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{"GladTeX__cache__version": "2.0", "\\begin{aligned}\n\\beta^2 E_x - \\tilde{\\partial}_x (\n \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)\n + \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)\n ) &= \\omega^2 \\mu_{yy} \\epsilon_{xx} E_x\n +\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}}\n(\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\\n-\\beta^2 E_y + \\tilde{\\partial}_y (\n \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)\n + \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)\n ) &= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y\n -\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}}\n(\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 111.7813692054657}, "path": "_doc_mathimg/eqn037.svg"}}, "\\begin{bmatrix}\n -\\imath \\omega \\epsilon & \\nabla \\times \\\\\n \\nabla \\times & \\imath \\omega \\mu\n \\end{bmatrix}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 77.3383153998754}, "path": "_doc_mathimg/eqn011.svg"}}, "\\beta^2 E_{xy} = A_E E_{xy}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn053.svg"}}, "\\mu_{yy}": {"false": {"pos": {"depth": 5.1432292047526005, "width": 23.4001260816635, "height": 12.701293015801}, "path": "_doc_mathimg/eqn050.svg"}}, "\\imath \\omega \\epsilon_{zz} E_z = \\hat{\\partial}_x H_y -\n\\hat{\\partial}_y H_x \\\\": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn043.svg"}}, "\\vec{E}": {"false": {"pos": {"depth": 0.6691693166041001, "width": 13.873412986498002, "height": 16.787190246986903}, "path": "_doc_mathimg/eqn018.svg"}}, "\\imath \\beta H_y": {"false": {"pos": {"depth": 5.1432292047526005, "width": 35.7014351074641, "height": 16.923482243579603}, "path": "_doc_mathimg/eqn024.svg"}}, "\\imath \\beta \\tilde{\\partial}_x": {"false": {"pos": {"depth": 3.780253238827, "width": 31.386291215342702, "height": 19.290690184399402}, "path": "_doc_mathimg/eqn022.svg"}}, "A_E": {"false": {"pos": {"depth": 3.0691532566045, "width": 22.653136767004902, "height": 14.6716502998754}, "path": "_doc_mathimg/eqn054.svg"}}, "(\\nabla \\times (\\frac{1}{\\epsilon} \\nabla \\times) - \\omega^2 \\mu) E =\n\\imath \\omega M": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.14021678816121}, "path": "_doc_mathimg/eqn014.svg"}}, "\\begin{aligned}\nE_z &= \\frac{1}{- \\omega \\beta \\epsilon_{zz}} ((\n \\hat{\\partial}_y \\hat{\\partial}_x H_z\n -\\hat{\\partial}_x \\hat{\\partial}_y H_z)\n + \\imath \\omega (\\hat{\\partial}_x \\epsilon_{xx} E_x +\n\\hat{\\partial}_y \\epsilon{yy} E_y))\n &= \\frac{1}{\\imath \\beta \\epsilon_{zz}} (\\hat{\\partial}_x\n\\epsilon_{xx} E_x + \\hat{\\partial}_y \\epsilon{yy} E_y)\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.61576210960591}, "path": "_doc_mathimg/eqn045.svg"}}, "\\begin{bmatrix}\n -\\imath \\omega \\epsilon & \\nabla \\times \\\\\n \\nabla \\times & \\imath \\omega \\mu\n \\end{bmatrix}\n \\begin{bmatrix} E \\\\\n H\n \\end{bmatrix}\n = \\begin{bmatrix} J \\\\\n -M\n \\end{bmatrix}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 77.3383153998754}, "path": "_doc_mathimg/eqn012.svg"}}, "\\hat{\\partial}_x": {"false": {"pos": {"depth": 3.0691532566045, "width": 16.6549675836258, "height": 19.0715941898768}, "path": "_doc_mathimg/eqn048.svg"}}, "\\begin{aligned}\n\\imath \\beta H_y &= \\imath \\omega \\epsilon_{xx} E_x -\n\\hat{\\partial}_y H_z \\\\\n\\imath \\beta H_x &= -\\imath \\omega \\epsilon_{yy} E_y -\n\\hat{\\partial}_x H_z \\\\\n\\imath \\omega E_z &= \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y -\n\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 123.44872758044839}, "path": "_doc_mathimg/eqn021.svg"}}, "\\begin{aligned}\n\\nabla \\times \\vec{E}(x, y, z) &= -\\imath \\omega \\mu \\vec{H} \\\\\n\\nabla \\times \\vec{H}(x, y, z) &= \\imath \\omega \\epsilon \\vec{E} \\\\\n\\vec{E}(x,y,z) &= (\\vec{E}_t(x, y) + E_z(x, y)\\vec{z}) e^{-\\imath\n\\beta z} \\\\\n\\vec{H}(x,y,z) &= (\\vec{H}_t(x, y) + H_z(x, y)\\vec{z}) e^{-\\imath\n\\beta z} \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 137.0006072416514}, "path": "_doc_mathimg/eqn016.svg"}}, "\\begin{aligned}\n \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &=\n \\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2} \\hat{B}_{\\vec{r}\n+ \\frac{1}{2}}\n - \\hat{M}_{\\vec{r}\n+ \\frac{1}{2}} \\\\\n \\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &=\n -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\n\\tilde{D}_{\\vec{r}}\n +\n\\tilde{J}_{\\vec{r}} \\\\\n \\tilde{\\nabla} \\cdot \\hat{B}_{\\vec{r} + \\frac{1}{2}} &= 0 \\\\\n \\hat{\\nabla} \\cdot \\tilde{D}_{\\vec{r}} &= \\rho_{\\vec{r}}\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 142.99240709185642}, "path": "_doc_mathimg/eqn002.svg"}}, "S = E \\times H": {"false": {"pos": {"depth": 2.0025026166041, "width": 79.28586601785331, "height": 13.605000993208302}, "path": "_doc_mathimg/eqn008.svg"}}, "H_x": {"false": {"pos": {"depth": 3.0691532566045, "width": 21.365172799204, "height": 14.6716502998754}, "path": "_doc_mathimg/eqn040.svg"}}, "\\begin{aligned}\n-\\imath \\omega \\mu_{xx} \\imath \\beta H_x &= -\\beta^2 E_y + \\imath\n\\beta \\tilde{\\partial}_y E_z \\\\\n-\\imath \\omega \\mu_{xx} \\imath \\beta H_x &= -\\beta^2 E_y +\n\\tilde{\\partial}_y (\n \\frac{1}{\\epsilon_{zz}}\n\\hat{\\partial}_x (\\epsilon_{xx} E_x)\n + \\frac{1}{\\epsilon_{zz}}\n\\hat{\\partial}_y (\\epsilon_{yy} E_y)\n )\\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 96.4900695877482}, "path": "_doc_mathimg/eqn031.svg"}}, "\\tilde{\\partial}_x": {"false": {"pos": {"depth": 3.0691532566045, "width": 16.6549675836258, "height": 18.5795888688436}, "path": "_doc_mathimg/eqn047.svg"}}, "E \\times H": {"false": {"pos": {"depth": 2.0025026166041, "width": 47.6454868088628, "height": 13.605000993208302}, "path": "_doc_mathimg/eqn007.svg"}}, "\\begin{aligned}\n \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &=\n \\imath \\omega \\hat{B}_{\\vec{r} + \\frac{1}{2}}\n - \\hat{M}_{\\vec{r} + \\frac{1}{2}} \\\\\n \\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &=\n -\\imath \\omega \\tilde{D}_{\\vec{r}}\n + \\tilde{J}_{\\vec{r}} \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 88.0262431326772}, "path": "_doc_mathimg/eqn005.svg"}}, "\\Delta_t \\to 0": {"false": {"pos": {"depth": 3.0691532566045, "width": 51.8751213697886, "height": 14.6716502998754}, "path": "_doc_mathimg/eqn003.svg"}}, "H_{yx}^\\star =\n\\begin{bmatrix}H_y^\\star \\\\ -H_x^\\star \\end{bmatrix}": {"false": {"pos": {"depth": 16.1172275970693, "width": 98.3070655423233, "height": 40.2344549941386}, "path": "_doc_mathimg/eqn058.svg"}}, "\\imath \\beta\n\\tilde{\\partial}_y": {"false": {"pos": {"depth": 5.1432292047526005, "width": 30.9912298918859, "height": 20.653666150325}, "path": "_doc_mathimg/eqn026.svg"}}, "\\beta^2 \\begin{bmatrix} E_x \\\\\n E_y \\end{bmatrix} =\n (\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\\n 0 & \\mu_{xx}\n\\epsilon_{yy} \\end{bmatrix} +\n \\begin{bmatrix} -\\mu_{yy} \\hat{\\partial}_y \\\\\n \\mu_{xx} \\hat{\\partial}_x \\end{bmatrix}\n\\mu_{zz}^{-1}\n \\begin{bmatrix} -\\tilde{\\partial}_y &\n\\tilde{\\partial}_x \\end{bmatrix} +\n \\begin{bmatrix} \\tilde{\\partial}_x \\\\\n \\tilde{\\partial}_y \\end{bmatrix}\n\\epsilon_{zz}^{-1}\n \\begin{bmatrix} \\hat{\\partial}_x \\epsilon_{xx} &\n\\hat{\\partial}_y \\epsilon_{yy} \\end{bmatrix})\n \\begin{bmatrix} E_x \\\\\n E_y \\end{bmatrix}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 79.13832468820851}, "path": "_doc_mathimg/eqn038.svg"}}, "\\imath \\beta\nH_x": {"false": {"pos": {"depth": 3.780253238827, "width": 36.096495097587606, "height": 15.560506277654}, "path": "_doc_mathimg/eqn023.svg"}}, "\\imath\n\\omega \\mu_{zz} H_z": {"false": {"pos": {"depth": 3.780253238827, "width": 57.574298560642504, "height": 15.382751615431202}, "path": "_doc_mathimg/eqn034.svg"}}, "\\begin{aligned}\n\\tilde{E}_{l, \\vec{r}} &\\to \\tilde{E}_{\\vec{r}} \\\\\n\\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to\n\\tilde{H}_{\\vec{r} + \\frac{1}{2}} \\\\\n\\tilde{J}_{l, \\vec{r}} &\\to \\tilde{J}_{\\vec{r}} \\\\\n\\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to\n\\tilde{M}_{\\vec{r} + \\frac{1}{2}} \\\\\n\\Omega &\\to \\omega \\\\\n\\tilde{\\partial}_t &\\to -\\imath \\omega \\\\\n \\hat{\\partial}_t &\\to -\\imath \\omega \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 215.7439719397339}, "path": "_doc_mathimg/eqn004.svg"}}, "(2 \\beta) \\partial_{\\epsilon_i}(\\beta) E_{xy} + \\beta^2\n\\partial_{\\epsilon_i} E_{xy}\n = \\partial_{\\epsilon_i}(A_E) E_{xy} + A_E \\partial_{\\epsilon_i}\nE_{xy}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn057.svg"}}, "\\begin{aligned}\n-\\imath \\omega \\mu_{yy} \\imath \\beta H_y &= \\beta^2 E_x - \\imath\n\\beta \\tilde{\\partial}_x E_z \\\\\n-\\imath \\omega \\mu_{yy} \\imath \\beta H_y &= \\beta^2 E_x -\n\\tilde{\\partial}_x (\n \\frac{1}{\\epsilon_{zz}}\n\\hat{\\partial}_x (\\epsilon_{xx} E_x)\n + \\frac{1}{\\epsilon_{zz}}\n\\hat{\\partial}_y (\\epsilon_{yy} E_y)\n )\\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 96.4900695877482}, "path": "_doc_mathimg/eqn032.svg"}}, "\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\\n 0 & \\mu_{xx}\n\\epsilon_{yy} \\end{bmatrix} +\n \\begin{bmatrix} -\\mu_{yy} \\hat{\\partial}_y \\\\\n \\mu_{xx} \\hat{\\partial}_x \\end{bmatrix}\n\\mu_{zz}^{-1}\n \\begin{bmatrix} -\\tilde{\\partial}_y &\n\\tilde{\\partial}_x \\end{bmatrix} +\n \\begin{bmatrix} \\tilde{\\partial}_x \\\\\n \\tilde{\\partial}_y \\end{bmatrix} \\epsilon_{zz}^{-1}\n \\begin{bmatrix} \\hat{\\partial}_x \\epsilon_{xx} &\n\\hat{\\partial}_y \\epsilon_{yy} \\end{bmatrix}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 79.13832468820851}, "path": "_doc_mathimg/eqn046.svg"}}, "\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot\n\\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})\n -\\omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = -\\imath\n\\omega \\tilde{J}_{\\vec{r}} \\\\": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn006.svg"}}, "\\beta": {"false": {"pos": {"depth": 3.780253238827, "width": 11.0697223899236, "height": 15.560506277654}, "path": "_doc_mathimg/eqn039.svg"}}, "(\\nabla \\times (\\frac{1}{\\mu} \\nabla \\times) - \\Omega^2 \\epsilon) E =\n-\\imath \\omega J": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.14021678816121}, "path": "_doc_mathimg/eqn010.svg"}}, "E_{xy} = \\begin{bmatrix} E_x \\\\\n E_y\n\\end{bmatrix}": {"false": {"pos": {"depth": 16.0024822666046, "width": 82.572691269016, "height": 40.0049629998759}, "path": "_doc_mathimg/eqn055.svg"}}, "\\begin{aligned}\n-\\imath \\omega \\mu_{yy} (\\imath \\beta H_y) &= \\omega^2 \\mu_{yy}\n\\epsilon_{xx} E_x\n +\\mu_{yy} \\hat{\\partial}_y\n\\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.61576210960591}, "path": "_doc_mathimg/eqn036.svg"}}, "\\epsilon_{xx}": {"false": {"pos": {"depth": 3.0691532566045, "width": 21.0902701394099, "height": 10.6272170676529}, "path": "_doc_mathimg/eqn049.svg"}}, "\\begin{aligned}\n\\imath \\beta \\tilde{\\partial}_x \\imath \\omega E_z &= \\imath \\beta\n\\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y\n - \\imath \\beta\n\\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\\n &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}}\n\\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x - \\hat{\\partial}_y\nH_z)\n - \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y\n(-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\\n &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}}\n\\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x)\n - \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y\n(-\\imath \\omega \\epsilon_{yy} E_y) \\\\\n\\imath \\beta \\tilde{\\partial}_x E_z &= \\tilde{\\partial}_x\n\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)\n + \\tilde{\\partial}_x\n\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 190.0466672488332}, "path": "_doc_mathimg/eqn025.svg"}}, "\\epsilon_i": {"false": {"pos": {"depth": 3.0691532566045, "width": 12.1894730285965, "height": 10.6272170676529}, "path": "_doc_mathimg/eqn056.svg"}}, "sens_{i} =\n\\frac{\\partial\\beta}{\\partial\\epsilon_i}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.9402034348282}, "path": "_doc_mathimg/eqn052.svg"}}, "\\begin{aligned}\n-\\imath \\omega \\mu_{xx} H_x &= \\tilde{\\partial}_y E_z + \\imath \\beta\nE_y \\\\\n-\\imath \\omega \\mu_{yy} H_y &= -\\imath \\beta E_x -\n\\tilde{\\partial}_x E_z \\\\\n-\\imath \\omega \\mu_{zz} H_z &= \\tilde{\\partial}_x E_y -\n\\tilde{\\partial}_y E_x \\\\\n\\imath \\omega \\epsilon_{xx} E_x &= \\hat{\\partial}_y H_z + \\imath\n\\beta H_y \\\\\n\\imath \\omega \\epsilon_{yy} E_y &= -\\imath \\beta H_x -\n\\hat{\\partial}_x H_z \\\\\n\\imath \\omega \\epsilon_{zz} E_z &= \\hat{\\partial}_x H_y -\n\\hat{\\partial}_y H_x \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 187.99558063344372}, "path": "_doc_mathimg/eqn020.svg"}}, "\\imath \\omega \\mu_{yy} H_y": {"false": {"pos": {"depth": 5.1432292047526005, "width": 58.3613465409663, "height": 16.7457275813568}, "path": "_doc_mathimg/eqn030.svg"}}, "\\begin{aligned}\n\\imath \\beta \\tilde{\\partial}_y E_z &= \\tilde{\\partial}_y\n\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)\n + \\tilde{\\partial}_y\n\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.260211451828}, "path": "_doc_mathimg/eqn027.svg"}}, "\\begin{aligned}\n-\\imath \\omega \\mu_{xx} (\\imath \\beta H_x) &= -\\imath \\omega\n\\mu_{xx} (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\\n &= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y + \\imath\n\\omega \\mu_{xx} \\hat{\\partial}_x (\n \\frac{1}{-\\imath \\omega \\mu_{zz}}\n(\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x)) \\\\\n &= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y\n -\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}}\n(\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 138.24802321046602}, "path": "_doc_mathimg/eqn035.svg"}}, "\\nabla \\times\n(\\frac{1}{\\epsilon} \\nabla \\times) - \\omega^2 \\mu": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.14021678816121}, "path": "_doc_mathimg/eqn013.svg"}}, "H_y": {"false": {"pos": {"depth": 5.1432292047526005, "width": 20.9701128090805, "height": 16.7457275813568}, "path": "_doc_mathimg/eqn041.svg"}}, "\\imath \\beta\n\\tilde{\\partial}_y E_z": {"false": {"pos": {"depth": 5.1432292047526005, "width": 48.9772934422343, "height": 20.653666150325}, "path": "_doc_mathimg/eqn028.svg"}}, "\\begin{aligned}\n\\tilde{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{-\\imath \\omega\n\\Delta_t / 2}\\\\\n \\hat{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{ \\imath \\omega\n\\Delta_t / 2}\\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 83.11297392217561}, "path": "_doc_mathimg/eqn001.svg"}}, "E_z": {"false": {"pos": {"depth": 3.0691532566045, "width": 19.324400850223302, "height": 14.6716502998754}, "path": "_doc_mathimg/eqn042.svg"}}, "\\vec{H}": {"false": {"pos": {"depth": 0.6691693166041001, "width": 15.554931611126701, "height": 16.787190246986903}, "path": "_doc_mathimg/eqn019.svg"}}, "\\begin{aligned}\n\\imath \\beta H_y &= \\imath \\omega \\epsilon_{xx} E_x -\n\\hat{\\partial}_y H_z \\\\\n\\imath \\beta H_x &= -\\imath \\omega \\epsilon_{yy} E_y -\n\\hat{\\partial}_x H_z \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 83.6049779098755}, "path": "_doc_mathimg/eqn044.svg"}}, "\\imath \\omega\n\\mu_{xx} H_x": {"false": {"pos": {"depth": 3.780253238827, "width": 59.5465278446701, "height": 15.382751615431202}, "path": "_doc_mathimg/eqn029.svg"}}, "\\nabla \\times (\\frac{1}{\\mu}\n\\nabla \\times) - \\Omega^2 \\epsilon": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.14021678816121}, "path": "_doc_mathimg/eqn009.svg"}}, "\\epsilon": {"false": {"pos": {"depth": 0.6691693166041001, "width": 7.664205141728201, "height": 8.2272331276525}, "path": "_doc_mathimg/eqn015.svg"}}, "\\omega^2 \\begin{bmatrix} \\epsilon_{yy} \\mu_{xx} & 0 \\\\\n 0 & \\epsilon_{xx}\n\\mu_{yy} \\end{bmatrix} +\n \\begin{bmatrix} -\\epsilon_{yy} \\tilde{\\partial}_y \\\\\n \\epsilon_{xx} \\tilde{\\partial}_x\n\\end{bmatrix} \\epsilon_{zz}^{-1}\n \\begin{bmatrix} -\\hat{\\partial}_y & \\hat{\\partial}_x\n\\end{bmatrix} +\n \\begin{bmatrix} \\hat{\\partial}_x \\\\\n \\hat{\\partial}_y \\end{bmatrix} \\mu_{zz}^{-1}\n \\begin{bmatrix} \\tilde{\\partial}_x \\mu_{xx} &\n\\tilde{\\partial}_y \\mu_{yy} \\end{bmatrix}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 79.13832468820851}, "path": "_doc_mathimg/eqn051.svg"}}, "\\imath\n\\beta H_x": {"false": {"pos": {"depth": 3.780253238827, "width": 36.096495097587606, "height": 15.560506277654}, "path": "_doc_mathimg/eqn033.svg"}}, "(2 \\beta) \\partial_{\\epsilon_i}(\\beta) H_{yx}^\\star E_{xy} + \\beta^2\nH_{yx}^\\star \\partial_{\\epsilon_i} E_{xy}\n = H_{yx}^\\star \\partial_{\\epsilon_i}(A_E) E_{xy} + H_{yx}^\\star A_E\n\\partial_{\\epsilon_i} E_{xy}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn059.svg"}}, "\\begin{aligned}\n\\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l\n\\Delta_t} \\\\\n\\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &=\n\\tilde{H}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega (l - \\frac{1}{2})\n\\Delta_t} \\\\\n\\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l -\n\\frac{1}{2}) \\Delta_t} \\\\\n\\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &=\n\\tilde{M}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega l \\Delta_t} \\\\\n\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot\n\\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})\n -\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} &=\n-\\imath \\Omega \\tilde{J}_{\\vec{r}} e^{\\imath \\omega \\Delta_t / 2} \\\\\n\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 197.45816573021241}, "path": "_doc_mathimg/eqn000.svg"}}, "\\begin{aligned}\n-\\imath \\omega \\mu_{xx} H_x &= \\partial_y E_z - \\partial_z E_y \\\\\n-\\imath \\omega \\mu_{yy} H_y &= \\partial_z E_x - \\partial_x E_z \\\\\n-\\imath \\omega \\mu_{zz} H_z &= \\partial_x E_y - \\partial_y E_x \\\\\n\\imath \\omega \\epsilon_{xx} E_x &= \\partial_y H_z - \\partial_z H_y\n\\\\\n\\imath \\omega \\epsilon_{yy} E_y &= \\partial_z H_x - \\partial_x H_z\n\\\\\n\\imath \\omega \\epsilon_{zz} E_z &= \\partial_x H_y - \\partial_y H_x\n\\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 172.0049796998754}, "path": "_doc_mathimg/eqn017.svg"}}, "H_{yx}^\\star \\cdot\nE_{xy} = H^\\star \\times E": {"false": {"pos": {"depth": 6.6987104991989, "width": 137.2477592354726, "height": 18.3012075424698}, "path": "_doc_mathimg/eqn062.svg"}}, "A_H": {"false": {"pos": {"depth": 3.0691532566045, "width": 23.880919402977, "height": 14.6716502998754}, "path": "_doc_mathimg/eqn061.svg"}}, "\\partial_{\\epsilon_i} A_E": {"false": {"pos": {"depth": 4.69533321595, "width": 40.4030949899226, "height": 16.4755849214437}, "path": "_doc_mathimg/eqn066.svg"}}, "i": {"false": {"pos": {"depth": 0.6691693166041001, "width": 6.6828811662613, "height": 11.8277783709722}, "path": "_doc_mathimg/eqn065.svg"}}, "f": {"false": {"pos": {"depth": 3.780253238827, "width": 10.7688183974462, "height": 15.560506277654}, "path": "_doc_mathimg/eqn070.svg"}}, "H_{yx}^\\star": {"false": {"pos": {"depth": 6.6987104991989, "width": 27.349812649588003, "height": 18.3012075424698}, "path": "_doc_mathimg/eqn060.svg"}}, "\\vec{sens} =\n\\vec{v}_{left} \\star \\vec{v}_{right}": {"false": {"pos": {"depth": 5.1432292047526005, "width": 130.3767460739146, "height": 17.2168169029129}, "path": "_doc_mathimg/eqn068.svg"}}, "[\\tilde{\\partial}f]_{1 + \\frac{1}{2}}": {"false": {"pos": {"depth": 8.8576051118932, "width": 51.8502187037445, "height": 24.3680407241323}, "path": "_doc_mathimg/eqn078.svg"}}, "[\\tilde{\\partial}_x f]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m\n+ 1} - f_m)": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.14021678816121}, "path": "_doc_mathimg/eqn069.svg"}}, "\\partial_{\\epsilon_i}(\\beta)\n = \\frac{1}{2 \\beta} \\frac{H_{yx}^\\star \\partial_{\\epsilon_i}(A_E)\nE_{xy} }{H_{yx}^\\star E_{xy}}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 78.2744527098053}, "path": "_doc_mathimg/eqn064.svg"}}, "m +\n\\frac{1}{2}": {"false": {"pos": {"depth": 6.1866251786677, "width": 43.20469091988271, "height": 20.0295808325938}, "path": "_doc_mathimg/eqn082.svg"}}, "(m \\pm \\frac{1}{2},n,p)": {"false": {"pos": {"depth": 6.1866251786677, "width": 86.9006738274831, "height": 20.0295808325938}, "path": "_doc_mathimg/eqn090.svg"}}, "sens_i =\n\\vec{v}_{left} \\partial_{\\epsilon_i} (\\epsilon_{xyz})\n\\vec{v}_{right}": {"false": {"pos": {"depth": 5.1432292047526005, "width": 174.97193962570142, "height": 17.812398221356702}, "path": "_doc_mathimg/eqn067.svg"}}, "[\\hat{\\partial}_x f ]_{m - \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m}\n- f_{m - 1})": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.14021678816121}, "path": "_doc_mathimg/eqn074.svg"}}, "(m,n,p)": {"false": {"pos": {"depth": 4.6691692166041, "width": 58.4784532047053, "height": 17.3383382332082}, "path": "_doc_mathimg/eqn089.svg"}}, "m": {"false": {"pos": {"depth": 0.6691693166041001, "width": 15.041890290619401, "height": 8.2272331276525}, "path": "_doc_mathimg/eqn071.svg"}}, "[\\hat{\\nabla} f]_{m,n,p} =\n\\vec{x} [\\hat{\\partial}_x f]_{m + \\frac{1}{2},n,p} +\n \\vec{y} [\\hat{\\partial}_y f]_{m,n +\n\\frac{1}{2},p} +\n \\vec{z} [\\hat{\\partial}_z f]_{m,n,p +\n\\frac{1}{2}}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn084.svg"}}, "\\Delta_{x, m}": {"false": {"pos": {"depth": 5.1432292047526005, "width": 34.613137801338205, "height": 16.7457275813568}, "path": "_doc_mathimg/eqn072.svg"}}, "\\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x,\nm + 1})": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.14021678816121}, "path": "_doc_mathimg/eqn081.svg"}}, "m + \\frac{1}{2}": {"false": {"pos": {"depth": 6.1866251786677, "width": 43.20469091988271, "height": 20.0295808325938}, "path": "_doc_mathimg/eqn080.svg"}}, "[\\tilde{\\nabla} f]_{m,n,p} = \\vec{x}\n[\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} +\n \\vec{y} [\\tilde{\\partial}_y f]_{m,n +\n\\frac{1}{2},p} +\n \\vec{z} [\\tilde{\\partial}_z f]_{m,n,p\n+ \\frac{1}{2}}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn083.svg"}}, "f_1": {"false": {"pos": {"depth": 3.780253238827, "width": 15.3939209484853, "height": 15.560506277654}, "path": "_doc_mathimg/eqn076.svg"}}, "\\vec{r} +\n\\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2})": {"false": {"pos": {"depth": 6.1866251786677, "width": 199.6987656741974, "height": 20.0295808325938}, "path": "_doc_mathimg/eqn102.svg"}}, "[\\tilde{\\partial}f]_{0 + \\frac{1}{2}}": {"false": {"pos": {"depth": 8.8576051118932, "width": 51.8502187037445, "height": 24.3680407241323}, "path": "_doc_mathimg/eqn077.svg"}}, "\\hat{g}": {"false": {"pos": {"depth": 3.780253238827, "width": 9.4141837646454, "height": 15.560506277654}, "path": "_doc_mathimg/eqn093.svg"}}, "\\begin{aligned}\n \\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=\n\\\\\n [\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n +\n\\frac{1}{2}, p + \\frac{1}{2}} &=\n \\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} -\n\\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\\n &+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} -\n\\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\\n &+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} -\n\\tilde{\\partial}_y g^z_{m + \\frac{1}{2},n,p})\n \\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 151.20033355332492}, "path": "_doc_mathimg/eqn091.svg"}}, "\\Delta_{x, m},\n\\Delta_{x, m+1}, ...": {"false": {"pos": {"depth": 5.1432292047526005, "width": 109.4619119301188, "height": 16.7457275813568}, "path": "_doc_mathimg/eqn073.svg"}}, "\\hat{h}": {"false": {"pos": {"depth": 0.6691693166041001, "width": 10.356770407747401, "height": 16.6716102498764}, "path": "_doc_mathimg/eqn095.svg"}}, "\\hat{\\nabla} \\times \\hat{H}": {"false": {"pos": {"depth": 2.0025026166041, "width": 48.443785455572, "height": 17.8271888876536}, "path": "_doc_mathimg/eqn110.svg"}}, "H_{yx}^\\star A_E \\partial_{\\epsilon_i} E_{xy} = \\beta^2 H_{yx}^\\star\n\\partial_{\\epsilon_i} E_{xy}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn063.svg"}}, "(m, n \\pm\n\\frac{1}{2}, p \\pm \\frac{1}{2})": {"false": {"pos": {"depth": 6.1866251786677, "width": 115.3228931169276, "height": 20.0295808325938}, "path": "_doc_mathimg/eqn098.svg"}}, "\\hat{g}_{m,n,p} = \\vec{x} g^x_{m - \\frac{1}{2},n,p} +\n \\vec{y} g^y_{m,n - \\frac{1}{2},p} +\n \\vec{z} g^z_{m,n,p - \\frac{1}{2}}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn086.svg"}}, "\\vec{r} = (m, n, p)": {"false": {"pos": {"depth": 4.6691692166041, "width": 87.0475644904775, "height": 17.3383382332082}, "path": "_doc_mathimg/eqn101.svg"}}, "f_0": {"false": {"pos": {"depth": 3.780253238827, "width": 15.3939209484853, "height": 15.560506277654}, "path": "_doc_mathimg/eqn075.svg"}}, "\\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p\n- \\frac{1}{2}} =\n [\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p\n- \\frac{1}{2}}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn092.svg"}}, "\\hat{M}": {"false": {"pos": {"depth": 0.6691693166041001, "width": 18.1660155458496, "height": 16.493855587653602}, "path": "_doc_mathimg/eqn106.svg"}}, "\\hat{\\nabla} \\cdot \\tilde{J} + \\hat{\\partial}_t \\rho\n= 0": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn108.svg"}}, "\\begin{aligned}\n\\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}\n &= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}}) -\n\\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\\n &= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\\n &= - \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}}\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 109.04632260717521}, "path": "_doc_mathimg/eqn118.svg"}}, "d_{n,m,p} = [\\hat{\\nabla} \\cdot\n\\tilde{g}]_{n,m,p}\n = [\\hat{\\partial}_x g^x]_{m,n,p} +\n [\\hat{\\partial}_y g^y]_{m,n,p} +\n [\\hat{\\partial}_z g^z]_{m,n,p}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn088.svg"}}, "d_{n,m,p} = [\\tilde{\\nabla} \\cdot\n\\hat{g}]_{n,m,p}\n = [\\tilde{\\partial}_x g^x]_{m,n,p} +\n [\\tilde{\\partial}_y g^y]_{m,n,p} +\n [\\tilde{\\partial}_z g^z]_{m,n,p}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn087.svg"}}, "\\tilde{h}": {"false": {"pos": {"depth": 0.6691693166041001, "width": 10.356770407747401, "height": 16.179606262176502}, "path": "_doc_mathimg/eqn096.svg"}}, "\\begin{aligned}\n \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=\n -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}\n - \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\\n \\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times\n\\tilde{E}_{l,\\vec{r}} &=\n -\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} +\n\\frac{1}{2}} \\\\\n \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot\n\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=\n \\hat{\\nabla} \\times (-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2},\n\\vec{r} + \\frac{1}{2}}) \\\\\n \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot\n\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=\n -\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2},\n\\vec{r} + \\frac{1}{2}} \\\\\n \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot\n\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=\n -\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\tilde{E}_{l,\n\\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\\n \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot\n\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}})\n + \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}}\n\\cdot \\tilde{E}_{l, \\vec{r}}\n &= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2},\n\\vec{r}}\n \\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 215.8903759360738}, "path": "_doc_mathimg/eqn111.svg"}}, "[\\hat{\\partial}f]_{0 -\n\\frac{1}{2}}": {"false": {"pos": {"depth": 8.8576051118932, "width": 51.8502187037445, "height": 24.8600460451655}, "path": "_doc_mathimg/eqn079.svg"}}, "\\tilde{g}_{m,n,p} = \\vec{x} g^x_{m + \\frac{1}{2},n,p}\n+\n \\vec{y} g^y_{m,n + \\frac{1}{2},p} +\n \\vec{z} g^z_{m,n,p + \\frac{1}{2}}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn085.svg"}}, "\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}} = 0": {"false": {"pos": {"depth": 3.0691532566045, "width": 72.8270861793228, "height": 18.8938181943212}, "path": "_doc_mathimg/eqn117.svg"}}, "(m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm\n\\frac{1}{2})": {"false": {"pos": {"depth": 6.1866251786677, "width": 143.7451137397054, "height": 20.0295808325938}, "path": "_doc_mathimg/eqn097.svg"}}, "\\tilde{E}": {"false": {"pos": {"depth": 0.6691693166041001, "width": 13.873412986498002, "height": 16.0018515999537}, "path": "_doc_mathimg/eqn103.svg"}}, "\\begin{aligned}\n\\tilde{\\partial}_t &\\Rightarrow (e^{ \\imath \\omega \\Delta_t} - 1) /\n\\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2)\ne^{-\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{-\\imath \\omega\n\\Delta_t / 2}\\\\\n \\hat{\\partial}_t &\\Rightarrow (1 - e^{-\\imath \\omega \\Delta_t}) /\n\\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{\n\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{ \\imath \\omega \\Delta_t\n/ 2}\\\\\n\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 135.02581395768792}, "path": "_doc_mathimg/eqn113.svg"}}, "\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot\n\\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})\n -\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = -\\imath\n\\Omega \\tilde{J}_{\\vec{r}} e^{\\imath \\omega \\Delta_t / 2} \\\\": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn114.svg"}}, "\\hat{H}": {"false": {"pos": {"depth": 0.6691693166041001, "width": 15.554931611126701, "height": 16.493855587653602}, "path": "_doc_mathimg/eqn104.svg"}}, "\\tilde{g}": {"false": {"pos": {"depth": 3.780253238827, "width": 9.4141837646454, "height": 15.0685009566208}, "path": "_doc_mathimg/eqn094.svg"}}, "\\begin{aligned}\n \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t\n\\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}\n -\n\\hat{M}_{l, \\vec{r} + \\frac{1}{2}} \\\\\n \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2},\\vec{r} + \\frac{1}{2}}\n&= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}\n +\n\\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\\n \\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}\n&= 0 \\\\\n \\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 143.1701630874125}, "path": "_doc_mathimg/eqn099.svg"}}, "\\tilde{J}": {"false": {"pos": {"depth": 0.6691693166041001, "width": 11.4790437130239, "height": 16.0018515999537}, "path": "_doc_mathimg/eqn105.svg"}}, "\\begin{aligned}\n\\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l\n\\Delta_t} \\\\\n\\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l -\n\\frac{1}{2}) \\Delta_t}\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 83.7246272402176}, "path": "_doc_mathimg/eqn112.svg"}}, "\\begin{aligned}\n\\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\\n\\epsilon_{\\vec{r}} &= \\epsilon \\\\\n\\tilde{J}_{\\vec{r}} &= 0 \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 108.19024396191051}, "path": "_doc_mathimg/eqn115.svg"}}, "\\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}} +\n\\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn119.svg"}}, "\\tilde{\\nabla} \\times\n\\tilde{E}": {"false": {"pos": {"depth": 2.0025026166041, "width": 46.7622668309433, "height": 17.3351848999537}, "path": "_doc_mathimg/eqn109.svg"}}, "c = \\sqrt{\\mu \\epsilon}": {"false": {"pos": {"depth": 6.0602825151596, "width": 58.239723877340204, "height": 17.9782915505427}, "path": "_doc_mathimg/eqn126.svg"}}, "\\mu": {"false": {"pos": {"depth": 3.780253238827, "width": 10.764179730895501, "height": 11.338317049875402}, "path": "_doc_mathimg/eqn107.svg"}}, "\\begin{aligned}\n \\hat{B}_{\\vec{r}} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot\n\\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\\n \\tilde{D}_{\\vec{r}} &= \\epsilon_{\\vec{r}} \\cdot\n\\tilde{E}_{\\vec{r}}\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 85.145851204687}, "path": "_doc_mathimg/eqn100.svg"}}, "(\\tilde{\\nabla}^2 + K^2) \\phi_{\\vec{r}} = 0": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn120.svg"}}, "K^2 = \\Omega^2 \\mu \\epsilon": {"false": {"pos": {"depth": 3.780253238827, "width": 76.6230447510905, "height": 17.1297809050888}, "path": "_doc_mathimg/eqn121.svg"}}, "\\hat{\\nabla} \\times \\tilde{\\nabla} \\times\n\\tilde{E}_{\\vec{r}} - \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn116.svg"}}, "= \\nabla_f\n\\times E": {"false": {"pos": {"depth": 5.1432292047526005, "width": 70.6787902330302, "height": 16.7457275813568}, "path": "_doc_mathimg/eqn138.svg"}}, "\\phi_{\\vec{r}} = A e^{\\imath (k_x m \\Delta_x\n+ k_y n \\Delta_y + k_z p \\Delta_z)}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn122.svg"}}, "c \\Delta_t < \\Delta_x / \\sqrt{3}": {"false": {"pos": {"depth": 4.6691692166041, "width": 96.07104293155722, "height": 19.8538208369878}, "path": "_doc_mathimg/eqn130.svg"}}, "(k_x, k_y, k_z), \\omega": {"false": {"pos": {"depth": 5.1432292047526005, "width": 89.4390444306905, "height": 17.812398221356702}, "path": "_doc_mathimg/eqn127.svg"}}, "K_y, K_z": {"false": {"pos": {"depth": 5.1432292047526005, "width": 48.0542587986435, "height": 16.7457275813568}, "path": "_doc_mathimg/eqn124.svg"}}, "c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu\n\\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} +\n\\frac{1}{\\Delta_z^2})": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 77.5094793955963}, "path": "_doc_mathimg/eqn128.svg"}}, "\\begin{aligned}\n\\tilde{\\partial}_x &\\Rightarrow (e^{ \\imath k_x \\Delta_x} - 1) /\n\\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{ \\imath\nk_x \\Delta_x / 2} = \\imath K_x e^{ \\imath k_x \\Delta_x / 2}\\\\\n \\hat{\\partial}_x &\\Rightarrow (1 - e^{-\\imath k_x \\Delta_x}) /\n\\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{-\\imath\nk_x \\Delta_x / 2} = \\imath K_x e^{-\\imath k_x \\Delta_x / 2}\\\\\nK_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 135.02581395768792}, "path": "_doc_mathimg/eqn123.svg"}}, "\\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_{\\vec{r}} \\\\\n K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 72.00500353320821}, "path": "_doc_mathimg/eqn125.svg"}}, "\\Delta_x / \\sqrt{3}": {"false": {"pos": {"depth": 4.6691692166041, "width": 50.4400014056666, "height": 19.8538208369878}, "path": "_doc_mathimg/eqn132.svg"}}, "r +\n\\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2},\np + \\frac{1}{2})": {"false": {"pos": {"depth": 6.1866251786677, "width": 199.6987656741974, "height": 20.0295808325938}, "path": "_doc_mathimg/eqn134.svg"}}, "r = (m, n,\np)": {"false": {"pos": {"depth": 4.6691692166041, "width": 87.0475644904775, "height": 17.3383382332082}, "path": "_doc_mathimg/eqn133.svg"}}, "c\n\\Delta_t": {"false": {"pos": {"depth": 3.0691532566045, "width": 25.8955686859441, "height": 14.6716502998754}, "path": "_doc_mathimg/eqn131.svg"}}, "\\Delta_x = \\Delta_y = \\Delta_z": {"false": {"pos": {"depth": 5.1432292047526005, "width": 102.73884809819539, "height": 16.7457275813568}, "path": "_doc_mathimg/eqn129.svg"}}, "\\mu = \\begin{bmatrix} \\mu_{xx} & 0 & 0 \\\\\n 0 & \\mu_{yy} & 0 \\\\\n 0 & 0 & \\mu_{zz} \\end{bmatrix}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 94.00494164987641}, "path": "_doc_mathimg/eqn136.svg"}}, "\\begin{aligned}\nU_l &= \\epsilon \\tilde{E}^2_l + \\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot\n\\hat{H}_{l - \\frac{1}{2}} \\\\\nU_{l + \\frac{1}{2}} &= \\epsilon \\tilde{E}_l \\cdot \\tilde{E}_{l + 1}\n+ \\mu \\hat{H}^2_{l + \\frac{1}{2}} \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 89.49096442939249}, "path": "_doc_mathimg/eqn149.svg"}}, "= \\nabla_b\n\\times H": {"false": {"pos": {"depth": 3.0691532566045, "width": 70.5881129019638, "height": 14.6716502998754}, "path": "_doc_mathimg/eqn137.svg"}}, "\\otimes": {"false": {"pos": {"depth": 2.0025026166041, "width": 13.7828156554296, "height": 12.005005033208201}, "path": "_doc_mathimg/eqn141.svg"}}, "\\begin{aligned}\n \\tilde{S}_{l, l', \\vec{r}} &=& &\\tilde{E}_{l, \\vec{r}}\n\\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\\n &=& &\\vec{x} (\\tilde{E}^y_{l,m+1,n,p}\n\\hat{H}^z_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^z_{l,m+1,n,p}\n\\hat{H}^y_{l', \\vec{r} + \\frac{1}{2}}) \\\\\n & &+ &\\vec{y} (\\tilde{E}^z_{l,m,n+1,p}\n\\hat{H}^x_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^x_{l,m,n+1,p}\n\\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}}) \\\\\n & &+ &\\vec{z} (\\tilde{E}^x_{l,m,n,p+1}\n\\hat{H}^y_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^y_{l,m,n,p+1}\n\\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}})\n \\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 151.9653282008667}, "path": "_doc_mathimg/eqn140.svg"}}, "S": {"false": {"pos": {"depth": 0.6691693166041001, "width": 11.904906369044001, "height": 12.2716676932083}, "path": "_doc_mathimg/eqn158.svg"}}, "\\begin{aligned}\n \\tilde{E}_l &= \\tilde{E}_{l, \\vec{r}} \\\\\n \\hat{H}_l &= \\tilde{H}_{l, \\vec{r} + \\frac{1}{2}} \\\\\n \\tilde{\\epsilon} &= \\tilde{\\epsilon}_{\\vec{r}} \\\\\n \\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 109.8124972546875}, "path": "_doc_mathimg/eqn144.svg"}}, "l, l'": {"false": {"pos": {"depth": 3.780253238827, "width": 22.1345007799708, "height": 16.181639595459}, "path": "_doc_mathimg/eqn143.svg"}}, "l' = l - \\frac{1}{2}": {"false": {"pos": {"depth": 6.1866251786677, "width": 64.6120997180308, "height": 20.0295808325938}, "path": "_doc_mathimg/eqn147.svg"}}, "c \\Delta_t < \\frac{\\Delta_x}{\\sqrt{3}}": {"false": {"pos": {"depth": 8.7757971139384, "width": 65.8639303534017, "height": 23.033548757494604}, "path": "_doc_mathimg/eqn139.svg"}}, "J": {"false": {"pos": {"depth": 0.6691693166041001, "width": 11.4790437130239, "height": 12.2716676932083}, "path": "_doc_mathimg/eqn151.svg"}}, "H": {"false": {"pos": {"depth": 0.6691693166041001, "width": 15.554931611126701, "height": 12.2716676932083}, "path": "_doc_mathimg/eqn154.svg"}}, "\\begin{aligned}\n \\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}}\n &= (\\mu \\hat{H}^2_{l - \\frac{1}{2}}\n +\\epsilon \\tilde{E}_{l-1} \\cdot \\tilde{E}_l) / \\Delta_t \\\\\n -(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}\n +\\epsilon \\tilde{E}^2_l) / \\Delta_t \\\\\n - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\\\\n - \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\\n \\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 146.3535323411616}, "path": "_doc_mathimg/eqn148.svg"}}, "\\epsilon = \\begin{bmatrix} \\epsilon_{xx} & 0 & 0 \\\\\n 0 & \\epsilon_{yy} & 0 \\\\\n 0 & 0 & \\epsilon_{zz} \\end{bmatrix}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 94.00494164987641}, "path": "_doc_mathimg/eqn135.svg"}}, "M": {"false": {"pos": {"depth": 0.6691693166041001, "width": 18.1660155458496, "height": 12.2716676932083}, "path": "_doc_mathimg/eqn153.svg"}}, "l' = l + \\frac{1}{2}": {"false": {"pos": {"depth": 6.1866251786677, "width": 64.3526797245163, "height": 20.0295808325938}, "path": "_doc_mathimg/eqn145.svg"}}, "E": {"false": {"pos": {"depth": 0.6691693166041001, "width": 13.873412986498002, "height": 12.2716676932083}, "path": "_doc_mathimg/eqn152.svg"}}, "\\begin{aligned}\n (U_{l+\\frac{1}{2}} - U_l) / \\Delta_t\n &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\\\\n - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\\\\n - \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\\n (U_l - U_{l-\\frac{1}{2}}) / \\Delta_t\n &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\\\\n - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\\\\n - \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\\n\\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 201.75133762288309}, "path": "_doc_mathimg/eqn150.svg"}}, "\\sim 10^{-3}": {"false": {"pos": {"depth": 0.6691693166041001, "width": 49.042052107282004, "height": 14.018696982865901}, "path": "_doc_mathimg/eqn157.svg"}}, "\\begin{aligned}\n \\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}}\n &= \\hat{H}_{l + \\frac{1}{2}} \\cdot\n (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l -\n\\frac{1}{2}}) -\n \\tilde{E}_l \\cdot (\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} -\n\\tilde{E}_l)\n - \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l +\n\\frac{1}{2}} \\\\\n &= (-\\mu / \\Delta_t)(\\hat{H}^2_{l + \\frac{1}{2}} - \\hat{H}_{l +\n\\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}) -\n (\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} \\cdot \\tilde{E}_l -\n\\tilde{E}^2_l)\n - \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l +\n\\frac{1}{2}} \\\\\n &= -(\\mu \\hat{H}^2_{l + \\frac{1}{2}}\n +\\epsilon \\tilde{E}_{l+1} \\cdot \\tilde{E}_l) / \\Delta_t \\\\\n +(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}\n +\\epsilon \\tilde{E}^2_l) / \\Delta_t \\\\\n - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\\\\n - \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\\n \\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 205.17282553734591}, "path": "_doc_mathimg/eqn146.svg"}}, "\\tau > \\frac{2 * \\pi}{\\omega}": {"false": {"pos": {"depth": 6.1866251786677, "width": 52.4271426893214, "height": 20.0295808325938}, "path": "_doc_mathimg/eqn156.svg"}}, "f_r(t) = (1 - \\frac{1}{2} (\\omega (t -\n\\tau))^2) e^{-(\\frac{\\omega (t - \\tau)}{2})^2}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 75.14021678816121}, "path": "_doc_mathimg/eqn155.svg"}}, "\\begin{aligned}\n \\hat{\\nabla} \\cdot \\tilde{S}_{l, l', \\vec{r}}\n &= \\hat{\\nabla} \\cdot (\\tilde{E}_{l, \\vec{r}} \\otimes\n\\hat{H}_{l', \\vec{r} + \\frac{1}{2}}) \\\\\n &= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla}\n\\times \\tilde{E}_{l, \\vec{r}} -\n \\tilde{E}_{l, \\vec{r}} \\cdot \\hat{\\nabla} \\times \\hat{H}_{l',\n\\vec{r} + \\frac{1}{2}} \\\\\n &= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot\n (-\\tilde{\\partial}_t \\mu_{\\vec{r} + \\frac{1}{2}} \\hat{H}_{l -\n\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} -\n \\hat{M}_{l, \\vec{r} + \\frac{1}{2}}) -\n \\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t\n\\tilde{\\epsilon}_{\\vec{r}} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} +\n \\tilde{J}_{l', \\vec{r}}) \\\\\n &= \\hat{H}_{l'} \\cdot (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}}\n- \\hat{H}_{l - \\frac{1}{2}}) -\n \\tilde{E}_l \\cdot (\\epsilon / \\Delta_t\n)(\\tilde{E}_{l'+\\frac{1}{2}} - \\tilde{E}_{l'-\\frac{1}{2}})\n - \\hat{H}_{l'} \\cdot \\hat{M}_{l} - \\tilde{E}_l \\cdot\n\\tilde{J}_{l'} \\\\\n \\end{aligned}": {"true": {"pos": {"depth": 0.6691693166041001, "width": 598.8462583621765, "height": 145.558569694369}, "path": "_doc_mathimg/eqn142.svg"}}} |