Introduction

WAVE NATURE OF LIGHT

Huygens' Wave Theory

(i) Each point on a wavefront acts as a source of new disturbance and emits its own set of spherical waves called secondary wavelets. The secondary wavelets travel in all directions with the velocity of light so long as they move in the same medium.

(ii) The envelope or the locus of these wavelets in the forward direction gives the position of new wavefront at any subsequent time.

A surface on which the wave disturbance is in the same phase at all points is called a wavefront.

Wave optics involves effects that depend on the wave nature of light. In fact, it is the results of interference and diffraction that prove that light behaves as a wave rather than a stream of particles (as Newton believed).

Like other waves, light waves are also associated with a disturbance, which one consists of oscillating electric and magnetic field. The electric field associated with a plane wave propagating along the x-direction can be expressed in the form:

$\vec{E}$= $\vec{E}$_o[sin($\omega$t - kx + $\varphi$_o)]

where $\omega$, k and $\varphi$_o bearing their usual meanings.

Points to remember regarding Interference

$\bullet$ When two waves with amplitude A₁ and A₂ superimpose at a point, the amplitude of resultant wave is given by

A = $\sqrt{A_1^2+A_2^2+2A_1{A}_2\cos \phi }$

Where $\varphi$ is the phase difference between the two waves at that point.

$\bullet$ Intensity (I) = $\frac{1}{2{\mu }_{0}C}{E}_0^2$. C = speed of light, E₀ = electric field amplitude

$\bullet$ Intensity (I) = I₁ + I₂ + 2$\sqrt{I_1{I}_2}$cos$\varphi$.

Hence for I to be constant, $\varphi$ must be constant.

$\bullet$ When $\varphi$ changes randomly with time, the intensity = I₁ + I₂.

$\bullet$ When $\varphi$ does not change with time, we get an intensity pattern and the sources are said to be coherent. Coherent sources have a constant phase relationship i.e. one that does not change with time.

$\bullet$ The intensity at a point becomes a maximum when $\varphi$ = 2n$\pi$ (n = 0, 1, 2, . .) and there is constructive interference.

$\bullet$ If $\varphi$ = (2n - 1)$\pi$, there is destructive interference. (Here n is a non-negative integer)

Determination of Phase Difference

The phase difference between two waves at a point will depend upon

(a) the difference in path lengths of the two waves from their respective sources.

(b) the refractive index of the medium

(c) initial phase difference, between the source, if any.

(d) Reflections, if any, in the path followed by waves.

$\bullet$ In case of light waves, the phase difference on account of path difference

= $\left[\frac{Opticalpathdifference}{\lambda }\right]\times 2\pi$ = $\left[\frac{\mu (Geometricalpathdifference)}{\lambda }\right]2\pi$

where $\lambda$ is the wavelength in free space.

$\bullet$ In case of reflection, the reflected disturbance differs in phase by $\pi$ with respect to the incident one if the wave is incident on a denser medium from a rarer medium. No such change of phase occurs when the wave is reflected in going from a denser medium to a rarer medium.