SAT Physics Physical Optics - Diffraction

SAT Physics Physical Optics - Diffraction

DIFFRACTION
When waves pass near a barrier or through an opening (slit), they bend and spread out to fill the space behind the barrier or the slit. The bending of a wave due to a barrier or opening is known as diffraction. The amount of bending has to do with the size of the obstacle or opening compared with the wavelength of the wave. In the diagrams in Figure 17.1, parallel wave fronts are shown approaching various openings. The bending due to diffraction increases as the openings become smaller.

Figure 17.1. Diffraction

   When the opening is large as compared with the wavelength of the waves, the waves move through the opening with little diffraction. This creates large shadow regions with no wave activity to the left and right of the opening. When light waves are diffracted, the absence of light in the shadow regions leaves these areas dark. However, when the size of the opening is similar to the size of the wavelength of the waves, the diffraction is so pronounced that the spreading wave fronts form a circular pattern with no shadow regions. Since light is composed of very small wavelengths, openings that cause significant diffraction must be incredibly narrow.
A geometric explanation for the circular nature of the resulting diffraction pattern was proposed by Christian Huygens and is known as the Huygens’ principle. There are two main aspects of his principle.

1. Every oscillator in a wave creates spherical wavelets that propagate outward.
2. The wave front created by these oscillators is due to the combined interference of the wavelets.

   Figure 17.2 shows a diagram of a portion of a linear wave. The dots lying along the crest of the wave represent the individual oscillators each creating circular wave fronts.

Figure 17.2. Wave fronts

   This provides an explanation for the circular wave fronts seen when waves move through narrow openings. If the opening is very small, only one, or very few, oscillator(s) propagate the wave through the opening. As a result, circular wave fronts are generated on the other side of the opening.

INTERFERENCE OF LIGHT
When light waves interact, they can interfere constructively or destructively. If identical wave crests (represented by wave fronts) meet, constructive interference adds them together to create a single larger wave. Constructive interference causes water waves to increase in height. It causes sound to become louder and light to become brighter. If, on the other hand, a wave crest meets a wave trough of identical size, destructive interference will cancel these waves entirely. This creates an absence of wave activity. It causes water to be flat, quiet instead of loudness, and darkness instead of brightness.

Young’s Double-Slit Experiment
Thomas Young constructed an experiment, known as Young’s double-slit experiment, involving the interference of light. He shined monochromatic light, which is light composed of only one wavelength, on two incredibly narrow openings (slits) that were near each other. The light diffracted through each slit, causing circular wave fronts to spread outward. These patterns overlapped each other and created both constructive and destructive interference. The resulting pattern was projected onto a screen where constructive interference created bright regions (bright fringes or maximums) and destructive interference created dark regions (dark fringes or minimums). Figure 17.3 shows the interference pattern created by the circular wave fronts spreading from each slit.

Figure 17.3. Young’s double-slit experiment

   The expanding circular wave fronts represent the wave crests. Where wave crests intersect, they interfere constructively. In Figure 17.3, lines through successive wave crest intersections have been added. They extend from a point midway between the slits toward the screen. Where these lines of constructive interference hit the screen is where bright bands (maximums, m) of light will be seen. In the regions in between, destructive interference creates bands that are dark (minimums, m). The maximums and minimums are numbered starting with the central maximum, m = 0. The bright maximums are numbered as whole numbers (m = 1, 2, or 3 ...), while the minimums are numbered with halves (m = 0.5,1.5, or 2.5 ...). Be very careful with the values assigned to dark minimums. The value for the third dark minimum (3rd dark fringe) is m = 2.5 and not 3.5 as students often incorrectly believe. The value representing the maximums and minimums have no units, and the reason behind the numbering system will be discussed below.
    A mathematical relationship describes the resulting bright and dark fringes. This will be easier to explain if the diagram is simplified to solve for one specific maximum. Figure 17.4 shows the mathematical relationships for the first bright maximum, m = 1. Only the lines extending to the central (reference) maximum and the first maximum are shown.

Figure 17.4. First bright fringe position for the double-slit experiment

   Various key variables have been labeled in Figure 17.4. These include the number for the maximum to be analyzed m, the distance from the central maximum to the maximum being investigated x_m, the spacing between the slits d, the distance from the slits to the screen L, and the angle 6. The actual width of the slits themselves is not needed. The variable m has no units, and the angle is measured in degrees. All other variables are lengths measured in meters. Students often confuse m and x_m. Remember that m is the number for the maximum and x_m is the distance to that maximum. There are two mathematical relationships for Young’s double-slit experiment:

The formula on the left is actually an approximation that works well as long as the screen distance, L, is very large, which is typically the case on exams.
    When the diagram is viewed differently, an explanation for the numerical values of m is apparent. In Figure 17.5, two rays of light are shown moving from each slit toward the maximum, m.

Figure 17.5. Path difference

   Light ray B must travel a longer distance than light ray A The difference between the distance the two rays travel is known as the path difference. Light reaching the first maximum, m = 1, has a path difference equal to 1 wavelength. Light reaching the second maximum, m = 2, has a path difference equal to 2 wavelengths. In other words the maximum numbers, m, represent the number of wavelengths by which the paths differ. Bright maximums occur when crests meet and interfere constructively. At maximum m = 1, a wave following path A must meet up with a wave following path B that is exactly 1 wavelength off. Likewise, all the other maximums must occur when the paths of light differ by whole numbers of wavelengths. Otherwise, their crests will not match. The dark minimums appear when the waves are off by half of a wavelength. This is when crests meet troughs and destructively interfere.
    Young’s double-slit experiment is considered experimental evidence that light possesses a wave characteristic. The double-slit effects can be seen with water waves, and it can be heard with sound waves. Sometimes on an exam, the two closely spaced slits may be replaced with two closely spaced loudspeakers. The effect is the same with sound as it is for light.
    A key element in a double-slit experiment is using waves that have a single wavelength. If more than one wavelength is present, a more complex and less symmetrical interference pattern would occur. For sound waves, the experiment can be conducted using two speakers both playing a tone with the same frequency. For light waves, the experiment must be conducted using monochromatic light, which is light with a single constant wavelength and a specific color.

Young’s Double-Slit Experiment

    Monochromatic light passes through two narrow slits and is projected onto a screen, creating a double-slit interference pattern. The first bright maximum occurs at a distance of 0.004 meters from the central maximum. The slit spacing is 3.0 x 10^-6 meters, and the distance to the screen is 2.0 meters. Determine the wavelength of light used in this experiment
 
WHAT'S THE TRICK?

If no angle is given, then the formula solving for double-slit interference is:

Trends in Double-Slit Interference Patterns

   Light incident on two narrow slits is used to project an interference pattern onto a screen. How will the interference pattern change if the distance between the slits is doubled?
 
WHAT'S THE TRICK?

The appearance of the interference pattern is tied to the distance between the maximums, x_m. If the distance between the maximums increases, the maximums are moving farther apart. The bright and dark regions will spread out. If the distance between maximums decreases, the maximums will move closer together, compressing the appearance of the interference pattern. Analyze how the change in slit spacing, d, affects the interference pattern spacing, x_m.

 The interference pattern spacing, x_m, is halved. Therefore, the pattern compresses.

Single-Slit Interference
A pattern similar to Young’s double-slit experiment can be seen even when only one slit is present. Huygens’ principle offers an explanation. Two points at opposite ends of the slit are sources of circular wavelets that will interfere with each other in the same manner as two points in separate slits. However, when a single slit is involved, a large amount of wave activity moves straight through the center of the slit with minimal interference. The result is a very large central maximum. This maximum is not only wider but is also more intense than the central maximum produced with a double slit. The intensity of light is associated with its brightness. The wider and brighter central maximum is surrounded by much smaller maximums that are dimmer.
    The intensity of light is often indicated as a curve superimposed on the double- and single¬slit diagrams as shown in Figure 17.6.
    Intensity graphs have been superimposed on the diagrams in Figure 17.6. They indicate where bright maximums and dark minimums occur. For both patterns, the central maximum is the brightest and largest. However, the maximums for a double-slit interference pattern are more evenly spaced and are more similar in size. The single-slit pattern has a distinctly oversized central maximum.
    Another important difference is that the double-slit pattern depends on the space between the slits, d, and not on the actual width of either slit. However, the single-slit pattern depends on the actual width of the slit opening. Increasing the slit width in the single-slit pattern has the same effect as increasing the slit spacing for the double-slit pattern. In both cases, the interference pattern compresses so that the space between maximums is decreased. The reverse is true if the double slits are moved closer together or if the single-slit width is reduced. This will cause both patterns to spread out, moving the maximums away from each other.

Figure 17.6. Double - and single - slit interference patterns