** **

**Wave Optics :**

**It is the branch of Physics that deals with the production , emission and propagation of light ,its nature and study about the phenomena ****of interference, diffraction and polarization.**

**The nature of light has been a subject of controversy. Therefore Variousvarious theories have been proposed to explain the nature of light.**

**1. Newton’s corpuscular theory of light :-
**

**- Newton put forward this theory of light in 1678.
**

**- This theory was based on rectilinear propagation of light.
**

**- According to this theory light consist of tiny particles called corpuscules which shout out by luminous objects.**

**-This theory could explain colours of light ( due to different size of ****corpuscles),rectilinear propagation of light, reflection and refraction but failed to ****explain interference, diffraction and polarization.**

**Failure of this theory :-
**

**-This theory predicts higher velocity of light of light in denser medium as ****compared to that in a rarer medium. Which is not true practically.**

**-With the emission of corpuscles, the mass of the source of light should decrease ****but no such change in mass is detected.**

**2. Huygen’s wave theory of light:-
**

**- This theory was proposed by Huygen in 1690.**

**-According to this theory light propagates from the source in the form of wave.**

**-According to wave theory, for the propagation of wave a medium is necessary. ****So, it was assumed that all the space including vacuum is filled with a ****hypothetical medium called ‘ether’, which had the properties of elasticity and i****nertia.**

**-This theory could explain rectilinear propagation of light, reflection, ****interference, diffraction etc. But failed to explain photoelectric effect, Compton ****effect etc.**

**Failure of this theory :-
**

**-Huygen assumed that light waves are longitudinal in nature but later on it was ****proved that light waves are transverse in nature.**

**-This theory could not prove the existence of ether.**

**3. Maxwell’s electromagnetic wave theory of light:-
**

**-In 1860 Maxwell gave the mathematical theory of electromagnetism which ****predicted the existence of electromagnetic waves which require no material ****medium for propagation so the difficulty of ether was removed.**

**-The electromagnetic wave consist of electric and magnetic field.
**

**-Variations in both electric and magnetic fields occur simultaneously. Therefore, t****hey attain their maxima and minima at the same place and at the same time.**

**-The direction of electric and magnetic fields are mutually perpendicular to each ****other and as well as to the direction of propagation of wave.**

**-Light travels in free space with a constant velocity and is given by the relation,
**

**C = E/b**

**-The speed of electromagnetic wave depends entirely on the electric and ****magnetic properties of the medium, in which the wave travels and not on the ****amplitudes of their variations.**

**𝐶 =1/ √𝜇𝑟εr= 3 x 108 m/s **

**where 𝜇𝑟 = permeability of the medium**

**εr = permittivity of the medium**

Fig:- Propagation of electromagnetic wave

Fig:- Propagation of electromagnetic wave

**-This theory could explain reflection , refraction, interference, diffraction,polarization but could not explain photoelectric effect, Compton effect etc.**

**Note:-**

**-For discussion of optical property of EM wave, more significance is given to Electric Field, E. Therefore, Electric Field is called ‘light vector’.**

**-Maxwell’s theory was conformed by Hertz in 1886,who produced and detected electromagnetic waves in Laboratory.**

**4. Quantum theory of light:-
**

**-This theory was proposed by Einstein in 1905.He made the use of Planck’s Quantum theory of black body radiations as the basis for his theory of nature of light.**

**-According to this theory, the light is produced, absorbed and propagated as packets of energy called photons.**

**-The energy associated with each photon is E= hf= ℎ𝑐/𝜆**

**-This theory could explain Photoelectric effect, Compton effect, but failed to explain interference, difference and polarization.**

**5. Dual theory of light:-
**

**-This theory was given by de Broglie in 1924 according to which light behaves both as particles and waves.**

**-The wave nature of light dominates when light interacts with itself.
**

**-The particle nature of light dominates when light interacts with matter.
**

**-This theory of light successfully explains all the phenomena connected with light.**

**Wavefront:-
**

**According to Huygen’s wave theory of light, from the source of light periodic disturbances are produced which propagate in the form of wave. As the wave ****propagates in the medium, particles of the medium execute simple harmonic ****motion. If we join all the particles which are vibrating in the same phase and are ****equal distance from the source , a wave front is obtained.**

**Thus the wave front at any instant is defined as the locus of the particles of the ****medium which are vibrating in the same phase.**

Wavelets:-

Wavelets:-

**Each point on a wavefront acts as a new source of disturbance. The ****disturbances from these points are called wavelets. These wavelets ****spread out in all directions in the medium with the velocity of light.**

**Note:-
**

**-A line perpendicular to wavefront is known as ray.
**

**-The disturbance from real source of light is known as wavefront.
**

**-The disturbances from imaginary sources of light are called wavelets.**

**Types of Wavefront:
**

**On the basis of nature of source of light , there are following types of ****wavefront.**

(i) Spherical wavefront:

(i) Spherical wavefront:

**This type of wave front is originated from a point source of light placed at near to ****the observer and the intensity varies as inverse of square of distance from the S****ource.**

**i.e. I ∝ 1/r^2 ( ∴ I =p/4𝜋𝑟^2)**

**(ii) Cylindrical wavefront:
**

**This type of wave front is originated from a linear source of light placed at near to ****the observer and the intensity varies as inverse of distance from the source.**

**i.e. I ∝ 1/r ( ∴ I =p/2πr.l)**

**(iii) Plane wavefront:
**

**This type of wave front is originated from a point or a linear source of light placed ****at very far distance from the observer and the intensity does not vary with distance.**

**i.e. I ∝1/r^0 ( ∴ I =p/𝑙𝑥𝑏)**

**Note: A part of spherical or cylindrical wavefront is plane wavefront.**

**Huygen’s Principle:
**

**Huygen’s principle is a geometrical method for determining the position and shape ****of given wavefront at any instant in the future if its present position and shape are ****known.**

**In order to explain how the wave front is propagated forward through a ****homogenous isotropic medium ( velocity of light is same in all directions), Huygen ****made following assumptions:**

**I. Each point on a wavefront acts as a new source of disturbance. The ****disturbances from these points are called wavelets (secondary ****wavelets). These wavelets spread out in all directions in the medium ****with the velocity of light.**

**II. The new wavefront (called secondary wavefront) is then obtained by ****constructing a tangential plane to all these wavelets (secondary ****wavelets). Thus the new wavefront is an envelope of all the ****secondary wavelets at that instant.**

**Huygen’s Construction:**

**Ilustration:**

**To illustrate the Huygen’s principle, let us consider a part of wavefront ****AB called Primary wavefront at t=0. We have to find the position the ****wave front at time ‘t’ using Huygen’s principle.**

**Contd…
**

**-Take the number of points a, b, c, d………on the primary wavefront AB. These ****points are the sources of secondary wavelets.**

**-At time ‘t’ the radius of these secondary wavelets is ‘ct’, where ‘c’ is velocity of l****light. Taking each point as center, draw circles of radius ‘ct’.**

**-Draw a tangent A’B’ common to all these circles in forward direction. This gives t****he position of new wavefront called secondary wavefront at the instant ‘t’.**

**-The Huygen’s construction gives a backward wavefront also shown by dotted line ****A”B” which is contrary to observation.**

**Note: Backward wavefront is rejected. Why?
**

**Amplitude of secondary wavelet is proportional to ½ (1+cosθ). Obviously, for the ****backward wavelet θ = 180° and (1+cosθ) is 0.**

**How can we convert plane wavefront into spherical and spherical into Plane?**

**Ans: Each can be achieved using Convex lens.**

**Behaviour of a Plane Wavefront in a Concave Mirror, Convex Mirror,Convex Lens, Concave Lens and Prism:**

**Verification of Laws of Reflection at a Plane Surface (On the basis of**

**Huygen’s Principle):**

**The laws of reflection are:**

**I. The angle of incidence is equal to**

**angle of reflection for all wavelength**

**and for any pair of materials.**

**II. The incident rays , the reflected rays**

**and the normal to the reflecting**

**surface all lie on the same plane.**

**To verify the laws of reflection on the basis of wave theory of light, let us consider a plane**

**wavefront AB is incident on reflecting plane surface XY as in figure. The lines PB,QE and**

**RA which are perpendicular to the wavefront AB represent incident rays. According to**

**Huygen’s principle each point on a plane wavefront AB acts as a new source of**

**disturbance. Therefore the disturbance at B after time ‘t’ reaches to the point B’. At the**

**same time ‘t’ the disturbance at A reaches to the point A’ .**

**A’B’ – Reflected wavefront**

**XY – Reflecting surface**

**Contd……..**

**Similarly the disturbance at E reaches to the point E’. Hence, AA’=BB’= ct , where c**

**is speed of light. If we construct a tangential plane to all the reflected secondary**

**wavelets, a new wavefront A’B’ is obtained called reflected plane wavefront. And**

**perpendicular lines A’P’, E’Q’ and B’R’ represents the reflected rays.**

**As in figure, < 𝑅AN = <i (Angle of incidence)**

**<NAB = 90-i**

**<NAB’= 90**

**< 𝐵AB’ = 90-(90-i)= <i**

**Similarly,< 𝑅′𝐵’N’=< 𝐴′𝐵’A = r ( Angle of reflection)**

**In Δ’s ABB’ and AA’B’**

**I. AA’=BB’ = ct [ distance travelled by the disturbance in time t]**

**II. <ABB’ = <AA’B’ [ both being 900**

**]**

**III. AB’= AB’ [ Common sides ]**

**Hence, Δ ABB’ ≅ Δ AA’B’ [ By S.A.S axiom]**

**Contd…….**

**Therefore, <BAB’ = <A’B’A [ Corresponding angles of congruent triangles]**

**i.e < 𝑖 = < 𝑟**

**Hence angle of incidence is equal to angle of reflection which is the first law**

**of reflection.**

**Further , the incident plane wavefront AB, reflected wave front A’B’ and the**

**reflecting surface XY are perpendicular to plane of paper. So, the incident ray,**

**reflected ray and normal to the reflecting surface all lie on the same plane.**

**This proves the second law of reflection.**

**Verification of Laws of Refraction at a Plane Surface (On the basis of Huygen’s**

**Principle):**

**When the light enters from rarer medium to**

**denser medium with the velocities c and v**

**respectively, then the laws of refraction are:**

**a. The ratio of sine of the angle of incidence to**

**sine of angle of refraction is constant for any**

**two given media. Therefore, sin i/ sinv= μ**

**Where μ is constant called the refractive index**

**of the medium with respect to air ( here).(Snell’s**

**law)**

**AB – Incident wavefront**

**CD – Refracted wavefront**

**XY – Refracting surface**

**b. The incident rays , the refracted rays and the**

**normal at the point of incidence on the**

**refracting surface all lie on the same plane.**

**To verify the laws of refraction on the basis of wave theory of light, let us consider a plane**

**wavefront AB is incident on refracting plane surface XY separating two different media as in**

**figure. The lines PB,QE and RA which are perpendicular to the wavefront AB represent incident**

**rays.**

**Contd………….**

**According to Huygen’s principle each point on a plane wavefront AB acts as a new**

**source of disturbance. Therefore the disturbance at B after time ‘t’ reaches to the**

**point B’. Therefore BB’ = ct ,where c is velocity of light in rarer medium. At the same**

**time ‘t’ the disturbance at A reaches to the point A’ inside the denser medium with**

**the velocity v. Therefore, AA’= vt. Similarly the disturbance at E reaches to the point**

**E’. If we construct a tangential plane to all the refracted secondary wavelets, a new**

**wavefront A’B’ is obtained inside the denser medium called refracted plane**

**wavefront. And perpendicular lines A’P’, E’Q’ and B’R’ on the refracted plane wave**

**front A’B’ represent the refracted rays.**

**As in figure, < 𝑅AN = <i (Angle of incidence)**

**< 𝑁AB = (90-i)**

**<NAB’= 90**

**𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, < 𝐵AB’ = 90-(90-i)= <i**

**Similarly,< 𝑃′𝐵’N’=< 𝐴′𝐵’A = < r ( Angle of reflection)**

**Contd….**

**In right angled Δ ABB’**

**𝑠𝑖𝑛 𝑖=**

**𝐵𝐵′**/

**𝐴𝐵′**

**=**

**𝑐𝑡**/

**𝐴𝐵′**

**……………………(i)**

**Similarly, In right angled Δ AA’B’**

**𝑠𝑖𝑛 𝑟=**

**𝐴𝐴′**/

**𝐴𝐵′**

**=**

**𝑣𝑡**/

**𝐴𝐵′**

**……………………(ii)**

**Dividing Equation (i) by (ii) we get,**

**𝑠𝑖𝑛 𝑖**/

**𝑠𝑖𝑛 𝑟**

**=**

**𝑐**/

**𝑣**

**………………………………..(iii)**

**Where, 𝑐**/

**𝑣**

**=μ , is refractive index of the denser medium with respect to rarer**

**medium.**

**Therefore, 𝑠𝑖𝑛 𝑖**/

**𝑠𝑖𝑛 𝑟**

**=**

**𝑐**/

**𝑣**

**= μ**

**i.e the ratio of sine of angle of incidence to sine of angle of refraction is constant**

**which is snell’s law.**

**Contd...**

**- Further, the incident plane wavefront AB, refracted plane wavefront A'B' and the refracting surface XY are perpendicular to plane of paper. So, the incident ray, refracted ray and normal to the refracting surface all lie on the same plane. This proves the second law of refraction.**