# Huygens principle of secondary wavelets

## Huygens principle of Wave Theory

Huygens Principle helps us to find the position of a given wavefront at any instant if we know its present position. A wavefront is defined as the continuous locus of all particles of a medium vibrating in the same phase.

For example, when a stone is dropped into a Pound of water, circular ripples are produced on the surface of the water. These ripples spread outward. Similarly, a point source of light emits a spherical wave with the source at the center, and these waves spread outward in all directions.

if the medium is isotropic, then the disturbance after time t will lie on a sphere of radius vt, where v is the velocity of the wave in the medium. Since the disturbance from the source moves in all directions, therefore, the particles of the medium at a distance vt from the center of the source will vibrate in the same phase. Any surface passing through these points or particles is called a wavefront. The wavefront is spherical as shown in figure 1.

At a very large distance from the source, any small part of the spherical wavefront appears plane, and part of the spherical wavefront is called plane wavefront as shown in figure 2.

According to Huygens principle of secondary wavelets :
All the points on the primary wavefront can be regarded as a point of new disturbance or source. Each point gives rise to secondary wavelets which spread out in all directions with the velocity of the wave.

The surface obtained by joining the tangents drawn on each secondary wavelet will be the new position of the wavefront after a time t.

This can be understood with the following example. Consider a point source S which gives the spherical wavefront AB at the instant t = 0. According to Huygens principle, every point on this wavefront is considered as the center of secondary wavelets. Since the medium is homogeneous and isotropic, the medium has the same property at all points and the speed of propagation of the wave is the same in all the directions, so the light wave travels an equal distance in equal time in all directions.
After time t, each secondary wavelet will travel a distance equal to vt. By taking every point as the center, draw a sphere of radius equal to vt. Now draw the tangents on every sphere and join them by freehand. Thus, we get the curve CD which is again a sphere with S as its Centre. Hence the new position of the wavefront AB after time t is CD.

## Drawbacks of Huygens principle

Huygens principle suffers from two serious drawbacks:
Why is the wavefront EF which shows backward propagation of light not considered?
Why is only the effective portion of the wavelets are touched by the enveloping surface and a large portion untouched by the surface is taken as ineffective?
The explanation for the first drawback was given on the basis of Stokes’s law. According to this, the amplitude and hence the intensity due to the secondary wavelets depend upon the obliquity ( 1+ cosθ ), where θ is the angle between the line drawn from the origin of the secondary wavelets to the point where the intensity is to be determined, and the normal to the wavefront drawn outwards.

Thus, along the forward direction or along the direction of the normal, θ=0, and hence the intensity has some specific value. Along a direction opposite to the normal, along the backward direction, θ=180 and the obliquity factor ( 1+cosθ ) = ( 1+cos 180 ) = 1 – 1 = 0. Hence the resultant intensity in the backward direction is zero and, therefore, back wavefront EF does not exist.

The second drawback can be explained on the basis of the principle of superposition. Due to the superposition of secondary wavelets, their major portions cancel Each Other effect by interference. Hence only the effective portions cancel Each Other effect by interference. Therefore, the only effective portions left are those where these secondary wavelets do not superpose each other.

## Diffraction

The phenomenon by virtue of which the light bends into the region of Geometrical shadow and there is a periodic variation in the intensity of light outside the Geometrical shadow is called diffraction of light.

To understand the diffraction more clearly, let us consider a monochromatic source (S) of light placed near the narrow slit AB as shown in the figure. According to geometrical optics, the region between P and Q on the screen should be uniformly illuminated and the portion above point P and below point Q should be completely dark. The portion above point P and below point Q is known as the geometrical shadow.

However, observations taken minutely show that the region between P and Q is not uniformly illuminated and there is also some intensity of light inside the geometrical Shadow if the aperture of slit AB is small as compared to the wavelength of light used. It is observed that if the aperture AB is made sufficiently narrow, the reason PQ is marked by equidistant dark and bright fringes where there exist a few and unequally spaced fringes above P and below Q (but near these points), in the regions of geometrical shadow.

Thus bending of lightwave when it passes through a narrow slit is known as the phenomena of diffraction and the intensity distribution on the screen is known as the diffraction pattern.

## The phenomena of diffraction are classified into two categories

1. Fresnel class of diffraction.
2. Fraunhofer class of diffraction.

Fresnel class of diffraction.

In the fresnel class of diffraction, the source and the screen are placed at a finite distance from the slit. This is also known as near field diffraction. In this case, incident wavefronts and diffracted wavefronts are taken as spherical wavefronts.

Fraunhofer class of diffraction.

In Fraunhofer diffraction, the source and the screen are placed at an infinite distance from the slit. This can be achieved by placing the screen on the focal plane of another convex lens as shown in fig. Since the source and the screen are infinitely distant apart, so this is also known as far-field diffraction. In Fraunhofer diffraction, incident and reflected wavefronts are planes.

## Huygens Fresnel diffraction theory

According to Huygens principle of wave theory, the future position of the wavefront may be derived from a past position by considering every point of the wavefront to be a source of secondary waves. The progress of the wavefront can often be described by ray optics. We now turn to situations where rays provide totally inadequate description.

Diffraction effects occur when part of the wavefront is removed by an obstacle in an otherwise infinite wavefront, or when all but a part of the wavefront is removed by an aperture or stop. The importance of the diffraction effect depends on the size of the obstacle or aperture compared with the wavelength.

The wavefront passing through the aperture is initially divided into equal elements small compared with the wavefront. Each such element is considered to be a source of secondary Huygens wave. To determine the amplitude at any point beyond the aperture these waves are summed. The phase of wavelets is behind that of the portion of the original wavefront emitting it by an amount determined by its distance and the wavelength λ.

This summation of Huygens wavelets taking account of their phase is referred to as Huygens Fresnel diffraction theory. It was Huygens who first saw the new wavefront as the envelope of the wavelets. Fresnel contributed the essential idea of the interference of the Huygens wavelets.

Suppose W is an incident wavefront and AB is the aperture. P is a point where we want to study the effect. C is any point in the aperture as shown in the figure.

let CP = L

The amplitude and phase at P may be considered to be the sum of Huygens wavelets from points such as C. The phase at P of the wavelet originating at C is behind that at B by 2πl/λ radians.

To calculate the amplitude at P all such contributions from each equal element have to be taken. Such a process first involves the calculation of the phase of each contribution, relative to some phase reference, often the face of the contribution from the center of the aperture. Then follows the summation of all such contributions either by an integral, a series, or possibly by the application of the trigonometry.