Optics 004 · Ray worlds, boundaries, and natural optics

Curved Mirror Observatory

How does mirror shape move the focus and change off-axis image quality? The curvature at the mirror's center sets where rays near the axis focus. A paraboloid sends every ray parallel to the axis to one point. A sphere focuses its outer rays closer to the mirror, so the focal spot smears out. How well the mirror images off-axis points depends on its exact conic shape.

A labeled cutaway of a conic mirror keeps the vertex radius of curvature separate from the clear aperture. The conic mesh, a ruler to the center of curvature, the focus marker, a parallel source and detector samples show each change in space.

Mean axis crossing

\bar x_f

$0.50$

Longitudinal spread

\sigma_x

$50\%$

Curvature-radius focus

f_{\mathrm{paraxial}}

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Curved Mirror Observatory

BackgroundThe mirror is a conic of revolution, defined by its sag: how far the surface dips at each distance from the axis. The vertex radius of curvature and the clear aperture are set independently, and the surface follows $z(r)=\frac{r^2}{R\left(1+\sqrt{1-(1+K)r^2/R^2}\right)}$ . Each ray reflects about the normal given by the local slope. This keeps the exact geometric trace separate from the paraxial thin-mirror reference.

Why it mattersHow does mirror shape move the focus and change off-axis image quality?

Start with the essentials

Focus question: How does mirror shape move the focus and change off-axis image quality?
One-sentence intuition: A paraboloid sends rays parallel to its axis to a single focus. On a sphere, where a ray crosses the axis depends on how far from the axis it lands. The detector therefore reports the spot width $\sigma_x$ ; a single nominal focal length would hide the spread of the marginal rays.

Core mathematical model

Conic sag

z(r)=\frac{r^2}{R\left(1+\sqrt{1-(1+K)r^2/R^2}\right)}

The vertex radius and the conic constant set the drawn surface and its slope at every point.

Paraxial focal length of a spherical mirror

f_{\mathrm{paraxial}}=\frac{R}{2}

This value holds for a spherical mirror with a small aperture. Marginal and off-axis rays still have to be traced.

Common difficulties

Applying the mirror equation to every ray

Typical misconceptionAll rays from a spherical mirror must cross at the paraxial focal length.

Better mental modelThe mirror equation is a paraxial approximation. Reflecting each ray about the true surface normal shows that the marginal rays cross elsewhere: this is longitudinal spherical aberration.

Run the experiment

01
Scene 1: Spherical conjugates
Keep the clear aperture fixed and reduce the vertex radius of curvature; watch the same opening become visibly deeper.
What to observe: With the aperture fixed, a smaller radius of curvature makes the sag larger. On the sphere, the marginal rays also cross the axis closer to the mirror than the paraxial focus.
02
Scene 2: Parabolic focus
Switch to the parabolic scene without changing the vertex radius.
What to observe: Rays parallel to the axis meet at the parabolic focus to within numerical precision.
03
Scene 3: Off-axis conic spot
Use the off-axis conic scene and inspect the three-dimensional detector spot.
What to observe: Off-axis light lands on the detector as a spread-out spot, not at a single focus.