add waveguide eigenproblem derivation
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@ -12,6 +12,14 @@ Submodules:
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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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# 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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@ -5,7 +5,144 @@ The propagation direction is chosen to be along the z axis, and all fields
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are given an implicit z-dependence of the form `exp(-1 * wavenumber * z)`.
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As the z-dependence is known, all the functions in this file assume a 2D grid
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(i.e. `dxes = [[[dx_e_0, dx_e_1, ...], [dy_e_0, ...]], [[dx_h_0, ...], [dy_h_0, ...]]]`).
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(i.e. `dxes = [[[dx_e[0], dx_e[1], ...], [dy_e[0], ...]], [[dx_h[0], ...], [dy_h[0], ...]]]`).
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===============
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Consider Maxwell's equations in continuous space, in the frequency domain. Assuming
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a structure with some (x, y) cross-section extending uniformly into the z dimension,
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with a diagonal \\( \\epsilon \\) tensor, we have
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$$
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\\begin{align*}
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\\nabla \\times \\vec{E}(x, y, z) &= -\\imath \\omega \\mu \\vec{H} \\\\
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\\nabla \\times \\vec{H}(x, y, z) &= \\imath \\omega \\epsilon \\vec{E} \\\\
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\\vec{E}(x,y,z) = (\\vec{E}_t(x, y) + E_z(x, y)\\vec{z}) e^{-\\gamma z} \\\\
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\\vec{H}(x,y,z) = (\\vec{H}_t(x, y) + H_z(x, y)\\vec{z}) e^{-\\gamma z} \\\\
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\\end{align*}
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$$
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Expanding the first two equations into vector components, we get
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$$
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\\begin{align*}
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-\\imath \\omega \\mu_{xx} H_x &= \\partial_y E_z - \\partial_z E_y \\\\
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-\\imath \\omega \\mu_{yy} H_y &= \\partial_z E_x - \\partial_x E_z \\\\
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-\\imath \\omega \\mu_{zz} H_z &= \\partial_x E_y - \\partial_y E_x \\\\
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\\imath \\omega \\epsilon_{xx} E_x &= \\partial_y H_z - \\partial_z H_y \\\\
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\\imath \\omega \\epsilon_{yy} E_y &= \\partial_z H_x - \\partial_x H_z \\\\
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\\imath \\omega \\epsilon_{zz} E_z &= \\partial_x H_y - \\partial_y H_x \\\\
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\\end{align*}
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$$
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Substituting in our expressions for \\( \\vec{E}, \\vec{H} \\) and discretizing:
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$$
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\\begin{align*}
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-\\imath \\omega \\mu_{xx} H_x &= \\tilde{\\partial}_y E_z + \\gamma E_y \\\\
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-\\imath \\omega \\mu_{yy} H_y &= -\\gamma E_x - \\tilde{\\partial}_x E_z \\\\
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-\\imath \\omega \\mu_{zz} H_z &= \\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x \\\\
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\\imath \\omega \\epsilon_{xx} E_x &= \\hat{\\partial}_y H_z + \\gamma H_y \\\\
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\\imath \\omega \\epsilon_{yy} E_y &= -\\gamma H_x - \\hat{\\partial}_x H_z \\\\
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\\imath \\omega \\epsilon_{zz} E_z &= \\hat{\\partial}_x H_y - \\hat{\\partial}_y H_x \\\\
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\\end{align*}
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$$
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Rewrite the last three equations as
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$$
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\\begin{align*}
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\\gamma H_y &= \\imath \\omega \\epsilon_{xx} E_x - \\hat{\\partial}_y H_z \\\\
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\\gamma H_x &= -\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z \\\\
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\\imath \\omega E_z &= \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y - \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\
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\\end{align*}
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$$
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Now apply \\( \\gamma \\tilde{\\partial}_x \\) to the last equation,
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then substitute in for \\( \\gamma H_x \\) and \\( \\gamma H_y \\):
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$$
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\\begin{align*}
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\\gamma \\tilde{\\partial}_x \\imath \\omega E_z &= \\gamma \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y
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- \\gamma \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\
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&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon{xx} E_x - \\hat{\\partial}_y H_z)
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- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon{yy} E_y - \\hat{\\partial}_x H_z) \\\\
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&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon{xx} E_x)
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- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon{yy} E_y) \\\\
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\\gamma \\tilde{\\partial}_x E_z &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
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\\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
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\\end{align*}
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$$
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With a similar approach (but using \\( \\tilde{\\partial}_y \\) instead), we can get
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$$
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\\begin{align*}
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\\gamma \\tilde{\\partial}_y E_z &= \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
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\\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
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\\end{align*}
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$$
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We can combine this equation for \\( \\gamma \\tilde{\\partial}_y E_z \\) with
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the unused \\( \\imath \\omega \\mu_{xx} H_z \\) and \\( \\imath \\omega \\mu_{yy} H_y \\) equations to get
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$$
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\\begin{align*}
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-\\imath \\omega \\mu_{xx} \\gamma H_x &= \\gamma^2 E_y + \\tilde{\\partial}_y (
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\\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
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+ \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
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) \\\\
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-\\imath \\omega \\mu_{yy} \\gamma H_y &= -\\gamma^2 E_x - \\tilde{\\partial}_x (
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\\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
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+ \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
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)\\\\
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\\end{align*}
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$$
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However, based on our rewritten equation for \\( \\gamma H_x \\) and the so-far unused
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equation for \\( \\imath \\omega \\mu_{zz} H_z \\) we can also write
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$$
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\\begin{align*}
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-\\imath \\omega \\mu_{xx} (\\gamma H_x) &= -\\imath \\omega \\mu_{xx} (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
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&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
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-\\imath \\omega \\mu_{xx} \\hat{\\partial}_x (
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\\frac{1}{-\\imath \\omega \\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x)) \\\\
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&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
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+\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
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\\end{align*}
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$$
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and, similarly,
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$$
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\\begin{align*}
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-\\imath \\omega \\mu_{yy} (\\gamma H_y) &= -\\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
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+\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
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\\end{align*}
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$$
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Using these, we can construct the eigenvalue problem
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$$ \\beta^2 \\begin{bmatrix} E_x \\\\
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E_y \\end{bmatrix} =
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(\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\
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0 & \\mu_{xx} \\epsilon_{yy} \\end{bmatrix} +
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\\begin{bmatrix} -\\mu_{yy} \\hat{\\partial}_y \\\\
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\\mu_{xx} \\hat{\\partial}_x \\end{bmatrix} \\mu_{zz}^{-1}
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\\begin{bmatrix} -\\tilde{\\partial}_y & \\tilde{\\partial}_x \\end{bmatrix} +
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\\begin{bmatrix} \\tilde{\\partial}_x \\\\
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\\tilde{\\partial}_y \\end{bmatrix} \\epsilon_{zz}^{-1}
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\\begin{bmatrix} \\hat{\\partial}_x \\epsilon_{xx} & \\hat{\\partial}_y \\epsilon_{yy} \\end{bmatrix})
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\\begin{bmatrix} E_x \\\\
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E_y \\end{bmatrix}
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$$
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An equivalent eigenvalue problem can be formed using the \\( H_x, H_y \\) fields, if those are more convenient.
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Note that \\( E_z \\) was never discretized, so \\( \\gamma \\) and \\( \\beta \\) will need adjustment
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to account for numerical dispersion if the result is introduced into a space with a discretized z-axis.
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"""
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# TODO update module docs
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