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Chapter 4 Diffraction of Light Waves

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Diffraction Huygen’s principle requires that the waves spread out after they pass through slits This spreading out of light from its initial line of travel is called diffraction In general, diffraction occurs when waves pass through small openings, around obstacles or by sharp edges

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A single slit placed between a distant light source and a screen produces a diffraction pattern It will have a broad, intense central band The central band will be flanked by a series of narrower, less intense secondary bands Called secondary maxima The central band will also be flanked by a series of dark bands Called minima

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The results of the single slit cannot be explained by geometric optics Geometric optics would say that light rays traveling in straight lines should cast a sharp image of the slit on the screen

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Fraunhofer Diffraction Fraunhofer Diffraction occurs when the rays leave the diffracting object in parallel directions Screen very far from the slit Converging lens (shown) A bright fringe is seen along the axis (θ = 0) with alternating bright and dark fringes on each side

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Single Slit Diffraction According to Huygen’s principle, each portion of the slit acts as a source of waves The light from one portion of the slit can interfere with light from another portion The resultant intensity on the screen depends on the direction θ

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All the waves that originate at the slit are in phase Wave 1 travels farther than wave 3 by an amount equal to the path difference (a/2) sin θ If this path difference is exactly half of a wavelength, the two waves cancel each other and destructive interference results

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In general, destructive interference occurs for a single slit of width a when sin θ dark = mλ / a m = 1, 2, 3, … Doesn’t give any information about the variations in intensity along the screen

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The general features of the intensity distribution are shown A broad central bright fringe is flanked by much weaker bright fringes alternating with dark fringes The points of constructive interference lie approximately halfway between the dark fringes

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Resolution of Single-Slit and Circular Apertures The resolution is the ability of optical systems to distinguish between closely spaced objects, which are limited because of the wave nature of light If no diffraction occurred, two distinct bright spots would be observed on the viewing screen. However, because of diffraction, each source is imaged as a bright central region flanked by weaker bright and dark bands.

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If the two sources are separated enough to keep their central maxima from overlapping, their images can be distinguished and are said to be resolved. If the sources are close together, however, the two central maxima overlap and the images are not resolved.

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Rayleigh's criterion To decide when two images are resolved, the following criterion is used: When the central maximum of one image falls on the first minimum another image, the images are said to be just resolved. This limiting condition of resolution is known as Rayleigh's criterion.

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The diffraction patterns of two point sources (solid curves) and the resultant pattern (dashed curves) for various angular separations of the sources

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From Rayleigh's criterion, we can determine the minimum angular separation, θ min, subtended by the sources at the slit so that their images are just resolved. the first minimum in a single-slit diffraction pattern occurs at the angle for which sin θ = λ / a where a is the width of the slit. According to Rayleigh's criterion, this expression gives the smallest angular separation for which the two images are resolved.

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Because λ « a in most situations, sin θ is small and we can use the approximation sin θ ≈ θ. Therefore, the limiting angle of resolution for a slit of width a is θ min = λ / a where θ min is expressed in radians. Hence, the angle subtended by the two sources at the slit must be greater than λ / a if the images are to be resolved.

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The diffraction pattern of a circular aperture consists of a central circular bright disk surrounded by progressively fainter rings. The limiting angle of resolution of the circular aperture is: Where D is the diameter of the aperture.

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Diffraction Grating The diffracting grating consists of many equally spaced parallel slits A typical grating contains several thousand lines per centimeter The intensity of the pattern on the screen is the result of the combined effects of interference and diffraction

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Diffraction Grating The condition for maxima is d sin θ bright = m λ m = 0, 1, 2, … The integer m is the order number of the diffraction pattern If the incident radiation contains several wavelengths, each wavelength deviates through a specific angle

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All the wavelengths are focused at m = 0 This is called the zeroth order maximum The first order maximum corresponds to m = 1 Note the sharpness of the principle maxima and the broad range of the dark area This is in contrast to the broad, bright fringes characteristic of the two- slit interference p attern

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diffraction grating spectrometer. The collimated beam incident on the grating is spread into its various wavelength components with constructive interference for a particular wavelength occurring at the angles that satisfy the equation

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Resolving power of the diffraction grating The diffraction grating is useful for measuring wavelengths accurately. Like the prism, the diffraction grating can be used to disperse a spectrum into its components. Of the two devices, the grating may be more precise if one wants to distinguish between two closely spaced wavelengths.

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If λ 1 and λ 2 are the two nearly equal wavelengths between which the spectrometer can barely distinguish, the resolving power R is defined as where λ = ( λ 1 + λ 2 ) / 2, and Δ λ = λ 2 - λ 1 a grating that has a high resolving power can distinguish small differences in wavelength.

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if N lines of the grating are illuminated, it can be shown that the resolving power in the m th order diffraction equals the product N m : R = N m Thus, resolving power increases with increasing order number. R is large for a grating that has a large number of illuminated slits.

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Consider the second-order diffraction pattern (m = 2) of a grating that has 5000 rulings illuminated by the light source. The resolving power of such a grating in second order is: R = 5000 x 2 = 10,000. The minimum wavelength separation between two spectral lines that can be just resolved, assuming a mean wavelength of 600 nm, is Δλ = λ / R = 6.00 X nm. For the third-order principal maximum, R = and Δλ = 4.00 x 2 nm, and so on.

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