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Diffraction ___________________________________________ Fraunhofer and Fresnel diffraction. Fraunhofer diffraction through a single slit. Diffraction pattern due to a Plane transmission diffraction grating, absent spectra. Dispersive and Resolving Power. Rayleigh’s criterion for resolution. I. Introduction What is Diffraction? When light waves pass through a small aperture, an interference pattern is observed rather than a sharp spot of light. This behavior indicates that light, once it has passed through the aperture, spreads beyond the narrow path defined by the aperture into regions that would be in shadow if light traveled in straight lines. Other waves, such as sound waves and water waves, also have this property of spreading when passing through apertures or by sharp edges. This phenomenon, known as diffraction, can be described only with a wave model for light. 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. Very similar effects are observed when there is an alteration in the properties of the medium in which the wave is travelling, for example a variation in refractive index for light waves or in acoustic impedance for sound waves and these can also be referred to as diffraction effects. 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. In general, diffraction occurs when waves pass through small openings, around obstacles, or past sharp edges, as shown in Figure 1. When an opaque object is placed between a point source of light and a screen, no sharp boundary exists on the screen between a shadowed region and an illuminated region. The illuminated region above the shadow of the object contains alternating light and dark fringes. Such a display is called a diffraction pattern. Figure 1 Light from a small source passes by the edge of an opaque object. We might expect no light to appear on the screen below the position of the edge of the object. In reality, a light bend around the top edge of the object and enters this region. Because of these effects, a diffraction pattern consisting of bright and dark fringes appears in the region above the edge of the object. II. Fraunhofer and Fresnel diffraction Diffraction divided into two types. Fresnel and Fraunhoffer. Fraunhofer was born in Straubing, Bavaria. He became an orphan at the age of 11, and he started working as an apprentice to a harsh glassmaker named Philipp Anton Weichelsberger. In 1801 the workshop in which he was working collapsed and he was buried in the rubble. The rescue operation was led by Maximilian IV Joseph, Prince Elector of Bavaria (the future Maximilian I Joseph). The prince entered Fraunhofer's life, providing him with books and forcing his employer to allow the young Joseph Fraunhofer time to study. After eight months of study, Fraunhofer went to work at the Optical Institute at Benediktbeuern, a secularised Benedictine monastery devoted to glass making. There he discovered how to make the world's finest optical glass and invented incredibly precise methods for measuring dispersion. In 1818 he became the director of the Optical Institute. Due to the fine optical instruments he had developed, Bavaria overtook England as the centre of the optics industry. Even the likes of Michael Faraday were unable to produce glass that could rival Fraunhofer's. His illustrious career eventually earned him an honorary doctorate from the University of Erlangen in 1822. In 1824, he was awarded the order of merit, became a noble, and made an honorary citizen of Munich. Like many glassmakers of his era who were Joseph von Fraunhofer poisoned by heavy metal vapours, Fraunhofer died young, in 1826 at the (1787–1826) German age of 39. His most valuable glassmaking recipes are thought to have gone to the grave with him. Fraunhofer diffraction: Light can occur as plane waves, which we can imagine as the waves that come rolling in over the ocean. These plane waves can hit an obstruction, like when ocean waves hit a dock, and travel on in a very different pattern. At a large (compared to the size of the obstruction) distance away from the obstruction, there will be an illumination pattern of light and dark depending on the direction from the obstruction. This pattern is a Fraunhofer diffraction pattern. The (effective) source of light and the place where you're recieving the light must be relatively _far_ from the obstruction (e.g. >1 meter from a 0.1 millimeter slit or hole), hence the alternate name of "far-field" diffraction. These distances must be large enough so that the lights that arrives and leaves the obstruction and reaches the wall or screen are nearly plane waves. Fraunhofer diffraction deals with the limiting cases where the light approaching the diffracting object is parallel and monochromatic, and where the image plane is at a distance large compared to the size of the diffracting object. The more general case where these restrictions are relaxed is called Fresnel diffraction. Figure 2 Fresnel was the son of an architect, born at Broglie (Eure). His early progress in learning was slow, and he still could not read when he was eight years old. At thirteen he entered the École Centrale in Caen, and at sixteen and a half the École Polytechnique, where he acquitted himself with distinction. From there he went to the École des Ponts et Chaussées. He served as an engineer successively in the departments of Vendée, Drôme and Ille-et-Vilaine; but having supported the Bourbons in 1814 he lost his appointment on Napoleon's return to power. On the second restoration of the monarchy, he obtained a post as engineer in Paris, where much of his life from that time onwards was spent. His researches in optics, which were continued until his death, appear to have begun about the year 1814, when he prepared a paper on the aberration of light, which, however, was not published. In 1818 he wrote a memoir on diffraction for which in the ensuing year he received the prize of the Académie des Sciences at Paris. He was in 1823 unanimously elected a member of the academy, and in 1825 he became a member of the Royal Society of London, which in 1827, at the time of his last illness, awarded him the Rumford Medal. In 1819 he was nominated a commissioner of lighthouses, for which he was the first to construct a special type of lens, now called a Fresnel lens, as substitutes for mirrors. Fresnel died of tuberculosis at Ville-d'Avray, near Paris. His labours in the cause of optical science received during his lifetime only scant public recognition, and some of his papers were not printed by the Académie des Sciences till many years after Augustin-Jean Fresnel his death. But as he wrote to Young in 1824: in himself "that sensibility, or that vanity, which people call love of glory" had been blunted. "All the (1788–1827) French compliments," he says, "that I have received from Arago, Laplace and Biot never gave me so much pleasure as the discovery of a theoretic truth, or the confirmation of a calculation by experiment." Fresnel diffraction: Light can also occur as spherical waves, which is analogous to the circular waves expanding from where we just dropped a pebble in water. A point source of light produces spherical waves. After these spherical waves pass by an obstruction, they will produce a Fresnel diffraction pattern at the screen or wall where they arrive. In general, the distance between the source and obstruction and the distance between the obstruction and the screen can be arbitrarily close, but the interesting distances are on the order of centimeters to 1 meter from a millimeter sized obstruction. Because these distances are not too far, this phenomenon is also called "near-field" diffraction. Fraunhofer diffraction is the special case where the incoming light is assumed to be parallel and the image plane is assumed to be at a very large distance compared to the diffracting object. Fresnel diffraction refers to the general case where those restrictions are relaxed. This makes it much more complex mathematically. Some cases can be treated in a reasonable empirical and graphical manner to explain some observed phenomena. Figure 3 III. Fraunhofer diffraction through a single slit In this section, we abandon that assumption and see how the finite width of slits is the basis for understanding Fraunhofer diffraction. We can deduce some important features of this phenomenon by examining waves coming from various portions of the slit, as shown in Figure 4. According to Huygens’s principle, each portion of the slit acts as a source of light waves. Hence, light from one portion of the slit can interfere with light from another portion, and the resultant light intensity on a viewing screen depends on the direction θ. To analyze the diffraction pattern, it is convenient to divide the slit into two halves, as shown in Figure 4. Keeping in mind that all the waves are in phase as they leave the slit, consider rays 1 and 3. As these two rays travel toward a viewing screen far to the right of the figure, ray 1 travels farther than ray 3 by an amount equal to the path difference (a/2) sin θ, where a is the width of the slit. Figure 4 Diffraction of light by a narrow slit of width a. Each portion of the slit acts as a point source of light waves. The path difference between rays 1 and 3 or between rays 2 and 4 is (a/2) sin θ (drawing not to scale). Similarly, the path difference between rays 2 and 4 is also (a/2) sin q. If this path difference is exactly half a wavelength (corresponding to a phase difference of 180°), then the two waves cancel each other and destructive interference results. This is true for any two rays that originate at points separated by half the slit width because the phase difference between two such points is 180°. Therefore, waves from the upper half of the slit interfere destructively with waves from the lower half when a λ sin θ = 2 2 or when λ sin θ = a If we divide the slit into four equal parts and use similar reasoning, we find that the viewing screen is also dark when 2λ sin θ = a Likewise, we can divide the slit into six equal parts and show that darkness occurs on the screen when 3λ sin θ = a Therefore, the general condition for destructive interference is ⎛λ ⎞ sin θ = m⎜ ⎟ ⎝a⎠ m = ±1,±2,±3.............. This equation gives the values of θ for which the diffraction pattern has zero light intensity—that is, when a dark fringe is formed. However, it tells us nothing about the variation in light intensity along the screen. The general features of the intensity distribution are shown in Figure 5. A broad central bright fringe is observed; this fringe is flanked by much weaker bright fringes alternating with dark fringes. The various dark fringes occur at the values of θ that satisfy above Equation. Each bright-fringe peak lies approximately halfway between its bordering dark fringe minima. Note that the central bright maximum is twice as wide as the secondary maxima. Figure 5a Figure 5b IV. Plane transmission Diffraction Grating If we prepare an aperture with thousands of adjacent slits, we have a so-called transmission-diffraction grating. The width of a single slit? the opening? is given by d, and the distance between slit centers is given by (see Figure 6). For clarity, only a few of the thousands of slits normally present in a grating are shown. Note that the spreading of light occurs always in a direction perpendicular to the direction of the long edge of the slit opening? That is, since the long edge of the slit opening is vertical in Figure 6, the spreading is in the horizontal direction? Along the screen. Figure 6 Diffraction of light through a grating under Fraunhofer conditions The resulting diffraction pattern is a series of sharply defined, widely spaced fringes, as shown. The central fringe, on the symmetry axis, is called the zeroth-order fringe. The successive fringes on either side are called lst order, 2nd order, etc., respectively. They are numbered according to their positions relative to the central fringe, as denoted by the letter p. The intensity pattern on the screen is a superposition of the diffraction effects from each slit as well as the interference effects of the light from all the adjacent slits. The combined effect is to cause overall cancellation of light over most of the screen with marked enhancement over only limited regions, as shown in Figure 4-22. The location of the bright fringes is given by the following expression, called the grating equation, assuming that Fraunhofer conditions hold. =L L(sin α + sin θp) = pλ (a) where = distance between slit centers =L α = angle of incidence of light measured with respect to the normal to the grating surface θp = angle locating the pth-order fringe p = an integer taking on values of 0, ±1, ±2, etc. λ = wavelength of light Note that, if the light is incident on the grating along the grating normal (a = 0), the grating equation, above equation, reduces to the more common form shown in Equation below L(sin θp) = pλ (b) If, for example, you shine a HeNe laser beam perpendicularly onto the surface of a transmission grating, you will see a series of brilliant red dots, spread out as shown in Figure 6. A complete calculation would show that less light falls on each successively distant red dot or fringe, the p = 0 or central fringe being always the brightest. Nevertheless, the location of each bright spot, or fringe, is given accurately by Equation (a) for either normal incidence (α = 0) or oblique incidence (α ? 0). If light containing a mixture of wavelengths (white light, for example) is directed onto the transmission grating, Equation (a) holds for each component color or wavelength. So each color will be spread out on the screen according to Equation (a), with the longer wavelengths (red) spreading out farther than the shorter wavelengths (blue). In any case, the central fringe (p = 0) always remains the same color as the incident beam, since all wavelengths in the p = 0 fringe have θp = 0, hence all overlap to re-form the "original" beam and therefore the original "color." V. Diffraction Pattern due to a Plane transmission Diffraction Grating The progression to a larger number of slits shows a pattern of narrowing the high intensity peaks and a relative increase in their peak intensity. This progresses toward the diffraction grating, with a large number of extremely narrow slits. This gives very narrow and very high intensity peaks that are separated widely. Since the positions of the peaks depends upon the wavelength of the light, this gives high resolution in the separation of wavelengths. This makes the diffraction grating like a "super prism". Plane transmission diffraction grating When there is a need to separate light of different wavelengths with high resolution, then a diffraction grating is most often the tool of choice. This "super prism" aspect of the diffraction grating leads to application for measuring atomic spectra in both laboratory instruments and telescopes. A large number of parallel, closely spaced slits constitutes a diffraction grating. The condition for maximum intensity is the same as that for the double slit or multiple slits, but with a large number of slits the intensity maximum is very sharp and narrow, providing the high resolution for spectroscopic applications. The peak intensities are also much higher for the grating than for the double slit. The two aspects of the grating intensity relationship can be illustrated by the diffraction from five slits. The intensity is given by the interference intensity expression modulated by the single slit diffraction envelope for the slits which make up the grating: Total intensity expression: Grating Intensity Comparison VI. Absent Spectra. For normal incidence principal maxima in case of grating is given by: (a + b) sin θ = nλ Equation (1) n = 0,1,2........... And for single slit the minima is given by a sin θ = mλ Equation (2) m = 1,2,3........... And when both the are simultaneously satisfied the principal maxima of order n will not be present in the grating spectrum this is also know as condition of absent spectra. Equation (1)/ Equation (2) (a + b) sin θ nλ = a sin θ mλ ( a + b) n = a m ( a + b) m n= a (so for given a and b nth maxima is absent in the grating spectrum) VII. Rayleigh’s criterion for resolution. Resolving power The resolving power of an optical instrument is its ability to separate the images of two objects, which are close together. Some binary stars in the sky look like one single star when viewed with the naked eye, but the images of the two stars are clearly resolved when viewed with a telescope. Why? The merging of the images in the eye is caused by diffraction. If you look at a far-away object, then the image of the object will form a diffraction pattern on your retina. For two far-away objects, separated by a small angle q, the diffraction patterns will overlap. You are able to resolve the two objects as long as the central maxima of the two diffraction patterns do not overlap. The two images are just resolved when one central maximum falls onto the first minimum of the other diffraction pattern. This is known as the Rayleigh criterion. If the two central maxima overlap the two objects look like one The width of the central maximum in a diffraction pattern depends on the size of the aperture, (i.e. the size of the slit). The aperture of your eye is your pupil. A telescope has a much larger aperture, and therefore has a greater resolving power. The minimum angular separation of two objects which can just be resolved is given by qmin = 1.22l/D, where D is the diameter of the aperture. The factor of 1.22 applies to circular apertures like the pupil of your eye or the apertures in telescopes and cameras. The closer you are to two objects, the greater is the angular separation between them. Up close, two objects are easily resolved. As your distance from the objects increases, their images become less well resolved and eventually merge into one image. The Rayleigh criterion is the generally accepted criterion for the minimum resolvable detail - the imaging process is said to be diffraction-limited when the first diffraction minimum of the image of one source point coincides with the maximum of another. Overlapping images from two apertures are just resolved when the center of one Airy disk falls on the first minimum of the other. Resolution, Example • Pluto and its moon, Charon • Left: Earth-based telescope is blurred • Right: Hubble Space Telescope clearly resolves the two objects VIII. Dispersive and Resolving Power As shown in the left side of the figure, when the wavelength (λ) is much smaller than the aperture width (d), the wave simply travels onward in a straight line, just as it would if it were a particle or no aperture were present. However, when the wavelength exceeds the size of the aperture, we experience diffraction of the light according to the equation: sinθ = λ/d Where θ is the angle between the incident central propagation direction and the first minimum of the diffraction pattern. The experiment produces a bright central maximum which is flanked on both sides by secondary maxima, with the intensity of each succeeding secondary maximum decreasing as the distance from the center increases. Figure 4 illustrates this point with a plot of beam intensity versus diffraction radius. Note that the minima occurring between secondary maxima are located in multiples of π. Diffraction of light plays a paramount role in limiting the resolving power of any optical instrument (for example: cameras, binoculars, telescopes, microscopes, and the eye). The resolving power is the optical instrument's ability to produce separate images of two adjacent points. This is often determined by the quality of the lenses and mirrors in the instrument as well as the properties of the surrounding medium (usually air). The wave- like nature of light forces an ultimate limit to the resolving power of all optical instruments. Our discussions of diffraction have used a slit as the aperture through which light is diffracted. However, all optical instruments have circular apertures, for example the pupil of an eye or the circular diaphragm and lenses of a microscope. Circular apertures produce diffraction patterns similar to those described above, except the pattern naturally exhibits a circular symmetry. Mathematical analysis of the diffraction patterns produced by a circular aperture is described by the equation: sinθ(1) = 1.22(λ/d) where θ(1) is the angular position of the first order diffraction minima (the first dark ring), λ is the wavelength of the incident light, d is the diameter of the aperture, and 1.22 is a constant. Under most circumstances, the angle θ(1) is very small so the approximation that the sin and tan of the angle are almost equal yields: θ(1) ≅ 1.22(λ/d) From these equations it becomes apparent that the central maximum is directly proportional to λ/d making this maximum more spread out for longer wavelengths and for smaller apertures. The secondary mimina of diffraction set a limit to the useful magnification of objective lenses in optical microscopy, due to inherent diffraction of light by these lenses. No matter how perfect the lens may be, the image of a point source of light produced by the lens is accompanied by secondary and higher order maxima. This could be eliminated only if the lens had an infinite diameter. Two objects separated by a distance less than θ(1) can not be resolved, no matter how high the power of magnification. While these equations were derived for the image of a point source of light an infinite distance from the aperture, it is a reasonable approximation of the resolving power of a microscope when d is substituted for the diameter of the objective lens. Thus, if two objects reside a distance D apart from each other and are at a distance L from an observer, the angle (expressed in radians) between them is: θ=D/L which leads us to be able to condense the last two equations to yield: D(0) = 1.22(λL/d) Where D(0) is the minimum separation distance between the objects that will allow them to be resolved. Using this equation, the human eye can resolve objects separated by a distance of 0.056 millimeters, however the photoreceptors in the retina are not quite close enough together to permit this degree of resolution, and 0.1 millimeters is a more realistic number under normal circumstances. The resolving power of optical microscopes is determined by a number of factors including those discussed, but in the most ideal circumstances, this number is about 0.2 micrometers. This number must take into account optical alignment of the microscope, quality of the lenses, as well as the predominant wavelengths of light used to image the specimen. While it is often not necessary to calculate the exact resolving power of each objective (and would be a waste of time in most instances), it is important to understand the capabilities of the microscope lenses as they apply to the real world.

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