Diffraction

Type of physical science: Classical physics

Field of study: Optics

The study of geometric optics is built on the assumption that in a homogeneous transparent medium, light travels in a straight line. When light propagation was treated as wave motion, it was realized that light can bend around sharp edges to a small extent; any such deviation from geometrical optics is called diffraction.

Overview

Light is normally considered to be traveling in straight lines, and for all practical purposes it does. Because of this property, any obstacle in the path of light casts a shadow. If light travels exactly along straight lines, the shadow must have the exact geometric shape of the obstacle. On closer examination, however, it is found that even the sharpest shadow is blurred slightly at the edges. This happens as a result of the small bending of light around the sharp edges of the obstacle, known as diffraction.

The diffraction effect is common to all radiations that propagate in the form of waves.

Radio waves and sound waves bend around obstacles quite a bit. A person cannot be seen behind a screen, but one can hear that person talk. The bending effect depends on the wavelength of the radiation under consideration. The wavelength is the combined length of a wave crest and trough.

The greater the wavelength, the higher the bending will be. The length of radio waves is of the order of hundreds of meters. Hence, radio waves can bend easily around even buildings, as long as they are not too big. On the other hand, the wavelength of light is less than a millionth of a meter; therefore, light bends around sharp edges only by a small amount. When one looks at a distant street lamp through a thin curtain, some interesting patterns can be seen around the lamp, which are caused by diffraction.

To investigate this phenomenon more carefully in a laboratory environment, place a monochromatic source that emits light of a single wavelength or color behind a screen having a very small round hole. The light from this point source is allowed to fall on a dime, and the shadow of the dime is observed on a white screen. It will be seen that the edge of the shadow is fuzzy and its size is less than the expected geometric shadow. More interesting, however, the shadow is surrounded by circular fringes, which are alternately bright and dark circular rings in this case. As one moves away from the edge, the shadow of these fringes decreases in width.

Also, the bright fringes become less bright, and the dark fringes become less dark. Hence, within a short distance, uniform illumination can be seen. Even more strange, when a circular object such as the dime is used, a bright spot can appear at the center of the geometric shadow. These phenomena, which involve deviations from the straight-line behavior of radiation, are called diffraction.

An adequate explanation for the diffraction phenomenon can be given on the basis of the Huygens-Fresnel principle. As light from a source travels through a medium, the different particles of the medium are set into vibration. The locus of all points that are in the same state of vibration is called a wavefront. In geometric optics, where light is studied on the basis of some geometric laws, light may be considered to be traveling in the form of rays and beams. In physical optics, where the focus is on the physical nature of light, light can be thought of as traveling in the form of wavefronts. A point source sends out a spherical wavefront in a homogeneous transparent medium and an illuminated slit sends out a cylindrical wavefront. Both of these wavefronts at a very large distance from the source become plane wavefronts.

According to Christiaan Huygens, every point on a wavefront acts as a source of secondary spherical wavelets. Augustin-Jean Fresnel added to this assumption that the actual light field at any point beyond the wavefront is a superposition of all these secondary wavelets, taking into account both their amplitude and phase. When the Huygens-Fresnel principle is used, every point of a finite-width light source can be considered as a source of secondary wavelets.

Hence, diffraction is concerned with a continuous array of sources in contrast to the discrete number of sources in the interference phenomenon. There may be confusion between interference and diffraction, as they are closely related phenomena. Interference occurs when two or more beams of light--separately divided from the same source--are superimposed to produce interference fringes. In diffraction, the observed phenomenon is caused by the interference of light from a continuous portion of the wavefront, which is obstructed by an obstacle or limited by an aperture. The diffraction effects will be significant only when the size of the obstacle or aperture is not too big, compared to the wavelength of the source.

As an example, one may consider a plane wavefront incident on a narrow rectangular aperture: a slit. The plane wavefront is obtained when the source is effectively at a very large distance from the slit. If the source is placed at the focal point of a lens, the light emerging through the lens is parallel or the wavefront is plane. If the light coming through the slit falls on a distant white screen, a rectangular bright patch of light is seen, with alternately bright and dark diffraction bands on either side of the central maximum. The central maximum basically represents the image of the slit on a distant screen. The angular width of the central maximum is defined as the angle subtended by the first dark bands on either side of the central maximum at the center of the slit. This angular width can be shown to be directly proportional to the wavelength of the incident light and inversely proportional to the width of the slit. Hence, the central maximum will spread as the slit width is narrowed. As the length of the slit is very large compared to its width, the diffraction pattern caused by points of the wavefront along the length of the slit has a very small angular width and is barely visible on the screen.

The diffraction pattern caused by a rectangular aperture can be accounted for easily on the basis of the Huygens-Fresnel principle. The standard approach divides the portion of the wavefront passing through the aperture into equal elements, small in size compared to the wavelength of the source. Each such element is then considered to be the source of secondary wavelets. These wavelets reaching the different regions of the screen are summed up, taking their phases into account. If both the source of light and the observation screen are far enough from the diffraction aperture, the diffraction is classified as Fraunhofer (named for Joseph von Fraunhofer), or far-field, diffraction. In this case, the phase of the contribution from an element on the wavefront is a linear function of the distance across the aperture. When the source and the screen are nearby, the curvature of the wavefront must be taken into account; it is then called Fresnel, or near-field, diffraction.

The mathematical treatment of near-field diffraction is more complex, but it is simpler as an experiment, because collimating lenses, which produce an effective far-field situation, are unnecessary. When a spherical wavefront is limited by an aperture, Fresnel devised a method for dealing with the contributions from various parts of the wavefront by dividing the wavefront into what are called Fresnel's half-period zones. The contributions from successive zones are out of phase, on the average. In the case of a point source, the half-period zones are annular rings around a central circle. If the alternate zones are blocked, a zone plate is obtained, which acts like a lens, but with multiple images. Fresnel's theoretical treatment was given a more rigorous basis by Gustav Robert Kirchhoff, a German physicist.

The Fraunhofer diffraction pattern caused by a double slit--two narrow slits separated by a small distance--consisted of the overlapping diffraction patterns caused by the two single slits. The diffraction maxima were filled with the equal-width interference fringes of the double slit. The pattern agreed with the Huygens-Fresnel principle. The theory was then extended to three, four, and so on, to a large number of slits. The many-slit device is called a diffraction grating, which has found wide applications in spectroscopic studies and X-ray crystallography.

The diffraction pattern caused by circular aperture is of considerable importance in observational astronomy. A theoretical investigation of the diffraction caused by a circular aperture was carried out as early as 1835 by the Royal astronomer, Sir George B. Airy. The diffraction pattern deduced by Airy proved to be of special importance to astronomers because it is the same pattern produced in the focal plane of an ideal telescope with a circular lens (or mirror) when the plane wavefront from a distant star is incident on its objective lens. The circular edge of the objective lens of the telescope acts as a circular aperture, and it is the diffraction ring systems caused by two nearby stars that determine whether or not they can be distinguished as separate.

Applications

It is common experience that two close objects--like two letters of the alphabet in fine print, which can be distinguished at normal reading distance--cannot be seen as separate at greater distances. This effect is a result of diffraction. When a millimeter ruler or two lines drawn on a paper with one millimeter separation are viewed at increasing distances from the eye, the lines become blurred together and are thus unresolved at a distance of about 3 meters or 3,000 millimeters. Thus, for the human eye directly looking at the lines, the angular resolution limit is about 1/3,000. The diffraction limit of angular resolution of the human eye is given by the angular size of the image spot on the retina from an incident plane wave emitted by a distant point source. When this is calculated from the diameter of the pupil of the eye (width of the circular aperture) and the mean wavelength of the incident light (about 550 nanometers), it is found to agree with the angular resolution limit calculated with a millimeter ruler. Hence, the limits of resolution of the eye is understood on the basis of the diffraction phenomenon by treating the pupil of the eye as a circular diffraction aperture and the retina as a screen. The same idea is applicable to the resolving power of an astronomical telescope, where the objective lens acts as the diffraction aperture.

In the case of Fraunhofer (or far-field) diffraction by a circular aperture, the pattern obtained is a central circular patch with alternately bright and dark circular rings around it. The central patch of light is commonly known as the Airy disk. In an astronomical telescope, the image of a star seen at the field of view of the eyepiece is the Airy disk of the objective. In a telescope, the lens facing the object is known as the objective lens, and the lens facing the eye is the eyepiece. An eyepiece viewing the primary image and providing further magnification merely enlarges the details of the diffraction pattern formed by the objective. The limit of resolution is set already by the primary image. A somewhat arbitrary, but useful criterion known as Rayleigh's (named for Lord Rayleigh) criterion for the resolution of two diffraction images requires that the centers of the image patterns be no nearer than the radius of the Airy disk. In other words, the maximum of one pattern falls directly over the first minimum of the other.

Hence, the resolving power of the telescope works out to be the ratio of the diameter of the objective to the wavelength of the source. The greater the diameter of the objective, the better will be the resolution. As stars are at extremely large distances, a good astronomical telescope should have a high resolving power to see two nearby stars as separate. The telescope, according to diffraction principles, must have a large objective lens. The famous Palomar astronomical telescope in Southern California has an objective diameter of 508 centimeters. The resolving power can be increased by decreasing the wavelength of the light used. Thus, if the ultraviolet light is used from the stars, a better resolution is obtained. The ultraviolet light is absorbed by the ozone layer of the atmosphere. The Hubble Space Telescope can view the stars in ultraviolet light.

The most important application of diffraction is the diffraction grating. A diffraction grating is a multiple-slit device that takes advantage of the sensitivity of its diffraction pattern to the wavelength of the incident light. To be effective, an optical diffraction grating must have thousands of slits of equal width separated by equal spaces with about six thousand slits per centimeter. Hence, the construction of a diffraction grating is a technological feat. Usually, lines are drawn on a plane glass plate with a diamond-head so that the lines act as opaque spaces and the space between the lines act as slits. When composite light is incident normally on the grating, it can be shown that the different wavelengths are diffracted at different angles. Therefore, the spectral composition of the source can be determined, and the dispersion and resolution produced by the grating is much better than that of a prism. The angle of dispersion depends on the grating element, which is the width of a slit and an opaque space together. This grating is of the transmission type.

In the reflection grating, lines are ruled on a reflecting surface and the periodic reflection of the incident light operates like the periodic transmission of waves from a transmission grating. Research quality gratings are usually of the reflection type. Several other types of gratings have been developed, also. In a concave grating, the grooves are drawn on a concave metallic reflector, which is very useful in calibrated spectrographs that measure the wavelength of spectral lines. In blazed gratings, the grooves are drawn in appropriate shapes so that most of the light goes into one particular order of the spectra. These gratings are used often for spectroscopic studies, and several types of mounts have been developed to facilitate these studies.

The diffraction phenomenon is common to all type of radiation that exhibit wave nature. The antenna arrays for radio waves are constructed taking into account diffraction principles. Several attempts were made after the discovery of optical gratings to produce ruled gratings for the diffraction of X rays. There are two problems in achieving this objective: X rays are highly penetrating and are not reflected and refracted easily. Further, the lines have to be ruled much closer so that the grating element is comparable to the much lower wavelength of X rays. X-ray wavelengths are about one thousand times smaller than that of visible light. Hence, about one thousand times more lines have to be ruled for the X-ray gratings, which is practically impossible. These problems were solved finally when, in the early part of the twentieth century, it was suggested that the crystal lattices are spaced at distances comparable to the wavelength of X rays; therefore, the crystal lattice can be used as a three-dimensional grating for the X rays.

The study of diffraction of X rays in crystals has led to the understanding of complex crystal structures. X-ray crystallography is now a well-developed technology having applications in a variety of fields such as solid-state physics, mineralogy, and molecular chemistry.

Context

The understanding of the physical nature of light was the pursuit of many eminent physicists in the seventeenth and eighteenth centuries. In the seventeenth century, Sir Isaac Newton and Huygens pioneered two different theories for the nature of light. According to Newton, light is of particle nature, whereas Huygens viewed light to be in the form of waves.

This controversy raged for nearly three centuries until the advent of quantum electrodynamics, which reconciled the arguments with the wave-particle duality of light. The diffraction phenomenon can be accounted for only on the basis of the wave theory of radiation. Therefore, the development and understanding of this subject through the important works of scientists such as Fresnel, Fraunhofer, Kirchhoff, and others gave strong support to the wave theory. The wave theory was developed further by James Clerk Maxwell, who showed that light is in the form of electromagnetic waves. These waves are made up of an electric component and a magnetic component; these components are locked in step at right angles to each other and fluctuate together.

The study of the diffraction phenomenon underscored the defects in image formation and the limitations of optical instruments in this regard. It also led to the construction of new astronomical telescopes with huge objectives, which led to enormous progress in the field of astronomy. The development of the optical diffraction grating and its extension to X-ray diffraction and crystallography is one of the greatest contributions to the advancement of science.

All solids can be classified as crystalline or amorphous, and there are more than twenty thousand crystalline materials known as of the 1990's. The study of these solid-state materials has been greatly enhanced by X-ray diffraction techniques. The first X-ray diffraction experiment was carried out in 1912 by Max von Laue, Walter Friedrich, and Paul Knipping. The theoretical basis for their experimental work was given by Sir William Lawrence Bragg in 1912. Inside a crystal, a simple geometrical structure of atoms called the basis is repeated in all directions without change in composition, dimension, or orientation. When each basis is replaced by a point, the crystal lattice is obtained. The X-ray diffraction pattern of a crystal, both its geometry and the relative intensity of its spots, gives very valuable information on the lattice structure of the crystal and the composition of its basis. Through this technique, the structure of the crystal can be determined completely, even if it contains some tens or hundreds of thousands of atoms, as in a protein molecule. As such, the understanding of the diffraction phenomenon has led to the development of a wide variety of scientific disciplines, such as physics, chemistry, geology, and biology.

The concept of Fresnel's zone plate was used by Dennis Gabor in the late 1940's in his development of the basic principles of holography. As scientific discoveries are continually made, their complete understanding will continue to depend on an understanding of such basic phenomenon as the diffraction of radiation.

Principal terms

AMPLITUDE: the extent of vibration on either side of the equilibrium position; in wave motion, the height of a crest or the depth of a trough gives the amplitude

FRINGES: alternately bright and dark or colored bands

INTENSITY: the brightness of light, which is directly proportional to the square of the amplitude

MONOCHROMATIC SOURCE: a light source of single wavelength

PHASE: being in step; when two waves proceed in step, they are said to be in phase

RESOLVING POWER: the capacity to show two nearby objects as separate

WAVEFRONT: the locus of all points that are in the same state of vibration

WAVELENGTH: the length of a wave crest and trough together

Bibliography

Bailey, John M. LIBERAL ARTS PHYSICS: INVARIANCE AND CHANGE. San Francisco: W. H. Freeman, 1969. This book gives a clear description in nontechnical terms, with practically no mathematics. Chapters 13 and 14 contain relevant information. Bailey has done an excellent job of addressing the general reader without any omission of important concepts.

Graham-Smith, Francis, and J. H. Thomson. OPTICS. New York: John Wiley & Sons, 1971. A highly readable presentation. Includes some mathematical equations, but the description and figures are clear enough for easy comprehension. Chapters 1, 2, 5, 9, and 11 are recommended.

Hewitt, Paul G. CONCEPTUAL PHYSICS. 6th ed. Glenview, Ill.: Scott, Foresman, 1989. A widely used introductory physics text for nonscience majors. The wave nature of light, light emission, and the diffraction phenomenon are described in easy-to-understand language with many examples.

Jenkins, Francis A., and Harvey E. White. FUNDAMENTALS OF OPTICS. 4th ed. New York: McGraw-Hill, 1976. A classic text on optics that follows a traditional approach. Chapters 12 and 15 to 18 discuss diffraction and its applications, with necessary background material. The language is a bit terse. Contains useful material.

Kittel, Charles. BERKELEY PHYSICS COURSE. 5 vols. New York: McGraw-Hill, 1965-1971. A widely acclaimed series. Although a bit technical, the thoughtful presentation of the material makes it worth reading even for the general reader. Includes interesting home experiments. Chapters 4, 7, and 9 are particularly relevant.

Leighton, Robert B. PRINCIPLES OF MODERN PHYSICS. New York: McGraw-Hill, 1959. A concise account of X-ray diffraction. Chapter 12 explains the principle of X-ray diffraction and its application to the study of crystal structure.

Pedrotti, Frank L., and Leno S. Pedrotti. INTRODUCTION TO OPTICS. Englewood Cliffs, N.J.: Prentice-Hall, 1987. A very well written and modern textbook. Chapters 1 and 19 to 21 contain applicable information. The technical material can be overlooked without impacting the descriptive parts. Good material on applications.

Van Heel, A. C. S., and C. H. F. Velzel. WHAT IS LIGHT? New York: McGraw-Hill, 1968. Chapters 1 and 4 give a good account of diffraction and the background needed for its understanding. Written with minimal mathematics. Excellent figures and photographs.

X-Ray Determination of Molecular Structure