Chapter 34 Wave Optics PDF

60 Wave Optics 117 Light Propagation. E3 Light is a form of energy which generally gives the sensation of sight. (1) Different theories Huygen’s wave theory Maxwell’s EM wave theory Einstein’s quantum theory de-Broglie’s dual theory of light (i) Based on Rectilinear propagation of light (i) Light tr...

60 Wave Optics 117 Light Propagation. E3 Light is a form of energy which generally gives the sensation of sight. (1) Different theories Huygen’s wave theory Maxwell’s EM wave theory Einstein’s quantum theory de-Broglie’s dual theory of light (i) Based on Rectilinear propagation of light (i) Light travels in a hypothetical medium ether (high elasticity very low density) as waves (ii) He proposed that light waves are of longitudinal nature. Later on it was found that they are transverse (i) Light travels in the form of EM waves with speed in free space 1 c 0 0 (i) Light is produced, absorbed and propagated as packets of energy called photons (i) Light propagates both as particles as well as waves (ii) EM waves consists of electric and magnetic field oscillation and they do not require material medium to travel (ii) Energy associated with each photon hc E  h  (ii) Wave nature of light dominates when light interacts with light. The particle nature of light dominates when the light interacts with matter (microscopic particles ) U D YG (ii) Light propagates in the form of tiny particles called Corpuscles. Colour of light is due to different size of corpuscles ID Newtons corpuscular theory h = planks constant  6.6  10 34 J - sec  = frequency  = wavelength U ST  (2) Optical phenomena explained () or not explained () by the different theories of light S. No. Phenomena Corpuscul ar Wave Theory E.M. wave Quantum Dual (i) Rectilinear Propagation      (ii) Reflection      (iii) Refraction      (iv) Dispersion ×   ×  (v) Interference ×   ×  118 Wave Optics Diffraction ×   ×  (vii) Polarisation ×   ×  (viii) Double refraction ×   ×  (ix) Doppler’s effect ×   ×  (x) Photoelectric effect × × ×   (3) Wave front E3 (i) Suggested by Huygens 60 (vi) (ii) The locus of all particles in a medium, vibrating in the same phase is called Wave Front (WF) (iii) The direction of propagation of light (ray of light) is perpendicular to the WF. Spherical WF ID (iv) Types of wave front Plane WF Light ray Light rays Line source D YG Point source U Cylindrical WF (v) Every point on the given wave front acts as a source of new disturbance called secondary wavelets. Which travel in all directions with the velocity of light in the medium. U A surface touching these secondary wavelets tangentially in the forward direction at any instant gives the new wave front at that instant. This is called secondary Secondary wavelets Primary wave front Secondary wave front ST wave front Point source Note :  Wave front always travels in the forward direction of the medium.  Light rays is always normal to the wave front.  The phase difference between various particles on the wave front is zero. Principle of Super Position. When two or more than two waves superimpose over each other at a common particle of the medium then the resultant displacement (y) of the particle is equal to the vector sum of the displacements (y1 and y2) produced by individual waves. i.e. y  y 1  y 2 Wave Optics 119 (1) Graphical view : (i) Resultant 1 Waves are meeting at a point in same phase 2 + y2 = y = y1 + y2 (ii) 1 60 y1 Resultant y = y1 – y2 + Waves are meeting at a point with out of phase y2 = 2 E3 y1 ID (2) Phase / Phase difference / Path difference / Time difference (i) Phase : The argument of sine or cosine in the expression for displacement of a wave is defined as the phase. For displacement y = a sin  t ; term  t = phase or instantaneous phase (ii) Phase difference () : The difference between the phases of two waves at a point is called phase difference i.e. if y 1  a1 sin  t and y 2  a 2 sin ( t   ) so phase difference =  U (iii) Path difference () : The difference in path length’s of two waves meeting at a point is  2  D YG called path difference between the waves at that point. Also   (iv) Time difference (T.D.) : Time difference between the waves meeting at a point is T T.D.   2 (3) Resultant amplitude and intensity If suppose we have two waves y 1  a1 sin  t and y 2  a 2 sin ( t   ) ; where a1 , a 2  Individual amplitudes,  = Phase difference between the waves at an instant when they are meeting a point. I1, I2 = Intensities of individual waves U Resultant amplitude : After superimposition of the given waves resultant amplitude (or the amplitude of resultant wave) is given by A  a 12  a 22  2 a 1 a 2 cos  ST For the interfering waves y1 = a1 sin t and y2 = a2 cos t, Phase difference between them is 90. So resultant amplitude A  a12  a 22 o Resultant intensity : As we know intensity  (Amplitude)2  I1  ka12 , I 2  ka22 and I  kA 2 (k is a proportionality constant). Hence from the formula of resultant amplitude, we get the following formula of resultant intensity I  I1  I 2  2 I1 I 2 cos  Note :  The term 2 I1 I 2 cos  is called interference term. For incoherent interference this term is zero so resultant intensity I  I1  I 2 (4) Coherent sources The sources of light which emits continuous light waves of the same wavelength, same frequency and in same phase or having a constant phase difference are called coherent sources. 120 Wave Optics Two coherent sources are produced from a single source of light by adopting any one of the following two methods Division of wave front Division of amplitude The coherent sources obtained are imaginary e.g. Fresnel's biprism, Llyod's mirror Youngs' double slit etc. 60 The wave front emitted by a narrow source is divided in two parts by reflection of refraction. Light sources is extended. Light wave partly reflected (50%) and partly transmitted (50%) The amplitude of wave emitted by an extend source of light is divided in two parts by partial reflection and partial refraction. The coherent sources obtained are real e.g. Newtons rings, Michelson's interferrometer colours in thin films M E3 The light source is narrow S1 S S L ID S2 Two waves superimpos e 1 Reflection coating M2 U Note :  Laser light is highly coherent and monochromatic. D YG  Two sources of light, whose frequencies are not same and phase difference between the waves emitted by them does not remain constant w.r.t. time are called non-coherent.  The light emitted by two independent sources (candles, bulbs etc.) is non-coherent and interference phenomenon cannot be produced by such two sources. U  The average time interval in which a photon or a wave packet is emitted from an L Distance of coherence atom is defined as the time of coherence. It is  c   , it's c Velocity of light value is of the order of 10–10 sec. ST Interference of Light. When two waves of exactly same frequency (coming from two coherent sources) travels in a medium, in the same direction simultaneously then due to their superposition, at some points intensity of light is maximum while at some other points intensity is minimum. This phenomenon is called Interference of light. (1) Types : It is of following two types Constructive interference Destructive interference (i) When the waves meets a point with same phase, constructive interference is obtained at that point (i.e. maximum light) (i) When the wave meets a point with opposite phase, destructive interference is obtained at that point (i.e. minimum light) Wave Optics 121 (ii) Phase difference between the waves at the point of observation   0 o or 2n (ii)   180 o or (2n  1) ; n = 1, 2,... or (2n  1) ; n  0,1,2.....  (iii) Path difference between the waves at the point of observation   n  (i.e. even multiple of /2) (iii)   (2n  1) (iv) Resultant amplitude at the point of observation will be maximum (iv) Resultant amplitude at the point of observation will be minimum I max  I1  I 2  2 I1 I 2 the point of I min  I1  I 2  2 I1 I 2  2 I min  I1  I 2  I0  I max  2I0 If (v) Resultant intensity at observation will be minimum  ID I1  I 2 a1  a 2  A min  0 E3 If (v) Resultant intensity at the point of observation will be maximum  60 a1  a 2  a0  A max  2a 0 I max  (i.e. odd multiple of /2) A min  a1  a 2 a1  a 2  A min  0 If 2 If I1  I 2  2 I1  I 2  I 0  I min  0 U (2) Resultant intensity due to two identical waves : For two coherent sources the resultant intensity is given by I  I1  I 2  2 I1 I 2 cos   2 cos 2  2 ] Note :  D YG For identical source I1  I 2  I0  I  I 0  I 0  2 I 0 I 0 cos   4 I0 cos 2  [1 + cos 2 In interference redistribution of energy takes place in the form of maxima and minima. Average intensity : Iav  U  Ratio of maximum and minimum intensities : ST  I max  I min  I1  I 2  a12  a 22 2 2 2  I1 / I 2  1  I max  I1  I 2     a1  a 2    a1 / a 2  1  also   a a   a /a 1  I / I 1 I min  I1  I 2  2   1  1 2   1 2   2 2 I1 I2   a1    a2     I max 1  I min  I max  1  I min  If two waves having equal intensity (I1 = I2 = I0) meets at two locations P and Q with path difference 1 and 2 respectively then the ratio of resultant intensity at point P and Q will be IP IQ    cos 2  1     2       cos 2 2 cos 2  2  2    cos 2 1 122 Wave Optics Young’s Double Slit Experiment (YDSE) Monochromatic light (single wavelength) falls on two narrow slits S1 and S2 which are very close together acts as two coherent sources, when waves coming from two coherent sources (S 1 , S 2 ) superimposes on each other, an interference pattern is obtained on the screen. In YDSE 2 Bright S1 d = Distance between slits D = Distance between slits and screen  = Wavelength of monochromatic light emitted from source 1 Bright d S 1 Bright S2 2 Bright 3 Bright 3 Dark 2 Dark 1 Dark Central bright fringe (or Central maxima) 1 Dark 2 Dark 3 Dark 4 Dark ID D Screen 4 Dark E3 3 Bright 60 alternate bright and dark bands obtained on the screen. These bands are called Fringes. (1) Central fringe is always bright, because at central position   0 o or   0 U (2) The fringe pattern obtained due to a slit is more bright than that due to a point. D YG (3) If the slit widths are unequal, the minima will not be complete dark. For very large width uniform illumination occurs. (4) If one slit is illuminated with red light and the other slit is illuminated with blue light, no interference pattern is observed on the screen. (5) If the two coherent sources consist of object and it’s reflected image, the central fringe is dark instead of bright one. (6) Path difference P U Path difference between the interfering waves meeting at a point P on the screen x xd  d sin  D S1 ST is given by   where x is the position of point P from central maxima.   n  ; where n = 0,  1,  2, ……. For maxima at P :  and For minima at P :  d S2 (2n  1) ; where n =  1,  2, ……. 2  C M Screen D Note :  If the slits are vertical, the path difference () is d sin , so as  increases,  also increases. But if slits are horizontal path difference is d cos , so as  increases,  decreases. P P S1  d  C S1 d S2 D C S2 Wave Optics 123 (7) More about fringe (i) All fringes are of D  β d D d (ii) If the whole YDSE set up is taken in another medium then  changes so  changes   3 e.g. in water w  a   w  a   a w w 4  60 and angular fringe width   E3 equal width. Width of each fringe is   1 i.e. with increase in separation between the sources,  decreases. d nD (iv) Position of nth bright fringe from central maxima x n   n ; n  0, 1, 2.... d (2 n  1) D (2 n  1)  (v) Position of nth dark fringe from central maxima x n  ; n  1, 2,3....  ID (iii) Fringe width   2d 2 U (vi) In YDSE, if n 1 fringes are visible in a field of view with light of wavelength 1 , while n 2 with light of wavelength  2 in the same field, then n1 1  n 2  2. (vii) Separation (x ) between fringes D YG Between nth bright and mth bright fringes (n  m) x  (n  m ) Between nth bright and mth dark fringe 1  (a) If n  m then x   n  m    2  1  (b) If n  m then x   m  n    2  (8) Identification of central bright fringe U To identify central bright fringe, monochromatic light is replaced by white light. Due to overlapping central maxima will be white with red edges. On the other side of it we shall get a few coloured band and then uniform illumination. ST (9) Condition for observing sustained interference (i) The initial phase difference between the interfering waves must remain constant : Otherwise the interference will not be sustained. (ii) The frequency and wavelengths of two waves should be equal : If not the phase difference will not remain constant and so the interference will not be sustained. (iii) The light must be monochromatic : This eliminates overlapping of patterns as each wavelength corresponds to one interference pattern. (iv) The amplitudes of the waves must be equal : This improves contrast with I max  4 I0 and I min  0. 124 Wave Optics  (v) The sources must be close to each other : Otherwise due to small fringe width     1  d the eye can not resolve fringes resulting in uniform illumination. (10) Shifting of fringe pattern in YDSE 60 If a transparent thin film of mica or glass is put in the path of one of the waves, then the whole fringe pattern gets shifted. t  D Fringe shift  (  1) t  (  1) t d   Additional path difference  (  1)t  S1 d E3 If film is put in the path of upper wave, fringe pattern shifts upward and if film is placed in the path of lower wave, pattern shift downward. C S2 (  1) t  or t  n (  1) ID  If shift is equivalent to n fringes then n  Screen D D YG U  Shift is independent of the order of fringe (i.e. shift of zero order maxima = shift of nth order maxima.  Shift is independent of wavelength. (11) Fringe visibility (V) With the help of visibility, knowledge about coherence, fringe contrast an interference pattern is obtained. V I1 I 2 I max  I min 2 If I min  0 , V  1 (maximum) i.e., fringe visibility will be I max  I min (I1  I 2 ) best. Also if I max  0, V  1 and If I max  I min , V  0 (12) Missing wavelength in front of one of the slits in YDSE From figure S2P = D 2  d 2 and S 1 P  D U So the path difference between the waves reaching at P ST  d2    S 2 P  S 1 P  D  d  D  D 1  2  D   2 For Dark at P   1/ 2 2  1 d2 From binomial expansion   D  1  2 D2  S1 P d D S2  d2 D   2D  D (2n  1)  d2   Missing wavelength at P 2D 2 By putting n  1, 2, 3.... Missing wavelengths are   Central position  d2 (2 n  1) D d2 d2 d2 , ,.... D 3D 5D Illustrations of Interference Interference effects are commonly observed in thin films when their thickness is comparable to wavelength of incident light (If it is too thin as compared to wavelength of light Wave Optics 125 it appears dark and if it is too thick, this will result in uniform illumination of film). Thin layer of oil on water surface and soap bubbles shows various colours in white light due to interference of waves reflected from the two surfaces of the film. Air Air 60 Oil Air Water Soap bubble in air E3 Oil film on water surface (1) Thin films : In thin films interference takes place between the waves reflected from it’s two surfaces and waves refracted through it. t  r r ID Reflected rays U Refracted rays Interference in refracted light Condition of constructive interference (maximum intensity) Condition of constructive interference (maximum intensity) D YG Interference in reflected light   2  t cos r  (2n  1)  2   2  t cos r  (2n)  2 For normal incidence so 2  t  (2n  1)  2 2  t  n U For normal incidence r = 0 Condition of destructive interference (minimum intensity)  ST   2  t cos r  (2n) 2 For normal incidence 2  t  n Condition of destructive interference (minimum intensity)   2  t cos r  (2n  1)  2 For normal incidence 2  t  (2n  1)  2 Note :  The Thickness of the film for interference in visible light is of the order of 10 ,000 Å. (2) Lloyd's Mirror A plane glass plate (acting as a mirror) is illuminated at almost grazing incidence by a light from a slit S1. A virtual image S2 of S1 is formed closed to S1 by reflection and these two act as coherent sources. The expression giving the fringe width is the same as for the double slit, but the fringe system differs in one important respect. 126 Wave Optics In Lloyd's mirror, if the point P, for example, is such that the path difference S 2 P  S 1 P is a 60 whole number of wavelengths, the fringe at P is dark not bright. This is due to 180 o phase change which occurs when light is reflected from a denser medium. This is equivalent to adding an extra half wavelength to the path of the reflected wave. At grazing incidence a fringe is formed at O, where the geometrical path difference between the direct and reflected waves is zero and it follows that it will be dark rather than bright. P E3 S1 d O ID S2 x  (2n  1) / 2 and (for maximum intensity) D YG Doppler’s Effect in Light U Thus, whenever there exists a phase difference of a  between the two interfering beams of light, conditions of maximas and minimas are interchanged, i.e., x  n  (for minimum intensity) The phenomenon of apparent change in frequency (or wavelength) of the light due to relative motion between the source of light and the observer is called Doppler’s effect. If   actual frequency,  '  Apparent frequency, v = speed of source w.r.t stationary observer, c = speed of light U Source of light moves towards stationary observer (v