Adding some formulae to the text
Former-commit-id: c244a6681365147a17017dbb1f0f75192d99c351
This commit is contained in:
Marek Nečada
parent
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commit
fe0ccb92b8
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@ -79,8 +79,21 @@
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\begin_body
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\begin_layout Standard
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\begin_inset FormulaMacro
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\newcommand{\vect}[1]{\mathbf{#1}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\ud}{\mathrm{d}}
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\end_inset
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\end_layout
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\begin_layout Title
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Electromagnetic multiple scattering, spherical waves and shit
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Electromagnetic multiple scattering, spherical waves and ****
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\end_layout
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\begin_layout Author
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@ -95,33 +108,562 @@ Zillion conventions for spherical vector waves
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Legendre polynomials and spherical harmonics: messy from the very beginning
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\end_layout
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\begin_layout Subsection
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Kristensson
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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P_{l}^{-m}=\left(-1\right)^{m}\frac{\left(l-m\right)!}{\left(l+m\right)!}P_{l}^{m}\left(\cos\theta\right),\quad m\ge0
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\]
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\end_inset
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Kristensson uses the Condon-Shortley phase, so (sect.
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[K]D.2)
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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Y_{lm}\left(\hat{\vect r}\right)=\left(-1\right)^{m}\sqrt{\frac{2l+1}{4\pi}\frac{\left(l-m\right)!}{\left(l+m\right)!}}P_{l}^{m}\left(\cos\theta\right)e^{im\phi}
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\]
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\end_inset
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\begin_inset Formula
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\[
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Y_{lm}^{\dagger}\left(\hat{\vect r}\right)=Y_{lm}^{*}\left(\hat{\vect r}\right)
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\]
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\end_inset
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\begin_inset Formula
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\[
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Y_{l,-m}\left(\hat{\vect r}\right)=\left(-1\right)^{m}Y_{lm}^{\dagger}\left(\hat{\vect r}\right)
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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Orthonormality:
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\begin_inset Formula
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\[
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\int Y_{lm}\left(\hat{\vect r}\right)Y_{l'm'}^{\dagger}\left(\hat{\vect r}\right)\,\ud\Omega=\delta_{ll'}\delta_{mm'}
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\]
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\end_inset
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\end_layout
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\begin_layout Section
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Pi and tau (?), spherical Bessel functions
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Pi and tau
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\end_layout
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\begin_layout Subsection
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Taylor
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{eqnarray*}
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\tilde{\pi}_{mn}\left(\cos\theta\right) & = & \sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}\frac{m}{\sin\theta}P_{n}^{m}\left(\cos\theta\right)\\
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\tilde{\tau}_{mn}\left(\cos\theta\right) & = & \sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}\frac{\ud}{\ud\theta}P_{n}^{m}\left(\cos\theta\right)
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\end{eqnarray*}
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\end_inset
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\end_layout
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\begin_layout Section
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Vector spherical harmonics (?)
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\end_layout
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\begin_layout Section
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All the conventions
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\begin_layout Subsection
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Kristensson
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\end_layout
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\begin_layout Standard
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Original formulation, sect.
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[K]D.3.3
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{eqnarray*}
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\vect A_{1lm}\left(\hat{\vect r}\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}\left(\hat{\vect{\theta}}\frac{1}{\sin\theta}\frac{\partial}{\partial\phi}Y_{lm}\left(\hat{\vect r}\right)-\hat{\vect{\phi}}\frac{\partial}{\partial\theta}Y_{lm}\left(\hat{\vect r}\right)\right)\\
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\vect A_{2lm}\left(\hat{\vect r}\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}\left(\hat{\vect{\theta}}\frac{\partial}{\partial\phi}Y_{lm}\left(\hat{\vect r}\right)-\hat{\vect{\phi}}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}Y_{lm}\left(\hat{\vect r}\right)\right)\\
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\vect A_{3lm}\left(\hat{\vect r}\right) & = & \hat{\vect r}Y_{lm}\left(\hat{\vect r}\right)
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\end{eqnarray*}
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\end_inset
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Normalisation:
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\begin_inset Formula
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\[
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\int\vect A_{n}\left(\hat{\vect r}\right)\cdot\vect A_{n'}^{\dagger}\left(\hat{\vect r}\right)\,\ud\Omega=\delta_{nn'}
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\]
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\end_inset
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Here
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\begin_inset Formula $\mbox{ }^{\dagger}$
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\end_inset
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means just complex conjugate, apparently (see footnote on p.
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89).
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\end_layout
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\begin_layout Section
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Spherical Bessel functions
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Spherical-Bessel-functions"
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\end_inset
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\end_layout
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\begin_layout Standard
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The radial dependence of spherical vector waves is given by the spherical
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Bessel functions and their first derivatives.
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Commonly, the following notation is adopted
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\begin_inset Formula
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\begin{eqnarray*}
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z_{n}^{(1)}(x) & = & j_{n}(x),\\
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z_{n}^{(2)}(x) & = & y_{n}(x),\\
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z_{n}^{(3)}(x) & = & h_{n}^{(1)}(x)=j_{n}(x)+iy_{n}(x),\\
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z_{n}^{(4)}(x) & = & h_{n}^{(2)}(x)=j_{n}(x)-iy_{n}(x).
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\end{eqnarray*}
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\end_inset
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Here,
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\begin_inset Formula $j_{n}$
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\end_inset
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is the spherical Bessel function of first kind (regular),
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\begin_inset Formula $y_{j}$
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\end_inset
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is the spherical Bessel function of second kind (singular), and
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\begin_inset Formula $h_{n}^{(1)},h_{n}^{(2)}$
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\end_inset
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are the Hankel functions a.k.a.
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spherical Bessel functions of third kind.
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In spherical vector waves,
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\begin_inset Formula $j_{n}$
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\end_inset
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corresponds to regular waves,
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\begin_inset Formula $h^{(1)}$
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\end_inset
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corresponds (by the usual convention) to outgoing waves, and
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\begin_inset Formula $h^{(2)}$
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\end_inset
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corresponds to incoming waves.
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To describe scattering, we need two sets of waves with two different types
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of spherical Bessel functions
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\begin_inset Formula $z_{n}^{(J)}$
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\end_inset
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.
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Most common choice is
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\begin_inset Formula $J=1,3$
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\end_inset
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, because if we decompose the field into spherical waves centered at
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\begin_inset Formula $\vect r_{0}$
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\end_inset
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, the field produced by other sources (e.g.
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spherical waves from other scatterers or a plane wave) is always regular
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at
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\begin_inset Formula $\vect r_{0}$
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\end_inset
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.
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Second choice which makes a bit of sense is
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\begin_inset Formula $J=3,4$
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\end_inset
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as it leads to a nice expression for the energy transport.
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\end_layout
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\begin_layout Section
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Spherical vector waves
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\end_layout
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\begin_layout Standard
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TODO
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\begin_inset Formula $M,N,\psi,\chi,\widetilde{M},\widetilde{N},u,v,w,\dots$
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\end_inset
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, sine/cosine convention (B&H), ...
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\end_layout
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\begin_layout Standard
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There are two mutually orthogonal types of divergence-free (everywhere except
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in the origin for singular waves) spherical vector waves, which I call
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electric and magnetic, given by the type of multipole source to which they
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correspond.
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This is another distinction than the regular/singular/ingoing/outgoing
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waves given by the type of the radial dependence (cf.
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section
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "sec:Spherical-Bessel-functions"
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\end_inset
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).
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Oscillating electric current in a tiny rod parallel to its axis will generate
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electric dipole waves (net dipole moment of magnetic current is zero) moment
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, whereas oscillating electric current in a tiny circular loop will generate
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magnetic dipole waves (net dipole moment of electric current is zero).
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\end_layout
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\begin_layout Standard
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In the usual cases we encounter, the part described by the magnetic waves
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is pretty small.
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\end_layout
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\begin_layout Subsection
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Taylor
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\end_layout
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\begin_layout Standard
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Definition [T](2.40);
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\begin_inset Formula $\widetilde{\vect N}_{mn}^{(j)},\widetilde{\vect M}_{mn}^{(j)}$
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\end_inset
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are the electric and magnetic waves, respectively:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{eqnarray*}
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\widetilde{\vect N}_{mn}^{(j)} & = & \frac{n(n+1)}{kr}\sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}z_{n}^{j}\left(kr\right)\hat{\vect r}\\
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& & +\left[\tilde{\tau}_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}+i\tilde{\pi}_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}z_{n}^{j}\left(kr\right)\\
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\widetilde{\vect M}_{mn}^{(j)} & = & \left[i\tilde{\pi}_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}-\tilde{\tau}_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}z_{n}^{j}\left(kr\right)
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\end{eqnarray*}
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\end_inset
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\end_layout
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\begin_layout Subsection
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Kristensson
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\end_layout
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\begin_layout Standard
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Definition [K](2.4.6);
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\begin_inset Formula $\vect u_{\tau lm},\vect v_{\tau lm},\vect w_{\tau lm}$
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\end_inset
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are the waves with
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\begin_inset Formula $j=3,1,4$
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\end_inset
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respectively, i.e.
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outgoing, regular and incoming waves.
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The first index distinguishes between the electric (
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\begin_inset Formula $\tau=2$
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\end_inset
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) and magnetic (
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\begin_inset Formula $\tau=1$
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\end_inset
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).
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Kristensson uses a multiindex
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\begin_inset Formula $n\equiv(\tau,l,m)$
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\end_inset
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to simlify the notation.
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\begin_inset Formula
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\begin{eqnarray*}
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\left(\vect{u/v/w}\right)_{2lm} & = & \frac{1}{kr}\frac{\ud\left(kr\, z_{l}^{(j)}\left(kr\right)\right)}{\ud\, kr}\vect A_{2lm}\left(\hat{\vect r}\right)+\sqrt{l\left(l+1\right)}\frac{z_{l}^{(j)}(kr)}{kr}\vect A_{3lm}\left(\hat{\vect r}\right)\\
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\left(\vect{u/v/w}\right)_{1lm} & = & z_{l}^{(j)}\left(kr\right)\vect A_{1lm}\left(\hat{\vect r}\right)
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\end{eqnarray*}
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\end_inset
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\end_layout
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\begin_layout Subsection
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Relation between Kristensson and Taylor
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\begin_inset CommandInset label
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LatexCommand label
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name "sub:Kristensson-v-Taylor"
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\end_inset
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\end_layout
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\begin_layout Standard
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Kristensson's and Taylor's VSWFs seem to differ only by an
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\begin_inset Formula $l$
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\end_inset
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-dependent normalization factor, and notation of course (n.b.
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the inverse index order)
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\begin_inset Formula
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\begin{eqnarray*}
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\left(\vect{u/v/w}\right)_{2lm} & = & \frac{\widetilde{\vect N}_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}}\\
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\left(\vect{u/v/w}\right)_{1lm} & = & \frac{\widetilde{\vect M}_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}}
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\end{eqnarray*}
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\end_inset
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\end_layout
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\begin_layout Section
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Plane wave expansion
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\end_layout
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\begin_layout Subsection
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Taylor
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\end_layout
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\begin_layout Standard
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\begin_inset Formula $x$
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\end_inset
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-polarised,
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\begin_inset Formula $z$
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\end_inset
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-propagating plane wave,
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\begin_inset Formula $\vect E=E_{0}\hat{\vect x}e^{i\vect k\cdot\hat{\vect z}}$
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\end_inset
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(CHECK):
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\begin_inset Formula
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\begin{eqnarray*}
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\vect E & = & -i\left(p_{mn}\widetilde{\vect N}_{mn}^{(1)}+q_{mn}\widetilde{\vect M}_{mn}^{(1)}\right)\\
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p_{mn} & = & E_{0}\frac{4\pi i^{n}}{n(n+1)}\tilde{\tau}_{mn}(1)\\
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q_{mn} & = & E_{0}\frac{4\pi i^{n}}{n(n+1)}\tilde{\pi}_{mn}(1)
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\end{eqnarray*}
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\end_inset
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while it can be shown that
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\begin_inset Formula
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\begin{eqnarray*}
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\tilde{\pi}_{mn}(1) & = & -\frac{1}{2}\sqrt{\frac{\left(2n+1\right)\left(n\left(n+1\right)\right)}{4\pi}}\left(\delta_{m,1}+\delta_{m,-1}\right)\\
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\tilde{\tau}_{mn}(1) & = & -\frac{1}{2}\sqrt{\frac{\left(2n+1\right)\left(n\left(n+1\right)\right)}{4\pi}}\left(\delta_{m,1}-\delta_{m,-1}\right)
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\end{eqnarray*}
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\end_inset
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\end_layout
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\begin_layout Subsection
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Kristensson
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\end_layout
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\begin_layout Standard
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\begin_inset Formula $x$
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\end_inset
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-polarised,
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\begin_inset Formula $z$
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\end_inset
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-propagating plane wave,
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\begin_inset Formula $\vect E=E_{0}\hat{\vect x}e^{i\vect k\cdot\hat{\vect z}}$
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\end_inset
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(CHECK, ):
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\begin_inset Formula
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\[
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\vect E=\sum_{n}a_{n}\vect v_{n}
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\]
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\end_inset
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\begin_inset Formula
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\begin{eqnarray*}
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a_{1lm} & = & E_{0}i^{l+1}\sqrt{\left(2l+1\right)\pi}\left(\delta_{m,1}+\delta_{m,-1}\right)\\
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a_{2lm} & = & E_{0}i^{l+1}\sqrt{\left(2l+1\right)\pi}\left(\delta_{m,1}+\delta_{m,-1}\right)
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\end{eqnarray*}
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\end_inset
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\end_layout
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\begin_layout Section
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Radiated energy
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\end_layout
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\begin_layout Standard
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In this section I summarize the formulae for power
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\begin_inset Formula $P$
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\end_inset
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radiated from the system.
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For an absorbing scatterer, this should be negative (n.b.
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sign conventions can be sometimes confusing).
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If the system is excited by a plane wave with intensity
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\begin_inset Formula $E_{0}$
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\end_inset
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, this can be used to calculate the absorption cross section,
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\begin_inset Formula
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\[
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\sigma_{\mathrm{abs}}=-\frac{P}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}}.
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\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Kristensson
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\begin_inset CommandInset label
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LatexCommand label
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name "sub:Radiated enenergy-Kristensson"
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\end_inset
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\end_layout
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\begin_layout Standard
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Sect.
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[K]2.6.2; here this form of expansion is assumed:
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\begin_inset Formula
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\begin{equation}
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\vect E\left(\vect r,\omega\right)=k\sqrt{\eta_{0}\eta}\sum_{n}\left(a_{n}\vect v_{n}\left(k\vect r\right)+f_{n}\vect u_{n}\left(k\vect r\right)\right).\label{eq:power-Kristensson-E}
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\end{equation}
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\end_inset
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Here
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\begin_inset Formula $\eta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}$
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\end_inset
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is the wave impedance of free space and
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\begin_inset Formula $\eta=\sqrt{\mu/\varepsilon}$
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\end_inset
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is the relative wave impedance of the medium.
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\end_layout
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||||
\begin_layout Standard
|
||||
The radiated power is then (2.28):
|
||||
\begin_inset Formula
|
||||
\[
|
||||
P=\frac{1}{2}\sum_{n}\left(\left|f_{n}\right|^{2}+\Re\left(f_{n}a_{n}^{*}\right)\right)
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsection
|
||||
Taylor
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Here I derive the radiated power in Taylor's convention by applying the
|
||||
relations from subsection
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "sub:Kristensson-v-Taylor"
|
||||
|
||||
\end_inset
|
||||
|
||||
to the Kristensson's formulae (sect.
|
||||
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "sub:Radiated enenergy-Kristensson"
|
||||
|
||||
\end_inset
|
||||
|
||||
).
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Assume the external field decomposed as (here I use tildes even for the
|
||||
expansion coefficients in order to avoid confusion with the
|
||||
\begin_inset Formula $a_{n}$
|
||||
\end_inset
|
||||
|
||||
in
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:power-Kristensson-E"
|
||||
|
||||
\end_inset
|
||||
|
||||
)
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\vect E\left(\vect r,\omega\right)=\sum_{mn}\left[-i\left(\tilde{p}_{mn}\vect{\widetilde{N}}_{mn}^{(1)}+\tilde{q}_{mn}\widetilde{\vect M}_{mn}^{(1)}\right)+i\left(\tilde{a}_{mn}\widetilde{\vect N}_{mn}^{(3)}+\tilde{b}_{mn}\widetilde{\vect M}_{mn}^{(3)}\right)\right]
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
(there is minus between the regular and outgoing part!).
|
||||
The coefficients are related to those from
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:power-Kristensson-E"
|
||||
|
||||
\end_inset
|
||||
|
||||
as
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\tilde{p}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{-i\sqrt{n(n+1)}}a_{2nm},\quad\tilde{q}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{-i\sqrt{n(n+1)}}a_{1nm},
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\tilde{a}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{i\sqrt{n(n+1)}}f_{2nm},\quad\tilde{b}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{i\sqrt{n(n+1)}}f_{1nm}.
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
The radiated power is then
|
||||
\begin_inset Formula
|
||||
\[
|
||||
P=\frac{1}{2}\sum_{m,n}\frac{n\left(n+1\right)}{k^{2}\eta_{0}}\left(\left|a_{mn}\right|^{2}+\left|b_{mn}\right|^{2}-\Re\left(a_{mn}p_{mn}^{*}\right)-\Re\left(b_{mn}q_{mn}^{*}\right)\right).
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Limit solutions
|
||||
\end_layout
|
||||
|
|
@ -183,11 +725,11 @@ Mie decomposition of Green's function for single nanoparticle
|
|||
\end_layout
|
||||
|
||||
\begin_layout Chapter
|
||||
Translation of spherical waves: shit gets insane
|
||||
Translation of spherical waves: getting insane
|
||||
\end_layout
|
||||
|
||||
\begin_layout Chapter
|
||||
Multiple scattering: nice linear algebra born from all the shit
|
||||
Multiple scattering: nice linear algebra born from all the mess
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
|
|
|||
Loading…
Reference in New Issue