`color{blue}{star}` INTRODUCTION

`color{blue}{star}` HUYGENS PRINCIPLE

`color{blue}{star}` HUYGENS PRINCIPLE

`color{blue} ✍️`In 1637 Descartes gave the corpuscular model of light and derived Snell’s law. It explained the laws of reflection and refraction of light at an interface.

`color{blue} ✍️`The corpuscular model predicted that if the ray of light (on refraction) bends towards the normal then the speed of light would be greater in the second medium.

`color{blue} ✍️`The wave theory was not readily accepted primarily because of Newton’s authority and also because light could travel through vacuum and it was felt that a wave would always require a medium to propagate from one point to the other.

`color{blue} ✍️`However, when Thomas Young performed his famous interference experiment in 1801, it was firmly established that light is indeed a wave phenomenon. The wavelength of visible light was measured and found to be extremely small; for example, the wavelength of yellow light is about `0.5 μm.`

`color{blue} ✍️`Because of the smallness of the wavelength of visible light (in comparison to the dimensions of typical mirrors and lenses), light can be assumed to approximately travel in straight lines.

`color{blue} ✍️`Thus, around the middle of the nineteenth century, the wave theory seemed to be very well established. The only major difficulty was that since it was thought that a wave required a medium for its propagation, how could light waves propagate through vacuum.

`color{blue} ✍️`This was explained when Maxwell put forward his famous electromagnetic theory of light. Maxwell had developed a set of equations describing the laws of electricity and magnetism and using these equations he derived what is known as the wave equation from which he predicted the existence of electromagnetic waves*.

`color{blue} ✍️`From the wave equation, Maxwell could calculate the speed of electromagnetic waves in free space and he found that the theoretical value was very close to the measured value of speed of light. From this, he propounded that light must be an electromagnetic wave.

`color{blue} ✍️`Thus, according to Maxwell, light waves are associated with changing electric and magnetic fields; changing electric field produces a time and space varying magnetic field and a changing magnetic field produces a time and space varying electric field.

`color{blue} ✍️`The changing electric and magnetic fields result in the propagation of electromagnetic waves (or light waves) even in vacuum.

`color{blue} ✍️`The corpuscular model predicted that if the ray of light (on refraction) bends towards the normal then the speed of light would be greater in the second medium.

`color{blue} ✍️`The wave theory was not readily accepted primarily because of Newton’s authority and also because light could travel through vacuum and it was felt that a wave would always require a medium to propagate from one point to the other.

`color{blue} ✍️`However, when Thomas Young performed his famous interference experiment in 1801, it was firmly established that light is indeed a wave phenomenon. The wavelength of visible light was measured and found to be extremely small; for example, the wavelength of yellow light is about `0.5 μm.`

`color{blue} ✍️`Because of the smallness of the wavelength of visible light (in comparison to the dimensions of typical mirrors and lenses), light can be assumed to approximately travel in straight lines.

`color{blue} ✍️`Thus, around the middle of the nineteenth century, the wave theory seemed to be very well established. The only major difficulty was that since it was thought that a wave required a medium for its propagation, how could light waves propagate through vacuum.

`color{blue} ✍️`This was explained when Maxwell put forward his famous electromagnetic theory of light. Maxwell had developed a set of equations describing the laws of electricity and magnetism and using these equations he derived what is known as the wave equation from which he predicted the existence of electromagnetic waves*.

`color{blue} ✍️`From the wave equation, Maxwell could calculate the speed of electromagnetic waves in free space and he found that the theoretical value was very close to the measured value of speed of light. From this, he propounded that light must be an electromagnetic wave.

`color{blue} ✍️`Thus, according to Maxwell, light waves are associated with changing electric and magnetic fields; changing electric field produces a time and space varying magnetic field and a changing magnetic field produces a time and space varying electric field.

`color{blue} ✍️`The changing electric and magnetic fields result in the propagation of electromagnetic waves (or light waves) even in vacuum.

`color{blue} ✍️`We would first define a `"wavefront"` : when we drop a small stone on a calm pool of water, waves spread out from the point of impact. Every point on the surface starts oscillating with time. At any instant, a photograph of the surface would show circular rings on which the disturbance is maximum.

`color{blue} ✍️`Clearly, all points on such a circle are oscillating in phase because they are at the same distance from the source. Such a locus of points, which oscillate in phase is called a `"wavefront."` Thus a wavefront is defined as a surface of constant phase.

`color{blue} ✍️`The speed with which the wavefront moves outwards from the source is called the speed of the wave. The energy of the wave travels in a direction perpendicular to the wavefront.

`color{blue} ✍️`If we have a point source emitting waves uniformly in all directions, then the locus of points which have the same amplitude and vibrate in the same phase are spheres and we have what is known as a spherical wave as shown in Fig. 10.1(a).

`color{blue} ✍️`At a large distance from the source, a small portion of the sphere can be considered as a plane and we have what is known as a plane wave [Fig. 10.1(b)].

`color{blue} ✍️`Now, if we know the shape of the wavefront at `t = 0,` then Huygens principle allows us to determine the shape of the wavefront at a later time `τ.`

`color{blue} ✍️`Thus, Huygens principle is essentially a geometrical construction, which given the shape of the wafefront at any time allows us to determine the shape of the wavefront at a later time.

`color{blue} ✍️`Let us consider a diverging wave and let `F_1F_2` represent a portion of the spherical wavefront at t = 0 (Fig. 10.2).

`color{blue} ✍️`Now, according to Huygens principle, each point of the wavefront is the source of a secondary disturbance and the wavelets emanating from these points spread out in all directions with the speed of the wave.

`color{blue} ✍️`These wavelets emanating from the wavefront are usually referred to as secondary wavelets and if we draw a common tangent to all these spheres, we obtain the new position of the wavefront at a later time.

`color{blue} ✍️`Thus, if we wish to determine the shape of the wavefront at `t = τ`, we draw spheres of radius vτ from each point on the spherical wavefront where v represents the speed of the waves in the medium. If we now draw a common tangent to all these spheres, we obtain the new position of the wavefront at `t = τ`. The new wavefront shown as `G_1G_2` in Fig. 10.2 is again spherical with point `O` as the centre.

`color{blue} ✍️`The above model has one shortcoming: we also have a backwave which is shown as `D_1D_2` in Fig. 10.2.

Huygens argued that the amplitude of the secondary wavelets is maximum in the forward direction and zero in the backward direction; by making this adhoc assumption, Huygens could explain the absence of the backwave. However, this adhoc assumption is not satisfactory and the absence of the backwave is really justified from more rigorous wave theory.

`color{blue} ✍️`In a similar manner, we can use Huygens principle to determine the shape of the wavefront for a plane wave propagating through a medium (Fig. 10.3).

`color{blue} ✍️`Clearly, all points on such a circle are oscillating in phase because they are at the same distance from the source. Such a locus of points, which oscillate in phase is called a `"wavefront."` Thus a wavefront is defined as a surface of constant phase.

`color{blue} ✍️`The speed with which the wavefront moves outwards from the source is called the speed of the wave. The energy of the wave travels in a direction perpendicular to the wavefront.

`color{blue} ✍️`If we have a point source emitting waves uniformly in all directions, then the locus of points which have the same amplitude and vibrate in the same phase are spheres and we have what is known as a spherical wave as shown in Fig. 10.1(a).

`color{blue} ✍️`At a large distance from the source, a small portion of the sphere can be considered as a plane and we have what is known as a plane wave [Fig. 10.1(b)].

`color{blue} ✍️`Now, if we know the shape of the wavefront at `t = 0,` then Huygens principle allows us to determine the shape of the wavefront at a later time `τ.`

`color{blue} ✍️`Thus, Huygens principle is essentially a geometrical construction, which given the shape of the wafefront at any time allows us to determine the shape of the wavefront at a later time.

`color{blue} ✍️`Let us consider a diverging wave and let `F_1F_2` represent a portion of the spherical wavefront at t = 0 (Fig. 10.2).

`color{blue} ✍️`Now, according to Huygens principle, each point of the wavefront is the source of a secondary disturbance and the wavelets emanating from these points spread out in all directions with the speed of the wave.

`color{blue} ✍️`These wavelets emanating from the wavefront are usually referred to as secondary wavelets and if we draw a common tangent to all these spheres, we obtain the new position of the wavefront at a later time.

`color{blue} ✍️`Thus, if we wish to determine the shape of the wavefront at `t = τ`, we draw spheres of radius vτ from each point on the spherical wavefront where v represents the speed of the waves in the medium. If we now draw a common tangent to all these spheres, we obtain the new position of the wavefront at `t = τ`. The new wavefront shown as `G_1G_2` in Fig. 10.2 is again spherical with point `O` as the centre.

`color{blue} ✍️`The above model has one shortcoming: we also have a backwave which is shown as `D_1D_2` in Fig. 10.2.

Huygens argued that the amplitude of the secondary wavelets is maximum in the forward direction and zero in the backward direction; by making this adhoc assumption, Huygens could explain the absence of the backwave. However, this adhoc assumption is not satisfactory and the absence of the backwave is really justified from more rigorous wave theory.

`color{blue} ✍️`In a similar manner, we can use Huygens principle to determine the shape of the wavefront for a plane wave propagating through a medium (Fig. 10.3).