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SpECTRE
v2026.04.01
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The solution for a half-space mirror deformed by a laser beam. More...
#include <HalfSpaceMirror.hpp>
Classes | |
| struct | AbsoluteTolerance |
| struct | BeamWidth |
| struct | IntegrationIntervals |
| struct | Material |
| struct | RelativeTolerance |
Public Types | |
| using | constitutive_relation_type |
| using | options |
Public Member Functions | |
| HalfSpaceMirror (const HalfSpaceMirror &)=default | |
| HalfSpaceMirror & | operator= (const HalfSpaceMirror &)=default |
| HalfSpaceMirror (HalfSpaceMirror &&)=default | |
| HalfSpaceMirror & | operator= (HalfSpaceMirror &&)=default |
| std::unique_ptr< elliptic::analytic_data::AnalyticSolution > | get_clone () const override |
| HalfSpaceMirror (double beam_width, constitutive_relation_type constitutive_relation, size_t integration_intervals=350, double absolute_tolerance=1e-12, double relative_tolerance=1e-10) | |
| double | beam_width () const |
| size_t | integration_intervals () const |
| double | absolute_tolerance () const |
| double | relative_tolerance () const |
| const constitutive_relation_type & | constitutive_relation () const |
| template<typename DataType, typename... RequestedTags> | |
| tuples::TaggedTuple< RequestedTags... > | variables (const tnsr::I< DataType, 3 > &x, tmpl::list< RequestedTags... >) const |
| void | pup (PUP::er &p) override |
| NOLINTNEXTLINE(google-runtime-references) | |
Static Public Attributes | |
| static constexpr Options::String | help |
The solution for a half-space mirror deformed by a laser beam.
This solution is mapping (via the fluctuation dissipation theorem) thermal noise to an elasticity problem where a normally incident and axisymmetric laser beam with a Gaussian beam profile acts on the face of a semi-infinite mirror. Here we assume the face to be at \(z = 0\) and the material to extend to \(+\infty\) in the z-direction as well as for the mirror diameter to be comparatively large to the beam width. The mirror material is characterized by an isotropic homogeneous constitutive relation \(Y^{ijkl}\) (see Elasticity::ConstitutiveRelations::IsotropicHomogeneous). In this scenario, the auxiliary elastic problem has an applied pressure distribution equal to the laser beam intensity profile \(p(r)\) (see Eq. (11.94) and Eq. (11.95) in [203] with F = 1 and the time dependency dropped)
\begin{align}T^{zr} &= T^{rz} = 0 \\ T^{zz} &= p(r) = \frac{e^{-\frac{r^2}{r_0^2}}}{\pi r_0^2}\text{.} \end{align}
in the form of a Neumann boundary condition to the face of the mirror. We find that this stress in cylinder coordinates is produced by the displacement field
\begin{align}\xi_{r} &= \frac{1}{2 \mu} \int_0^{\infty} dk J_1(kr)e^{(-kz)}\left(1 - \frac{\lambda + 2\mu}{\lambda + \mu} + kz \right) \tilde{p}(k) \\ \xi_{\phi} &= 0 \\ \xi_{z} &= \frac{1}{2 \mu} \int_0^{\infty} dk J_0(kr)e^{(-kz)}\left(1 + \frac{\mu}{\lambda + \mu} + kz \right) \tilde{p}(k) \end{align}
and the strain
\begin{align}\Theta &= \frac{1}{2 \mu} \int_0^{\infty} dk J_0(kr) k e^{(-kz)}\left(\frac{-2\mu}{\lambda + \mu}\right) \tilde{p}(k) \\ S_{rr} &= \Theta - S_{\phi\phi} - S_{zz} \\ S_{\phi\phi} &= \frac{\xi_{r}}{r} \\ S_{(rz)} &= -\frac{1}{2 \mu} \int_0^{\infty} dk J_1(kr) k e^{(-kz)}\left(kz \right) \tilde{p}(k) \\ S_{zz} &= \frac{1}{2 \mu} \int_0^{\infty} dk J_0(kr) k e^{(-kz)}\left(-\frac{\mu}{\lambda + \mu} - kz \right) \tilde{p}(k) \end{align}
(see Eqs. (11 a) - (11 c) and (13 a) - (13 e), with (13 c) swapped in favor of (12 c) in [139]), where \(\tilde{p}(k)= \frac{1}{2\pi} e^{-(\frac{kr_0}{2})^2}\) is the Hankel-Transform of the lasers intensity profile and \( \Theta = \mathrm{Tr}(S)\) the materials expansion.
| using Elasticity::Solutions::HalfSpaceMirror::options |
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inlineoverridevirtual |
Implements elliptic::analytic_data::AnalyticSolution.
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staticconstexpr |