## Wave Nature of Light

$Wavefront and Wavelet$

Wavefront: The locus of all vibrating particles at which the planes of vibration of a physical quantity associated with the wave is same.

Wavelet: The disturbance which travels in all direction due to each point in wavefront is wavelet

$\textbf{Huygen’s wave principle}$

i) Each point on primary wavefront is source of secondary wavelets.

ii) Secondary wavelets travel forward with speed of light.

iii) The surface of tangency to the secondary wavelets in the forward direction at any time gives the position of new wavefront i.e. secondary wavefront.

Note: This principle can explain rectilinear propagation of light, reflection, refraction, interference, diffraction and polarization but fails to explain photoelectric effect, Compton effect and Raman effect.

$\textbf{Types of wavefronts:}$

$\underline{Spherical}$

$A∝\frac{1}{r}$

$I∝\frac{1}{r^2}$

Where, A is amplitude and I is Intensity

$\underline{Cylindrical}$

$A∝\frac{1}{\sqrt{r}}$

$I∝\frac{1}{r}$

Where, A is amplitude and I is Intensity

$\underline{Plane}$

$A∝r^0$

$I∝r^0$

Where, A is amplitude and I is Intensity

$\textbf{Superposition of waves}$

$\to$ When two or more waves overlap, the resultant displacement at any point and at any instant is given by adding the instantaneous displacements that would be produced at any point by the individual waves if each were present alone.

i.e. $y = y_1 + y_2 + ..... + y_n$

Let, $y_1 = a_1 cos \omega t$ and $y_2 = a_2 cos ( \omega t$ + $\phi )$ superimpose. Then,

i) Phase difference = $\phi$

ii) Resultant amplitude , $a = \sqrt{a_1^2 + a_2^2 + 2 a_1 a_2 cos \phi}$

-$a_{max} = a_1 + a_2 (for \phi = 0^0)$

-$a_{min} = a_1 - a_2 (for \phi = 180^0)$

iii) Resultant intensity , $I = I_1 + I_2+ 2 \sqrt{I_1} \sqrt{I_ 2} cos \phi$

-$I_{max} = (I_1 + I_2)^2 (for \phi = 0^0)$

-$I_{min} = (I_1 - I_2 )^2 (for \phi = 180^0)$

## Interference

When two waves from coherent sources superimpose it results into variation of intensity at different points. This phenomenon is called interference.

It results into redistribution of intensity of light waves.

It is based on conservation of energy.

Condition for sustained interference:

$\to$ Two sources emitting waves must be coherent.

$\to$ Sources should be monochromatic and they shall emit light continuously.$\\ \to$ The separation between two sources should be small (but not less than wavelength of light) and distance of screen from sources must be large.

## Young's Double Slit Experiment

It is due to superposition of two waves reaching at point in screen (P) from two coherent sources (i.e. slits S1 at A and S2 at B)

### Types of Interference:

Constructive Interference Destructive Interference 1. Wavefronts arrive in phase and bright fringes are formed. 1. Wavefronts arrive out of phase and dark fringes are formed. 2. Path difference: $(\Delta x) = n\lambda$ 2.Path difference:$(\Delta x) = (2n − 1)\dfrac{\lambda}{2}$ 3. Phase difference (∆∅) = 2nπ (n = 0, 1, 2, .....) 3.Phase difference:(∆∅) = (2n − 1)π (n = 1, 2, 3, .....) 4. Distance of nth bright fringe from central fringe $(y_n) = \Delta x \times \dfrac{D}{d} = n\lambda \dfrac{D}{d}$ 4. Distance of nth bright fringe from central fringe $(y_n) = \Delta x \times \dfrac{D}{d} = (2n-1)\lambda \dfrac{D}{2d}$ 5. Fringe width $(\beta)$ = distance between two consecutive bright fringe$\\$$= y_n – y_{n-1} = \lambda\dfrac{D}{d}$ 5. Fringe width $(\beta)$ = distance between two consecutive bright fringe$\\$$= y_n – y_{n-1} = \lambda\dfrac{D}{d}$ - So in YDSE, all fringe have same width i.e. $\beta = \lambda \dfrac{D}{d}$
- Angular thickness of bright\dark fringes can be calculated from path difference equation i.e. $\delta x = d \sin \theta$
- When a transparent slab of thickness ‘t’ and refractive index ‘μ’ is placed in path of one of waves from slit then it causes shift of entire fringe pattern without affecting fringe width.$\\$Fringe shift = $\dfrac{D(\mu−1)t}{d} = \dfrac{\beta (\mu−1)t}{\lambda} \\$