Optics 031 · Imaging, instruments, and visual systems

Build a Telescope

How do aperture, optical layout, and obstruction determine what a telescope can resolve? The aperture diameter sets the finest detail through the diffraction limit: double it and the smallest resolvable angle halves. The layout sets the focal length and the actual path between the mirrors. A central obstruction, such as a secondary mirror, moves light out of the central spot and into the rings around it, so a smaller central spot does not by itself mean better contrast.

Compare four telescope layouts with exact reflections from opaque mirrors, a Cassegrain primary with a central hole, diffraction from an annular pupil and a binary star viewed in a fixed field.

Angular magnification and pupil diameter

M_\theta, D_{\mathrm{exit}}

$0.50$

Unobstructed reference and binary separation

\theta_{R,0}, \Delta\theta

$50\%$

Clear collecting area and obstruction

A_{\mathrm{clear}}, \epsilon

$0.00\pi$

Model regime $\text{valid model regime}$

The model uses ideal on-axis geometry and monochromatic scalar diffraction. Atmospheric seeing, diffraction spikes from the spider, the fields of real eyepieces, off-axis aberrations and coating losses are not simulated.

Start with the essentials

Focus question: How do aperture, optical layout, and obstruction determine what a telescope can resolve?
One-sentence intuition: A central obstruction moves light out of the central core and into the rings, following $I(q)=\left[\frac{2[J_1(q)-\epsilon J_1(\epsilon q)]}{q(1-\epsilon^2)}\right]^2$ . A smaller core alone does not prove better image contrast; the halo of rings and the lost collecting area both count.

Core mathematical model

Afocal magnification and pupil diameter

M_\theta=-\frac{f_o}{f_e},\qquad D_{\mathrm{exit}}=\frac{D}{|M_\theta|}

The eyepiece focal length has a sign: negative for Galilean, positive for Keplerian. The pupil-diameter relation is paraxial, and the Galilean exit pupil is virtual. A shorter eyepiece makes things look bigger, but it does not narrow the diffraction PSF measured in angle on the sky.

Unobstructed reference and clear area

\theta_{R,0}=1.22\frac{\lambda}{D},\qquad A_{\mathrm{clear}}=\frac{\pi D^2}{4}(1-\epsilon^2)

The Rayleigh value is a reference for an unobstructed circular pupil. For the annular pupil, look at the computed image of the two stars; do not treat that number as the real resolving limit. The clear area subtracts the shadow of the secondary but leaves out losses from coatings and the spider.

Reflection at the real mirror surfaces

\hat{\mathbf k}_{\mathrm{out}}=\hat{\mathbf k}_{\mathrm{in}}-2(\hat{\mathbf k}_{\mathrm{in}}\cdot\hat{\mathbf n})\hat{\mathbf n}

Each reflected segment starts at a point on the drawn surface and uses the normal computed there. The Newtonian folds the converging beam with an opaque tilted flat. The Cassegrain sends the light back through the opening in the primary. No light passes through either mirror.

Confocal Cassegrain geometry

x_p=p-\frac{r^2}{4f_1},\qquad x_s=c-a\sqrt{1+\frac{r^2}{b^2}},\qquad f_{\mathrm{eff}}=f_1\frac{F_2-v_s}{v_s-F_1}

One focus of the hyperboloid sits at the focus of the primary, and the other at the final focus behind it. Reflection from its surface turns a beam converging toward the first focus into one converging toward the second. In this family of designs, changing the secondary obstruction moves the vertex of the secondary, so it changes the effective focal length as well as the diffraction.

Common difficulties

More magnification or a narrower core means more detail

Typical misconceptionA shorter eyepiece or larger central stop always resolves a closer binary.

Better mental modelThe eyepiece only changes the angular scale. An obstruction moves energy into the rings and lets less light through. Compare the same sky field at equal aperture and read the linear intensity profile. The picture cards show the square root of brightness, each scaled to its own peak.

Run the experiment

01
Scene 1: Afocal refractor
Choose the refractor preset, then set the layout to zero for Galilean or one for Keplerian. Check whether the rays cross before the eyepiece, and compare the sign of the angular magnification.
What to observe: Only the Keplerian forms a real intermediate focus between the objective and the eyepiece; the Galilean eyepiece catches the beam before that focus.
02
Scene 2: Newtonian reflector
Open Newtonian and orbit the bench. Follow the cyan rays to the primary, the amber rays to the tilted secondary and the green rays through the side eyepiece. Pink rays are blocked and stop at the secondary.
What to observe: With the obstruction fixed, doubling the aperture halves the angular resolution scale and gives four times the collecting area.
03
Scene 3: Confocal Cassegrain
Open Cassegrain and vary the obstruction, aperture and binary separation. Check that the mirrors and the hole always match the folded ray paths, then compare the actual image with an unobstructed reference of the same diameter.
What to observe: At each opaque mirror the beam reverses its direction along the axis. Light reaches the Cassegrain rear focus through a hole; it is never drawn passing through solid mirror material.