Ch31-A Diffraction 2 北邮大二物理光学

4 Diffraction in the Double Slit Experiment In our previous analysis of Young s double slit experiment we assumed that the slits were infinitesimally narrow so that the central diffraction maximum was spread out over the whole screen For the case of real slits the diffraction reduces the intensity of the bright interference fringes to the side of center so they are not all of the same height Intensity of double slit diffraction The diffraction pattern is the interference fringes modulated by the single slit diffraction 1 2cos 2 EE 120 sin 2 2 EEE 0 sin 2 2cos 22 EE 2 2 0 sin 2 cos 22 II interference factor Intensity of double slit diffraction 2 sina 2 sind diffraction factor 2 00 2 IE 2 2 0 sin 2 cos 22 II 2 sin 2 sin a d Intensity of double slit diffraction The diffraction pattern is the interference fringes modulated by the single slit diffraction 2 2 0 sin 2 cos 22 II Intensity of double slit diffraction 22 sin sinad Diffraction minimum sinn a Interference maximum sinm d 2 2 0 sin 2 cos 22 II Intensity of double slit diffraction A diffraction pattern is produced by a light of wavelength 580nm from a distant source incident on two identical parallel slits separated by a distance of 0 530mm a What is the angular positions of the first and second order maxima b Let the slits have width 0 320mm In terms of the intensity I0 at the center of the central maximum what is the intensity at each of the angular positions in part a Example Example A diffraction pattern is produced by a light of wavelength 580nm from a distant source incident on two identical parallel slits separated by a distance of 0 530mm a What is the angular positions of the first and second order maxima b Let the slits have width 0 320mm In terms of the intensity I0 at the center of the central maximum what is the intensity at each of the angular positions in part a Solution a 580nm d 0 530mm For the first maximum for the second maximum 933 1 sin 580 10 0 530 101 09 10dmm 3 11 sin1 09 10 rad 933 22 sin2 2 580 10 0 530 102 19 10 raddmm Example b Considering the effect of single slit diffraction 2 0 sin 2 2 II 3 sin1 73 10 2 aa 2 33 100 33 sin 1 73 101 09 10 0 252 1 73 101 09 10 III 2 33 200 33 sin 1 73 102 19 10 0 0253 1 73 102 19 10 III Example Cont d 3 1 81 101 66 ad An array of a large number of parallel slits all with the same width a and spaced equal distance d between centers is called a diffraction grating The spacing d between centers of adjacent slits is called the grating spacing For grating what we have been calling slits are often called rulings 划线划线 or lines 5 Diffraction Grating Slit width a Grating spacing d Number of slit N Constructive interference occurs at the angle when the optical path length difference between rays from any pair of adjacent slits is equal to an integer number of wavelengths Principal maxima principal max sin 0 1 2 dm m That is the same direction as for the double slit situation The other two parameters aand N of grating do not affect the locations of principal maxima The important difference between a double slit and multiple slit diffraction pattern is that the principal maxima are much sharper and narrower for a grating principal max sin 0 1 2 dmm two slit six slit Principal maxima For the principal maxima the rays from adjacent slits are in phase 2m 2 1 2 1 1 21 mmmN N mN NN Minima and secondary maxima The minima occur when phase difference between the rays from adjacent slits is integer multiple of 2 N The number of the minima between two adjacent maxima is N 1 The number of the secondary maxima between two adjacent maxima is N 2 2 mmmN N 2m principal maxima minima Minima and secondary maxima 2 sin 2 N ER 1 2 sin 2 ER 1 sin 2 sin 2 N EE 10 sin 2 2 EE 22 let sin sinda 0 sin 2sin 2 2sin 2 N EE 22 0 sin 2sin 2 2sin 2 N II diffraction factor interference factor The intensity of diffraction pattern for a grating 22 0 sin 2sin 2 2sin 2 N II 2sin 0 1 2 mdmm 22 2 sin sincos 2 222 limlim 1 sincos sin 222 2 mm m N NNN N 2 2 0 sin 2 2 IN I The locations of principal maxima maxima 2 but 2Nmm 2 m mmN N 0 1 2 sin 1 2 1 m m dm mNN There are N 1 minima between two adjacent principal maxima and N 2 secondary maxima 22 0 sin 2sin 2 2sin 2 N II The locations of minima minima Width of the principal maxima The angular separation between the principal maximum and its nearest minimum cos mm d N cos m m Nd sin m dm 1 sin mm dm N Width of the principal maxima The greater the number of slits N the sharper the principal maximum The multiple slit interference pattern is modulated by the diffraction factor So the heights of the principal maxima are not same sin sinsin an dkakn missing ordermissing order d 3a max sin1 d mm d The effect of diffraction factor The diffraction factor may induce the missing order The highest order which can be seen in the range from 90 to 90 If d ka The k th principal maxima will miss because an interference maximum and a diffraction minimum corresponds to the same value Principal maximaMinima Secondary maxima LocationIntensity Half width Missing order LocationNumberLocationNumber d a N Summary of grating Principal maximaMinima Secondary maxima LocationIntensity Half width Missing order LocationNumberLocationNumber d a Modulation N N 2I0 N 1 N 2 sinm d cosNd cosNd dm an dm an sin 0 2 m N d mNN sin 0 2 m N d mNN Summary of grating For each diffraction pattern 1 how many slits are there for the grating 2 Which pattern corresponds to largest slit width 3 what is d a for each pattern Example A light of wavelength 600nm falls normally on a grating The second and third order maxima appear at the angular positions of sin 0 20 and sin 0 30 It is found that the fourth order maximum is missing from the diffraction pattern a Determine the slit separation of the grating b and the width of each slit c List the all order of maxima that will appear on the viewing screen Example Example A light of wavelength 600nm falls normally on a grating The second and third order maxima appear at the angular positions of sin 0 20 and sin 0 30 It is found that the fourth order maximum is missing from the diffraction pattern Determine a the slit separation of the grating b the width of each slit c List the all order of maxima that will appear on the viewing screen Solution For the second order maximum a b c On the viewing screen the order of 0 1 2 3 5 6 7 9 maxima will appear sin0 202 d 2 600nm 0 206000nm 6 00 md 4da 1 50 ma max 6000nm 10 600nm d m Example 6 Grating Spectrometers The principle of grating spectrometer The angular position of the diffraction maxima or diffraction lines of a grating is proportional to the light wavelength sinm d sinm d Gratings are therefore used as spectrometers to determine the wavelength For a grating with large number of slits the maxima are very narrow and brighter which allows their position to be measured with great precision The principle of grating spectrometer Dispersion the angular separation per unit wavelength interval D sin cos mm dmdm cos m m D d minmin coscos m mm m dNd min RmN min mN The resolving power for a diffraction grating Resolving power using Rayleigh s criterion min R 589 0 589 6 nm 546 2nm 579 2nm 656 2nm 486 1nm 410 1nm 434nm Balm