4: Interference
- Page ID
- 128460
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.1: Reflections and phase
- We revise reflection and transmission at the junction between a light string and a heavy string. Analogously, going from a transparent medium with low refractive index to one with high n, the reflected light wave has a 90° phase change. From high n to low, the reflected light has no phase change. In both cases, there is no phase change in the transmitted wave. We briefly view reflections in a thin film and revisit Young’s experiment.
- 4.2: Newton's Rings
- Newton’s interference rings can be used to measure lens geometry when grinding them. A plano-convex lens lies on a flat, leaving a thin layer of air between them. Monochromatic light produces a pattern of dark and bright rings. Reflections inside the glass have phase change 0 and going from air to glass they have π phase change. These phase changes plus those due to travelling through the air layer determine conditions for destructive and constructive interference: dark and bright rings.
- 4.3: Thin Films I
- We demonstrate interference colours in soapy water-glycerol films in air. In reflection, π phase change at air-water and 0 at water-air interfaces gives destructive interference for all colours if the film is much thinner than wavelength ; a black film results. Where slightly thicker, interference is approximately constructive for most wavelengths, giving a nearly white band.
- 4.4: Thin Films II
- Non-reflective coatings on lenses and spectacles have n<nglass. This gives phase change π on reflection at both reflections. If thickness t=λ/4n for mid-range , there is destructive interference in reflection and therefore maximum transmission: nearly all light enters the lens. This increases sensitivity of optical instruments. It raises puzzles for the particle caricature of photons. Coherence length is the path distance over which phases are strongly correlated and so interference is possible.
- 4.5: Young's experiment
- Two small, coherent sources (size ≪λ) separated by a distance d (small, but somewhat greater than ). At points with distances from the sources differing by mλ, where m is an integer, constructive interference gives maxima (bright bands); differing by (m+1/2)λ gives minima (dark). At angle , phase difference ϕ=2πdsin θ /λ . Phasor addition gives the amplitude and intensity as a function of θ. We return to the photon puzzle: comparing the pattern for two slits with that for one, why does
- 4.6: Single Photon
- We demonstrate Young's experiment with light so weak that only one photon is in the apparatus at a time. (Via amplification, we hear the photon arrivals as clicks.) Nevertheless, a histogram of photon arrivals vs angle gives the classic Young’s experiment interference pattern. Again, this raises questions associated with the particle caricature of photons: Which slit did the photon ‘go through’? and How is it possible that adding more photons makes the pattern darker in some regions?
- 4.7: Electron Interference
- Young’s experiment with electrons or neutrons also gives the classic interference pattern. Instead of interference (which is technically quite difficult), here we produce a diffraction pattern for electrons. (Diffraction is analysed in the next chapter.) The size of the pattern depends on the accelerating voltage: larger accelerating voltage gives faster electrons, which produce a smaller interference or diffraction pattern, indicating a shorter wavelength.
- 4.8: Appendix
- 4.8.1: Introduction to Young's experiment
- 4.8.2: Young's experiment with single photons.
- 4.8.3: Molecules and photons
- 4.8.4: Reflections and phases
- 4.8.5: Thin film interference and reflections
- 4.8.6: Soap bubbles
- 4.8.7: Non-reflective coatings.
- 4.8.8: Coherence length
- 4.8.9: Newton's rings
- 4.8.10: Interference Experiments
- 4.8.10.1: Double slit