CHPTR 36 Notes: Diffraction

Diffraction Phenomenon

• Definition of Diffraction

  • Occurs when light passes through a small aperture or opening, causing it to spread into regions that would remain in shadow if light traveled in straight lines.

  • This phenomenon is not exclusive to light; it also occurs with other waves such as sound and water waves.

  • General conditions for diffraction:

  • Waves passing through small openings.

  • Waves bending around obstacles.

  • Waves passing past sharp objects.

Types of Diffraction

• Near-field Diffraction (Fresnel Diffraction)

  • Occurs when both the source of the waves and the screen detecting these waves are relatively close to the obstacle causing diffraction.

• Far-field Diffraction (Fraunhofer Diffraction)

  • Occurs when the source and screen are far enough from the obstacle that all relevant rays can be considered parallel.

  • Fraunhofer diffraction patterns consist of:

  • A central maxima that is the brightest.

  • Weaker maxima flanking the central maxima.

  • Alternating dark fringes in between the maxima.

Diffraction Patterns from Single Slits

• When light passes through a single, narrow slit, it produces a diffraction pattern similar to the interference pattern seen in Young’s double slit experiment.

• Key features of this pattern:

  • The intensity of the fringes is variable.

  • The central fringe is the brightest among all the fringes.

Mathematical Descriptions of Diffraction Patterns

• Conditions for Dark Fringes

  • Formula:
    $a imes heta = m imes rac{ ext{λ}}{1, 2, ext{…}}$

  • Here, 'a' represents slit width, 'm' indicates the order of the dark fringe, and 'λ' is the wavelength of light.

  • This formula gives the vertical location of the dark fringes on the screen.

• Intensity Distribution

  • Total phase difference between waves from the top and bottom portions of the slit can be described mathematically.

  • Intensity distribution on the screen is given by:
    $I = I_0 rac{ ext{sin}^2\bigg( rac{β}{2}\bigg)}{β^2}$
    where:

  • $β = rac{2πa ext{sin}( heta)}{λ}$

Diffraction Gratings

• Definition and Characteristics

  • Diffraction grating consists of a large number of equally spaced parallel slits.

  • Example:

  • A grating with 5000 lines/cm has a slit spacing of approximately 2µm.

  • The diffraction pattern observed on a screen results from the combined effects of interference and diffraction.

Conditions for Maxima in Diffraction Patterns

• Mathematical Condition for Bright Fringes (Maxima)

  • The condition for maxima (bright fringes) in the interference pattern can be expressed as:
    $d imes ext{sin}( heta) = m imes ext{λ}$

  • Where 'd' is the slit spacing, and 'm' represents the order of maxima (0, 1, 2,…).

  • This expression can be employed to calculate the wavelength of light, given the grating spacing and angle.

Applications of Diffraction

• Wavelength Determination

  • The wavelength of any electromagnetic wave can be determined if a grating with proper spacing (where d is approximately equal to λ) is available.

  • Atomic layers in solids, which have a spacing of about 0.1nm, can serve as gratings to diffract X-rays, as they match wavelengths of the order of 0.1nm.

• Bragg’s Law

  • For constructive interference in diffraction, the condition given by Bragg’s law is:
    $2d imes ext{sin}(θ) = m imes ext{λ}$

  • Where the angle θ is measured from the plane of the crystal lattice.

  • The significance of this law is highlighted with an example:

  • The Laue pattern produced by a single beryllium aluminum silicate crystal.

  • X-ray diffraction techniques have been used to elucidate the double helix structure of DNA.

Examples of Diffraction Patterns

• Example of a diffraction pattern made by four point sources through circular apertures.

Resolution of the Human Eye

• The resolution of the human eye for light of wavelength 500nm, assuming a pupil size of 2mm is calculated with the formula: $D_{res} = rac{1.22 imes ext{λ}}{D imes θ}$

  • In this case:

  • $D_{res} = 3.1 imes 10^{-1}$, which corresponds to 1.22 radians.

  • This value equates to the thickness of a human hair measured at a distance of 25cm.