diffraction

[New Latin diffrāctiō, diffrāctiōn-, from Latin diffrāctus, past participle of diffringere : dis-, apart; see dis- + frangere, to break.]

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Concept

Diffraction is the bending of waves around obstacles, or the spreading of waves by passing them through an aperture, or opening. Any type of energy that travels in a wave is capable of diffraction, and the diffraction of sound and light waves produces a number of effects. (Because sound waves are much larger than light waves, however, diffraction of sound is a part of daily life that most people take for granted.) Diffraction of light waves, on the other hand, is much more complicated, and has a number of applications in science and technology, including the use of diffraction gratings in the production of holograms.

How It Works

Comparing Sound and Light Diffraction

Imagine going to a concert hall to hear a band, and to your chagrin, you discover that your seat is directly behind a wide post. You cannot see the band, of course, because the light waves from the stage are blocked. But you have little trouble hearing the music, since sound waves simply diffract around the pillar. Light waves diffract slightly in such a situation, but not enough to make a difference with regard to your enjoyment of the concert: if you looked closely while sitting behind the post, you would be able to observe the diffraction of the light waves glowing slightly, as they widened around the post.

Suppose, now, that you had failed to obtain a ticket, but a friend who worked at the concert venue arranged to let you stand outside an open door and hear the band. The sound quality would be far from perfect, of course, but you would still be able to hear the music well enough. And if you stood right in front of the doorway, you would be able to see light from inside the concert hall. But, if you moved away from the door and stood with your back to the building, you would see little light, whereas the sound would still be easily audible.

Wavelength and Diffraction

The reason for the difference—that is, why sound diffraction is more pronounced than light diffraction—is that sound waves are much, much larger than light waves. Sound travels by longitudinal waves, or waves in which the movement of vibration is in the same direction as the wave itself. Longitudinal waves radiate outward in concentric circles, rather like the rings of a bull's-eye.

The waves by which sound is transmitted are larger, or comparable in size to, the column or the door—which is an example of an aperture—and, hence, they pass easily through apertures and around obstacles. Light waves, on the other hand, have a wavelength, typically measured in nanometers (nm), which are equal to one-millionth of a millimeter. Wavelengths for visible light range from 400 (violet) to 700 nm (red): hence, it would be possible to fit about 5,000 of even the longest visible-light wavelengths on the head of a pin!

Whereas differing wavelengths in light are manifested as differing colors, a change in sound wavelength indicates a change in pitch. The higher the pitch, the greater the frequency, and, hence, the shorter the wavelength. As with light waves—though, of course, to a much lesser extent—short-wavelength sound waves are less capable of diffracting around large objects than are long-wave length sound waves. Chances are, then, that the most easily audible sounds from inside the concert hall are the bass and drums; higher-pitched notes from a guitar or other instruments, such as a Hammond organ, are not as likely to reach a listener outside.

Observing Diffraction in Light

Due to the much wider range of areas in which light diffraction has been applied by scientists, diffraction of light and not sound will be the principal topic for the remainder of this essay. We have already seen that wavelength plays a role in diffraction; so, too, does the size of the aperture relative to the wavelength. Hence, most studies of diffraction in light involve very small openings, as, for instance, in the diffraction grating discussed below.

But light does not only diffract when passing through an aperture, such as the concert-hall door in the earlier illustration; it also diffracts around obstacles, as, for instance, the post or pillar mentioned earlier. This can be observed by looking closely at the shadow of a flagpole on a bright morning. At first, it appears that the shadow is "solid," but if one looks closely enough, it becomes clear that, at the edges, there is a blurring from darkness to light. This "gray area" is an example of light diffraction.

Where the aperture or obstruction is large compared to the wave passing through or around it, there is only a little "fuzziness" at the edge, as in the case of the flagpole. When light passes through an aperture, most of the beam goes straight through without disturbance, with only the edges experiencing diffraction. If, however, the size of the aperture is close to that of the wavelength, the diffraction pattern will widen. Sound waves diffract at large angles through an open door, which, as noted, is comparable in size to a sound wave; similarly, when light is passed through extremely narrow openings, its diffraction is more noticeable.

Early Studies in Diffraction

Though his greatest contributions lay in his epochal studies of gravitation and motion, Sir Isaac Newton (1642-1727) also studied the production and propagation of light. Using a prism, he separated the colors of the visible light spectrum—something that had already been done by other scientists—but it was Newton who discerned that the colors of the spectrum could be recombined to form white light again.

Newton also became embroiled in a debate as the nature of light itself—a debate in which diffraction studies played an important role. Newton's view, known at the time as the corpuscular theory of light, was that light travels as a stream of particles. Yet, his contemporary, Dutch physicist and astronomer Christiaan Huygens (1629-1695), advanced the wave theory, or the idea that light travels by means of waves. Huygens maintained that a number of factors, including the phenomena of reflection and refraction, indicate that light is a wave. Newton, on the other hand, challenged wave theorists by stating that if light were actually a wave, it should be able to bend around corners—in other words, to diffract.

Grimaldi Identifies Diffraction

Though it did not become widely known until some time later, in 1648—more than a decade before the particle-wave controversy erupted—Johannes Marcus von Kronland (1595-1667), a scientist in Bohemia (now part of the Czech Republic), discovered the diffraction of light waves. However, his findings were not recognized until some time later; nor did he give a name to the phenomenon he had observed. Then, in 1660, Italian physicist Francesco Grimaldi (1618-1663) conducted an experiment with diffraction that gained widespread attention.

Grimaldi allowed a beam of light to pass through two narrow apertures, one behind the other, and then onto a blank surface. When he did so, he observed that the band of light hitting the surface was slightly wider than it should be, based on the width of the ray that entered the first aperture. He concluded that the beam had been bent slightly outward, and gave this phenomenon the name by which it is known today: diffraction.

Fresnel and Fraunhofer Diffraction

Particle theory continued to have its adherents in England, Newton's homeland, but by the time of French physicist Augustin Jean Fresnel (1788-1827), an increasing number of scientists on the European continent had come to accept the wave theory. Fresnel's work, which he published in 1818, served to advance that theory, and, in particular, the idea of light as a transverse wave.

In Memoire sur la diffraction de la lumiere, Fresnel showed that the transverse-wave model accounted for a number of phenomena, including diffraction, reflection, refraction, interference, and polarization, or a change in the oscillation patterns of a light wave. Four years after publishing this important work, Fresnel put his ideas into action, using the transverse model to create a pencil-beam of light that was ideal for lighthouses. This prism system, whereby all the light emitted from a source is refracted into a horizontal beam, replaced the older method of mirrors used since ancient times. Thus Fresnel's work revolutionized the effectiveness of lighthouses, and helped save lives of countless sailors at sea.

The term "Fresnel diffraction" refers to a situation in which the light source or the screen are close to the aperture; but there are situations in which source, aperture, and screen (or at least two of the three) are widely separated. This is known as Fraunhofer diffraction, after German physicist Joseph von Fraunhofer (1787-1826), who in 1814 discovered the lines of the solar spectrum (source) while using a prism (aperture). His work had an enormous impact in the area of spectroscopy, or studies of the interaction between electromagnetic radiation and matter.

Real-Life Applications

Diffraction Studies Come of Age

Eventually the work of Scottish physicist James Clerk Maxwell (1831-1879), German physicist Heinrich Rudolf Hertz (1857-1894), and others confirmed that light did indeed travel in waves. Later, however, Albert Einstein (1879-1955) showed that light behaves both as a wave and, in certain circumstances, as a particle.

In 1912, a few years after Einstein published his findings, German physicist Max Theodor Felix von Laue (1879-1960) created a diffraction grating, discussed below. Using crystals in his grating, he proved that x rays are part of the electromagnetic spectrum. Laue's work, which earned him the Nobel Prize in physics in 1914, also made it possible to measure the length of x rays, and, ultimately, provided a means for studying the atomic structure of crystals and polymers.

Scientific Breakthroughs Made Possible By Diffraction Studies

Studies in diffraction advanced during the early twentieth century. In 1926, English physicist J. D. Bernal (1901-1971) developed the Bernal chart, enabling scientists to deduce the crystal structure of a solid by analyzing photographs of x-ray diffraction patterns. A decade later, Dutch-American physical chemist Peter Joseph William Debye (1884-1966) won the Nobel Prize in Chemistry for his studies in the diffraction of x rays and electrons in gases, which advanced understanding of molecular structure. In 1937, a year after Debye's Nobel, two other scientists—American physicist Clinton Joseph Davisson (1881-1958) and English physicist George Paget Thomson (1892-1975)—won the Prize in Physics for their discovery that crystals can bring about the diffraction of electrons.

Also, in 1937, English physicist William Thomas Astbury (1898-1961) used x-ray diffraction to discover the first information concerning nucleic acid, which led to advances in the study of DNA (deoxyribonucleic acid), the building-blocks of human genetics. In 1952, English biophysicist Maurice Hugh Frederick Wilkins (1916-) and molecular biologist Rosalind Elsie Franklin (1920-1958) used x-ray diffraction to photograph DNA. Their work directly influenced a breakthrough event that followed a year later: the discovery of the double-helix or double-spiral model of DNA by American molecular biologists James D. Watson (1928-) and Francis Crick (1916-). Today, studies in DNA are at the frontiers of research in biology and related fields.

Diffraction Grating

Much of the work described in the preceding paragraphs made use of a diffraction grating, first developed in the 1870s by American physicist Henry Augustus Rowland (1848-1901). A diffraction grating is an optical device that consists of not one but many thousands of apertures: Rowland's machine used a fine diamond point to rule glass gratings, with about 15,000 lines per in (2.2 cm). Diffraction gratings today can have as many as 100,000 apertures per inch. The apertures in a diffraction grating are not mere holes, but extremely narrow parallel slits that transform a beam of light into a spectrum.

Each of these openings diffracts the light beam, but because they are evenly spaced and the same in width, the diffracted waves experience constructive interference. (The latter phenomenon, which describes a situation in which two or more waves combine to produce a wave of greater magnitude than either, is discussed in the essay on Interference.) This constructive interference pattern makes it possible to view components of the spectrum separately, thus enabling a scientist to observe characteristics ranging from the structure of atoms and molecules to the chemical composition of stars.

X-Ray Diffraction

Because they are much higher in frequency and energy levels, x rays are even shorter in wavelength than visible light waves. Hence, for x-ray diffraction, it is necessary to have gratings in which lines are separated by infinitesimal distances. These distances are typically measured in units called an angstrom, of which there are 10 million to a millimeter. Angstroms are used in measuring atoms, and, indeed, the spaces between lines in an x-ray diffraction grating are comparable to the size of atoms.

When x rays irradiate a crystal—in other words, when the crystal absorbs radiation in the form of x rays—atoms in the crystal diffract the rays. One of the characteristics of a crystal is that its atoms are equally spaced, and, because of this, it is possible to discover the location and distance between atoms by studying x-ray diffraction patterns. Bragg's law—named after the father-andson team of English physicists William Henry Bragg (1862-1942) and William Lawrence Bragg (1890-1971)—describes x-ray diffraction patterns in crystals.

Though much about x-ray diffraction and crystallography seems rather abstract, its application in areas such as DNA research indicates that it has numerous applications for improving human life. The elder Bragg expressed this fact in 1915, the year he and his son received the Nobel Prize in physics, saying that "We are now able to look ten thousand times deeper into the structure of the matter that makes up our universe than when we had to depend on the microscope alone." Today, physicists applying x-ray diffraction use an instrument called a diffractometer, which helps them compare diffraction patterns with those of known crystals, as a means of determining the structure of new materials.

Holograms

A hologram—a word derived from the Greek holos, "whole," and gram, "message"—is a three-dimensional (3-D) impression of an object, and the method of producing these images is known as holography. Holograms make use of laser beams that mix at an angle, producing an interference pattern of alternating bright and dark lines. The surface of the hologram itself is a sort of diffraction grating, with alternating strips of clear and opaque material. By mixing a laser beam and the unfocused diffraction pattern of an object, an image can be recorded. An illuminating laser beam is diffracted at specific angles, in accordance with Bragg's law, on the surfaces of the hologram, making it possible for an observer to see a three-dimensional image.

Holograms are not to be confused with ordinary three-dimensional images that use only visible light. The latter are produced by a method known as stereoscopy, which creates a single image from two, superimposing the images to create the impression of a picture with depth. Though stereoscopic images make it seem as though one can "step into" the picture, a hologram actually enables the viewer to glimpse the image from any angle. Thus, stereoscopic images can be compared to looking through the plate-glass window of a store display, whereas holograms convey the sensation that one has actually stepped into the store window itself.

Developments in Holography

While attempting to improve the resolution of electron microscopes in 1947, Hungarian-English physicist and engineer Dennis Gabor (1900-1979) developed the concept of holography and coined the term "hologram." His work in this area could not progress by a great measure, however, until the creation of the laser in 1960. By the early 1960s, scientists were using lasers to create 3-D images, and in 1971, Gabor received the Nobel Prize in physics for the discovery he had made a generation before.

Today, holograms are used on credit cards or other identification cards as a security measure, providing an image that can be read by an optical scanner. Supermarket checkout scanners use holographic optical elements (HOEs), which can read a universal product code (UPC) from any angle. Use of holograms in daily life and scientific research is likely to increase as scientists find new applications: for instance, holographic images will aid the design of everything from bridges to automobiles.

Holographic Memory

One of the most fascinating areas of research in the field of holography is holographic memory. Computers use a binary code, a pattern of ones and zeroes that is translated into an electronic pulse, but holographic memory would greatly extend the capabilities of computer memory systems. Unlike most images, a hologram is not simply the sum of its constituent parts: the data in a holo-graphic image is contained in every part of the image, meaning that part of the image can be destroyed without a loss of data.

To bring the story full-circle, holographic memory calls to mind an idea advanced by a scientist who, along with Huygens, was one of Newton's great professional rivals, German mathematician and philosopher Gottfried Wilhelm Leibniz (1646-1716). Though Newton is usually credited as the father of calculus, Leibniz developed his own version of calculus at around the same time.

As a philosopher, Leibniz had apparently had a number of strange ideas, which made him the butt of jokes among some sectors of European intellectual society: hence, the French writer and thinker Voltaire (François-Marie Arouet; 1694-1778) satirized him with the character Dr. Pangloss in Candide (1759). Few of Leibniz's ideas were more bizarre than that of the monad: an elementary particle of existence that reflected the whole of the universe.

In advancing the concept of a monad, Leibniz was not making a statement after the manner of a scientist: there was no proof that monads existed, nor was it possible to prove this in any scientific way. Yet, a hologram appears to be very much like a manifestation of Leibniz's imagined monads, and both the hologram and the monad relate to a more fundamental aspect of life: human memory. Neurological research in the late twentieth century suggested that the structure of memory in the human mind is holo-graphic. Thus, for instance, a patient suffering an injury affecting 90% of the brain experiences only a 10% memory loss.

Where to Learn More

Barrett, Norman S. Lasers and Holograms. New York: F. Watts, 1985.

"Bragg's Law and Diffraction: How Waves Reveal the Atomic Structure of Crystals" (Web site). <http://www.journey.sunysb.edu/ProjectJava/Bragg/home.html> (May 6, 2001).

Burkig, Valerie. Photonics: The New Science of Light. Hillside, N.J.: Enslow Publishers, 1986.

"Diffraction of Sound" (Web site). <http://hyperphysics.phy-astr.gsu.edu/hbase/sound/diffrac.html> (May 6, 2001).

Gardner, Robert. Experimenting with Light. New York: F. Watts, 1991.

Graham, Ian. Lasers and Holograms. New York: Shooting Star Press, 1993.

Holoworld: Holography, Lasers, and Holograms (Web site). <http://www.holoworld.com> (May 6, 2001).

Proffen, T. H. and R. B. Neder. Interactive Tutorial About Diffraction (Web site). <http://www.uniwuerzburg.de/mineralogie/crystal/teaching/teaching.html> (May 6, 2001).

Snedden, Robert. Light and Sound. Des Plaines, IL: Heinemann Library, 1999.

"Wave-Like Behaviors of Light." The Physics Classroom (Web site). <http://www.glenbrook.k12.il.us/gbssci/phys/Class/light/u12l1a.html> (May 6, 2001).

The bending of light, or other waves, into the region of the geometrical shadow of an obstacle. More exactly, diffraction refers to any redistribution in space of the intensity of waves that results from the presence of an object that causes variations of either the amplitude or phase of the waves. Most diffraction gratings cause a periodic modulation of the phase across the wavefront rather than a modulation of the amplitude. Although diffraction is an effect exhibited by all types of wave motion, this article will deal only with electromagnetic waves, especially those of visible light. For discussion of the phenomenon as encountered in other types of waves See also Electron diffraction; Neutron diffraction; Sound.

Diffraction is a phenomenon of all electromagnetic radiation, including radio waves; microwaves; infrared, visible, and ultraviolet light; and x-rays. The effects for light are important in connection with the resolving power of optical instruments. See also Radio-wave propagation; X-ray diffraction.

There are two main classes of diffraction, which are known as Fraunhofer diffraction and Fresnel diffraction. The former concerns beams of parallel light, and is distinguished by the simplicity of the mathematical treatment required and also by its practical importance. The latter class includes the effects in divergent light, and is the simplest to observe experimentally.

To illustrate the difference between the methods of observation of the two types of diffraction, Fig. 1 shows the experimental arrangements required to observe them for a circular hole in a screen S. The light originates at a very small source O, which can conveniently be a pinhole illuminated by sunlight. In Fraunhofer diffraction, the source lies at the principal focus of a lens L₁ which renders the light parallel as it falls on the aperture. A second lens L₂ focuses parallel diffracted beams on the observing screen F, situated in the principal focal plane of L₂. In Fresnel diffraction, no lenses intervene. The diffraction effects occur chiefly near the borders of the geometrical shadow, indicated by the broken lines. An alternative way of distinguishing the two classes, therefore, is to say that Fraunhofer diffraction concerns the effects near the focal point of a lens or mirror, while Fresnel diffraction concerns those effects near the edges of shadows. Photographs of diffraction patterns are shown in Fig. 2.

Observation of the two principal types of diffraction, in the case of a circular aperture. (a) Fraunhofer and (b) Fresnel diffraction.

slit; (b) Fresnel pattern, circular aperture.">
Diffraction patterns, photographed with visible light. (a) Fraunhofer pattern, for a slit; (b) Fresnel pattern, circular aperture.

Fraunhofer diffraction

This class of diffraction is characterized by a linear variation of the phases of the Huygens secondary waves with distance across the wavefront, as they arrive at a given point on the observing screen. At the instant that the incident plane wave occupies the plane of the diffracting screen, it may be regarded as sending out, from each element of its surface, a multitude of secondary waves, the joint effect of which is to be evaluated in the focal plane of the lens L₂. The analysis of these secondary waves involves taking account of both their amplitudes and their phases. The simplest way to do this is to use a graphical method, the method of the so-called vibration curve, which can readily be extended to cases of Fresnel diffraction. See also Huygens' principle.

The vibration curve results from the addition of a large (really infinite) number of infinitesimal vectors, each representing the contribution of the Huygens secondary waves from an element of surface of the wavefront. If these elements are assumed to be of equal area, the magnitudes of the amplitudes to be added will all be equal. They will, however, generally differ in phase, so that if the elements were small but finite each would be drawn at a small angle with the preceding one, as shown in Fig. 3a. The resultant of all elements would be the vector A. When the individual vectors represent the contributions from infinitesimal surface elements (as they must for the Huygens wavelets), the diagram becomes a smooth curve, the vibration curve, shown in Fig. 3b. The intensity on the screen is then proportional to the square of this resultant amplitude. In this way, the distribution of the intensity of light in any Fraunhofer diffraction pattern may be determined.

Vibration curves. (a) Addition of many equal amplitudes differing in phase by equal amounts. (b) Equivalent curve, when amplitudes and phase differences become infinitesimal.

The intensity distribution for Fraunhofer diffraction by a slit as a function of the angle θ from the center may be simply calculated by the method of the vibration curve. The intensity at any angle is given by Eq. (1), where I₀ is the intensity at the center of the pattern, and β is given by Eq. (2), where b is the width of the
1.

2.
slit and λ is the wavelength. The central maximum is twice as wide as the subsidiary ones, and is about 21 times as intense as the strongest of these. A photograph of this pattern is shown in Fig. 2a.

Fraunhofer diffraction by a circular aperture determines the resolving power of instruments such as telescopes, cameras, and microscopes, in which the width of the light beam is usually limited by the rim of one of the lenses. The method of the vibration curve may be extended to find the angular width of the central diffraction maximum for this case. An exact construction of the curve or, better, a mathematical calculation shows that the extreme phase differences required are ± 1.220π, yielding Eq. (3)
3.
for the angle θ at the first zero of intensity. Here d is the diameter of the circular aperture. This pattern has circular symmetry and consists of a diffuse central disk, called the Airy disk, surrounded by faint rings. The angular radius of the disk, given by Eq. (3), may be extremely small for an actual optical instrument, but it sets the ultimate limit to the sharpness of the image, that is, to the resolving power. See also Resolving power (optics).

Fresnel diffraction

The diffraction effects obtained when the source of light or the observing screen are at a finite distance from the diffracting aperture or obstacle come under the classification of Fresnel diffraction. This type of diffraction requires for its observation only a point source, a diffracting screen of some sort, and an observing screen. The latter is often advantageously replaced by a magnifier or a low-power microscope. The observed diffraction patterns generally differ according to the radius of curvature of the wave and the distance of the point of observation behind the screen. If the diffracting screen has circular symmetry, such as that of an opaque disk or a round hole, a point source of light must be used. If it has straight, parallel edges, it is desirable from the standpoint of brightness to use an illuminated slit parallel to these edges. In the latter case, it is possible to regard the wave emanating from the slit as a cylindrical one. For the purpose of deriving the vibration curve, the appropriate way of dividing the wavefront into infinitesimal elements is to use annular rings in the first case, and strips parallel to the axis of the cylinder in the second case.

The zone plate is a special screen designed to block off the light from every other half-period zone, and represents an interesting application of Fresnel diffraction. The Fresnel half-period zones are drawn, with radii proportional to the square roots of whole numbers, and alternate ones are blackened. The drawing is then photographed on a reduced scale. When light from a point source is sent through the negative, an intense point image is produced, much like that formed by a lens.

The bending of electromagnetic waves as they pass around corners or through holes smaller than the wavelengths of the waves themselves. See diffraction grating and refraction.

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The breaking up of an incoming wave by some sort of geometrical structure — for example, a series of slits — followed by reconstruction of the wave by interference. Diffraction of light is characterized by alternate bands of light and dark or bands of different colors.

The bending or breaking up of a ray of light into its component parts.

• x-ray d. — a method used to determine the three-dimensional structure of the single object, e.g. protein molecule, that composes the crystal. Based on recording and analyzing the diffraction pattern of an x-ray beam passing through a crystalline structure, either organic or inorganic.

The intensity pattern formed on a screen by diffraction from a square aperture

Diffraction

Colors seen in a spider web are partially due to diffraction, according to some analyses.^[1]

Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Similar effects are observed when light waves travel through a medium with a varying refractive index or a sound wave through one with varying acoustic impedance. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, x-rays and radio waves. As physical objects have wave-like properties (at the atomic level), diffraction also occurs with matter and can be studied according to the principles of quantum mechanics.

While diffraction occurs whenever propagating waves encounter such changes, its effects are generally most pronounced for waves where the wavelength is on the order of the size of the diffracting objects. If the obstructing object provides multiple, closely-spaced openings, a complex pattern of varying intensity can result. This is due to the superposition, or interference, of different parts of a wave that traveled to the observer by different paths (see diffraction grating).

The formalism of diffraction can also describe the way in which waves of finite extent propagate in free space. For example, the expanding profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer are all explained by diffraction theory.

Contents 1 Examples 2 History 3 The mechanism of diffraction 4 Diffraction systems 4.1 Single-slit diffraction 4.2 Diffraction grating 4.3 Diffraction by a circular aperture 4.4 Propagation of a laser beam 4.5 Diffraction-limited imaging 4.6 Speckle patterns 5 Common features of diffraction patterns 6 Particle diffraction 7 Bragg diffraction 8 Coherence 9 See also 10 References 11 External links

Examples

Solar glory at the steam from hot springs. A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly-sized water droplets.

The effects of diffraction can be regularly seen in everyday life. The most colorful examples of diffraction are those involving light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern we see when looking at a disk. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example. Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges. The speckle pattern which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. All these effects are a consequence of the fact that light propagates as a wave.

Diffraction can occur with any kind of wave. Ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree.^[2] Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope.

History

Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803

The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665.^[3][4][5] Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating to be discovered.^[6] Thomas Young performed a celebrated experiment in 1803 demonstrating interference from two closely spaced slits.^[7] Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, made public in 1815^[8] and 1818,^[9] and thereby gave great support to the wave theory of light that had been advanced by Christiaan Huygens^[10] and reinvigorated by Young, against Newton's particle theory.

The mechanism of diffraction

Photograph of single-slit diffraction in a circular ripple tank

Diffraction arises because of the way in which waves propagate; this is described by the Huygens–Fresnel principle. The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary radial wave. The subsequent propagation and addition of all these radial waves form the new wavefront. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves, an effect which is often known as wave interference. The summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima.

The form of a diffraction pattern can be determined from the sum of the phases and amplitudes of the Huygens wavelets at each point in space. There are various analytical models which can be used to do this including the Fraunhofer diffraction equation for the far field and the Fresnel Diffraction equation for the near field. Most configurations cannot be solved analytically, but can yield numerical solutions through finite element and boundary element methods.

Diffraction systems

It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another out.

The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem. For water waves, this is already the case, water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three dimensional nature of the problem.

Some of the simpler cases of diffraction are considered below.

Single-slit diffraction

Main article: Diffraction formalism

Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.

Graph and image of single-slit diffraction

A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity.

A slit which is wider than a wavelength has a large number of point sources spaced evenly across the width of the slit. The light at a given angle is made up of contributions from each of these point sources and if the relative phases of these contributions vary by 2π or more, we expect to find minima and maxima in the diffracted light.

We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to λ/2. Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is given by so that the minimum intensity occurs at an angle θ_min given by

where d is the width of the slit.

A similar argument can be used to show that if we imagine the slit to be divided into four, six, eight parts, etc, minima are obtained at angles θ_n given by

where n is an integer other than zero.

There is no such simple argument to enable us to find the maxima of the diffraction pattern. The intensity profile can be calculated using the Fraunhofer diffraction integral as

where the sinc function is given by sinc(x) = sin(πx)/(πx) if x ≠ 0, and sinc(0) = 1.

It should be noted that this analysis applies only to the far field, that is, at a distance much larger than the width of the slit.

2-slit (top) and 5-slit diffraction of red laser light

Diffraction of a red laser using a diffraction grating

A diffraction pattern of a 633 nm laser through a grid of 150 slits

Diffraction grating

Main article: Diffraction grating

A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θ_m which are given by the grating equation

where θ_i is the angle at which the light is incident, d is the separation of grating elements and m is an integer which can be positive or negative.

The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a convolution of diffraction and interference patterns.

The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.

A computer-generated image of an Airy disk

Computer generated light diffraction pattern from a circular aperture of diameter 0.5micron at a wavelength of 0.6micron (red-light) at distances of 0.1cm – 1cm in steps of 0.1cm. One can see the image moving from the Fresnel region into the Fraunhofer region where the Airy pattern is seen.

Diffraction by a circular aperture

Main article: Airy disk

The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the Airy Disk. The variation in intensity with angle is given by

where a is the radius of the circular aperture, k is equal to 2π/λ and J₁ is a Bessel function. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.

Propagation of a laser beam

The way in which the profile of a laser beam changes as it propagates is determined by diffraction. The output mirror of the laser is an aperture, and the subsequent beam shape is determined by that aperture. Hence, the smaller the output beam, the quicker it diverges. Diode lasers have much greater divergence than He–Ne lasers for this reason.

Paradoxically, it is possible to reduce the divergence of a laser beam by first expanding it with one convex lens, and then collimating it with a second convex lens whose focal point is coincident with that of the first lens. The resulting beam has a larger aperture, and hence a lower divergence.

Diffraction-limited imaging

Main article: Diffraction-limited system

The Airy disk around each of the stars from the 2.56m telescope aperture can be seen in this lucky image of the binary star zeta Boötis.

The ability of an imaging system to resolve detail is ultimately limited by diffraction. This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an Airy disk having a central spot in the focal plane with radius to first null of

where λ is the wavelength of the light and N is the f-number (focal length divided by diameter) of the imaging optics. In object space, the corresponding angular resolution is

where D is the diameter of the entrance pupil of the imaging lens (e.g., of a telescope's main mirror).

Two point sources will each produce an Airy pattern – see the photo of a binary star. As the point sources move closer together, the patterns will start to overlap, and ultimately they will merge to form a single pattern, in which case the two point sources cannot be resolved in the image. The Rayleigh criterion specifies that two point sources can be considered to be resolvable if the separation of the two images is at least the radius of the Airy disk, i.e. if the first minimum of one coincides with the maximum of the other.

Thus, the larger the aperture of the lens, and the smaller the wavelength, the finer the resolution of an imaging system. This is why telescopes have very large lenses or mirrors, and why optical microscopes are limited in the detail which they can see.

Speckle patterns

Main article: speckle pattern

The speckle pattern which is seen when using a laser pointer is another diffraction phenomenon. It is a result of the superpostion of many waves with different phases, which are produced when a laser beam illuminates a rough surface. They add together to give a resultant wave whose amplitude, and therefore intensity varies randomly.

Common features of diffraction patterns

The upper half of this image shows a diffraction pattern of He-Ne laser beam on an elliptic aperture. The lower half is its 2D Fourier transform approximately reconstructing the shape of the aperture.

Several qualitative observations can be made of diffraction in general:

• The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction. In other words: The smaller the diffracting object, the 'wider' the resulting diffraction pattern, and vice versa. (More precisely, this is true of the sines of the angles.)
• The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.
• When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.

Particle diffraction

See also: neutron diffraction and electron diffraction

Quantum theory tells us that every particle exhibits wave properties. In particular, massive particles can interfere and therefore diffract. Diffraction of electrons and neutrons stood as one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the de Broglie wavelength

where h is Planck's constant and p is the momentum of the particle (mass × velocity for slow-moving particles) . For most macroscopic objects, this wavelength is so short that it is not meaningful to assign a wavelength to them. A sodium atom traveling at about 3000 m/s would have a De Broglie wavelength of about 5 pico meters.

Because the wavelength for even the smallest of macroscopic objects is extremely small, diffraction of matter waves is only visible for small particles, like electrons, neutrons, atoms and small molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids and large molecules like proteins.

Relatively larger molecules like buckyballs were also shown to diffract.^[11]

Bragg diffraction

Following Bragg's law, each dot (or reflection), in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.

For more details on this topic, see Bragg diffraction.

Diffraction from a three dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. It is similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes. The condition of constructive interference is given by Bragg's law:

where

λ is the wavelength,

d is the distance between crystal planes,

θ is the angle of the diffracted wave.

and m is an integer known as the order of the diffracted beam.

Bragg diffraction may be carried out using either light of very short wavelength like x-rays or matter waves like neutrons (and electrons) whose wavelength is on the order of (or much smaller than) the atomic spacing^[12]. The pattern produced gives information of the separations of crystallographic planes d, allowing one to deduce the crystal structure. Diffraction contrast, in electron microscopes and x-topography devices in particular, is also a powerful tool for examining individual defects and local strain fields in crystals.

Coherence

Main article: Coherence (physics)

The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent.

The length over which the phase in a beam of light is correlated, is called the coherence length. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence, as it is related to the presence of different frequency components in the wave. In the case of light emitted by an atomic transition, the coherence length is related to the lifetime of the excited state from which the atom made its transition.

If waves are emitted from an extended source, this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double slit experiment, this would mean that if the transverse coherence length is smaller than the spacing between the two slits, the resulting pattern on a screen would look like two single slit diffraction patterns.

In the case of particles like electrons, neutrons and atoms, the coherence length is related to the spatial extent of the wave function that describes the particle.

See also

• Atmospheric diffraction
• Bragg diffraction
• Cloud iridescence
• Diffraction formalism
• Diffraction limit
• Diffractometer
• Dynamical theory of diffraction
• Diffraction grating
• Electron diffraction
• Fraunhofer diffraction
• Fresnel diffraction
• Fresnel imager
• Fresnel number
• Fresnel zone
• Neutron diffraction
• Prism
• Powder diffraction
• Refraction
• Schaefer–Bergmann diffraction
• Thinned array curse
• X-ray scattering techniques

References

1. ^ Dietrich Zawischa. "Optical effects on spider webs". http://www.itp.uni-hannover.de/%7Ezawischa/ITP/spiderweb.html. Retrieved 2007-09-21. 
2. ^ Andrew Norton (2000). Dynamic fields and waves. CRC Press. p. 102. ISBN 9780750307192. http://books.google.com/books?id=XRRMxjr24pwC&pg=PA102&dq=sound+wave+diffraction+behind+tree&lr=&as_drrb_is=q&as_minm_is=1&as_miny_is=2009&as_maxm_is=12&as_maxy_is=2009&as_brr=3&as_pt=ALLTYPES&ei=EjjDSbulDJG4kwSBy6yADg. 
3. ^ Francesco Maria Grimaldi, Physico mathesis de lumine, coloribus, et iride, aliisque annexis libri duo (Bologna ("Bonomia"), Italy: Vittorio Bonati, 1665), pages 1-11. Available on-line (in Latin) at: http://fermi.imss.fi.it/rd/bdv?/bdviewer/bid=300682# .
4. ^ Jean Louis Aubert (1760). Memoires pour l'histoire des sciences et des beaux arts. Paris: Impr. de S. A. S.; Chez E. Ganeau. pp. 149. http://books.google.com/books?id=OCLC58901501&id=3OgDAAAAMAAJ&pg=PP151&lpg=PP151&dq=grimaldi+diffraction+date:0-1800&as_brr=1. 
5. ^ Sir David Brewster (1831). A Treatise on Optics. London: Longman, Rees, Orme, Brown & Green and John Taylor. pp. 95. http://books.google.com/books?vid=OCLC03255091&id=opYAAAAAMAAJ&pg=RA1-PA95&lpg=RA1-PA95&dq=grimaldi+diffraction+date:0-1840&as_brr=1. 
6. ^ Letter from James Gregory to John Collins, dated 13 May 1673. Reprinted in: Correspondence of Scientific Men of the Seventeenth Century...., ed. Stephen Jordan Rigaud (Oxford, England: Oxford University Press, 1841), vol. 2, pages 251-255; see especially page 254. Available on-line at: http://books.google.com/books?id=0h45L_66bcYC&pg=PA254&dq=feather+ovals&ei=5jlaSsLQKJnkygTi1Lz8CA&ie=ISO-8859-1&output=html .
7. ^ Young, Thomas (1804-01-01), "The Bakerian Lecture: Experiments and calculations relative to physical optics", Philosophical Transactions of the Royal Society of London 94: 1–16, http://books.google.com/books?id=7AZGAAAAMAAJ&pg=PA1&ie=ISO-8859-1&output=html&source=gbs_search_r&cad=1 . (Note: This lecture was presented before the Royal Society on 24 November 1803.)
8. ^ Augustin-Jean Fresnel (1816) "Mémoire sur la diffraction de la lumière … ," Annales de la Chemie et de Physique, 2nd series, vol. 1, pages 239-281. (Presented before l'Académie des sciences on 15 October 1815.) Available on-line (in French) at: http://www.bibnum.education.fr/physique/optique/premier-memoire-sur-la-diffraction-de-la-lumiere .
9. ^ Augustin-Jean Fresnel (1826) "Mémoire sur la diffraction de la lumière," Mémoires de l'Académie des Sciences (Paris), vol. 5, pages 33-475. (Summitted to l'Académie des sciences of Paris on 20 April 1818.)
10. ^ Christiaan Huygens, Traité de la lumiere (Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1. (Note: Huygens published his Traité in 1690; however, in the preface to his book, Huygens states that in 1678 he first communicated his book to the French Royal Academy of Sciences.)
11. ^ Brezger, B.; Hackermüller, L.; Uttenthaler, S.; Petschinka, J.; Arndt, M.; Zeilinger, A. (February 2002). "Matter–Wave Interferometer for Large Molecules" (reprint). Physical Review Letters 88 (10): 100404. doi:10.1103/PhysRevLett.88.100404. http://homepage.univie.ac.at/Lucia.Hackermueller/unsereArtikel/Brezger2002a.pdf. Retrieved 2007-04-30. 
12. ^ John M. Cowley (1975) Diffraction physics (North-Holland, Amsterdam) ISBN 0 444 10791 6

External links

Wikimedia Commons has media related to: Diffraction

The Wikibook Nanotechnology has a page on the topic of Nano-optics

• Do Sensors “Outresolve” Lenses?; on lens and sensor resolution interaction.
• Diffraction and acoustics.
• Diffraction in photography.
• On Diffraction at MathPages.
• Diffraction pattern calculators at The Wolfram Demonstrations Project
• Wave Optics – A chapter of an online textbook.
• 2-D wave Java applet – Displays diffraction patterns of various slit configurations.
• Diffraction Java applet – Displays diffraction patterns of various 2-D apertures.
• Diffraction approximations illustrated – MIT site that illustrates the various approximations in diffraction and intuitively explains the Fraunhofer regime from the perspective of linear system theory.
• Gap Obstacle Corner – Java simulation of diffraction of water wave.
• Google Maps – Satellite image of Panama Canal entry ocean wave diffraction.

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