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Constructive and destructive interference, path difference, Young's double slit, coherence, and thin-film colors and coatings.
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By the end of this lesson you will be able to predict interference patterns from two sources and thin films.
From the waves lessons you know $v = f\lambda$, that waves add where they overlap, and that standing waves come from two waves traveling in opposite directions. This lesson follows waves from two sources overlapping in space, which produces patterns of reinforcement and cancellation that reveal light to be a wave.
| Term | What it means |
|---|---|
| Superposition | Overlapping waves add their displacements. |
| Constructive interference | Waves in step reinforce: path difference $m\lambda$. |
| Destructive interference | Waves half a cycle apart cancel: path difference $(m + \tfrac{1}{2})\lambda$. |
| Coherent sources | Sources with a fixed phase relationship, such as two slits lit by one laser. |
| Fringe spacing | $\Delta y = \lambda L/d$ for two slits a distance $d$ apart, screen at $L$. |
| Thin-film interference | Colors from light reflecting off the two surfaces of a thin layer. |
When two waves overlap, their displacements add. If they arrive in step, crest on crest, they reinforce: constructive interference. If they arrive half a cycle apart, crest on trough, they cancel: destructive interference. For two sources in step, what matters is the path difference:
$$\Delta r = m\lambda \ \text{(bright)}, \qquad \Delta r = \left(m + \tfrac{1}{2}\right)\lambda \ \text{(dark)}.$$
In Young's double-slit experiment, light from two narrow slits a distance $d$ apart lands on a screen at distance $L$, making bright fringes spaced $\Delta y = \lambda L/d$ apart. Only coherent waves, with a steady phase relationship, make a steady pattern.
Another way: picture
Picture two stones dropped into a pond at the same moment. Their ripples spread and cross. Along some lines crests always meet crests and the water heaves high; along others crests meet troughs and the water stays flat. Those lines of calm fan out from between the stones, and that fan is an interference pattern.
Another way: steps
In 1801, Thomas Young in London passed sunlight through a pinhole and then through two closely spaced slits. On the wall beyond he saw not two bright lines but a row of bright and dark bands. Particles would simply have made two bright strips; only waves, overlapping and canceling, could make the dark bands.
Young's experiment convinced physicists that light is a wave, and from the fringe spacing he measured its wavelength, a few ten-thousandths of a millimeter. The same experiment done with electrons, one at a time, a century and a half later, showed that matter waves interfere too.
From two slits a distance $d$ apart, light reaching a point on the screen at angle $\theta$ travels an extra distance $d\sin\theta$ from the farther slit. Bright fringes appear where this is a whole number of wavelengths: $d\sin\theta = m\lambda$. For a distant screen and small angles, $\sin\theta \approx y/L$, so the bright fringes sit at $y = m\lambda L/d$.
The spacing $\lambda L/d$ grows with wavelength and screen distance and shrinks as the slits move apart. Red light makes wider fringes than blue. Measuring the spacing, with a ruler and known slits, gives the wavelength directly.
Two ordinary light bulbs never make a visible interference pattern. Each emits light in short, random bursts, so the phase relationship between them changes billions of times a second, and any pattern washes out. Interference needs coherent sources, locked in phase.
Young got coherence by splitting light from one pinhole into two slits. Lasers, invented in 1960 by Theodore Maiman at Hughes Research Laboratories in California, emit highly coherent light, which is why a laser pointer through two slits makes such crisp fringes.
Light striking a soap film or an oil slick reflects from both its top and bottom surfaces. The two reflections recombine, and whether they reinforce or cancel depends on the extra path $2t$ inside the film, the wavelength in the film, and whether either reflection is inverted, which happens when light reflects from a material of higher index.
For a soap film in air, only the top reflection is inverted, so a film a quarter wavelength thick reflects that color strongly. Because the thickness varies, different colors are reinforced in different places, painting swirling bands on bubbles.
Checking an answer. Fringe spacing must grow with wavelength and screen distance and shrink with slit separation. The central fringe must be bright. Visible wavelengths must come out between about $400$ and $700$ nm.
Superposition holds because the wave equation is linear: the sum of two solutions is also a solution. For light this is exact in vacuum and excellent in ordinary materials.
The fringe formula uses the approximation $\sin\theta \approx \tan\theta \approx y/L$, valid when the fringes are close to the center compared with the screen distance. Energy is not destroyed at dark fringes; it is redistributed to the bright ones, which are four times as intense as either slit alone.
Two speakers playing the same tone set up an interference pattern in a room. Walk across it and the sound swells and fades as the path difference changes. Audio engineers place speakers carefully in theaters and arenas to avoid dead spots.
Noise-canceling headphones use destructive interference on purpose. A microphone picks up outside noise, and the electronics play back the same sound inverted, half a cycle out of step. The two cancel inside the ear cup, which works best for steady, low-pitched sounds like airplane engines.
Uncoated glass reflects about four percent of light at each surface, which causes glare on eyeglasses and ghost images in cameras. A thin coating a quarter wavelength thick makes the reflections from its top and bottom cancel.
With magnesium fluoride, index $1.38$, a coating about $100$ nm thick cancels reflected green light, the middle of the visible spectrum. Red and blue are canceled less, which is why coated lenses show a faint purple reflection. Multilayer coatings on good camera lenses cut reflections to a fraction of a percent across all colors.
An interferometer splits a light beam, sends the parts along two paths, and recombines them. A change in either path of a fraction of a wavelength shifts the fringes. Albert Michelson, at the Case School in Cleveland, used one in 1887 with Edward Morley to search for an ether wind, and found none.
Today the Laser Interferometer Gravitational-Wave Observatory, with detectors in Hanford, Washington, and Livingston, Louisiana, uses four-kilometer arms to detect changes in length a thousand times smaller than a proton, caused by gravitational waves from colliding black holes.
Many of nature's brightest colors come from interference, not pigment. The blue of a morpho butterfly's wings comes from layered nanostructures that reinforce blue reflections. Peacock feathers, beetle shells and the inside of an abalone shell shimmer with colors that shift as you tilt them, a sign of interference.
Engineers imitate these structures for anti-counterfeiting features. The color-shifting ink on United States currency, where the numeral in the corner changes from copper to green as you tilt the bill, uses thin-film interference that is very hard to copy.
After rain, a thin film of oil on a wet road shows rainbow bands. The oil is typically a few hundred nanometers thick and varies from place to place, and each thickness reinforces a different color in the reflected light. Where the film is thinnest, all colors cancel and it looks dark.
Scientists use such interference colors to measure film thicknesses. In chip factories, instruments shine light on silicon wafers and analyze the reflected colors to check coating layers a few nanometers thick, without touching the wafer.
At a dark fringe, the waves from the two slits cancel, and at first it looks as if energy has vanished. It has not. At a bright fringe, the two waves add to twice the amplitude, and since intensity grows as the square of the amplitude, the brightness there is four times what one slit alone would give. Averaged across the pattern, the light is exactly twice one slit's, as energy conservation requires: it is simply moved out of the dark bands and piled into the bright ones.
That redistribution is useful. Radio engineers arrange several antennas so their signals interfere constructively toward a city and destructively toward open ocean, sending power where listeners are. Cell towers and Wi-Fi routers with several antennas steer their beams the same way, a technique called beamforming, adjusting the phases between antennas many times a second to follow a moving phone.
Most American eyeglass lenses are sold with an anti-reflective coating. Without it, each lens surface reflects about four percent of the light, making glare for the wearer and hiding the wearer's eyes in photographs and video calls.
A layer of magnesium fluoride, index $1.38$, deposited to a thickness of $\lambda/(4n)$, about $100$ nm for $550$ nm green light, makes the reflections from its top and bottom surfaces cancel. Both reflections are inverted, since each goes from lower to higher index, so the quarter-wave thickness gives exactly half a wavelength of extra path. Premium coatings stack several layers of different materials and thicknesses to cancel reflections across the whole visible spectrum.
On September 14, 2015, the Laser Interferometer Gravitational-Wave Observatory detected, for the first time, gravitational waves from two black holes merging over a billion light-years away. Each LIGO detector, in Hanford, Washington, and Livingston, Louisiana, is an interferometer with two $4$ km arms at right angles.
A laser beam is split down both arms, reflected back and recombined so that the light cancels at the output. A passing gravitational wave stretches one arm and squeezes the other by about $10^{-18}$ m, shifting the interference and letting light through. Rainer Weiss, Kip Thorne and Barry Barish won the 2017 Nobel Prize for the discovery.
It is natural to think two beams of light always add up to more light. Coherent light waves arriving half a cycle apart cancel, leaving dark fringes where both beams shine. The energy is not lost; it is redistributed to the bright fringes.
A related error is to think the fringes are shadows or images of the slits. There are many fringes from just two slits, evenly spaced, and their spacing depends on the wavelength, which no shadow could do.
Red laser light, $650$ nm, passes through slits $0.20$ mm apart onto a wall $3.0$ m away. Convert units.
$\lambda = 6.50 \times 10^{-7}\ \text{m}, \ d = 2.0 \times 10^{-4}\ \text{m}$
SI units.
Find the fringe spacing.
$\Delta y = \dfrac{6.50 \times 10^{-7} \times 3.0}{2.0 \times 10^{-4}} = 9.75 \times 10^{-3}\ \text{m}$
About a centimeter.
Find the third bright fringe's position.
$y_3 = 3 \times 9.75 = 29.3\ \text{mm}$
From the center.
Find the first dark fringe.
$y = \tfrac{1}{2} \times 9.75 = 4.9\ \text{mm}$
Halfway.
Switch to a green $532$ nm laser. Find the new spacing.
$\Delta y = 9.75 \times \dfrac{532}{650} = 7.98\ \text{mm}$
Shorter wavelength, closer fringes.
Two speakers play $343$ Hz in step. Find the wavelength.
$\lambda = \dfrac{343}{343} = 1.00\ \text{m}$
Speed over frequency.
A listener is $5.0$ m from one and $7.5$ m from the other. Find the path difference.
$\Delta r = 2.5\ \text{m}$
Extra distance.
Count the wavelengths.
$\dfrac{2.5}{1.00} = 2.5$
A half-integer.
Decide what is heard.
$\text{quiet: destructive}$
Crest meets trough.
Change the tone to $686$ Hz. Recount.
$\dfrac{2.5}{0.50} = 5.0$
A whole number.
Decide what is heard now.
$\text{loud: constructive}$
Same place, different pitch.
A soap film, index $1.33$, is $100$ nm thick. Find the extra path for light reflected from the back.
$2t = 200\ \text{nm}$
Down and back.
Note the reflection inversions.
$\text{top inverted, bottom not}$
Air to film inverts; film to air does not.
Write the condition for bright reflection.
$2t = \left(m + \tfrac{1}{2}\right)\dfrac{\lambda}{n}$
The inversion adds half a wavelength.
Solve for the brightest wavelength with $m = 0$.
$\lambda = 4nt = 4 \times 1.33 \times 100 = 532\ \text{nm}$
Green.
Try $m = 1$.
$\lambda = \dfrac{4nt}{3} = 177\ \text{nm}$
Ultraviolet, invisible.
State the film's color.
$\text{green}$
Where it is $100$ nm thick.
Write the fringe spacing.
$\Delta y = \dfrac{\lambda L}{d}$
Two slits.
Substitute in SI units.
$\Delta y = \dfrac{5.00 \times 10^{-7} \times 2.0}{5.0 \times 10^{-4}}$
Meters.
Evaluate the spacing.
Light of wavelength $405$ nm passes through two slits $0.15$ mm apart onto a screen $2.5$ m away. How far apart are neighboring bright fringes, in mm?
Complete the worked solution: two speakers play the same $171.5$ Hz tone in step. A listener is $6.0$ m from one and $9.0$ m from the other. With sound at $343$ m/s, find the wavelength in m, the path difference in m, and the path difference in wavelengths.
Find the wavelength.
$\lambda = \dfrac{v}{f} =$ l
Speed over frequency.
Find the path difference.
$\Delta r = r_2 - r_1 =$ p
Extra distance from the far speaker.
Count the wavelengths.
$\dfrac{\Delta r}{\lambda} =$ n
Whole or half-integer?
Decide what the listener hears.
$\text{whole: loud; half-integer: quiet}$
Constructive or destructive.
Match each term to its meaning.
| path difference a whole number of wavelengths | path difference a whole number plus half a wavelength | waves that keep a steady phase relationship | light reflected from two surfaces of a film interfering | |
|---|---|---|---|---|
| constructive interference | ||||
| destructive interference | ||||
| coherent sources | ||||
| colors on a soap bubble |
Light of wavelength $700$ nm falls on two slits $0.21$ mm apart; the screen is $1.2$ m away. Fill in the distances from the central bright fringe, in mm, to the first bright fringe, the second bright fringe, and the first dark fringe.
| value | |
|---|---|
| first bright fringe (mm) | |
| second bright fringe (mm) | |
| first dark fringe (mm) |
Light of wavelength $600$ nm falls on two slits onto a screen $2$ m away. Write the distance from the center to the first bright fringe, in mm, as a function of the slit separation $d$ in mm.
Answer:
In a lab, a laser shines through slits $0.4$ mm apart onto a wall $2.0$ m away, and the bright fringes are $2.8$ mm apart. What is the laser's wavelength, in nm?
Answer: nm
Eyeglass lenses are coated with silica, index $1.46$, to cancel reflections of $550$ nm light. Both reflections, from the top and bottom of the coating, are inverted, so the coating should be a quarter wavelength thick. How thick, in nm?
Answer: nm
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Light of wavelength $650$ nm falls on two slits onto a screen $1.0$ m away. Write the distance from the center to the first bright fringe, in mm, as a function of the slit separation $d$ in mm.
Answer:
You can analyze interference. Explain to someone how two beams of light can combine to make darkness.
23. Your turn: light of $500$ nm through slits $0.50$ mm apart falls on a screen $2.0$ m away. What is the fringe spacing?, step 3
$\Delta y = 2.0 \times 10^{-3}\ \text{m} = 2.0\ \text{mm}$
Two millimeters.