Interference and Diffraction Notes

Waves In Phase and 180° Out of Phase

  • For stable interference, waves must have a constant phase difference.
  • Individual waves must remain unchanged relative to one another.
  • In-Phase Waves:
    • Crest of one wave overlaps the crest of another.
    • Phase difference of 0°.
  • Out-of-Phase Waves:
    • Crest of one wave overlaps the trough of another wave.
    • Phase difference of 180°.
  • Coherence: Phase difference between two waves is constant; waves don't shift relative to each other with time. Sources of such waves are coherent.
  • Incoherent Light:
    • Light waves from separate bulbs are emitted independently.
    • Random changes in one bulb's light don't necessarily occur in the other.
    • Phase difference between light waves is not constant.
    • Conditions for interference change with each phase change, so no single interference pattern is observed.

Demonstrating Interference

  • Experiment Setup:
    • Light from a single source passes through a narrow slit.
    • Then passes through two narrow parallel slits.
    • The slits act as coherent light sources because the waves come from the same source.
    • Any random change affects both beams simultaneously.
  • Monochromatic Light:
    • Produces bright and dark parallel bands (fringes) on a distant viewing screen.
    • Constructive interference: Bright fringe.
    • Destructive interference: Dark fringe.
  • Coherence:
    • Correlation between the phases of two or more waves.

White-Light Interference

  • Using a white-light source to observe interference makes the situation more complicated because white light includes waves of many wavelengths.
  • Interference pattern is stable or well-defined at positions where there is constructive interference between light waves of the same wavelength.
  • This explains the color bands on either side of the center band of white light.

Conditions for Interference of Light Waves

  • Waves from two coherent sources can combine at the viewing screen.
  • Central Point:
    • Waves travel equal distances.
    • Arrive in phase.
    • Constructive interference occurs, forming a bright fringe.
  • Off-Center Point:
    • Wave from the more distant slit travels one wavelength farther.
    • Waves are in phase.
    • Constructive interference occurs, forming a second bright fringe.
  • Midway Point:
    • One wave travels half a wavelength farther.
    • Trough of one wave overlaps the crest of the other.
    • Destructive interference occurs.
    • A dark fringe appears.

Predicting the Location of Interference Fringes

  • Setup: Two narrow slits separated by distance d, with coherent, monochromatic light waves l1 and l2 projected onto a screen.
  • If the distance from the slits to the screen is very large compared with the distance between the slits, then l1 and l2 make the same angle, \theta, with the horizontal dotted lines.
  • Angle \theta also indicates the position where waves combine with respect to the central point of the screen.
  • Path difference: The difference in distance traveled by the two waves.
  • The path difference between the two waves is equal to d \sin \theta.
  • The value for the path difference varies with angle\theta, which defines a specific position on the screen.
  • The path difference determines whether the two waves are in or out of phase when they arrive at the viewing screen.
  • Constructive Interference (Bright Fringes):
    • Path difference is zero or a whole-number multiple of the wavelength.
    • Condition: d \sin \theta = \pm m\lambda, where m = 0, 1, 2, 3,…
    • m is the order number of the fringe.
    • Central bright fringe at \theta = 0 (m = 0) is the zeroth-order maximum or central maximum.
    • First maximum on either side, when m = 1, is the first-order maximum, and so forth.
  • Destructive Interference (Dark Fringes):
    • Path difference is an odd multiple of half the wavelength.
    • Condition: d \sin \theta = \pm (m + \frac{1}{2}) \lambda, where m = 0, 1, 2, 3,…
    • If m = 0, the path difference is \pm \frac{\lambda}{2}, which is required for the first dark fringe on either side of the central maximum.
    • If m = 1, the path difference is \pm \frac{3\lambda}{2}, which is for the second dark fringe on each side of the central maximum.
  • Path difference: The difference in the distance traveled by two beams when they are scattered in the same direction from different points.
  • Order number: The number assigned to interference fringes with respect to the central bright fringe.

Interference

  • Colors on a soap bubble are a result of light waves combining.
  • Interference takes place only between waves with the same wavelength.
  • When two waves with identical wavelengths interact, they combine to form a resultant wave.
  • The resultant wave has the same wavelength as the component waves.
  • The displacement at any instant equals the sum of the displacements of the component waves (superposition principle).
  • Monochromatic Light: Light source with a single wavelength.
  • Constructive Interference:
    • Component waves combine to form a resultant wave with the same wavelength but with a greater amplitude.
    • Result is brighter light.
  • Destructive Interference:
    • Resultant amplitude is less than the amplitude of the larger component wave.
    • Result is dimmer light or dark spots.

Position of Higher-Order Interference Fringes

  • Representation of the interference pattern formed by double-slit interference.
  • Numbers indicate the two maxima that form on either side of the central (zeroth-order) maximum.
  • Darkest areas indicate the positions of the dark fringes, or minima, that also appear in the pattern.
  • Double-slit interference provides a method of measuring the wavelength of light because the separation between interference fringes varies for light of different wavelengths.
  • This technique was used to make the first measurement of the wavelength of light.