Answer any one question. a) Define coherent sources of light. Prove that the dark and bright fringes are equally spaced in Young's double slit experiment. b) What is diffraction grating? Discuss the…

a) Define coherent sources of light. Prove that the dark and bright fringes are equally spaced in Young's double slit experiment.

Definition of Coherent Sources

Coherent sources of light are two or more sources that emit light waves with a constant phase difference and same frequency/wavelength. This means the waves maintain a fixed relationship over time, producing a stable interference pattern. In Young’s double-slit experiment, a single light source is split into two secondary sources (slits) using a narrow slit, ensuring coherence.

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Proof that Fringes are Equally Spaced

In Young’s double-slit experiment, light from two coherent slits $S_1$ and $S_2$ interferes on a screen. The path difference $\Delta x$ between waves from the two slits at a point $P$ on the screen is given by: $$\Delta x=d\sin \theta$$ where:

• $d$ = separation between the slits,
• $\theta$ = angle between the central line and the line joining the slits to point $P$.

For constructive interference (bright fringes), the path difference is an integer multiple of the wavelength $\lambda$: $$d\sin \theta =m\lambda (m=0,\pm 1,\pm 2,\dots )$$ For destructive interference (dark fringes), the path difference is an odd multiple of $\frac{\lambda }{2}$: $$d\sin \theta =\left(m+\frac{1}{2}\right)\lambda (m=0,\pm 1,\pm 2,\dots )$$

For small angles ($\theta \approx \sin \theta$), the position $y$ of the $m$-th fringe from the central maximum is: $$y=\frac{m\lambda D}{d}\text{(bright fringes)}$$ $$y=\frac{\left(m+\frac{1}{2}\right)\lambda D}{d}\text{(dark fringes)}$$ where $D$ is the distance between the slits and the screen.

Spacing between adjacent fringes (bright or dark): The distance between two consecutive bright fringes ($m$ and $m+1$) is: $$\Delta y=y_{m+1}-y_m=\frac{(m+1)\lambda D}{d}-\frac{m\lambda D}{d}=\frac{\lambda D}{d}$$ Similarly, the spacing between two consecutive dark fringes is also $\frac{\lambda D}{d}$.

Since $\frac{\lambda D}{d}$ is constant for a given setup, both bright and dark fringes are equally spaced.

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b) What is a diffraction grating? Discuss the formation of diffraction pattern due to a diffraction grating.

Definition of Diffraction Grating

A diffraction grating is an optical component with a large number of parallel, closely spaced slits (or grooves) that diffract light into several beams traveling in different directions. These slits are typically separated by distances comparable to the wavelength of light.

Formation of Diffraction Pattern

When monochromatic light (single wavelength $\lambda$) is incident normally on a diffraction grating with $N$ slits separated by distance $d$, the light waves from each slit interfere constructively or destructively, producing a diffraction pattern on a screen.

Condition for Constructive Interference (Bright Fringes): For maxima (bright fringes), the path difference between waves from adjacent slits must be an integer multiple of the wavelength: $$d\sin \theta =m\lambda (m=0,\pm 1,\pm 2,\dots )$$ where:

• $\theta$ = angle of diffraction,
• $m$ = order of the fringe (central maximum is $m=0$).

Intensity Distribution:

• The central maximum ($m=0$) is the brightest because all waves interfere constructively.
• Higher-order maxima ($m=\pm 1,\pm 2,\dots$) become progressively dimmer due to the interference of multiple waves (the intensity $I$ of the $m$-th order is proportional to ${\left(\frac{\sin \beta }{\beta }\right)}^2$, where $\beta =\frac{\pi d\sin \theta }{\lambda }$).
• Minima (dark fringes) occur when the path difference causes destructive interference for all but one slit, given by: $$d\sin \theta =\left(m+\frac{1}{N}\right)\lambda \text{(for }N\text{ slits)}$$ where $N$ is the total number of slits.

Key Features of the Pattern:

1. Sharp and well-defined maxima due to the large number of slits.
2. Higher resolving power than a double slit (can distinguish between closely spaced wavelengths).
3. Multiple orders of diffraction (unlike double-slit, which produces only two first-order maxima).
4. Angular dispersion increases with order $m$, allowing spectral analysis (used in spectrometers).

The diffraction pattern consists of bright lines (spectra) at specific angles, with the spacing between fringes depending on $\lambda$ and $d$. This property is exploited in spectroscopy to analyze light sources by decomposing them into their constituent wavelengths.