Wave Optics

 Wave Nature of Light

$\text{Wavefront and Wavelet}$

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 = $\dfrac{D(\mu−1)t}{d} = \dfrac{\beta (\mu−1)t}{\lambda} \\$

(Since, $\beta = \lambda \dfrac{D}{d})$

Newton's Ring Experiment

When light ray incidents on plane convex lens with varying thickness of film, due to superposition of reflected rays, central dark and alternate bright-dark circular fringes are observed.

Rings are fringes of equal width.

Diameter of $n^{th}$ bright ring Dn bright = $\sqrt{(2n − 1) \times 2\lambda R}$

Diameter of $n^{th}$ dark ring Dn dark = $4n\lambda R$

Wavelength of monochromatic light used is: $\lambda = \dfrac{D^2_{n+p}− D^2_n}{4pR}$ ($D_{n+p}$ and $D_n$ are diameter of $(n+p)^{th}$ and $n^{th}$ bright\dark ring respectively ‘R’ is radius of Plano convex lens used.)

Diffraction

• The bending of wave around the corner of obstacle of size of order of wavelength is called diffraction.
• Diffraction is basically interference effects due to superposition of many light waves.

Fraunhofer diffraction

• When source, obstacle and screen are far enough apart that wavefronts to split are plane in nature, it is called fraunhofer diffraction.

The diffraction pattern consists of central bright fringe (central maxima) and alternate dark and bright fringes (secondary minima and maxima).

Secondary Maxima	Secondary Minima
Path difference $(\delta x) =(2n + 1)\dfrac{\lambda}{2} \\$(n = 1, 2, 3, .....)	Path difference $(\delta x) = n \lambda \\$(n = 1, 2, 3, .....)
Angular position of $n^{th}$ maxima$\\ = \sin\theta n \equiv \theta n \\= (2n + 1)\dfrac{\lambda}{2d}$	Angular position of $n^{th}$ minima$\\ = \sin\theta n ≈ \theta n \\= (n)\dfrac{\lambda}{2d}$
Distance of nth maxima from central maxima$\\(y_n) = (2n+1)\dfrac{\lambda D}{2d} \\ ≈ (2n+1)\dfrac{\lambda f}{2d} \\$(F is focal length of converging lens.)	Distance of nth minima from central maxima$\\(y_n) = n\dfrac{\lambda D}{2d} \\ ≈ n \dfrac{\lambda f}{2d} \\$(F is focal length of converging lens.)

Central Maxima:

• It is the bright band in which most of light is diffracted.

• Angular width of central maxima (θc) =2λ/d

• Linear width of central maxima = 2λD/d ≈ 2λf/d

Note

Central maxima lies between first minima on both sides from central point.$\\$The intensity of all fringes in interference is same whereas the intensity of fringes goes on decreasing from central maxima to secondary maxima in diffraction.

Diffraction Grating

• An array of large number of parallel slits, all with the same width ‘a’ and spaced equal distance ‘d’ is called diffraction grating.
• Grating spacing (d) = grating element = a + b
• If ‘N’ be number of lines drawn per unit length then, grating constant $\dfrac{1}{N} = a + b$
• When monochromatic wave is incident normally on the grating, the position of maxima are given as $d \sin\theta = n\lambda , (n = 0, ±1, ±2, ±3, ......)$

Polarization

• Restrictions of vibration or oscillation of a wave in a plane perpendicular to wave, in one direction is called polarization.
• Only transverse wave can be polarized.
• The device used for achieving polarization are called polarizer, For example; Tourmaline crystal, Polaroid, etc.

Brewster's Law:

When light is incident on boundary of two media such that angle between reflected ray and refracted ray is $90^o$ then reflected rays are fully polarized and angle of incidence is equal to angle of polarization and under this condition $\mu = \tan i_p$

(μ is refractive index of medium and ip is angle of polarization.)

Malu's Law:

“If polarized light is passed through analyzer, the intensity of light transmitted is directly proportional to square of cosine of angle between planes of polarizer and analyzer”

• i.e.$I \alpha \cos^2\theta \to I = I_o \cos^2\theta ; I_o$ is intensity of light after passing through polarizer.
• If incident light is unpolarized, intensity of emergent light after passing through polarizer is reduced to half.

Some Shortcut Tricks to Solve the problem

• In YDSE, if n1 fringes are visible in a given field of view with light of wavelength $\lambda_1$ and $n_2$ with light of wavelength $λ_2$, then $n_1λ_1 = n_2λ_2 \to nλ =$constant
• When YDSE is performed in medium of refractive index ′μ′, fringe width becomes $\dfrac{1}{μ}$ times as that in air.
• For a small movement $\Delta D$ of screen in YDSE, small change in fringe width $\Delta \beta$ is given as $\Delta \beta = \lambda \Delta \dfrac{D}{d}$
• The highest order spectrum which may be seen with monochromatic light of wavelength $\lambda$ on diffraction pattern with slit width d is $n = \dfrac{d}{λ}$(As $\theta_{max} = 900$ ).