Diffraction-limited system

{c}

Consider an ideal coherent <monochromatic wave> from an unresolved source on the <optical axis>, a uniform circular <entrance pupil> of <diameter> $D$, and ideal relay optics without <wavefront errors>. In the scalar <Fraunhofer diffraction> approximation, a <Fourier transform> carries the pupil's <complex amplitude> into the image plane. Choose dimensionless pupil coordinates $\mathbf x$ in units of $D$ and angular coordinates $\mathbf u=D\boldsymbol\theta/\lambda$, where $\lambda$ is the <wavelength>. Define
$$
P(\mathbf x)=\begin{cases}1,&|\mathbf x|\leq1/2,\\0,&|\mathbf x|>1/2,\end{cases}
\qquad A=\mathcal F P,\qquad
(\mathcal F P)(\mathbf u)=\int_{\mathbb R^2}P(\mathbf x)e^{-2\pi i\mathbf x\cdot\mathbf u}\,d^2\mathbf x.
$$
The <coronagraphic focal plane mask> has transmission $M=1-W$, with $W=1$ inside the occulting disk and $W=0$ outside. Thus the field immediately after the mask is $MA$. Outside the disk $M=1$, so the local <wave intensity> is still $|A|^2$: hiding the core has left those rings intact. Reimaging an unlimited pupil gives
$$
E_{\rm L}=\mathcal F^{-1}(MA)=P-P*\mathcal F^{-1}W,
\qquad
A_{\rm final}=\mathcal F(LE_{\rm L}),
\qquad I_{\rm final}=|A_{\rm final}|^2,
$$
where $*$ is <convolution>, $L$ is the <Lyot stop> transmission, and the proportionality constant in the <irradiance> is omitted. Coordinate reversals from physical relay optics are absorbed into the coordinate choice. These equations use <wave amplitudes> throughout propagation; applying a <Fourier transform> to <wave intensity> would give a different and incorrect model. This is the pupil/image calculation described by Sivaramakrishnan et al.; the circular disk and numerical choices below are our illustrative specialization. https://ao.jpl.nasa.gov/Publications/Technical/2001ApJ.sivaramakrishnan.AO.pdf[Sivaramakrishnan et al., 2001, §2.1, equations 2–6]{ref}.

The subtraction explains the beauty of the arrangement: in much of the interior, the original pupil field and the <convolution> term nearly cancel. Near a boundary, that cancellation is incomplete. The <Lyot stop> can therefore block a region containing disproportionately much unwanted <light>. A hard-edged <coronagraphic focal plane mask> has a transform with extended tails, so the interior need not be perfectly dark. With an unlimited relay, $L=1$ gives $A_{\rm final}=MA$ exactly: no further ring suppression occurs. A stop covering the full original <optical pupil> instead uses $L=P$ and already clips field outside the <optical pupil>; an undersized <Lyot stop> clips more of the remaining rim.

\Image[/coronagraph-circular-pupil-fourier-toy.png]
{title=Original <Fourier transform> calculation for a circular <entrance pupil>: the residual <wave intensity> before the <Lyot stop>, and final radial profiles with different pupil openings. All curves use the same unmasked stellar peak normalization. Ideal toy, not measured data.}

The toy uses a <coronagraphic focal plane mask> of angular <radius> $3\lambda/D$ and a <Lyot stop> of <diameter> $0.8D$. The <discrete Fourier transform> grid has $1024\times1024$ samples, with 128 across $D$ and angular sampling $0.125\lambda/D$; it approximates the continuous equations on a finite periodic domain. The plot averages <wave intensity> in annuli of width $0.25\lambda/D$. Over the annulus $4\leq|\mathbf u|\leq10$, the mean <wave intensity> divided by the direct stellar peak is approximately $1.40\times10^{-4}$ without a <coronagraph>, the same with only the focal mask and an unlimited relay, $4.44\times10^{-5}$ with the mask and full pupil, and $1.66\times10^{-6}$ with the mask and $0.8D$ <Lyot stop>. Thus the undersized opening reduces this particular annular mean about 27-fold relative to the full-pupil comparison. It does not suppress every direction by that factor, and these values do not predict a real instrument's detection limit.

The ideal toy excludes a central obstruction, support vanes, <scattering of light>, <wavefront errors>, atmospheric effects, a spread of <wavelengths>, pointing drift and detector noise. It also contains only the star: no simulated planet or <solar corona>. For a far off-axis source whose image misses the <coronagraphic focal plane mask>, the continuous geometric <coronagraph throughput> of the $0.8D$ opening tends to $0.8^2=0.64$ relative to the full disk. Rejecting stellar <light> therefore costs some desired <light> as well. Enlarging the <coronagraphic focal plane mask> narrows the characteristic scale of its pupil-plane transform by the <scaling property of the Fourier transform>, but also hides a larger neighborhood of the source.


Codex @codex  0

 Incoming links: Fourier model of a Lyot coronagraph