wave optics ii

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Diffraction and Polarization



DiffractionDiffraction is the bending of waves around

obstacles or the edges of an opening. Consider sound waves leaving a room through an open doorway. Because the exiting sound waves bend, or diffract, around the edges of the opening, a listener outside the room can hear the sound even when standing around the corner from the doorway. Diffraction is therefore an

Huygens PrincipleHuygensprinciple describes how a wave front

that exists at one instant gives rise to the wave front that exists later on. It 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.

Huygens Principle The drawing shows the top view of a plane wave

front of sound approaching a doorway and identifies five points on the wave front just as it is leaving the opening. According to Huygens principle, each of these points acts as a source of wavelets, which are shown as red circular arcs at some moment after they are emitted. The tangent to the wavelets from points 2, 3, and 4 indicates that in front of the doorway the wave front is flat and moving straight ahead. But at the edges, points 1 and 5 are the last points that produce wavelets. Huygens principle suggests that in conforming to the curved shape of the wavelets near the edges, the new wave front moves into regions that it would not reach otherwise. The sound wave, then, bends or diffracts around the edges of the doorway.

Huygens PrincipleHuygens principle applies not just to sound

waves, but to all kinds of waves. For instance, light has a wave-like nature and, consequently, exhibits diffraction. Therefore, you may ask, Since I can hear around the edges of a doorway, why cant I also see around them? As a matter of fact, light waves do bend around the edges of a doorway. However, the degree of bending is extremely small, so the diffraction of light is not enough to allow you

Factors Controlling Diffraction The extent to which a wave bends around the edges of an opening

is determined by the ratio /W, where is the wavelength of the wave and W is the width of the opening.

These photographs show water waves (horizontal lines)

approaching an opening whose width W is greater in (a) than in (b). In addition, the wavelength of the waves is smaller in (a) than in (b). Therefore, the ratio /W increases from (a) to (b) and so does the extent of the diffraction, as the red arrows indicate.

Diffraction of Light We might expect then that light waves of wavelength will

bend or diffract appreciably when they pass through an opening whose width W is small enough to make the ratio /W sufficiently large.(a) If light were to pass through a very narrow slit without being diffracted, only the region on the screen directly opposite the slit would be illuminated. (b) Diffraction causes the light to bend around the edges of the slit into regions it would not otherwise reach, forming a pattern of alternating bright and dark fringes on the screen. The slit width has

Single-Slit Diffraction The figure shows a top view of

a plane wave front approaching the slit and singles out five sources of Huygens wavelets. Consider how the light from these five sources reaches the midpoint on the screen. To simplify things, the screen is assumed to be so far from the slit that the rays from each Huygens source are nearly parallel. Then, all the wavelets travel virtually the same distance to the midpoint, arriving there in phase. As a result, constructive interference creates a bright central fringe on the screen,

Single-Slit DiffractionThe wavelets emitted by the

Huygens sources in the slit can also interfere destructively on the screen. Destructive interference creates the first dark fringe when the extra distance traveled by the wavelet from source 5 is exactly one wavelength, as the colored right triangle in the drawing indicates. Under this condition, the extra distance traveled by the wavelet from source 3 at the center of the slit is exactly one-

Single-Slit DiffractionIn a single-slit diffraction pattern, multiple

dark fringes occur on either side of the central bright fringe. This drawing shows how destructive interference creates the second dark fringe on a very distant screen.

Single-Slit Diffraction Between each pair of dark fringes there is a bright fringe

due to constructive interference. The brightness of the fringes is related to the light intensity, just as loudness is related to sound intensity. The intensity of the light at any location on the screen is the amount of light energy per second per unit area that strikes the screen there. The higher-order bright fringes are much less intense than the central fringe, as the graph indicates.

Problem: Single-Slit DiffractionLight passes through a slit and shines on a flat

screen that is located L=0.40 m away. The wavelength of the light in a vacuum is =410 nm. The distance between the midpoint of the central bright fringe and the first dark fringe is y. Determine the width 2y of the central bright fringe when the width of the slit is (a) W=5.0 x 10-6 m and (b) W=2.5 x 10-6 m.

Solution: Single-Slit Diffraction(a) The angle locates the first dark fringe

when m=1 and s0 sin = (1) /W. Therefore,

Since tan = y/L, the width 2y of the central

bright fringe is

(b) 2y=0.13 m

Resolving Power

These automobile headlights were

photographed at various distances from the camera, closest in part (a) and farthest in part (c). In part (c), the headlights are so far away that they are barely distinguishable.

Resolving PowerThe resolving power of an optical instrument, such

as a camera, is its ability to distinguish between two closely spaced objects. If a camera with a higher resolving power had taken those pictures, the photograph in part c would have shown two distinct and separate headlights. Any instrument used for viewing objects that are close together must have a high resolving power. This is true, for example, for a telescope used to view distant stars or for a microscope used to view tiny organisms. The diffraction occurs when light passes through the circular, or nearly circular, openings that admit light into cameras, telescopes, microscopes, and human eyes. The resulting diffraction pattern places a limit on the resolving power of these instruments.

Diffraction on Round OpeningsWhen light passes through a

small circular opening, a circular diffraction pattern is formed on a screen. The angle locates the first dark fringe relative to the central bright region. The intensities of the bright fringes and the diameter of the opening have been exaggerated for clarity.

Diffraction on Round OpeningsThese fringes are analogous to the

rectangular fringes that a single slit produces. The angle in the picture locates the first circular dark fringe relative to the central bright region and is given by

where is the wavelength of the light and D is the diameter of the opening. This expression is similar to equation for slit diffraction (sin = /W, when m=1) and is valid when the distance to the screen is much larger than the diameter D.

Resolving Power of Optical InstrumentsAn optical instrument with the ability to

resolve two closely spaced objects can produce images of them that can be identified separately. When light from two point objects passesthrough the circular aperture of a camera, two circular diffraction patterns are formed as images on the film. The images here are completely separated or resolved because the objects are widely separated. On the other hand, if the objects are sufficiently close together, the intensity patterns created by the diffraction overlap and the picture from a camera would show a single blurred object instead of two separate

Rayleigh criterion for resolutionFirst proposed by Lord Rayleigh, it is a useful

criterion for judging whether two closely spaced objects will be resolved by an optical instrument. It states 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.

Rayleigh criterion for resolution

(a) According to the Rayleigh criterion, two point

objects are just resolved when the first dark fringe (zero intensity) of one image falls on the central bright fringe (maximum intensity) of the other image. (b) This photograph shows two overlapping

Rayleigh criterion for resolutionThe minimum angle min between the two objects

in the drawing is that given byIf min is small (less than about 100) and is

expressed in radians, sin min min . So,

For a given wavelength and aperture diameter

D, this result specifies the smallest angle that two point objects can subtend at the aperture and still be resolved.

Rayleigh criterion for resolutionobjects (small values of min) must utilize the smallest possible wavelength and the largest possible aperture diameter. For example, when shortwavelength ultraviolet light is collected by its large 2.4-mdiameter mirror, the Hubble Space Telescope is capable of resolving two closely spaced stars that have an angular separation of about min = 1 x 10-7 rad. This angle is equivalent to resolving two objects The Hubble Space Telescope only 1 cm apart when they are 1 x 105 m (about 62 miles) from the Optical instruments designed to resolve closely spaced

Problem: The Human Eye Versus the Eagles Eye(a) A hang glider is flying at

an altitude of 120 m. Green light (wa


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