM | hybrid/asymptotic for huge electrical size |
| Charged particles | Lorentz ODE | collisions / space charge / PIC |
| Atomic two-level system | optical Bloch equations | hyperfine/Zeeman multilevel density matrix |
| Strong periodic atomic drive | dressed-state / Floquet | full dissipative Floquet master equation |
1. Plane wave
A minimal complex representation is
import numpy as np
z = np.linspace(0, 2, 1000)
f = 1e9
c = 299_792_458
beta = 2*np.pi*f/c
E = np.exp(-1j*beta*z)
| Checks include linear phase advance, constant $ | E | $ in a lossless medium, and wavelength $2\pi/\beta$. |
2. Transmission line and reflection
For a lossless line,
\[Z_{in}=Z_0\frac{Z_L+jZ_0\tan\beta l}{Z_0+jZ_L\tan\beta l}.\]Plots of $Z_{in}(l)$ and $\Gamma(l)$ on a Smith chart show the effect of electrical length directly.
3. Antenna array factor
For a linear array,
\[AF(\theta)=\sum_{n=0}^{N-1}w_ne^{jn(kd\sin\theta+\beta_s)}.\]Useful parameters include element count, spacing, steering phase, amplitude taper, phase/amplitude errors, grating lobes, and scan loss after multiplication by an element pattern.
A full-wave model becomes important when mutual coupling and finite element patterns matter.
4. Radar waveform
An FMCW model contains
transmit chirp → delayed/Doppler-shifted echo → mixer → beat signal → FFT → range/velocity estimate.
Range resolution can be checked against $c/(2B)$ and Doppler against $2v/\lambda$.
5. Lorentz-force particle tracking
Integrate
\[m\dot{\mathbf v}=q(\mathbf E+\mathbf v\times\mathbf B),\qquad \dot{\mathbf r}=\mathbf v.\]A Boris pusher is useful for long charged-particle trajectories in magnetic fields because it preserves gyromotion better than naive Euler stepping.
6. Monte Carlo transport + Shockley–Ramo
For particle $j$,
\[i_j=q_j\mathbf v_j\cdot\mathbf E_w.\]A complete transport calculation can include sampled creation positions and velocities, actual $\mathbf E$ and $\mathbf B$ fields, stochastic collisions, wall/electrode interactions, weighting-field interpolation, particle summation, and the measured $Z_T(\omega)$ of the readout electronics.
This separates transport physics from signal induction.
7. Optical Bloch equations
For a two-level atom,
\[\dot\rho=-\frac{i}{\hbar}[H,\rho]+\mathcal L(\rho).\]Useful consistency checks include the zero-drive limit, weak-drive Lorentzian response, saturation/power broadening, expected detuning symmetry, $\mathrm{Tr}\rho=1$, and physical populations.
8. Three-level EIT
| For a ladder system $ | g\rangle\rightarrow | e\rangle\rightarrow | r\rangle$, calculate steady-state $\rho_{ge}$ and use |
Realistic models may include Doppler averaging, transit time, laser linewidth, spatial Rabi variation, Zeeman structure and RF coupling.
9. Floquet / Shirley model
For a periodic Hamiltonian $H(t+T)=H(t)$, expand in photon replicas and diagonalize the enlarged Floquet Hamiltonian.
Useful outputs include quasienergies, bare-state overlap, target-state shift, avoided-crossing gaps, replica index $q$, pathway amplitudes, and eigenvector continuity by overlap between neighboring field points.
10. Magnetostatics and Halbach arrays
Solve
\[\nabla\cdot\mathbf B=0,\qquad \mathbf B=\mu_0(\mathbf H+\mathbf M).\]| For atomic sensing, useful post-processing includes both $ | B | $ and the distribution of Zeeman shifts across the illuminated vapor volume. |
11. FEM, FDTD and MoM
- FEM: bounded complex geometries, waveguides, cavities and dielectric structures.
- FDTD: broadband time-domain propagation and transients.
- MoM: open-region conducting surfaces, wires, radiation and scattering.
See Comparison Tables and Computational Methods.