Wave Optics

Introduction : 

In 1637;Descattes gave cospuscular theory of light and derived Snell's law . The cospuscular theory predicted that if a ray of light bends towards the normal the speed of light in the second medium would be greater than the first medium .Corpuscular theory was further developed by Newton . In 1678, the Dutch physicist Christian Huygens put forward the wave theory of light .It predicted that if light bends towards the normal its speed in second medium would be less than the first , a sharp contrast to the Corpuscular theory .In 1850, Foucault experimentally proved that speed of light in water is less than speed in air confirming the wave nature of light . Wave theory was not firstly accepted because of great authority of Newton . However Thomas Young in 1801 performed his famous interference experiment firmly establishing the wave theory .

The phenomena of interference , diffraction ,reflection and refraction are well explained by the wave model of light . The only difficulty is that for a wave to propage a medium is needed. But light can travel in vaccum also . This was explained by Maxwell when he proved the electromagnetic nature of light .Maxwell developed a wave equation by assuming light is a propagation electric and magnetic vector mutually perpendicular to each other.From this he calculated the speed of light in vacuum . Value so obtained is very close to the experimentally obtained value of speed of light . So he confirmed light as electromagnetic wave able to propagate on free space .Light waves are associated with changing electric and magnetic field .Changing electric field produces time and space varying magnetic field and changing magnetic field produces time and space varying electric field .The changing electric and magnetic field result in propagation of electromagnetic waves in vacuum .

Huygens Principle : 

When we drop a small stone on the surface of calm pond ,waves are generated and spread out from the point of impact .Waves are seen as circular ring spreading out from point of origin . All points on such a circle are oscillating in phase because there are at same distance from the source . So locus of all points oscillating in phase is called wave front . A wave front is defined as surface of constant phase .The speed with which the wavefront moves outwards from the source is called speed of the wave . The energy of the wave travels in a direction perpendicular to the wave front . If the wave front is a sphere then it is called spherical wavefront . At a large distance a portion of the sphere can be considered as plane and we get plane wave front .

If we know the shape of wavefront at a time t= 0 we can construct the wave front at later time using Huygens principle .It is essentially a geometrical construction. According to Huygens principle , each point on the wavefront is the source of secondary waves and the  secondary wavelets generating from these points spread out in all directions with the speed of the wave .If we draw a common tangent to all these spheres we get the position of wavefront at later time .

If we want to draw the shape of wavefront at time t , we draw spheres with radius vt  at each point on the wavefront( t =0) and if we draw a common tangent to all the spheres we get spherical wavefront at time t .The new wavefront G1G2 is again a spherical wavefront. The above model has a short coming , there is a backward wave .Huygens argued that amplitude of backward wave is zero .

With use of Huygens principle we can construct plane wavefronts also 

Refraction of a plane wave : 

We can derive laws of refraction using Huygens principle .

We will now use Huygens principle to derive laws of refraction.Let v1 and v2  are the speed of light in medium 1 and 2 respectively.We assume a plane wavefront A'A incident on the interface at an angle I as shown in the figure. Let t be the time taken by the wavefront to travel the distance BC . Thus BC= v1t . In order to determine the shape of refracted wavefront we draw a sphere of radius v2t  from the point A .Let CE represent a tangent plane drawn from point C to the sphere . Then AE = v2 t and CE is the refracted wave front . From the triangle ABC and AEC we readily get , 

Sin i = BC/AC = v1t/AC 

Sin r = AE/AC = v2t/AC , where i and r are the angle of incidence and refraction respectively . From the above relations we get ,

Sin I/Sin r = v1/v2 

From the above relation we get the result , if i > r then then v2< v1 implies if velocity is less in second medium rays bend towards the normal . If experimentally proved contrasting the corpuscular theory of light . Now , n1 = C/v1 ( n1 is the refractive index of first medium) Similarly for the 2nd medium 

n2 = C/v2  So

 n2/n1 = v1/v2

Giving Sin i /Sinr = n2/n1

Or , n1 Sin i = n2 Sin r , this is Snell's law of refraction.

If we take BC = one  wavelength in medium 1(lamda1) then AE will also be one wave length(lamda2) in medium 2 

So v1/v2 = lamda1 /lamda2

This implies v1/lamda1 = v2/lamda2

Gives the frequency is same in both the media .

So when light travels from one to another medium its wavelength is changed but not its frequency .

Reflection of a plane wave front by a plane surfac : 

We consider a plane wavefront AB incident at an angle I on a reflecting surfac MN . If v represents the speed of light and t is the time of taken by the  wavefront to reach from B to C then the distanc BC = v t In order to construct the reflected wavefront  we draw a circle with radius v t  from the point A . CE represents the tangent to the circle from C . Obviously AE = BC = v t 

. The triangle AEC and ABC  are congruent.Therefore i = r . This is law of reflection.

With the laws of reflection and refraction we can understand behaviour of lenses ,prism and mirror . When a plane wavefront passes through a thin prism emergent wavefront remain plane with a tilt as shown in the figure .But a plane wavefront passing through a lens it becomes spherical with radius equal to focal length and when reflected by a concave mirror it also becomes spherical wavefront with radius R/2 .

Coherent and Incoherent Addition of waves : 

Here we will discuss about interference pattern produced by superposition of two waves . The superposition principle :At a particular point in a medium the resultant displacement produced by a number of waves is the vector sum of displacements produced by each of the waves . 

We consider two niddles on the surface of water moving up and down periodically and identically .They produce two  water waves and at a particular point the phase difference produced by the two waves remain constant . When this happen we say the sources are coherent .We consider the points where S1P = S2P .Since the distances of the point P is same from both the points The two waves will reach the point P at the same times . So  the two displacement can be represented by ,

y1 = a cos wt  

and  y2 = a cos wt 

So the resultant displacement y is ,

y = y1 + y2 = 2a cos wt , so the resultant intensity is given by ,

I = 4 Io , Io is the intensity produced by individual waves and it is proportional to a^2 , amplitude square . Here the two waves interfere constructively and what we call it as constructive interference .

If the path difference at point P of waves coming from the two sources is 2 x wave length then the the waves can be written as

 y1 = a cos wt 

and y2 = a cos (wt - 4 pi)= a cos wt  , where path difference of 2 x wave length implies a phase difference of 4 pi . The two waves are once again in phase so the resultant intensity is 4 Io giving constructive interference .

But if the path difference is 2.5 x wavelength then the waves may be written as y1 = a cos wt and

y2 = a cos ( wt + 5 pi) = - a cos wt 

The two waves are out of phase calcelling each other giving intensity zero. Here the waves interfere destructively .

To summarise: If we have two coherent sources S1 and S2 vibrating in phase and for arbitrary point P where the path difference, 

S1P - S2P = n x lamda ( n = 0 , 1 ,2, 3 .......) we will have constructive interference  and the resulttant intensity will be 4 Io .

On the otherhand if the path difference at P is (n+1/2) multiple of wavelength i.e 

S1P - S2P = ( n +1/2)x lamda  ( n =0,  1 , 2 , 3 ........) then the waves will interfere destructively giving intensity zero at the point P.

For any arbitrary point G let the difference between two waves is ዋ .The can be represented by ,

y1 = a cos ω t and y2 = a cos( ധt + φ)

And the resultant displacement is given by , 

y = y1 + y2 

    = a [ cos ωt + cos( ω t + φ) 

    = 2 a cos( φ /2) cos ( ω t + φ/2)

The amplitude is given by 2 a cos (φ/2) and therefore intensity is given by 

I = 4 Io cos^2( φ/2) 

If  φ = 0 , 土 2π ,土4 π,  ........

We have constructive interference.

On the other hand if 

φ = 土 π, 土 3 π, 土 5 π .... We have destructive interference leading to zero intensity .

Now if the two sources are coherent the phsase difference fi will not change with time at any  point i.e maxima and minima will not change with time and we will get stable interference pattern . If the phase difference of the two sources changes rapidly with time we say the sources are incoherent. This is what happens when two separate sources illumanate a wall .