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@ -74,7 +74,7 @@ $$
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\\end{align*}
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$$
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With a similar approach (but using \\( \\tilde{\\partial}_y \\) instead), we can get
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With a similar approach (but using \\( \\gamma \\tilde{\\partial}_y \\) instead), we can get
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$$
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\\begin{align*}
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@ -90,7 +90,7 @@ $$
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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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+ \\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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@ -175,26 +175,21 @@ def operator_e(omega: complex,
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for use with a field vector of the form `cat([E_x, E_y])`.
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More precisely, the operator is
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$$ \\omega^2 \\mu_{yx} \\epsilon_{xy} +
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\\mu_{yx} \\begin{bmatrix} -D_{by} \\\\
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D_{bx} \\end{bmatrix} \\mu_z^{-1}
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\\begin{bmatrix} -D_{fy} & D_{fx} \\end{bmatrix} +
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\\begin{bmatrix} D_{fx} \\\\
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D_{fy} \\end{bmatrix} \\epsilon_z^{-1}
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\\begin{bmatrix} D_{bx} & D_{by} \\end{bmatrix} \\epsilon_{xy} $$
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where
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\\( \\epsilon_{xy} = \\begin{bmatrix}
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\\epsilon_x & 0 \\\\
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0 & \\epsilon_y
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\\end{bmatrix} \\),
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\\( \\mu_{yx} = \\begin{bmatrix}
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\\mu_y & 0 \\\\
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0 & \\mu_x
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\\end{bmatrix} \\),
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\\( D_{fx} \\) and \\( D_{bx} \\) are the forward and backward derivatives along x,
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and each \\( \\epsilon_x, \\mu_y, \\) etc. is a diagonal matrix representing
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$$
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\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\
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0 & \\mu_{xx} \\epsilon_{yy} \\end{bmatrix} +
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\\begin{bmatrix} -\\mu_{yy} \\hat{\\partial}_y \\\\
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\\mu_{xx} \\hat{\\partial}_x \\end{bmatrix} \\mu_{zz}^{-1}
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\\begin{bmatrix} -\\tilde{\\partial}_y & \\tilde{\\partial}_x \\end{bmatrix} +
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\\begin{bmatrix} \\tilde{\\partial}_x \\\\
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\\tilde{\\partial}_y \\end{bmatrix} \\epsilon_{zz}^{-1}
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\\begin{bmatrix} \\hat{\\partial}_x \\epsilon_{xx} & \\hat{\\partial}_y \\epsilon_{yy} \\end{bmatrix}
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$$
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\\( \\tilde{\\parital}_x} \\) and \\( \\hat{\\partial}_x \\) are the forward and backward derivatives along x,
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and each \\( \\epsilon_{xx}, \\mu_{yy}, \\) etc. is a diagonal matrix containing the vectorized material
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property distribution.
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This operator can be used to form an eigenvalue problem of the form
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`operator_e(...) @ [E_x, E_y] = wavenumber**2 * [E_x, E_y]`
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@ -246,26 +241,21 @@ def operator_h(omega: complex,
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for use with a field vector of the form `cat([H_x, H_y])`.
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More precisely, the operator is
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$$ \\omega^2 \\epsilon_{yx} \\mu_{xy} +
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\\epsilon_{yx} \\begin{bmatrix} -D_{fy} \\\\
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D_{fx} \\end{bmatrix} \\epsilon_z^{-1}
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\\begin{bmatrix} -D_{by} & D_{bx} \\end{bmatrix} +
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\\begin{bmatrix} D_{bx} \\\\
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D_{by} \\end{bmatrix} \\mu_z^{-1}
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\\begin{bmatrix} D_{fx} & D_{fy} \\end{bmatrix} \\mu_{xy} $$
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where
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\\( \\epsilon_{yx} = \\begin{bmatrix}
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\\epsilon_y & 0 \\\\
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0 & \\epsilon_x
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\\end{bmatrix} \\),
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\\( \\mu_{xy} = \\begin{bmatrix}
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\\mu_x & 0 \\\\
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0 & \\mu_y
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\\end{bmatrix} \\),
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\\( D_{fx} \\) and \\( D_{bx} \\) are the forward and backward derivatives along x,
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and each \\( \\epsilon_x, \\mu_y, \\) etc. is a diagonal matrix.
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$$
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\\omega^2 \\begin{bmatrix} \\epsilon_{yy} \\mu_{xx} & 0 \\\\
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0 & \\epsilon_{xx} \\mu_{yy} \\end{bmatrix} +
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\\begin{bmatrix} -\\epsilon_{yy} \\tilde{\\partial}_y \\\\
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\\epsilon_{xx} \\tilde{\\partial}_x \\end{bmatrix} \\epsilon_{zz}^{-1}
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\\begin{bmatrix} -\\hat{\\partial}_y & \\hat{\\partial}_x \\end{bmatrix} +
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\\begin{bmatrix} \\hat{\\partial}_x \\\\
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\\hat{\\partial}_y \\end{bmatrix} \\mu_{zz}^{-1}
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\\begin{bmatrix} \\tilde{\\partial}_x \\mu_{xx} & \\tilde{\\partial}_y \\mu_{yy} \\end{bmatrix}
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$$
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\\( \\tilde{\\parital}_x} \\) and \\( \\hat{\\partial}_x \\) are the forward and backward derivatives along x,
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and each \\( \\epsilon_{xx}, \\mu_{yy}, \\) etc. is a diagonal matrix containing the vectorized material
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property distribution.
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This operator can be used to form an eigenvalue problem of the form
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`operator_h(...) @ [H_x, H_y] = wavenumber**2 * [H_x, H_y]`
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Jan Petykiewicz
4 years ago