Wave Optics: One Mark Questions, Book Back Answers and Notes

About Wave Optics

Medium ~150 min study

Ray optics handles mirrors, lenses and prisms well, but it cannot explain the coloured bands on a soap bubble or the way light spreads after passing through a very narrow slit. For these you need to treat light as a wave. That is what this unit does.

Everything rests on the principle of superposition. When two waves meet, their displacements add, and this gives interference and diffraction. The unit starts with Huygens' principle, uses it to prove the laws of reflection and refraction, and then works out the fringe width in Young's double slit experiment and the positions of minima in single slit diffraction. Polarisation then shows that light is a transverse wave, which fits Maxwell's electromagnetic theory.

Expect derivations and numericals. The usual ones are proving the laws of reflection and refraction with wavefronts, finding fringe width, and finding the wavelength of light using a diffraction grating. Also revise Brewster's law, Malus' law and the magnification formulas for microscopes and telescopes, which suit short answers.

What you'll learn

• Describe spherical and plane wavefronts and how they are formed
• Use Huygens' principle to prove the laws of reflection and refraction
• Calculate fringe width in Young's double slit experiment
• Distinguish between Fresnel and Fraunhofer diffraction
• Explain the resolving power of microscopes and telescopes
• Use Brewster's law to find the polarising angle
• Solve problems on the diffraction grating to find wavelength

Before you start

• Reflection and refraction from ray optics
• Wave terms such as frequency, wavelength and phase
• The principle of superposition of waves
• Basic trigonometry, including small angle approximations

Wave Optics explained

Light as a Wave

Theories of Light

The textbook traces how ideas about light changed. Newton's corpuscular theory treated light as tiny particles. Huygens' wave theory described light as waves moving outward as wavefronts. Maxwell's electromagnetic theory showed that light consists of oscillating electric and magnetic fields. Einstein's quantum theory then explained the photoelectric effect by treating light as photons. This unit works with the wave picture.

Wavefronts and Huygens' Principle

A wavefront is a surface on which all points vibrate in the same phase. A point source gives spherical wavefronts, and a very distant source gives plane wavefronts. Huygens' principle says that every point on a wavefront acts as a source of secondary wavelets. The new wavefront, a short time later, is the surface that touches all these wavelets (their envelope). With this construction you can prove both the law of reflection and Snell's law of refraction.

Interference

When two coherent waves overlap, the energy is redistributed. Where crests meet crests the interference is constructive and you see a bright fringe. Where crests meet troughs it is destructive and you see a dark fringe. Coherent sources have the same frequency and a constant phase difference. In Young's double slit experiment the fringe width is $\beta = \frac{\lambda D}{d}$, so measuring $\beta$, D and d gives the wavelength $\lambda$. For example, with $\lambda = 600$ nm, D = 1 m and d = 1 mm, the fringe width is 0.6 mm.

Diffraction

Diffraction is the bending of light around the edges of an obstacle or aperture whose size is comparable to the wavelength. It is interference between secondary wavelets from different parts of the same wavefront. In Fresnel diffraction the source or screen is at a finite distance and the wavefronts are spherical or cylindrical. In Fraunhofer diffraction they are effectively at infinity and the wavefronts are plane. For a single slit of width a, minima occur at $a \sin\theta = n\lambda$. A diffraction grating, with many equally spaced slits, gives sharp maxima that are used to measure wavelengths.

Polarisation

In unpolarised light the electric vector vibrates in all directions perpendicular to the direction of travel. In plane polarised light it vibrates in only one direction. Only transverse waves can be polarised, so polarisation proves that light is transverse. Light can be polarised by selective absorption (polaroids), reflection, double refraction and scattering. Brewster's law says the reflected light is completely polarised when the angle of incidence $i_p$ satisfies $n = \tan(i_p)$. At this angle the reflected and refracted rays are perpendicular to each other.

Optical Instruments

The last part applies ray and wave ideas to instruments. The simple microscope, compound microscope and astronomical telescope are studied for magnification. The wave nature of light also sets a limit on how much detail any instrument can show: because of diffraction, a point object gives a small disc rather than a point image. Two objects can be resolved only if their diffraction patterns are separated enough, so resolving power depends on the wavelength used and the aperture of the lens.

Common mistakes to avoid

• Saying the central fringe in Young's experiment is dark. It is bright, because the path difference there is zero.
• Treating interference and diffraction as unrelated. Diffraction is interference of wavelets from different parts of one wavefront.
• Using degrees in small angle approximations. sin θ ≈ θ only when θ is in radians.
• Saying sound can be polarised. Only transverse waves can be polarised, and sound in air is longitudinal.
• Swapping wavelength and aperture in resolution formulas. A larger aperture or a shorter wavelength gives better resolution.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

What is the difference between a ray and a wavefront?

A wavefront is a surface on which all points vibrate in the same phase. A ray is a line drawn perpendicular to the wavefront, showing the direction in which the wave, and its energy, travels. Rays are enough for mirrors and lenses, but you need wavefronts to explain interference and diffraction.

Why are two independent light bulbs not coherent sources?

Light from a bulb comes from a huge number of atoms emitting randomly and independently, so its phase changes rapidly and unpredictably. Two separate bulbs therefore do not keep a constant phase difference, and the interference pattern shifts too fast to see. That is why Young used two slits lit by the same source.

How does the fringe width change if the apparatus is immersed in water?

Fringe width is proportional to wavelength. In water, light travels slower and its wavelength becomes λ/n, where n is about 1.33. So the fringe width also becomes β/n and the fringes move closer together.

What is the limit of resolution of an optical instrument?

It is the smallest distance or angle between two objects at which they can still be seen as separate. Because of diffraction, even a perfect lens forms a small disc for a point object. If two objects are closer than this limit, their discs overlap and they look like one.

Does polarisation prove that light is a transverse wave?

Yes. In a longitudinal wave such as sound, particles vibrate along the direction of travel, so there is nothing to restrict to one direction. Light can be plane polarised, which is only possible if its vibrations are perpendicular to the direction of travel. So light must be transverse.

What happens when white light is used in Young's experiment?

The central fringe is white, because the path difference is zero there for every colour. Away from the centre, each colour has a different fringe width, so the fringes separate into colours. Only a few coloured fringes are seen near the centre, after which the colours overlap and the screen looks uniformly lit.