The lens frame

LensFrameSimulation propagates the field on a grid that expands from the cell centre to the mirrors exactly like the cell eigenmode. It is the recommended solver.

Why

A fixed grid must hold the beam on the mirrors and resolve it at the waist. The ratio of the two is

\[s_m = \frac{w_\text{mirror}}{w_0} = \sqrt{1 + \left(\frac{d}{2 z_R}\right)^2},\qquad z_R = \frac{\sqrt{d(2R-d)}}{2},\]

which grows without bound towards the concentric limit, i.e. for the high-energy cells. A fixed grid needs 480 points per axis for the tutorial cell (\(s_m = 6.4\)) and 1125 for the 18 mJ Kaumanns cell (\(s_m = 14.6\)); the lens frame needs 75 for both. Its size, about box_factor × points_per_waist × \(\sqrt{\lambda_\max/\lambda_\min}\) points per axis, does not depend on \(d\).

The transformation

With \(z\) measured from the cell centre,

\[A(x, y, z) = \frac{1}{s}\, B\!\left(\frac{x}{s}, \frac{y}{s}, \zeta\right) \exp\!\left(\frac{i k r^2}{2\mathcal{R}}\right),\qquad s = \sqrt{1 + z^2/z_R^2},\quad \mathcal{R} = z + \frac{z_R^2}{z},\quad \zeta = z_R\arctan\frac{z}{z_R}.\]

  • Exact for every colour: all wavelengths share the same \(z_R\), so the same \(s\) and \(\mathcal{R}\).

  • The mirror is the identity: at the mirror \(\mathcal{R} = R\), and the reflection turns the frame wavefront into the frame wavefront of the way back. A bounce is only the spectral response of the mirror and its aperture, taken at the physical radius \(s_m\rho\).

  • The centre is physical: there \(s = 1\) and \(B = A\), so every diagnostic recorded at the centre is unchanged. Mirror records are in frame units (sim.mirror_scale converts them).

In the frame the paraxial envelope equation becomes, exactly,

\[\frac{\partial B}{\partial \zeta} = \frac{i}{2k}\nabla_\perp^2 B - \frac{i k \rho^2}{2 z_R^2} B + i\beta(\Omega)\,s^2 B + i\gamma(\Omega)\,\mathcal{F}_t\!\left[|B|^2 B\right].\]

Diffraction and the harmonic potential have the Hermite–Gauss modes of the cell as eigenstates, so the Gouy phase per pass is built in. Dispersion keeps its physical length (\(s^2 d\zeta = dz\)) and the Kerr term, and with it the B-integral, is unchanged.

Scheme

Each step \(h\) in \(\zeta\):

[diffraction + dispersion](h/2)  [potential(h/2)  Kerr-RK4(h)  potential(h/2)]  [diffraction + dispersion](h/2)

with \(h = \min(\texttt{gouy\_step}\cdot z_R,\ \varphi_\text{max}/(k_0 n_2 I_\text{peak}))\). gouy_step (0.05 rad) sets the accuracy of diffraction, phi_max (0.02 rad) that of the Kerr step. The phase factors are rebuilt from 1D arrays, so no full-size operator is stored.

Limits

  • The frame equation is paraxial; the standard solver is not. For MPC beams (a few mrad) the difference is negligible.

  • It assumes the symmetric two-mirror cell, unfolded and on axis.

  • The box absorber acts in the frame: box_factor waists at the centre, \(s_m\) times more on the mirror.

  • tests/test_lensframe.py checks that one pass leaves the eigenmode unchanged (overlap > 0.9999) and that, for a Kerr-matched 1 mJ pulse in a near-concentric cell, both solvers agree within 0.5 % on the B-integral, the energy, the HG₀₀ fraction and the beam size.