Wave optics is the study of the wave nature of light. Interference and diffraction are two main phenomena giving convincing evidence that light is a wave.

Wavefront

A wavefront is a surface joining the points of same phase.  The speed with which the wavefronts move away from the source is called phase velocity or wave velocity.  The wave transports energy along the lines perpendicular to the wavefronts, with wave velocity.  There lines are called rays.

For a point source, the wavefronts are spherical extending in three dimension space.  If the source is far off, the wavefront becomes almost a plane.

Huygens’ Principle

1. All points on a wavefront vibrate in same phase with same frequency.
2. Every point on the wavefront acts as a point source of spherical secondary wavelets.
3. After a time t, the new wavefront is the surface tangent to these secondary wavelets.
4. Wavefronts move with the velocity of wave in that medium.

Interference

Two waves (whether sound or light) of equal frequencies travelling almost in the same direction show interference.  Consider two waves coming from sources S₁ and S₂.  These reach point P with a path difference Dx, having amplitude A₁ and A₂,

y₁ = A₁ sin [ωt – kx]                                                                …(1)                        y₂ = A₂ sin [ωt – k(x + Δx)]
or y₂ = A₂ sin [ωt – kx – ϕ]                                                         …(2)
where ϕ=kΔx=2πλ(Δx)

By principle of superposition, the resultant wave at P is
y = y₁ + y₂ = A₁ sin [ωt – kx] + A₂ sin [wt – kx – Φ]
= [A₁ + A₂ cos ϕ] sin (ωt – kx) – [A₂ sin ϕ] cos (ωt – kx)      …(3)
Putting  A₁ + A₂ cos ϕ = A cosθ   and    A₂ sin ϕ = A sinθ
so that    A² = [2A₁A₂ cos ϕ]                                                 …(4)
Eq. (3) becomes
y = A sin [ωt – kx – θ]                                                             …(5)
The intensity of the resultant wave is
I=12ρυω2A2=KA2
=K[A21+A22+2A1A2cosϕ]
or I=I1+I2+2I1I2−−−−√cosϕInterferenceterm                                              …(6)
Thus, we find that the resultant intensity I at point P is not just the sum of individual intensities, but has an interference term too.

Maxima and Minima

From Eq. (6), I is maximum, when
cosϕ = +1
or   ϕ = 0, 2π, 4π, …   or   ϕ=2nπ

Imax=I1+I2+2I1I2−−−−√=(I1−−√+I2−−√)2=K(A1+A2)2

I will be minimum, when cosϕ = –1
or ϕ = π, 3π, 5π, …      or ϕ=(2n−1)π

Imin=I1+I2−2I1I2−−−−√=(I1−−√∼I2−−√)2=K(A1∼A2)2

The ratios

ImaxImin=(I1√+I2√)2(I1√∼I2√)2=(A1+A2)2(A1∼A2)2

If  I₁ = I₂ = I₀   (i.e.,  A₁ = A₂),  we have
I_max = 4I₀      and     I_min = 0
Thus, when interference of two waves of equal intensities occur, the intensity of maxima  becomes 4 times that of single wave and that of minima  becomes zero.

Coherent Sources

Two sources are said to be coherent if they produces waves of the same frequency and a constant phase difference.  For observing the interference pattern, the two sources S₁ and S₂ must be coherent.  If it is not so, the interference term 2I1I2−−−−√cosϕ in Eq. (6) becomes zero when averaged over many cycles.  For such incoherent sources,  the intensity I = I₁ + I₂.  There are no observable maxima and minima, as the eye cannot follow such rapid and random changes in intensity at a point.

Unlike sound waves, two independent sources of light cannot be coherent.  Sound is a bulk property of matter.  So, two independent sources of sound can produce coherent waves.  However, light is emitted from a source due to the vibrations of its atoms.  The individual atoms emit light randomly and independent of each other.

In practice, coherent sources are obtained either by dividing the wavefront (as in the case of Young’s Double Slit Experiment,  Fresnel’s biprism,  Lloyd mirror, etc.) or by dividing the amplitude (as in the case of thin films, Newton rings,  etc.) of the incoming waves from a single source.

A laser discovered in 1960, is different from common light sources.  Its atoms act in a cooperative manner so as to produce intense, monochromatic, unidirectional and coherent light.  Thus, two independent laser beams can produce observable interference on a screen.

YOUNG’S DOUBLE SLIT EXPERIMENT (YDSE)

This experiment was done by English scientist Thomas Young in 1801 to prove that light is a wave.

Light from a distant monochromatic source illuminates slit S₀ in screen A.  Emerging light from S₀ spreads to illuminate two parallel, closely spaced slits S₁ and S₂ in screen B.  The slit’s length extends into and out of the page.  These slits behave like two coherent sources.  Light wave from these slits spread out and fall on the screen C.  An interference pattern of alternate bright (maxima) and dark (minima) bands or fringes is seen on the screen.In the experiment, the distance D between screens B and C is very large compared to the distance d between the slits S₁ and S₂ (D >> d).  Therefore, the angle q is very small, and hence

sinθ≈tanθ=yD

where y is the distance of point P from the central point O.
The path difference between the two waves arriving at point P is

p=S1P∼S2P=x∼(x+Δx)=Δx=dsinθ

For nth maxima,

p=nλ
or  (yn)max=nλDd ,  n = 0, 1, 2, 3, …

For nth minima,

p=(2n−1)λ/2
or (yn)min=(2n−1)λ2Dd,           n = 1, 2, 3, …

Note that
(1)  For the central point O, path difference p = 0, hence it is a maxima.  We can call it zeroth maxima.
(2)  The nth minima comes before the nth maxima.

Fringe Width

It is the distance between two consecutive maxima (or minima) on the screen,

b=(yn+1)max−(yn)max=(n+1)λDd−nλDd
or β=λDd

Note that fringe width b is independent of n.  That is, the interference fringes have same width throughout.

Angular Fringe-Width

θ0=βD=λd

Fringe Shift (y₀)

Suppose that a transparent sheet of refractive index μ and thickness t is introduced in one of the paths of interfering waves.  Its optical path becomes μt instead of t.  That is,  these occurs an increase in optical path by (μ – 1)t.  Because of this, a given fringe shifts from its present position

y=Dd(Δx) to its new position   y′=Dd[Δx+(μ−1)t].
⸫   Lateral shift, y0=y′−y=Dd(μ−1)t=βλ(μ−1)t

As shown in the figure, entire fringe–pattern is displaced by y₀ towards the side in which the plate is introduced, without any change in fringe width.Conditions for Observing Sustained Interference

(1)  The frequencies of the two interfering waves must equal.
(2) The initial phase difference between the two interfering waves must remain constant with time.
(3) The light must be monochromatic; otherwise there will be overlapping of patterns corresponding to each wavelength.
(4) The intensity of the two waves should be same so as to improve contrast.
(5) The two sources should be closely spaced; otherwise the fringes will be too close for the eye to resolve.
(6) The sources should be narrow.