Interference and Diffraction

Summary

Summary
 
Key terms
Interference: When two waves of the same type meet, they combine to create a larger or smaller wave.
Interference Pattern: Pattern resulting from interference. Light interference patterns are alternating light (minima) and dark bands (maxima).
Phase: It denotes the particular point in the cycle of a waveform, measured as an angle in degrees.
Diffraction: Occurs when a wave hits an obstacle or a barrier.
Fresnel Diffraction: An approximation that can be used to estimate propagation of waves when very close to an antenna emitting electromagnetic waves.
Fraunhofer Diffraction: An approximation that can be used to estimate the propagation of waves when very far from an antenna emitting electromagnetic waves.
Dispersion: The splitting of a ray into its component colors is known as dispersion of light. The band of colors into which the light splits is known as a spectrum.
Polarization: Polarization describes the orientation of the oscillations of certain types of waves, most commonly electromagnetic waves. Plane–polarized light indicates that the electric field lies consistently within a single plane either vertically or horizontally.
Key concepts
  • Interference: A phenomenon in which two waves superimpose to form a resultant wave of greater or lower amplitude. Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency.
  • Huygens' Principle: Every point on a propagating wavefront serves as the source of spherical secondary wavelets, such that the wavefront at some later time is the envelope of these wavelets. If the propagating wave has a frequency f, and is transmitted through the medium at a speed v, then the secondary wavelets will have the same frequency and speed. Laws of Reflection and Refraction can be derived using this principle.
  • Young's Double Slit Experiment: The two slits are illuminated by a single laser beam. If the width of the slits is small enough (less than the wavelength of the laser light), the slits diffract the light into cylindrical waves. These two cylindrical wavefronts are superimposed, and the amplitude, and therefore the intensity, at any point in the combined wavefronts depends on both the magnitude and the phase of the two wavefronts. The difference in phase between the two waves is determined by the difference in the distance travelled by the two waves. The path difference between two waves travelling at an angle θ is given by: dsinθ = n λ


    where n is the order of the fringe. When the two waves are in phase, i.e. the path difference is equal to an integral number of wavelengths, the summed amplitude, and therefore the summed intensity is maximum, and when they are in anti–phase, i.e. the path difference is equal to half a wavelength, one and a half wavelengths, etc., then the two waves cancel and the summed intensity is zero. This effect is known as interference. The interference fringe maxima occur at angles dθ= n λ   where n = 0,1,2,... and ? is the wavelength of the light.
  • Phase of the waves: The relationship between rays travelling along the same path and the interference between the rays is given by:
    • If retardation is a whole number (i.e., 0, 1, 2, 3, etc.) of wavelengths. The two waves, A and B, are IN PHASE, and they constructively interfere with each other. The resultant wave (R) is the sum of wave A and B.
    • When retardation is = 1/2 , 1 1/2, 2 1/2 . . . wavelengths. The two waves are OUT OF PHASE they destructively interfere, cancelling out each other.
    • If the retardation is an intermediate value, the the two waves will:
      (1) be partially in phase, with the interference being partially constructive
      (2) be partially out of phase, partially destructive.
  • Newton's rings: The interference rings caused by the reflection of light between these two surfaces – a spherical surface and an adjacent flat surface are called Newton's rings. Newton's rings appear as a series of concentric, alternating light and dark rings centered at the point of contact between the two surfaces. The light rings are caused by constructive interference between the incident and reflected light rays, while the dark rings are caused by destructive interference. Counting the rings and knowing the wavelength of the illumination can give an accurate measure of the size of any hollows and heights on a surface; that characterizes the lens.

  • Diffraction: Diffraction of light occurs when a light wave passes by a corner or through an opening or slit that is physically the approximate size of, or even smaller than that light's wavelength. Fresnel was able to explain the phenomenon of diffraction. He postulated that diffraction was due to interference between waves starting from different parts of same wavefront. Fresnel Diffraction is the special case where the incoming light is assumed to be parallel and the image plane is assumed to be at a very large distance compared to the diffracting object. Fraunhofer and Fresnel diffraction are two important classes of diffraction. Fraunhofer diffraction involves coherent plane waves incident upon an obstruction. Fresnel diffraction is the same, except that the waves are spherical, effectively originating from a point source. If the Fresnel Diffraction through a slit occurs, then the angle at which the dark fringes occur is given by:
    sin θ = m λ/d   where θ is the angle between the central incident propagation direction and the first minimum of the diffraction pattern and m is an integer that labels the order of each minimum; λ is the wavelength of the light and d is the width of the slit.
  • Dispersion: Visible light is actually made up of different colors. Each color bends by a different amount when refracted by glass. That's why visible light is split, or dispersed, into different colors when it passes through a lens or prism. Shorter wavelengths, like purple and blue light, bend the most. Longer wavelengths, like red and orange light, bend the least.

  • Polarization: Light waves can vibrate in many directions. If they are vibrating in one direction –– in a single plane such as up and down , then the light is said to be polarized light. Those waves that are vibrating in more than one direction –– in more than one plane such as both up/down and left/right –– are called unpolarized light. If two polaroid filters are used and placed so that one is rotated 90 degrees to the other, no light will be able to pass. Some polarization will also occur during reflection, refraction, and scattering of light. When reflecting off non–metallic surfaces, the resulting light will be polarized parallel to the reflected surface. During refraction, a beam of light will be split up into two polarized beams, one polarized parallel and one perpendicular to the boundary. Scattering also causes partial polarization.

  • Selective Absorption: A number of crystalline materials absorb more light in one incident plane than another, so that light progressing through the material become more and more polarized as they proceed. This anisotropy in absorption is called dichroism. There are several naturally occurring dichroic materials, and the commercial material polaroid also polarizes by selective absorption.
key formulae
  • Amplitude(R) of interference of two waves is given by

    where
    a = amplitude of each wave
    λ = wavelength of each wave
    x = path difference between the two
  • Intensity :
  • I = I1 + I2 + 2(I1.I2 cos θ)1/2
  • Fresnel's Diffraction :
    • The angle at which dark fringes occur is given by
      sin θ = (mλ)/d
      where
      m = integer that labels the order of each minimum
      λ = wavelength of the light
      d = width of the slit
  • Fraunhofer Diffraction :
  • Brewster's law :
    μ = tan ip