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    "from geoscilabs.em.PlanewaveWidgetFD import PlanewaveWidget, PolarEllipse, InteractivePlaneProfile\n",
    "from geoscilabs.em.DipoleWidgetFD import InteractiveDipoleProfile\n",
    "from geoscilabs.em.VolumeWidgetPlane import InteractivePlanes, plotObj3D"
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    "# Planewave propagation in a Whole-space (frequency-domain)"
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    "# Purpose\n",
    "\n",
    "We visualizae downward propagating planewave in the homogeneous earth medium. With the three apps: a) Plane wave app, b) Profile app, and c) Polarization ellipse app, we understand fundamental concepts of planewave propagation. \n"
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    "# Set up\n",
    "\n",
    "Planewave EM equation can be written as \n",
    "\n",
    "$$\\frac{\\partial^2 \\mathbf{E}}{\\partial z^2} + k^2 \\mathbf{E} = 0,$$\n",
    "\n",
    "For homogeneous earth, solution can be simply derived:\n",
    "\n",
    "\n",
    "$$\\mathbf{E} = \\mathbf{E}_0 e^{ikz}$$\n",
    "\n",
    "$$\\mathbf{H} = - i \\omega \\mu \\nabla \\times (\\mathbf{E}_0 e^{ikz}).$$\n",
    "\n",
    "where complex wavenumber $k$ is \n",
    "\n",
    "$$ k = \\sqrt{\\mu \\epsilon \\omega^2 - i \\mu \\sigma \\omega}.$$\n",
    "\n",
    "In time domain, the wave travelling in the negative z-direction has the form:\n",
    "\n",
    "$$ \\mathbf{e} = \\mathbf{e}_0^- e^{i(k z + \\omega t)}.$$"
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    "ax = plotObj3D()"
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    "# Planewave app\n",
    "\n",
    "## Parameters:\n",
    "\n",
    "- Field: Type of EM fields (\"Ex\": electric field, \"Hy\": magnetic field)\n",
    "- AmpDir: Type of the vectoral EM fields \n",
    "\n",
    "    None: $F_x$ or $F_y$ or $F_z$\n",
    "    \n",
    "    Amp: $\\mathbf{F} \\cdot \\mathbf{F}^* = |\\mathbf{F}|^2$\n",
    "    \n",
    "    Dir: Real part of a vectoral EM fields, $\\Re[\\mathbf{F}]$\n",
    "    \n",
    "- ComplexNumber: Type of complex data (\"Re\", \"Im\", \"Amp\", \"Phase\")    \n",
    "- Frequency: Transmitting frequency (Hz)\n",
    "- Sigma: Conductivity of homogeneous earth (S/m)\n",
    "- Scale: Choose \"log\" or \"linear\" scale \n",
    "- Time: "
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    "dwidget = PlanewaveWidget()\n",
    "dwidget.InteractivePlaneWave()"
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    "# Profile app\n",
    "\n",
    "We visualize EM fields at vertical profile (marked as red dots in the above app). \n",
    "\n",
    "## Parameters:\n",
    "\n",
    "- **Field**: Ex, Hy, and Impedance \n",
    "- ** $\\sigma$ **: Conductivity (S/m)\n",
    "- **Scale**: Log10 or Linear scale\n",
    "- **Fixed**: Fix the scale or not\n",
    "- **$f$**: Frequency\n",
    "- **$t$**: Time\n"
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    "# Polarization Ellipse app"
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    "Polarwidget = PolarEllipse(); \n",
    "Polarwidget.Interactive()"
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