Optics 067 · Diffraction, Fourier optics, and computational imaging

Pupil Engineering & Coronagraph

Reveal a faint companion with a four-plane Fourier coronagraph, then trade stellar rejection against planet throughput, pupil apodization and wavefront errors.

Total stellar / planet throughput

T_\star,T_p

$0.50$

Relative separation / brightness

\rho_p,c

$50\%$

Lyot radius / entrance power

r_L/R,P_A

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Pupil Engineering & Coronagraph

Mode index (named buttons below)

h

2

Opaque focal disk radius

r

2.50\lambda/D

Planet separation from star

\rho_p

5\lambda/D

Lyot outer radius / entrance radius

r_L/R

0.82

Central obstruction radius / entrance radius

\varepsilon

0

Cosine amplitude exponent (zero is clear)

a

0

Edge-to-centre defocus waves (not RMS)

W

0\lambda

Common horizontal pointing error

\delta_\star

0\lambda/D

Planet faintness in decades below star

e_c

4

A star hides its planet in a diffraction halo. Compare ordinary imaging, the opaque focal disk, then the Lyot stop: blocking the core alone leaves halo light. All images share one fixed logarithmic scale. Drag the planet marker, reduce separation, or disturb pointing and focus. Shrinking a real stop may reveal the planet—but also throws away its light. The apparatus and every record use the same masks and complex fields.

Ideal monochromatic scalar optics and synthetic point sources. Quantized finite-grid masks, illustrative cosine apodizer, fixed clear-pupil normalization; not a telescope design or detection claim.

Physics tutorial

Audit the physics of Pupil Engineering & Coronagraph

BackgroundBlocking a star's focus does not remove its diffraction halo. Alternating pupil and focal planes propagate a complex wave: $I_D=|\mathcal F\{L\mathcal F^{-1}[M\mathcal F(E_A)]\}|^2$ . This sky-coordinate expression includes pupil reversal. The hardware has three positive lenses.

Why it mattersHow can masks and apodizers suppress a star without erasing a nearby planet?

Start with the essentials

Focus question: How can masks and apodizers suppress a star without erasing a nearby planet?
One-sentence intuition: Incoherent star and planet intensities—not amplitudes—add: $I_{\rm sky}=I_\star+cI_p$ . Planet signals retain their faintness; no image is normalized to its own maximum.

Core mathematical model

Complex-amplitude masks

\begin{aligned}E_B&=M\mathcal F[E_A]\\E_C&=L\mathcal F^{-1}[E_B]\\E_D&=\mathcal F[E_C]\end{aligned}

Sky coordinates absorb pupil inversion. The disk is opaque, the stop an open annulus. Propagate fields; do not fit an arbitrary sigmoid.

Radius and conjugates

\begin{aligned}r_B&=\rho\lambda f/D\\z_A=0,\quad z_B&=2f\\z_C=4f,\quad z_D&=6f\\\mathbf r_C&=-\mathbf r_A\end{aligned}

250 mm lenses, a 10 mm pupil and 550 nm light fix the disk radius. Scene focal and pupil scales differ. Physical D is inverted; the sky record corrects this once.

Illustrative apodization

\begin{aligned}A(r)&=\cos^a\!\left(\frac{\pi r}{2R}\right)\\\phi&=2\pi[\delta x/D+W r^2/R^2]\\E_A&=P A e^{i\phi}\end{aligned}

Power is amplitude squared. Defocus is edge-to-centre waves, not RMS. The cosine family is not an optimized prolate mask.

Signal is not certainty

\begin{aligned}I_{\rm sky}&=I_\star+cI_p\\T_p&=P_{D,p}/P_{0,p}\\C_{\rm ap}&=\frac{\sum_{\mathcal A}cI_p}{\sum_{\mathcal A}I_\star}\end{aligned}

Actual samples within radius 0.7 diffraction unit of the planet. No noise, broadband light or finite star: not a real detection prediction.

Common difficulties

A dark core is not a clean planet image

Typical misconceptionA bigger disk or smaller stop always improves detection.

Better mental modelThe disk redistributes the remaining coherent field. A pupil stop rejects residual light but also reduces planet throughput and broadens its response. Pointing and defocus can increase contamination. A noiseless aperture ratio is not SNR or a detection guarantee.

Run the experiment

01
Scene 1: Planet buried in a stellar halo
Compare ordinary imaging, disk alone and full Lyot on one fixed scale. Inspect the stellar pupil before the stop.
What to observe: Focal disk is opaque; Lyot stop is a pupil aperture. They occupy different conjugate planes.
02
Scene 2: Block the core: halo survives
Drag the planet inward. Change both masks, comparing throughput and contamination.
What to observe: Total throughput includes entrance losses, referred to clear-pupil incident power.
03
Scene 3: Lyot stop: rejection costs light
Add obstruction or apodization, then pointing and defocus. Clear the wavefront to separate error leakage from pupil design.
What to observe: Mask edges are quantized. Finite-grid suppression is not a certified extreme-contrast prediction.