1 00:00:19,020 --> 00:00:25,320 In the last lecture we had understood the propagation characteristics or light guidance 2 00:00:25,320 --> 00:00:29,320 in an optical fiber using ray theory. 3 00:00:29,320 --> 00:00:38,370 We had seen that using ray theory we can explain the light guidance in terms of successive 4 00:00:38,370 --> 00:00:40,110 total internal reflections. 5 00:00:40,110 --> 00:00:50,980 We have also seen that only a particular guidance cone is possible, which has a maximum light 6 00:00:50,980 --> 00:00:52,079 acceptance angle. 7 00:00:52,079 --> 00:01:02,570 So, only those rays which fall in this light guidance cone are guided, but in ray theory 8 00:01:02,570 --> 00:01:09,509 there are certain limitations, these limitations are number one when we go down to the fiber 9 00:01:09,509 --> 00:01:16,250 core dimensions which are comparable to the wavelength of light, then ray theory does 10 00:01:16,250 --> 00:01:19,040 not work it is not valid. 11 00:01:19,040 --> 00:01:28,370 So, if we talk about telecom fiber which has co dimensions of about 10 micron diameter. 12 00:01:28,370 --> 00:01:32,910 So, our ray theory would not work. 13 00:01:32,910 --> 00:01:38,880 Second thing is there are certain intricacies of light guidance, which ray theory cannot 14 00:01:38,880 --> 00:01:39,880 explain. 15 00:01:39,880 --> 00:01:52,390 In these intricacies, one is that when we launch light into an optical fiber, then light 16 00:01:52,390 --> 00:02:02,220 basically propagates or the light guidance in an optical fiber can be understood in the 17 00:02:02,220 --> 00:02:07,350 form of certain modes of propagation. 18 00:02:07,350 --> 00:02:12,740 These modes of propagation correspond to certain discrete angles, at which the light should 19 00:02:12,740 --> 00:02:23,190 be launched into the fiber, and these modes are discrete ray theory cannot explain these 20 00:02:23,190 --> 00:02:26,959 discrete modes. 21 00:02:26,959 --> 00:02:34,930 Another thing is that when light propagates in an optical fiber, in certain amount of 22 00:02:34,930 --> 00:02:38,660 light also extends to the cladding. 23 00:02:38,660 --> 00:02:47,560 So, how light penetrates into the cladding region, ray theory is not able to give insight 24 00:02:47,560 --> 00:02:50,340 into that also. 25 00:02:50,340 --> 00:02:59,310 See in order to understand more accurately the light guidance in an optical fiber, we 26 00:02:59,310 --> 00:03:06,420 should resort to a more accurate theory and that is wave theory. 27 00:03:06,420 --> 00:03:18,290 So, in this lecture we will extend our understanding of light guidance in an optical fiber to wave 28 00:03:18,290 --> 00:03:19,500 theory. 29 00:03:19,500 --> 00:03:28,370 So, the outline of this lecture is in this way let first we will understand what is the 30 00:03:28,370 --> 00:03:34,150 meaning of modes of an optical fiber, and why after all we need to understand modes 31 00:03:34,150 --> 00:03:40,350 of an optical fiber, what is the necessity for modal analysis of an optical fiber. 32 00:03:40,350 --> 00:03:51,860 When a natural question arises how many modes of fiber supports, and is they are some bearing 33 00:03:51,860 --> 00:04:00,480 on the number of modes on fiber parameters, and also what would be the effect of fiber 34 00:04:00,480 --> 00:04:05,540 parameters on the number of modes possible in an optical fiber. 35 00:04:05,540 --> 00:04:12,400 We will also see that we can integrate all the parameters of the fiber and the wavelength 36 00:04:12,400 --> 00:04:19,959 of light into a single parameter, which is known as normalized frequency. 37 00:04:19,959 --> 00:04:26,590 Then we will try to understand what are single modes and what are multimode fibers and then 38 00:04:26,590 --> 00:04:29,130 graded index fibers. 39 00:04:29,130 --> 00:04:35,020 Let us first understand modes for that let me first take a laser beam a typical beam 40 00:04:35,020 --> 00:04:43,499 from a laser pointer for example, or typical beam from a single mode helium neon laser. 41 00:04:43,499 --> 00:04:54,781 If I look at this beam here when it just exits from the laser the spot looks like this, if 42 00:04:54,781 --> 00:05:05,029 I capture this beam just at the exit of this laser as soon as it comes out then this spot 43 00:05:05,029 --> 00:05:07,939 looks like this. 44 00:05:07,939 --> 00:05:17,689 And if I see how the intensity of this spot varies along a diameter it looks like this. 45 00:05:17,689 --> 00:05:26,810 Now I like it propagate over a large distance and then after few meters, I intercept this 46 00:05:26,810 --> 00:05:34,559 beam, then I see that the spot size has become a bigger now and this is how the intensity 47 00:05:34,559 --> 00:05:36,190 profile would look like. 48 00:05:36,190 --> 00:05:43,289 If I go even further the spot size becomes even larger and the beam looks like this, 49 00:05:43,289 --> 00:05:51,340 and if I know the intensity pattern here I would be able to tell what would be the intensity 50 00:05:51,340 --> 00:05:57,270 pattern here or there, I know how this beam of light propagates in a medium. 51 00:05:57,270 --> 00:06:05,360 But if I have in the same way an optical fiber and I launch that laser beam which is not 52 00:06:05,360 --> 00:06:12,680 optimized perhaps for this fiber, I just take any laser beam or any intensity pattern and 53 00:06:12,680 --> 00:06:18,620 I launch it into this input end of the fiber. 54 00:06:18,620 --> 00:06:26,050 Then can you predict how would this intensity pattern change as it will go along the fiber 55 00:06:26,050 --> 00:06:35,711 or would it change at all or not, how would this propagate, how would this intensity pattern 56 00:06:35,711 --> 00:06:45,590 evolve as it goes along the length of the fiber, and another question is why after all 57 00:06:45,590 --> 00:06:54,009 it is important to know how does this intensity profile propagate in this fiber. 58 00:06:54,009 --> 00:07:02,449 Why it is necessary to know the evolution of intensity pattern in the fiber. 59 00:07:02,449 --> 00:07:10,659 So, let me see how does it happen, these are some stimulated patterns. 60 00:07:10,659 --> 00:07:20,889 See if I launch some arbitrary intensity pattern in the fiber and I capture that at different 61 00:07:20,889 --> 00:07:27,550 lengths of the fiber, then I see that these intensity patterns at different lengths do 62 00:07:27,550 --> 00:07:33,150 not have any correlation with another. 63 00:07:33,150 --> 00:07:44,189 However, if I launch this particular intensity pattern in the fiber then what happens did 64 00:07:44,189 --> 00:07:54,039 it goes as it is, it does not change its shape similarly another one like this and yet another 65 00:07:54,039 --> 00:07:55,039 one. 66 00:07:55,039 --> 00:08:01,530 So, what I see that here are certain intensity patterns and corresponding amplitude patterns, 67 00:08:01,530 --> 00:08:12,099 which are sustained when they propagate through the fiber, their shapes do not change. 68 00:08:12,099 --> 00:08:22,809 And I also find out that each of such field patterns has certain definite velocity, and 69 00:08:22,809 --> 00:08:27,939 the velocities of different patterns are different. 70 00:08:27,939 --> 00:08:36,440 And I call these field patterns which are sustained throughout the length of propagation 71 00:08:36,440 --> 00:08:40,779 of fiber as modes of the fiber. 72 00:08:40,779 --> 00:08:55,960 Now, let me take a laser beam and send it through a thick block of glass, which is let 73 00:08:55,960 --> 00:09:02,310 us say 2 centimeter by 2 centimeter block of glass and this laser beam has dimensions 74 00:09:02,310 --> 00:09:11,350 maybe 5 millimeter across and the refractive index of this class is n. 75 00:09:11,350 --> 00:09:18,840 Then I know the velocity of light in this block of glass of refractive index n can be 76 00:09:18,840 --> 00:09:20,410 given by c by n. 77 00:09:20,410 --> 00:09:27,590 Where c is the velocity of light in free space. 78 00:09:27,590 --> 00:09:37,000 My question is if I have a certain field pattern which is propagating in the fiber, what velocity 79 00:09:37,000 --> 00:09:38,510 would it have. 80 00:09:38,510 --> 00:09:41,660 So, let us see that. 81 00:09:41,660 --> 00:09:47,520 This is a transverse cross section of a fiber we are the core is 10 micron across and then 82 00:09:47,520 --> 00:09:58,350 you have a cladding, and this is one intensity pattern which I launch into this fiber, and 83 00:09:58,350 --> 00:09:59,560 which is the mode. 84 00:09:59,560 --> 00:10:12,050 So, it sustains its shape as it propagates, what do I see here that certain part of the 85 00:10:12,050 --> 00:10:14,650 intensity extends into cladding. 86 00:10:14,650 --> 00:10:20,340 So, this some proportion of power in the core and some proportion of power in the cladding, 87 00:10:20,340 --> 00:10:31,070 unlike in this case where the whole spot is in the refractive index region n, but here 88 00:10:31,070 --> 00:10:37,070 it is distributed in the regions of refractive indices n1and n2. 89 00:10:37,070 --> 00:10:43,950 So, what would be the velocity what would be the effective refractive index it would 90 00:10:43,950 --> 00:10:55,470 see intuitively I can say that the refractive index that this field pattern will experience 91 00:10:55,470 --> 00:11:05,090 would neither be n1 nor it would be n2, but it would be somewhere between n2 and n1 and 92 00:11:05,090 --> 00:11:13,150 this effective refractive index felt by or experienced by this particular shape is also 93 00:11:13,150 --> 00:11:16,060 known in short as effective index. 94 00:11:16,060 --> 00:11:24,680 So, from now onwards I will just call it effective index and I represent it by n effective. 95 00:11:24,680 --> 00:11:33,140 So, this neff lies now between n2 and n1 and I know that there are different such patterns, 96 00:11:33,140 --> 00:11:40,380 and for different patterns or different modes the proportion of field in the core and in 97 00:11:40,380 --> 00:11:48,350 the cladding is different, and that is why they would experience different effective 98 00:11:48,350 --> 00:11:57,990 refractive index of this composite medium or they should have different effective indices. 99 00:11:57,990 --> 00:12:04,440 So, different modes would now travel with different velocities, they will have different 100 00:12:04,440 --> 00:12:08,840 effective indices, they will have different propagation constants. 101 00:12:08,840 --> 00:12:20,990 So, in a fiber these kind of intensity patterns I have seen that they extent to cladding, 102 00:12:20,990 --> 00:12:24,920 but energy does not propagate out up to infinity into cladding. 103 00:12:24,920 --> 00:12:31,670 The energies still guided in the core, it certain part of the energy is only extended 104 00:12:31,670 --> 00:12:40,920 to the cladding and these modes are called guided modes of the fiber, and for these modes 105 00:12:40,920 --> 00:12:49,440 neff lies between n2 and n1, and there are only certain discrete patterns certain discrete 106 00:12:49,440 --> 00:12:50,440 field patterns. 107 00:12:50,440 --> 00:12:52,990 So, these are discrete modes. 108 00:12:52,990 --> 00:13:00,010 Apart from these there are certain modes which carry energy in the cladding up to infinity. 109 00:13:00,010 --> 00:13:11,300 So, energy radiates out these modes are called radiation modes and for these modes the effective 110 00:13:11,300 --> 00:13:18,060 index or neff is less than n2 and they form a continuum. 111 00:13:18,060 --> 00:13:26,970 So, they are continue of modes radiation modes, they are not discrete; and what I also see 112 00:13:26,970 --> 00:13:36,990 is that these modes are orthogonal and they form a complete set what does it mean that 113 00:13:36,990 --> 00:13:45,850 if I have any arbitrary field pattern then that arbitrary field pattern can be expressed 114 00:13:45,850 --> 00:13:57,590 as the superposition of all these modes, and since it is possible to completely define 115 00:13:57,590 --> 00:14:02,550 the propagation of these modes, I know the propagation characteristics of these modes 116 00:14:02,550 --> 00:14:09,480 how do they propagate in a fiber, what are the field patterns, what are their respective 117 00:14:09,480 --> 00:14:19,550 effective indices and correspondingly velocities, then it is possible to predict the evolution 118 00:14:19,550 --> 00:14:25,500 of any arbitrary field pattern when it propagates in an optical fiber. 119 00:14:25,500 --> 00:14:36,450 So, that is why understanding of modes is very important. 120 00:14:36,450 --> 00:14:47,970 Let us now see if I have a fiber with core refractive index 1.45, cladding refractive 121 00:14:47,970 --> 00:15:01,280 index 1.44 and I make a fiber of core radius 20 micrometer and find out the modes of the 122 00:15:01,280 --> 00:15:04,900 fiber at 1550 nanometer wavelength. 123 00:15:04,900 --> 00:15:11,190 If I launch a light of wavelength 1550 nanometer into this fiber then what are the possible 124 00:15:11,190 --> 00:15:13,600 guided modes of this fiber. 125 00:15:13,600 --> 00:15:21,490 We will learn how to calculate these modes how to find out these nodes as we go along 126 00:15:21,490 --> 00:15:32,230 in this course, but let me just give you that for this fiber at this wavelength, these are 127 00:15:32,230 --> 00:15:39,520 the possible intensity patterns and some of their orientations. 128 00:15:39,520 --> 00:15:51,620 So, these have different orientations also, when I change now the core radius I bring 129 00:15:51,620 --> 00:16:01,590 down the core radius from 20 to 15, I see that in such a fiber these 2 patterns are 130 00:16:01,590 --> 00:16:03,860 missing they are not supported. 131 00:16:03,860 --> 00:16:13,570 If I further bring down the core radius, then only these 2 patterns are supported. 132 00:16:13,570 --> 00:16:18,790 Even further bringing down the radius gives me only one pattern which is possible. 133 00:16:18,790 --> 00:16:30,440 So, what do I see that if for a given core and cladding refractive index and giving wavelength, 134 00:16:30,440 --> 00:16:38,090 if I change the radius of the core if I decrease the radius of the core, then the number of 135 00:16:38,090 --> 00:16:40,950 modes become smaller and smaller. 136 00:16:40,950 --> 00:16:47,040 So, a fiber with smaller core has less number of modes. 137 00:16:47,040 --> 00:16:50,779 Now, what I do? 138 00:16:50,779 --> 00:17:02,170 I fix the cladding refractive index to 1.44 and the core radius 28 micrometer, again I 139 00:17:02,170 --> 00:17:09,909 launch light of wavelength 1550, but now I change the refractive index of the core and 140 00:17:09,909 --> 00:17:13,370 hence the numerical aperture of the fiber. 141 00:17:13,370 --> 00:17:21,260 So, for this combination of n1 and n2 the numerical aperture of the fiber is 0.42 and 142 00:17:21,260 --> 00:17:30,170 I see 7 patterns propagating in this fiber. 143 00:17:30,170 --> 00:17:41,430 If I change n1 from 1.5 to 1.7, and correspondingly I change numerical aperture from 0.42 to 0.29 144 00:17:41,430 --> 00:17:50,660 I decrease the number of modes, and for 0.2 numerical aperture there are only 2 and then 145 00:17:50,660 --> 00:17:55,990 if I bring down the numerical aperture to 0.17 there is only one mode. 146 00:17:55,990 --> 00:18:03,120 So, I see that a fiber with lower numerical aperture has less number of modes. 147 00:18:03,120 --> 00:18:13,240 So, I have seen that the number of modes depend upon the core radius, and the index contrast 148 00:18:13,240 --> 00:18:15,570 or numerical aperture. 149 00:18:15,570 --> 00:18:24,880 Index contrast between core and cladding refractive indices or numerical aperture the question 150 00:18:24,880 --> 00:18:33,390 is this wavelength also affect the number of modes that can propagate in an optical 151 00:18:33,390 --> 00:18:41,990 fiber for that let me perform a very simple experiment. 152 00:18:41,990 --> 00:18:43,820 So, what I do? 153 00:18:43,820 --> 00:18:47,910 I take a fiber which is given which is fixed. 154 00:18:47,910 --> 00:18:59,040 So, n1, n2 and a they are fixed, now I launch light into this fiber monochromatic light 155 00:18:59,040 --> 00:19:00,790 from tunable light source. 156 00:19:00,790 --> 00:19:12,280 Tunable laser whose wavelength I can change, and I capture the output from the fiber using 157 00:19:12,280 --> 00:19:16,700 a microscope objective and image it onto a CCD. 158 00:19:16,700 --> 00:19:25,490 So, that I can see the pattern what I wish to do is when I change the wavelength I would 159 00:19:25,490 --> 00:19:31,490 like to see how the output pattern changes. 160 00:19:31,490 --> 00:19:42,180 Let me perform this let me see the CCD output, when I launch wavelength 200 nanometer light 161 00:19:42,180 --> 00:19:49,150 of wavelength 200 nanometer, then this is the kind of pattern I see. 162 00:19:49,150 --> 00:19:56,660 I see a speckle pattern this is nothing, but the interference between all the possible 163 00:19:56,660 --> 00:20:06,270 modes of the fiber and my calculations tell me that this fiber supports 39 modes, at this 164 00:20:06,270 --> 00:20:09,309 wavelength at 200 nanometer wavelength. 165 00:20:09,309 --> 00:20:17,020 So, this is the interference of 39 intensity patterns, of field patterns. 166 00:20:17,020 --> 00:20:27,929 When I change the wavelength from 200 to 250 this changes the grain size of speckle here 167 00:20:27,929 --> 00:20:31,610 increases slightly number of modes they are decreased. 168 00:20:31,610 --> 00:20:39,800 I further change it to 300 nanometer the number of modes are now 17, I increase it to 350 169 00:20:39,800 --> 00:20:44,880 nanometer number of modes are 13, 400 nanometer number of modes are 10. 170 00:20:44,880 --> 00:20:55,210 And now you can see the grain size in the speckle pattern; 500 nanometer 7 modes 600 171 00:20:55,210 --> 00:20:56,210 nanometers. 172 00:20:56,210 --> 00:21:04,370 5 modes 70 nanometers 4 modes 900 nanometers only 2 modes. 173 00:21:04,370 --> 00:21:12,940 And you just see the pattern here 1200 nanometers again 2 modes and when I have 1500 nanometer 174 00:21:12,940 --> 00:21:20,990 I see a very clean pattern here, which is the fundamental mode, which means that there 175 00:21:20,990 --> 00:21:25,390 are no speckles here which means that there is only one mode. 176 00:21:25,390 --> 00:21:32,340 So, when I increase the wavelength, I see the number of modes decrease. 177 00:21:32,340 --> 00:21:44,390 How can I integrate all these observations? 178 00:21:44,390 --> 00:21:58,929 What I can see is that the number of modes depends upon core radius, numerical aperture 179 00:21:58,929 --> 00:21:59,929 and wavelength. 180 00:21:59,929 --> 00:22:09,760 And I can integrate all these parameters into a single parameter which is known as normalized 181 00:22:09,760 --> 00:22:19,050 frequency because that it goes as 1 over lambda, which is defined as V is equal to 2 pi over 182 00:22:19,050 --> 00:22:26,420 lambda naught times a times numerical aperture, and since numerical aperture is defined as 183 00:22:26,420 --> 00:22:29,870 a square root of n1 square minus n2 square. 184 00:22:29,870 --> 00:22:36,920 So, V is defined as 2 pi over lambda naught times a times square root of n1 square minus 185 00:22:36,920 --> 00:22:40,020 n2 square. 186 00:22:40,020 --> 00:22:49,059 So, how many modes would be supported by a fiber will depend upon what is the value of 187 00:22:49,059 --> 00:22:58,290 V. And we find that when V is less than a certain number 2.4048, we will find out from 188 00:22:58,290 --> 00:23:09,160 where this number has come later on, but when V is less than 2.4048 then fiber supports 189 00:23:09,160 --> 00:23:19,280 only one mode, and such a fiber is called single mode fiber and where it is larger than 190 00:23:19,280 --> 00:23:24,190 2.4048 then the number of modes are more. 191 00:23:24,190 --> 00:23:31,170 So, for V less than 2.4048 it is single mode fiber, and if V is much larger than 2.4048 192 00:23:31,170 --> 00:23:41,710 then we call it multimode fiber, for intermediate values of V we can call it few mode fiber. 193 00:23:41,710 --> 00:23:52,010 Let us look at single mode fiber typical single mode fiber; it is used in long haul optical 194 00:23:52,010 --> 00:23:54,450 telecommunication. 195 00:23:54,450 --> 00:24:02,480 The fiber which is laid on sea bed is single mode fiber, it is a small core diameter typically 196 00:24:02,480 --> 00:24:11,290 10 micrometer, and you can couple light into this fiber or the light source which we use 197 00:24:11,290 --> 00:24:16,690 with this fiber is usually laser diode. 198 00:24:16,690 --> 00:24:18,940 Since its dimensions are very small. 199 00:24:18,940 --> 00:24:28,299 So, coupling light into this fiber is critical and since you are using it a transmission 200 00:24:28,299 --> 00:24:35,510 fiber to carry data over long distances, then you will have to joint 2 fibers at some point 201 00:24:35,510 --> 00:24:42,280 and since the dimensions of this fiber are very small. 202 00:24:42,280 --> 00:24:48,370 So, joining of 2 fibers is difficult. 203 00:24:48,370 --> 00:24:55,490 You require precision equipment’s while handling and working this fiber, and the components 204 00:24:55,490 --> 00:25:04,350 which are used in the system which employs this fiber are also should be very compact 205 00:25:04,350 --> 00:25:05,870 and precise. 206 00:25:05,870 --> 00:25:15,860 Typical parameters for a silica glass single mode fiber are relative index difference between 207 00:25:15,860 --> 00:25:23,250 the core and cladding is 0.003 or 0.3 percent which corresponds to numerical aperture of 208 00:25:23,250 --> 00:25:31,120 0.1 and core diameter of sorry core radius of about 5 micrometer. 209 00:25:31,120 --> 00:25:37,320 If I look at multimode fiber this multimode fiber, is suitable for local area networks. 210 00:25:37,320 --> 00:25:45,690 We will see why single mode fiber is used for long distance communication, while multimode 211 00:25:45,690 --> 00:25:52,120 fiber is suitable for local area network and not for long haul telecommunication. 212 00:25:52,120 --> 00:25:58,470 This fiber has a core diameter which is much larger 50 micron and we can use light emitting 213 00:25:58,470 --> 00:26:05,860 diode also to couple light into the a multimode fiber since it has got large dimensions. 214 00:26:05,860 --> 00:26:11,770 So, coupling of light is easier and system equipment’s and components are also cheaper. 215 00:26:11,770 --> 00:26:19,460 Typical parameters of a silica glass multimode fiber are: delta is about 1 percent which 216 00:26:19,460 --> 00:26:26,860 corresponds to 0.2 numerical aperture, and core radius is about 25 micron. 217 00:26:26,860 --> 00:26:36,640 Well it is single mode fiber or multimode fiber usually the cladding diameter is 125 218 00:26:36,640 --> 00:26:44,929 micron, what changes is only core diameter and on top of that you have acrylic coating. 219 00:26:44,929 --> 00:26:49,090 So, overall diameter is most of the time 250 micron. 220 00:26:49,090 --> 00:26:56,920 So, if you look at a single mode fiber or multimode fiber you would not be able to tell 221 00:26:56,920 --> 00:27:03,310 which fiber it is it single mode fiber or it is multimode fiber you can only find out 222 00:27:03,310 --> 00:27:09,110 whether it is single mode or multimode where you launch light into this and observe the 223 00:27:09,110 --> 00:27:10,110 output. 224 00:27:10,110 --> 00:27:17,799 This is the output from a single mode fiber and this is the output from a multimode fiber, 225 00:27:17,799 --> 00:27:24,600 the patterns the explanation of these patterns is obvious because it is single mode. 226 00:27:24,600 --> 00:27:30,480 So, that has got only one intensity pattern and this is the fundamental mode and it is 227 00:27:30,480 --> 00:27:31,480 multi mode. 228 00:27:31,480 --> 00:27:32,480 So, there are large number of modes. 229 00:27:32,480 --> 00:27:38,450 So, this is the interference of all the modes, this is a speckle pattern. 230 00:27:38,450 --> 00:27:48,041 Now, you can also have a step index and graded index fibers, this is step index single mode 231 00:27:48,041 --> 00:27:55,610 fiber where the core diameter is about 10 micron and the refractive index in the core 232 00:27:55,610 --> 00:28:00,450 and in the cladding is uniform. 233 00:28:00,450 --> 00:28:05,860 In the same dimensions, you can also have refractive index in the core which changes 234 00:28:05,860 --> 00:28:13,160 with radial position, then it is called graded index single mode fiber. 235 00:28:13,160 --> 00:28:19,620 If you have core dimensions of the order of 50 micron core diameter typically 50 micron, 236 00:28:19,620 --> 00:28:26,970 and the refractive index in the core and in the cladding is uniform then it is a step 237 00:28:26,970 --> 00:28:34,470 index multimode fiber, and the number of modes supported by this fiber can be estimated by 238 00:28:34,470 --> 00:28:40,370 V square by 2, where V is the normalized frequency. 239 00:28:40,370 --> 00:28:46,150 In such a situation you can also have index variation refractive index variation in the 240 00:28:46,150 --> 00:28:47,930 core something like this. 241 00:28:47,930 --> 00:29:00,290 So, this is then called graded index multimode fiber, and if this variation is parabolic 242 00:29:00,290 --> 00:29:09,250 then the number of modes can be approximated by V square by 4 in such a fiber. 243 00:29:09,250 --> 00:29:17,430 Let us take an example, I have step index silica glass fiber which has n1 is equal to 244 00:29:17,430 --> 00:29:26,090 1.45 delta is equal to 1 percent and core radius 3 micrometer. 245 00:29:26,090 --> 00:29:29,770 What would be the range of wavelengths in which the fiber would be single mode it if 246 00:29:29,770 --> 00:29:31,650 I want to find that out. 247 00:29:31,650 --> 00:29:37,010 Then I know that from these parameters I can immediately calculate the numerical aperture 248 00:29:37,010 --> 00:29:45,960 of the fiber, which comes out to be 0.2025, and I also know that for fiber to be single 249 00:29:45,960 --> 00:29:53,059 mode it V should be less than 2.4048, and since V is defined as 2 pi over lambda naught 250 00:29:53,059 --> 00:29:59,630 times a times NA, then lambda naught should be greater than this number. 251 00:29:59,630 --> 00:30:06,980 So, for all the wavelengths which are longer than 1.607 micrometer, this fiber would be 252 00:30:06,980 --> 00:30:13,880 single model and for the wavelength shorter than this the fiber will support more than 253 00:30:13,880 --> 00:30:16,360 one modes. 254 00:30:16,360 --> 00:30:23,540 Now, can I find out how many modes will this fiber support, if I change the wavelength 255 00:30:23,540 --> 00:30:26,419 to 0.3 micrometer. 256 00:30:26,419 --> 00:30:35,740 So, it is easy I can immediately find the value of V at this value of lambda naught 257 00:30:35,740 --> 00:30:44,480 and it comes out to be 12.88, and since it is much larger than 2.4048 then the number 258 00:30:44,480 --> 00:30:52,610 of modes can be approximated by V square by 2 let me tell you that these number of modes 259 00:30:52,610 --> 00:30:58,610 n is equal to V square by 2 for a step index and n is equal to V square by 4 for graded 260 00:30:58,610 --> 00:31:05,190 index fiber spray parabolic index fiber, they are valid only when V is much larger than 261 00:31:05,190 --> 00:31:10,080 2.4048 typically more than 10 ok. 262 00:31:10,080 --> 00:31:14,340 So, what we have learnt in this lecture? 263 00:31:14,340 --> 00:31:24,630 We have seen that wave theory is required to address the limitations posed by ray theory 264 00:31:24,630 --> 00:31:27,650 in particular for the case of small core fibers. 265 00:31:27,650 --> 00:31:33,570 Where it completely breaks down ray theory completely breaks down, wave theory predict 266 00:31:33,570 --> 00:31:42,400 certain discrete modes of propagation in a fiber and any arbitrary field distribution 267 00:31:42,400 --> 00:31:49,010 can be expressed in terms of superposition of all the modes of a fiber. 268 00:31:49,010 --> 00:31:56,050 Number of modes in a fiber depends on the value of core radius refractive index difference 269 00:31:56,050 --> 00:32:01,460 between the core and the cladding, and the wavelength, but all these parameters can be 270 00:32:01,460 --> 00:32:07,810 integrated into a single parameter which is called V the normalized frequency. 271 00:32:07,810 --> 00:32:15,870 If V is less than 2.4048 the fiber is single mode and if it is much larger than 2.4048 272 00:32:15,870 --> 00:32:19,080 it is a multimode fiber. 273 00:32:19,080 --> 00:32:25,850 A single mode fiber has a small core it supports high data rate and it is used in long haul 274 00:32:25,850 --> 00:32:34,000 telecommunication system, while a multimode fiber has a large core and it is suitable 275 00:32:34,000 --> 00:32:37,460 only for local area networks. 276 00:32:37,460 --> 00:32:49,360 Apart from other applications other than telecom applications, and single mode fiber and multimode 277 00:32:49,360 --> 00:32:53,750 fiber both can be step index fibers or graded index fibers. 278 00:32:53,750 --> 00:32:54,659 Thank you.