Notes from Toppers
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: $$\theta = 1.22 \frac{\lambda}{D}$$