81 Young’s Double Slit Experiment

In the previous chapter, we learned that light exhibits wave-like behaviors such as diffraction. However, for much of scientific history there was considerable debate over whether light was truly a wave or instead consisted of tiny particles. Two of the greatest scientists of the seventeenth century reached different conclusions.

Christiaan Huygens argued that light behaved as a wave and developed a mathematical description of wave propagation using what is now known as Huygens's principle. In contrast, Isaac Newton proposed that light consisted of particles, or corpuscles. Because of Newton's enormous influence and the many successes of his mechanics, the particle theory dominated scientific thinking for more than a century.

The debate was finally settled in 1801 when the English physician and physicist Thomas Young performed one of the most famous experiments in physics: the double-slit experiment. The pattern produced by this experiment could only be explained if light behaved as a wave. Today it remains one of the clearest demonstrations of wave interference and serves as the foundation for many modern optical technologies.

Coherence and Interference

If light is a wave, why do we not observe interference patterns everywhere? The answer is that wave effects become noticeable only under specific conditions.

First, the light must interact with objects whose dimensions are comparable to its wavelength, such as the narrow slits used in Young's experiment. Second, the waves reaching the screen must maintain a constant relationship with one another. This property is called coherence.

Young cleverly solved the coherence problem by first passing sunlight through a single narrow slit. This slit acted as a common source for the two slits that followed, ensuring that the light emerging from both slits remained coherent. Each slit therefore behaved as a new wave source with a well-defined phase relationship.

To simplify the discussion, we will consider monochromatic light, meaning light with a single wavelength. Although white light contains many wavelengths, each wavelength forms its own interference pattern. Using monochromatic light allows us to clearly understand the physics before considering more complex sources.

Constructive and Destructive Interference

Whenever two or more waves overlap, their amplitudes combine according to the principle of superposition. The resulting wave depends on the relative phases of the individual waves.

If two identical waves arrive with their crests and troughs aligned, their amplitudes reinforce one another, producing a larger wave. This phenomenon is called constructive interference.

If one wave arrives exactly half a wavelength behind the other, the crest of one wave coincides with the trough of the other. Their amplitudes cancel, producing destructive interference.

Interference from Two Slits

When monochromatic light passes through two narrow slits, each slit acts as a new source of coherent waves. Because the slits are narrow, the light emerging from them spreads out by diffraction, allowing the two waves to overlap.

Where the waves meet crest-to-crest or trough-to-trough, they reinforce one another and produce bright regions on the screen. Where crests meet troughs, they cancel, creating dark regions. The alternating bright and dark bands are called an interference pattern.

The same phenomenon occurs for many different types of waves. Water waves generated by two vibrating sources produce nearly identical regions of reinforcement and cancellation, demonstrating that interference is a universal property of waves rather than a behavior unique to light.

Path Difference and the Origin of Bright and Dark Fringes

Why are some points on the screen bright while others are dark? The answer depends on the distance each wave travels from its slit to a particular point on the screen.

Although the waves leave the two slits in phase, one wave may travel slightly farther than the other before reaching a given location. This difference in travel distance is called the path difference. A path difference changes the relative phase of the waves when they arrive at the screen.

• If the path difference is exactly one wavelength, two wavelengths, three wavelengths, or any other whole-number multiple of the wavelength, the waves arrive in phase and produce constructive interference.
• If the path difference is one-half wavelength, three-halves wavelengths, five-halves wavelengths, or any other half-integer multiple of the wavelength, the waves arrive exactly out of phase and produce destructive interference.

This simple idea—comparing the difference in distance traveled by the two waves—is the foundation for all of the mathematical relationships developed in the remainder of this chapter. In the next section, we will determine how this path difference depends on the geometry of the double-slit experiment and derive the equations that predict the locations of every bright and dark fringe.

Path Difference Determines the Interference Pattern

We now have all the ingredients needed to explain why some regions of the screen appear bright while others appear dark. Although the waves leave the two slits in phase, they generally do not travel the same distance before reaching a particular point on the screen.

This difference in the distances traveled is called the path difference. The path difference determines the relative phase of the two waves when they arrive at the screen and therefore determines whether they reinforce or cancel each other.

Calculating the Path Difference

To predict where bright and dark fringes will appear, we must determine the path difference between the two waves. Figure 81.5 shows the geometry of the double-slit experiment.

If the screen is much farther from the slits than the separation between the slits, the two rays reaching a point on the screen make nearly the same angle, θ, with the original direction of the beam. Under this approximation, simple trigonometry shows that the path difference is

where

• d is the distance between the two slits,
• θ is the angle measured from the center of the pattern, and
• ΔL is the difference in the distances traveled by the two waves.

Conditions for Constructive and Destructive Interference

Once the path difference is known, determining the type of interference is straightforward.

In these equations,

• λ is the wavelength of the light,
• d is the slit separation,
• θ is the angle from the central axis, and
• m is called the order of the interference pattern.

The central bright fringe corresponds to m = 0. The first bright fringes on either side are m = ±1, followed by the second-order fringes (m = ±2), and so on.

Interference Fringes

The constructive and destructive interference conditions predict a series of alternating bright and dark bands known as interference fringes. For vertical slits, the fringes appear as vertical bright and dark lines that spread horizontally away from the center of the screen.

The central bright fringe is always the most intense because it corresponds to zero path difference. Fringes farther from the center generally become dimmer because diffraction from each slit causes the overall light intensity to decrease with angle.

The equations also show an important trend: if the slit separation d becomes smaller, the angles at which the bright fringes occur become larger. In other words, bringing the slits closer together causes the interference pattern to spread out over a wider region of the screen. This is another example of the general principle that wave effects become more pronounced when the size of an object is comparable to the wavelength of the wave.

The equation for constructive interference predicts the location of every bright fringe in the pattern:

This equation also explains how the spacing of the fringes depends on the slit separation. Suppose the wavelength and the interference order remain fixed. Solving for the angle gives

As the slit separation d becomes smaller, the value of sin θ must become larger. Consequently, the bright fringes appear at larger angles from the center of the pattern, causing the entire interference pattern to spread out.

This result illustrates an important principle that applies throughout wave physics: wave effects become more pronounced when the size of the object interacting with the wave is comparable to the wavelength. Narrower slit separations therefore produce wider interference patterns.

Section Summary

• Young's double-slit experiment provided convincing experimental evidence that light behaves as a wave by producing an interference pattern that cannot be explained by ray optics alone.
• Interference occurs when two or more coherent waves overlap. The resulting wave is determined by the principle of superposition, in which the amplitudes of the individual waves add together.
• In a double-slit experiment, each narrow slit acts as a coherent source of diffracted light waves that interfere with one another.
• Bright fringes (constructive interference) occur when the path difference between the two waves is an integer multiple of the wavelength:

• Dark fringes (destructive interference) occur when the path difference is an odd half-integer multiple of the wavelength:

• The interference order m identifies the position of each bright fringe relative to the central maximum.
• Reducing the distance between the slits causes the interference pattern to spread over a larger angle, making the fringes farther apart on the screen.

Conceptual Questions

8. Young's double-slit experiment divides a single light beam into two coherent sources. Would two completely independent light sources, such as the headlights of a distant car, produce the same interference pattern? Explain why or why not.
9. If you perform Young's double-slit experiment first in air and then repeat it in water using the same wavelength source and the same slit separation, how will the angles of the interference fringes change? Will the observed color of the light change? Explain your reasoning.
10. Is it possible to arrange a situation in which only destructive interference occurs everywhere? Why or why not?
11. Figure 81.7 shows the central region of a double-slit interference pattern produced by monochromatic red light. The bright spots are evenly spaced, but their brightness gradually decreases away from the center.
    • Which feature is characteristic of double-slit interference?
    • Which feature is caused by single-slit diffraction?
    • Based on the pattern, is the slit width larger or smaller than the distance between the two slits? Explain your reasoning.

Problems & Exercises

6. At what angle does the first-order maximum occur when 450-nm blue light passes through two slits separated by 0.0500 mm?
7. Calculate the angle of the third-order maximum for 580-nm yellow light passing through two slits separated by 0.100 mm.
8. What slit separation will produce a first-order maximum at an angle of [latex]30.0^\circ[/latex] for 610-nm orange light?
9. Find the separation between two slits that produces the first minimum for 410-nm violet light at an angle of [latex]45.0^\circ[/latex].
10. Light passing through two slits separated by [latex]3.00\ \mu\text{m}[/latex] produces its third minimum at an angle of [latex]30.0^\circ[/latex]. Calculate the wavelength of the light. Clearly identify the appropriate interference condition and show the steps in your calculation.
11. What is the wavelength of light passing through two slits separated by [latex]2.00\ \mu\text{m}[/latex] if the third-order maximum occurs at an angle of [latex]60.0^\circ[/latex]?
12. What is the highest-order maximum possible for 400-nm light passing through two slits separated by [latex]25.0\ \mu\text{m}[/latex]?
13. Find the largest wavelength of light for which a first-order maximum can occur when the slit separation is [latex]1.20\ \mu\text{m}[/latex]. Is this wavelength within the visible spectrum?
14. What is the smallest slit separation that can produce a second-order maximum for 720-nm red light?
15. 1. What is the smallest slit separation that can produce a second-order maximum for at least one visible wavelength?
    2. What is the smallest slit separation that can produce a second-order maximum for every wavelength in the visible spectrum?
16. The first-order maximum for monochromatic light passing through a double slit occurs at an angle of [latex]10.0^\circ[/latex].
    1. At what angle does the second-order maximum occur?
    2. At what angle does the first minimum occur?
    3. What is the highest-order maximum possible for this arrangement?
17. Figure 81.8 shows a double slit located a distance [latex]x[/latex] from a screen. The distance from the center of the screen to a bright fringe is [latex]y[/latex], and the separation between the slits is [latex]d[/latex]. For small angles, [latex]\sin\theta\approx\tan\theta\approx\theta[/latex], where [latex]\theta[/latex] is measured in radians. Use these approximations and the constructive interference condition to show that the distance between adjacent bright fringes is
    [latex]\Delta y=\frac{x\lambda}{d}.[/latex]

    

18. Use the fringe-spacing equation to calculate the distance between adjacent bright fringes for 633-nm light passing through two slits separated by 0.0800 mm. The screen is located 3.00 m from the slits, as shown in Figure 81.8.
19. Light passing through two slits separated by 0.120 mm produces adjacent bright fringes 7.50 mm apart on a screen located 2.00 m from the slits. Determine the wavelength of the light.