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## Coordinate Reference systems
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In the following, for any coordinate $`j`$ of a reference system, the corresponding versor is defined as $`\hat{\bf e}_j = \nabla j / \lvert \nabla j \rvert`$. A few **right-handed** coordinate systems are used in the code:
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In this documentation, the following notation is used:
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- a bold font is used for vector quantities, e.g. $`{\bf v}`$, a normal font for their magnitude, e.g. $`v = \lVert {\bf v} \rVert`$, and for scalar values in general.
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- for any coordinate $`j`$ of a reference system, the corresponding versor is defined as $`\hat{\bf e}_j = \nabla j / \lVert \nabla j \rVert`$.
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A few **right-handed** coordinate systems are used in the code:
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- The **tokamak** reference system, sed to define the beam launch point, the magnetic equilibrium, the first wall shape, and the beam trajectory.
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It can be alternatively expressed:
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- In **Cartesian** coordinates $`(x, y, z)`$.
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- In **cylindrical** coordinates $`(R, φ, Z)`$, sharing the origin the vertical axis $`z = Z`$ with the Cartesian system
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- In **cylindrical** coordinates $`(R, φ, Z)`$, sharing the origin the vertical axis $`z = Z`$ with the Cartesian system.
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The transformation from the cylindrical to the Cartesian system given by:
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x = R \cos\phi \quad y = R \sin\phi \quad z = Z
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```math
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x = R \cos\phi, \quad y = R \sin\phi, \quad z = Z
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```
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The $`z`$-axis is the tokamak symmetry axis. and $`z = 0`$ is typically at the tokamak mid-plane.
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The $`z`$-axis is the tokamak symmetry axis, and $`z = 0`$ is typically at the tokamak mid-plane.
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For the purpose of the physics analysis it may be convenient to rotate this coordinate system around the $`z`$-axis, so that the launching point is at $`\phi = 0`$, i.e., in the $`xz`$-plane. In this case, at the launching point $`\hat{\bf e}_x`$ points radially outward, and $`\hat{\bf e}_y = \hat{\bf e}_\phi`$ is pointing in the counter clockwise direction when viewed from above.
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... | ... | @@ -23,11 +30,39 @@ In the following, for any coordinate $`j`$ of a reference system, the correspond |
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- beam polarisation angles
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This Cartesian system $`(\tilde x, \tilde y, \tilde z)`$ is co-moving with the beam. At any point along the beam path, it has:
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- origin on the beam axis
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- $`\tilde z`$-axis aligned with the on-axis wavevector $`k`$
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- $`\tilde x`$-axis in the **tokamak**
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- $`\tilde z`$-axis aligned with the on-axis wavevector $`{\bf k}`$
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- $`\tilde x`$-axis in the **tokamak** horizontal plane
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```math
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\hat{\bf e}_{\tilde x} = (\hat{\bf e}_{z} \times {\bf k})/
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\lVert \hat{\bf e}_{z} \times {\bf k} \rVert, \quad
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\hat{\bf e}_{\tilde y} = \hat{\bf e}_{\tilde z} \times \hat{\bf e}_{\tilde x}, \quad
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\hat{\bf e}_{\tilde z} = {\bf k}/k
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```
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In case $`{\bf k} \parallel \hat{\bf e}_z`$ the above expression for $`\hat{\bf e}_{\tilde x}`$ is undefined, and $`\hat{\bf e}_{\tilde x} = -\hat{\bf e}_{\phi}`$ is used instead.
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> Check sign of $`\hat{\bf e}_{\tilde x}`$ (both cases)
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- The local **plasma** system, where the local plasma dispersion relation is solved. In this Cartesian system $`(\bar x, \bar y, \bar z)`$:
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- the local magnetic field $`B`$ is along the $`\bar z`$-axis
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- the wavevector is in the $`\bar{x}\bar{z}`$-plane
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```math
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\hat{\bf e}_{\bar x} = \hat{\bf e}_{\bar y} \times \hat{\bf e}_{\bar z}, \quad
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\hat{\bf e}_{\bar y} = ({\bf B} \times {\bf k})/
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\lVert {\bf B} \times {\bf k} \rVert, \quad
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\hat{\bf e}_{\bar z} = {\bf B}/B
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```
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In case $`{\bf k} \parallel {\bf B}`$ the above expression for $`\hat{\bf e}_{\tilde y}`$ is undefined, and ...
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> Missing definition for the case $`{\bf k} \parallel {\bf B}`$
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- The local **plasma-wave** system, where the local plasma dispersion relation is solved.
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> Check sign of $`\hat{\bf e}_{\tilde y}`$
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## Quasi-optical approximation
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... | ... | |