Double-slit interference Knowledge-Based Study

Study notes

1. Wave Nature of Light and Coherence

   In 1801, Thomas Young provided definitive evidence for the wave nature of light by demonstrating that light waves can produce an interference pattern. When light passes through two closely spaced parallel slits, each slit acts as a secondary source of coherent cylindrical waves, as described by Huygens' principle.

   For a sustained and observable interference pattern to form on a screen, the two light sources must satisfy two essential conditions:

   • Monochromaticity: The waves must have the same wavelength (and frequency).
   • Coherence: The waves must maintain a constant phase difference over time.

   In Young's arrangement, coherence is achieved by illuminating both slits from a single primary light source. The light diffracted by the two slits overlaps in the region between the slits and the viewing screen, resulting in alternating bright and dark bands known as interference fringes.

   Derivation of the Optical Path Difference

   Consider two parallel slits separated by a distance $d$. Light of wavelength $\lambda$ emerges from both slits and travels to an arbitrary point $P$ on a screen placed at a perpendicular distance $L$ from the slits, where $L \gg d$.

   Let $y$ be the distance along the screen from the central maximum (the point directly opposite the midpoint of the slits) to point $P$. The lines connecting the two slits to point $P$ make an angle $\theta$ with the central axis.

   Because the distance to the screen is vastly greater than the slit separation, the two rays heading toward $P$ are essentially parallel. The extra distance traveled by the wave from the lower slit compared to the upper slit is the path difference, denoted by $\Delta L$:

   $$\Delta L = d \sin\theta$$

   For small angles ($\theta \ll 1\text{ rad}$), we apply the small-angle approximation:

   $$\sin\theta \approx \tan\theta = \frac{y}{L}$$

   Substituting this relation gives the path difference in terms of measurable laboratory dimensions:

   $$\Delta L \approx \frac{d \cdot y}{L}$$

   Bright and Dark Fringe Formations

   The type of interference occurring at point $P$ depends solely on the phase relationship dictated by the path difference $\Delta L$.

   1. Constructive Interference (Bright Fringes / Maxima)

   Constructive interference occurs when the waves arrive at the screen in phase. This happens when the path difference is an integer multiple of the wavelength:

   $$d \sin\theta = m\lambda \quad \text{or} \quad y_m = \frac{m\lambda L}{d}, \quad m = 0, \pm 1, \pm 2, \dots$$

   Here, $m$ is the order number of the fringe. The central bright fringe corresponds to $m = 0$.

   2. Destructive Interference (Dark Fringes / Minima)

   Destructive interference occurs when the two waves arrive completely out of phase (a phase difference of $\pi$ radians or $180^\circ$). This occurs when the path difference is an odd half-integer multiple of the wavelength:

   $$d \sin\theta = \left(m + \frac{1}{2}\right)\lambda \quad \text{or} \quad y'_m = \left(m + \frac{1}{2}\right)\frac{\lambda L}{d}, \quad m = 0, \pm 1, \pm 2, \dots$$

   Fringe Width ($\Delta y$) and Parametric Variations

   The distance between any two consecutive bright fringes (or two consecutive dark fringes) is called the fringe width (or fringe spacing), denoted by $\Delta y$:

   $$\Delta y = y_{m+1} - y_m = \frac{\lambda L}{d}$$

   This equation illustrates how experimental parameters directly influence the fringe spacing:

   • Wavelength ($\lambda$): Red light has a longer wavelength than blue light. Therefore, using red light produces wider, more spread-out fringes than blue light.
   • Screen Distance ($L$): Moving the screen farther away increases the fringe width proportionally.
   • Slit Separation ($d$): Bringing the two slits closer together increases the fringe width. If the slits are moved farther apart, the fringes become narrower and pack more tightly.
   • Medium Effect: If the entire apparatus is submerged in a liquid with refractive index $n$, the wavelength of light decreases to $\lambda_n = \frac{\lambda}{n}$. Consequently, the fringe spacing decreases by a factor of $n$: $$\Delta y_n = \frac{\Delta y}{n}$$