Lenses and Dispersion
The W. M. Keck Observatory on Mauna Kea, Hawaii reached full operation in 2003 with its 10 m segmented primary mirror — the world's largest at the time. Its diffraction limit is $7.2 \times 10^{-8}$ rad (0.015 arcseconds). The telescope's glass has $n_{violet} = 1.530$ and $n_{red} = 1.515$: that difference of 0.015 is enough to cause chromatic aberration if uncorrected, which is precisely why Keck uses a mirror (not a lens) as its primary — mirrors have zero chromatic aberration regardless of wavelength.
A convex lens and a concave lens both bend light. Which one converges parallel rays to a single point and which one diverges them? Predict and sketch or describe in words.
Warm-up — dispersion of light through a prism occurs because:
Core Content
Hold a magnifying glass (convex lens) up to sunlight and move a piece of paper toward and away from it: at one particular distance the sunlight converges to a bright, hot spot on the paper that can scorch it — that is the focal point. Now hold the same lens in front of your eye and look at text through it: the text appears larger and upright. These two behaviours — real convergence and virtual magnification — both come from the same lens, just by changing where the object is relative to the focal point.
| Property | Convex (converging) | Concave (diverging) |
|---|---|---|
| Shape | Thicker in middle | Thinner in middle |
| Parallel rays | Converge to focal point $F$ | Diverge (appear to come from $F$) |
| Image type | Real or virtual (depends on object distance) | Always virtual, upright, smaller |
| Applications | Camera, magnifying glass, projector, eye | Spectacles for short-sightedness |
Convex (converging) lens: parallel rays converge to focal point $F$; forms real images when object is beyond $F$, virtual images inside $F$. Concave (diverging) lens: parallel rays diverge; always forms a virtual, upright, smaller image regardless of object distance.
Pause — write the highlighted lens image rules into your book before moving on.
We just saw that convex and concave lenses differ in how they converge or diverge light. That raises a question: if white light passes through a glass prism, why doesn't it just refract as a single beam — why does it split into colours? This card answers it → glass has a slightly different refractive index for each wavelength, so different colours bend by different amounts (dispersion).
The refractive index of glass is not the same for all wavelengths. Violet light has the highest $n$ (bends the most); red light has the lowest $n$ (bends the least). This difference is called dispersion.
In a triangular prism, white light is split into the spectrum (ROYGBIV) on exiting. In raindrops, white sunlight is dispersed and reflected internally, producing a rainbow. The observer sees violet at the inner arc and red at the outer arc.
Dispersion: refractive index $n$ varies with wavelength in glass — violet has highest $n$ and bends most; red has lowest $n$ and bends least. In a prism or raindrop, white light separates into a spectrum (ROYGBIV). Chromatic aberration in lenses is caused by dispersion.
Add the highlighted dispersion rule to your notes before the check below.
A concave (diverging) lens always produces an image that is:
Violet light refracts more than red light when entering glass because it has a higher refractive index.
A convex lens always produces a real image.
Activities
Describe the image formed in each case (type, orientation, relative size):
- Object at 3f from a convex lens ($f = 10$ cm)
- Object at $f/2$ from a convex lens ($f = 10$ cm)
- Any object position in front of a concave lens
Explain the formation of a rainbow. Your answer should refer to: refraction on entry, total/partial internal reflection, refraction on exit, and dispersion. State which colour appears on the outside of the arc.
Match each application to the correct lens type and explain why:
- Spectacles for short-sightedness (myopia)
- A camera objective lens
- A magnifying glass
Which of these does NOT demonstrate dispersion?
In a rainbow, which colour appears at the outer edge of the arc?
Chromatic aberration in a lens is caused by:
UnderstandBand 3(3 marks) 1. Distinguish between a convex and concave lens in terms of how they affect parallel rays and the type of image they can produce.
ApplyBand 4(3 marks) 2. Explain why a glass prism separates white light into a spectrum. In your answer, state which colour bends the most and why.
AnalyseBand 5(4 marks) 3. Explain chromatic aberration in a refracting telescope and how an achromatic doublet lens corrects it. Use the word "dispersion" in your answer.
Show all answers
Short Answer — Model Answers
Q1 (3 marks): Convex (converging): thicker at the centre; parallel rays are refracted toward the principal axis and converge at the focal point $F$. Can form real (inverted) or virtual (upright, magnified) images depending on object position. Concave (diverging): thinner at centre; parallel rays are refracted away from axis, appearing to diverge from the focal point. Always forms virtual, upright, smaller images regardless of object position.
Q2 (3 marks): White light contains all visible wavelengths (colours). Glass has slightly different refractive indices for each wavelength — this is dispersion. Violet light ($n \approx 1.343$) refracts more than red ($n \approx 1.331$) because it slows more in glass. On passing through a prism with two refractions, the angular separation of colours builds up, producing a visible spectrum. Violet bends most, red bends least.
Q3 (4 marks): In a single converging lens, dispersion causes violet light to focus at a shorter distance than red light. Different colours focus at different points, blurring the image with coloured fringes — chromatic aberration. An achromatic doublet uses a convex crown glass lens combined with a concave flint glass lens. Crown glass has less dispersion; flint glass has more. The flint glass element introduces an opposite dispersion that partially cancels the crown glass dispersion, bringing red and violet to approximately the same focal length and greatly reducing chromatic aberration.
The Keck Observatory (Mauna Kea, 2003) crystallises the core insight of this lesson: its glass has $n_{violet} = 1.530$ and $n_{red} = 1.515$, a difference of 0.015. That tiny difference means a refracting lens of 10 m aperture would focus violet light at a noticeably shorter distance than red light — chromatic aberration — blurring every astronomical image. Keck uses a 10 m mirror instead (diffraction limit $7.2 \times 10^{-8}$ rad = 0.015 arcsec), because mirrors reflect all wavelengths identically. Convex lenses converge to a focal point; concave always diverge — and the concave never produces a real image.