3.S: Interference (Summary)
- Page ID
- 8073
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Key Terms
| coherent waves | waves are in phase or have a definite phase relationship |
| fringes | bright and dark patterns of interference |
| incoherent | waves have random phase relationships |
| interferometer | instrument that uses interference of waves to make measurements |
| monochromatic | light composed of one wavelength only |
| Newton’s rings | circular interference pattern created by interference between the light reflected off two surfaces as a result of a slight gap between them |
| order | integer m used in the equations for constructive and destructive interference for a double slit |
| principal maximum | brightest interference fringes seen with multiple slits |
| secondary maximum | bright interference fringes of intensity lower than the principal maxima |
| thin-film interference | interference between light reflected from different surfaces of a thin film |
Key Equations
| Constructive interference | \(\Delta l = m\lambda\), for m = 0, ±1, ±2, ±3… |
| Destructive interference | \(\Delta l = (m + \frac{1}{2})\lambda\), for m = 0, ±1, ±2, ±3… |
| Path length difference for waves from two slits to a common point on a screen | \(\Delta l = d \, sin \, \theta\) |
| Constructive interference | \(d \, sin \, \theta = m \lambda\), for m = 0, ±1, ±2, ±3… |
| Destructive interference | \(d \, sin \, \theta = (m + \frac{1}{2})\lambda\), for m = 0, ±1, ±2, ±3… |
| Distance from central maximum to the m-th bright fringe | \(y_m = \frac{m\lambda D}{d}\) |
| Displacement measured by a Michelson interferometer | \(\Delta d = m \frac{\lambda_0}{2}\) |
Summary
3.1: Young's Double-Slit Interference
- Young’s double-slit experiment gave definitive proof of the wave character of light.
- An interference pattern is obtained by the superposition of light from two slits.
3.2: Mathematics of Interference
- In double-slit diffraction, constructive interference occurs when \(dsinθ=mλ\) (for \(m=0,±1,±2,±3…\)), where d is the distance between the slits, \(θ\) is the angle relative to the incident direction, and m is the order of the interference.
- Destructive interference occurs when \(dsinθ=(m+\frac{1}{2})λ\) for \(m=0,±1,±2,±3,…\)
3.3: Multiple-Slit Interference
- Interference from multiple slits (\(N>2\)) produces principal as well as secondary maxima.
- As the number of slits is increased, the intensity of the principal maxima increases and the width decreases.
3.4: Interference in Thin Films
- When light reflects from a medium having an index of refraction greater than that of the medium in which it is traveling, a \(180°\) phase change (or a \(λ/2\) shift) occurs.
- Thin-film interference occurs between the light reflected from the top and bottom surfaces of a film. In addition to the path length difference, there can be a phase change.
3.5: The Michelson Interferometer
- When the mirror in one arm of the interferometer moves a distance of \(λ/2\) each fringe in the interference pattern moves to the position previously occupied by the adjacent fringe.
Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).