WAVE OPTICS

                   1. Introduction

As we know that a very small particle which is in a state of motion can show two nature -particle and wave nature. So, in this topic we will know about the wave nature of matter. 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. 

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. This corpuscular model of light was further developed by Isaac Newton in his famous book entitled OPTICS and because of 

the tremendous popularity of this book, the corpuscular model is very often attributed to Newton. In 1678, the Dutch physicist Christian Huygens put forward the  wave theory of light – it is this wave model of light. As we will see, the wave model could satisfactorily explain the phenomena of reflection and refraction; however, it predicted that on refraction if the wave bends towards the normal then the speed of light would be less in the second medium. This is in contradiction to the prediction made by using the corpuscular model of light. It was much later confirmed by experiments where it was shown that the speed of light in water is less than the speed in air confirming the prediction of the 

wave model; Foucault carried out this experiment in 1850. 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. 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 0.6 µ m (micro metre). 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. This is the field of geometrical optics. Indeed, the branch of optics in which one completely neglects the finiteness of the wavelength is called geometrical optics and a ray is defined as the path of energy 

propagation in the limit of wavelength tending to zero. After the interference experiment of Young in 1801, for the next 40 years or so, many experiments were carried out involving the 

interference and diffraction of light waves; these experiments could only be satisfactorily explained by assuming a wave model of light. 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. 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. 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 electro magnetic wave. 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. The changing electric and magnetic fields result inthe propagation of 

electromagnetic waves (or light waves) even in vacuum. The original formulation of the Huygens principle and derive the laws of reflection and refraction. The phenomenon of interference which is based on the principle of superposition. The phenomenon of diffraction which is based on Huygens- Fresnel principle. The phenomenon of polarisation which is based on the fact that the light waves are transverse electromagnetic waves.

 

          2. H U Y GENS PRINCIPLE

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. 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. 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. 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. 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. 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. Thus, Huygens principle is essentially a geometrical construction, which given the shape of the wavefront at any time allows us to determine 

the shape of the wavefront at a later time. Let us consider a diverging wave and let F1 F 2 represent a portion of the spherical wavefront at t = 0. Now, according to Huygens principle, each point of the wavefront is the source of a secondary emanating disturbance and the wavelets  from these points spread out in all directions with the speed of the wave. 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. Thus, if we wish to determine the shape of the wavefront at t = τ, we draw spheres of radius v τ 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 G1 G2 in Fig. 10.2 is again

spherical with point O as the centre.The model has one shortcoming: we also have a back wave which is shown as D1 D2 . Huygens argued that the amplitude of the secondary wavelets is maximum in the forward direction and zero in the backward direction; by making this a d hoc assumption, Huygens could explain the absence of the back wave. However, this a d hoc assumption is not satisfactory and the absence of the back wave is really justified from more rigorous wave theory. In a similar manner, we can use Huygens principle to determine the

shape of the wavefront for a plane wave propagating through a medium.

 

 3. The doppler effect

We should mention here that one should be careful in constructing the wavefronts if the source (or the observer) is moving. For example, if there is no medium and the source moves away from the observer, then later wavefronts have to travel a greater distance to reach the observer and hence take a longer time. The time taken between the arrival of two successive wavefronts is hence longer at the observer than it is at the source. Thus, when the source moves away from the observer the frequency as measured by the source will be smaller. This is known as the Doppler effect. Astronomers call the increase in wavelength due to doppler effect as red shift since a wavelength in the middle of the visible region of the spectrum moves towards the red end of the spectrum. When waves are received from a source moving towards the observer, there is an apparent decrease in wavelength, this is referred to as blue shift.For velocities small compared to the speed of light, we can use the same formulae which we use for sound waves. The fractional change in frequency ∆ν/ν is given by –v radial/c, where v radial is the component of the source velocity along the line joining the observer to the source relative to the observer; v radial is considered positive 

when the source moves away from the observer.

The formula is valid only when the speed of the source is small compared to that of light. A more accurate formula for the Doppler effect which is valid even when the speeds are close to that of light, requires the use of Einstein’s special theory of relativity. The Doppler effect for light is very important in astronomy. It is the basis for the measurements of the radial velocities of distant galaxies.

 

4. COHERENT AND INCOHERENT ADDITION OF WAVES

The interference pattern produced by the superposition of two waves. Indeed the entire field of interference is based on the super position principle according to which at a particular point in the medium, the resultant displacement produced by a number of waves is the vector sum of the displacements produced by each of the waves. Consider two needles S1 and S2 moving periodically up and down in an identical fashion in a trough of water. They produce two water waves, and at a particular point, the phase difference between the displacements produced by each of the waves does not change with time; when this happens the two sources are said to be coherent. The position of crests (solid circles) and troughs (dashed circles) at a given instant of time. Consider a point P for which S1 P = S2 P 

Since the distances S1 P and S2 P are equal, waves from S1 and S2 will take the same time to travel to the point P and waves that emanate 

from S1 and S2 in phase will also arrive, at the point P, in phase. Thus, if the displacement produced by the source S1 at the point P is given by  y 1 = a cos ω t 

then, the displacement produced by the source S2 (at the point P) will also be given by 

y 2 = a cos ω t

Thus, the resultant of displacement at P would be given by    y = y 1 + y 2 = 2 a cos ω t 

Since the intensity is proportional to the square of the amplitude, the resultant intensity will be given by I = 4 I 0 

where I 0 represents the intensity produced by each one of the individual sources; I 0 is proportional to a2. In fact at any point on the perpendicular bisector of S1 S2, the intensity will be 4I 0. The two sources are said to interfere constructively and we have what is referred to as constructive interference.

 

5. INTERFERENCE OF LIGHT WAVES AND YOUNG’S EXPERIMENT

If we use two sodium lamps illuminating two pinholes we will not observe any interference fringes. This is because of the fact that the light wave emitted from an ordinary source (like a sodium lamp) undergoes abrupt phase changes in times of the order of 10–10 seconds. Thus

the light waves coming out from two independent  sources of light will not have any fixed phase relationship and would be incoherent, when this happens, as discussed in the previous section, the intensities on the screen will add up. The British physicist Thomas Young used an ingenious technique to “lock” the phases of the waves emanating from S1 and S2. He made two pinholes S1 and S2 (very close to each other) on an opaque screen. These were illuminated by another pinholes that was in turn, lit by a bright source. Light waves spread out from S and fall on both S1 and S2. S1 and S2 then behave like two coherent sources because light waves coming out from S1 and S2 are derived from the same original source and any abrupt phase change in S will manifest in exactly similar phase changes in the light coming out from S1 and S2. Thus, the two sources S1 and S2 will be locked in phase; i.e., they will be coherent like the two vibrating needle in our water wave example. Thus spherical waves emanating from S1 and S2 will produce interference fringes on the screen GG′. The  positions of maximum and minimum intensities can be calculated by using the analysis where we had shown that for an 

arbitrary point P on the line GG′. Obviously, the central point O will be bright because S1 O = S2 O and it will correspond  to n = 0. If we consider the line perpendicular to the plane of the paper and passing through O [i.e., along the y-axis] then all points on this line will be equidistant from S1 and S2 and we will have a bright central fringe which is a straight line. In order to determine the shape of the interference pattern on the screen we note that a particular fringe would correspond to the locus of points with a constant value of S2 P – S1 P. Whenever this constant is an integral multiple of λ, the fringe will be bright and whenever it is an odd integral multiple of λ/2 it will be a dark fringe. Now, the locus of the point P lying in the x-y plane such that S2 P – S1 P (= ∆) is a constant, is a hyperbola. Thus the fringe pattern will strictly be a hyperbola; however, 

if the distance D is very large compared to the fringe width, the fringes will be very nearly straight lines. 

The wave nature of light was demonstrated convincingly for the first time in 1801 by Thomas Young by a wonderfully simple experiment. He let a ray of sunlight into a dark room, placed a dark screen in front of it, pierced with two small pinholes, and beyond this, at some distance, a white screen. He then saw two darkish lines at both sides of a bright line, which gave him sufficient encouragement to repeat the experiment, this time with spirit flame as light source, with a little salt in it to produce the bright yellow sodium light. This time he saw a number of dark lines, regularly spaced; the first clear proof that light added to light can produce darkness. This phenomenon is called interference. Thomas Young had expected it because he believed in the wave theory of light.

                  6. DIFFRACTION

If we look clearly at the shadow cast by an opaque object, close to the region of geometrical shadow, there are alternate dark and bright regions just like in interference. This happens due to the phenomenon of diffraction. Diffraction is a general characteristic exhibited by all types of waves, be it sound waves, light waves, water waves or matter waves. Since the wavelength of light is much smaller than the dimensions of most obstacles; we do not encounter diffraction effects of light in everyday observations. However, the finite resolution of our eye or of optical instruments such as telescopes or microscopes is limited due to the phenomenon of diffraction. Indeed the colours that you see when a CD is viewed is due to diffraction effects. We will now discuss the phenomenon of diffraction.