Concept Summary

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Topic | Discussion | Learning Aids
Principle of linear superposition | 27.1 THE PRINCIPLE OF LINEAR SUPERPOSITION The principle of linear superposition states that when two or more waves are present simultaneously in the same region of space, the resultant disturbance is the sum of the disturbances from the individual waves. |
Constructive interference | Constructive interference occurs at a point when two waves meet there crest-to-crest and trough-to-trough, thus reinforcing each other. When two waves that start out in phase and have traveled some distance meet at a point, constructive interference occurs whenever the travel distances are the same or differ by any integer number of wavelengths: − = m, where and are the distances traveled by the waves, and m = 0, 1, 2, 3, … . |
Destructive interference | Destructive interference occurs at a point when two waves meet there crest-to-trough, thus mutually canceling each other. When two waves that start out in phase and have traveled some distance meet at a point, destructive interference occurs whenever the travel distances differ by any odd integer number of half-wavelengths: where and are the distances traveled by the waves, and m = 0, 1, 2, 3, … . |
Coherent sources | Two sources are coherent if the waves they emit maintain a constant phase relation. In other words, the waves do not shift relative to one another as time passes. If constructive and destructive interference are to be observed, coherent sources are necessary. |
Double-slit experiment | 27.2 YOUNG’S DOUBLE-SLIT EXPERIMENT In Young’s double-slit experiment, light passes through a pair of closely spaced narrow slits and produces a pattern of alternating bright and dark fringes on a viewing screen. The fringes arise because of constructive and destructive interference. The angle that locates the mth-order bright fringe is given by |
Bright fringes of a double slit |

| (27.1)

where is the wavelength of the light, and d is the spacing between the slits. The angle that locates the mth dark fringe is given by
Dark fringes of a double slit |

| (27.2)

| 27.3 THIN-FILM INTERFERENCE Constructive and destructive interference of light waves can occur with thin films of transparent materials. The interference occurs between light waves that reflect from the top and bottom surfaces of the film. One important factor in thin-film interference is the thickness of a film relative to the wavelength of the light within the film. The wavelength within a film is |
Wavelength within a film |

| (27.3)

where is the wavelength in a vacuum, and n is the index of refraction of the film.
Phase change due to reflection | A second important factor is the phase change that can occur when light reflects at each surface of the film:

1. | When light travels through a material with a smaller index of refraction toward a material with a larger index of refraction, reflection at the boundary occurs along with a phase change that is equivalent to one-half a wavelength in the film.
2. | When light travels through a material with a larger index of refraction toward a material with a smaller index of refraction, there is no phase change upon reflection at the boundary.

Michelson interferometer | 27.4 THE MICHELSON INTERFEROMETER An interferometer is an instrument that can be used to measure the wavelength of light by employing interference between two light waves. The Michelson interferometer splits the light into two beams, one of which travels to a fixed mirror, reflects from it, and returns. The other beam travels to a movable mirror, reflects from it, and returns. When the two returning beams are combined, interference is observed, the amount of which depends on the travel distances. |

Use Self-Assessment Test 27.1 to evaluate your understanding of Sections 27.1, 27.2, 27.3 and 27.4 .

Diffraction Huygens’ principle | 27.5 DIFFRACTION Diffraction is the bending of waves around obstacles or around the edges of an opening. Diffraction is an interference effect that can be explained with the aid of Huygens’ principle. This principle states that every point on a wave front acts as a source of tiny wavelets that move forward with the same speed as the wave; the wave front at a later instant is the surface that is tangent to the wavelets.
When light passes through a single narrow slit and falls on a viewing screen, a pattern of bright and dark fringes is formed because of the superposition of Huygens wavelets. The angle that specifies the mth dark fringe on either side of the central bright fringe is given by
Dark fringes for single-slit diffraction |

| (27.4)

where is the wavelength of the light and W is the width of the slit.
Resolving power | 27.6 RESOLVING POWER The resolving power of an optical instrument is the ability of the instrument to distinguish between two closely spaced objects. Resolving power is limited by the diffraction that occurs when light waves enter an instrument, often through a circular opening. |
Rayleigh criterion | The Rayleigh criterion specifies that two point objects are just resolved when the first dark fringe in the diffraction pattern of one falls directly on the central bright fringe in the diffraction pattern of the other. According to this specification, the minimum angle (in radians) that two point objects can subtend at a circular aperture of diameter D and still be resolved as separate objects is |
Minimum angle for resolving two point objects |

| (27.6)

where is the wavelength of the light.
Diffraction grating | 27.7 THE DIFFRACTION GRATING A diffraction grating consists of a large number of parallel, closely spaced slits. When light passes through a diffraction grating and falls on a viewing screen, the light forms a pattern of bright and dark fringes. The bright fringes are referred to as principal maxima and are found at an angle such that |
Principal maxima of a diffraction grating |

| (27.7)

where is the wavelength of the light and d is the separation between two adjacent slits.
| 27.8 COMPACT DISCS, DIGITAL VIDEO DISCS, AND THE USE OF INTERFERENCE Compact discs and digital video discs depend on interference for their operation. |
| 27.9 X-RAY DIFFRACTION A diffraction pattern forms when X-rays are directed onto a crystalline material. The pattern arises because the regularly spaced atoms in a crystal act like a diffraction grating. Because the spacing is extremely small, on the order of 1 × l0−10m, the wavelength of the electromagnetic waves must also be very small—hence, the use of X-rays. The crystal structure of a material can be determined from its X-ray diffraction pattern. |

Use Self-Assessment Test 27.2 to evaluate your understanding of Sections 27.5, 27.6, 27.7, 27.8 and 27.9.

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