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< / li >
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< / nav >
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< div class = "md-content" data-md-component = "content" >
< article class = "md-content__inner md-typeset" >
< h1 id = "fdtd" > fdtd< a class = "headerlink" href = "#fdtd" title = "Permanent link" > ¶ < / a > < / h1 >
< div class = "doc doc-object doc-module" >
< h2 id = "meanas.fdtd" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-module" > < / code > < span class = "doc doc-object-name doc-module-name" > meanas.fdtd< / span >
< a href = "#meanas.fdtd" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h2 >
< div class = "doc doc-contents first" >
< p > Utilities for running finite-difference time-domain (FDTD) simulations< / p >
< p > See the discussion of < code > Maxwell's Equations< / code > in < code > meanas.fdmath< / code > for basic
mathematical background.< / p >
< h3 id = "meanas.fdtd--timestep" > Timestep< a class = "headerlink" href = "#meanas.fdtd--timestep" title = "Permanent link" > ¶ < / a > < / h3 >
< p > From the discussion of "Plane waves and the Dispersion relation" in < code > meanas.fdmath< / code > ,
we have< / p >
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< div class = "arithmatex" > \[ c^2 \Delta_t^2 = \frac{\Delta_t^2}{\mu \epsilon} < 1 / ( \ frac { 1 } { \ Delta_x ^ 2 } + \ frac { 1 } { \ Delta_y ^ 2 } + \ frac { 1 } { \ Delta_z ^ 2 } ) \ ] < / div >
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< p > or, if < span class = "arithmatex" > \(\Delta_x = \Delta_y = \Delta_z\)< / span > , then < span class = "arithmatex" > \(c \Delta_t < \frac{\Delta_x}{\sqrt{3}}\)< / span > .< / p >
< p > Based on this, we can set< / p >
< div class = "highlight" > < pre > < span > < / span > < code > dt = sqrt(mu.min() * epsilon.min()) / sqrt(1/dx_min**2 + 1/dy_min**2 + 1/dz_min**2)
< / code > < / pre > < / div >
< p > The < code > dx_min< / code > , < code > dy_min< / code > , < code > dz_min< / code > should be the minimum value across both the base and derived grids.< / p >
< h3 id = "meanas.fdtd--poynting-vector-and-energy-conservation" > Poynting Vector and Energy Conservation< a class = "headerlink" href = "#meanas.fdtd--poynting-vector-and-energy-conservation" title = "Permanent link" > ¶ < / a > < / h3 >
< p > Let< / p >
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< div class = "arithmatex" > \[
\begin{aligned}
\tilde{S}_{l, l', \vec{r}} & =& & \tilde{E}_{l, \vec{r}} \otimes \hat{H}_{l', \vec{r} + \frac{1}{2}} \\
& =& & \vec{x} (\tilde{E}^y_{l,m+1,n,p} \hat{H}^z_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^z_{l,m+1,n,p} \hat{H}^y_{l', \vec{r} + \frac{1}{2}}) \\
& & + & \vec{y} (\tilde{E}^z_{l,m,n+1,p} \hat{H}^x_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^x_{l,m,n+1,p} \hat{H}^z_{l', \vec{r} + \frac{1}{2}}) \\
& & + & \vec{z} (\tilde{E}^x_{l,m,n,p+1} \hat{H}^y_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^y_{l,m,n,p+1} \hat{H}^z_{l', \vec{r} + \frac{1}{2}})
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\end{aligned}
\]< / div >
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< p > where < span class = "arithmatex" > \(\vec{r} = (m, n, p)\)< / span > and < span class = "arithmatex" > \(\otimes\)< / span > is a modified cross product
in which the < span class = "arithmatex" > \(\tilde{E}\)< / span > terms are shifted as indicated.< / p >
< p > By taking the divergence and rearranging terms, we can show that< / p >
< div class = "arithmatex" > \[
\begin{aligned}
\hat{\nabla} \cdot \tilde{S}_{l, l', \vec{r}}
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& = \hat{\nabla} \cdot (\tilde{E}_{l, \vec{r}} \otimes \hat{H}_{l', \vec{r} + \frac{1}{2}}) \\
& = \hat{H}_{l', \vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l, \vec{r}} -
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\tilde{E}_{l, \vec{r}} \cdot \hat{\nabla} \times \hat{H}_{l', \vec{r} + \frac{1}{2}} \\
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& = \hat{H}_{l', \vec{r} + \frac{1}{2}} \cdot
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(-\tilde{\partial}_t \mu_{\vec{r} + \frac{1}{2}} \hat{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} -
\hat{M}_{l, \vec{r} + \frac{1}{2}}) -
\tilde{E}_{l, \vec{r}} \cdot (\hat{\partial}_t \tilde{\epsilon}_{\vec{r}} \tilde{E}_{l'+\frac{1}{2}, \vec{r}} +
\tilde{J}_{l', \vec{r}}) \\
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& = \hat{H}_{l'} \cdot (-\mu / \Delta_t)(\hat{H}_{l + \frac{1}{2}} - \hat{H}_{l - \frac{1}{2}}) -
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\tilde{E}_l \cdot (\epsilon / \Delta_t )(\tilde{E}_{l'+\frac{1}{2}} - \tilde{E}_{l'-\frac{1}{2}})
- \hat{H}_{l'} \cdot \hat{M}_{l} - \tilde{E}_l \cdot \tilde{J}_{l'} \\
\end{aligned}
\]< / div >
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< p > where in the last line the spatial subscripts have been dropped to emphasize
the time subscripts < span class = "arithmatex" > \(l, l'\)< / span > , i.e.< / p >
< div class = "arithmatex" > \[
\begin{aligned}
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\tilde{E}_l & = \tilde{E}_{l, \vec{r}} \\
\hat{H}_l & = \tilde{H}_{l, \vec{r} + \frac{1}{2}} \\
\tilde{\epsilon} & = \tilde{\epsilon}_{\vec{r}} \\
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\end{aligned}
\]< / div >
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< p > etc.
For < span class = "arithmatex" > \(l' = l + \frac{1}{2}\)< / span > we get< / p >
< div class = "arithmatex" > \[
\begin{aligned}
\hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}}
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& = \hat{H}_{l + \frac{1}{2}} \cdot
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(-\mu / \Delta_t)(\hat{H}_{l + \frac{1}{2}} - \hat{H}_{l - \frac{1}{2}}) -
\tilde{E}_l \cdot (\epsilon / \Delta_t)(\tilde{E}_{l+1} - \tilde{E}_l)
- \hat{H}_{l'} \cdot \hat{M}_l - \tilde{E}_l \cdot \tilde{J}_{l + \frac{1}{2}} \\
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& = (-\mu / \Delta_t)(\hat{H}^2_{l + \frac{1}{2}} - \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}}) -
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(\epsilon / \Delta_t)(\tilde{E}_{l+1} \cdot \tilde{E}_l - \tilde{E}^2_l)
- \hat{H}_{l'} \cdot \hat{M}_l - \tilde{E}_l \cdot \tilde{J}_{l + \frac{1}{2}} \\
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& = -(\mu \hat{H}^2_{l + \frac{1}{2}}
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+\epsilon \tilde{E}_{l+1} \cdot \tilde{E}_l) / \Delta_t \\
+(\mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}}
+\epsilon \tilde{E}^2_l) / \Delta_t \\
- \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\
- \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\
\end{aligned}
\]< / div >
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< p > and for < span class = "arithmatex" > \(l' = l - \frac{1}{2}\)< / span > ,< / p >
< div class = "arithmatex" > \[
\begin{aligned}
\hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}}
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& = (\mu \hat{H}^2_{l - \frac{1}{2}}
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+\epsilon \tilde{E}_{l-1} \cdot \tilde{E}_l) / \Delta_t \\
-(\mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}}
+\epsilon \tilde{E}^2_l) / \Delta_t \\
- \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\
- \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\
\end{aligned}
\]< / div >
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< p > These two results form the discrete time-domain analogue to Poynting's theorem.
They hint at the expressions for the energy, which can be calculated at the same
time-index as either the E or H field:< / p >
< div class = "arithmatex" > \[
\begin{aligned}
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U_l & = \epsilon \tilde{E}^2_l + \mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}} \\
U_{l + \frac{1}{2}} & = \epsilon \tilde{E}_l \cdot \tilde{E}_{l + 1} + \mu \hat{H}^2_{l + \frac{1}{2}} \\
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\end{aligned}
\]< / div >
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< p > Rewriting the Poynting theorem in terms of the energy expressions,< / p >
< div class = "arithmatex" > \[
\begin{aligned}
(U_{l+\frac{1}{2}} - U_l) / \Delta_t
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& = -\hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}} \\
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- \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\
- \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\
(U_l - U_{l-\frac{1}{2}}) / \Delta_t
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& = -\hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}} \\
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- \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\
- \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\
\end{aligned}
\]< / div >
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< p > This result is exact and should practically hold to within numerical precision. No time-
or spatial-averaging is necessary.< / p >
< p > Note that each value of < span class = "arithmatex" > \(J\)< / span > contributes to the energy twice (i.e. once per field update)
despite only causing the value of < span class = "arithmatex" > \(E\)< / span > to change once (same for < span class = "arithmatex" > \(M\)< / span > and < span class = "arithmatex" > \(H\)< / span > ).< / p >
< h3 id = "meanas.fdtd--sources" > Sources< a class = "headerlink" href = "#meanas.fdtd--sources" title = "Permanent link" > ¶ < / a > < / h3 >
< p > It is often useful to excite the simulation with an arbitrary broadband pulse and then
extract the frequency-domain response by performing an on-the-fly Fourier transform
of the time-domain fields.< / p >
< p > < code > accumulate_phasor< / code > in < code > meanas.fdtd.phasor< / code > performs the phase accumulation for one
or more target frequencies, while leaving source normalization and simulation-loop
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policy to the caller. < code > temporal_phasor(...)< / code > and < code > temporal_phasor_scale(...)< / code >
apply the same Fourier sum to a scalar waveform, which is useful when a pulsed
source envelope must be normalized before being applied to a point source or
mode source. < code > real_injection_scale(...)< / code > is the corresponding helper for the
common real-valued injection pattern < code > numpy.real(scale * waveform)< / code > . Convenience
wrappers < code > accumulate_phasor_e< / code > , < code > accumulate_phasor_h< / code > , and < code > accumulate_phasor_j< / code >
apply the usual Yee time offsets. < code > reconstruct_real(...)< / code > and the corresponding
< code > reconstruct_real_e/h/j(...)< / code > wrappers perform the inverse operation, converting
phasors back into real-valued field snapshots at explicit Yee-aligned times.
For scalar < code > omega< / code > , the reconstruction helpers accept either a plain field
phasor or the batched < code > (1, *sample_shape)< / code > form used by the accumulator helpers.
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The helpers accumulate< / p >
< div class = "arithmatex" > \[ \Delta_t \sum_l w_l e^{-i \omega t_l} f_l \]< / div >
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< p > with caller-provided sample times and weights. In this codebase the matching
electric-current convention is typically < code > E -= dt * J / epsilon< / code > in FDTD and
< code > -i \omega J< / code > on the right-hand side of the FDFD wave equation.< / p >
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< p > For FDTD/FDFD equivalence, this phasor path is the primary comparison workflow.
It isolates the guided < code > +\omega< / code > response that the frequency-domain solver
targets directly, regardless of whether the underlying FDTD run used real- or
complex-valued fields.< / p >
< p > For exact pulsed FDTD/FDFD equivalence it is often simplest to keep the
injected source, fields, and CPML auxiliary state complex-valued. The
< code > real_injection_scale(...)< / code > helper is instead for the more ordinary one-run
real-valued source path, where the intended positive-frequency waveform is
injected through < code > numpy.real(scale * waveform)< / code > and any remaining negative-
frequency leakage is controlled by the pulse bandwidth and run length.< / p >
< p > < code > reconstruct_real(...)< / code > is for a different question: given a phasor, what late
real-valued field snapshot should it produce? That raw-snapshot comparison is
stricter and noisier because a monitor plane generally contains both the guided
field and the remaining orthogonal content,< / p >
< div class = "arithmatex" > \[ E_{\text{monitor}} = E_{\text{guided}} + E_{\text{residual}} . \]< / div >
< p > Phasor/modal comparisons mostly validate the guided < code > +\omega< / code > term. Raw
real-field comparisons expose both terms at once, so they should be treated as
secondary diagnostics rather than the main solver-equivalence benchmark.< / p >
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< p > The Ricker wavelet (normalized second derivative of a Gaussian) is commonly used for the pulse
shape. It can be written< / p >
< div class = "arithmatex" > \[ f_r(t) = (1 - \frac{1}{2} (\omega (t - \tau))^2) e^{-(\frac{\omega (t - \tau)}{2})^2} \]< / div >
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< p > with < span class = "arithmatex" > \(\tau > \frac{2 * \pi}{\omega}\)< / span > as a minimum delay to avoid a discontinuity at
t=0 (assuming the source is off for t< 0 this gives < span class = "arithmatex" > \(\sim 10^{-3}\)< / span > error at t=0).< / p >
< h3 id = "meanas.fdtd--boundary-conditions" > Boundary conditions< a class = "headerlink" href = "#meanas.fdtd--boundary-conditions" title = "Permanent link" > ¶ < / a > < / h3 >
< p > < code > meanas.fdtd< / code > exposes two boundary-related building blocks:< / p >
< ul >
< li > < code > conducting_boundary(...)< / code > for simple perfect-electric-conductor style field
clamping at one face of the domain.< / li >
< li > < code > cpml_params(...)< / code > and < code > updates_with_cpml(...)< / code > for convolutional perfectly
matched layers (CPMLs) attached to one or more faces of the Yee grid.< / li >
< / ul >
< p > < code > updates_with_cpml(...)< / code > accepts a three-by-two table of CPML parameter blocks:< / p >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > cpml_params[axis][polarity_index]
< / code > < / pre > < / div >
< p > where < code > axis< / code > is < code > 0< / code > , < code > 1< / code > , or < code > 2< / code > and < code > polarity_index< / code > corresponds to < code > (-1, +1)< / code > .
Passing < code > None< / code > for one entry disables CPML on that face while leaving the other
directions unchanged. This is how mixed boundary setups such as "absorbing in x,
periodic in y/z" are expressed.< / p >
< p > When comparing an FDTD run against an FDFD solve, use the same stretched
coordinate system in both places:< / p >
< ol >
< li > Build the FDTD update with the desired CPML parameters.< / li >
< li > Stretch the FDFD < code > dxes< / code > with the matching SCPML transform.< / li >
< li > Compare the extracted phasor against the FDFD field or residual on those
stretched < code > dxes< / code > .< / li >
< / ol >
< p > The electric-current sign convention used throughout the examples and tests is< / p >
< div class = "arithmatex" > \[
E \leftarrow E - \Delta_t J / \epsilon
\]< / div >
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< p > which matches the FDFD right-hand side< / p >
< div class = "arithmatex" > \[
-i \omega J.
\]< / div >
< div class = "doc doc-children" >
< / div >
< / div >
< / div > < h2 id = "core-update-and-analysis-helpers" > Core update and analysis helpers< a class = "headerlink" href = "#core-update-and-analysis-helpers" title = "Permanent link" > ¶ < / a > < / h2 >
< div class = "doc doc-object doc-module" >
< h2 id = "meanas.fdtd.base" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-module" > < / code > < span class = "doc doc-object-name doc-module-name" > meanas.fdtd.base< / span >
< a href = "#meanas.fdtd.base" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h2 >
< div class = "doc doc-contents first" >
< p > Basic FDTD field updates< / p >
< div class = "doc doc-children" >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.base.maxwell_e" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > maxwell_e< / span >
< a href = "#meanas.fdtd.base.maxwell_e" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > maxwell_e< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span > < span class = "n" > dxes< / span > < span class = "p" > :< / span > < span class = "n" > dx_lists_t< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > fdfield_updater_t< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Build a function which performs a portion the time-domain E-field update,< / p >
< div class = "highlight" > < pre > < span > < / span > < code > E += curl_back(H[t]) / epsilon
< / code > < / pre > < / div >
< p > The full update should be< / p >
< div class = "highlight" > < pre > < span > < / span > < code > E += (curl_back(H[t]) + J) / epsilon
< / code > < / pre > < / div >
< p > which requires an additional step of < code > E += J / epsilon< / code > which is not performed
by the generated function.< / p >
< p > See < code > meanas.fdmath< / code > for descriptions of< / p >
< ul >
< li > This update step: "Maxwell's equations" section< / li >
< li > < code > dxes< / code > : "Datastructure: dx_lists_t" section< / li >
< li > < code > epsilon< / code > : "Permittivity and Permeability" section< / li >
< / ul >
< p > Also see the "Timestep" section of < code > meanas.fdtd< / code > for a discussion of
the < code > dt< / code > parameter.< / p >
< p > < span class = "doc-section-title" > Parameters:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th >
< th > Type< / th >
< th > Description< / th >
< th > Default< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > dt< / code >
< / td >
< td >
< code > < span title = "float" > float< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Timestep. See < code > meanas.fdtd< / code > for details.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dxes< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " dx_lists_t
module-attribute
(meanas.fdmath.dx_lists_t)" href="../fdmath/#meanas.fdmath.types.dx_lists_t">dx_lists_t< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Grid description; see < code > meanas.fdmath< / code > .< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< / tbody >
< / table >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield_updater_t
module-attribute
(meanas.fdmath.fdfield_updater_t)" href="../fdmath/#meanas.fdmath.types.fdfield_updater_t">fdfield_updater_t< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Function < code > f(E_old, H_old, epsilon) -> E_new< / code > .< / p >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.base.maxwell_h" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > maxwell_h< / span >
< a href = "#meanas.fdtd.base.maxwell_h" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > maxwell_h< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span > < span class = "n" > dxes< / span > < span class = "p" > :< / span > < span class = "n" > dx_lists_t< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > fdfield_updater_t< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Build a function which performs part of the time-domain H-field update,< / p >
< div class = "highlight" > < pre > < span > < / span > < code > H -= curl_forward(E[t]) / mu
< / code > < / pre > < / div >
< p > The full update should be< / p >
< div class = "highlight" > < pre > < span > < / span > < code > H -= (curl_forward(E[t]) + M) / mu
< / code > < / pre > < / div >
< p > which requires an additional step of < code > H -= M / mu< / code > which is not performed
by the generated function; this step can be omitted if there is no magnetic
current < code > M< / code > .< / p >
< p > See < code > meanas.fdmath< / code > for descriptions of< / p >
< ul >
< li > This update step: "Maxwell's equations" section< / li >
< li > < code > dxes< / code > : "Datastructure: dx_lists_t" section< / li >
< li > < code > mu< / code > : "Permittivity and Permeability" section< / li >
< / ul >
< p > Also see the "Timestep" section of < code > meanas.fdtd< / code > for a discussion of
the < code > dt< / code > parameter.< / p >
< p > < span class = "doc-section-title" > Parameters:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th >
< th > Type< / th >
< th > Description< / th >
< th > Default< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > dt< / code >
< / td >
< td >
< code > < span title = "float" > float< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Timestep. See < code > meanas.fdtd< / code > for details.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dxes< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " dx_lists_t
module-attribute
(meanas.fdmath.dx_lists_t)" href="../fdmath/#meanas.fdmath.types.dx_lists_t">dx_lists_t< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Grid description; see < code > meanas.fdmath< / code > .< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< / tbody >
< / table >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield_updater_t
module-attribute
(meanas.fdmath.fdfield_updater_t)" href="../fdmath/#meanas.fdmath.types.fdfield_updater_t">fdfield_updater_t< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Function < code > f(E_old, H_old, epsilon) -> E_new< / code > .< / p >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< / div >
< / div >
< / div >
< div class = "doc doc-object doc-module" >
< h2 id = "meanas.fdtd.pml" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-module" > < / code > < span class = "doc doc-object-name doc-module-name" > meanas.fdtd.pml< / span >
< a href = "#meanas.fdtd.pml" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h2 >
< div class = "doc doc-contents first" >
< p > Convolutional perfectly matched layer (CPML) support for FDTD updates.< / p >
< p > The helpers in this module construct per-face CPML parameters and then wrap the
standard Yee updates with the additional auxiliary < code > psi< / code > fields needed by the
CPML recurrence.< / p >
< p > The intended call pattern is:< / p >
< ol >
< li > Build a < code > cpml_params[axis][polarity_index]< / code > table with < code > cpml_params(...)< / code > .< / li >
< li > Pass that table into < code > updates_with_cpml(...)< / code > together with < code > dt< / code > , < code > dxes< / code > , and
< code > epsilon< / code > .< / li >
< li > Advance the returned < code > update_E< / code > / < code > update_H< / code > closures in the simulation loop.< / li >
< / ol >
< p > Each face can be enabled or disabled independently by replacing one table entry
with < code > None< / code > .< / p >
< div class = "doc doc-children" >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.pml.cpml_params" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > cpml_params< / span >
< a href = "#meanas.fdtd.pml.cpml_params" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > cpml_params< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > axis< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > polarity< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "n" > thickness< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "o" > =< / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "n" > ln_R_per_layer< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "o" > =< / span > < span class = "o" > -< / span > < span class = "mf" > 1.6< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > epsilon_eff< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "o" > =< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "n" > mu_eff< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "o" > =< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "n" > m< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "o" > =< / span > < span class = "mf" > 3.5< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "n" > ma< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "o" > =< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "n" > cfs_alpha< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "nb" > dict< / span > < span class = "p" > [< / span > < span class = "nb" > str< / span > < span class = "p" > ,< / span > < span class = "n" > Any< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Construct the parameter block for one CPML face.< / p >
< p > < span class = "doc-section-title" > Parameters:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th >
< th > Type< / th >
< th > Description< / th >
< th > Default< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > axis< / code >
< / td >
< td >
< code > < span title = "int" > int< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Which Cartesian axis the CPML is normal to (< code > 0< / code > , < code > 1< / code > , or < code > 2< / code > ).< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > polarity< / code >
< / td >
< td >
< code > < span title = "int" > int< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Which face along that axis (< code > -1< / code > for the low-index face,
< code > +1< / code > for the high-index face).< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dt< / code >
< / td >
< td >
< code > < span title = "float" > float< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Timestep used by the Yee update.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > thickness< / code >
< / td >
< td >
< code > < span title = "int" > int< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Number of Yee cells occupied by the CPML region.< / p >
< / div >
< / td >
< td >
< code > 8< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > ln_R_per_layer< / code >
< / td >
< td >
< code > < span title = "float" > float< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Logarithmic attenuation target per layer.< / p >
< / div >
< / td >
< td >
< code > -1.6< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > epsilon_eff< / code >
< / td >
< td >
< code > < span title = "float" > float< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Effective permittivity used when choosing the CPML scaling.< / p >
< / div >
< / td >
< td >
< code > 1< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > mu_eff< / code >
< / td >
< td >
< code > < span title = "float" > float< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Effective permeability used when choosing the CPML scaling.< / p >
< / div >
< / td >
< td >
< code > 1< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > m< / code >
< / td >
< td >
< code > < span title = "float" > float< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Polynomial grading exponent for < code > sigma< / code > and < code > kappa< / code > .< / p >
< / div >
< / td >
< td >
< code > 3.5< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > ma< / code >
< / td >
< td >
< code > < span title = "float" > float< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Polynomial grading exponent for the complex-frequency shift < code > alpha< / code > .< / p >
< / div >
< / td >
< td >
< code > 1< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > cfs_alpha< / code >
< / td >
< td >
< code > < span title = "float" > float< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Maximum complex-frequency shift parameter.< / p >
< / div >
< / td >
< td >
< code > 0< / code >
< / td >
< / tr >
< / tbody >
< / table >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > < span title = "dict" > dict< / span > [< span title = "str" > str< / span > , < span title = "typing.Any" > Any< / span > ]< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Dictionary with:< / p >
< / div >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > < span title = "dict" > dict< / span > [< span title = "str" > str< / span > , < span title = "typing.Any" > Any< / span > ]< / code >
< / td >
< td >
< div class = "doc-md-description" >
< ul >
< li > < code > param_e< / code > : < code > (p0, p1, p2)< / code > arrays for the E update< / li >
< / ul >
< / div >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > < span title = "dict" > dict< / span > [< span title = "str" > str< / span > , < span title = "typing.Any" > Any< / span > ]< / code >
< / td >
< td >
< div class = "doc-md-description" >
< ul >
< li > < code > param_h< / code > : < code > (p0, p1, p2)< / code > arrays for the H update< / li >
< / ul >
< / div >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > < span title = "dict" > dict< / span > [< span title = "str" > str< / span > , < span title = "typing.Any" > Any< / span > ]< / code >
< / td >
< td >
< div class = "doc-md-description" >
< ul >
< li > < code > region< / code > : slice tuple selecting the CPML cells on that face< / li >
< / ul >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.pml.updates_with_cpml" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > updates_with_cpml< / span >
< a href = "#meanas.fdtd.pml.updates_with_cpml" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > updates_with_cpml< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > cpml_params< / span > < span class = "p" > :< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > dict< / span > < span class = "p" > [< / span > < span class = "nb" > str< / span > < span class = "p" > ,< / span > < span class = "n" > Any< / span > < span class = "p" > ]< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "p" > ]],< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "n" > dxes< / span > < span class = "p" > :< / span > < span class = "n" > dx_lists_t< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "n" > epsilon< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > *< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dtype< / span > < span class = "p" > :< / span > < span class = "n" > DTypeLike< / span > < span class = "o" > =< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > float32< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "nb" > tuple< / span > < span class = "p" > [< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "n" > Callable< / span > < span class = "p" > [[< / span > < span class = "n" > fdfield_t< / span > < span class = "p" > ,< / span > < span class = "n" > fdfield_t< / span > < span class = "p" > ,< / span > < span class = "n" > fdfield_t< / span > < span class = "p" > ],< / span > < span class = "kc" > None< / span > < span class = "p" > ],< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "n" > Callable< / span > < span class = "p" > [[< / span > < span class = "n" > fdfield_t< / span > < span class = "p" > ,< / span > < span class = "n" > fdfield_t< / span > < span class = "p" > ,< / span > < span class = "n" > fdfield_t< / span > < span class = "p" > ],< / span > < span class = "kc" > None< / span > < span class = "p" > ],< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Build Yee-step update closures augmented with CPML terms.< / p >
< p > < span class = "doc-section-title" > Parameters:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th >
< th > Type< / th >
< th > Description< / th >
< th > Default< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > cpml_params< / code >
< / td >
< td >
< code > < span title = "collections.abc.Sequence" > Sequence< / span > [< span title = "collections.abc.Sequence" > Sequence< / span > [< span title = "dict" > dict< / span > [< span title = "str" > str< / span > , < span title = "typing.Any" > Any< / span > ] | None]]< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Three-by-two sequence indexed as < code > [axis][polarity_index]< / code > .
Entries are the dictionaries returned by < code > cpml_params(...)< / code > ; use
< code > None< / code > to disable CPML on one face.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dt< / code >
< / td >
< td >
< code > < span title = "float" > float< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Timestep.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dxes< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " dx_lists_t
module-attribute
(meanas.fdmath.dx_lists_t)" href="../fdmath/#meanas.fdmath.types.dx_lists_t">dx_lists_t< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Yee-grid spacing lists < code > [dx_e, dx_h]< / code > .< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > epsilon< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Electric material distribution used by the E update.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dtype< / code >
< / td >
< td >
< code > < span title = "numpy.typing.DTypeLike" > DTypeLike< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Storage dtype for the auxiliary CPML state arrays.< / p >
< / div >
< / td >
< td >
< code > < span title = "numpy.float32" > float32< / span > < / code >
< / td >
< / tr >
< / tbody >
< / table >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > < span title = "collections.abc.Callable" > Callable< / span > [[< span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > ], None]< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > < code > (update_E, update_H)< / code > closures with the same call shape as the basic< / p >
< / div >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > < span title = "collections.abc.Callable" > Callable< / span > [[< span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > ], None]< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Yee updates:< / p >
< / div >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > < span title = "tuple" > tuple< / span > [< span title = "collections.abc.Callable" > Callable< / span > [[< span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > ], None], < span title = "collections.abc.Callable" > Callable< / span > [[< span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > ], None]]< / code >
< / td >
< td >
< div class = "doc-md-description" >
< ul >
< li > < code > update_E(e, h, epsilon)< / code > < / li >
< / ul >
< / div >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > < span title = "tuple" > tuple< / span > [< span title = "collections.abc.Callable" > Callable< / span > [[< span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > ], None], < span title = "collections.abc.Callable" > Callable< / span > [[< span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > ], None]]< / code >
< / td >
< td >
< div class = "doc-md-description" >
< ul >
< li > < code > update_H(e, h, mu)< / code > < / li >
< / ul >
< / div >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > < span title = "tuple" > tuple< / span > [< span title = "collections.abc.Callable" > Callable< / span > [[< span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > ], None], < span title = "collections.abc.Callable" > Callable< / span > [[< span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > , < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > ], None]]< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > The closures retain the CPML auxiliary state internally.< / p >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< / div >
< / div >
< / div >
< div class = "doc doc-object doc-module" >
< h2 id = "meanas.fdtd.boundaries" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-module" > < / code > < span class = "doc doc-object-name doc-module-name" > meanas.fdtd.boundaries< / span >
< a href = "#meanas.fdtd.boundaries" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h2 >
< div class = "doc doc-contents first" >
< p > Boundary conditions< / p >
< h3 id = "meanas.fdtd.boundaries--todo-conducting-boundary-documentation" > TODO conducting boundary documentation< a class = "headerlink" href = "#meanas.fdtd.boundaries--todo-conducting-boundary-documentation" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc doc-children" >
< / div >
< / div >
< / div >
< div class = "doc doc-object doc-module" >
< h2 id = "meanas.fdtd.energy" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-module" > < / code > < span class = "doc doc-object-name doc-module-name" > meanas.fdtd.energy< / span >
< a href = "#meanas.fdtd.energy" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h2 >
< div class = "doc doc-contents first" >
< div class = "doc doc-children" >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.energy.poynting" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > poynting< / span >
< a href = "#meanas.fdtd.energy.poynting" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > poynting< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > e< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span > < span class = "n" > h< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span > < span class = "n" > dxes< / span > < span class = "p" > :< / span > < span class = "n" > dx_lists_t< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > fdfield_t< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Calculate the poynting vector < code > S< / code > (< span class = "arithmatex" > \(S\)< / span > ).< / p >
< p > This is the energy transfer rate (amount of energy < code > U< / code > per < code > dt< / code > transferred
between adjacent cells) in each direction that happens during the half-step
bounded by the two provided fields.< / p >
< p > The returned vector field < code > S< / code > is the energy flow across +x, +y, and +z
boundaries of the corresponding < code > U< / code > cell. For example,< / p >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > mx = numpy.roll(mask, -1, axis=0)
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > my = numpy.roll(mask, -1, axis=1)
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > mz = numpy.roll(mask, -1, axis=2)
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > u_hstep = fdtd.energy_hstep(e0=es[ii - 1], h1=hs[ii], e2=es[ii], **args)
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > u_estep = fdtd.energy_estep(h0=hs[ii], e1=es[ii], h2=hs[ii + 1], **args)
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > delta_j_B = fdtd.delta_energy_j(j0=js[ii], e1=es[ii], dxes=dxes)
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > du_half_h2e = u_estep - u_hstep - delta_j_B
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > s_h2e = -fdtd.poynting(e=es[ii], h=hs[ii], dxes=dxes) * dt
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > planes = [s_h2e[0, mask].sum(), -s_h2e[0, mx].sum(),
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > s_h2e[1, mask].sum(), -s_h2e[1, my].sum(),
< a id = "__codelineno-0-13" name = "__codelineno-0-13" href = "#__codelineno-0-13" > < / a > s_h2e[2, mask].sum(), -s_h2e[2, mz].sum()]
< a id = "__codelineno-0-14" name = "__codelineno-0-14" href = "#__codelineno-0-14" > < / a >
< a id = "__codelineno-0-15" name = "__codelineno-0-15" href = "#__codelineno-0-15" > < / a > assert_close(sum(planes), du_half_h2e[mask])
< / code > < / pre > < / div >
< p > (see < code > meanas.tests.test_fdtd.test_poynting_planes< / code > )< / p >
2026-04-19 16:40:30 -07:00
< p > The full relationship is< / p >
< div class = "arithmatex" > \[
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\begin{aligned}
(U_{l+\frac{1}{2}} - U_l) / \Delta_t
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& = -\hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}} \\
- \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\
- \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\
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(U_l - U_{l-\frac{1}{2}}) / \Delta_t
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& = -\hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}} \\
- \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\
- \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\
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\end{aligned}
2026-04-19 16:40:30 -07:00
\]< / div >
2026-04-19 00:22:42 -07:00
< p > These equalities are exact and should practically hold to within numerical precision.
No time- or spatial-averaging is necessary. (See < code > meanas.fdtd< / code > docs for derivation.)< / p >
< p > < span class = "doc-section-title" > Parameters:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th >
< th > Type< / th >
< th > Description< / th >
< th > Default< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > e< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > E-field< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > h< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > H-field (one half-timestep before or after < code > e< / code > )< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dxes< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " dx_lists_t
module-attribute
(meanas.fdmath.dx_lists_t)" href="../fdmath/#meanas.fdmath.types.dx_lists_t">dx_lists_t< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Grid description; see < code > meanas.fdmath< / code > .< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< / tbody >
< / table >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th > < th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td > < code > s< / code > < / td > < td >
< code > < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Vector field. Components indicate the energy transfer rate from the
corresponding energy cell into its +x, +y, and +z neighbors during
the half-step from the time of the earlier input field until the
time of later input field.< / p >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.energy.poynting_divergence" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > poynting_divergence< / span >
< a href = "#meanas.fdtd.energy.poynting_divergence" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > poynting_divergence< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > s< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "o" > *< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "n" > e< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "n" > h< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "n" > dxes< / span > < span class = "p" > :< / span > < span class = "n" > dx_lists_t< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > fdfield_t< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Calculate the divergence of the poynting vector.< / p >
< p > This is the net energy flow for each cell, i.e. the change in energy < code > U< / code >
per < code > dt< / code > caused by transfer of energy to nearby cells (rather than
absorption/emission by currents < code > J< / code > or < code > M< / code > ).< / p >
< p > See < code > poynting< / code > and < code > meanas.fdtd< / code > for more details.
Args:
s: Poynting vector, as calculated with < code > poynting< / code > . Optional; caller
can provide < code > e< / code > and < code > h< / code > instead.
e: E-field (optional; need either < code > s< / code > or both < code > e< / code > and < code > h< / code > )
h: H-field (optional; need either < code > s< / code > or both < code > e< / code > and < code > h< / code > )
dxes: Grid description; see < code > meanas.fdmath< / code > .< / p >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th > < th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td > < code > ds< / code > < / td > < td >
< code > < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Divergence of the poynting vector.
Entries indicate the net energy flow out of the corresponding
energy cell.< / p >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.energy.energy_hstep" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > energy_hstep< / span >
< a href = "#meanas.fdtd.energy.energy_hstep" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > energy_hstep< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > e0< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > h1< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "n" > e2< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "n" > epsilon< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "n" > mu< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dxes< / span > < span class = "p" > :< / span > < span class = "n" > dx_lists_t< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > fdfield_t< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Calculate energy < code > U< / code > at the time of the provided H-field < code > h1< / code > .< / p >
< p > TODO: Figure out what this means spatially.< / p >
< p > < span class = "doc-section-title" > Parameters:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th >
< th > Type< / th >
< th > Description< / th >
< th > Default< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > e0< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > E-field one half-timestep before the energy.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > h1< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > H-field (at the same timestep as the energy).< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > e2< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > E-field one half-timestep after the energy.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > epsilon< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Dielectric constant distribution.< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > mu< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Magnetic permeability distribution.< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dxes< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " dx_lists_t
module-attribute
(meanas.fdmath.dx_lists_t)" href="../fdmath/#meanas.fdmath.types.dx_lists_t">dx_lists_t< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Grid description; see < code > meanas.fdmath< / code > .< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< / tbody >
< / table >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Energy, at the time of the H-field < code > h1< / code > .< / p >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.energy.energy_estep" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > energy_estep< / span >
< a href = "#meanas.fdtd.energy.energy_estep" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > energy_estep< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > h0< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > e1< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "n" > h2< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "n" > epsilon< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "n" > mu< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dxes< / span > < span class = "p" > :< / span > < span class = "n" > dx_lists_t< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > fdfield_t< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Calculate energy < code > U< / code > at the time of the provided E-field < code > e1< / code > .< / p >
< p > TODO: Figure out what this means spatially.< / p >
< p > < span class = "doc-section-title" > Parameters:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th >
< th > Type< / th >
< th > Description< / th >
< th > Default< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > h0< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > H-field one half-timestep before the energy.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > e1< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > E-field (at the same timestep as the energy).< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > h2< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > H-field one half-timestep after the energy.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > epsilon< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Dielectric constant distribution.< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > mu< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Magnetic permeability distribution.< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dxes< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " dx_lists_t
module-attribute
(meanas.fdmath.dx_lists_t)" href="../fdmath/#meanas.fdmath.types.dx_lists_t">dx_lists_t< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Grid description; see < code > meanas.fdmath< / code > .< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< / tbody >
< / table >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Energy, at the time of the E-field < code > e1< / code > .< / p >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.energy.delta_energy_h2e" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > delta_energy_h2e< / span >
< a href = "#meanas.fdtd.energy.delta_energy_h2e" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > delta_energy_h2e< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > e0< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "n" > h1< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "n" > e2< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "n" > h3< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > epsilon< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "n" > mu< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "n" > dxes< / span > < span class = "p" > :< / span > < span class = "n" > dx_lists_t< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > fdfield_t< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Change in energy during the half-step from < code > h1< / code > to < code > e2< / code > .< / p >
< p > This is just from (e2 * e2 + h3 * h1) - (h1 * h1 + e0 * e2)< / p >
< p > < span class = "doc-section-title" > Parameters:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th >
< th > Type< / th >
< th > Description< / th >
< th > Default< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > e0< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > E-field one half-timestep before the start of the energy delta.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > h1< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > H-field at the start of the energy delta.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > e2< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > E-field at the end of the energy delta (one half-timestep after < code > h1< / code > ).< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > h3< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > H-field one half-timestep after the end of the energy delta.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > epsilon< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Dielectric constant distribution.< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > mu< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Magnetic permeability distribution.< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dxes< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " dx_lists_t
module-attribute
(meanas.fdmath.dx_lists_t)" href="../fdmath/#meanas.fdmath.types.dx_lists_t">dx_lists_t< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Grid description; see < code > meanas.fdmath< / code > .< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< / tbody >
< / table >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Change in energy from the time of < code > h1< / code > to the time of < code > e2< / code > .< / p >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.energy.delta_energy_e2h" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > delta_energy_e2h< / span >
< a href = "#meanas.fdtd.energy.delta_energy_e2h" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > delta_energy_e2h< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > h0< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "n" > e1< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "n" > h2< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "n" > e3< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > epsilon< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "n" > mu< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "n" > dxes< / span > < span class = "p" > :< / span > < span class = "n" > dx_lists_t< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > fdfield_t< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Change in energy during the half-step from < code > e1< / code > to < code > h2< / code > .< / p >
< p > This is just from (h2 * h2 + e3 * e1) - (e1 * e1 + h0 * h2)< / p >
< p > < span class = "doc-section-title" > Parameters:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th >
< th > Type< / th >
< th > Description< / th >
< th > Default< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > h0< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > E-field one half-timestep before the start of the energy delta.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > e1< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > H-field at the start of the energy delta.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > h2< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > E-field at the end of the energy delta (one half-timestep after < code > e1< / code > ).< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > e3< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > H-field one half-timestep after the end of the energy delta.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > epsilon< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Dielectric constant distribution.< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > mu< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Magnetic permeability distribution.< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dxes< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " dx_lists_t
module-attribute
(meanas.fdmath.dx_lists_t)" href="../fdmath/#meanas.fdmath.types.dx_lists_t">dx_lists_t< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Grid description; see < code > meanas.fdmath< / code > .< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< / tbody >
< / table >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Change in energy from the time of < code > e1< / code > to the time of < code > h2< / code > .< / p >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.energy.delta_energy_j" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > delta_energy_j< / span >
< a href = "#meanas.fdtd.energy.delta_energy_j" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > delta_energy_j< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > j0< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span > < span class = "n" > e1< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span > < span class = "n" > dxes< / span > < span class = "p" > :< / span > < span class = "n" > dx_lists_t< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > fdfield_t< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Calculate the electric-current work term < span class = "arithmatex" > \(J \cdot E\)< / span > on the Yee grid.< / p >
< p > This is the source contribution that appears beside the flux divergence in
the discrete Poynting identities documented in < code > meanas.fdtd< / code > .< / p >
< p > Note that each value of < code > J< / code > contributes twice in a full Yee cycle (once per
half-step energy balance) even though it directly changes < code > E< / code > only once.< / p >
< p > < span class = "doc-section-title" > Parameters:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th >
< th > Type< / th >
< th > Description< / th >
< th > Default< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > j0< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Electric-current density sampled at the same half-step as the
current work term.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > e1< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Electric field sampled at the matching integer timestep.< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dxes< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " dx_lists_t
module-attribute
(meanas.fdmath.dx_lists_t)" href="../fdmath/#meanas.fdmath.types.dx_lists_t">dx_lists_t< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Grid description; defaults to unit spacing.< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< / tbody >
< / table >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Per-cell source-work contribution with the scalar field shape.< / p >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.energy.dxmul" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > dxmul< / span >
< a href = "#meanas.fdtd.energy.dxmul" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > dxmul< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > ee< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > hh< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "n" > epsilon< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "n" > mu< / span > < span class = "p" > :< / span > < span class = "n" > fdfield< / span > < span class = "o" > |< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "n" > dxes< / span > < span class = "p" > :< / span > < span class = "n" > dx_lists_t< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > fdfield_t< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Multiply E- and H-like field products by material weights and cell volumes.< / p >
< p > < span class = "doc-section-title" > Parameters:< / span > < / p >
< table >
< thead >
< tr >
< th > Name< / th >
< th > Type< / th >
< th > Description< / th >
< th > Default< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > ee< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Three-component electric-field product, such as < code > e0 * e2< / code > .< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > hh< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Three-component magnetic-field product, such as < code > h1 * h1< / code > .< / p >
< / div >
< / td >
< td >
< em > required< / em >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > epsilon< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > | < span title = "float" > float< / span > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Electric material weight; defaults to < code > 1< / code > .< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > mu< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " fdfield (meanas.fdmath.fdfield)" href = "../fdmath/#meanas.fdmath.types.fdfield" > fdfield< / a > | < span title = "float" > float< / span > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Magnetic material weight; defaults to < code > 1< / code > .< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > dxes< / code >
< / td >
< td >
< code > < a class = "autorefs autorefs-internal" title = " dx_lists_t
module-attribute
(meanas.fdmath.dx_lists_t)" href="../fdmath/#meanas.fdmath.types.dx_lists_t">dx_lists_t< / a > | None< / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Grid description; defaults to unit spacing.< / p >
< / div >
< / td >
< td >
< code > None< / code >
< / td >
< / tr >
< / tbody >
< / table >
< p > < span class = "doc-section-title" > Returns:< / span > < / p >
< table >
< thead >
< tr >
< th > Type< / th >
< th > Description< / th >
< / tr >
< / thead >
< tbody >
< tr class = "doc-section-item" >
< td >
< code > < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > Scalar field containing the weighted electric plus magnetic contribution< / p >
< / div >
< / td >
< / tr >
< tr class = "doc-section-item" >
< td >
< code > < span title = "meanas.fdmath.fdfield_t" > fdfield_t< / span > < / code >
< / td >
< td >
< div class = "doc-md-description" >
< p > for each Yee cell.< / p >
< / div >
< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< / div >
< / div >
< / div >
< div class = "doc doc-object doc-module" >
< h2 id = "meanas.fdtd.phasor" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-module" > < / code > < span class = "doc doc-object-name doc-module-name" > meanas.fdtd.phasor< / span >
< a href = "#meanas.fdtd.phasor" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h2 >
< div class = "doc doc-contents first" >
< p > Helpers for extracting single- or multi-frequency phasors from FDTD samples.< / p >
< p > These helpers are intentionally low-level: callers own the accumulator arrays and
the sampling policy. The accumulated quantity is< / p >
< div class = "highlight" > < pre > < span > < / span > < code > dt * sum(weight * exp(-1j * omega * t_step) * sample_step)
< / code > < / pre > < / div >
< p > where < code > t_step = (step + offset_steps) * dt< / code > .< / p >
< p > The usual Yee offsets are:< / p >
< ul >
< li > < code > accumulate_phasor_e(..., step=l)< / code > for < code > E_l< / code > < / li >
< li > < code > accumulate_phasor_h(..., step=l)< / code > for < code > H_{l + 1/2}< / code > < / li >
< li > < code > accumulate_phasor_j(..., step=l)< / code > for < code > J_{l + 1/2}< / code > < / li >
< / ul >
2026-04-19 16:40:30 -07:00
< p > < code > temporal_phasor(...)< / code > and < code > temporal_phasor_scale(...)< / code > apply the same Fourier
sum to a 1D scalar waveform. They are useful for normalizing a pulsed source
before that scalar waveform is applied to a point source or spatial mode source.
< code > real_injection_scale(...)< / code > is a companion helper for the common real-valued
injection pattern < code > numpy.real(scale * waveform)< / code > , where < code > waveform< / code > is the
analytic positive-frequency signal and the injected real current should still
produce a desired phasor response.
< code > reconstruct_real(...)< / code > and its < code > E/H/J< / code > wrappers perform the inverse operation:
they turn one or more phasors back into real-valued field snapshots at explicit
Yee-aligned sample times. For a scalar target frequency they accept either a
plain field phasor or the batched < code > (1, *sample_shape)< / code > form used elsewhere in
this module.< / p >
2026-04-19 00:22:42 -07:00
< p > These helpers do not choose warmup/accumulation windows or pulse-envelope
normalization. They also do not impose a current sign convention. In this
codebase, electric-current injection normally appears as < code > E -= dt * J / epsilon< / code > ,
which matches the FDFD right-hand side < code > -1j * omega * J< / code > .< / p >
< div class = "doc doc-children" >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.phasor.accumulate_phasor" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > accumulate_phasor< / span >
< a href = "#meanas.fdtd.phasor.accumulate_phasor" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > accumulate_phasor< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > accumulator< / span > < span class = "p" > :< / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > complexfloating< / span > < span class = "p" > ],< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > omegas< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "o" > |< / span > < span class = "nb" > complex< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "o" > |< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "nb" > complex< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > |< / span > < span class = "n" > NDArray< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "n" > sample< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "n" > step< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "o" > *< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "n" > offset_steps< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "o" > =< / span > < span class = "mf" > 0.0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > < span class = "n" > weight< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "o" > =< / span > < span class = "mf" > 1.0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-13" name = "__codelineno-0-13" href = "#__codelineno-0-13" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > complexfloating< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Add one time-domain sample into a phasor accumulator.< / p >
< p > The added quantity is< / p >
< div class = "highlight" > < pre > < span > < / span > < code > dt * weight * exp(-1j * omega * t_step) * sample
< / code > < / pre > < / div >
< p > where < code > t_step = (step + offset_steps) * dt< / code > .< / p >
< details class = "note" open >
< summary > Note< / summary >
< p > This helper already multiplies by < code > dt< / code > . If the caller's normalization
factor was derived from a discrete sum that already includes < code > dt< / code > , pass
< code > weight / dt< / code > here.< / p >
< / details >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
2026-04-19 16:40:30 -07:00
< h3 id = "meanas.fdtd.phasor.temporal_phasor" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > temporal_phasor< / span >
< a href = "#meanas.fdtd.phasor.temporal_phasor" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > temporal_phasor< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > samples< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > omegas< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "o" > |< / span > < span class = "nb" > complex< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "o" > |< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "nb" > complex< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > |< / span > < span class = "n" > NDArray< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "o" > *< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "n" > start_step< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "n" > offset_steps< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "o" > =< / span > < span class = "mf" > 0.0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > complexfloating< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Fourier-project a 1D temporal waveform onto one or more angular frequencies.< / p >
< p > The returned quantity is< / p >
< div class = "highlight" > < pre > < span > < / span > < code > dt * sum(exp(-1j * omega * t_step) * samples[step_index])
< / code > < / pre > < / div >
< p > where < code > t_step = (start_step + step_index + offset_steps) * dt< / code > .< / p >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.phasor.temporal_phasor_scale" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > temporal_phasor_scale< / span >
< a href = "#meanas.fdtd.phasor.temporal_phasor_scale" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > temporal_phasor_scale< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > samples< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > omegas< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "o" > |< / span > < span class = "nb" > complex< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "o" > |< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "nb" > complex< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > |< / span > < span class = "n" > NDArray< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "o" > *< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "n" > start_step< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "n" > offset_steps< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "o" > =< / span > < span class = "mf" > 0.0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "n" > target< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "o" > =< / span > < span class = "mf" > 1.0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > complexfloating< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Return the scalar multiplier that gives a desired temporal phasor response.< / p >
< p > The returned scale satisfies< / p >
< div class = "highlight" > < pre > < span > < / span > < code > temporal_phasor(scale * samples, omegas, dt, ...) == target
< / code > < / pre > < / div >
< p > for each target frequency. The result keeps a leading frequency axis even
when < code > omegas< / code > is scalar.< / p >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.phasor.real_injection_scale" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > real_injection_scale< / span >
< a href = "#meanas.fdtd.phasor.real_injection_scale" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > real_injection_scale< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > samples< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > omegas< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "o" > |< / span > < span class = "nb" > complex< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "o" > |< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "nb" > complex< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > |< / span > < span class = "n" > NDArray< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "o" > *< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "n" > start_step< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "n" > offset_steps< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "o" > =< / span > < span class = "mf" > 0.0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "n" > target< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "o" > =< / span > < span class = "mf" > 1.0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > complexfloating< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Return the scale for a real-valued injection built from an analytic waveform.< / p >
< p > If the time-domain source is applied as< / p >
< div class = "highlight" > < pre > < span > < / span > < code > numpy.real(scale * samples[step])
< / code > < / pre > < / div >
< p > then the desired positive-frequency phasor is obtained by compensating for
the 1/2 factor between the real-valued source and its analytic component:< / p >
< div class = "highlight" > < pre > < span > < / span > < code > scale = 2 * target / temporal_phasor(samples, ...)
< / code > < / pre > < / div >
< p > This helper normalizes only the intended positive-frequency component. Any
residual negative-frequency leakage is controlled by the waveform design and
the accumulation window.< / p >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.phasor.reconstruct_real" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > reconstruct_real< / span >
< a href = "#meanas.fdtd.phasor.reconstruct_real" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > reconstruct_real< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > phasors< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > omegas< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "o" > |< / span > < span class = "nb" > complex< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "o" > |< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "nb" > complex< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > |< / span > < span class = "n" > NDArray< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "n" > step< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "o" > *< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "n" > offset_steps< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "o" > =< / span > < span class = "mf" > 0.0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > floating< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Reconstruct a real-valued field snapshot from one or more phasors.< / p >
< p > The returned quantity is< / p >
< div class = "highlight" > < pre > < span > < / span > < code > real(phasor * exp(1j * omega * t_step))
< / code > < / pre > < / div >
< p > where < code > t_step = (step + offset_steps) * dt< / code > .< / p >
< p > For multi-frequency inputs, the leading frequency axis is preserved. For a
scalar < code > omega< / code > , callers may pass either < code > (1, *sample_shape)< / code > or
< code > sample_shape< / code > ; the return shape matches that choice.< / p >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
2026-04-19 00:22:42 -07:00
< h3 id = "meanas.fdtd.phasor.accumulate_phasor_e" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > accumulate_phasor_e< / span >
< a href = "#meanas.fdtd.phasor.accumulate_phasor_e" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > accumulate_phasor_e< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > accumulator< / span > < span class = "p" > :< / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > complexfloating< / span > < span class = "p" > ],< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > omegas< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "o" > |< / span > < span class = "nb" > complex< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "o" > |< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "nb" > complex< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > |< / span > < span class = "n" > NDArray< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "n" > sample< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "n" > step< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "o" > *< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "n" > weight< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "o" > =< / span > < span class = "mf" > 1.0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > complexfloating< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Accumulate an E-field sample taken at integer timestep < code > step< / code > .< / p >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.phasor.accumulate_phasor_h" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > accumulate_phasor_h< / span >
< a href = "#meanas.fdtd.phasor.accumulate_phasor_h" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > accumulate_phasor_h< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > accumulator< / span > < span class = "p" > :< / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > complexfloating< / span > < span class = "p" > ],< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > omegas< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "o" > |< / span > < span class = "nb" > complex< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "o" > |< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "nb" > complex< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > |< / span > < span class = "n" > NDArray< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "n" > sample< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "n" > step< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "o" > *< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "n" > weight< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "o" > =< / span > < span class = "mf" > 1.0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > complexfloating< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Accumulate an H-field sample corresponding to < code > H_{step + 1/2}< / code > .< / p >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.phasor.accumulate_phasor_j" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > accumulate_phasor_j< / span >
< a href = "#meanas.fdtd.phasor.accumulate_phasor_j" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > accumulate_phasor_j< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > accumulator< / span > < span class = "p" > :< / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > complexfloating< / span > < span class = "p" > ],< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > omegas< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "o" > |< / span > < span class = "nb" > complex< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "o" > |< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "nb" > complex< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > |< / span > < span class = "n" > NDArray< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "n" > sample< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "n" > step< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "o" > *< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "n" > weight< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "o" > =< / span > < span class = "mf" > 1.0< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > complexfloating< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Accumulate a current sample corresponding to < code > J_{step + 1/2}< / code > .< / p >
< / div >
< / div >
2026-04-19 16:40:30 -07:00
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.phasor.reconstruct_real_e" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > reconstruct_real_e< / span >
< a href = "#meanas.fdtd.phasor.reconstruct_real_e" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > reconstruct_real_e< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > phasors< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > omegas< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "o" > |< / span > < span class = "nb" > complex< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "o" > |< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "nb" > complex< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > |< / span > < span class = "n" > NDArray< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "n" > step< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > floating< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Reconstruct a real E-field snapshot taken at integer timestep < code > step< / code > .< / p >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.phasor.reconstruct_real_h" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > reconstruct_real_h< / span >
< a href = "#meanas.fdtd.phasor.reconstruct_real_h" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > reconstruct_real_h< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > phasors< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > omegas< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "o" > |< / span > < span class = "nb" > complex< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "o" > |< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "nb" > complex< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > |< / span > < span class = "n" > NDArray< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "n" > step< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > floating< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Reconstruct a real H-field snapshot corresponding to < code > H_{step + 1/2}< / code > .< / p >
< / div >
< / div >
< div class = "doc doc-object doc-function" >
< h3 id = "meanas.fdtd.phasor.reconstruct_real_j" class = "doc doc-heading" >
< code class = "doc-symbol doc-symbol-heading doc-symbol-function" > < / code > < span class = "doc doc-object-name doc-function-name" > reconstruct_real_j< / span >
< a href = "#meanas.fdtd.phasor.reconstruct_real_j" class = "headerlink" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "doc-signature highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nf" > reconstruct_real_j< / span > < span class = "p" > (< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "n" > phasors< / span > < span class = "p" > :< / span > < span class = "n" > ArrayLike< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > omegas< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "o" > |< / span > < span class = "nb" > complex< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "o" > |< / span > < span class = "n" > Sequence< / span > < span class = "p" > [< / span > < span class = "nb" > float< / span > < span class = "o" > |< / span > < span class = "nb" > complex< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "o" > |< / span > < span class = "n" > NDArray< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "n" > dt< / span > < span class = "p" > :< / span > < span class = "nb" > float< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "n" > step< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "n" > NDArray< / span > < span class = "p" > [< / span > < span class = "n" > numpy< / span > < span class = "o" > .< / span > < span class = "n" > floating< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< div class = "doc doc-contents " >
< p > Reconstruct a real current snapshot corresponding to < code > J_{step + 1/2}< / code > .< / p >
< / div >
< / div >
2026-04-19 00:22:42 -07:00
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