# Wave Optics

## Young’s Double Slit Experiment


### Learning Objectives

By the end of this section, you will be able to:
1. Explain the phenomena of interference.
2. Define constructive interference for a double slit and destructive interference for a double slit.
Although Christiaan Huygens thought that light was a wave, Isaac Newton did not. Newton felt that there were other explanations for color, and for the interference and diffraction effects that were observable at the time. Owing to Newton’s tremendous stature, his view generally prevailed. The fact that Huygens’s principle worked was not considered evidence that was direct enough to prove that light is a wave. The acceptance of the wave character of light came many years later when, in 1801, the English physicist and physician Thomas Young (1773–1829) did his now-classic double slit experiment (see ).
Why do we not ordinarily observe wave behavior for light, such as observed in Young’s double slit experiment? First, light must interact with something small, such as the closely spaced slits used by Young, to show pronounced wave effects. Furthermore, Young first passed light from a single source (the Sun) through a single slit to make the light somewhat coherent. By coherent, we mean waves are in phase or have a definite phase relationship. Incoherent means the waves have random phase relationships. Why did Young then pass the light through a double slit? The answer to this question is that two slits provide two coherent light sources that then interfere constructively or destructively. Young used sunlight, where each wavelength forms its own pattern, making the effect more difficult to see. We illustrate the double slit experiment with monochromatic (single ) light to clarify the effect.  shows the pure constructive and destructive interference of two waves having the same wavelength and amplitude.
When light passes through narrow slits, it is diffracted into semicircular waves, as shown in (a). Pure constructive interference occurs where the waves are crest to crest or trough to trough. Pure destructive interference occurs where they are crest to trough. The light must fall on a screen and be scattered into our eyes for us to see the pattern. An analogous pattern for water waves is shown in (b). Note that regions of constructive and destructive interference move out from the slits at well-defined angles to the original beam. These angles depend on wavelength and the distance between the slits, as we shall see below.
To understand the double slit interference pattern, we consider how two waves travel from the slits to the screen, as illustrated in . Each slit is a different distance from a given point on the screen. Thus different numbers of wavelengths fit into each path. Waves start out from the slits in phase (crest to crest), but they may end up out of phase (crest to trough) at the screen if the paths differ in length by half a wavelength, interfering destructively as shown in (a). If the paths differ by a whole wavelength, then the waves arrive in phase (crest to crest) at the screen, interfering constructively as shown in (b). More generally, if the paths taken by the two waves differ by any half-integral number of wavelengths [, , , etc.], then destructive interference occurs. Similarly, if the paths taken by the two waves differ by any integral number of wavelengths (, , , etc.), then constructive interference occurs.
 shows how to determine the path length difference for waves traveling from two slits to a common point on a screen. If the screen is a large distance away compared with the distance between the slits, then the angle  between the path and a line from the slits to the screen (see the figure) is nearly the same for each path. The difference between the paths is shown in the figure; simple trigonometry shows it to be , where  is the distance between the slits. To obtain constructive interference for a double slit, the path length difference must be an integral multiple of the wavelength, or
Similarly, to obtain destructive interference for a double slit, the path length difference must be a half-integral multiple of the wavelength, or
where  is the wavelength of the light,  is the distance between slits, and  is the angle from the original direction of the beam as discussed above. We call  the order of the interference. For example,  is fourth-order interference.
The equations for double slit interference imply that a series of bright and dark lines are formed. For vertical slits, the light spreads out horizontally on either side of the incident beam into a pattern called interference fringes, illustrated in . The intensity of the bright fringes falls off on either side, being brightest at the center. The closer the slits are, the more is the spreading of the bright fringes. We can see this by examining the equation 
For fixed  and , the smaller  is, the larger  must be, since


.

 This is consistent with our contention that wave effects are most noticeable when the object the wave encounters (here, slits a distance  apart) is small. Small  gives large , hence a large effect.

### Test Prep for AP Courses


### Section Summary

1. Young’s double slit experiment gave definitive proof of the wave character of light.
2. An interference pattern is obtained by the superposition of light from two slits.
3. There is constructive interference when , where  is the distance between the slits,  is the angle relative to the incident direction, and  is the order of the interference.
4. There is destructive interference when .

### Conceptual Questions


### Problems & Exercises
