35-1 Light as a Wave

35-1 Light as a Wave#

Prompts

  • Using a sketch, explain Huygens’ principle. How does it predict where a wavefront will be at a future time?

  • Explain refraction in terms of wavefronts: why does a wavefront bend when it crosses an interface at an angle? What happens to the wavelength and speed in each medium?

  • Define the index of refraction \(n\). When light refracts through an interface, does the frequency change? What about the wavelength and speed?

  • Two light waves (same wavelength in vacuum) travel through different materials of length \(L\) with indices \(n_1\) and \(n_2\), then meet. How do you find their phase difference? What is the “effective” phase difference, and why does only the fractional part matter for interference?

  • How do rainbows and butterfly-wing colors relate to optical interference? What is the role of path differences within a raindrop?

Lecture Notes#

Overview#

  • Optical interference—the combining of light waves—requires treating light as a wave, not just rays (geometrical optics).

  • Huygens’ principle predicts wave propagation: each point on a wavefront acts as a source of spherical secondary wavelets; the new wavefront is tangent to them.

  • Refraction follows from Huygens: wavefronts change speed at an interface, so wavelength and direction change; frequency stays constant.

  • When two waves travel through different materials, their phase difference can change because wavelength depends on index of refraction—this drives interference patterns.

Huygens’ principle#

Huygens’ principle (1678, Christian Huygens): All points on a wavefront serve as point sources of spherical secondary wavelets. After a time \(t\), the new position of the wavefront is the surface tangent to these wavelets.

  • Plane wave in vacuum: A plane wavefront (e.g., plane \(ab\)) propagates by having points on it emit spherical wavelets. At time \(\Delta t\), each wavelet has radius \(c\,\Delta t\). The new wavefront is the plane tangent to them—parallel to the original, displaced by \(c\,\Delta t\).

  • This construction predicts reflection and refraction without Maxwell’s equations; it gives physical meaning to the index of refraction.

Wave vs. ray

Geometrical optics (rays) is valid when features are much larger than the wavelength. Interference requires the wave picture—phase matters.

Refraction and index of refraction#

When a plane wave crosses an interface at an angle, the part of the wavefront that enters first slows down (or speeds up). The wavefront bends, changing the direction of propagation.

Index of refraction \(n\):

(267)#\[ n = \frac{c}{v} \]

where \(c\) is the speed of light in vacuum and \(v\) is the speed in the medium. Vacuum: \(n = 1\); air: \(n \approx 1\); glass: \(n \approx 1.5\).

Snell’s law (from Huygens geometry):

(268)#\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \]

  • Wavelengths in the two media satisfy \(\lambda_1/v_1 = \lambda_2/v_2\) (same time for one wavelength to cross).

  • The refracted wavefront is tangent to Huygens wavelets in the second medium; the geometry yields Snell’s law.

Wavelength in a medium#

The wavelength in a medium depends on the index of refraction:

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

where \(\lambda\) is the wavelength in vacuum. Since \(v = \lambda f\) and \(f\) is the same in vacuum and in the medium:

(270)#\[ f = \frac{c}{\lambda} = \frac{v}{\lambda_n} \quad \Rightarrow \quad \lambda_n = \frac{v}{c}\lambda = \frac{\lambda}{n} \]

Important

The frequency of light does not change when it enters a medium. Only the speed and wavelength change.

Phase difference from different materials#

When two waves (same wavelength \(\lambda\) in vacuum) travel through different materials of length \(L\) with indices \(n_1\) and \(n_2\), they accumulate different phase.

Number of wavelengths in length \(L\):

(271)#\[ N = \frac{L}{\lambda_n} = \frac{nL}{\lambda} \]

For two materials of the same length \(L\):

(272)#\[ N_1 = \frac{n_1 L}{\lambda}, \qquad N_2 = \frac{n_2 L}{\lambda} \]

Phase difference (in wavelengths):

(273)#\[ |N_2 - N_1| = \frac{|n_2 - n_1| L}{\lambda} \]

Effective phase difference: Only the fractional part matters. A shift of \(m\) full wavelengths (\(m\) integer) leaves the waves in the same relative phase.

Effective phase difference

Interference

0 or 1 wavelength

Fully constructive (brightness)

0.5 wavelength

Fully destructive (darkness)

Other

Intermediate (dim)

Path length difference

Same logic as section 17-3 (sound): if two waves travel through paths of different lengths \(\Delta L\), the phase difference is \(\Delta L/\lambda\) (in wavelengths). For light, combine path length difference with phase from different materials.

Example: Phase difference from different materials

Two light waves (wavelength 550.0 nm in vacuum) are in phase and have equal amplitudes. One travels through air (\(n_1 = 1.000\)); the other through a plastic layer of thickness \(L = 2.600\) mm and \(n_2 = 1.600\). Find the phase difference in wavelengths and the effective phase difference. What type of interference occurs if they meet at a common point?

Solution: Use Eq. (273): \(|N_2 - N_1| = (n_2 - n_1)L/\lambda = (1.600 - 1.000)(2.600\times10^{-3})/(550.0\times10^{-9}) \approx 2.84\) wavelengths. Effective phase difference = fractional part = 0.84 wavelength. This is between 0.5 (destructive) and 1.0 (constructive), closer to 1.0 → intermediate brightness, closer to fully constructive.

Rainbows and optical interference#

Rainbows and supernumerary arcs (faint secondary arcs below the primary rainbow) are natural examples of optical interference.

  • Light enters a raindrop along different paths; waves emerging at a given angle have traveled different distances inside the drop.

  • At certain angles, waves emerge in phase → constructive interference → bright color.

  • At other angles, waves have different phases → dim or washed out.

  • Butterfly wings (e.g., Morpho blue) and anti-counterfeit inks use similar interference for color shifting with viewing angle.

Summary#

  • Huygens’ principle: All points on a wavefront emit spherical secondary wavelets; new wavefront = surface tangent to them.

  • Refraction: \(n = c/v\); Snell’s law \(n_1\sin\theta_1 = n_2\sin\theta_2\); frequency unchanged, wavelength \(\lambda_n = \lambda/n\).

  • Phase difference: Two waves through materials of length \(L\) with indices \(n_1\), \(n_2\) differ by \(|n_2 - n_1|L/\lambda\) wavelengths; use fractional part for interference.

  • Rainbows and supernumeraries: Optical interference from path differences in raindrops.