Huygens

Reflection and Refraction of plane waves

Interference

Superposition of light waves

Young's interference experiment

Diffraction

Single slit diffraction

Diffraction by a circular aparture

Resolving power of optical instruments

Polarisation

Polarisation by Reflection

Brewster Angle

1621 - Snell's Law (Empirical relation)

1637 - Explanation of Snell's law by Descartes based on a "Corpuscular model of light," later established by Newton.

1678 - Huygens put forward the "Wave Theory", but "corpuscular theory" prevailed.

1801 - Young interference experiment.

Convincing experimental evidence that "light is a wave".

1864 - Maxwell's Theory of Electromagnetic Waves.

$\psi(x, t)=A \cos (k x-\omega t)$

$k=\frac{2 \pi}{\lambda}, \omega=2 \pi \nu$

At any instant $t=t_1$, "surface of constant phase".

$\left(k x-\omega t_1\right)=$ constant $\Rightarrow k x=$ constant

$\Rightarrow x=$ constant.

i.e. planes perpendicular to the $x$-axis.

As t increases x has to increase, for a given "wave front".

$k$ is the "propagation constant" or" phase constant".

Propagating in an arbitrary direction

$\psi=A \cos (\vec{k} \cdot \vec{r}-\omega t)$

$\vec{k}= \hat{i} k_x+ \hat{j} k_y+ \hat{k} k_z$

$\vec{r}=\hat{i} x+\hat{j} y+\hat{k} z$

$\vec{k} \cdot \vec{r}=k_x x+k_y y+k_z z$

$\left(\vec{k} \cdot \vec{r}-\omega t_1\right)$= constant

$\Rightarrow \vec{k} \cdot \vec{r}$ = constant

$\Rightarrow k_x x+k_y y+k_z z$ = constant

Propagating in an arbitrary direction

$\vec{k}$ is perpendicular to the planes.

Its direction represents direction of propagation of wave, $|\vec{k}|=(2 \pi / \lambda)$ is the propagation constant.

$\psi(r, t)=\frac{A}{r} \cos (k r-\omega t)$

Surface of constant phase, $k r$ = constant

$\Rightarrow r$ = constant, represents surface of a sphere of radius.

As t increases r increases, for a given wave front.

$\frac{A}{r}$ takes care of the decrease in Intensity.

All points on a wavefront act like point sources that give out secondary wavelets, which propagate outward with the speed of the wave.

The wavefront at a later time $\Delta t$ is given by the surface. tangent to these secondary wavelets.

Using Huygens Principle

$t=t_1+\Delta t$

$r=v \Delta t$

$t = t_1+2 \Delta t$

Using Huygens Principle

PLANE WAVES

"The new wavefront at a later time $\Delta t$ is the envelop tangent to all the secondary wavelets".

Amplitude of the wave at the tangent only.

Using Huygens Principle

$\frac{A}{r} \cos (k r-\omega t)$

Propagation of waves - by Huygens principle

"New wavefront at a later time $\Delta t$ is the envelop tangent to the secondary wavelets".

$t=t_1$

$t=t_1+\Delta t$

$n_2 > n_1$

$V_2 < V_1$

Wavefront in the second medium is tangent to all the secondary wavelets.

$B C=v_1 \Delta t$

$A D =v_2 \Delta t$

In $\Delta A B C, \quad \sin i=\frac{B C}{A C}=\frac{v_1 \Delta t}{A C}$

In $\Delta A D C, \sin r=\frac{A D}{A C}=\frac{v_2 \Delta t}{A C}$

$\frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$(Snell's law)

Law of refraction.

$v_2 < v_1$ if $n_2 > n_1$

Application of Huygens Principle to light passing through apertures

Plane waves are incident on this aperture.

Rectilinear propagation of light.

Geometrical shadow: the region where the light rays do not reach.

Huygens principle was able to explain diffraction.

Secondary wavelets are spheres originating from the point sources.

Wave is also propagating into the geometrical shadow.

We almost have spherical waves which are emanating from the holes.

There were no experiments which could prove that light is a wave.

The two points are $\pi$ difference in phase.

The maxima and minima coinciding.

Here is an intensity variation.

First - With sunlight from a small opening in the roof.

Then - With Sodium light.

The wave nature of light was convincingly demonstrated for the first time by Young' Experiment.

Explained the formation of "fringes" by Wave Theory.

Again, in 1802, he explained formation of "Newton's rings" by the Wave Theory.

Sunlight from the roof.

A bright intensity at the center.

If you plot the intensity, you could see a bright intensity peak, a bright peak at the center here and then you saw some colours here and then there is uniform illumination far away from here.