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1 Fraunhofer Diffraction: Single, multiple slit(s) & Circular aperture Fri. Nov. 22, 2002
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2 Fraunhofer diffraction limit If aperture is a square - X The same relation holds in azimuthal plane and 2 ~ measure of the area of the aperture Then we have the Fraunhofer diffraction if, Fraunhofer or far field limit
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3 Fraunhofer, Fresnel limits The near field, or Fresnel, limit is See 10.1.2 of text
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4 Fraunhofer diffraction Typical arrangement (or use laser as a source of plane waves) Plane waves in, plane waves out S f1f1 f2f2 screen
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5 Fraunhofer diffraction 1. Obliquity factor Assume S on axis, so Assume small ( < 30 o ), so 2. Assume uniform illumination over aperture r’ >> so is constant over the aperture 3. Dimensions of aperture << r r will not vary much in denominator for calculation of amplitude at any point P consider r = constant in denominator
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6 Fraunhofer diffraction Then the magnitude of the electric field at P is,
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7 Single slit Fraunhofer diffraction y = b y dy P roro r r = r o - ysin dA = L dy where L ( very long slit)
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8 Single slit Fraunhofer diffraction Fraunhofer single slit diffraction pattern
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9 Single Slit Fraunhofer diffraction: Effect of slit width Minima for sin = 0 = p = k(b/2)sin or, sin = p( /b) First minima at sin = /b b
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10 Single Slit Fraunhofer diffraction: Effect of slit width Width of central max 2 ( /dimension of aperture) This relation is characteristic of all Fraunhofer diffraction If b is very large 0 and a point source is imaged as a point If b is very small (~ ) /2 and light spreads out across screen (diminishes at large angles for to F( )
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11 Diffraction from an array of N slits, separated by a distance a and of width b y=0 y=a y=a+b y=2a y=2a+b y=3a y=3a+b y=(N-1)a y=(N-1)a + b y=b P
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12 Diffraction from an array of N slits It can be shown that, where,
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13 Diffraction and interference for N slits The diffraction term Minima for sin = 0 = p = k(b/2)sin or, sin = p( /b) The interference term Amplitude due to N coherent sources Can see this by adding N phasors that are 2 out of phase. See Hecht Problem 10.2
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14 Interference term Maxima Maxima occur at = m (m = 0,1, 2, 3,..) To see this use L’Hopital’s rule _______ Thus maxima occur at sin = m /a This is the same result we have derived for Young’s double slit Intensity of principal maxima, I = N 2 I o i.e. N times that due to one slit
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15 Interference term Minima Minima occur for = /N, 2 /N, … (N-1) /N and when we add m For example, _______________________ Thus principal maxima have a width determined by zeros on each side Since = ( / )a sin = /N The angular width is determined by sin = /(Na) Thus peaks are N times narrower than in a single slit pattern (also a > b)
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16 Interference term Subsidiary or Secondary Maximum Subsidiary or Secondary Maximum Now between zeros must have secondary maxima Assume these are approximately midway Then first at [ m+3/(2N) ] Then it can be shown that
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17 Single slit envelope Now interference term or pattern is modulated by the diffraction term which has zeros at =( b/ )sin = p or, sin = p /b But, sin = m /a locate the principal maxima of the interference pattern
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18 Single slit envelope Thus at a given angle a/b=m/p Then suppose a/b = integer For example, a = 3b Then m = 3, 6, 9, interference maxima are missing
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19 Diffraction gratings Composed of systems with many slits per unit length – usually about 1000/mm Also usually used in reflection Thus principal maxima vary sharp Width of peaks Δ = (2/N) As N gets large the peak gets very narrow For example, _________________
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20 Diffraction gratings Resolution Resolution Imagine trying to resolve two wavelengths 1 2 Assume resolved if principal maxima of one falls on first minima of the other See diagram___________
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21 Diffraction gratings m 1 = a sin m 2 = a sin ’ But must have Thus m( 2 - 1 )= a (sin ’ - sin ) = ( 1 /N) Or mΔ = /N Resolution, R = /Δ = mN E.g.
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22 Fraunhofer diffraction from a circular aperture x y P Lens plane r
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23 Fraunhofer diffraction from a circular aperture Do x first – looking down Path length is the same for all rays = r o Why?
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24 Fraunhofer diffraction from a circular aperture Do integration along y – looking from the side -R +R y=0 roro r = r o - ysin P
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25 Fraunhofer diffraction from a circular aperture Let Then
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26 Fraunhofer diffraction from a circular aperture The integral where J 1 ( ) is the first order Bessell function of the first kind.
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27 Fraunhofer diffraction from a circular aperture These Bessell functions can be represented as polynomials: and in particular (for p = 1),
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28 Fraunhofer diffraction from a circular aperture Thus, where = kRsin and I o is the intensity when =0
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29 Fraunhofer diffraction from a circular aperture Now the zeros of J 1 ( ) occur at, = 0, 3.832, 7.016, 10.173, … = 0, 1.22 , 2.23 , 3.24 , … =kR sin = (2 / ) sin Thus zero at sin = 1.22 /D, 2.23 /D, 3.24 /D, …
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30 Fraunhofer diffraction from a circular aperture The central Airy disc contains 85% of the light
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31 Fraunhofer diffraction from a circular aperture D sin = 1.22 /D
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32 Diffraction limited focussing sin = 1.22 /D The width of the Airy disc W = 2fsin 2f = 2f(1.22 /D) = 2.4 f /D W = 2.4(f#) > f# > 1 Cannot focus any wave to spot with dimensions < D f
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33 Fraunhofer diffraction and spatial resolution Suppose two point sources or objects are far away (e.g. two stars) Imaged with some optical system Two Airy patterns If S 1, S 2 are too close together the Airy patterns will overlap and become indistinguishable S1S1 S2S2
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34 Fraunhofer diffraction and spatial resolution Assume S 1, S 2 can just be resolved when maximum of one pattern just falls on minimum (first) of the other Then the angular separation at lens, e.g. telescope D = 10 cm = 500 X 10 -7 cm e.g. eye D ~ 1mm min = 5 X 10 -4 rad
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