Diffraction Knowledge-Based Study

Study notes

1. Wave Nature and Diffraction

   Diffraction is the bending and spreading of waves around obstacles or through apertures whose dimensions are comparable to the wavelength of the wave.

   According to Huygens' Principle, every point on a primary wavefront acts as a source of secondary spherical wavelets. As a wavefront passes through an opening or past an edge, these secondary wavelets spread out into the geometrical shadow region, superimposing on one another to produce a characteristic intensity pattern.

   Condition for Observable Diffraction

   Diffraction is most noticeable when the aperture width or obstacle size $a$ is on the order of the wavelength $\lambda$:

   • If $a \gg \lambda$, waves propagate in nearly straight lines (rectilinear propagation, described well by ray optics), and diffraction effects are minimal.
   • If $a \approx \lambda$ or $a < \lambda$, significant spreading occurs, and the wave nature of light becomes dominant.

   Single-Slit Fraunhofer Diffraction

   Consider monochromatic light of wavelength $\lambda$ incident normally on a long slit of width $a$. When viewed on a distant screen (or using a converging lens), the resulting pattern consists of a broad, bright central maximum flanked by alternating dark fringes (minima) and weaker secondary maxima.

   Condition for Minima (Destructive Interference)

   To determine where destructive interference occurs on the screen, we divide the slit of width $a$ into pairs of zones whose wavelets cancel out in pairs. The dark fringes occur at diffraction angles $\theta$ satisfying:

   $$a \sin\theta = m\lambda \quad \text{for } m = \pm 1, \pm 2, \pm 3, \dots$$

   Here, $m$ is the order of the minimum. Notice that $m = 0$ does not represent a minimum; instead, $\theta = 0$ corresponds to the center of the central maximum.

   Secondary Maxima

   Between successive minima lie secondary maxima, occurring approximately at:

   $$a \sin\theta \approx \left(m + \frac{1}{2}\right)\lambda \quad \text{for } m = \pm 1, \pm 2, \dots$$

   The intensity drops sharply away from the center: the first secondary maximum has an intensity of only about 4.5% of the central peak.

   Angular and Linear Spread

   The central maximum extends from the first minimum on one side ($m = -1$) to the first minimum on the other side ($m = +1$).

   Angular Width

   Using the small-angle approximation $\sin\theta \approx \theta$ (valid when $\lambda \ll a$), the first minimum occurs at an angular position:

   $$\theta_1 \approx \frac{\lambda}{a}$$

   The total angular width $\Delta\theta$ of the central maximum is twice this angle:

   $$\Delta\theta = 2\theta_1 \approx \frac{2\lambda}{a}$$

   Linear Width on a Distant Screen

   If the screen is located at a distance $L$ from the slit, the linear half-width is $y_1 = L \tan\theta_1 \approx L\theta_1 = \frac{L\lambda}{a}$. Therefore, the total linear width $w$ of the central maximum is:

   $$w = 2y_1 = \frac{2L\lambda}{a}$$

   Key dependencies:

   • Effect of Slit Width ($a$): Narrowing the slit causes the central maximum to broaden. Widening the slit makes the central maximum narrower and sharper.
   • Effect of Wavelength ($\lambda$): Longer wavelengths (e.g., red light) diffract more widely than shorter wavelengths (e.g., blue light).

   Diffraction Limit of Optical Instruments

   When light from a point source passes through a circular aperture of diameter $D$ (such as a camera lens, telescope aperture, or human pupil), it forms an Airy disk—a central bright disk surrounded by faint concentric rings.

   The angular radius $\theta_R$ of the first dark ring is given by:

   $$\theta_R = 1.22 \frac{\lambda}{D}$$

   The Rayleigh Criterion

   Two point sources are considered just resolved when the central maximum of one Airy disk falls directly on the first minimum (dark ring) of the other. Thus, the minimum angular separation $\theta_{\text{min}}$ required to resolve two sources is:

   $$\theta_{\text{min}} = 1.22 \frac{\lambda}{D}$$

   • To improve optical resolution (achieve a smaller $\theta_{\text{min}}$), an instrument must use a larger aperture diameter $D$ or operate at a shorter wavelength $\lambda$.

   Comparing Interference and Diffraction

   While both phenomena originate from wave superposition, they differ fundamentally in how the interfering waves arise:

   • Interference: Involves the superposition of waves originating from a discrete, finite number of distinct sources or slits (e.g., Young's double-slit experiment).
   • Diffraction: Involves the mutual superposition of an infinite number of continuous wavelets emerging from different parts of the same wavefront through a single opening or around an obstacle.

   Combined Double-Slit Pattern

   In realistic double-slit experiments, each slit has a finite width $a$, while the slits are separated by distance $d$ ($d > a$). The resulting intensity pattern consists of closely spaced double-slit interference fringes modulated by the broader single-slit diffraction envelope. Where a double-slit interference maximum coincides with a single-slit diffraction minimum, a missing order occurs.