5: Diffraction
- Page ID
- 128466
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 5.1: Diffraction, shadows etc.
- We compare shadows and diffraction using light, sound and water waves. When \(\lambda\) is much smaller than objects and apertures, we observe sharp shadows and beams; these are analysed with geometrical optics. When \(\lambda\)and size are comparable, diffraction is observed; this is analysed using Huygens’ construction. The Poisson-Arago dot is a small, bright spot observed at the centre of the shadow of a very precisely circular or spherical object: strong support for the wave theory
- 5.2: 1D single slit
- 1D: a single slit produces a bright central maximum, with subsidiary maxima and minima. We divide the single slit (width \(a\) ) into Huygens sources. Cancellation produces minima at \(\sin \theta=m \lambda / a\), where \(m=1,2\) etc. In the limit of small Huygens sources, phasor addition quantifies the intensity as function of angle \(\theta\). In Young's experiment, the diffraction term modulates the interference term. It is worth revising the previous page to see the resultant interference-di
- 5.3: Diffraction gratings
- A diffraction grating has many equally spaced slits; it can measure \(\lambda\) of sources precisely in transmission or reflection. Incandescent sources have continuous spectra. Electrons in atoms have discrete energies governed by quantum mechanics. Consequently, atoms have spectra with discrete emission and absorption lines. Helium was discovered by its spectral lines in light from the Sun before it was observed on Earth.
- 5.4: Diffraction- 2D and 3D
- 2D \& 3D: X-rays have \(\lambda\) comparable with atomic spacing in condensed matter. X-ray diffraction reveals crystal structure: equidistant atomic planes in a crystal at spacing \(d\) produce diffraction maxima according to Bragg's law: \(2 d \sin \theta=n \lambda\) where \(n=1,2\) etc. Analogously, neutron diffraction requires neutrons with similar range of \(\lambda\).
- 5.5: Aperture and resolution
- Diffraction of light or other radiation through a circular aperture with diameter \(a\) produces the Airy pattern with a first circular minimum at \(\sin \theta=1.22 \lambda / a\), then subsidiary maxima and minima. The Rayleigh criterion for diffraction limited resolution has one central maximum coincident with a neighbouring first minimum, hence again \(\sin \theta=1.22 \lambda / a\). Diffraction often limits the resolution of optical and radio telescopes.
- 5.6: Electrons and neutrons
- From quantum mechanics, the wavelength of electrons and neutrons, like photons, is \(\lambda =\) Planck's constant \(/\) momentum \(=\mathrm{h} / \mathrm{p}\). Through a potential difference \(V\), non-relativistic electrons gain kinetic energy \(\frac{1}{2} m v^2=e V\). Thus potentials of several kV provide electron wavelengths of tens of pm, and hence diffraction patterns from interactions with crystals.
- 5.7: Holography
- A coherent beam of light is split into two paths. The first beam scatters off the object and interferes with the second beam at the recording medium (e.g. a holographic plate). The resulting interference pattern encodes three-dimensional information about the object. This is the hologram. A projection of the original object can be recovered by illuminating and viewing the recorded hologram appropriately. Using digital techniques, interference patterns can be made to project virtual objects, such
- 5.8: Appendix
- 5.8.1: Diffraction, shadows, beams, Huygens' construction.
- 5.8.2: Diffraction from a single slit. Young's experiment with finite slits.
- 5.8.3: The quantum mechanics of cricket
- 5.8.4: Diffraction gratings and spectroscopy.
- 5.8.5: Poisson-Arago dot
- 5.8.6: Rayleigh criterion and the Airy disc- Aperture and resolution
- 5.8.7: X-ray diffraction- Bragg's law
- 5.8.8: Holography
- 5.8.9: Diffraction Experiments
- 5.8.9.1: Single slit
- 5.8.9.2: Diffraction grating