Summary

General Principles

Huygens’ principle says that each point on a wave front is the source of a spherical wavelet. The wave front at a later time is tangent to all the wavelets.

The figure illustrates a diagrammatic representation of Huygens principle.

Full Alternative Text

Diffraction is the spreading of a wave after it passes through an opening.

Constructive and destructive interference are due to the overlap of two or more waves as they spread behind openings.

The figure consists of two parts that illustrates the diffraction pattern of a light wave after passing through a single slit and interference pattern of a light wave after passing through a double slit.

Important Concepts

The wave model of light considers light to be a wave propagating through space. Diffraction and interference are important.

The ray model of light considers light to travel in straight lines like little particles. Diffraction and interference are not important.

Diffraction is important when the width of the diffraction pattern of an aperture equals or exceeds the size of the aperture.

For a circular aperture, the crossover between the ray and wave models occurs for an opening of diameter $D_{c} \approx \sqrt{2 λ L}$ .

In practice, $D_{c} \approx 1 mm$ for visible light. Thus

Use the wave model when light passes through openings $< 1 mm$ in size. Diffraction effects are usually important.
Use the ray model when light passes through openings $> 1 mm$ in size. Diffraction is usually not important.

Applications

Diffraction

Single slit of width $a$ :

A bright central maximum of width

w = \frac{2 λ L}{a}

is flanked by weaker secondary maxima.

Dark fringes are located at angles such that

\begin{matrix} a sin θ_{p} = p λ & p = 1, 2, 3, \dots \end{matrix}

If $λ / a ≪ 1$ , then from the small-angle approximation

\begin{matrix} θ_{p} = \frac{p λ}{a} & y_{p} = \frac{p λ L}{a} \end{matrix}

Interference due to wave-front division

Waves overlap as they spread out behind slits. Constructive interference occurs along antinodal lines. Bright fringes are seen where the antinodal lines intersect the viewing screen.

A photograph of the diffraction pattern obtained from a double slit on a screen. The diffraction pattern consists of alternating bright and dark fringes. The bright fringes have a larger width compared to the dark fringes.

Double slit with separation $d$ :

Equally spaced bright fringes are located at

\begin{matrix} θ_{m} = \frac{m λ}{d} & y_{m} = \frac{m λ L}{d} & m = 0, 1, 2, \dots \end{matrix}

The fringe spacing is $Δ y = \frac{λ L}{d}$

Diffraction grating with slit spacing $d$ :

A photograph of the diffraction pattern obtained from a single slit on a screen. The diffraction pattern consists of alternating bright and dark fringes. The dark fringes have a larger width compared to the bright fringes.

Very bright and narrow fringes are located at angles and positions

\begin{matrix} d sin θ_{m} = m λ & y_{m} = L tan θ_{m} \end{matrix}

Circular aperture of diameter $D$ :

A photograph of the diffraction pattern obtained from a circular opening on a screen. The diffraction pattern consists of a bright circle at the center surrounded by alternating bright and dark concentric circles of reduced thickness and intensity.

A bright central maximum of diameter

w = \frac{2.44 λ L}{D}

is surrounded by circular secondary maxima.

The first dark fringe is located at

\begin{matrix} θ_{1} = \frac{1.22 λ}{D} & y_{1} = \frac{1.22 λ L}{D} \end{matrix}

For an aperture of any shape, a smaller opening causes a more rapid spreading of the wave behind the opening.

Interference due to amplitude division

An interferometer divides a wave, lets the two waves travel different paths, then recombines them. Interference is constructive if one wave travels an integer number of wavelengths more or less than the other wave. The difference can be due to an actual path-length difference or to a different index of refraction.

Michelson interferometer

The number of bright-dark-bright fringe shifts as mirror $M_{2}$ moves distance $Δ L_{2}$ is

Δ m = \frac{Δ L_{2}}{λ / 2}