Definitions of pulse statistics#

Energy, peak power and width are different measurements of a propagated field. This chapter gives their conventions and explains why broadband Poynting flux, pulse shape and spatial representation matter. The RDW lesson introduces these measurements on an actual run.

The definitions#

The analytic signal, once#

Modal run power is the cycle-averaged Hermitian flux \(P_m(t)=\sum_p\operatorname{Re}[a_{mp}(t)a_{w,mp}(t)^*]\), with \(a_w\) the analytic signal formed from \(w a\). The real instantaneous sum-frequency term is excluded. Standalone array functions use \(|b(t)|^2\); see the power definitions for their broadband difference. On an envelope grid the carrier is removed before the inverse transform; on a field grid analytic_signal() constructs the positive-frequency signal. Both members of the flux product use this construction.

That is one construction, from one place, and it follows SciPy’s convention: the zero-frequency bin is not doubled and the Nyquist bin is kept. Only that convention makes \(\operatorname{Re} E_a = E\) an identity and the two energy functionals an exact Parseval pair. A construction that doubles the zero-frequency bin and discards the Nyquist bin agrees with it on any apodised grid, where the spectral collar has driven both end bins to zero, and differs at the \(1/N\) level otherwise.

Energy, and windows that tile#

energy() is energy_omega(), the exact discrete form of the energy functional of conventions.md §4, which agrees with the time-domain functional to round-off rather than to a quadrature error.

energy_window() applies a hard gate, \(\omega_{lo} \le \omega < \omega_{hi}\). The half-open edge is what makes windows laid end to end assign every bin exactly once, so their energies sum to the total exactly — which is a test. There is no tapered edge, because a windowed energy never returns to the time domain and has nothing to gain from a smooth one.

Peak power#

For modal run statistics the reductions are:

\[ P^{\text{peak}}_m = \max_t P_m(t) , \qquad P^{\text{peak}}_{\text{tot}} = \max_t \sum_m P_m(t) . \]

The polarisation components of one mode are summed before the maximum because they are two components of one field and their powers add at each instant. The modes are summed before the maximum too, so peak_power_total is the peak of the total power and not the largest of the per-mode peaks, nor their sum: modes that peak at different times never carry their combined power at any single instant.

One reduction serves all three statistics, the windowed one included, and a regression test checks it on a two-mode counter-example where the largest row of per-mode powers, [5.0, 0.5], is not the peak of the sum, 5.5.

Two temporal widths#

The total power is formed, the half-maximum level taken from its global peak with no baseline subtraction, and the crossings found by linear interpolation between the two samples that bracket them. Which two samples is the whole question when a trace has structure:

  • fwhm_t() walks outwards from the peak and stops at the first sample below half maximum on either side. A dispersive wave, a soliton that has walked off, a noise pedestal — none of them widens the answer, which is why this is the number to read as a pulse duration.

  • fwhm_t_outer() takes the first and last sample anywhere in the window above half maximum, so it spans every satellite.

On a clean single pulse the two agree; the gap between them says how much sits outside the central feature.

The interpolation is written as \(x_1 + (L - y_1)(x_2 - x_1)/(y_2 - y_1)\). Forming the slope \(m\) and intercept \(c\) of the line and evaluating \((L - c)/m\) instead would, on a time axis of order \(10^{-12}\) with an intercept of order \(10^{12}\), throw away most of the significand.

Note

A trace that never falls to half its maximum — a continuous wave, an empty mode, a pulse that fills the window — has no width, and the answer is nan. Likewise a mode carrying no light has no spectral centroid. These are the two places in the catalogue where a value is genuinely undefined, and each docstring says so; a missing dependency, by contrast, raises.

Profiles and dispersion along z#

density, pressure and core_radius are one implementation, profile_value(), reading three different ZProfile paths at the plane’s position; the catalogue gives each its own units.

zdw() root-finds \(\beta_2 = 0\) for each mode, which is why it is off by default. collect() narrows the search window to the grid’s own band with inband_wavelengths(), because a zero-dispersion point outside the simulated band says nothing about the run and hunting for one there drives every material model into extrapolation. Each call stands alone – no search starts from the previous plane’s answer, so one failure cannot poison the later planes – and a mode with no zero in the window raises rather than returning a missing value that propagates.

Far-field spectra#

The five far-field statistics of a Cartesian or line run (far-field apertures) are one pass over the stored spectrum \(E_{c,q,k}\) of column \(c\), polarisation \(q\) and frequency bin \(k\), with the geometry’s Parseval weights \(w_c\) (spectral_weights), its on-axis weights \(\alpha_c\) (on_axis) and the spectral density \(\operatorname{sd}(E)\) of spectral_density() (\(2\delta t^2|E|^2/\pi\) on a field grid, \(\delta t^2|E|^2/2\pi\) on an envelope grid), and \(\operatorname{sd}(A,B)\) the same factor times \(\operatorname{Re}(A\bar B)\). Each column and bin carries the dimensionless axial admittance of the medium at the plane, \(\zeta_{c,k}=ck_z(c,k)/\omega_k\), with \(k_z\) the principal root of the run’s compensated square, the root the stages use, and \(\eta=\operatorname{Re}\zeta\); both are zero on inactive and evanescent bins and at \(\omega_k\le0\) (energy and flux in a medium). Column \(c\) carries the stored axes \(\mathbf k^{(s)}_c\) and the physical transverse wavevector \(\boldsymbol\kappa_c=-\mathbf k^{(s)}_c\). Aperture \(a\), of centre \(\mathbf X_a\), half-width \(r_a\) (\(D/2\), or \(D_q/2\) under quadrature weighting) and distance \(z_a\), sees the column’s paraxial image at the distance

\[ \rho_a(c,k)=\sqrt{\bigl(s_ak^{(s)}_{x,c}+X_a\bigr)^2+\bigl(s_ak^{(s)}_{y,c}+Y_a\bigr)^2}, \qquad s_a=\frac{z_ac}{\omega_k}, \]

evaluated as written, without hypot or a fused multiply-add, so a hard edge decides identically on every backend. The weight \(W_a=W(\rho_a)\) of aperture_profile() is exactly zero at \(\omega_k\le0\) and where \(s_a|\mathbf k^{(s)}_c|\ge z_a\). Then

\[\begin{split} \begin{aligned} S_a(\omega_k)&=\frac{\varepsilon_0c}{2}\sum_cw_cW_a^p\eta_{c,k}\sum_q\operatorname{sd}(E_{c,q,k}), &p&=2\ \text{(amplitude)},\ 1\ \text{(quadrature)},\\ S^{\rm ax}_a(\omega_k)&=\frac{\varepsilon_0c}{2}\sum_q\operatorname{sd}\Bigl(\sum_c\alpha_cW_aE_{c,q,k},\ \sum_c\alpha_cW_a\zeta_{c,k}E_{c,q,k}\Bigr),\\ S^R_j(\omega_k)&=\frac{\varepsilon_0c}{2}\sum_{\boldsymbol\kappa_c\in R_j}w_c\eta_{c,k}\sum_q\operatorname{sd}(E_{c,q,k}),\\ S^{\rm beam}(\omega_k)&=\frac{\varepsilon_0c}{2}\sum_cw_c\eta_{c,k}\sum_q\operatorname{sd}(E_{c,q,k}),\\ P_j&=\frac{\varepsilon_0c}{2}\sum_cw_c\sum_q\sum_k\nu_k\,\eta_{c,k}\,r_j(c,k)\,|E_{c,q,k}|^2, \end{aligned} \end{split}\]

for aperture_spectrum, aperture_on_axis_spectrum, region_spectrum (a region \(R_j\) is a half-open box of \(\boldsymbol\kappa\)), beam_spectrum and absorbed_power. In \(P_j\), \(\nu_k\) are the exact energy weights of energy_weights() and the rates are the spectral absorber’s \(\alpha_{\rm abs}(\omega_k)\) on every bin, the transverse-wavevector rate (zero until the operator carries one), \(2\min(|\operatorname{Im}k_z|,\alpha_{\rm ev})\) on active evanescent bins and \(2\operatorname{Im}k_z\) on active propagating bins. With \(W\equiv1\), \(S_a\) is \(S^{\rm beam}\) and \(S^{\rm ax}_a\) is on_axis_spectrum; \(\delta\omega \sum_kS^{\rm beam}_k\) is energy on every interior bin, and \(-\sum_jP_j\) is the derivative of energy along \(z\) under the linear flow.