document the wave polarisation convention
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Michele Guerini Rocco
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@ -42,7 +42,7 @@ in
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makefile2graph graphviz
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makefile2graph graphviz
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# documentation
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# documentation
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help2man pandoc
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help2man pandoc librsvg
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haskellPackages.pandoc-crossref
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haskellPackages.pandoc-crossref
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(texlive.combine {
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(texlive.combine {
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inherit (texlive) scheme-small xetex fontspec;
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inherit (texlive) scheme-small xetex fontspec;
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@ -5,7 +5,7 @@ author:
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- L. Figini[^1]
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- L. Figini[^1]
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- A. Mariani[^1]
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- A. Mariani[^1]
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- M. Guerini Rocco[^2]
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- M. Guerini Rocco[^2]
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date: 'November 30, 2012. Updated: November 30, 2021'
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date: 'November 30, 2012. Updated: November 9, 2024'
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lang: en-GB
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lang: en-GB
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language: english
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language: english
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@ -124,8 +124,8 @@ header-includes: |
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```{=latex}
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```{=latex}
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% Set font only if available
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% Set font only if available
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\IfFontExistsTF{Libertinus Serif}{\setmainfont{Libertinus Serif}}{}
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\IfFontExistsTF{Libertinus Serif}{\setmainfont{Libertinus Serif}}{}
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\IfFontExistsTF{Libertinus Math}{\setmonofont{Libertinus Math}}{}
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\IfFontExistsTF{Libertinus Math}{\setmathfont{Libertinus Math}}{}
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\IfFontExistsTF{blabla}{\setsansfont{blablabla}}{}
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\IfFontExistsTF{Julia Mono}{\setmathfont{Julia Mono}}{}
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```
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```
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...
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...
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118
doc/2.physics.md
118
doc/2.physics.md
@ -3,22 +3,22 @@
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## Coordinate Reference systems
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## Coordinate Reference systems
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A few sets of coordinate systems are used in the code. The reference system is
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A few sets of coordinate systems are used in the code. The reference system is
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the right handed cartesian orthogonal system $(x, y, z)$ with $z$ axis being
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the right handed Cartesian orthogonal system $(x, y, z)$ with $z$ axis being
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the tokamak symmetry axis. For the purpose of the physics analysis this
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the tokamak symmetry axis. For the purpose of the physics analysis this
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coordinate system may be rotated around the $z-$axis so that the $x z$ plane
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coordinate system may be rotated around the $z-$axis so that the $x z$ plane
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contains the launching point, i.e., $z$ vertical, $x$ radially outward through
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contains the launching point, i.e., $z$ vertical, $x$ radially outward through
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the port center, and $y$ pointing in the counter clockwise direction when
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the port center, and $y$ pointing in the counter clockwise direction when
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viewed from above.
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viewed from above.
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In addition to the right handed cartesian orthogonal system specified above, we
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In addition to the right handed Cartesian orthogonal system specified above, we
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introduce also a right-handed cylindrical system $(R,φ,Z)$ with transformation
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introduce also a right-handed cylindrical system $(R,φ,Z)$ with transformation
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from the cylindrical to the cartesian system given by $x= R\cosφ$, $y=R\sinφ$,
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from the cylindrical to the Cartesian system given by $x= R\cosφ$, $y=R\sinφ$,
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$z=Z$.
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$z=Z$.
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## Quasi-optical approximation
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## Quasi-optical approximation
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In the complex eikonal framework, the a solution of the wave equation for the
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In the complex eikonal framework, a solution of the wave equation for the
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electric field is looked for in the form
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electric field is looked for in the form
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$$
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$$
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@ -28,9 +28,9 @@ $$
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$$ {#eq:eikonal-ansatz}
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$$ {#eq:eikonal-ansatz}
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such that it allows for Gaussian beam descriptions.
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such that it allows for Gaussian beam descriptions.
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In [@eq:eikonal-ansatz], $ω$ is the real frequency, $k_0 = ω/c$ the
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In [@eq:eikonal-ansatz], $ω$ is the real frequency, $k_0 = ω/c$ the wavevector
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wavevector amplitude in vacuum, ${\bf e}({\bf x})$ the polarisation versor and
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amplitude in vacuum, ${\bf e}({\bf x})$ the normalised polarisation (Jones)
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$E_0({\bf x})$ the slowly varying wave amplitude.
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vector and $E_0({\bf x})$ the slowly varying wave amplitude.
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The function $S({\bf x})$ is the complex eikonal, $S = S_R({\bf x}) + i S_I
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The function $S({\bf x})$ is the complex eikonal, $S = S_R({\bf x}) + i S_I
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({\bf x})$, in which the real part $S_R({\bf x})$ is related to the beam
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({\bf x})$, in which the real part $S_R({\bf x})$ is related to the beam
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@ -456,16 +456,53 @@ $$
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$$ {#eq:pjgauss}
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$$ {#eq:pjgauss}
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## Reflection at inner wall and polarisation
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## Mode coupling and reflection at inner wall
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The polarisation of the beam is used to compute the coupling to the Ordinary
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(O) and Extraordinary (X) plasma modes when the beam crosses the
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vacuum-plasma interface. The fraction of power converted into a mode is given
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by the coupling coefficient
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$$
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c_\text{mode} = (\hat{\mathbf e}_\text{mode}⋅\hat{\mathbf e})²,
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$$
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where $\hat{\mathbf e}_\text{mode},\hat{\mathbf e}$ are the plasma mode and
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beam Jones vectors, respectively. The mode vector is defined as the
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eigenvector of the cold plasma dielectric tensor in the low density limit.
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The beam vector at launch is computed from the polarisation ellipse parameters
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using the formula:
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$$
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\begin{aligned}
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\hat{e}₁ &= \cosχ\cosψ + i\sinχ\sinψ \\
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\hat{e}₂ &= \cosχ\sinψ - i\sinχ\cosψ
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\end{aligned}
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$$ {#eq:ellipse2field}
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The following convention is assumed (illustrated in [@fig:ellipse]):
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- $ψ$ is the angle between the $x$ axis and the major axis.
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- $χ = \tan(b/a)$ where $a,b$ are ellipse major and minor semi-axes, respectively.
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- A positive $ψ$ corresponds to an ellipse rotated counterclockwise in the
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$x∧y$ plane.
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- A positive $χ$ corresponds to an ellipse traced clockwise in the $x∧y$ plane
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with the $z$ axis in the direction of the wave propagation. In other words:
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negative helicity (projection of spin angular momentum unto wavevector) or
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left handed wave (IEEE convention).
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If the initial polarisation is not specified, 100% coupling to a given mode
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is assumed.
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{#fig:ellipse}
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A model for wave reflection on a smooth surface is included in GRAY. This is
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A model for wave reflection on a smooth surface is included in GRAY. This is
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used to describe beam reflection on the inner wall of the tokamak in the cases
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used to describe the beam reflection on the inner wall of the tokamak in the
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where only partial absorption occurs at the first pass in the plasma. An ideal
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cases where only partial absorption occurs at the first pass in the plasma.
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conductor is assumed for the reflecting surface, so that the full power of the
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An ideal conductor is assumed for the reflecting surface, so that the full
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incident beam is transferred to the reflected one. The vector refractive index
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power of the incident beam is transferred to the reflected one. The vector
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${\bf N}_{\rm{refl}}$ and the unit electric field $\hat {\bf e}_{\rm{refl}}$ of
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refractive index ${\bf N}_{\rm{refl}}$ and the Jones vector $\hat {\bf
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the reflected wave are
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e}_{\rm{refl}}$ of the reflected beam are
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$$
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$$
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{\bf N}_{\rm{refl}} =
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{\bf N}_{\rm{refl}} =
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{\bf N}_{\rm{in}} - 2 ({\bf N}_{\rm{in}}
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{\bf N}_{\rm{in}} - 2 ({\bf N}_{\rm{in}}
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@ -474,50 +511,23 @@ $$
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-\hat {\bf e}_{\rm{in}}
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-\hat {\bf e}_{\rm{in}}
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+ 2 (\hat {\bf e}_{\rm{in}} \cdot \hat {\bf n}) \hat {\bf n},
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+ 2 (\hat {\bf e}_{\rm{in}} \cdot \hat {\bf n}) \hat {\bf n},
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$$
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$$
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being ${\bf N}_{\rm{in}}$ and $\hat {\bf e}_{\rm{in}}$ the vector refractive
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being ${\bf N}_{\rm{in}}$ and $\hat {\bf e}_{\rm{in}}$ the vector refractive
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index and the unit electric field of the incoming wave, and $\hat {\bf n}$ the
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index and the Jones vector of the incoming wave, and $\hat {\bf n}$ the
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normal unit vector to the wall at the beam incidence point.
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normal unit vector to the wall at the beam incidence point.
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The Stokes parameter for the unit electric vector $\hat {\bf e}$ in vacuum are
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The reflected beam Jones vector is again used to compute the coupling
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defined in the beam reference system $({\bar x},{\bar y},{\bar z})$ as
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to the plasma modes at the second and successive pass, with potentially
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$2^{n-1}$ independent modes being traced after n reflections.
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Note that the Jones vectors of the ordinary and extraordinary modes are
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orthogonal w.r.t. the standard Hermitian product: $\hat{\mathbf e}_{\rm
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O}⋅\hat{\mathbf e}_{\rm X}^* = 0$.
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From [@eq:ellipse2field] it then follows that these relations hold:
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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I &= \vert \hat e_{\bar x} \vert^2 + \vert \hat e_{\bar y} \vert^2 = 1 \\
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ψ_{\rm O} &= ψ_{\rm X} + \frac{π}{2} \\
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Q &= \vert \hat e_{\bar x} \vert^2 - \vert \hat e_{\bar y} \vert^2 \\
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χ_{\rm O} &= -χ_{\rm X} \\
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U &= 2 \cdot {\rm Re} (\hat e_{\bar x} \hat e_{\bar y}^*) \\
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1 &= c_{\rm O} + c_{\rm X}
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V &= 2 \cdot {\rm Im} (\hat e_{\bar x} \hat e_{\bar y}^*).
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\end{aligned}
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$$ {#eq:stokes}
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Alternatively, the two angles $\psi_p$ and $\chi_p$ can be used:
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$$
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\begin{aligned}
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Q &= \cos {2 \psi_p} \cos {2 \chi_p} \\
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U &= \sin {2 \psi_p} \cos {2 \chi_p} \\
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V &= \sin {2 \chi_p}
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\end{aligned}
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\end{aligned}
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$$
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$$
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with the latter meaning all the incoming power is coupled to the plasma.
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which define respectively the major axis orientation and the ellipticity of the
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polarisation ellipse. The polarisation parameters of the reflected wave are
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used to compute the coupling with the Ordinary (OM) and Extraordinary (XM)
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modes at the vacuum-plasma interface before the calculation of the second pass
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in the plasma. At the second pass both modes are traced, taking into account
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that the power fraction coupled to each mode is
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$$
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P_{\rm O,X} =
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\frac{P_{\rm in}}{2}
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(1 + Q_{\rm in} Q_{\rm O,X}
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+ U_{\rm in} U_{\rm O,X}
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+ V_{\rm in} V_{\rm O,X}).
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$$
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Note that the polarisation vectors of OM and XM form an orthogonal base:
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$\psi_{p{\rm O}}=\psi_{p{\rm X}}+\pi/2$, $\chi_{p{\rm O}}=-\chi_{p{\rm X}}$ and
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as a consequence $Q_{\rm O}=-Q_{\rm X}$, $U_{\rm O}=-U_{\rm X}$, and $V_{\rm
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O}=-V_{\rm X}$, so that $P_{\rm O} + P_{\rm X} = P_{\rm in}$, i.e. all the
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incoming power is coupled to the plasma.
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@ -113,14 +113,21 @@ Antenna/beam launcher parameters
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- *MODE_X*, extraordinary (X)
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- *MODE_X*, extraordinary (X)
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**psi** (default: **0.0**)
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**psi** (default: **0.0**)
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: ψ (deg), angle between the principal axes of the polarisation
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: ψ (deg), angle between the x and the major axis of the
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ellipse and the (x,y) axes
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polarisation ellipse. ψ∈[-90, 90] and is positive for
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rotating counterclockwise.
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Note: only used in alternative to *iox* if *ipol=.true.*.
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Note: only used in alternative to *iox* if *ipol=.true.*.
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**chi** (default: **0.0**)
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**chi** (default: **0.0**)
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: χ=atan(ε) (deg), where ε is the ellipticity of the polarisation
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: χ (deg), angle between the principal axes of the polarisation
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ellipse
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ellipse. χ∈[-45, 45] and is defined by tan(χ) = b/a, where
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a,b are the major,minor semi-axis, respective.
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χ>0 means the ellipse is traced clockwise in the x∧y plane
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with the z axis in the direction of the wave propagation.
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In other words: negative helicity (projection of spin angular
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momentum unto wavevector) or left handed wave (IEEE convention).
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Note: only used in alternative to *iox* if *ipol=.true.*.
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Note: only used in alternative to *iox* if *ipol=.true.*.
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367
doc/res/ellipse.svg
Normal file
367
doc/res/ellipse.svg
Normal file
@ -0,0 +1,367 @@
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|
style="font-variant-caps:normal;font-variant-east-asian:normal;font-variant-ligatures:normal;font-variant-numeric:normal"
|
||||||
|
id="tspan6">ψ</tspan></text>
|
||||||
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
||||||
|
style="font-variant-caps:normal;font-variant-east-asian:normal;font-variant-ligatures:normal;font-variant-numeric:normal"
|
||||||
|
id="tspan7">χ</tspan></text>
|
||||||
|
<text
|
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|
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|
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id="tspan8">b</tspan></text>
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|
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|
stroke="#000"
|
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|
stroke-linecap="round"
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|
stroke-opacity=".88235"
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|
stroke-width=".96"
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stroke-width="1.44"
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font-size="16px"
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id="tspan19">x</tspan></text>
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transform="scale(.99979 1.0002)"
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id="tspan20">y</tspan></text>
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marker-end="url(#marker2-7)"
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stroke="#01709c"
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stroke-linecap="round"
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stroke-linejoin="bevel"
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|
After Width: | Height: | Size: 11 KiB |
@ -3,6 +3,12 @@
|
|||||||
box-sizing: border-box;
|
box-sizing: border-box;
|
||||||
}
|
}
|
||||||
|
|
||||||
|
/* Justify all text */
|
||||||
|
body {
|
||||||
|
text-align: justify;
|
||||||
|
hypens: auto;
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
/* Make headings smaller */
|
/* Make headings smaller */
|
||||||
h1 { font-size: 1.8em }
|
h1 { font-size: 1.8em }
|
||||||
@ -39,10 +45,15 @@ h3:hover > .header-section-number { opacity: 0; }
|
|||||||
/* Fix spacing of numbered equations */
|
/* Fix spacing of numbered equations */
|
||||||
td .katex-display { margin: 0 }
|
td .katex-display { margin: 0 }
|
||||||
|
|
||||||
|
/* Fix equations width */
|
||||||
|
div[id^="eq"] table { width: 100%; }
|
||||||
|
|
||||||
/* Center the title */
|
/* Center the title */
|
||||||
header { text-align: center }
|
header { text-align: center }
|
||||||
|
|
||||||
|
/* Center figures */
|
||||||
|
figure { text-align: center }
|
||||||
|
|
||||||
/* Inline the authors */
|
/* Inline the authors */
|
||||||
header .author {
|
header .author {
|
||||||
display: inline-block;
|
display: inline-block;
|
||||||
@ -95,7 +106,7 @@ nav {
|
|||||||
color: #d0d6e2;
|
color: #d0d6e2;
|
||||||
padding: 1.2em;
|
padding: 1.2em;
|
||||||
padding-left: 0;
|
padding-left: 0;
|
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overflow-y: scroll;
|
overflow-y: auto;
|
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}
|
}
|
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nav a:link { text-decoration: none }
|
nav a:link { text-decoration: none }
|
||||||
nav a:link, a:visited { color: #d0d6e2 }
|
nav a:link, a:visited { color: #d0d6e2 }
|
||||||
|
|||||||
@ -17,33 +17,41 @@ contains
|
|||||||
! polarisation ellipse angles ψ, χ
|
! polarisation ellipse angles ψ, χ
|
||||||
!
|
!
|
||||||
! Notes:
|
! Notes:
|
||||||
! - ψ∈[-π/2, π/2] is the angle between the x and the major axis
|
! - ψ∈[-π/2, π/2] is the angle between the x and the major axis.
|
||||||
|
!
|
||||||
! - χ∈[-π/4, π/4] is defined by tan(χ) = b/a, where a,b are the
|
! - χ∈[-π/4, π/4] is defined by tan(χ) = b/a, where a,b are the
|
||||||
! major,minor semi-axis, respectively; χ>0 for positive helicity
|
! major,minor semi-axis, respectively.
|
||||||
! (left-handed wave), χ<0 for negative helicity (right-handed wave).
|
!
|
||||||
|
! - χ>0 means the ellipse is traced clockwise in the x∧y plane
|
||||||
|
! with the z axis in the direction of the wave propagation.
|
||||||
|
! In other words: negative helicity (projection of spin angular
|
||||||
|
! momentum unto wavevector) or left handed wave (IEEE convention).
|
||||||
|
!
|
||||||
|
! - ψ>0 rotates the ellipse counterclockwise
|
||||||
|
|
||||||
! subroutine arguments
|
! subroutine arguments
|
||||||
real(wp_), intent(in) :: psi, chi
|
real(wp_), intent(in) :: psi, chi
|
||||||
complex(wp_), intent(out) :: e_x(:), e_y(:)
|
complex(wp_), intent(out) :: e_x(:), e_y(:)
|
||||||
|
|
||||||
! The Eikonal ansatz is:
|
! Consider a plane wave with the electric field given as
|
||||||
!
|
!
|
||||||
! E̅(r̅, t) = Re e̅(r̅) exp(-ik₀S(r̅) + iωt)
|
! E̅(r̅, t) = Re e̅(r̅) exp(ik̅⋅r̅ - iωt)
|
||||||
!
|
!
|
||||||
! where e̅(r̅) = [|e₁|exp(iφ₁), |e₂|exp(iφ₂), 0], since the wave
|
! where k̅ = k₀z and e̅(r̅) = [|e₁|exp(iφ₁), |e₂|exp(iφ₂), 0],
|
||||||
! is transversal in vacuum. At a fixed position r̅=0, ignoring
|
! since the wave is transversal in a vacuum. At a fixed position
|
||||||
! the third component, we have:
|
! r̅=0, ignoring the third component, we have:
|
||||||
!
|
!
|
||||||
! E̅(0, t) = [|e₁|cos(φ₁ + ωt), |e₂|cos(φ₂ + ωt)]
|
! E̅(0, t) = [|e₁|cos(φ₁ - ωt), |e₂|cos(φ₂ - ωt)]
|
||||||
! = [|e₁|cos(φ₁)cos(ωt) - |e₁|sin(φ₁)sin(ωt),
|
! = [|e₁|cos(φ₁)cos(ωt) + |e₁|sin(φ₁)sin(ωt),
|
||||||
! |e₂|cos(φ₂)cos(ωt) - |e₂|sin(φ₂)sin(ωt)]
|
! |e₂|cos(φ₂)cos(ωt) + |e₂|sin(φ₂)sin(ωt)]
|
||||||
!
|
!
|
||||||
! Then, we compare this to the parametric equation of
|
! Then, we compare this to the parametric equation of an ellipse
|
||||||
! an ellipse rotated by ψ through the origin,
|
! rotated by ψ through the origin (traced in the same direction
|
||||||
|
! as for the electric field),
|
||||||
!
|
!
|
||||||
! P̅(t) = R(ψ) [acos(ωt), bsin(ωt)]
|
! P̅(t) = R(ψ) [a⋅cos(ωt), -b⋅sin(ωt)]
|
||||||
! = [cos(ψ)a⋅cos(ωt), -sin(ψ)b⋅sin(ωt),
|
! = [cos(ψ)a⋅cos(ωt) +sin(ψ)b⋅sin(ωt),
|
||||||
! sin(ψ)a⋅cos(ωt), cos(ψ)b⋅sin(ωt)]
|
! sin(ψ)a⋅cos(ωt) -cos(ψ)b⋅sin(ωt)]
|
||||||
!
|
!
|
||||||
! at ωt=0 and ωt=π/2, so:
|
! at ωt=0 and ωt=π/2, so:
|
||||||
!
|
!
|
||||||
@ -76,6 +84,7 @@ contains
|
|||||||
!
|
!
|
||||||
e_x = cosd(chi)*cosd(psi) + im * sind(chi)*sind(psi)
|
e_x = cosd(chi)*cosd(psi) + im * sind(chi)*sind(psi)
|
||||||
e_y = cosd(chi)*sind(psi) - im * sind(chi)*cosd(psi)
|
e_y = cosd(chi)*sind(psi) - im * sind(chi)*cosd(psi)
|
||||||
|
|
||||||
end subroutine ellipse_to_field
|
end subroutine ellipse_to_field
|
||||||
|
|
||||||
|
|
||||||
@ -118,17 +127,27 @@ contains
|
|||||||
!
|
!
|
||||||
chi = asind(imag(2 * e_x * conjg(e_y))) / 2
|
chi = asind(imag(2 * e_x * conjg(e_y))) / 2
|
||||||
psi = atan2d(real(2 * e_x * conjg(e_y)), abs(e_x)**2 - abs(e_y)**2) / 2
|
psi = atan2d(real(2 * e_x * conjg(e_y)), abs(e_x)**2 - abs(e_y)**2) / 2
|
||||||
|
|
||||||
end subroutine field_to_ellipse
|
end subroutine field_to_ellipse
|
||||||
|
|
||||||
|
|
||||||
pure subroutine pol_limit(N, B, Bres, sox, e_x, e_y)
|
pure subroutine pol_limit(N, B, Bres, sox, e_x, e_y)
|
||||||
! Computes the Jones vectors of the cold plasma dispersion
|
! Computes the Jones vectors of the cold plasma dispersion
|
||||||
! relation in the limit of vanishing electron density
|
! relation in the limit of vanishing electron density
|
||||||
!
|
!
|
||||||
! Note: the Jones vectors are given in the local beam frame,
|
! Notes:
|
||||||
! that is, the z axis is aligned with the wave vector and x axis
|
! - The Jones vectors are given in the local beam frame,
|
||||||
! lies in the tokamak equatorial plane.
|
! that is, the z axis is aligned with the wave vector
|
||||||
! This allows to directly compare the beam polarisation with
|
! and the x axis lies in the tokamak equatorial plane.
|
||||||
! the plasma modes Jones vectors to obtain the power couplings.
|
! This allows to directly compare the beam polarisation with
|
||||||
|
! the plasma modes Jones vectors to obtain the power couplings.
|
||||||
|
!
|
||||||
|
! - The dielectric tensor is obtained using the convention
|
||||||
|
!
|
||||||
|
! E̅(r̅, t) = ∫ d³k dω/(2π)⁴ E̅(k̅, ω) exp(ik̅⋅r̅ - iωt)
|
||||||
|
!
|
||||||
|
! for the Fourier transform. This is commonplace, but it's
|
||||||
|
! the opposite of the eikonal ansatz, so the Jones vector
|
||||||
|
|
||||||
! subroutine arguments
|
! subroutine arguments
|
||||||
real(wp_), intent(in) :: N(3) ! N̅ refractive index
|
real(wp_), intent(in) :: N(3) ! N̅ refractive index
|
||||||
@ -287,6 +306,7 @@ contains
|
|||||||
e = matmul(R, e)
|
e = matmul(R, e)
|
||||||
e_x = e(1)
|
e_x = e(1)
|
||||||
e_y = e(2)
|
e_y = e(2)
|
||||||
|
|
||||||
end subroutine pol_limit
|
end subroutine pol_limit
|
||||||
|
|
||||||
end module polarization
|
end module polarization
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user