Additional: Wavefronts and Rays

Types of Wavefronts (Spherical, Plane, Cylindrical)

A wavefront is defined as the locus of all points in a medium (or space) that are in the same phase of oscillation at a given instant. Wavefronts are always perpendicular to the direction of propagation of the wave (represented by rays). As a wave propagates, the wavefront moves through the medium or space.

The shape of the wavefront depends on the shape of the source of the waves and the properties of the medium (whether it is homogeneous and isotropic). Here are the common types of wavefronts:

Spherical Wavefront

A spherical wavefront is produced by a point source of waves radiating uniformly in all directions in a homogeneous and isotropic medium. As the waves propagate outwards from the point source, all points at the same distance from the source will be in the same phase, forming a spherical surface. The radius of the spherical wavefront increases as the wave propagates outwards.

The rays associated with a spherical wavefront are radial lines emanating outwards from the point source.

(Image Placeholder: A point source in the center. Show concentric circles or spheres around the source representing spherical wavefronts. Show radial lines (rays) emanating outwards from the source, perpendicular to the wavefronts.)

As a spherical wavefront travels very far from the point source, the curvature of the wavefront becomes very small, and a small portion of the distant spherical wavefront can be approximated as a plane wavefront.

Plane Wavefront

A plane wavefront is a wavefront that is a flat surface (a plane). Plane wavefronts are produced by a distant point source (where a small part of a large spherical wavefront appears flat) or by a line source of infinite length. Plane wavefronts are also produced when a spherical wave passes through a converging lens or reflects from a concave mirror, adjusted to make the emergent rays parallel.

The rays associated with a plane wavefront are parallel lines, perpendicular to the plane wavefront.

(Image Placeholder: Several parallel lines representing plane wavefronts moving in a specific direction (arrow). Show parallel lines (rays) perpendicular to the wavefronts, pointing in the direction of propagation.)

Treating waves as plane waves simplifies many analyses in wave optics.

Cylindrical Wavefront

A cylindrical wavefront is produced by a linear source of waves (like a long, thin slit or a linear antenna) radiating uniformly along its length in a homogeneous and isotropic medium. All points at the same perpendicular distance from the line source will be in the same phase, forming a cylindrical surface whose axis coincides with the line source.

The rays associated with a cylindrical wavefront are lines radiating outwards from the line source in a plane perpendicular to the line source.

(Image Placeholder: A line representing the source. Show concentric cylinders around the line representing cylindrical wavefronts. Show rays emanating radially outwards from the line source in planes perpendicular to the line source.)

These are the basic shapes of wavefronts encountered in optics. Understanding the type of wavefront helps in applying Huygens' principle and analysing wave propagation.

Principle of Superposition applied to Wavefronts

The Principle of Superposition states that when two or more waves overlap, the resultant disturbance is the vector sum of the individual disturbances. This principle applies to the displacements or field values at points in the medium or space. However, the principle can also be applied conceptually to wavefronts, although it is not as straightforward as simply adding wavefront shapes.

Superposition in terms of Wavefronts

When two waves interact, their wavefronts influence each other. The principle of superposition dictates that the disturbance at any point is the sum of the disturbances arriving from the individual wavefronts of each wave. The resulting interference pattern (like bright and dark fringes) arises from the superposition of the individual waves at different points in space, which in turn relates to the relative phases of the wavelets arriving at those points.

Consider two sets of wavefronts from two coherent sources. At points where a crest from one wave meets a crest from the other (or trough meets trough), the waves are in phase, and constructive interference occurs, resulting in maximum amplitude. On wavefront diagrams, this corresponds to where wavefronts of the same phase from both sources intersect. At points where a crest from one wave meets a trough from the other, the waves are out of phase, and destructive interference occurs, resulting in minimum amplitude. On wavefront diagrams, this corresponds to where wavefronts that are out of phase from both sources intersect.

The interference pattern is essentially the spatial distribution of points where constructive and destructive interference occurs, resulting from the superposition of the wave amplitudes based on their relative phases. The shape of the resultant wavefront in the region of interference is not simply the sum of the shapes of the individual wavefronts, but rather describes the surface of constant phase of the resultant wave, which is formed by the superposition process.

For example, in Young's double-slit experiment, spherical wavelets from the two slits ($S_1$ and $S_2$) spread out and overlap. On the screen, the interference pattern (alternating bright and dark fringes) is formed by the constructive and destructive superposition of these wavelets at different points. The points of constructive interference occur where the path difference from $S_1$ and $S_2$ is an integer multiple of the wavelength, meaning the wavefronts from $S_1$ and $S_2$ arrive in phase. The bright fringes represent lines (or surfaces in 3D) where the resultant amplitude is maximum due to this in-phase superposition.

Huygens' Principle and Superposition

Huygens' principle itself implicitly relies on superposition. When the new wavefront is constructed as the envelope of secondary wavelets, this construction is valid because the wavelets from different points on the preceding wavefront interfere constructively to form the new wavefront in the forward direction. The wavelets interfering in other directions are assumed to cancel out due to destructive interference (as explained by Fresnel). So, the propagation described by Huygens' principle is a result of the constructive superposition of secondary wavelets along their common tangent.

In summary, while we don't typically draw the resultant wavefront by simply adding the individual wavefronts, the principle of superposition is the underlying physical principle that governs how waves from multiple sources (or multiple points on a wavefront) combine to produce the observed pattern or the propagation of the wavefront itself.