C.3.3 Diffraction and interference patterns (HL only)

Single-Slit Diffraction

Understanding Single-Slit Diffraction

1. When light passes through a single narrow slit, it spreads out and forms a diffraction pattern on a screen.
2. This pattern consists of a central bright fringe flanked by dimmer andn arrower fringes.

Mathematical Description of Diffraction

The position of the first minimum in the diffraction pattern is determined by the equation:

$$\theta =\frac{\lambda }{b}$$

where:

• $\theta$ is the angle of the first minimum
• $\lambda$ is the wavelength of the light
• $b$ is the width of the slit

This equation shows that the angle of the first minimum depends on the ratio of the wavelength to the slit width.

Intensity Patterns

1. The intensity of the diffraction pattern is highest at the central maximum and decreases for the secondary maxima.
2. The first secondary maximum is about 4.5% as bright as the central maximum.

Multiple Slits and Diffraction Gratings

Double-Slit Interference

1. In a double-slit experiment, light passing through two slits creates an interference pattern of bright and dark fringes.
2. The condition for constructive interference (bright fringes) is:

$$d\sin \theta =n\lambda$$

where:

• $d$ is the distance between the slits
• $\theta$ is the angle of the fringe
• $n$ is the order of the fringe (an integer)
• $\lambda$ is the wavelength of the light

Diffraction Gratings

1. A diffraction grating consists of many closely spaced slits, producing a more defined interference pattern.
2. The condition for constructive interference is the same as for double slits:

$$d\sin \theta =n\lambda$$

Combined Patterns: Single-Slit Diffraction Modulating Double-Slit Interference

In a real double-slit experiment, the slits have a finite width, causing the single-slit diffraction pattern to modulate the double-slit interference pattern.

This means the intensity of the interference fringes is enveloped by the diffraction pattern.

Intensity and Wavelength: Relationship Between Slit Width, Wavelength, and Diffraction Pattern Brightness

The intensity of the diffraction pattern depends on the slit width and the wavelength of the light.

Key Relationships

• Wider Slits:
  • Allow more light to pass through, increasing the overall intensity of the pattern.
  • However, they also decrease the angular width of the central maximum.
• Longer Wavelengths:
  • Result in wider diffraction patterns, spreading the light over a larger area and reducing the intensity of the central maximum.
• Narrower Slits:
  • Produce wider diffraction patterns but reduce the overall intensity because less light passes through.

Reflection