Diffraction Patterns Due to a ‘Single-Slit’ and a ‘Circular Aperture’

1. Single-Slit Diffraction:

• Huygens’ Principle and secondary wavelets:

• Each point on a wavefront acts as a source of secondary wavelets that spread out in all directions. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Diffraction of light:

• The bending of light waves around the edges of an obstacle or through a narrow aperture. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Single-slit diffraction pattern:

• When a beam of monochromatic light passes through a single narrow slit, the light spreads out and produces a pattern of alternating bright and dark bands on a screen placed behind the slit. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Diffraction minima and maxima:

• Dark bands (minima) occur when waves from different parts of the slit interfere destructively, while bright bands (maxima) occur when waves interfere constructively. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Diffraction bands:

• The pattern of alternating bright and dark bands produced by single-slit diffraction. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Intensity distribution in single-slit diffraction:

• The intensity of light at a point on the screen varies as $$I = I_0 \frac{\sin^2 \left(\frac{\pi a \sin \theta}{\lambda}\right)}{\left(\frac{\pi a \sin \theta}{\lambda}\right)^2}$$ where $$I_0$$ is the intensity of light at the central maximum, $$a$$ is the slit width, $$\lambda$$ is the wavelength of light, and $$\theta$$ is the angle of diffraction. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

2. Circular Aperture Diffraction:

• Diffraction of light by a circular aperture:

• When a beam of monochromatic light passes through a circular aperture, the light spreads out and produces a pattern of concentric bright and dark rings on a screen placed behind the aperture. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Airy disk:

• The central bright spot in the circular aperture diffraction pattern. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Diffraction rings:

• The concentric bright and dark rings surrounding the Airy disk in the circular aperture diffraction pattern. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Intensity distribution in circular aperture diffraction:

• The intensity of light at a point on the screen varies as $$I = I_0 \left[ \frac{2J_1 \left(\frac{\pi a \sin \theta}{\lambda}\right)}{\frac{\pi a \sin \theta}{\lambda}}\right]^2$$ where $$I_0$$ is the intensity of light at the center of the Airy disk, $$a$$ is the radius of the aperture, $$\lambda$$ is the wavelength of light, $$\theta$$ is the angle of diffraction, and $$J_1$$ is the Bessel function of the first order. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

3. Comparison of Single-Slit and Circular Aperture Diffraction:

• Similarities and differences in diffraction patterns:

• Both single-slit and circular aperture diffraction patterns exhibit alternating bright and dark bands/rings due to interference of light waves.

• The central maximum is brightest in both patterns.

• Effect of aperture size on diffraction:

• As the aperture size decreases, the diffraction bands/rings become wider.

• Smaller apertures produce wider diffraction patterns. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Effect of wavelength on diffraction:

• Diffraction is more pronounced for shorter wavelengths. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Rayleigh criterion for resolution:

• The minimum angular separation between two point sources that can be resolved by an optical instrument is given by the Rayleigh criterion: $$\theta = 1.22 \frac{\lambda}{D}$$ where $$\theta$$ is the angular separation, $$\lambda$$ is the wavelength of light, and $$D$$ is the diameter of the aperture. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

4. Applications of Diffraction:

• Diffraction grating:

• A grating consists of a large number of parallel slits or lines closely spaced together. It produces sharp and well-defined diffraction spectra, used in spectrometers and lasers. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Spectrometer:

• An instrument used to measure the wavelength of light by analyzing its diffraction pattern. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Resolving power of optical instruments:

• The ability of an optical instrument to distinguish between two closely spaced objects is determined by its resolving power, which is related to the diffraction limit. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Diffraction-limited imaging:

• The limit to the resolution of an optical system imposed by diffraction. (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

5. Mathematical Treatment:

• Mathematical analysis of single-slit diffraction:

• Using Huygens’ principle and considering the interference of secondary wavelets, the intensity distribution $$I$$ in single-slit diffraction is derived as $$I = I_0 \frac{\sin^2 \left(\frac{\pi a \sin \theta}{\lambda}\right)}{\left(\frac{\pi a \sin \theta}{\lambda}\right)^2}$$ (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

• Mathematical analysis of circular aperture diffraction:

• Using Huygens’ principle and considering the interference of secondary wavelets, the intensity distribution $$I$$ in circular aperture diffraction is derived as $$I = I_0 \left[ \frac{2J_1 \left(\frac{\pi a \sin \theta}{\lambda}\right)}{\frac{\pi a \sin \theta}{\lambda}}\right]^2$$ (Reference: NCERT Class 12, Chapter 10, ‘Waves’)

6. Important Formulas:

• Single-slit diffraction minima: $$d\sin \theta = m\lambda \text{, } m = \pm 1, \pm 2, \pm 3,…$$
• Circular aperture diffraction minima: $$J_1\left( \frac{\pi a\sin\theta}{\lambda} \right) = 0$$
• Rayleigh criterion for resolution: $$\theta = 1.22 \frac{\lambda}{D}$$