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clarify that an Er dipole at r=0 and m=±1 has circular polarization
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oskooi committed Dec 21, 2024
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Expand Up @@ -668,7 +668,7 @@ The example uses the same setup as the [previous tutorial](#radiation-pattern-of

$$P(\theta) \approx \int_0^R P(r,\theta) s(r) 2\pi rdr = \sum_{n=0}^{N-1} P(r_n,\theta) s(r_n) 2\pi r_n \Delta r$$,

where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note that a dipole placed exactly at $r = 0$ would have *zero* contribution to the total radiation pattern because its area ($2\pi rdr$) is zero. In this example, the $E_r$ dipole at $r = 0$ is actually placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the radiation pattern in vacuum for a set of dipoles at different radial positions. The radiation pattern of an $E_r$ dipole in vacuum is $cos^2(\theta) + 1$ which is independent of its position in $r$. This criteria is used to obtain $s(r_0) = 1$ and $s(r > 0) = 0.5(r_0/r)^2$. (A $1/r^2$ dependence is expected because a cylindrical delta function should really include a $1/r$ factor in order to integrate to 1 with $\int r \, dr$, but Meep currently does not include this $1/r$ in sources that have zero radial width, and power goes like the square of the current amplitude—therefore, we must include an additional $1/r^2$ factor to obtain the correct relative power for dipoles at different radii.) This weighting function is also used to sum the flux emitted by each dipole (obtained using using the LDOS feature). This quantity is the denominator in the expression for the extraction efficiency.
where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note that a dipole placed exactly at $r = 0$ would have *zero* contribution to the total radiation pattern because its area ($2\pi rdr$) is zero. In this example, the $E_r$ dipole at $r = 0$ is actually placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). Note that an $E_r$ dipole at $r = 0$ with $m = \pm 1$ has circular (*not* linear) polarization. $s(r)$ can be determined empirically by computing the radiation pattern in vacuum for a set of dipoles at different radial positions. The radiation pattern of an $E_r$ dipole with circular polarization in vacuum is $cos^2(\theta) + 1$ which is independent of its position in $r$. This criteria is used to obtain $s(r_0) = 1$ and $s(r > 0) = 0.5(r_0/r)^2$. (A $1/r^2$ dependence is expected because a cylindrical delta function should really include a $1/r$ factor in order to integrate to 1 with $\int r \, dr$, but Meep currently does not include this $1/r$ in sources that have zero radial width, and power goes like the square of the current amplitude—therefore, we must include an additional $1/r^2$ factor to obtain the correct relative power for dipoles at different radii.) This weighting function is also used to sum the flux emitted by each dipole (obtained using using the LDOS feature). This quantity is the denominator in the expression for the extraction efficiency.

This figure shows the radiation pattern from $N=11$ dipoles with $\lambda$ of 1.0 $\mu$m in the middle of a disc of height 0.29 $\mu$m, radius 1.2 $\mu$m, and refractive index 2.4. The extraction efficiency for this setup is 0.933517. The runtime is about two hours using two Intel 4.2 GHz Xeon cores.

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