When a stone is dropped into still water, it generates a series of waves from the point at which the stone came into contact with the water. Such waves oscillate in phase because they are at the same distance from the point of impact. A locus of points that oscillate in phase is known as a wavefront. The energy of the waves is propagated perpendicular to the wavefront. Huygens Principle is used to determine the shape of a wavefront at a later time t = τ using the shape of the wavefront at t = 0. When waves are generated in a wavefront, each point on each wave acts as a new source of secondary waves. According to Huggen's Principle, if a tangent is drawn through these points, the shape of the wavefront at a later time can be found.

a. Refraction of a Plane Wave

This section of the NCERT Solutions for Chapter 10 Physics explores what happens when a plane wavefront travels from a rarer medium to a denser medium. Using Huygens Principle, Snell’s laws of refraction are proved.

b. Refraction at a Rarer Medium

This section explores what happens when a wavefront is refracted at a rarer medium, i.e. v2 > v1. The angle of refraction is larger than the angle of incidence. Critical angle ic is defined by the equation:

sin ic = n2/n1.

If i = ic then sin r = 1 and r = 90°. If i is greater than ic, there is no refracted wave, and total internal reflection occurs.

c. Reflection of a Plane Wave by a Plane Surface

A wavefront appears when all the points of an incident wave are in phase. After reaching the plane surface, the incident waves are reflected back. Lighter waves reach the surface before the denser waves. Let the time taken by some portion of the wavefront to interact with the boundary be τ.m As the incident wavefront touches the surface, every point on the incident wavefront results in secondary wavelets. The wavefront for the secondary wavelets is spherical. The radius of the wavefront is given by r = vτ. As the other points of the wavelets reach the surface, both τ and the radius decrease eventually. The tangent of all these points gives the reflected wavefront.

d. The Doppler Effect

If there is no medium and the observer is far away from the source of light, the time taken between two consecutive wavefronts to reach the observer is much longer at the observer than at the source. The successive wavefronts travel a larger distance at the observer than at the source. If the source moves further away from the observer, the frequency as measured by the source will be smaller. This is called the Doppler effect.

An increase in the wavelength due to the Doppler effect is called a red shift while a decrease in the wavelength is called the blue shift. The Doppler equation is given by: Δv/v = –vradial/c, where vradial is the source velocity along the line joining the observer to the source, and is taken as positive when the source moves further from the observer. The Doppler formula can be applied only when the speed of the source is small compared to that of light.

Consider two sources of light S1 and S2 producing light waves. At a particular point, the phase difference between the displacements produced by the waves remains constant. Such sources are coherent. If the displacement by S1 at the point P is , and the displacement produced by S2 is , the net displacement at P is given by: . The corresponding intensity is given by: , where I0 is the intensity of each of the sources.

Coherent sources generate waves of the same frequency, and have a constant phase difference. 

Constructive interference occurs when the crest of one wave coincides with the crest of another wave, generating maximum amplitude. Their displacement is the same and phase difference constant. If there are two coherent sources S1 and S2 vibrating in phase, then for any point P when the path difference is S1P ~ S2P = nƛ, where ~ represents the difference between S1P and S2P, the intensity is , and the interference is constructive.

Destructive interference occurs when the crest of one wave coincides with the trough of another wave, producing minimum amplitude. Their displacement is different and the phase difference is not constant. If the path difference is S1P ~ S2P = (n + ½)ƛ, the net intensity is zero, and the interference is destructive.

If two sodium lamps are used to illuminate two pinholes, there will not be any interference fringes because the light wave generated by a lamp changes phase as quickly as 10–10 seconds, meaning that these two sources of light are incoherent. The British physicist Thomas Young designed an experiment to ‘lock’ the phases of the waves emitted by S1 and S2. He took an opaque screen and made two pinholes S1 and S2 very close to each other. These pinholes were illuminated by other pinholes that were lit by a bright source. S1 and S2 then behave as if they are two coherent sources because any phase change in the main source results in the same phase change in the lightwaves emitted by S1 and S2, locking the two sources in phase.

Light can illuminate objects around corners. This is due to the bending of light, which is known as diffraction. This section of the NCERT Solutions for Class 12 Wave Optics chapter covers the diffraction of light in detail.

a. The Single Slit

In Young’s experiment, it was evident that a narrow slit behaves as a new source from which light bends and illuminates the area around. The light appears to bend around corners and illuminate areas where a shadow is expected to exist. When the two slits in Young’s experiment are replaced by one narrow slit, an illuminated region appears with the intensity of light being the strongest at the centre and fading away towards the edges.

b. Seeing the Single Slit Diffraction Pattern

To see the single-slit diffraction pattern at home, two razor blades and one electric bulb with a straight filament can be used. The two blades have to be positioned in such a way that the edges are parallel and have a narrow slit between them. The slit has to be placed parallel to the filament and directly in front of the eye. The pattern should now appear with its bright and dark regions. The pattern may show some colours depending on the wavelength of the bands. The fringes can be seen more clearly by placing a blue or red filter.

c. Resolving Power of Optical Instruments

Consider a beam of light hitting a convex lens. If the lens is well adjusted for aberrations, then the beam will focus on a single point. However, due to diffraction, the beam gets focused to a region of the finite area instead of a point. In such circumstances, the effects caused by diffraction are considered by taking a plane wave incident on a circular aperture and then on a convex lens. The analysis of the corresponding diffraction pattern is similar to the analysis obtained from the single-slit diffraction pattern. The pattern on the focal plane consists of a bright region in the centre with concentric dark and bright rings surrounding it. The resolving power of an optical instrument is the reciprocal of the distance or angular separation between two objects that can be resolved when looked at through the optical instrument.

The resolving power of a telescope is . A large diameter d gives a better resolution of the image.

The resolving power of a microscope is , where n is the refractive index of the medium between the object and the aperture. A large value of the numerical aperture nsinθ results in a high resolution of the image.

d. The Validity of Ray Optics

Consider a slit with aperture a. The angular size of the central bright region in the diffraction pattern is given by λ/a. After covering a distance z, the diffracted beam reaches a width zλ/a. Equating zλ/a with a, the distance is obtained beyond which divergence of the beam becomes significant. Therefore, the distance z = a2/λ is known as the Fresnel distance, defined as the distance at which the spreading due to diffraction becomes comparable to the width of the slit. If the distance is lesser than zF, the spreading due to diffraction is smaller than the beam. If the distance is larger than zF, the spreading due to diffraction begins to dominate because of ray optics.

Light is an electromagnetic wave, which means that its electric field moves in one direction while the magnetic field moves in another. However, both vibrate in planes perpendicular to each other. A light wave vibrating in more than one plane is unpolarised light. Sunlight, tubelight, light from light bulbs, etc. are examples of unpolarized light. Light waves vibrating in a single plane are known as polarised waves. The waves of plane-polarised light vibrate in the same direction. The process of polarizing unpolarized light is called polarization.

a. Polarisation by Scattering

When white light enters a medium, the light gets scattered. The energy carried by light is transferred to the atoms in the medium. The atoms then vibrate at the same frequency as the incident beam and further transfer the energy to the surrounding atoms in the medium. These vibrations generate electromagnetic radiation in the form of light. This process continues throughout the medium and is called scattering of light. Light scattered in a direction perpendicular to the incident beam is always plane polarised.

b. Polarisation by Reflection

When a light ray is incident on a denser medium, some light gets reflected while the remaining gets refracted. As per Snell’s law, if the incident angle increases, the angle of refraction also increases. As per the laws of reflection, the angle incidence is the same as the angle of reflection. Therefore, when the angle of incidence is equal to the angle of polarization, the reflected light will be completely plane-polarized. Brewster’s Law states that when the angle of incidence is equal to the angle of polarisation, the reflected ray and refracted ray will be perpendicular to each other.

1 . How is a wavefront different from a ray?

A ray shows the direction in which a wave travels. A wavefront is the locus of all the points of a wave in a constant phase. A ray is normal to the wavefront.

2. What are the uses of wave optics?

Wave optics are useful in many areas of science such as in astronomy, engineering, fibre optics and more.

3. What are wave optics?

Wave optics is the study of the behaviour of light such as diffraction, polarization and more.