Optics 031 · Imaging, instruments, and visual systems

Build a Telescope

Four telescope layouts couple exact opaque-mirror reflections, an open Cassegrain primary, annular diffraction and a fixed-field binary-star comparison.

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 $\text{valid model regime}$

On-axis ideal geometry and monochromatic scalar diffraction. Seeing, spider spikes, real eyepiece fields, 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 redistributes light from the core into rings according to $I(q)=\left[\frac{2[J_1(q)-\epsilon J_1(\epsilon q)]}{q(1-\epsilon^2)}\right]^2$ . A smaller central core does not by itself demonstrate better image contrast; both the halo and the lost collecting area matter.

Core mathematical model

Afocal magnification and pupil diameter

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

Eyepiece focal length is signed: negative for Galilean and positive for Keplerian. The diameter relation is paraxial; the Galilean exit pupil is virtual. Shortening the eyepiece increases apparent size without narrowing the sky-space diffraction PSF.

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 number is an unobstructed circular-pupil reference. For the annular pupil, inspect the computed two-star image rather than treating that number as the actual resolving threshold. Clear area excludes the secondary shadow but omits coatings and spider losses.

Actual mirror reflection

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

Every reflected segment starts at a point on the displayed surface and uses its computed normal. Newtonian folds the converging beam at an opaque tilted flat. Cassegrain returns through the primary opening; there is no physical transmitted branch 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}

The hyperboloid shares the primary focus and the final rear focus. Its normal converts a beam converging toward the first into one converging toward the second. Secondary obstruction changes vertex position in the stated family, hence changes effective focal length as well as 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 changes angular scale only. Obstruction shifts energy into rings and lowers throughput. Compare identical sky fields, equal apertures and the linear intensity profile; the picture cards use square-root brightness and individual peak normalization.

Run the experiment

01
Scene 1: Afocal refractor
Choose the refractor preset, then select layout zero or one for Galilean or Keplerian. Follow whether rays cross before the eyepiece, and compare the sign of angular magnification.
What to observe: Only Keplerian forms a real intermediate focus between the objective and eyepiece; Galilean intercepts it early.
02
Scene 2: Newtonian reflector
Open Newtonian and orbit the bench. Follow cyan rays to the primary, amber rays to the tilted secondary and green rays through the side eyepiece. Pink blocked rays stop at the secondary.
What to observe: Doubling aperture doubles angular resolving scale improvement and quadruples collecting area at fixed obstruction.
03
Scene 3: Confocal Cassegrain
Open Cassegrain, vary obstruction, aperture and binary separation. Check that the mirror and hole geometry remain consistent with the folded rays, then compare the actual image with its same-diameter unobstructed reference.
What to observe: The reflected beams reverse axial direction at each opaque mirror. A Cassegrain rear focus is reached through a hole, never by drawing light through solid mirror material.