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96 lines
3.7 KiB
Python
96 lines
3.7 KiB
Python
"""
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Tools for finite difference frequency-domain (FDFD) simulations and calculations.
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These mostly involve picking a single frequency, then setting up and solving a
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matrix equation (Ax=b) or eigenvalue problem.
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Submodules:
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- `operators`, `functional`: General FDFD problem setup.
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- `solvers`: Solver interface and reference implementation.
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- `scpml`: Stretched-coordinate perfectly matched layer (scpml) boundary conditions
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- `waveguide_2d`: Operators and mode-solver for waveguides with constant cross-section.
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- `waveguide_3d`: Functions for transforming `waveguide_2d` results into 3D.
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================================================================
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From the "Frequency domain" section of `meanas.fdmath`, we have
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$$
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\\begin{aligned}
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\\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
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\\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= \\tilde{H}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\
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\\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\
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\\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= \\tilde{M}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega l \\Delta_t} \\\\
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
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-\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} &= -\\imath \\Omega \\tilde{J}_{\\vec{r}} e^{\\imath \\omega \\Delta_t / 2} \\\\
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\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
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\\end{aligned}
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$$
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resulting in
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$$
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\\begin{aligned}
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\\tilde{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
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\\hat{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
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\\end{aligned}
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$$
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Maxwell's equations are then
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$$
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\\begin{aligned}
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\\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &=
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\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2} \\hat{B}_{\\vec{r} + \\frac{1}{2}}
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- \\hat{M}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &=
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-\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2} \\tilde{D}_{\\vec{r}}
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+ \\tilde{J}_{\\vec{r}} \\\\
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\\tilde{\\nabla} \\cdot \\hat{B}_{\\vec{r} + \\frac{1}{2}} &= 0 \\\\
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\\hat{\\nabla} \\cdot \\tilde{D}_{\\vec{r}} &= \\rho_{\\vec{r}}
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\\end{aligned}
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$$
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With $\\Delta_t \\to 0$, this simplifies to
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$$
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\\begin{aligned}
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\\tilde{E}_{l, \\vec{r}} &\\to \\tilde{E}_{\\vec{r}} \\\\
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\\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to \\tilde{H}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\tilde{J}_{l, \\vec{r}} &\\to \\tilde{J}_{\\vec{r}} \\\\
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\\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to \\tilde{M}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\Omega &\\to \\omega \\\\
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\\tilde{\\partial}_t &\\to -\\imath \\omega \\\\
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\\hat{\\partial}_t &\\to -\\imath \\omega \\\\
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\\end{aligned}
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$$
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and then
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$$
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\\begin{aligned}
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\\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &=
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\\imath \\omega \\hat{B}_{\\vec{r} + \\frac{1}{2}}
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- \\hat{M}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &=
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-\\imath \\omega \\tilde{D}_{\\vec{r}}
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+ \\tilde{J}_{\\vec{r}} \\\\
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\\end{aligned}
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$$
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$$
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
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-\\omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = -\\imath \\omega \\tilde{J}_{\\vec{r}} \\\\
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$$
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# TODO FDFD?
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# TODO PML
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"""
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from . import solvers, operators, functional, scpml, waveguide_2d, waveguide_3d
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# from . import farfield, bloch TODO
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