Geometric and Physical Optics

Optics carries about 4 questions (our estimate). The two halves are ray optics (mirror and lens formulae, refraction, prism, total internal reflection) and wave optics (Young's fringe width, grating, Brewster's law, Malus' law). Almost every numerical question uses one of five formulae.

3.1 Reflection

Laws: the incident ray, reflected ray and normal are coplanar; \angle i=\angle r. A plane mirror forms a virtual image as far behind as the object is in front; rotating the mirror by \theta rotates the reflected ray by 2\theta; two mirrors at angle \theta form \frac{360^\circ}{\theta}-1 images. Minimum mirror length to see oneself fully is half one's height.

Spherical mirrors (Cartesian sign convention: distances measured from the pole, positive in the direction of the incident light; concave mirror has negative f, R): \dfrac{1}{v}+\dfrac{1}{u}=\dfrac{1}{f}=\dfrac{2}{R}, m=-\dfrac{v}{u}=\dfrac{h_i}{h_o}. With the "real-is-positive" shortcut for a concave mirror: object at u=30 cm with f=20 cm gives \frac{1}{v}=\frac{1}{20}-\frac{1}{30}, v=60 cm real, inverted, m=-2. Convex mirror always gives a virtual, erect, diminished image (rear-view mirrors, wide field).

3.2 Refraction

Snell's law \dfrac{\sin i}{\sin r}={}_1n_2=\dfrac{n_2}{n_1}=\dfrac{v_1}{v_2}; n=\frac{c}{v}; the frequency is unchanged, so \lambda_{medium}=\frac{\lambda}{n}. Apparent depth =\frac{\text{real}}{n} (a 12 cm column of water, n=\frac{4}{3}, looks 9 cm). Lateral shift through a slab of thickness t: d=t\sin(i-r)/\cos r.

Critical angle: \sin C=\dfrac{1}{n} (glass 1.5: 41.8^\circ; n=\sqrt{2}: 45^\circ; diamond 2.42: 24^\circ). Total internal reflection needs light going from the denser to the rarer medium at i>C. Applications: prisms in binoculars, mirages, sparkle of diamonds, endoscope, and the optical fibre (core n_1> cladding n_2): numerical aperture \text{NA}=\sqrt{n_1^2-n_2^2} (core 1.5, cladding 1.4: 0.54); used in telecommunication with very low loss.

Prism: \delta=i+e-A; minimum deviation when the ray passes symmetrically, n=\dfrac{\sin\frac{A+\delta_m}{2}}{\sin\frac{A}{2}}; for a thin prism \delta=(n-1)A. For A=60^\circ, n=\sqrt{3}: \delta_m=60^\circ.

Thin lens: \dfrac{1}{v}-\dfrac{1}{u}=\dfrac{1}{f} (Cartesian), m=\dfrac{v}{u}; lens maker's formula \dfrac{1}{f}=(n-1)\Big(\dfrac{1}{R_1}-\dfrac{1}{R_2}\Big) (for n relative to the surrounding medium: a glass lens in water has longer focal length, f_w=f_a\frac{(n_g-1)}{(n_g/n_w-1)}; f_{water}\approx4f_{air}). Power P=\frac{1}{f} (metres) in dioptres; lenses in contact P=P_1+P_2, \frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2}; separated by d: \frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2}-\frac{d}{f_1f_2}. A convex lens gives a real image with m=-1 at u=2f. Eye defects: myopia (concave lens), hypermetropia (convex lens). Microscope M\approx\frac{v_o}{u_o}\cdot\frac{D}{f_e}, telescope M=\frac{f_o}{f_e}.

3.3 Dispersion

White light splits into a spectrum because n depends on wavelength (n_{violet}>n_{red}; violet bends most). Dispersive power \omega=\dfrac{n_v-n_r}{n_y-1}. Angular dispersion \delta_v-\delta_r=(n_v-n_r)A. Chromatic aberration (different foci for different colours) is removed by an achromatic doublet: convex crown with concave flint, \frac{\omega_1}{f_1}+\frac{\omega_2}{f_2}=0. Spherical aberration (marginal rays focus nearer) is reduced by a stop or a parabolic mirror. Rainbow: dispersion plus internal reflection in raindrops (primary bow: one internal reflection, red outside at 42^\circ). Scattering (Rayleigh) \propto\lambda^{-4}: the sky is blue, the Sun red at sunrise and sunset; red is used for danger signals.

3.4 Nature and propagation of light

Huygens' principle: every point of a wavefront is a source of secondary wavelets, and the envelope forms the new wavefront; it derives the laws of reflection and refraction and shows that light slows down in a denser medium. The speed of light in vacuum c=3\times10^8 m s⁻¹ (Römer, Fizeau's toothed wheel, Foucault's rotating mirror, Michelson). In a medium v=\frac{c}{n} (n=1.5: 2\times10^8 m s⁻¹). Wavefronts are plane (distant source), spherical (point source) or cylindrical (slit).

3.5 Interference

Coherent sources have a constant phase difference (produced by splitting one wavefront). Young's double slit: path difference \Delta=\frac{yd}{D}; bright fringe \Delta=n\lambda, dark (n+\frac{1}{2})\lambda; fringe width \beta=\dfrac{\lambda D}{d} (for \lambda=600 nm, D=1 m, d=1 mm: 0.6 mm). Immersing the apparatus in a medium of index n reduces \beta to \frac{\beta}{n}; a thin sheet of thickness t over one slit shifts the pattern by \frac{(n-1)tD}{d}. Intensity I=4I_0\cos^2\frac{\phi}{2}; for two sources of intensities I_1,I_2: I_{max}/I_{min}=\Big(\frac{\sqrt{I_1}+\sqrt{I_2}}{\sqrt{I_1}-\sqrt{I_2}}\Big)^2. Thin films (colours of soap bubbles) and Newton's rings are the same effect. Interference is energy redistribution, not creation.

3.6 Diffraction

Bending of waves around obstacles; significant when the aperture is comparable with \lambda. Fraunhofer single slit: minima at a\sin\theta=n\lambda; central maximum width \frac{2\lambda D}{a}. Diffraction grating: d\sin\theta=n\lambda with d=\frac{1}{N} (5000 lines cm⁻¹: d=2\,\mum; \lambda=500 nm gives \sin\theta_1=0.25); the maximum order is n<\frac{d}{\lambda}. Resolving power of a telescope: minimum angle \theta=\dfrac{1.22\lambda}{D} (Rayleigh); a larger aperture resolves better; microscope \frac{2n\sin\theta}{1.22\lambda}; grating \frac{\lambda}{\Delta\lambda}=nN.

3.7 Polarization

Polarisation proves light is a transverse wave (sound cannot be polarised). Malus' law: I=I_0\cos^2\theta; unpolarised light through a polariser loses half. Brewster's law: reflected light is completely polarised when \tan\theta_p=n, and the reflected and refracted rays are at 90^\circ (n=\sqrt{3}: 60^\circ). Polaroid (aligned molecules) is used in sunglasses (to cut glare) and 3-D cinema. Optical activity (sugar solution) and double refraction (calcite).

Formula and fact sheet

Practice questions

Set A — Recall (10 questions)

1. The focal length of a concave mirror of radius 40 cm isa10 cmb20 cmc40 cmd80 cm
2. The critical angle for a medium of refractive index \sqrt{2} isa30^\circb60^\circc45^\circd90^\circ
3. The power of a lens of focal length 50 cm isa2 Db0.5 Dc5 Dd50 D
4. Total internal reflection can occur when light travels fromaair to waterba rarer to a denser mediumca denser to a rarer mediumdair to glass
5. A rainbow is formed byainterferencebdiffractioncpolarisationddispersion with total internal reflection
6. Light is completely polarised on reflection from a medium of index \sqrt{3} at the anglea60^\circb30^\circc45^\circd90^\circ
7. In Young's experiment \lambda=600 nm, D=1 m and d=1 mm. The fringe width isa0.06 mmb0.6 mmc6 mmd0.3 mm
8. The transverse nature of light is shown byainterferencebdiffractioncreflectiondpolarisation
9. The speed of light in a medium of refractive index 1.5 isa3\times10^8 m s⁻¹b2\times10^8 m s⁻¹c4.5\times10^8 m s⁻¹d1.5\times10^8 m s⁻¹
10. A biconvex glass lens (n=1.5) has both radii 20 cm. Its focal length isa10 cmb40 cmc20 cmd30 cm

Set B — Understanding (7 questions)

1. An object is 30 cm from a concave mirror of focal length 20 cm. The image distance isa60 cmb12 cmc30 cmd15 cm
2. A convex lens of f=10 cm forms the image of an object placed 15 cm away ata6 cmb25 cmc30 cmd150 cm
3. A prism of angle 60^\circ and index \sqrt{3} has minimum deviationa60^\circb30^\circc45^\circd75^\circ
4. Two thin lenses of powers +10 D and -2.5 D are in contact. The combination has powera12.5 Db7.5 Dc25 Dd4 D
5. A fish is 12 cm below the surface of water (n=\frac{4}{3}). Its apparent depth isa16 cmb12 cmc8 cmd9 cm
6. A grating has 5000 lines per cm. For \lambda=500 nm the first-order \sin\theta isa0.5b0.1c0.25d0.75
7. When the whole Young's apparatus is immersed in water (n=\frac{4}{3}), the fringe widthaincreases by \frac{4}{3}bis unchangedcvanishesddecreases to \frac{3}{4} of its value

Set C — Application (5 questions)

1. A convex lens forms a real image twice the size of the object on a screen 60 cm from the lens. Its focal length isa15 cmb20 cmc40 cmd30 cm
2. An optical fibre has a core of index 1.5 and cladding of 1.4. Its numerical aperture isa0.54b0.1c0.29d1.0
3. A telescope objective has aperture 0.1 m. For \lambda=500 nm the minimum angular separation resolved isa5\times10^{-6} radb6.1\times10^{-7} radc6.1\times10^{-6} radd1.2\times10^{-5} rad
4. Unpolarised light of intensity I_0 passes through two polaroids whose axes are at 60^\circ. The emerging intensity isaI_0/2bI_0/4c0dI_0/8
5. A glass lens (n=1.5) has focal length 20 cm in air. In water (n=\frac{4}{3}) its focal length isa80 cmb40 cmc20 cmd60 cm