Concepts Reviewed

1. Wave properties in double slit intereference

2. Peculiarities of electromagnetic waves

Tunable Parameters

1. Distance between the screen and the slits

2. Distance between the slits

3. Wavelength of light

4. Polarization of the two beams

Light

$\left[\frac{1}{\sqrt{\mu_0 \epsilon_0}}\right]=L T^{-1}$

$\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ = Speed of Light

$\epsilon_0$ = Permitting

$\mu_0$ = Permeability

$\mu_0$ = Definition

$\vec{E}(t, \vec{r}) \rightarrow$ Electric field

$\vec{B}(t, \vec{r}) \rightarrow$ Magnetic field

$\vec{B}(t, \vec{r}) \vec{E}(t) \vec{r}) k=\frac{2 \pi}{\lambda}$

$\omega = 2 \pi \nu$

$\nu \quad \vec{n}$

$\vec{E}$ = $\vec{E}_{0} \cos (k ~\vec{n}. \vec{r} - \omega t)$

$| \vec{B}| = |\vec{E}|/c$

$\vec{E} \times \vec{B} \parallel \vec{n} ;$

$\vec{E} . \vec{B} = 0; \vec{E} . \vec{n} = 0$

$\vec{E_1}=\vec {E_{10}} \cos (k ~\hat{n}_1 \cdot \vec{r}-\omega t)$

$\vec{E_2}=\vec{E}_{20} \cos (k ~\hat{n}_2 \cdot \vec{r}-\omega t)$

$\hat{n}_1=\cos \alpha ~\hat{i}+\sin \alpha ~\hat{j}$

$\hat{n}_2=\cos \beta ~\hat{i}+\sin \beta ~\hat{j}$

$D \gg d$

$\cos \alpha=\frac{D}{\left[\left(y-\frac{d}{2}\right)^2+D^2\right]^{1 / 2}}$

$\sin \alpha=\frac{y-\frac{d}{2}}{\left[\left(y-\frac{d}{2}\right)^2+D^2\right]^{1 / 2}}$=$\frac{y}{D}$

$\sin \alpha \approx \tan \alpha$

$\frac{d}{D} \gg 1$

$\approx$ 1 + binomial corrections

$y + \frac{d}{2} \rightarrow \sin \beta$

$\vec{E_1}=\vec{E_{10}} \cos [\vec{k} \cdot (\vec{r}-\frac{\vec{d}}{2})-\omega t]$

$\vec{E_2}=\vec{E_{20}} \cos [\vec{k} \cdot (\vec{r}+\frac{\vec{d}}{2})-\omega t]$

$\vec{E_T}=\vec {E_1}+\vec{E_2}$

$\left|\vec{E}_ T\right|^2 \left|\vec{E}_ 1\right|^2+\left|\vec{E}_ 2\right|^2+ 2 {\vec{E}_ 1 \cdot \vec{E}_2}$

$D \gg d$

$\left\langle\cos ^2\right\rangle=\frac{1}{2}$

$\left\langle\vec{E^2_ {T}}\right\rangle = \frac{1}{2} \vec{E_ {10}^{2}} +\frac{1}{2} \vec{E_ {20}^2}+2\left\langle\vec{E_ 1} \cdot \vec{E_2}\right\rangle$

(Intereference Term)

$\vec{E_1}=\vec{E_{10}} \cos[\vec{k} \cdot(\vec{r}-\frac{d}{2})-\omega t]$

$\vec{E_2}=\vec{E_{20}} \cos [\vec{k} \cdot (\vec{r}+\frac{\vec{d}}{2})-{\omega t}]$

$\langle\vec{E_ 1}. \vec{E_ 2}\rangle=\vec{E_ {10}} \cdot \vec{E_{20}}\langle \cos ()\cos ()\rangle$

$\cos [\vec{k} \cdot (\vec{r}+\frac{\vec{d}}{2})-\omega t]$ $\cos [\vec{k} \cdot(\vec{r}-\frac{\vec{d}}{2})-\omega t]$

$\cos A \cos B =\frac{1}{2}[\cos (A-B)+\cos (A+B)]$

$\langle L H S\rangle=\left\langle\frac{1}{2}[\right. \cos 2(\vec{k} \cdot \vec{r}-\omega t) +\cos \vec{k} \cdot \vec{d}]\rangle$

$\langle | \vec {E_ 1}|^2 + |\vec{E_ 2^2}|^2 + 2 \vec{E_ 1}.\vec{E_ 2}\rangle=\frac{1}{2} \vec{E_ {10}^2} + \frac{1}{2} \vec{E_ {20}^2}+ \frac{1\times 2}{2} \cos {(\vec k.\vec d)}\vec{E_ {10}}.\vec{E_{20}}$

$\vec{k} \cdot \vec{d}=k d \sin \alpha=k d\left(\frac{y}{D}\right)$

$I(y)=\frac{1}{2} \vec{E_{10}^2}+\frac{1}{2} \vec{E_{20}^2}+\vec{E_{10}} \cdot \vec{E_{20}} \times \cos \left(k d \frac{y}{D}\right)$

$\vec{E_{10}} \cdot \vec{E_{20}}=\left|E_{10} ~E_{20}\right|$ (parallel)

$\rightarrow$ Standard Interference Condition.

$\vec{E_{10}} \cdot \vec{E_{20}}=\left|E_{10} ~E_{20}\right|$

Intensity will be a maximum if $\cos () = -1$

$\vec{E_{10}} \parallel \vec{E_{20}}$

$\hat{E_{10}}=-\hat{E_{20}}$

$\vec{E_{10}} \cdot \vec{E_{20}}=0$

$\vec{E_{10}} \parallel \vec{E_{20}}$ : Standard Conditions

$\vec{E_{10}}$ : Anti Parallel

$\vec{E_{20}}$ : Exactly Opposite Conditions

$\vec{E_{10}} \perp \vec{E_{20}}$: The Pattern Vanishes Completely

Michaelson Interferometer

Max. Zehnder Interferometer

Max Zehnder

AJP: Rodrigues

We can alter the conditions for maxima and minima! Energy is proportional to $\vec{E^2}$

Conclusion: Wave theory of light rests on secure experimental foundations

Time line