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Nature of Light and Huygens’ Wave Theory Paper: Optics Lesson: Nature of Light and Huygens’ Wave Theory Author: Dr. D. V. Chopra College/Department: Associate Professor (Retired), Department of Physics and Electronics, Rajdhani College, University of Delhi Institute of Life Long Learning, University of Delhi Page 1 Nature of Light and Huygens’ Wave Theory Objective: After studying this chapter you should: 1. Be able to know about light and its nature 2. Be able to study various theories regarding nature of light 3. Be able to understand Huygens ‘principle of wave propagation of light and its application to derive laws of reflection and refraction of light 4. Be able to derive thin lens formula on the basis of Huygens’ wave theory 5. Be able to relate Huygens’ wave theory to Fermat’s principle 6. Be able to solve problems involving nature of light and Huygens’ wave theory 1. What is light and its nature ? So far, we have considered the theorems which may be deduced from the axioms formulated for rays of light and the optical instruments based on them. Such a study is made in Geometrical Optics which is developed without going into detail with regard to the nature of light. We shall see that the knowledge of the nature of light has an important bearing on the design of optical instruments, e.g., resolving power of a telescope using the phenomenon of diffraction which is based on the wave nature of light. In geometrical optics, it is assumed that light travels in straight lines and we assume particle nature of light. We are going to discuss theories of light; that is, answers to the question: What is light; What is nature of light ? The answers to the question and the experimental evidence bearing on it are called Physical Optics to distinguish it from the preceding geometrical optics. In physical optics, we deal with wave nature of light. There are only two possible types of theory, for there are only two possible ways of transmitting energy through empty space. One is by a stream of moving material particles, when the energy is transmitted as the kinetic energy of the particles, and other is by means of waves, when the energy is transmitted as the energy of the waves without any motion of medium transmitting them. And so the following theories of light arise: (i) (ii) (iii) (iv) Newton’s corpuscular theory of light (1665) Huygens’ wave theory of light (1678) Maxwell’s electromagnetic theory of light ( or radiation) (1873) Planck’s quantum theory of radiation (1900) 2. Theories regarding Nature of Light (i) Newton’s Corpuscular theory : In 1665, Newton gave the corpuscular theory of light. It states that light consists of a large number of minute material corpuscles (i.e. particles) emitted by a luminous body which produces the sensation of sight when these corpuscles strike the eye. According to this theory, light travels with a tremendous speed (3 x 108 ms1) in straight lines is a homogeneous medium. Different colours of light were ascribed to different – sized corpuscles. The corpuscles travel with tremendous speed in straight lines, because of their insignificant masses pull of gravity on them is negligible. This theory was able to explain the principle of rectilinear propagation of light, laws of reflection as well as the law of refraction. Failures of this theory: Explanation of laws of refraction leads to the result that light travels with a higher speed in denser medium. Again, the experiment of Faucault on the direct determinal of the velocity of light in different media gives an opposite result. The phenomenon of diffraction, viz, a small bending of the rays round a small object shows that the propagation of light is not strictly rectilinear but approximately so. Thus a correct Institute of Life Long Learning, University of Delhi Page 2 Nature of Light and Huygens’ Wave Theory theoty should explain the approximate rectilinear propagation of light but Newton’s corpuscular theory failed to explain this. (ii) Huygen’s wave theory: This wave theory was given by Huygens in 1678. According to this theory, a source of light propagates spherical waves in hypothetical medium called the ether which pervades the whole of universe, in much the same way that a stone sets up circular ripples when it falls into a pond. But as light can travel even in vacuum, Huygens had to assume the existence of this medium ‘ether’ in much the similar way as sound water waves require medium for its propagation. On the basis of this theory, these light waves produce the sensation of sight when they enter the eyes. It explains that light travels more slowly in water than air, the precise opposite of the corpuscular theory. Failures: Huygens in 1678 considered these light waves as longitudinal and the phenomenon of reflection, refraction, interference and diffraction were successfully explained. Difficulty aread to explain the phenomenon of polarization of light, but that difficulty was surmounted by Fresnel, who assumed the light waves to be transverse. Fresnel explained the approximate rectilinear propagation of light on the basis of wave theory and Fresnel half-period zones vis-à-vis diffraction. Though the Huygens’ wave theory, as modified by Fresnel, successfully supplented Newton’s corpuscular theory, yet it had many drawbacks. It necessitated the adoption of a hypothetical medium called ether possessing an extraordinary property of elastic solid. The velocity of transverse wave in a solid medium is given by is the modulus rigidity and , the density of the medium. Hence, to account high velocity of light, ether must possess high rigidity and low density – the elasticity of ether must be many times, greater than that of steel and its density many times less than that of the best vacuum we can produce ! card of etis asumeher medium. Discard of ‘either’ medium: On the basis of Huygens’ wave theory, light waves are transverse in nature and a longitudinal light wave has not been detected so far. Since the ether is assumed to have property of an elastic solid and elastic solid is capable of transmitting both transverse and longitudinal waves, this difficulty is removed by supposing that ether is incompressible. For incompressible fluid, bulk modulus (k) is infinite. The velocity of longitudinal wave This makes the velocity of longitudinal wave infinitely small, but can a medium be stable under such conditions ! Michlson – Morely experiment, conducted in 1887, attempted to measure the velocity of the earth through the ether. This experiment proved that there is no relative velocity between the earth and the ether. Nor only this, all experiments designed to detect the presence of ether have, so far failed and hence its contradictory properties cannot be tested. Hence, ether concept was abandoned. Institute of Life Long Learning, University of Delhi Page 3 Nature of Light and Huygens’ Wave Theory It must be emphasized that Fresnel (1788-1827) gave a satisfactory interpretation of many of the diffraction phenomena, assuming Huygens’ wave theory to which he added one or two assumptions. (iii) Maxwell’s electromagnetic theory of light (1873): Before the Michelron – Morley experiment was performed in 1887, it was assumed by Faraday that a medium like ether was an essential feature of the electrical theory. It was however not a mechanical medium in which material displacements took place but rather an electromagnetic phenomenon in which displacement currents and magnetic fields occur. The periodic disturbances which are supposed to constitute these waves, were called displacement currents by Maxwell. In 1873, Clark Maxwell working on the hypothesis of displacement current in dielectrics arrived at the conclusion that an electromagnetic wave travelling in space has a velocity given by where = permeability of the medium, = permittivity of the medium. In vaccum, equals velocity of light c in free space This led Maxwell to put forward his famous Electro-magnetic Theory of light. Maxwell therefore concluded theoretically that light is an electromagnetic wave. In 1873 In 1888, fifteen years later, Hertz experimentally proved the existence of electromagnetic waves and thus verified Maxwell’s electromagnetic theory. These electromagnetic waves are transverse in nature and differ from light waves in wavelength. In Maxwel’s electromagnetic theory, the vibrations in wave theory is replaced by the oscillations of electric and magnetic fields which are mutually perpendicular to each other and also perpendicular to direction of waves propagation. In fact, it is the electric wave that really constitutes light, while the magnetic wave, though no less real, is less important. Further confirmation of the electromagnetic theory of light is obtained from (a) Faraday effect, (b) Kerr effect, (c) Zeeman effect, (d) Electrostatic double refraction (Kerr), and (e) Stark-effect. But it failed to explain Photoelectric effect, Compton effect. (iv) Plancks Quantum theory of light (1900): In 1900, Planck gave quantum theory of radiation according to which exchanges of energy between ether and matter, instead taking place continuously, can occur only in discrete steps i.e., in multiples of some small unit, called the quantum. His assumption is that an oscillator or a vibrating electron absorbs or emits radiations having integral multiples of energy quanta and energy of one quantum is where is Planck’s constant and is the frequency of radiation. This idea of Planck was verified by experiment. In 1904, Einstein gave quantum theory of light; and thus, there was again the revival of Newton’s corpuscular theory of light. These corpuscles corparticles are considered, quanta Institute of Life Long Learning, University of Delhi Page 4 Nature of Light and Huygens’ Wave Theory called photons. According to this Einstein’s theory of light, every radiation is to consist of indivisible ‘radiation quanta’ of energy . Einstein (1904) used the concept of black body radiation put forth by Planck (1900). Quantum theory of light was able to explain the photoelectric effect, the atomic structure, the Compton effect and Raman effect, but it failed to explain the phenomena of interference, diffraction, etc. The latter phenomena were explainable on the basis of wave theory of light. Thus for one phenomenon we regard light as quanta (or particles) while for another phenomenon we regard light as waves. A satisfactory theory must therefore combine these two conceptions into one general coordinating principle. This led to dual nature of light. Co-relation between wave theory and quantum theory of light (or Dual nature of light) In 1926, de Broglie worked out mathematically that material particle may be supposed to be consisting of certain wave frequencies. Soon after Schrödinger extended Broglie’s theory and formulated a system of mechanics known as wave mechanics. This is how the wave and particle aspect of matter are co-related. Mathematically it is given by where m is mass of particle and its velocity where as is the wavelength of the wave associated with the moving particles called de Broglie wave and h, the Planck’s constant. Conclusion: Various phenomena of light propagation are explainable on the basis of electromagnetic wave theory of light, while the interaction of light with matter in the processes of emission and absorption is explainable on the basis of quantum theory of light. The former theory of light has wave concept while the latter theory has particle concept. This is known as dual nature of light which is valid even to-day. Similarly, electron appears to have a dual nature; it is a wave as well as a particle. An electon in motion behaves as a wave, but when it lands anywhere it lands as a particle. In 1926, Schrödinger described matter waves by formulating the equation called Schrödinger equation. The de Broglie’s theoretical prediction of the existence of matter waves was first verified experimentally in 1927 by C.J. Davisson and L.H. Germer and by G.P. Thomson. Value Addition: Ether: As sound needs a medium to propagate, so light must also need a medium to propagate. Ether was suggested a medium for light to propagate but ether is now a discarded hypothetical medium once thought to fill all space and to be responsible for carrying light waves and other electromagnetic waves. Such a medium of ether was postulated with mechanical properties adjusted to provide a consistent theory. For propagation of electromagnetic radiation, it was assumed to pervade all space and matter, to be extremely elastic yet extremely light, to transmit transverse waves with the speed of light to have a greater density in matter than in free space. A search for “ether” medium was made by Michelson and Morley in their famous Michelson – Morley experiment to detect the presence of ether. But, they failed to detect its existence. On this result obtained by Michelson and Morley in 1887, Albert Einstein built up his revolutionary Relativity and Quantum theories. Further, Einstein concluded that light rays do not travel in straight lines but they curve under the influence of a gravitational field. Institute of Life Long Learning, University of Delhi Page 5 Nature of Light and Huygens’ Wave Theory 3. Definition of Wave-Front and Ray If a stone is dropped into a pond, circular ripples spread out from the point O where the stone entered the water, as shown in Fig 1. They are produced by the vibration of the water due to the entry of the stone and the ripples are circles with O as centre. Each one of these circles is called a wave front and the direction OA, in which any very small portion of a wave front travels, is normal to the wave front and is called a ray. If a graph of the displacement (y) of the particles, which normally lie on the line OA, against their distance (x) from O is drawn, it will resemble the wave form shown in Fig. 2 and the wave form travels through the medium without any bodily motion of the medium itself. But we are dealing with waves travelling in two A dimensions, and hence the concept of wave front comes in. O A wave front at any instant of time is defined as the locus of all the neighboring particles in the medium which are being just disturbed at that instant of time and are consequently in the same Fig. 1 Wave-front phase of vibration. The direction in which the disturbance is propagated in a Y homogeneous medium is called a ray. It is always normal to the wave front. A wave x O front is the locus of a particular phase of the vibration of the particles of the B medium. The phase most usually chosen is the position of maximum positive Fig. 2 Wave-form displacement and so, the locus is the line joining the crests. The wave front is clearly a circle in the case of the waves produced by dropping a stone in a pond. If a plank BC O A floating on the surface of a pond is moved up and down, it will produce waves roughly of the form shown in Fig. 3, in which the wave fronts are straight lines, a ray being straight line, OA, normal to the wave fronts. C Fig. 3 Plane Wave Fronts In the case of waves in three dimensional, a point source will produce spherical waves, the rays being radii of the spheres, while a plane source will produce plane waves in which the rays are lines normal to the plane. 4. Huygens’ principle of wave propagation of light : We know from simple experience that a circular wave front grows into a circle of ever increasing radius; a linear wave front moves as a line parallel to itself; a spherical wave front spreads as a sphere of ever increasing radius; a plane wave front moves as a plane parallel to itself. There is no propagation of waves (or energy) in the backward direction. Institute of Life Long Learning, University of Delhi Page 6 Nature of Light and Huygens’ Wave Theory The theory of predicting the future course of wave fronts was put forward by Huygens and is known as Huygens’ principle. Explanation of Huygens’ theory of wave propagation: According to Huygens’s wave theory of light, each point on a source of light is a centre of periodic disturbance (=wave) from which transverse waves spread out in all direction, the velocity of propagation of the disturbance varying with the nature of the medium. In a homogeneous medium the waves travel with the same velocity in all directions. For a point source of light, in a homogeneous medium, the wave front is a sphere. At a very large distance from the point source the radius of he sphere is very large so that a small position of its is sensibly plane. The wave front in such a case is called a plane wave front. Hugens’ principle is a geometrical method of ding the position of the new wave front after a certain time. It states that the position of a wave front t seconds after its present position is found by considering each point of the given wave front as the source of secondary wavelets. With each point on the given wave front as centre a sphere of radius vt is drawn, where v is the velocity of the waves in the medium. The required position of the wave front is the envelope of the spherical wave lets in the forward direction. We shall illustrate its principle by applying it to the propagation of spherical and plane wave fronts. vt Ray O B Fig. 4 Construction of New Spherical wave front A vt In Fig 4 let O be a point source of light, placed in a homogeneous isotropic medium. Let v be the velocity of the light in the medium. A point source emits spherical waves in all directions, with O as centre Ray AB be the portion of the spherical wave front at any instant in order to obtain the wave front after a time t. B Consider a number of points on AB. Fig. 5 Construction of New Plane Wave Each point is a source of secondary Front wavelets. With each point on the given wave front as centre a sphere of radius vt is drawn. The sections of these secondary waves of a plane will be circles as shown in the figure. The required position of the new wave front is the envelope or common tangent of the spherical wavelets in the forward direction. This is shown by A/B/ which is the front envelope of these circles. By this type of construction we can construct one wave-front from another. Institute of Life Long Learning, University of Delhi Page 7 Nature of Light and Huygens’ Wave Theory The rear envelope of these wavelets is shown by which cannot represent the next wave-front after the time t. This can be explained using Stokes law. According to this law, the amplitude at each point of the secondary wave is proportional to (1+cos) where is the angle between the wave normal (OA or OB) and the direction of propagation which is the line, joining the point of the secondary wave to its centre. For example, for the first wavelet, A is its centre, is the direction of the normal OA and for point the direction of propagation is along , therefore angle is . For the point of the secondary which is directly behind the wave, the value of = and hence (1+cos) is zero. Thus, the intensity of the secondary waves in the rear is zero and hence the rear envelope of the circles cannot represent the section of the wave front. Hence, in Fig 4, the rear envelope cannot represent the section of the wavefront after time t. Similarly, we can construct new plane wave front from a plane wave front AB at any time t, as shown in Fig 5, by drawing spherical wavelets of radius vt from a number of points on AB. The rear envelope cannot be new plane wave front after time t. The direction in which a small part of the wave front advances in a medium is called a ray. In a homogeneous isotropic medium, the rays are normal to the wave front, as shown by arrows in Fig. 4 and 5. Applications of Huygens’ Principle of wave theory of light: Huygens’ principle of wave propagation of light may be used to account for the phenomena of reflection and refraction from the plane and curved surfaces. One can prove (i) Laws of reflection (ii) Laws of refraction (and hence Snell’s law) (iii) The mirror formula in terms of u, v and f (iv) The lens formula in terms of u, v and f. In order to explain the above mentioned phenomena, we shall apply Huygens’ postulates. These postulates are as follows: 1. Each point on a wave-front acts as the centre of a new disturbance called secondary wavelet. These wavelets travel out with the same speed as the orginal wave so long as the medium remains the same. 2. The envelope, namely, the tangitial plane to these secondary wavelets constitute the wave-front. 3. Only forward wavefront exists and no backward wave-front exixts. Explanation of Absence of a back wave: It is explained on the basis of Stokes’ law. According to this law, the amplitude propagated in any direction due to secondary wave is proportional to where is the angle between the wave normal and the diretrion of wave propagation. For the backward wave-front, Institute of Life Long Learning, University of Delhi Page 8 Nature of Light and Huygens’ Wave Theory Hence no effect of secondary wavelets is observed in the backward direction because amplitude is zero. 5. Laws of Reflection derived from the Huygens’ wave theory: We shall consider refletion of a plane wave front from a plane reflecting surface (or a plane mirror). Consider the incidence of a plane wave front AB upon a plane reflecting surface XY. Assume rays PM, QN etc.(which are normal to AB) to be in the plane of paper. Hence the sections AB and XY will be perpendicular to the plane of paper. Let v be the velocity of light and the wave front travelling along AC, strikes the reflecting surface at A. The angle of incidence, i, between the normal and the incident ray is equal to which the wave front AB makes with the reflecting surface XY. Let t be the time taken by the disturbance from B to . Then, during this time interval, the spherical wavelet from A will have spread out in a sphere of radius vt which is equal to AC, and the plane wave front AB will have occupied its parallel position if the reflecting surface were not present. According to the Huygens’ principle, each point on the wave front AB is a source of disturbance, and gives rise to a secondary wavelet. Owing to the presence of the reflecting surface, the disturbances from the various points on AB, after the striking surface XY spread out in spheres and travel in the upper portion, the envelope of which gives rise to a new wave front called the reflected wave front. To find the position of the reflected wave front, with A as centre and radius = = AC = vt, describe a sphere. From draw a tangent plane to the sphere touching the latter at . Then is the trace of the reflected wave front . Let make an angle r with the reflecting surface XY , where r is the angle of reflection. B’ B i X vt vt Q S Vt2 R P vt 1 i A r r M Y N E D To prove that is the C common reflecltled wave-front, we are required to show that the disturbances from each point on AB, such as P,Q, etc Fig. 6 Laws of Reflection from the Huygens’ after being reflected from the wave theory points M, N, etc. respectively of the surface XY, reach the wavefront exactly at the same time. Let t 1 be the time taken by the disturbance from P to reach reflecting surface XY at M and also let t2 be the time taken by the disturbance from M to reach . Then PM = vt1, MS = vt2 With M as centre draw a sphere so that it touches the tangent plane Institute of Life Long Learning, University of Delhi at S. Join MS Page 9 Nature of Light and Huygens’ Wave Theory Let AM = x, and In , and PM is parallel to . Hence s APM and are similar Similary are similar Adding Eq.5-1 and 5-2, t1+ t2 = t Hence, the wavelet from P, after reflection at M, reaches the point S on the wave front exactly at the same time as the disturbance from A reaches . Similarly it can be proved that disturbances from other points, such as Q etc. reach at the same time t, or is the trace of the common reflected wave front. Hence, a plane wave front is reflected back from a plane reflecting surface as a plane wave front. [The reader will recall that Huygens’ principle cannot predict the way the wave front will travel] In . We have is common These triangles are congruent Hence This is the second law of reflection.(Proved) Institute of Life Long Learning, University of Delhi Page 10 Nature of Light and Huygens’ Wave Theory Also AB, and are all section of the plane perpendicular to the plane of paper; hence normal to the planes are in the same plane. But normal to AB and are incident and reflected rays, hence the incident and reflected rays and normal to the surface at the point of incidence lie in the same plane. This is the first law of reflection (Proved). 6. Laws of Refraction on the basis of the Huygens’ wave theory: Fig 7 shows the trace XY of a plane surface perpendicular to the plane of paper, separating the two media of refractive indices 1 and 2, such that 2>1; and v1 and v2 being the velocities of light in the upper and lower media respectively in such a way that v 2 < v1. Let AB be the trace of a plane wave front from a monochromatic source of light, perpendicular to the plane of paper, incident at an angle i with surface XY, and travelling in the direction AE from less dense to more dense medium (Fig. 7). 1, v1 According to Huygens’ principle, each point on the wave front AB is a source of secondary (Rarer) B wavelet. Assume the wave front AB Q at time t=0. Let t be the time taken v1t i P i by the disturbance from B to reach v 1t1 A i X Y the surface XY at C. Then during L r C this time t, the spherical wavelet v2t2 r L from A will have spread out in a v2t M sphere of radius v1t and the plane N 2 , v2 E wave front will have occupied its D parallel position CE if the refracting (Denser) surface XY were not present. But owing to the presence of the Fig.7 Laws of Refraction from Huygens’ wave theory refracting surface XY, during this time t, the wavelet from A which started earlier, instead of spreading into the lower medium 2 as a sphere of radius AE = BC, moves into it with a velocity v2 and spreads out as a sphere of radius AD = v2t. An envelope (or a tangent plane) perpendicular to the plane of paper drawn from C to the sphere, i.e., the tangent plane CD would represent the refracted wave front. To prove that CD is the common refracted wave front for all disturbances which originate from points such as P, Q etc. on AB, one has to show that during the time the wavelet from A travels to D or from B to C, the wavelet from P reaches the wave front CD. Let t1 be the time taken by the wavelet from P to reach XY at L and also let t 2 be the time taken by the disturbance from L to reach CD at M. Then Let Institute of Life Long Learning, University of Delhi Page 11 Nature of Light and Huygens’ Wave Theory have PL parallel to BC and . Since these triangles are similar, Similarly, since are similar, This gives , Adding Eqs. (6-1) and (6-2) t1+ t2 = t Hence, the wavelet from P after refraction at L, reaches the point M on the wave front CD exactly at the same time as the wavelet from A reaches D. Similarly, it can be proved that wavelets from other points, such as Q etc., reach the same wave front at the same time or CD is the trace of common rerfracted wave front. Let r be the angle if refraction corresponding to angle of incidence, i. From , in which From , in which These equations give We have, since lower medium is denser than upper medium i.e. Therefore . is greater than one. is refractive index of second medium with respect to first medium. Institute of Life Long Learning, University of Delhi Page 12 Nature of Light and Huygens’ Wave Theory This proves second law of refraction. This is Snell’s law of refraction, i.e. Since This means that the velocity of light in a denser medium is less than that in a rarer medium. This has been verified experimentally. Also AB, CE and CD aer all traces of the planes perpendicular to the plane of paper; hence normals to these planes are in the same plane. Since normals AB and CD are the incident and refracted rays, hence the incident ray, refracted ray and normal to the refracting surface at the point of incidence lie in the same plane. This proves the first law of refraction. Total internal reflection on the basis of Huygens’ wave theory: From Eq,(6-4), we have Putting the value of BC from Eq. (6-3), we get, When the lower medium (see Fig.7) is denser, 2>1 and because maximum value of sin i is one then AD is always less than AC. Here AD is the radius of refracted wavelet. The tangent CD can always be drawn. Since CD is refracted wave front, there is always refracted ray in the lower medium. From Eq.(6-5), (or Snell’s law), Since Suppose the lower medium is rarer, then 2<1 From Eq.(7-1) it follows that the following three cases arise, depending upon whether AD is less than, equal to or greater than AC. (i) Case I: AD<AC, (Fig. 8) X A (Dens (Denser) er) B i C r Y D Institute of Life Long Learning, University of Delhi Page 13 Fig.8. AD<AC (Rarer) Nature of Light and Huygens’ Wave Theory It means that the radius of the refracted spherical wavelet AD is less than AC. Therefore, the point C lies outside the sphere. Hence a real tangent CD which is the refracted wave front can be obtained. That is to say, refraction is possible. (ii). Case II AD = AC, (Fig.9) Refracted wave front AD lies on the surface XY which separates the two media and point D coincides with C. The tangent CE drawn from C to the wavelet sphere will be normal to the surface XY. Since this tangent is also refracted wave front, therefore the acted rays will be parallel to the surface, along AC as shown in Fig 9. From Eq.7-1, putting AD = AC, replacing i by ic, we have, (Denser) B ic X A C r=900 =AD where is refractive index of denser medium with respect to rarer medium. Here µ is always greater than 1. This angle i c of incidence at which r is 900, is called critical angle. Fig.9.AD=AC Y E (Rarer) (iii). Case III AD>AC This is shown in Fig. 10. Point C will lie inside the spherical wavelet drawn from A in the lower medium. In such a case, no real tangent from C to the wavelet is possible. Hence no refracted wave front i.e. refracted ray is possible. Since reflection is possible, the reflected wavelet from A propagates in the first medium and the radius of this spherical wavelet in AD’, as shown in Fig.10. The radius AD’ of this wavelet is less than AC; therefore a real tangent CD’ may be drawn from C to the reflected wavelet. In this case, no refracted wave front is possible but only reflected wavefront CD’ is possible when the incident wave front travels from denser medium to rarer medium. From Eq. (71) and (7-2), it follows that, since AD>AC, we have i > ic . This is the case of total internal reflection. The conditions of total reflection are : (Denser) i X A B Y D C (i) The incident ray must proceed from denser medium to rarer medium. (ii) The angle of incidence (i) must exceed the critical angle (ic) between the two media. (Rarer) Fig.10. AD>AC Institute of Life Long Learning, University of Delhi Page 14 Nature of Light and Huygens’ Wave Theory 8. Reflection of a spherical wave front a spherical surface (concave mirror) Consider reflection of a spherical wave from a concave mirror MPM’, as shown in Fig. 11. Let R be the radius of curvature of the mirror, C centre of curvature and P its pole. Let A be the point source, emitting spherical wave fronts. Let XLY be a spherical wave front at any instant. This wave front touches the concave mirror at point M and M’. By the time wavelet from L touches P, wavelet from M and M’ reach E and D respectively. LP=ME=M’D During this time, the secondary wavelets from all other points on MLM’ reach the surface EPD to produce reflected wave front of incident wave front MLM’. The reflected wave front EPD converges towards point B. Thus B is the image of A. X M E Let AP = u PB = v PC = R Assume small aperture as compared to its radius of curvature. Then we can write A C B G H L P From the geometry of the figure for spherical surface , we have D Y Similarly, for the surface Fig.11 Reflection from a concave Mirror MP Similarly for the surface EPD, To the first approximation, Substitute Eq. 8-1, 8-2, 8-3 in Eq 8-4. According to sign convention, v, u and R are all negative, Institute of Life Long Learning, University of Delhi Page 15 Nature of Light and Huygens’ Wave Theory This is the required expression. N.B. Reflection of a spherical wave front from a convex surface can be studied in the same way. In this case the incident spherical wave front XLY would meet pole P of convex mirror. 9. Refraction of a spherical wave front at a spherical source (using Huygens’ wave theory) Refraction at a convex surface. K air We are to obtain the relation D We assume small aperture of the convex surface. Consider a convex surface KPL where pole is P and is refractive index of the medium having convex surface. Let A be a point object at a distance u from P. Since A is a point object, it emits spherical wave front. GPH is a spherical wave front touching the convex surface at P. Obviously the radius of curvature of the spherical wave front will be u. Each point on GPH is a source of secondary wavelets, as follows from Huygens’ wave theory. By the time wavelet from point D in air reaches B on the convex distance PQ in the medium where B Q S is the J G I A B y E P L Q P u R v H C S L Fig.12. Refraction at a convex surface (using Huygens’ wave theory) surface, wavelet from P will have covered a refracted wave front. DB = .P Q The refracted spherical wave front BQS appears to come from point I where I is the centre of this spherical wave front. Thus I is point image of point object A, BJ is refracted ray and the distance IQ = v. In other words, the radius of curvature of the refracted wave front is v. Now, draw perpendicular DE and BL on the principal axis Let DE=BL=y, say DB=EL=.P Q……………………………(9-1) EL=EP+PL ……………………………….(9-2) PQ=PL+LQ ……………………………….(9-3) EP+PL=(PL+LQ) ; [Putting Eq.8-2 and 8-3 in Eq.8-1] Institute of Life Long Learning, University of Delhi Page 16 Nature of Light and Huygens’ Wave Theory EP+PL= .PL+.LQ EP.LQ=.PL.PL…………………………………(9-4) From Fig. 13, using the property of circle, BL.LS=PL.LT =PL(PTPL) neglecting small quantity PL2. Hence R is radius of curvature of convex surface. Since BL=LS=y and y2 = PL 2R B Similarly, we can write P L C T S Putting above values in Eq. 9-4, Fig. 13 According to sign convention, u and v are negative and R is positive This is the required relation between u, v and R. If object is in medium of refractive index and the convex surface has medium of refractive index , then the above relation becomes where Institute of Life Long Learning, University of Delhi Page 17 Nature of Light and Huygens’ Wave Theory Refraction at a concave surface: A is a point object: Spherical wave front from A is G D H which reaches concave surface kPL of a medium of refractive index . Then A is centre of spherical wave front GDL whose radius is u G Air K D y Each point on wave front GDH is a source of secondary wavelets. By the time, wavelet from point D reaches P, the wavelet from K and L reaches D and E in the medium . Distance KD in the medium is equivalent to a distance .KD in air A y M D N P I v u Th refracted wave front is DPE which appears to originate from point I (I is centre of spherical (reracted) wave front DPE). PI = v L E H Fig.14. Refraction at concave surface where I is virtual image of point object A and DD’ is refracted ray Draw perpendiculars KM and DN from K and D on the principle axis Using property of circle, K C M P R L (Neglecting small quantity PM2 in comparison to ) Fig.15 Institute of Life Long Learning, University of Delhi Page 18 Nature of Light and Huygens’ Wave Theory Similarly, we can prove that Substituting the above values in Eq. 8-5 According to sign convention, u, v and R are all negative, This is the required expression. 10. Thin Convex Lens Formula on the basis of Huygens’ wave theory of light: In Fig 16, if A is a point object on the axis of a converging lens, it will produce a spherical wave represented by K1P1, touching the surface of the lens at P1. Let v1 and v2 be the velocities of light wave in air (medium refractive index 1) and medium of the lens (of refractive index 2) respectively. Since the light travels more slowly in the material of lens than in the surrounding medium, the portion of the wave front round and about P1 will travel more slowly than the outer parts and the wave front will turn inside out to become spherical and concave in the direction of motion. It is represented by K 2P2 with centre B, which is therefore the point image of the point object A. According to Huygens’ principle, each point on K1P1 is source of secondary wavelet. Since the wave diverges from A and converges to B again, the time taken for the light from A to reach B is the same by all paths; otherwise no image will be formed. Hence, the time taken by light to travel the path AP1LP2B is the same as that taken to travel the pathAK 1MK2B. The refractive index of the material of the lens is Institute of Life Long Learning, University of Delhi Page 19 Nature of Light and Huygens’ Wave Theory 1 v1 K2 p Let AM = p, MB = q; AL = u, LB = v, P1L = a, LP2 = b, ML=y C2 1 M K1 q y P1 A v1 L P2 C1 B a b As stated above regarding the time taken being equal along the two paths; one has u v 2 Fig. 16 Thin convex lens formula From right angled triangle AML [By binomial theorem is small] Similarly, from right angled triangle MLB, Adding Eq (10-2) and (10-3), From Eq.(10-2) and (10-4), Institute of Life Long Learning, University of Delhi Page 20 Nature of Light and Huygens’ Wave Theory Let C1 be the centre of curvature of the surface of the lens at which the light enters and C2 that at which it leaves, as shown in Fig. 16. The corresponding radii of curvature of the two surfaces of the lens are M y P1C1 = R1 P1 P2C2 = R2 From the property of circle in geometry, one can write, as shown in Fig. 17, P1LLQ = MLL a L C1 y 2R1a Q M Fig.17 Property of circle a(2R1a) = y2 2R1aa2 = y2 For a thin lens, a is small; therefore a2 is negligibly small compared to 2R1a. Similarly, Substituting these values in Eq. (10-4), we have This, on simplification, gives Using sign conventions of optical distances, u and R 2 are negative whereas v and R1 are positive. Rewriting the above equation, This is the general formula for lenses. For object lying at infinity u = , (second) focal length of the lens. Institute of Life Long Learning, University of Delhi v = f, Page 21 Nature of Light and Huygens’ Wave Theory From Eq.(10-7) and (10-8), we have N.B. (i) A similar formula for refraction by a concave lens, on the basis of Huygens’ wave theory, can be derived (ii) The reader will recall that the above procedure uses the same principle(regarding the time taken along the two paths) which was used in section 2.17 in proving the lens formula by Fermat’s principle and so the above proof from Huygens’ principle follows just the same lines. 11. Thin concave lens formula on the basis of Huygens’ wave theory of light. Fig. 18 shows a thin concave lens where A is a point object being on the principal axis of the lens. is the portion of the wave front coming from A, which touches the lens surface at . The wavelet from K1 reaches K2 in a time interval equal to that time taken by the wavelet from D to E. The wavelet from D partly travels through air and partly through the medium of the lens. Assume small thickness in the middle of the lens. Velocity v1 of light in air is greater than velocity v2 of light inside the lens. Hence the refracted wave front will take the form being convex outside. It will appear to come from B, (see Fig. 18). K1 q p D B A C1 K2 G L1 L2 E C2 H Fig. 18 Thin Concave Lens formula Hence B is the virtual point image of the point object A. Draw K1D and K2E perpendicular to the principal axis. Suppose Time taken by wavelet to travel from K1 to K2 in the lens is Time taken by light to travel from D to G partly in air and partly in lens is Institute of Life Long Learning, University of Delhi Page 22 Nature of Light and Huygens’ Wave Theory Since t1 = t2 Now, Multiply throughout by v1, Cancelling Since , This gives Let R1= radius of curvature of curved surface curvature C1. having centre of R2= radius of curvature of curved surface curvature C2. having centre of AH = u BH = v Institute of Life Long Learning, University of Delhi Page 23 Nature of Light and Huygens’ Wave Theory Substituting the above values, Using sign convention, u, v, and R1 are negative A O P For a point object at infinity B This gives the required expression, namely, 12. Fig.19 Ray OP is perpendicular to wave front AB Relation between Huygens’ wave theory and Fermat’s principle: Consider a plane wave front AB at any instant of time. We are to find the effect of AB at any point P on a screen. From P draw perpendicular PO and AB. Each point on AB is a source of secondary wavelets which are superimposed on P. It can be proved that the resultant effect at P due to all the secondary waves from the wave front AB will be due to a small element around O whereas the effect due to the others parts of the wave front will cancel each others effect and its resultant effect is nil. This is due to their mutual cancellation by the phenomenon called interference of light. Thus, the light travels along the path OP which is the shortest (or the least) path between the wave front AB and the point P. In other words, disturbance travels along the minimum path. It shows that it follows Fermat’s principle which is also the principle of extreme path. Thus Fermat’s principle follows from the concept of the wave theory of light. The straight line OP along which the disturbance travels is called a ray. Thus, a ray is a straight line which is perpendicular to wave front. 13. Solved Examples: Examples 1: Institute of Life Long Learning, University of Delhi Page 24 Nature of Light and Huygens’ Wave Theory (a) Calculate the optical path for two wavelengths 1 and 2 of light propagating in a medium of refractive index . (b) Calculate also for two lights of frequencies 1 and 2 . Solution: In air, all the waves (or wave-fronts) of different wavelets travel with the same speed of light. Therefore, the frequencies of light are inversely proportional to the wavelengths. If v1 and v2 are the velocities of waves in a medium, and 1 and 2 are their wavelengths, If v1 is velocity of light in air and v2 in a medium of refractive index , Optical path is the distance traversed by light in a medium of refractive index in time t. Let d be the distance traversed. Where v is velocity of light in the medium of refractive index . Distance travelled by light in the same time t in air is This distance is the equivalent distance in vacuum. It is called optical path. Institute of Life Long Learning, University of Delhi Page 25 Nature of Light and Huygens’ Wave Theory If d1 and d2 are distances of light travelled in medium ‘1’ and ‘2’ , (c) Velocity of light for each wavelength is different, but the frequency for each light remains the same Example 2: Calculate the total optical path when light travels d1, d2 and d3 in three different media with refractive indices 1, 2 and 3. How much distance in the third medium 3 will have the same optical path ? Solution: Total optical path is This is the answer for initial part of the question. Let l be the distance in the third medium 3 with the same optical path d. Example 3. A plane wave front in a medium of refractive index 1 is incident on a convex refracting suface of refractive index 2. If R is the radius of curvature of convex surface, show that the incident plane wave front will turn into a spherical refracted wave front and its curvature will be A A A0 1 D Medium 1 N P M C F Solution: Let A0B0 be a plane wave front moving towards convex surface . At any instant A0B0 touches E B0 B Institute of Life Long Learning, University of Delhi B Page 26 Fig.20, Ex.3 Nature of Light and Huygens’ Wave Theory the convex surface. Let it be AB . As rays are perpendicular to plane wave front, and are the rays meeting convex surface on and . Let 1 and v1 be the refractive index and velocity of light in incident medium and the corresponding quantities be 2 and v2 in refracting medium such that v1 > v2 and 1<2. If the convex surface were absent, plane wave front AB would have occupied the position after a time t. Due to the presence of refracting medium in which v2 < v1 the wavelet from D and E reach and while the wavelet from P reaches M instead of N. From Fig.20, it follows Because Thus the plane wave front takes the shape whose radius of curvature is larger than R. is radius of curvature of , then from the property of circle. From Eq.(1) and (2) Thus, a plane wave front becomes a spherical wave front whose curvature is less than that of the convex refracting surface (2 > 1) and is of the same sign as that of the convex surface. Summary: Institute of Life Long Learning, University of Delhi Page 27 Nature of Light and Huygens’ Wave Theory Light is a form of energy which can be transferred/transported in vacuum as well as in medium. In Geometrical Optics, it is assumed that light travels in straight lines and we assume particle nature of light. In Physical Optics, we deal with wave nature of light. In order to study the nature of light, following theories of light were developed, viz, Newton’s corpuscular theory of light (in year 1665) Huygens’s wave theory of light (in year 1678) Maxwell’s electromagnetic theory of light (or radiation) (in year 1873) Planck’s quantum theory of radiation (in year 1900) Newton’s Corpuscular theory of light: According to this theory, light consists of a large number of minute material corpuscles (or particles) emitted by a luminous body and it travels with a tremendous speed in straight lines in a homogeneous medium. This theory of light could explain rectilinear propagation of light, laws of reflection and refraction. It led to wrong result that light travels with a higher speed in denser medium. Huygens’s wave theory states that a source of light propagates spherical waves in hypothetical medium called ether, which pervades the whole of universe. It led to correct result that light travels more slowly in water than air. Michelson- Morley experiment conducted in 1887, failed to detect the presence of ether. Hence ether concept was abandoned. Maxwell’s electromagnetic wave theory of light concluded theoretically that an electromagnetic wave travelling in space has a velocity given by where = permeability of the medium = permittivity of the medium In vacuum, v equals velocity of light c in free space i.e. c = v = 3108 ms-1 This led Maxwell to put forward his famous electromagnetic theory of light in 1873. According to this theory, light is an electromagnetic wave. In 1888, fifteen years later, Hertz experimentally proved the existence of electromagnetic waves and thus verified Maxwell’s electromagnetic theory that electromagnetic waves are transverse in nature. Maxwell’s electromagnetic theory failed to explain phenomena like photoelectric effect and Compton Effect. In 1900, Planck gave quantum theory of radiation. According to this theory, exchanges of energy between ether and matter can occur only in discrete steps, i.e. in multiples of some small unit, called the quantum and energy of one quantum is where h is Planck’s constant and is the frequency of radiation. This idea of Planck’s constant was verified by experiment. In 1904, Einstein gave quantum theory of light. He used the concept of black body radiation put forth by Planck in year 1900. Quantum theory of light was able to explain the photoelectric effect, the atomic structure, the Compton Effect and Raman Effect, but it failed to explain the phenomena of interference, diffraction etc. The latter phenomena were explainable on the basis of wave theory of light. In other words, for one phenomenon we regard light as waves while for another phenomenon we regard light as quanta (particles). This led to dual nature of light. Institute of Life Long Learning, University of Delhi Page 28 Nature of Light and Huygens’ Wave Theory In 1924, de Broglie worked out mathematically to correlate dual nature of light. Soon after, in year 1926, Schrödinger extended Broglie’s theory and formulated a system of mechanics known as wave mechanics. This is how the wave and particle aspect of matter are co-related. Mathematically it is given by, where m is mass of particle and is its velocity, whereas is the wavelength of the wave associated with moving particle called the de Broglie wavelength and h, the Planck’s constant. De Broglie’s theoretical prediction of the existence of matter waves was first verified experimentally in 1927 by C.J. Davisson and L.H. Germer and by G.P. Thomson. A wavefront at any instant of time is defined as the locus of all the neighboring particles in the medium which are being just disturbed at that instant of time and are consequently in the same phase of vibration. The direction in which the disturbance is propagated in a homogeneous medium is called a ray. A ray is always normal to the wave front. In the case of waves in three dimensional, a point source will produce spherical waves, the rays being radii of the spheres, while a plane source will produce plane waves in which the rays are lines normal to the plane. Huygens’s principle of wave propagation of light is a geometrical method of finding the position of the new wave front after a certain time. It states that the position of a wave front t seconds after its present position is found by regarding each point of the given wave front as the source of secondary wavelets. With each point on the given wave front as centre a sphere of radius vt is drawn, where v is the velocity of the waves in the medium. The required position of the wave front is the envelope of the spherical wavelets in the forward direction. Only forward wave front exists and no backward wavefront exists. Application of Huygens’ principle of wave theory of light: One can prove (i) Laws of reflection (ii) Laws of refraction (and hence Snell’s law) (iii) The mirror formula in terms of u, and f, i.e. (iv) Thin lens formula viz. Also (v) Fermat’s principle which is also the principle of extreme path Exercise: 1. 2. 3. 4. 5. State four theories regarding nature of light. Briefly outline the Huygens’ wave theory of light. What is a wave-front ? Ray ? How is wave front produced and propagated ? State Huygens’ theory of secondary wavelets. Explain clearly Huygens’ principle for the propagation of light. Sate Hugen’s principle and show how it be used to explain the laws of reflection of light. (a) Explain Huygens’ principle for propagation of light. Institute of Life Long Learning, University of Delhi Page 29 Nature of Light and Huygens’ Wave Theory (b) State the laws of reflection and refraction of light and obtain them form the wave theory of light. 6. Explain the laws of refraction of light on wave theory of light. What is meant by refractive index and how is it related to the velocities of light in different media. Derive Snell’s law on the basis of wave theory of light. 8. How is refraction of a plane wave explained on wave theory of light. When does total internal reflection occure ? 9. Explain the phenomenon of total internal reflection on the basis of Huygens’ principle and obtain the value of critical angle. 10. Derive a formula for refraction of a spherical wave at a spherical surface according to the wave theory of light ? 11. Derive a formula for reflection of a spherical wave at a spherical surface according to the wave theory of light. 12. Account for the formation of images by refraction through lenses on the wave theory and prove the relation where the symbols have their usual significance 13. What is a wave-front and what is the mode of its propagation in a medium ? Using the wave theory establish the thin lens formula 14. Using the concept of wave theory prove the thin lens formula 15. State Huygens’s principle for propagation of light. Using the same, deduce the formula connecting object and image distances with the constants of a thin lens. 16. Derive the refraction formula for a thin concave lens on the basis of Huygens’ wave theory of light. 1 State and explain Huygens’ principle of secondary waves. Apply this principle for explaining the simultaneous reflection and refraction of a plane light wave from a plane surface of separation of two optical media. 18. Explain how Huygens’ wave theory of light is related to Fermat’s principle. 19. write a short essay on the nature of light. 20. Give an elementary account of the wave theory of light, and prove by its means that the focal length of a concave mirror is equal to half its radius of curvature. Institute of Life Long Learning, University of Delhi Page 30 Nature of Light and Huygens’ Wave Theory 21. Explain, on the basis of wave theory of light, how a lens forms an image of a point on its axis. Use the ideas of the wave theory to deduce an expression for the focal length of the lens, pointing out very clearly the assumptions made. Point out which of these assumptions are not true is practice, and in what way an actual lens may differ form the simple theoretical lens as a result. 22. (a) Apply the principles of the wave theory to find the relation between the focal length of a thin lens, the radii of curvature of the surfaces, and the refractive index of the material. (b) If a convex lens of focal length 15 cm, made of glass of refractive index 1.52 is totally immersed in a liquid of refractive index 1.35, how will its focal length be affected ? [Ans: Increased to 61.9 cm.] 23. Apply the principles of the wave theory to establish the formula for a thin lens. A ray of light is incident in a direction parallel to the axis on one face of a thin double convex lens of refractive index 1.5, and after two internal reflections, emerges from the second face. Show that it will cut the axis at a point at a distance from the centre of the lens. Institute of Life Long Learning, University of Delhi Page 31