Rosalind Franklin made the first x ray diffraction imaging of DNA; her pictures were instrumental in the discovery of the double helix structure Lecture 4 Applications of interference and diffraction[.]
Lecture 4:Applications of interference and diffraction Rosalind Franklin made the first x-ray diffraction imaging of DNA; her pictures were instrumental in the discovery of the double-helix structure X-ray Diffraction for Crystallography If we know about the grating, we can use the diffraction pattern to learn about the light source If instead we know about the source, we can use the diffraction pattern to learn about the “grating” For this to work, we need to have a source wavelength that is less than the grating spacing (otherwise, there are no orders of diffraction) Crystals consist of regularly spaced atoms regular array of scattering centers Typical lattice spacing is angstroms = x 10-10 m = 0.5 nm use x-rays! Bragg Law for constructive interference: 2d sinq = l d = lattice spacing q = x-ray angle (with respect to plane of crystal) l = x-ray wavelength X-ray Crystallography The Braggs made so many discoveries that Lawrence described the first few years as ‘like looking for gold and finding nuggets lying around everywhere’: • showed that the sodium and chloride ions were not bonded into molecules, but arranged in a lattice • could distinguish different cubic lattices • discovered the crystal structure of diamond • Lawrence Bragg was the youngest Laureate ever (25) to receive a Nobel Prize (shared with his father in 1915) • now standardly used for all kinds of materials analysis, even biological samples! • The same multi-layer interference phenomenon is now used to make highly wavelength-specific mirrors for lasers (“distributed Bragg feedback” [DBF]) Diffraction-limited Optics Diffraction has important implications for optical instruments Lens-making is a craft Even for a perfectly designed lens, however, the image of a point source will be a little “blurry” due to diffraction in passing through the circular aperture of the lens D Image plane q I10 Image plane I diff( x) 0.5 Point object 00 10 00 12.56 x qo 10 12.56 The amount of ‘smearing’ of the image is determined by size of the aperture D, and wavelength of incident light, l q The image of a point source through a circular aperture is like a single-slit diffraction pattern But note the difference: q0 l a q0 1.22 Slit l Circular D aperture Transmission of light through slits and circular apertures I10 I diff( x) 0.5 Slit, width a Observation screen: 00 -l/a 10 12.56 Monochromatic light source at a great distance, or a laser 0x0 l/a 10 q 12.56 I10 I diff( x) 0.5 Pinhole, diameter D Observation screen: 00 -1.22l/D 0x 1.22l/D12.56q 10 10 12.56 Object at any distance: I10 I diff( x) 0.5 Lens, diameter D Image Plane: 00 -1.22l/D 0x 1.22l/D12.56q 10 10 12.56 Laser with pinholes Circular-aperture diffraction pattern =“the Airy disk” Central lobe contains 84% of power exercise 1: Expansion of a Laser beam In 1985, a laser beam with a wavelength of l = 500 nm was fired from the earth and reflected off the space shuttle Discovery, in orbit at a distance of L = 350 km away from d the laser D If the (circular) aperture of the laser was D = 4.7 cm, what was the beam diameter d at the space shuttle? to a b c To make a smaller spot on the shuttle, what should we the beam diameter at the source? reduce it increase it cannot be made smaller Exercise 1: Expansion of a Laser beam In 1985, a laser beam with a wavelength of l = 500 nm was fired from the earth and reflected off the space shuttle Discovery, in orbit at a distance of L = 350 km away from d the laser D If the (circular) aperture of the laser was D = 4.7 cm, what was the beam diameter d at the space shuttle? Half-angle-width of diffraction maximum: to a b c l 500 10-9 q o 1.22 1.22 1.3 10-5 radians -2 D 4.7 10 d 2qo L 2(1.3 10-5 )(350 103 m) 9.1 m 84% of power is in central lobe To make a smaller spot on the shuttle, what should we the beam diameter at the source? reduce it Counter-intuitive as this is, it is correct – you increase it reduce beam divergence by using a bigger cannot be made smaller beam (Note: this will work until D ~ d) exercise 2: Focusing of a laser beam There are many times you would like to focus a laser beam to as small a spot as possible However, diffraction limits this Dlens d Dlaser f The (circular) aperture of a laser (l = 780 nm) has Dlaser = mm What is the spot-size d of the beam after passing through a (perfect) lens with focal length f=5mm, diameter Dlens = mm? (Hint: light passing through lens center is undeflected.) a b c d Which of the following will reduce the spot size? increase l decrease l increase dlens decrease dlens Exercise 2: Focusing of a laser beam There are many times you would like to focus a laser beam to as small a spot as possible However, diffraction limits this Dlens d Dlaser f The (circular) aperture of a laser (l = 780 nm) has Dlaser = mm What is the spot-size d of the beam after passing through a (perfect) lens with focal length f=5mm, diameter Dlens = mm? (Hint: light passing through lens center is undeflected.) The angular spread of the beam is determined by the smaller of Dlaser and Dlens: q o 1.22l / Dlaser Light at this angle will intercept the focal plane at d/2 ~ f qo: d 2qo f 2.44lf / Dlaser 2.44(0.78m)(5mm) /(5mm) 1.9m a b c d Which of the following will reduce the spot size? Since the diffraction is already limited by D, increase l increasing dlens doesn’t help decrease l There is a huge industry devoted to developing increase dlens cheap blue diode lasers (l ~ 400 nm) for just decrease dlens this purpose, i.e., to increase DVD capacity Angular Resolution Diffraction also limits our ability to “resolve” (i.e., distinguish) two point sources Consider two point sources (e.g., stars) with angular separation a viewed through a circular aperture or lens of diameter D pt sources D a a diffraction disks (not interference maxima) a2ac I I 10 I I 10 12.56 2 2I0 x 10 12.56 Sum diff3 ( x) I 10 12.56 2 2I0 x 10 12.56 12.56 2I0 22 10 12.56 x 10 12.56 Sum diff3 ( x) 00 10 0 0 Sum y0 12.56 x images resolvable Rayleigh’s Criterion: define the images to be resolved if a ac , where I diff3 ( x) 10 y diff2 ( x) 0.5 diff1 diff1 ( x) 0.5 diff2 0 aac/3 1 1 diff2 ( x) 0.5 diff1 00 aac 10 y0 10 ‘Diffraction limit’ of resolution 12.56 x 12.56 00 10 y0 10 images not resolvable 12.56 x 12.56 a c 1.22 l D At ac the central max of one image falls on the first minimum of the second image ... amount of ‘smearing’ of the image is determined by size of the aperture D, and wavelength of incident light, l q The image of a point source through a circular aperture is like a single-slit diffraction. .. diffraction pattern =“the Airy disk” Central lobe contains 84% of power exercise 1: Expansion of a Laser beam In 1985, a laser beam with a wavelength of l = 500 nm was fired from the earth and. .. 1: Expansion of a Laser beam In 1985, a laser beam with a wavelength of l = 500 nm was fired from the earth and reflected off the space shuttle Discovery, in orbit at a distance of L = 350 km