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 $l<em>1$ and $l</em>2$ projected onto a screen.
• If the distance from the slits to the screen is very large compared with the distance between the slits, then $l<em>1$ and $l</em>2$ 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.