Astronomy · From collecting light to recording data

Mirrors, lenses and image formation

Follow ideal rays from one distant source direction to a focus, then build an image from many directions while retaining the limitations of geometric optics.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

Back to the chapter

After reading, you can

Explain how refraction and reflection can concentrate collected radiation.
Interpret a focus and focal plane for a distant source.
Distinguish an ideal ray construction from a real telescope’s finite image response.

A focus concerns rays from one source point

For a sufficiently distant compact source, rays arriving across a small telescope aperture are approximately parallel for the purpose of geometric imaging. An optical system can redirect those rays toward a small image region. In an ideal ray model, their common meeting point is a focus.

That statement refers to one source direction. An extended sky contains many directions, and an imaging system sends different nearby directions toward different image locations. An image is built from that mapping; it is not all the light from the whole sky squeezed into one point.

The focal plane is the image plane near which a detector can record that directional pattern for the intended field. The effective focal length relates small input angles to image positions. In a simple near-axis model, an angle θ in radians produces an image displacement approximately fθ, where f is the effective focal length.

Real optics can depart from a flat ideal focal plane over a wide field. The word “plane” in the introductory model does not promise perfect sharpness for every field angle, wavelength or detector placement.

Lenses refract; mirrors reflect

A converging lens uses refraction at its surfaces to redirect radiation. A thin-lens ray model replaces the finite lens with an ideal element and uses small angles near the optical axis. For a distant on-axis source, parallel input rays then converge a focal length behind that element.

A concave reflecting system redirects rays through reflection. An ideal paraboloidal mirror brings on-axis parallel geometric rays to one focus. A spherical surface does not produce that same exact all-aperture focus; near-axis small-angle approximations can hide its spherical aberration.

The figure’s mirror construction therefore uses a true parabolic section, not a circular arc with rays forced toward a chosen point. Incoming light approaches the mirror from the open side and reflects back toward the focus. That change of direction is a physical distinction from transmission through the lens.

The two panels are ideal models for one distant source direction. The lens silhouette is a symbol for a paraxial thin element. The mirror is an ideal on-axis parabolic geometric construction. Neither panel specifies a complete telescope package, obstruction, coating or finite-wave response.

Two ways to form an image point

Scroll sideways to see the full diagram. Its explanation is below.

Original ideal geometric diagrams for one distant on-axis source direction. In both panels, x and y have the same drawing scale of 2.2 pixels per model unit; all model distances are drawing units, not a real instrument prescription. Panel A is a thin converging lens at x = 0 with focal length f = 100. Incoming parallel rays travel from the left in the positive x direction at y = −10, 0 and +10. The linear paraxial model changes their slopes to −y/100; each passes through the model focus (100, 0) and continues to x = 120. The lens outline is a symbol for an ideal thin element, not a finite-thickness ray trace. Panel B is the parabolic section x = y²/400 for −80 ≤ y ≤ 80, opening to the right, with vertex (0, 0) and focus (100, 0). Incoming rays approach from the right in the negative x direction and strike (16, −80), (0, 0) and (16, +80). Specular reflection sends every drawn ray toward the focus on the open right-hand side. The central incoming and reflected ray share one line but have opposite arrow directions. The surface is a true parabola, not a circular arc. Blue incoming paths and orange redirected paths are also distinguished by labels and arrow directions. The focus is marked only as a location; no detector package blocks incoming light. Each panel describes one source direction, while an image maps different directions to different locations. Real finite-aperture images have diffraction and may have aberrations, atmosphere and sampling effects; a ray intersection is not a zero-size recorded star. No off-axis, broadband or complete telescope performance is implied.

The distant-source limit simplifies a more general relation

For a simple thin converging lens with a real object and real image, one commonly used positive-distance convention gives 1/f = 1/do + 1/di. Here do is object distance from the ideal lens and di is image distance. This convention and approximation must be retained when using the equation.

As do becomes very large compared with f, 1/do becomes small and di approaches f. That is why a distant-object focus is located approximately one focal length from the ideal lens. A nearby object can require a different image distance.

For an invented f = 100 mm lens and do = 1000 mm object, 1/di = 1/100 − 1/1000 = 0.009 mm⁻¹. Thus di is about 111.1 mm, farther from the ideal element than its distant-source focal position. This is a thin-lens calculation, not a recipe for building or using an observing instrument.

The existing Physics Foundations records provide the thin-lens and mirror sign conventions for readers who want to explore more cases. Mixing conventions or applying this simple relation to a complex optical train without defining its effective quantities can produce misleading results.

A ray intersection is not a zero-size recorded star

Diffraction gives even ideal finite-aperture wave optics a finite point response. Aberrations, wavelength dependence, atmosphere, focus and detector sampling can broaden or distort the recorded pattern further. Rays are a useful geometrical model, not the complete behaviour of electromagnetic waves.

A lens’s refractive response can depend on wavelength; real designs manage chromatic and other aberrations through their materials and element arrangement. Reflecting designs have different advantages and limitations, including coatings, field aberrations and possible obstructions. No single schematic establishes perfect performance across the spectrum.

A detector near the intended image surface samples the delivered pattern. Increasing focal length spreads a given small angular interval over a larger physical distance in the focal plane. It does not by itself increase collecting area or guarantee finer physical information.

The following detector lesson follows that image into a numerical record. Later resolution lessons examine what sets the smallest distinguishable structure. Keeping these steps separate helps explain why a telescope is an observing system rather than just a magnifying picture.

Worked examples

Locate the image in the declared thin-lens convention

For the synthetic thin lens with f = 100 mm and real object distance do = 1000 mm, calculate the real image distance.

Use 1/di = 1/f − 1/do.
The difference is 0.010 − 0.001 = 0.009 mm⁻¹.
Invert to obtain di ≈ 111.1 mm.

About 111.1 mm from the ideal lens on its image side, within the stated positive-distance thin-lens model.

Relate a small angle to focal-plane scale

In a near-axis ideal model, f = 1000 mm and an input angular offset is 0.001 rad. What approximate focal-plane displacement follows?

Use the small-angle relation y ≈ fθ.
Multiply 1000 mm by 0.001.
The result is 1 mm; the angle is in radians.

About 1 mm from the on-axis image location. The calculation specifies image scale, not a resolving-power guarantee.

Try the reasoning

Does an imaging telescope put every sky direction at the same focus point?
Show the answer and reasoning
No.
Different directions map to different image locations. The common-point construction describes one direction.
May an exact wide-aperture on-axis parabolic focus be illustrated with an unlabeled spherical arc and forced ray paths?
Show the answer and reasoning
No.
Those geometries differ. A spherical mirror needs its stated small-angle approximation or its aberration retained.
Does a geometric ray intersection prove that a real star image is a mathematical zero-size point?
Show the answer and reasoning
No.
Diffraction and additional optical, atmospheric and detector effects produce a finite response.
If effective focal length doubles while entrance aperture stays fixed, has collecting area doubled?
Show the answer and reasoning
No.
Focal length changes image scale; entrance geometry determines geometric collecting area.

What to carry forward

A focus groups radiation from one source direction; many directions form an image.
Thin-lens and parabolic-mirror ray models have different constructions and declared limits.
Focal length sets image scale, while real detail also depends on finite-aperture response, optics, atmosphere and sampling.

References

Teaching resources and research supporting this lesson’s explanations and examples.

OpenStax, Rice University3 sources