Interference of Light — Revision Notes

⚡ 30-Second Revision

Superposition Principle: — Resultant displacement is vector sum of individual displacements.
Coherent Sources: — Constant phase difference, same $\lambda$, same frequency.
Path Difference ($\Delta x$): — Difference in distances traveled.
Phase Difference ($\Delta\phi$): — $\Delta\phi = \frac{2\pi}{\lambda} \Delta x$.
Constructive Interference (Bright Fringes): — $\Delta x = n\lambda$, $\Delta\phi = 2n\pi$ ($n=0,1,2...$). $I_{max} = 4I_0$.
Destructive Interference (Dark Fringes): — $\Delta x = (n + \frac{1}{2})\lambda$, $\Delta\phi = (2n+1)\pi$ ($n=0,1,2...$). $I_{min} = 0$.
Fringe Width (YDSE): — $\beta = \frac{\lambda D}{d}$.
Position of Bright Fringes: — $y_n = \frac{n\lambda D}{d}$.
Position of Dark Fringes: — $y_n' = (n + \frac{1}{2})\frac{\lambda D}{d}$.
Effect of Medium (refractive index $\mu$): — $\lambda' = \frac{\lambda}{\mu}$, $\beta' = \frac{\beta}{\mu}$.
Shift due to Thin Sheet (thickness t, refractive index $\mu$): — $\Delta y = \frac{D}{d}(\mu - 1)t$. Shift is towards the side of the sheet.

2-Minute Revision

Interference of light is the phenomenon of redistribution of light energy due to the superposition of two or more coherent light waves. Coherent sources are crucial, meaning they must have a constant phase difference and the same wavelength.

Young's Double Slit Experiment (YDSE) is the primary demonstration. In YDSE, light from two narrow slits, separated by 'd', interferes on a screen 'D' distance away. Constructive interference (bright fringes) occurs when the path difference is an integral multiple of the wavelength ($\Delta x = n\lambda$), while destructive interference (dark fringes) occurs when the path difference is an odd multiple of half the wavelength ($\Delta x = (n + \frac{1}{2})\lambda$).

The distance between consecutive bright or dark fringes is the fringe width, $\beta = \frac{\lambda D}{d}$. If the apparatus is immersed in a medium of refractive index $\mu$, the wavelength and thus the fringe width decrease by a factor of $\mu$.

Introducing a thin transparent sheet in the path of one slit causes a shift in the entire pattern, given by $\Delta y = \frac{D}{d}(\mu - 1)t$, but the fringe width remains unchanged. White light produces a central white fringe and colored fringes on either side.

5-Minute Revision

Interference is a wave phenomenon where two or more waves combine to form a resultant wave of greater, lower, or the same amplitude. For light, this means a redistribution of light energy, creating a pattern of bright and dark fringes.

The key to observing a stable pattern is using coherent sources, which maintain a constant phase difference and have identical wavelengths and frequencies. This is typically achieved by splitting light from a single monochromatic source, as in Young's Double Slit Experiment (YDSE).

In YDSE, light from two slits $S_1$ and $S_2$ (separated by 'd') reaches a screen at distance 'D'. The path difference ($\Delta x$) between the waves reaching a point 'y' from the center of the screen is approximately $\frac{yd}{D}$. This path difference dictates the type of interference:

Constructive Interference (Bright Fringes): — Occurs when $\Delta x = n\lambda$ (where $n=0, 1, 2, \dots$). The position of the $n^{th}$ bright fringe is $y_n = \frac{n\lambda D}{d}$. The central fringe ($n=0$) is always bright.
Destructive Interference (Dark Fringes): — Occurs when $\Delta x = (n + \frac{1}{2})\lambda$ (where $n=0, 1, 2, \dots$). The position of the $n^{th}$ dark fringe is $y_n' = (n + \frac{1}{2})\frac{\lambda D}{d}$.

The fringe width ($\beta$), the distance between two consecutive bright or dark fringes, is given by $\beta = \frac{\lambda D}{d}$. This formula is critical for numerical problems.

Key variations and effects:

1
Effect of Medium: — If the YDSE setup is immersed in a medium with refractive index $\mu$, the wavelength changes to $\lambda' = \frac{\lambda}{\mu}$. Consequently, the fringe width also decreases: $\beta' = \frac{\beta}{\mu}$.
2
Effect of Thin Transparent Sheet: — Placing a thin sheet of thickness 't' and refractive index '$\mu$' in front of one slit introduces an additional path difference of $(\mu - 1)t$. This causes the entire fringe pattern to shift by $\Delta y = \frac{D}{d}(\mu - 1)t$ towards the side of the sheet. The fringe width, however, remains unchanged.
3
White Light Interference: — Using white light results in a central white bright fringe (as $\Delta x=0$ for all colors). On either side, colored fringes appear, with violet closer to the center and red farther away, due to their different wavelengths and thus different fringe widths. After a few fringes, the colors overlap, and the pattern becomes indistinct.
4
Intensity: — If individual slit intensities are $I_1$ and $I_2$, then $I_{max} = I_1 + I_2 + 2\sqrt{I_1 I_2}$ and $I_{min} = I_1 + I_2 - 2\sqrt{I_1 I_2}$. If $I_1 = I_2 = I_0$, then $I_{max} = 4I_0$ and $I_{min} = 0$. The intensity at any point is $I = 4I_0 \cos^2(\frac{\Delta\phi}{2})$.

Example: If $\lambda = 600\text{ nm}$, $D = 1\text{ m}$, $d = 0.2\text{ mm}$. Fringe width $\beta = \frac{(600 \times 10^{-9}) \times 1}{0.2 \times 10^{-3}} = 3 \times 10^{-3}\text{ m} = 3\text{ mm}$.

Position of 2nd bright fringe ($n=2$): $y_2 = \frac{2 \times 600 \times 10^{-9} \times 1}{0.2 \times 10^{-3}} = 6 \times 10^{-3}\text{ m} = 6\text{ mm}$. Position of 1st dark fringe ($n=0$): $y_0' = \frac{(0 + 0.

5) \times 600 \times 10^{-9} \times 1}{0.2 \times 10^{-3}} = 1.5 \times 10^{-3}\text{ m} = 1.5\text{ mm}$.

Prelims Revision Notes

Interference of light is the result of superposition of two or more coherent light waves. Coherent sources are paramount, meaning they must emit light of the same wavelength (monochromatic) and maintain a constant phase difference. This is usually achieved by deriving two sources from a single primary source, as in Young's Double Slit Experiment (YDSE).

Key Formulas for YDSE:

Path Difference ($\Delta x$): — For a point P at distance 'y' from the central axis on a screen 'D' away from slits separated by 'd', $\Delta x = \frac{yd}{D}$.
Phase Difference ($\Delta\phi$): — $\Delta\phi = \frac{2\pi}{\lambda} \Delta x$.
Constructive Interference (Bright Fringes): — Occurs when $\Delta x = n\lambda$ or $\Delta\phi = 2n\pi$, where $n = 0, \pm 1, \pm 2, \dots$. The position of the $n^{th}$ bright fringe is $y_n = \frac{n\lambda D}{d}$. The central bright fringe is at $y_0 = 0$.
Destructive Interference (Dark Fringes): — Occurs when $\Delta x = (n + \frac{1}{2})\lambda$ or $\Delta\phi = (2n+1)\pi$, where $n = 0, \pm 1, \pm 2, \dots$. The position of the $n^{th}$ dark fringe is $y_n' = (n + \frac{1}{2})\frac{\lambda D}{d}$.
Fringe Width ($\beta$): — The distance between two consecutive bright or dark fringes is $\beta = \frac{\lambda D}{d}$.
Angular Fringe Width: — $\theta_\beta = \frac{\beta}{D} = \frac{\lambda}{d}$.

Effects of Changing Parameters:

Medium Change: — If the apparatus is immersed in a medium of refractive index $\mu$, the wavelength changes to $\lambda' = \frac{\lambda}{\mu}$, and consequently, the fringe width becomes $\beta' = \frac{\beta}{\mu}$. The pattern shrinks.
Thin Transparent Sheet: — Introducing a sheet of thickness 't' and refractive index '$\mu$' in the path of one slit causes a shift in the entire pattern by $\Delta y = \frac{D}{d}(\mu - 1)t$. The shift is towards the slit where the sheet is placed. Fringe width remains unchanged.
White Light: — Central fringe is white. Subsequent fringes are colored, with violet closer and red farther from the center. Beyond a few fringes, the pattern becomes indistinct.

Intensity Distribution:

If $I_1$ and $I_2$ are individual intensities, $I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2$ and $I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2$.
If $I_1 = I_2 = I_{slit}$, then $I_{max} = 4I_{slit}$ and $I_{min} = 0$. The intensity at any point is $I = 4I_{slit} \cos^2(\frac{\Delta\phi}{2})$.

Important Note: Always ensure consistent units (SI units are preferred) for all quantities in calculations. Pay attention to the order of fringes (e.g., first dark fringe corresponds to $n=0$ in the dark fringe formula).

Vyyuha Quick Recall

For Wave Light, Double Slit Demonstrates Interference: Fringe Width is Lambda D over d.

Fringe Width ($\beta$) = Lambda ($\lambda$) * D (screen distance) / d (slit separation).