1 00:00:18,650 --> 00:00:26,920 We have seen that ray theory has its limitations. It cannot accurately predict the propagation 2 00:00:26,920 --> 00:00:35,540 characteristics of an optical fiber, particularly when the light confinement dimensions are 3 00:00:35,540 --> 00:00:43,150 of the order of or comparable to the wavelength of light. In that case we will have to use 4 00:00:43,150 --> 00:00:53,980 wave theory. In wave theory light is treated as an electromagnetic wave. And therefore, 5 00:00:53,980 --> 00:01:03,510 we need to understand how these electromagnetic waves propagate in an optical fiber. Before 6 00:01:03,510 --> 00:01:10,600 doing that we would like to first understand how an electromagnetic wave propagates you 7 00:01:10,600 --> 00:01:18,670 know free space or in a dielectric medium which is of infinite extent for example, this 8 00:01:18,670 --> 00:01:28,370 room. So, first we would understand how light propagates in this room or in an infinitely 9 00:01:28,370 --> 00:01:37,241 extended dielectric medium. So, we will do this in this lecture. Since 10 00:01:37,241 --> 00:01:48,860 we want to understand propagation of light in infinitely extended dielectric medium, 11 00:01:48,860 --> 00:01:57,060 and light is an electromagnetic wave. So, we should first consider the Maxwell’s equations 12 00:01:57,060 --> 00:01:59,150 in infinite dielectric medium. 13 00:01:59,150 --> 00:02:11,920 So, the simplest case if we take, then it is a homogenous linear isotropic charge free 14 00:02:11,920 --> 00:02:21,550 and current free dielectric medium. One example of such a medium is glass. What is homogenous 15 00:02:21,550 --> 00:02:34,269 medium? Homogenous medium means that the refractive index of the medium is the same at every point. 16 00:02:34,269 --> 00:02:45,999 It does not depend upon x, y and z. Linear medium means that if I propagate light of 17 00:02:45,999 --> 00:02:53,689 frequency omega then it propagates as frequency omega itself, it does not generate new frequencies. 18 00:02:53,689 --> 00:03:01,189 Or if I propagate light in a medium then the refractive index of the medium remains independent 19 00:03:01,189 --> 00:03:09,870 of the intensity of light. So, this is a linear medium. Isotropic means that if I have two 20 00:03:09,870 --> 00:03:17,370 independent orthogonal polarizations for example, this and this. Then these polarizations see 21 00:03:17,370 --> 00:03:28,029 the same refractive index of the medium. And since it is a dielectric medium, so there 22 00:03:28,029 --> 00:03:35,370 are no free charges and no free currents. So, let me write down the Maxwell’s equations, 23 00:03:35,370 --> 00:03:42,790 which is the starting point for us. So, the first Maxwell’s equation is del dot D is 24 00:03:42,790 --> 00:03:53,280 equal to 0, which is nothing but the differential form of Gauss’s Law. The second equation 25 00:03:53,280 --> 00:04:04,639 is del dot B is equal to 0, which simply tells that magnetic monopoles do not exist. Third 26 00:04:04,639 --> 00:04:11,819 is del cross E is equal to minus del B over del t, which is nothing but Faraday's law 27 00:04:11,819 --> 00:04:21,880 in differential form. And the last one is del cross H is equal to del D over del t, 28 00:04:21,880 --> 00:04:30,199 which is nothing but Ampere's law. Apart from these four Maxwell’s equations we also have 29 00:04:30,199 --> 00:04:40,550 constitutive relations which are D is equal to epsilon E. And B is equal to mu H if I 30 00:04:40,550 --> 00:04:47,569 consider a dielectric medium, such as glass then it is a non-magnetic medium and in a 31 00:04:47,569 --> 00:04:52,830 non-magnetic medium mu is approximately equal to mu naught. 32 00:04:52,830 --> 00:05:02,870 So, at some places I will use this approximation also. I will use the convention throughout 33 00:05:02,870 --> 00:05:11,940 my lectures that curly letters wherever I use curly letters it means that they are the 34 00:05:11,940 --> 00:05:19,449 functions of x, y, z and t, while the straight letters do not have any time dependence they 35 00:05:19,449 --> 00:05:33,159 are the functions of only the special coordinates. So, what I want to do? I want to find out 36 00:05:33,159 --> 00:05:41,150 how electromagnetic wave propagates in a medium. And electromagnetic wave has associated electric 37 00:05:41,150 --> 00:05:50,889 field and magnetic field. So, what I want to do is basically I want to find out how 38 00:05:50,889 --> 00:05:56,639 the electric and magnetic fields associated with the light waves, vary with special coordinates 39 00:05:56,639 --> 00:06:05,289 and with time. For that I need to form a differential equation in E and a differential equation 40 00:06:05,289 --> 00:06:15,449 in H, which can tell me how E and H vary with x, y, z and t. 41 00:06:15,449 --> 00:06:22,389 In order to form the differential equation, let me again look at the four Maxwell’s 42 00:06:22,389 --> 00:06:29,750 equations and then let me do some mathematical manipulations. So, what I do? I take the curl 43 00:06:29,750 --> 00:06:39,669 of equation 3. So, I get del cross del cross E is equal to minus del del cross B over del 44 00:06:39,669 --> 00:06:42,460 t. 45 00:06:42,460 --> 00:06:54,500 Now I use the vector identities which gives me del of del this is equal to del cross del 46 00:06:54,500 --> 00:07:05,319 cross is equal to gradient of divergence of E minus del square E. And if I look at this 47 00:07:05,319 --> 00:07:15,580 the del cross B and I look at equation 4, if I multiply here by mu and here also then 48 00:07:15,580 --> 00:07:22,210 it becomes del cross b. So, del cross B is mu times del D over del t, So I put it here. 49 00:07:22,210 --> 00:07:30,009 Now since this is a homogenous medium and in a homogenous medium what I have del dot 50 00:07:30,009 --> 00:07:36,530 D is equal to 0 and D is equal to epsilon E. So, in a homogenous medium since epsilon 51 00:07:36,530 --> 00:07:46,290 is not a function of x, y and z, then that epsilon will come out of this del divergence 52 00:07:46,290 --> 00:07:53,400 operator. Which means that in a homogenous medium del dot D is equal to 0 will translate 53 00:07:53,400 --> 00:08:01,069 to del dot E is equal to 0. So, this term goes to 0. And again I put D 54 00:08:01,069 --> 00:08:08,400 is equal to epsilon E then this equation simply becomes del square E is equal to mu epsilon 55 00:08:08,400 --> 00:08:16,830 del 2 E over del t square. So, here you can see that this del square is nothing but del 56 00:08:16,830 --> 00:08:23,090 square over del x square plus del over del y square plus del square over del z square. 57 00:08:23,090 --> 00:08:35,000 So, this simply tells me that how E varies with x y and z and with time. So, I need to 58 00:08:35,000 --> 00:08:49,510 solve this differential equation. And get the functional form of E. So, in order to 59 00:08:49,510 --> 00:08:58,070 solve this equation, first what I would do I will consider a very simple case, simple 60 00:08:58,070 --> 00:09:07,721 case of one dimension. Which means that I assume that E varies only with z and t, if 61 00:09:07,721 --> 00:09:15,260 I assume this then this equation will now become del 2 E over del z square is equal 62 00:09:15,260 --> 00:09:21,230 to mu epsilon del 2 E over del t square. 63 00:09:21,230 --> 00:09:40,130 And since E is the vector Ex x cap plus Ey y cap plus Ez z cap, which means that this 64 00:09:40,130 --> 00:09:49,110 is note single equation, but it constitutes 3 equations - one in Ex, one in Ey and one 65 00:09:49,110 --> 00:09:50,800 in Ez. 66 00:09:50,800 --> 00:09:58,540 And so, so I have equation 3 equations in scalar components. So, if I draw this vector 67 00:09:58,540 --> 00:10:04,910 sign and write the scalar equation then it would be del 2 E over del z square is equal 68 00:10:04,910 --> 00:10:12,690 to mu epsilon del 2 E over del t square where E is nothing but it can be either Ex or Ey 69 00:10:12,690 --> 00:10:22,960 or Ez. So now, I need to solve this equation. How do I solve it? Well since E is a function 70 00:10:22,960 --> 00:10:35,560 of z and t and I also see that mu epsilon is independent of z and t, then I can use 71 00:10:35,560 --> 00:10:40,910 the method of separation of variables. What is the method of separation of variables? 72 00:10:40,910 --> 00:10:51,040 Well, I can represent this E of z t is equal to E of z and t of t. So, I separate them 73 00:10:51,040 --> 00:11:01,820 out. If I put this E into this equation, then I get an equation t times del 2 E over del 74 00:11:01,820 --> 00:11:09,260 z square is equal to mu epsilon Ez del 2 t over del t square. 75 00:11:09,260 --> 00:11:19,740 I divide this with this E times t and then I transform this equation into this form. 76 00:11:19,740 --> 00:11:27,130 What I have done essentially? I have separated out the variables. On the left hand side I 77 00:11:27,130 --> 00:11:36,200 have terms which contain only z, which are the function of z only and on the right hand 78 00:11:36,200 --> 00:11:42,720 side I have terms which contain t which are the function of t only. 79 00:11:42,720 --> 00:11:55,840 Now, to solve this what I do? The most natural thing is that I quit each of them to some 80 00:11:55,840 --> 00:12:03,610 constant. And since it is a second order differential equation, then that constant I will take in 81 00:12:03,610 --> 00:12:09,600 the form of his of a square. So, that I avoid square roots. 82 00:12:09,600 --> 00:12:23,620 So, I quit it to some constant k square. Let us now first solve the t part with this please 83 00:12:23,620 --> 00:12:34,330 see that k, k is simply a constant, a purely a mathematical constant. What does it signify? 84 00:12:34,330 --> 00:12:41,560 Physically we will see later on. So, if I now take this t part, then it is mu epsilon 85 00:12:41,560 --> 00:12:49,740 del 2 t over del t square is equal to k square t, and let me write this mu epsilon as 1 over 86 00:12:49,740 --> 00:13:01,970 v square, mu epsilon is equal to 1 over v square. Why I am doing this? Because this 87 00:13:01,970 --> 00:13:07,140 will then come here and I am taking in the form of a square, so that I have everywhere 88 00:13:07,140 --> 00:13:13,910 square itself. So, this will become now this and v square 89 00:13:13,910 --> 00:13:23,350 will come this side. So, what I do? I have now v square k square I represented by another 90 00:13:23,350 --> 00:13:37,440 constant omega square. So, in this way the t equation becomes del 2 t over del t square 91 00:13:37,440 --> 00:13:49,260 is equal to omega square times t. So, please again look that this k, this v and this omega 92 00:13:49,260 --> 00:13:57,410 there. I write now do not have any physical and interpretation for them, but later on 93 00:13:57,410 --> 00:14:06,760 I will see what is the physical interpretation. What is the solution of this equation now? 94 00:14:06,760 --> 00:14:13,000 I can immediately see that the solution of this equation is of the form e to the power 95 00:14:13,000 --> 00:14:20,500 plus minus omega t. Which means that s t goes to plus or minus infinity the solution blows 96 00:14:20,500 --> 00:14:31,970 up which means that this is not a physically viable solution. It is a solution it is a 97 00:14:31,970 --> 00:14:40,360 mathematically correct solution, but I do not have any use for such kind of solution 98 00:14:40,360 --> 00:14:49,920 because it is not physically viable. What kind of solution I am looking for? Well, I 99 00:14:49,920 --> 00:15:00,650 am looking for a solution which represents a wave. And a wave we will have an oscillatory 100 00:15:00,650 --> 00:15:09,740 solution, which means sin omega t or cosine omega t or a combination of these. 101 00:15:09,740 --> 00:15:17,040 So, I can immediately see that if I get a solution which is of the form of e to the 102 00:15:17,040 --> 00:15:24,550 power plus minus i omega t then it is an oscillatory solution. And for that I need to have minus 103 00:15:24,550 --> 00:15:32,680 omega square here, which means that I should have minus k square here. So, the most natural 104 00:15:32,680 --> 00:15:41,190 choice that occurred to me to take this as plus k square, does not give me any physically 105 00:15:41,190 --> 00:15:51,040 a viable solution. So, I take this as minus k square. So, I take it as minus k square 106 00:15:51,040 --> 00:15:56,930 and now if I do the same thing and get the solution then it is of course, e to the power 107 00:15:56,930 --> 00:16:06,510 plus minus i omega t, which is an oscillatory solution. 108 00:16:06,510 --> 00:16:16,300 Now let us look at z part. So, z part is 1 over E del 2 E over del z square is equal 109 00:16:16,300 --> 00:16:24,860 to minus k square or del 2 E over del z square is equal to minus k square Ez. Then the solution 110 00:16:24,860 --> 00:16:30,740 of this equation is simply of the form e to the power plus minus i kz. 111 00:16:30,740 --> 00:16:42,450 So, if I now combine these z part and t parts, then the complete solution is Ez t is equal 112 00:16:42,450 --> 00:16:48,500 to E 0 e to the power i plus minus kz plus minus omega t. This is the most general solution, 113 00:16:48,500 --> 00:16:59,110 where E 0 is the constant. It does not depend upon x, y, z or t. So, this is the most general 114 00:16:59,110 --> 00:17:13,159 solution of this equation. And this represents a wave. Now I can choose the sign shear appropriately 115 00:17:13,159 --> 00:17:23,699 So that it can represent a wave in a particular fashion. For example, if I choose the signs 116 00:17:23,699 --> 00:17:31,470 like this I represent it like this E 0 e to the power i omega t minus kz then this is 117 00:17:31,470 --> 00:17:37,930 the wave propagating in positive z direction. How it is propagating in positive z direction? 118 00:17:37,930 --> 00:17:56,220 I will see. What else I now see here is if I take a particular position z. If I fix z, 119 00:17:56,220 --> 00:18:04,940 let us say z is equal to 0. And I look at the solution in time then I see that the solution 120 00:18:04,940 --> 00:18:14,660 is oscillating in time. And it is oscillating in time with frequency omega after a certain 121 00:18:14,660 --> 00:18:23,130 time which is 2 pi over omega, it comes back to the same position. 122 00:18:23,130 --> 00:18:30,040 So, omega is nothing but the frequency or angular frequency of oscillation. So now, 123 00:18:30,040 --> 00:18:41,720 I interpret this omega which I had represented here, as the frequency. Now let us take a 124 00:18:41,720 --> 00:18:51,480 snapshot of this, is snapshot of this means that I freeze frame in time. Let us I fix, 125 00:18:51,480 --> 00:18:57,450 let us take that time is t is equal to 0, then the solution is e to the power minus 126 00:18:57,450 --> 00:19:07,300 i kz. Now if I look at this I plot it. So, it would be an oscillatory wave like this. 127 00:19:07,300 --> 00:19:16,930 In z a sinusoidal function in z. And I see that it repeats itself after a distance 2 128 00:19:16,930 --> 00:19:27,110 pi over k. Then k is nothing but the wave vector. So, here the k which I had taken just 129 00:19:27,110 --> 00:19:32,590 this constant purely mathematical constant now I see that it is nothing but the wave 130 00:19:32,590 --> 00:19:43,681 vector. And the omega which came out as v times k is nothing but the frequency. What 131 00:19:43,681 --> 00:19:55,980 is v I still need to see? Let us examine the nature of this wave what kind of wave it is. 132 00:19:55,980 --> 00:20:03,420 If I look at this phase of this wave then I find out what are the surfaces of constant 133 00:20:03,420 --> 00:20:13,570 phase, then what I find that if I put omega t minus kz is equal to constant so that I 134 00:20:13,570 --> 00:20:22,250 get the surface of constant phase. Then at a particular time t I get the surface 135 00:20:22,250 --> 00:20:31,020 z is equal to constant. z is equal to constant is an equation of a plane, which means that 136 00:20:31,020 --> 00:20:38,890 the surface of constant phase is a plane. And so, this kind of wave is known as plane 137 00:20:38,890 --> 00:20:53,770 wave. Let me find out at what velocity this surface of constant phase is moving. So, for 138 00:20:53,770 --> 00:21:01,130 that what I do I take the derivative of this, I differentiate this. When I differentiate 139 00:21:01,130 --> 00:21:09,100 this then I get omega Dt minus kz is equal to 0, which means Dz over Dt is equal to omega 140 00:21:09,100 --> 00:21:22,390 over k and omega over k is nothing but v, which means that v which I had from here is 141 00:21:22,390 --> 00:21:33,640 nothing but the velocity of the wave, the phase velocity. So, this is a plane wave moving 142 00:21:33,640 --> 00:21:40,990 with phase velocity omega over and that omega over k is nothing but v; the constant which 143 00:21:40,990 --> 00:21:45,110 we represent it here. 144 00:21:45,110 --> 00:21:56,020 Now, let me generalized this for 3 D case. So, the 3 D wave equation is this. And so, 145 00:21:56,020 --> 00:22:03,850 the solution would be E x y z t is equal to E0, E0 is again a constant times e to the 146 00:22:03,850 --> 00:22:11,820 power i omega t. So, the t solution remains the same. And since now I have xyz all of 147 00:22:11,820 --> 00:22:19,970 them. So, the solution would you minus kx x minus ky y minus kz z. What is k? k is nothing 148 00:22:19,970 --> 00:22:35,790 but kx x cap plus ky y cap plus kz z cap. And if I represent the position vector in 149 00:22:35,790 --> 00:22:54,440 vector form then it is x x cap plus y y cap plus z z cap. So, this is nothing but this 150 00:22:54,440 --> 00:23:01,809 is nothing but k dot r. So, I represent it as E 0 e to the power i omega t minus k dot 151 00:23:01,809 --> 00:23:10,370 r. And from here you can immediately see that k square is kx square plus ky square plus 152 00:23:10,370 --> 00:23:21,940 kz square. Now, for a given k I can have infinite sets 153 00:23:21,940 --> 00:23:30,930 of kx ky kz. So, infinite sets of kx ky kz can give me the same value of k. What is the 154 00:23:30,930 --> 00:23:40,410 meaning of those infinite sets? If you look at this is k dot r, in what direction this 155 00:23:40,410 --> 00:23:49,720 is moving? It is moving in one particular direction, which is represented by the values 156 00:23:49,720 --> 00:23:57,880 of kx ky and kz. kx ky and kz are nothing but the projections of vector k on x y and 157 00:23:57,880 --> 00:24:11,260 z axis. So, the values of kx ky and kz will give me at what angle this wave is moving. 158 00:24:11,260 --> 00:24:20,940 And with the propagation constant k the magnitude of propagation constant is always this. So, 159 00:24:20,940 --> 00:24:29,390 for a given magnitude of propagation constant k, I can have various angles possible. So, 160 00:24:29,390 --> 00:24:38,320 this tells me that if a light wave is allowed to go in an infinitely extended dielectric 161 00:24:38,320 --> 00:24:45,750 medium, then it can go in any direction. There is no restriction on it, it can go in this 162 00:24:45,750 --> 00:24:49,550 direction, this direction, this direction, this direction, this direction, this direction 163 00:24:49,550 --> 00:24:58,520 any direction is possible for the same value of k. 164 00:24:58,520 --> 00:25:09,370 I can do the same analysis for the magnetic field. And I will get the solution as H x 165 00:25:09,370 --> 00:25:18,929 y z t is equal to H 0 e to the power i omega t minus kx x minus ky y minus kz z or H 0 166 00:25:18,929 --> 00:25:26,330 e to the power i omega t minus k dot r. So, for a light wave I have got now the associated 167 00:25:26,330 --> 00:25:32,450 electric field which is represented by this, and associated magnetic field which is represented 168 00:25:32,450 --> 00:25:45,840 by this. These are of course, by scalar components. The same solutions are for E x y z and Hx 169 00:25:45,840 --> 00:25:55,970 Hy Hz. Well, so let me now consider a wave which is going in certain direction and let 170 00:25:55,970 --> 00:26:03,410 me choose does that direction as z direction. If I choose the direction is z direction, 171 00:26:03,410 --> 00:26:10,940 then mathematically I can write the electric field associated with this as E 0 e to the 172 00:26:10,940 --> 00:26:18,640 power i omega t minus kz. Of course, this constitutes 3 equations, one is in Ex another 173 00:26:18,640 --> 00:26:23,270 is in Ey and yet another one is in Ez. 174 00:26:23,270 --> 00:26:38,320 Similarly, the magnetic field can be written like this, where H is Hx x Hy y and Hz z cap. 175 00:26:38,320 --> 00:26:45,510 Now let me do one thing with this, let me take the divergence of this and since this 176 00:26:45,510 --> 00:26:53,309 is a homogenous medium, then del dot E would be equal to 0. If I put del dot E is equal 177 00:26:53,309 --> 00:26:59,450 to 0 it means that del Ex over del x plus del Ey over del y plus del Ez over del z is 178 00:26:59,450 --> 00:27:08,710 equal to 0. If I pick up Ex components from here, then it would be E 0 x e to the power 179 00:27:08,710 --> 00:27:16,600 i omega t minus kz. E 0 x is the constant and here you do not see any term of x which 180 00:27:16,600 --> 00:27:24,820 means del Ex over del x is equal to 0. Similarly, del Ey over del y is equal to 0, which means 181 00:27:24,820 --> 00:27:33,809 that if these 2 are 0 then this implies that del Ez over del z should be equal to 0, one 182 00:27:33,809 --> 00:27:42,640 thing. Second I do del dot B is equal to 0 I use this Maxwell’s equation del dot B 183 00:27:42,640 --> 00:27:49,370 is equal to 0. So, similarly it will also give me the del Hz over del z is equal to 184 00:27:49,370 --> 00:27:54,740 0. So, these solutions of the differential equations 185 00:27:54,740 --> 00:28:03,890 the wave equations which I got earlier. Now if I put these solutions into these Maxwell’s 186 00:28:03,890 --> 00:28:10,630 equations then they give me del Ez over del z is equal to 0, and del Hz over del z is 187 00:28:10,630 --> 00:28:24,490 equal to 0. What does it mean? If I look at Ez, from here then Ez would be E 0 z E 2 the 188 00:28:24,490 --> 00:28:32,690 power i omega t minus kz. If I look at Hz from here then it would be H 0 z e to the 189 00:28:32,690 --> 00:28:43,520 power i omega t minus kz. Now for this to be 0 there are 2 possibilities. One is Ez 190 00:28:43,520 --> 00:28:52,679 is constant and another is that the amplitude of Ez itself is 0. Let me first look at the 191 00:28:52,679 --> 00:28:59,000 first possibility that Ez is constant. If Ez is constant which means this whole thing 192 00:28:59,000 --> 00:29:07,630 is constant, E 0 z itself is a constant. So, which means that this term has to be a constant, 193 00:29:07,630 --> 00:29:14,850 if this term is a constant then there is no wave. Because then even the x component and 194 00:29:14,850 --> 00:29:19,250 y components they will become constant they will not vary with z and t. 195 00:29:19,250 --> 00:29:25,701 So, there is no wave solution, which means that I cannot put e to the power i omega t 196 00:29:25,701 --> 00:29:32,740 minus kz is constant. So, in order to satisfy this equation the only possibilities E 0 z 197 00:29:32,740 --> 00:29:42,320 0 which means that Ez is equal to 0 that is there is no longitudinal component because 198 00:29:42,320 --> 00:29:49,610 z is the direction of propagation itself. So, Ez is nothing but the longitudinal component. 199 00:29:49,610 --> 00:29:57,790 So, there is no longitudinal component. Similarly, this equation gives me that H 0 H should be 200 00:29:57,790 --> 00:30:03,910 equal to 0, or in this way Hz is equal to 0. So, there is no longitudinal component 201 00:30:03,910 --> 00:30:14,520 of magnetic field also which means that the solutions of the wave equation that I got 202 00:30:14,520 --> 00:30:20,679 are the waves which do not have any longitudinal components, which means that they are the 203 00:30:20,679 --> 00:30:32,210 transverse waves. Now if the wave is propagating in z direction, 204 00:30:32,210 --> 00:30:43,880 then there is no Ez there is no Hz. Which means that E cannot vibrate along z and H 205 00:30:43,880 --> 00:30:52,821 cannot vibrate along z. So, they can only vibrate in x and y directions. That is they 206 00:30:52,821 --> 00:30:58,809 can vibrate in a plane perpendicular to the direction of propagation if z is the direction 207 00:30:58,809 --> 00:31:05,590 of propagation which is the direction of this pointer then and then this is the transverse 208 00:31:05,590 --> 00:31:13,049 plane. So, E and H can only vibrate in this plane. 209 00:31:13,049 --> 00:31:25,020 So, which means that your, if this is the plane, then E can vibrate along this or along 210 00:31:25,020 --> 00:31:32,590 this or along this or along this. And similarly H can vibrator along this along this along 211 00:31:32,590 --> 00:31:44,230 this, what does this direction of vibration represent? This direction of vibration represents 212 00:31:44,230 --> 00:31:51,070 nothing but the polarization. The direction of vibration of electric field vector represents 213 00:31:51,070 --> 00:31:58,530 the direction of polarization. So, if I again look at this wave which is propagating in 214 00:31:58,530 --> 00:32:10,160 positive z direction, then the non-vanishing components of E and H would be Ex, Ey and 215 00:32:10,160 --> 00:32:15,140 Hx, Hy like this. 216 00:32:15,140 --> 00:32:27,710 And now if I consider a particular case, where I say let my electric field of light vibrate 217 00:32:27,710 --> 00:32:36,490 along x. And so, the y component is 0 because x and y are orthogonal to each other. So, 218 00:32:36,490 --> 00:32:41,880 so if it is vibrating along x then there is no y component. 219 00:32:41,880 --> 00:32:50,740 So, electric field is vibrating along this, and the wave is propagating like this. Then 220 00:32:50,740 --> 00:32:57,679 such a wave is known as linearly polarized wave polarized in x direction. So, this is 221 00:32:57,679 --> 00:33:06,440 x polarized wave. You can understand it in a very simple way that if you tie a string, 222 00:33:06,440 --> 00:33:15,169 one end of the string to tie on that side of the wall. And one end you keep with you 223 00:33:15,169 --> 00:33:21,780 and shake it, let us say this is x direction, if you shake it like this then a wave propagates 224 00:33:21,780 --> 00:33:32,910 like this, in the string. So, wave is going in z direction while a point on the string 225 00:33:32,910 --> 00:33:40,480 or particle on the string will oscillate in x direction. If you take any point on the 226 00:33:40,480 --> 00:33:45,960 string the direction of oscillation of that point will always be in x direction. So, this 227 00:33:45,960 --> 00:33:55,040 is x polarized wave. If you vibrate it in y-direction then it goes like this. So, particle 228 00:33:55,040 --> 00:33:58,960 oscillates in y direction then it is y polarized wave ok. 229 00:33:58,960 --> 00:34:09,590 So, if I consider this then what is the corresponding magnetic field? To get the corresponding magnetic 230 00:34:09,590 --> 00:34:16,829 field I use the Maxwell’s equation which relates the electric field to magnetic field. 231 00:34:16,829 --> 00:34:30,190 Then if I take the x component here Hx then del Hx over del t will be 0, if I use this. 232 00:34:30,190 --> 00:34:43,069 And del Hy over del t times mu would be equal to minus del Ex over del z. This gives me 233 00:34:43,069 --> 00:34:53,220 that H 0 x is equal to 0 which means that if Ex is non 0 then Hx would be 0 first thing. 234 00:34:53,220 --> 00:34:59,880 So, if the wave is polarized along x that is electric field vector is vibrating along 235 00:34:59,880 --> 00:35:08,869 x, then there is no component of magnetic field vibration in x. Magnetic field vibrates 236 00:35:08,869 --> 00:35:17,999 along y with what amplitude it comes from here. So, if I put this Hy from here and Ex 237 00:35:17,999 --> 00:35:26,890 from here and simplify this then the corresponding magnetic field amplitude is k over omega mu 238 00:35:26,890 --> 00:35:37,049 times E 0 x, or omega epsilon over k E 0 x. 239 00:35:37,049 --> 00:35:48,459 So, in summary what are the electric and magnetic fields associated with a light beam? Well 240 00:35:48,459 --> 00:35:54,350 if I take the example of linearly polarized wave which is polarized in x direction, and 241 00:35:54,350 --> 00:36:04,150 propagating in z-direction. Then the electric field associated with this is x cap E 0 e 242 00:36:04,150 --> 00:36:09,220 to the power i omega t minus kz it is the same equation as in the previous slide I have 243 00:36:09,220 --> 00:36:19,630 just generalized it instead of putting E 0 x, I have now put E 0 and x cap. 244 00:36:19,630 --> 00:36:26,329 Then the corresponding magnetic field would be in y direction. So, it would be y cap H 245 00:36:26,329 --> 00:36:34,250 0 e to the power i omega t minus kz. And the amplitude of H would be related to the amplitude 246 00:36:34,250 --> 00:36:41,559 of E by this relation. I can also write it down in terms of B this would be simplify 247 00:36:41,559 --> 00:36:50,690 y cap B 0 e to the power i omega t minus kz where B 0 would be nothing but k over omega 248 00:36:50,690 --> 00:36:57,730 E 0 because if you take this mu on this site, and combine this mu with H then it will become 249 00:36:57,730 --> 00:37:03,349 B. What is k over omega? k over omega is nothing 250 00:37:03,349 --> 00:37:12,779 but 1 over v. So, B 0 is nothing but E 0 over v. v is the velocity of electromagnetic wave. 251 00:37:12,779 --> 00:37:20,200 If I consider the free space than this v is nothing but c. The velocity of light in free 252 00:37:20,200 --> 00:37:26,170 space, which is 3 into 10 to the power 8, which is very large. The amplitude B 0 would 253 00:37:26,170 --> 00:37:35,369 be much smaller than the amplitude E 0 which means that the amplitude of magnetic field 254 00:37:35,369 --> 00:37:41,400 is very, very small as compared to the electric field associated with light. And that is why 255 00:37:41,400 --> 00:37:49,960 it is the electric field that affects the retina of our eye. And so, the direction of 256 00:37:49,960 --> 00:37:57,220 polarization is associated with the direction of electric field vibration. 257 00:37:57,220 --> 00:38:07,609 In the next lecture we would see more carefully about the polarization and we would also see 258 00:38:07,609 --> 00:38:11,220 how much power is associated with this kind of wave. 259 00:38:11,220 --> 00:38:12,160 Thank you.