Geometrical optics
When the wavelength is much smaller than the objects, light can be treated as rays that travel in straight lines. On that approximation the lens, the image sensor and the optical fiber are built.
01When light can be treated as rays
Light is an electromagnetic wave, but when the objects it meets are much larger than its wavelength —half a micrometer for visible light— it behaves as if it traveled in straight-line rays. That approximation is geometrical optics, and it is used to design lenses, mirrors, cameras and fibers.
When that condition is not met, diffraction and interference appear, which already belong to physical optics. The limit is not arbitrary: it is what sets the maximum resolution of a microscope and the minimum size of a transistor made by photolithography.
It is how many times slower light travels in that medium than in vacuum. Air 1.0003; water 1.33; ordinary glass 1.5; diamond 2.42. Always n ≥ 1.
02Reflection and refraction
| Law | Statement |
|---|---|
| Reflection | The angle of incidence equals the angle of reflection, measured from the normal |
| Refraction (Snell) | \( n_{1}\cdot \operatorname{sin} \theta _{1} = n_{2}\cdot \operatorname{sin} \theta _{2} \) |
On entering a denser medium light bends toward the normal; on leaving into a less dense one, it bends away. And if the angle of incidence exceeds a certain value, the light no longer exits: it is all reflected. That is the critical angle:
A fiber is a glass core with an index only slightly higher than that of the cladding around it. Every ray that enters within the acceptance cone strikes the wall above the critical angle and is totally reflected, again and again, kilometer after kilometer, with almost no loss of energy. This is what makes fiber transmission possible.
03Mirrors and thin lenses
A thin lens or a spherical mirror is described by a single equation and a single definition of magnification:
Three rays are enough to construct the image of any point, and two already suffice:
- The one that arrives parallel to the axis leaves passing through the focal point.
- The one that passes through the center of the lens is not deflected.
- The one that passes through the object focal point leaves parallel to the axis.
Move the object and see how the image changes. The three rays are drawn automatically and the equation is solved alongside.
04What image results in each case
| Object at | Image | Application |
|---|---|---|
| Beyond 2f | Real, inverted and smaller | Camera |
| At 2f | Real, inverted and the same size | 1:1 copier |
| Between f and 2f | Real, inverted and larger | Projector |
| At the focal point | None forms: the rays leave parallel | Collimator, headlight |
| Closer than f | Virtual, upright and larger | Magnifying glass |
| Diverging lens | Always virtual, upright and smaller | Myopia correction, viewfinder |
A real image forms where the light actually converges: it can be captured on a screen or a sensor. A virtual image is located where the rays appear to come from when extended backward; it can be seen with the eye, but it cannot be projected. That is why a magnifying glass does not work as a projector.
The power of a lens is measured in diopters, D = 1/f with f in meters. An eyeglass prescription of −2.5 diopters is a diverging lens with a 40 cm focal length.
05Instruments and aberrations
- Camera. A lens forms the real image on the sensor; focusing means moving the lens so that s′ falls exactly on it. The f-number —the relative aperture— controls how much light enters and the depth of field.
- Magnifier and microscope. They increase the angle under which an object is seen.
- Telescope. An objective of long focal length and an eyepiece of short focal length: the magnification is their ratio.
- The eye. A lens of variable focal length in front of a fixed screen, the retina. Myopia is an eye that is too long, and it is corrected with a diverging lens.
No real lens is perfect. Spherical aberration makes rays far from the axis focus at a different point; chromatic aberration makes each color focus differently because n depends on λ; and distortion bends the straight lines at the edge. Good lenses combine several elements of different glasses to compensate for them, and the rest is corrected by digital image processing.
06Optics in electronics
- Image sensors. A CMOS sensor needs a lens that forms a real image on it: without geometrical optics there is no camera.
- Optical fiber. Numerical aperture, acceptance cone and modal dispersion all follow from Snell’s law.
- Optical sensors. Photoelectric gates, encoders, barcode readers and proximity sensors.
- Photolithography. Integrated circuits are made by projecting a mask onto a wafer: the minimum size is set by the wavelength used.
- Displays and projectors. Lenses, mirrors and diffusers calculated with these same rules.
- Lidar and time-of-flight measurement. A laser pulse, receiving optics and a fast photodiode.
07In the lab
Focus the image of a distant object onto a screen: the lens-to-screen distance is f. Then repeat with the object at several distances and verify 1/f = 1/s + 1/s′ with a table of values.
Shine a laser from inside a container of water onto the surface, varying the angle until the refracted ray disappears. Compare the measured angle with arcsin(1/1.33) = 48.8°.
Build a gate with an infrared LED and a phototransistor, and use a slotted disk to count revolutions with a microcontroller. It is the most direct application of rectilinear propagation.
08Common mistakes
- Measuring angles from the surface instead of from the normal.
- Forgetting the sign convention and mixing real images with virtual ones.
- Believing that total reflection occurs when passing into a denser medium. It is the other way around: from the denser to the less dense.
- Confusing magnification with power. Magnification has no units; power is measured in diopters.
- Thinking that covering half a lens cuts half the image. The whole image is formed, but with half the light.
- Using geometrical optics where it does not apply, for example with apertures of the order of the wavelength.
09Self-assessment
An object is 15 cm from a converging lens with f = 6 cm. Where does the image form?
1/s′ = 1/6 − 1/15 = 0.1, so s′ = 10 cm, with m = −10/15 = −0.67: real, inverted and smaller.
What is the critical angle between glass (n = 1.5) and air?
θc = arcsin(1/1.5) ≈ 41.8°.
What focal length does a −2.5 diopter lens have?
f = 1/(−2.5) = −0.4 m: a 40 cm diverging lens.
Why does a magnifying glass form a virtual image?
Because the object is closer than the focal length: the rays leave diverging and the image is seen on the same side as the object, upright and enlarged.
Light passes from air into water with a 30° angle of incidence. What is the refracted angle?
sin θ₂ = sin30/1.33 = 0.376, so θ₂ ≈ 22.1°: it bends toward the normal.
Why does an optical fiber not lose light through its walls?
Because the rays strike above the critical angle between core and cladding, and total internal reflection returns them without any loss by refraction.
10Further reading
- Eugene Hecht. Optics. 5th ed., Pearson, 2017 (Spanish edition, “Óptica”). The reference text; chapters 4 and 5 cover all of geometrical optics.
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics, Volume 2. 9th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage Learning, 2014. Chapters 35 and 36, at the level of the course and with a well-explained sign convention.
- Gerd Keiser. Optical Fiber Communications. 5th ed., McGraw-Hill, 2015. From Snell to real fiber links: the bridge to telecommunications.
- Richard P. Feynman. The Feynman Lectures on Physics, Volume I. Addison-Wesley, 1963 (Spanish edition, “Física, volumen I”, Addison-Wesley / Pearson). Chapter 26 derives reflection and refraction from Fermat’s principle of least time.