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LECTURE 27: Interference, Diffraction, Resolution 8/16/12 2 Diffraction
- Geometrical Optics: (d<<)
- Wave Optics: d
- Let us consider the case of light impinging on a small disk. We observe:
- bright spot in the center.
- diffraction rings outside and inside the geometrical shadow area
- The bright spot at the center was predicted by Fresnel in 1818 and observed by Arago. 8/16/12 3 Diffraction from a Disk 8/16/12 4 Fresnel Diffraction (view of the intensity at screen) plane wave center bright spot: consistent only with wave theory. intensity cylindrical obstacle
Counter-intuitive result!
8/16/12 5 Diffraction diffraction pattern plane wave obstacle 8/16/12 6 Diffraction from a Single Slit (screen far away) Consider a monochromatic wave incident on a place with a narrow slit. Geometrical optics predicts that the transmitted beam has the same cross section of the slits Experiments show that wave optics is correct and that:
- there is a central bright band that is wider than the width of the slit
- Alternating dark and bright fringes border the central bright band 8/16/12 7 Diffraction from a Single Slit Central bright fringe (the waves from all points in the slit travel the same distance to reach the center – and thus are in Phase) 8/16/12 8 Diffraction from a Single Slit Represent the slit as a number of point sources of equal amplitude. Divide the slit into two and pair a point from the upper half with its partner in the lower half. thepairedrays occurs. destructiveinterference between , 2 1 sin 2 1 2 sin^ θ^ When a^ θ^ =^ λ a (first minimum)
8/16/12 13 Diffraction from a Circular Aperture (DEMO) The diffraction pattern of a circular aperture of diameter d is similar to a single slit of width a. The central bright spot is called Airy disk. About 85% of the power is in this area. The dark fringes are found at: d d d λ θ λ θ λ θ sin 3. 24 sin 2. 23 sin 1. 22 3 2 1 =
8/16/12 14 Diffraction from a Circular Aperture The bright fringes are at: The Airy disk limits the resolvability of nearby objects d d d λ θ λ θ λ θ sin 3. 70 sin 2. 68 sin 1. 63 3 2 1 =
Image of two nearby binary stars but diffraction patterns overlap 8/16/12 15 Rayleigh Criteria The minimum angular separation c of two marginally resolvable points is such that the maximum of the diffraction pattern from one falls on the first minimum of the diffraction pattern of the other, The first minima is at
d
λ sin θ = 1. 22
d d
C
sin 1. 22 1. 22
1
− Therefore Not resolved Resolved Barely resolved^ 8/16/12 16 Rayleigh Criteria If > C objects can be resolved If < R objects can not be resolved To increase our ability to distinguish objects we must minimize the diffraction pattern. Because we can : increase d or decrease **1) Use ultraviolet light
- e- beam used in Scanning Electron Microscopes (SEM) have** (light)/10^5 3) place object under a microscope in a drop of oil = /n
d
C λ α ≈ 1. 22
8/16/12 17 Rayleigh Criteria 1 2 When (^) C 1. 22 d λ α ≈ 8/16/12 18 Diffraction Gratings d #^ #
- What happen if we go from 2 slits to N slits?
- The fringes become narrower and faint secondary maxima appear between the fringes ( half-width of central line).
- If N is large (N/ 104 /cm) the fringes are very sharp and the secondary maxima can be neglected and you have a grating. δ sin θ 2 10 5000 1 ( ) 5000 4 pathlengthdifference d d cm cm N cm d cm lines cm N lines rulings d grating spacing = = ⎟ = × ⎠ ⎞ ⎜ ⎝ ⎛ = = ⇒ = = − Δ θ hw =^ λ Nd 8/16/12 19 Sharp bright fringes occur if where m is the order of the maxima. Grating are used to measure By detecting maxima of the diffraction pattern with m=1,2,… Resolving power Diffraction Gratings (DEMO)
δ = d sin θ= m λ m = 0 , 1 , 2
d
m
λ
sin θ =
R mN
λ λ
8/16/12 20 Diffraction Gratings (DEMO) hw Nd λ Δ θ = For a given wavelength and d, if N increases the half-width decreases
