1.4 Intensity distribution

1.1 Wavefronts
1.2 Huygens Principle
1.3 Interference of light
1.4 Intensity distribution
1.5 Phase difference

2.1 Fringe width $(w)$

3.1 Diffraction due to a single slit

4.1 Malu's law
4.2 Brewster's law

If $A_1$ and $A_2$ are the amplitudes of interfering waves due to two coherent sources $S_1$ & $S_2$ and $\phi $ is constant phase difference between the two waves at any point $P$, then the resultant amplitude at point $P$ will be,

$$R = \sqrt {A_1^2 + A_2^2 + 2{A_1}{A_2}\cos \phi } \quad ...(i)$$

where $R$ is the resultant of the two interfering waves.

As we know, the relation between amplitudes and intensity we can write, \[\left. \begin{gathered} {I_1} = A_1^2 \hspace{1em} \\ {I_2} = A_2^2 \hspace{1em} \\ I = {R^2} \hspace{1em} \\ \end{gathered} \right\}\quad ...(ii)\]

From equation $(i)$ and $(ii)$ we get,

$$\begin{equation} \begin{aligned} {R^2} = A_1^2 + A_2^2 + 2{A_1}{A_2}\cos \phi \\ I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos \phi \\\end{aligned} \end{equation} $$

If $I_1=I_2=I_0$, $$\begin{equation} \begin{aligned} I = {I_0} + {I_0} + 2\sqrt {{I_0}{I_0}} \cos \phi \\ I = 2{I_0}\left( {1 + \cos \phi } \right) \\ I = 4{I_0}{\cos ^2}\left( {\frac{\phi }{2}} \right) \\\end{aligned} \end{equation} $$

$$I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos \phi $$ For constructive interference or maximum intensity,

$$\begin{equation} \begin{aligned} \cos \phi = 1 \\ \phi = 2n\pi \quad {\text{where }}n \in I \\\end{aligned} \end{equation} $$ So, $$\begin{equation} \begin{aligned} I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \\ I = {\left( {\sqrt {{I_1}} + \sqrt {{I_2}} } \right)^2} \\\end{aligned} \end{equation} $$

If $I_1=I_2=I_0$, then, $${I_{\max }} = 4{I_0}$$

$$I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos \phi $$ For destructive interference or minimum intensity,

$$\begin{equation} \begin{aligned} \cos \phi = - 1 \\ \phi = (2n + 1)\pi \quad {\text{where }}n \in I \\\end{aligned} \end{equation} $$ So, $$\begin{equation} \begin{aligned} I = {I_1} + {I_2} - 2\sqrt {{I_1}{I_2}} \\ I = {\left( {\sqrt {{I_1}} - \sqrt {{I_2}} } \right)^2} \\\end{aligned} \end{equation} $$

If $I_1=I_2=I_0$, then, $${I_{\min }} = 0$$