• $S_1$ and $S_2$ are point sources, drawn from the same wavefront (or phasefront )

  • $\Rightarrow S_1$ and $S_2$ are. "in phase"

  • $\psi_1=a_1 \cos \omega t, \psi_2=a_2 \cos \omega t$

• At $P$,

• $\psi_1=a_1 \cos \left(k r_1-\omega t\right)$

• $\psi_2=a_2 \cos \left(k r_2-\omega t\right)$

• $\delta$, Phase difference $=k\left(r_2-r_1\right)$

• $\left(r_2-r_1\right) \rightarrow \text { Path Difference }$

• $\left(r_2-r_1\right)= \pm n \lambda \rightarrow \text { Bright fringes }$

• $\pm\left(n+\frac{1}{2}\right) \lambda \rightarrow \text { Dark fringes }$

& n=0,1,2...

• At the point $O, r_1 \rightarrow r_{1}0, r_2 \rightarrow r_{2}0 ; r_{1}0=r_{2}0$

• $\Rightarrow$ Path Difference $=0$

• Zeroth order bright fringe.

• If ' $S$ ' is slightly off-axis:

• At $P$,

• $\psi_1=A_1 \cos \left(k r_1-\omega t\right)$

• $\psi_2=A_2 \cos \left(k_2-\omega t+\Delta \phi\right)$

• $\Delta \phi$ is the initial phase difference.

• $\delta=k\left(r_2-r_1\right)+\Delta \phi$

• At $O, r_1=r_2$ but $\delta \neq 0$; $\Rightarrow$ Zeroth order bright fringe will not appear at $O$.

• $S_1 = cos \omega t$

• $S_2 = cos \omega (t- \Delta t)$

• [ $\omega t - \omega \Delta t$]

• [$\omega \Delta t = \Delta \phi$]

• $\omega t - (\omega t - \Delta \phi)$

• For $\delta = 0 , k(r_2-r_1) =-\Delta \phi$

• $\Rightarrow r_2<r_1$

• $\delta = 0$ at the point O' such that

• $S' S_1 + S_1 O' = S' S_2 + S_2 O'$

• The position of the zeroth order maxima the central fringe will be shifted

• The condition for maxima

• $\delta = k(r_2-r_1) + \Delta \phi = \pm n 2 \pi$ {n=0,1,2..........}

• $(r_2-r_1)$ = Path difference = $n . 2\pi \times \frac{\lambda}{2\pi} - \Delta \phi \frac {\lambda}{2\pi}$

• i.e Path Difference = $n\lambda - \frac {\Delta \phi}{2\pi} \lambda$ for $n^n$ maxima

• For the $n^{\text {n }}$ Bright fringe, if $\Delta \phi$ is a constant, Path Difference $\left(r_2-r_1\right)=\frac{x_n^{\prime} d}{D}=n \lambda-C$, --(i)

• where $C=\left(\frac{\Delta \phi}{2 \pi}\right) \lambda$ is a constant.

• For the $(n+1)^{\text {th }}$ fringe, $\frac{x_{n+1}^{\prime} d}{D}=(n+1) \lambda-c-(ii)$

• Fringe Shift, $\Delta x$ (due to Phase Difference $\Delta \phi$ )

• If $x_n^{\prime}$ is the new position of the $n^{\text {th }}$ order Bright fringe, in the presence of a constant phase difference $\Delta \phi$ between $S_1$ and $S_2$,

• $\frac{x_n^{\prime} d}{D}=n \lambda-C=\frac{x_n d}{D}-c$

• $\therefore\left(x_n-x_n^{\prime}\right) \frac{d}{D}=C=\left(\frac{\Delta \phi}{2 \pi}\right)^\lambda$

• or Fringe shift (of the $n^n$ fringes) , $\Delta x_n = \frac{\Delta \phi}{2 \pi}) \frac{\lambda D}{d}=\frac{\Delta \phi}{2 \pi} \beta$

• which is independent of $n$.

• Thus, Fringe shift $\Delta x=\left(\frac{\Delta \phi}{2 \pi}\right) \beta$

-depends on $\Delta \phi$

• (Geometry of the problem)

• $s s^{\prime}=l$

• $O O^{\prime}=x^{\prime}$

• $s_1 s_2=d$

• For the central fringe(path difference =0)

• $s^{\prime} s_1+s_1 O^{\prime}=s^{\prime} s_2+s_2 O^{\prime}$

• $\rightarrow s_2 O^{\prime}-s_1 O^{\prime}=s^{\prime} s_1-s^{\prime} s_2$

• $\Rightarrow \frac{x'}{D}=\frac{l}{L} \rightarrow \theta^{\prime}=\theta$

• $\rightarrow \mid r_2-r_1=q_1-q_2$

• $\frac{x^{\prime} d}{D}=\frac{l}{L} d$

• $\rightarrow s^{\prime} o^{\prime}$ is a straight line passing through M.

• $l$ - Source offeat $\Delta x=x^{\prime}-0-$ Fringe shift
• $\frac{x^{\prime}}{D}=\frac{l}{L}$
• or $\Delta x=\left(\frac{D}{L}\right) l$ is the fringe shift doe to source offset by $l$.

• Example: $D=100 \mathrm{cm}$, L $=10 \mathrm{cm}$ For an offset of $1 \mathrm{mm}, \Delta x=\frac{109}{10} \times 1 \mathrm{mm}=10 \mathrm{mm}$

• Ans: Fringe shift will also occur if the aperture Plate QQ' is offset with respect to 50 . This type of fringe shift is called Lateral shift

• Summary: Fringe Shift $\Delta x$ due to $\Delta \phi$

• 1. If there is a constant Phase Difference $\Delta \phi$ (including $\Delta \phi=0$ ) between the interfering waves, then there can be sustained (observable) Interference fringes.

• 2. Waves with a constant $\Delta \phi$ among them are called "Coherent Waves"

• $\Delta \phi=0 \Rightarrow$ Waves are in phase"

• $\Delta \phi = \pi \rightarrow$ Waves are "out of phase"

• 2. The "central fringe" ( and all other fringes) would shift by an amount $\Delta x \propto \Delta \phi$, but the "fringe pattern" and "fringe width" does not change.

• $\Delta x=\left(\frac{\Delta \phi}{2 \pi}\right) \beta$

• Q: Then how to measure the fringe shift, in practice? (Particularly when $\Delta \phi$ is several times $2 \pi$ )

• Ans: Use of "white light interference"

• Q: What if $\Delta \phi$ varies randomly with time? $\Delta \phi=\Delta \phi(t)$ at any point. When do we have such a situation?

• If $S_1$ and $S_2$ are two independent sources or derived from an extended sources:

• hv = $E_2 - E_1$

• I = $4 I_0 (cos^2 \frac{\delta}{2})$

• $\delta \equiv \delta (t)$

• $k(r_2 - r_1) + \Delta \phi$

• I = $4 I_0 < (cos^2 \frac{\delta}{2})>$

• $(cos^2 \frac{\delta}{2}) = \frac {1}{2}$

• = $2 I_0$

• $\psi_1 = A_1 sin \omega_1 t$

• $\psi_2 = A_2 sin \omega_2 t$

• $\omega_2 = 2 \pi f_2$

• $A_1 sin \omega_1 t$

• $A_2 sin \omega_2 t$

• $\Delta \phi (t)$

• $\Delta \phi =(\omega_2 - \omega_1)$ t

• $f_1 , f_2 \approx 10^{14} H_2 \Delta \phi (t)$

• Interference is not possible between two different wavelength.

• Consider 3 wavelengths:

• $\lambda_1=400 \mathrm{nm} \text { (near Blue) }$

• $\lambda_2=500 \mathrm{nm} \text { (near Green) }$

• $\lambda_3=600 \mathrm{nm} \text { (near Orange) }$

• Reveal : 'Blue' interferes with'Blue'. only,

• Interference in the Young's double -hole arrangement,

• $\beta=\left(\frac{D}{d}\right)^\lambda$. Assume $D=1 \mathrm{m}, d=1 \mathrm{mm}$

• For,

• $\lambda_1=400 \mathrm{nm}$

• $\lambda_2=500 \mathrm{km}$

• $\lambda_3=600 \mathrm{nm},$

• $\beta_1=0.4 \mathrm{mm}$

• $\beta_2=0.5 \mathrm{mm}$

• $\beta_3=0.6 \mathrm{mm}$

• $\lambda$ = 400nm

• $\lambda$ = 500nmn

• $\lambda$ = 600nm

• Maxima and minima occur at different positions for different wave length, except the central maximum - some position for all colours.

• What to expect if the source is white light?

• Thus the central fringe can be easily identified and the fringe shift $\Delta x$ , due to a phase change $\Delta \phi$ can be measured accuratily using white light interference.