Sections#

Review & Summary#

Huygens’ Principle#

The three-dimensional transmission of waves, including light, may be predicted by Huygens’ principle: all points on a wavefront serve as point sources of spherical secondary wavelets. After a time \(t\), the new wavefront is the surface tangent to these secondary wavelets. The law of refraction can be derived by assuming the index of refraction of any medium is \(n = c/v\), where \(v\) is the speed of light in the medium and \(c\) is the speed in vacuum.

Wavelength and Index of Refraction#

The wavelength \(\lambda_n\) of light in a medium depends on the index of refraction \(n\):

(265)#\[ \lambda_n = \frac{\lambda}{n} \]

where \(\lambda\) is the wavelength in vacuum. Phase difference between two waves can change if they pass through different materials.

Young’s Experiment#

In Young’s interference experiment, light passing through a single slit falls on two slits. The light leaving these slits flares out (by diffraction), and interference occurs in the region beyond. The light intensity at any point depends on the path length difference from the slits. Constructive interference (intensity maximum) occurs when the difference is an integer number of wavelengths; destructive (minimum) when it is an odd number of half-wavelengths:

(266)#\[ d\sin\theta = m\lambda \quad \text{(maximum)}, \qquad d\sin\theta = \left(m + \frac{1}{2}\right)\lambda \quad \text{(minimum)} \]

where \(\theta\) is the angle from the central axis and \(d\) is the slit separation.

Interference from Thin Films#

Reflection at a boundary with \(n_2 > n_1\) adds a phase change of \(\pi\) (equivalent to half-wavelength). For a thin film of thickness \(L\) and index \(n\), constructive or destructive interference depends on the number of half-wavelengths in the path and the phase changes at the interfaces.

Michelson’s Interferometer#

A Michelson interferometer splits light into two beams that reflect from mirrors and recombine. Moving one mirror by \(\lambda/2\) shifts the fringe pattern by one fringe. It is used to measure wavelengths and small distances with high precision.