19.1 Interference Notes

19.1 Interference

Agenda

• Types of light
• Thomas Young
• Coherent light
• Incoherent light
• Thin film interference
• Color Reinforcement

Interference of Light

• Interference is the phenomenon in which two waves superpose to form a resultant wave of lower, higher, or same amplitude.
• Constructive interference:
  • Occurs when two or more waves are identical and build a new wave of higher amplitude.
  • Crest on crest & trough on trough.
• Destructive interference:
  • Occurs when waves come together and cancel each other.
  • Crest on trough.

Coherent Light

• Light from two or more sources combines in superposition to produce smooth wavefronts (synchronous).
• The wave is in phase; wavelength, speed, & amplitude are the same for all waves.
• Can be created by one or many point sources due to superposition.
• Example: Laser.
• Monochromatic.

Incoherent Light

• Produced by unsynchronized wavefronts that are not in phase.
• They have different wavelengths, speeds, & amplitudes.
• Example: Falling rain drops on a swimming pool.
• Not always Monochromatic.

Monochromatic Light

• Mono means one, chromatic derives from chrome, which means colors, so monochromatic light in physics means the light that has one wavelength or one frequency (one color).
• Monochromatic light waves could produce coherent or incoherent light.
• Monochromatic coherent light: same wavelength (same color) released at the same velocity from the same narrow slit source (in phase).
• Monochromatic incoherent light: same wavelength (same color) released at different velocity (out of phase).

Thomas Young's Experiment

• Thomas Young did an experiment to prove that light is a wave, not a particle.
• He produced an interference pattern of light from a single coherent source through two slits.
• When the coherent light was directed through these two narrow, closely spaced slits in a barrier, light created overlap patterns of bright and dark bands called INTERFERENCE FRINGES.
• The interference fringes were produced by the constructive and destructive interference of waves; thus, the experiment showed that light has wave properties.

Double-Slit Interference (Interference of Coherent Light)

• Coherent light is monochromatic (same wavelength); bright fringes are produced by constructive interference.
• The intensity of the band decreases the farther it is from the central band.
• The dark fringes are produced by destructive interference due to the changes in wavelength.
• When white light was used in a double-slit experiment, a spectrum of colors was produced due to the differences in wavelength; this spectrum is known as Polychromatic light (where no dark bands appear).

Generation of Coherent Light from Noncoherent Light

• Light from monochromatic sources produces incoherent light, so placing a barrier with a narrow slit in front of the monochromatic source will produce coherent light (only tiny wavelengths will pass through the slit).
• Diffraction by the slit will produce nearly cylindrical wavefronts.
• These wavefronts, when passing through a second barrier with two narrow and closely spaced slits, will stay in phase.
• The new wavefronts from barrier 2 will interfere, forming areas of constructive (1 whole wavelength difference) and destructive (half wavelength phase difference) interference; their positions depend on the wavelength of the light used.
• Coherent waves can experience both constructive and destructive interferences.
• The dark & bright bands are separated by the same spacing and have equal widths.

Measuring the Wavelength of Light

• Waves from $S<em>1$ & $S</em>2$ follow a different path reaching point P, so they will have phase & path differences forming a high-intensity fringe.
• Since waves are in phase, they will interfere constructively on the screen creating the central band at $P_0$ or m = 0, where m = 0, 1, 2, 3, and so on.
• $M_1$ = first-order band and so on.
• Constructive interference occurs at locations $(X_m)$ on both sides of the central band using the formula:
• $n = frac{c}{v}$, where
  • $n$ is the index of refraction, inversely proportional to the speed of light in a medium,
  • $c$ is the speed of light in a vacuum ($3 \times 10^8 m/s$), and
  • $v$ is the speed of light in the medium.

Thin-Film Interference

• It’s the phenomenon of a spectrum of colors resulting from constructive and destructive interference of light waves due to reflection in a thin film (it results from incoherent light changing into reflected coherent light).
• Examples: Soap bubbles, oily film, Butterflies, peacock, etc.
• If a soap film is held vertically, its weight makes it thicker at the bottom than at the top.
• When the wave hits the film, it’s partially reflected (ray 1) and partially transmitted (ray 2).
• Both the reflected and transmitted rays have the same frequency as the original.

Color Reinforcement

• How can reflection of one color be enhanced?
  • When two reflected waves are in phase.
• If the thickness of the soap film is one-fourth the wavelength, so the round trip path is half wavelength, you may expect that ray 2 will return to the surface causing destructive interference.
• When a transverse wave is reflected from a low index into a high index medium (ray 1), it’s reflected & inverted. When the wave goes from a high index into a low index medium (ray 2), it’s transmitted and not inverted, which makes ray 1 & 2 in phase.

Extra Practice Problems and Solutions

• Problem 35: Light falls on a pair of slits 19.0 μm apart and 80.0 cm from a screen. The first-order bright band is 1.90 cm from the central bright band. What is the wavelength of the light?
  • Solution: $\lambda = \frac{xd}{L} = \frac{1.9 \times 10^{-2} \times 19 \times 10^{-6}}{80 \times 10^{-2}} = 4.51 \times 10^{-7} m$
• Problem 36: Oil Slick: A thin film of oil (n = 1.45) on a puddle of water produces different colors. What is the minimum thickness where the oil creates constructive interference for light with a wavelength equal to 545 nm?
  • Solution: $d = \frac{(m + \frac{1}{2})\lambda}{2n} = \frac{(\frac{1}{2}) \times 545}{2 \times 1.45} = 93.96 nm$
• Problem 37: Film Thickness: A plastic reflecting film (n = 1.83) is placed on an auto glass window (n = 1.52). What is the thinnest film that will reflect yellow-green light ($\lambda$ = 555 nm)? What is the next-thinnest film that will produce the same effect?
  • Solution:
    • Thinnest film: $d_1 = \frac{(m + \frac{1}{2})\lambda}{2n} = \frac{(\frac{1}{2}) \times 555}{2 \times 1.83} = 75.81 nm$
    • Next-thinnest film: $d<em>2 = 3 \times d</em>1 = 3 \times 75.81 = 227.43 nm$
• Problem 38: Insulation Film: Clear plastic (n = 1.81) covers windows. A blue stripe of color is observed with a wavelength of 445 nm. What are three possible thicknesses of the plastic where the blue stripe is produced?
  • Solution:
    • $d_1 = \frac{(m + \frac{1}{2})\lambda}{2n} = \frac{(\frac{1}{2}) \times 445}{2 \times 1.81} = 61.46 nm$
    • $d<em>2 = 3 \times d</em>1 = 184.38 nm$
    • $d<em>3 = 5 \times d</em>1 = 307.30 nm$
• Problem 39: Ranking Task: Rank lasers by wavelength from shortest to longest.
  • Solution: A < D < C < B < E
• Problem 63: A glass lens has an antireflective coating with n = 1.2 and a thickness of 125 nm. For which color(s) of light does complete destructive interference occur?
  • Solution: $\lambda = \frac{2nd}{m} = 300 nm$

Key Concepts from Answer Key

• Constructive interference leads to bright bands or strongly reflected colors.
• Destructive interference results in dark bands or non-reflective surfaces.
• Thin-film interference involves incoherent light turning into coherent light.
• The thickness of the film and the index of refraction are vital in determining interference patterns.
• The index of refraction is inversely proportional to the speed of light in a medium.