RAYMUND B. BOLALINDLSU Physics Department

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 interference effect.

Huygen’s PrincipleHuygens’principle 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.”

Huygen’s 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.

Huygen’s 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 can’t 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 to see around the corner.

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 LightWe 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 been exaggerated for clarity.

Single-Slit DiffractionThe 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, directly opposite the slit.

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-half of a wavelength.

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 DiffractionBetween 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 passes

through 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 objects.

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 but still resolvable diffraction patterns.

Rayleigh criterion for resolutionThe minimum angle min between the two objects in

the drawing is that given by

If 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 resolutionOptical instruments designed to resolve closely spaced

objects (small values of min) must utilize the smallest possible wavelength and the largest possible aperture diameter.

For example, when short-wavelength ultraviolet light is collected by its large 2.4-m-diameter 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 only 1 cm apart when they are 1 x 105 m (about 62 miles) from the telescope.

The Hubble Space Telescope

Problem: The Human Eye Versus the Eagle’s Eye

(a) A hang glider is flying at an altitude of 120 m. Green light (wavelength = 555 nm in vacuum) enters the pilot’s eye through a pupil that has a diameter of 2.5 mm. Determine how far apart two point objects must be on the ground if the pilot is to have any hope of distinguishing between them. (b) An eagle’s eye has a pupil with a diameter of 6.2 mm. Repeat part (a) for an eagle flying at the same altitude as the glider.

Solution: The Human Eye Versus the Eagle’s Eye

(a) For the pilot to have any hope of distinguishing between the objects, the separation distance must be at least

(b) For the eagle:

(Since the pupil of the eagle’s eye is larger than that of a human eye, diffraction creates less of a limitation for the eagle; the two objects can be closer together and still be resolved by the eagle’s eye.)

What You See is Not What You Get The French postimpressionist artist Georges Seurat developed a

technique of painting in which dots of color are placed close together on the canvas. From sufficiently far away the individual dots are not distinguishable, and the images in the picture take on a more normal appearance. The figure shows a person in an art museum, looking at one of Seurat’s paintings. Suppose that this person stands close to the painting, then backs up until the dots just become indistinguishable to his eyes and takes a picture from this position. When he gets home and has the film developed, however, he can see the individual dots in the enlarged photograph. Why does the camera resolve the individual dots, while his eyes do not? Assume that light enters his eyes through pupils that have diameters of 2.5 mm and enters the camera through an aperture, or opening, with a diameter of 25 mm.

What You See is Not What You GetThe Rayleigh criterion for resolving two point objects

specifies that min≈1.22/D. According to this criterion, a larger value of the diameter D of the opening means that the value for min is smaller, so that an optical instrument can be farther away and still resolve the objects. In other words, a larger value of D means a greater resolving power. For the eye and the camera the diameters are Deye=2.5 mm and Dcamera=25 mm , so the diameter for the camera is ten times larger than that for the eye. Thus, at the distance at which the eye loses its ability to resolve the individual dots in the painting, the camera can easily resolve the dots. We conclude, then, that the camera succeeds in resolving the dots in the painting while the eye fails because the camera gathers light through a much larger opening.

Diffraction GratingDiffraction patterns of bright and dark fringes occur

when monochromatic light passes through a single or double slit.

Fringe patterns also result when light passes through more than two slits, and an arrangement consisting of a large number of parallel, closely spaced slits called a diffraction grating, has proved very useful.

Gratings with as many as 40 000 slits per centimeter can be made, depending on the production method.

In one method a diamond-tipped cutting tool is used to inscribe closely spaced parallel lines on a glass plate, the spaces between the lines serving as the slits.

In fact, the number of slits per centimeter is often quoted as the number of lines per centimeter.

Diffraction GratingWhen light passes through a

diffraction grating, a central bright fringe (m=0) and higher-order bright fringes (m=1,2,3…) form when the light falls on a distant viewing screen.

These bright fringes are sometimes called the principal fringes or principal maxima, since they are places where the light intensity is a maximum. The term “principal” distinguishes them from other, much less bright fringes that are referred to as secondary fringes or secondary maxima.

Diffraction GratingThe bright fringes

produced by a diffraction grating are much narrower than those produced by a double slit.

Note the three small secondary bright fringes between the principal bright fringes of the grating.

For a large number of slits, these secondary fringes become very small.

Problem: Separating Colors with a Diffraction Grating

A mixture of violet light (=410 nm in vacuum) and red light (=660 nm in vacuum) falls on a grating that contains 1.0 x 104 lines/cm. For each wavelength, find the angle that locates the first-order maxima.

Solution: Separating Colors with a Diffraction Grating

The separation d between the slits is

For violet light, the angle locating the first-order maxima is

Similar calculation for red light red=660 x 10-9 m leads to red=410.

Because red and violet are different, separate first-order bright fringes are seen for violet and red light on a viewing screen.

Diffraction GratingIf the light used had been sunlight, the angles for

the first-order maxima would cover all values in the range between 24° and 41°, since sunlight contains all colors or wavelengths between violet and red. Consequently, a rainbow-like dispersion of the colors would be observed to either side of the central fringe on a screen. This drawing shows that the spectrum of colors associated with the m=2 order is completely separate from that of the m=1 order. For higher orders, however, the spectra from adjacent orders may overlap. The central maximum is white because all the colors overlap there.

A Compact Disc Is a Diffraction GratingLight reflected from the surface of a compact disc

is multicolored. The colors and their intensities depend on the orientation of the disc relative to the eye and relative to the light source.

The surface of a compact disc has a spiral grooved track (with adjacent grooves having a separation on the order of 1 m). Thus, the surface acts as a reflection grating.The light reflecting from the regions between theseclosely spaced grooves interferes constructively only in certain directions that depend on the wavelength and on the direction of the incident light. Any one section of the disc serves as a diffraction grating for white light, sending different colors in different directions. The different colors you see when viewing one section change as the light source, the disc, or you move to change the angles of incidence or diffraction.

PolarizationPolarization is a property of waves that

describes the orientation of their oscillations. For transverse waves, it describes the

orientation of the oscillations in the plane perpendicular to the wave's direction of travel.

Polarization of LightAn ordinary beam of light consists of

a large number of waves emitted by the atoms of the light source.

Each atom produces a wave having some particular orientation of the electric field vector E, corresponding to the direction of atomic vibration.

The direction of polarization of each individual wave is defined to be the direction in which the electric field is vibrating.

In the figure, this direction happens to lie along the y axis. However, an individual electromagnetic wave could have its E vector in the yz plane, making any possible angle with the y axis.

Polarization of LightBecause all directions of vibration from a wave

source are possible, the resultant electromagnetic wave is a superposition of waves vibrating in many different directions.

The result is an unpolarized light beam.The direction of wave propagation in this figure is

perpendicular to the page. The arrows show a few possible directions of the electric field vectors for the individual waves making up the resultant beam. At any given point and at some instant of time, all these individual electric field vectors add to give one resultant electric field vector.

Polarization of LightA wave is said to be linearly polarized if the

resultant electric field E vibrates in the same direction at all times at a particular point, as shown in the figure. (Sometimes, such a wave is described as plane-polarized, or simply polarized.)

The plane formed by E and the direction of propagation is called the plane of polarization of the wave.

It is possible to obtain a linearly polarized beam from an unpolarized beam by removing all waves from the beam except those whose electric field vectors oscillate in a single plane.

Polarization Processes Polarization by Selective AbsorptionPolarization by ReflectionPolarization by Double RefractionPolarization by Scattering

Polarization by Selective AbsorptionThe most common technique for producing polarized light is to

use a material that transmits waves whose electric fields vibrate in a plane parallel to a certain direction and that absorbs waves whose electric fields vibrate in all other directions.

In 1938, E. H. Land (1909–1991) discovered a material, which he called polaroid, that polarizes light through selective absorption by oriented molecules.

This material is fabricated in thin sheets of long-chain hydrocarbons. The sheets are stretched during manufacture so that the long-chain molecules align. After a sheet is dipped into a solution containing iodine, the molecules become good electrical conductors. However, conduction takes place primarily along the hydrocarbon chains because electrons can move easily only along the chains. As a result, the molecules readily absorb light whose electric field vector is parallel to their length and allow light through whose electric field vector is perpendicular to their length.

Polarization by Selective Absorption

Two polarizing sheets whose transmission axes make an angle with each other. Only a fraction of the polarized light incident on the analyzer is transmitted through it.

Effect of a polarizer on reflection from mud flats. In the picture on the left, the polarizer is rotated to transmit the reflections as well as possible; by rotating the polarizer by 90° (picture on the right) almost all specularly reflected sunlight is blocked.

The effects of a polarizing filter on the sky in a photograph. The picture on the right uses the filter.

Polarization by ReflectionWhen an unpolarized light beam is reflected

from a surface, the reflected light may be completely polarized, partially polarized, or unpolarized, depending on the angle of incidence.

If the angle of incidence is 0°, the reflected beam is unpolarized. For other angles of incidence, the reflected light is polarized to some extent, and for one particular angle of incidence, the reflected light is completely polarized.

Polarization by Reflection(a) When unpolarized

light is incident on a reflecting surface, the reflected and refracted beams are partially polarized.

(b) The reflected beam is completely polarized when the angle of incidence equals the polarizing angle p, which satisfies the equation n = tan p

(Brewster’s Law).

Polarization by ReflectionPolarization by reflection is a common

phenomenon. Sunlight reflected from water, glass, and snow is

partially polarized. If the surface is horizontal, the electric field vector of the reflected light has a strong horizontal component.

Sunglasses made of polarizing material reduce the glare of reflected light. The transmission axes of the lenses are oriented vertically so that they absorb the strong horizontal component of the reflected light. If you rotate sunglasses 90°, they will not be as effective at blocking the glare from shiny horizontal surfaces.

Polarization by Double RefractionSolids can be classified on the basis of internal

structure. Those in which the atoms are arranged in a specific order are called crystalline( e.g. NaCl structure). Those solids in which the atoms are distributed randomly are called amorphous.

When light travels through an amorphous material, such as glass, it travels with a speed that is the same in all directions. That is, glass has a single index of refraction.

In certain crystalline materials, however, such as calcite and quartz, the speed of light is not the same in all directions. Such materials are characterized by two indices of refraction. Hence, they are often referred to as double-refracting or birefringent materials.

Polarization by Double RefractionUpon entering a calcite crystal, unpolarized light splits

into two plane-polarized rays that travel with different velocities, corresponding to two angles of refraction, as shown in the figure. The two rays are polarized in two mutually perpendicular directions, as indicated by the dots and arrows. One ray, called the ordinary (O) ray, is characterized by an index of refraction nO that is the same in all directions. This means that if one could place a point source of light inside the crystal, the ordinary waves would spread out from the source as spheres.

Polarization by Double RefractionThe second plane-polarized ray, called the

extraordinary (E) ray, travels with different speeds in different directions and hence is characterized by an index of refraction nE that varies with the direction of propagation.

Polarization by ScatteringWhen light is incident on any material, the

electrons in the material can absorb and reradiate part of the light.

Such absorption and reradiation of light by electrons in the gas molecules that make up air is what causes sunlight reaching an observer on the Earth to be partially polarized. You can observe this effect—called scattering—by looking directly up at the sky through a pair of sunglasses whose lenses are made of polarizing material. Less light passes through at certain orientations of the lenses than at others.

Polarization by Scattering The figure on the right illustrates how sunlight becomes polarized

when it is scattered. An unpolarized beam of sunlight traveling in the horizontal

direction (parallel to the ground) strikes a molecule of one of the gases that make up air, setting the electrons of the molecule into vibration.

These vibrating charges act like the vibrating charges in an antenna. The horizontal component of the electric field vector in the incident wave results in a horizontal component of the vibration of the charges, and the vertical component of the vector results in a vertical component of vibration.

If the observer in the figure is looking straight up (perpendicular to the original direction of propagation of the light), the vertical oscillations of the charges send no radiation toward the observer. Thus, the observer sees light that is completely polarized in the horizontal direction, as indicated by the red arrows.

If the observer looks in other directions, the light is partially polarized in the horizontal direction.

Polarization by ScatteringPolarization by scattering is observed as light

passes through the atmosphere. The scattered light produces the brightness

and color in clear skies.

Why does the red color appear at the horizon during sunset?The red color in the sky at sunset

(and sunrise) is due to scattering. (Scattering is also responsible for the white color of clouds.)

Particles in our atmosphere that are approximately the same size as the wavelength of visible light cause the white light from the sun to scatter and split into individual components. Oxygen and Nitrogen (the main components of our atmosphere) scatter violet and blue light due to their small size. This is why the sky appears to be blue in the day time, especially at midday when the Sun is closest to us.

During sunrise and sunset the distance that the light has to travel from the Sun to an observer is at its greatest. This means the a large amount of blue and violet light has been scattered so the light that is received by an observer is mostly of a longer wavelength and therefore appears to be red.

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ATOMIC STRUCTURE