Comprehensive theory, key formulas, diagrams, and memory aids for Optics.
Refraction is the change in direction of a wave when it crosses the boundary between two media in which it travels at different speeds. It occurs because different parts of the wavefront change speed at different times.
Snell's Law quantifies refraction at a boundary: $$n_1 \sin\theta_1 = n_2 \sin\theta_2$$
The refractive index ($n$) of a medium is the ratio of the speed of light in a vacuum ($c$) to the speed of light in the medium ($v$): $$n = \frac{c}{v}$$
Since $c > v$ in all materials, $n \geq 1$ always. Higher refractive index means light slows down more in that medium. The refractive index of vacuum = 1.000, air ≈ 1.0003 (treated as 1 in calculations), water = 1.33, glass ≈ 1.5, diamond = 2.42.
When light moves from a medium of refractive index $n_1$ to $n_2$, the ratio: $$\frac{\sin\theta_1}{\sin\theta_2} = \frac{n_2}{n_1} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2}$$
Note: the frequency of light does not change on refraction — only wavelength and speed change.
When light travels from a denser medium (higher $n$) to a less dense medium (lower $n$), and the angle of incidence exceeds the critical angle $\theta_c$, all the light is reflected back into the denser medium. This is total internal reflection (TIR).
The critical angle satisfies: $$\sin\theta_c = \frac{n_2}{n_1} = \frac{1}{n}$$ (if medium 2 is air, $n_2 = 1$)
For glass ($n = 1.5$): $\theta_c = \sin^{-1}(1/1.5) = 41.8°$ For diamond ($n = 2.42$): $\theta_c = 24.4°$ — explains why diamonds sparkle so brilliantly.
Applications of TIR: - Optical fibres: Light travels along a glass fibre by repeated TIR at the glass–cladding interface. The cladding has lower $n$ than the core. Used in telecommunications (data transmission as light pulses) and endoscopy. - Bicycle reflectors and road markings - Periscopes and binoculars (using prisms) - Endoscopes: Flexible bundles of optical fibres that allow surgeons to image inside the body.
Dispersion in optical fibres: Different wavelengths travel at slightly different speeds in glass (different refractive indices). This causes pulse broadening over long distances, limiting data transmission rates. Step-index fibres have more dispersion than graded-index fibres.
A converging (convex) lens brings parallel rays together at the principal focus $F$. The focal length $f$ is the distance from the optical centre to $F$.
The thin lens equation: $$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$$
or using the real-is-positive convention: $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$
Where $u$ = object distance, $v$ = image distance. Sign convention varies by syllabus — always state which convention you use.
Magnification $m = \frac{v}{u} = \frac{\text{image height}}{\text{object height}}$
Power of a lens: $P = \frac{1}{f}$, measured in dioptres (D). Converging lenses have positive power; diverging lenses have negative power.
graph TD
A[Converging Lens] --> B{Object position}
B -->|Beyond 2F| C[Real, inverted, diminished image]
B -->|At 2F| D[Real, inverted, same size image]
B -->|Between F and 2F| E[Real, inverted, magnified image]
B -->|At F| F[Image at infinity - no image formed]
B -->|Inside F| G[Virtual, upright, magnified image]
A diffraction grating consists of thousands of equally spaced parallel slits. When monochromatic light passes through, each slit acts as a source of secondary wavelets. Constructive interference produces bright diffraction maxima at angles given by: $$d\sin\theta = n\lambda$$
Where: - $d$ = grating spacing = $\frac{1}{N}$ (where $N$ = number of lines per metre) - $\theta$ = angle of diffraction - $n$ = order of diffraction (0, ±1, ±2, ...) - $\lambda$ = wavelength of light
The zeroth order ($n=0$) is always straight ahead ($\theta = 0$). Higher orders appear at increasingly large angles.
Advantages over double slit: The diffraction grating produces much sharper, brighter maxima, making it ideal for accurate measurement of wavelengths. Used in spectrometers to analyse the emission spectra of elements, identifying them by their unique wavelength signatures.
All electromagnetic (EM) waves travel at $c = 3.00 \times 10^8$ m s⁻¹ in a vacuum. They are transverse waves consisting of oscillating electric and magnetic fields perpendicular to each other and to the direction of travel.
| Type | Wavelength Range | Typical Source |
|---|---|---|
| Radio | >0.1 m | Radio transmitters |
| Microwave | 1 mm – 0.1 m | Magnetrons, mobile phones |
| Infrared | 700 nm – 1 mm | Hot objects, IR LEDs |
| Visible | 400 – 700 nm | LEDs, lamps, stars |
| Ultraviolet | 10 – 400 nm | Sun, UV lamps |
| X-rays | 0.01 – 10 nm | X-ray tubes |
| Gamma rays | <0.01 nm | Radioactive nuclei |
The photon model treats light as discrete packets of energy: $$E = hf = \frac{hc}{\lambda}$$
Where $h = 6.63 \times 10^{-34}$ J s is Planck's constant.
The intensity of a beam equals the number of photons per second per unit area multiplied by the energy per photon: $$I = \frac{nhf}{At}$$
Intensity is proportional to the square of the amplitude for wave behaviour. In the photon model, higher intensity means more photons per second.
Light from ordinary sources vibrates in all planes perpendicular to the direction of travel. A polaroid filter transmits only the component of electric field oscillation in one plane (the transmission axis), producing plane-polarised light.
When a second polaroid (analyser) is placed after the first: - At 0° (parallel): Maximum transmission. - At 90° (crossed): Zero transmission. - At angle $\theta$: Intensity $I = I_0\cos^2\theta$ (Malus's Law).
Polarisation is used in: LCD screens, camera filters, stress analysis of materials (photoelasticity), and 3D cinema glasses.