AcademyLight Propagation
Academy
Huygens Principle
Level 1 - Physics topic page in Light Propagation.
Principle
Huygens principle models each point on a wavefront as a source of secondary wavelets. The next wavefront is the envelope of those wavelets.
Notation
\(v\)
wave speed in the medium
\(\mathrm{m\,s^{-1}}\)
\(\Delta t\)
small time interval
\(\mathrm{s}\)
\(v\Delta t\)
wavelet radius after time \(\Delta t\)
\(\mathrm{m}\)
\(\theta_i\)
incident angle
\(\mathrm{rad}\)
\(\theta_r\)
reflected angle
\(\mathrm{rad}\)
\(\theta_2\)
refracted angle
\(\mathrm{rad}\)
Method
Derivation 1: Construct a new wavefront
Start with a known wavefront. After time \(\Delta t\), draw secondary wavelets of radius \(v\Delta t\) from points on the old wavefront. The tangent envelope is the new wavefront.
Wavelet radius
\[r=v\Delta t\]
Derivation 2: Explain reflection
At a reflecting boundary, the Huygens construction makes the reflected wavefront leave with the same angle that the incident wavefront arrived.
Reflection law
\[\theta_r=\theta_i\]
Derivation 3: Explain refraction
If the wave speed changes across a boundary, wavelets grow at different speeds in the two media. This rotation of the wavefront gives Snell's law.
Speed form of Snell's law
\[\frac{\sin\theta_1}{\sin\theta_2}=\frac{v_1}{v_2}\]
Index form
\[n_1\sin\theta_1=n_2\sin\theta_2\]
Rules
Wavelet radius
\[r=v\Delta t\]
Reflection from wavefronts
\[\theta_r=\theta_i\]
Refraction from wavefronts
\[\frac{\sin\theta_1}{\sin\theta_2}=\frac{v_1}{v_2}\]
Examples
Question
A wavefront travels at
\[2.0\times10^8\,\mathrm{m\,s^{-1}}\]
How far do its Huygens wavelets expand in \[1.0\,\mathrm{ns}\]
?Answer
\[r=v\Delta t=(2.0\times10^8)(1.0\times10^{-9})=0.20\,\mathrm m\]
Checks
- Rays are perpendicular to wavefronts in isotropic media.
- A slower second medium bends rays toward the normal.
- A faster second medium bends rays away from the normal.
- Huygens principle gives a wavefront construction behind reflection and refraction laws.