1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high-quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,650 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,650 --> 00:00:17,550 at ocw.mit.edu. 8 00:00:21,293 --> 00:00:23,710 GEORGE BARBASTATHIS: The key point of the previous lecture 9 00:00:23,710 --> 00:00:29,740 was to simplify the equations for refraction 10 00:00:29,740 --> 00:00:34,060 from a spherical surface, such that we can write them 11 00:00:34,060 --> 00:00:35,050 in a simple form. 12 00:00:37,930 --> 00:00:40,430 OK, what I just said was redundancy. 13 00:00:40,430 --> 00:00:42,258 But we want to simplify them. 14 00:00:42,258 --> 00:00:44,050 I should have said we want to simplify them 15 00:00:44,050 --> 00:00:47,380 so that we can write them in a linearlized form. 16 00:00:47,380 --> 00:00:52,600 And the linearlized form is actually very useful 17 00:00:52,600 --> 00:00:57,800 because with this form, as I'm about to show, 18 00:00:57,800 --> 00:01:01,000 we're going to use a very familiar form of math, namely 19 00:01:01,000 --> 00:01:03,010 linear algebra matrices. 20 00:01:03,010 --> 00:01:06,430 And we can retrace through almost arbitrarily 21 00:01:06,430 --> 00:01:08,410 complex optical systems. 22 00:01:08,410 --> 00:01:15,030 Of course, there's a downside that if write trace this way, 23 00:01:15,030 --> 00:01:17,320 we do not really get very accurate results. 24 00:01:17,320 --> 00:01:19,640 Our results, as we discussed last time, 25 00:01:19,640 --> 00:01:22,890 are only accurate within the paraxial approximation. 26 00:01:22,890 --> 00:01:28,780 On the other hand, if we do this analysis even approximately, 27 00:01:28,780 --> 00:01:32,870 we can get an idea of how the optical system behaves. 28 00:01:32,870 --> 00:01:40,630 And then if we get this approximate design, 29 00:01:40,630 --> 00:01:43,360 this approximate idea, then we can plug the system 30 00:01:43,360 --> 00:01:47,080 into a more sophisticated numerical tool, 31 00:01:47,080 --> 00:01:50,260 for example, Zemax or CODE V. There's 32 00:01:50,260 --> 00:01:52,770 a number of different optical design 33 00:01:52,770 --> 00:01:56,350 and software suites available. 34 00:01:56,350 --> 00:02:00,700 And then we can actually get a more accurate analysis 35 00:02:00,700 --> 00:02:03,070 of how the optical systems would behave. 36 00:02:03,070 --> 00:02:06,741 So the other reason we write these matrices 37 00:02:06,741 --> 00:02:08,199 is that they give us actually quite 38 00:02:08,199 --> 00:02:10,389 a bit of information about the basic properties 39 00:02:10,389 --> 00:02:11,630 of optical systems. 40 00:02:11,630 --> 00:02:15,610 So it is really worthwhile to do this first, every time we 41 00:02:15,610 --> 00:02:17,800 face an optics problem, before we jump 42 00:02:17,800 --> 00:02:19,590 into the numerical analysis. 43 00:02:19,590 --> 00:02:23,110 And for those of you who really have a real-life problems 44 00:02:23,110 --> 00:02:26,400 to solve and you really need to go to software, 45 00:02:26,400 --> 00:02:28,470 we will tell you later during the class 46 00:02:28,470 --> 00:02:32,950 where we'll give you some information about what 47 00:02:32,950 --> 00:02:38,470 types of software are available and what you can do with those. 48 00:02:38,470 --> 00:02:42,250 But generally, this idea of using paraxial optics 49 00:02:42,250 --> 00:02:46,990 is very powerful in order to understand optical system. 50 00:02:46,990 --> 00:02:50,700 So the summary then of what paraxial optics standard is 51 00:02:50,700 --> 00:02:51,280 summarized. 52 00:02:51,280 --> 00:02:54,100 These are shown on this slide over here. 53 00:02:54,100 --> 00:02:55,960 The most important is the results 54 00:02:55,960 --> 00:03:02,020 on the right that allows you to write the angle of departure 55 00:03:02,020 --> 00:03:07,240 of a ray to the right-hand side of the card interface 56 00:03:07,240 --> 00:03:09,550 and the elevation of the ray, with respect 57 00:03:09,550 --> 00:03:13,223 to the right-hand side interface. 58 00:03:13,223 --> 00:03:14,890 It allows you to write them as functions 59 00:03:14,890 --> 00:03:18,340 of the corresponding quantities on the left-hand side 60 00:03:18,340 --> 00:03:19,620 of the interface. 61 00:03:19,620 --> 00:03:21,340 And as you can see there, it actually 62 00:03:21,340 --> 00:03:25,060 looks like it two by two set of vectors, which are 63 00:03:25,060 --> 00:03:30,880 related by a two by two matrix. 64 00:03:30,880 --> 00:03:32,810 And the matrix are not very interesting. 65 00:03:32,810 --> 00:03:39,190 As you can see, the most relevant, the most interesting 66 00:03:39,190 --> 00:03:43,520 element is the element on the first row, second column, 67 00:03:43,520 --> 00:03:46,690 which equals the index of refraction 68 00:03:46,690 --> 00:03:50,190 to the right of the interface, minus the index of refraction 69 00:03:50,190 --> 00:03:53,390 to the left, divided by the radius of curvature 70 00:03:53,390 --> 00:03:55,315 of the spherical interface. 71 00:03:55,315 --> 00:03:56,940 And, of course, the radius of curvature 72 00:03:56,940 --> 00:03:59,440 is simply the radius of the sphere. 73 00:03:59,440 --> 00:04:01,540 We're talking about the sphere. 74 00:04:01,540 --> 00:04:03,730 And then the other result that we put here and looks 75 00:04:03,730 --> 00:04:06,360 like a propagation through free space, it 76 00:04:06,360 --> 00:04:09,185 looks kind of too much trouble, really, for what it's worth. 77 00:04:09,185 --> 00:04:11,560 But you will see in a second that this is actually a very 78 00:04:11,560 --> 00:04:13,720 useful tool to have in mind. 79 00:04:13,720 --> 00:04:17,380 One of the thing that says is that the ray, as it propagates 80 00:04:17,380 --> 00:04:22,360 from the left over a distance d, if it propagates 81 00:04:22,360 --> 00:04:24,470 through free space, then what happens 82 00:04:24,470 --> 00:04:27,910 is the elevation of the ray changes, 83 00:04:27,910 --> 00:04:30,290 but the angle of propagation does not change. 84 00:04:30,290 --> 00:04:34,900 So this verifies what we know from Fermat's principle 85 00:04:34,900 --> 00:04:39,060 that a ray propagating in uniform space, 86 00:04:39,060 --> 00:04:40,670 it minimizes its path. 87 00:04:40,670 --> 00:04:45,560 Therefore, it must propagate in a straight line 88 00:04:45,560 --> 00:04:48,080 because the straight line is the minimum path between two 89 00:04:48,080 --> 00:04:50,650 points. 90 00:04:50,650 --> 00:04:52,310 Now, you might be wondering. 91 00:04:52,310 --> 00:04:57,200 We talked last time about ellipsoidal surfaces, 92 00:04:57,200 --> 00:05:00,460 hyperboloid surfaces, and spherical surfaces. 93 00:05:00,460 --> 00:05:03,500 You might wonder, why did we do this analysis 94 00:05:03,500 --> 00:05:05,090 in the context of the sphere? 95 00:05:05,090 --> 00:05:07,650 Well, that says it doesn't matter very much because all 96 00:05:07,650 --> 00:05:10,370 of these three possibilities, ellipsoids, 97 00:05:10,370 --> 00:05:14,180 hyperboloids, and spheres, near the axis, 98 00:05:14,180 --> 00:05:17,540 they actually all look kind of the same. 99 00:05:17,540 --> 00:05:19,810 If you compare a sphere-- 100 00:05:19,810 --> 00:05:23,180 so I should specify now what I mean by axis. 101 00:05:23,180 --> 00:05:26,030 So in the case of a sphere, as Professor Sheppard pointed out 102 00:05:26,030 --> 00:05:27,690 last time, in the case of a sphere, 103 00:05:27,690 --> 00:05:30,770 really, any lines going through the center of the sphere 104 00:05:30,770 --> 00:05:32,210 acts as the optical axis. 105 00:05:32,210 --> 00:05:36,200 But for the hyperboloid and the ellipsoid, that's not true. 106 00:05:36,200 --> 00:05:40,100 There's a very well-defined major axis and minor axis 107 00:05:40,100 --> 00:05:42,680 for the surfaces. 108 00:05:42,680 --> 00:05:48,670 And it turns out you can match the curvature of these surfaces 109 00:05:48,670 --> 00:05:51,950 with the curvature of a sphere so that [INAUDIBLE] 110 00:05:51,950 --> 00:05:53,750 a common axis, they all look the same, 111 00:05:53,750 --> 00:05:55,900 as I've done in this diagram. 112 00:05:55,900 --> 00:05:58,220 It's a little bit of an algebraic mess 113 00:05:58,220 --> 00:06:01,390 to actually work out how to do this. 114 00:06:01,390 --> 00:06:02,930 So I didn't do it in this slide. 115 00:06:02,930 --> 00:06:08,390 The idea is basically to do a Taylor series expansion 116 00:06:08,390 --> 00:06:13,410 on the explanations that describe all those surfaces. 117 00:06:13,410 --> 00:06:15,590 And then when you do that, each one of those 118 00:06:15,590 --> 00:06:17,803 is described by a different set of coefficients. 119 00:06:17,803 --> 00:06:19,220 So what you can do then is you can 120 00:06:19,220 --> 00:06:22,440 match the coefficients in order to match the curvature. 121 00:06:22,440 --> 00:06:25,460 So I will let the graduate version of the class, those 122 00:06:25,460 --> 00:06:28,380 of you who are taking 2710, as a homework, 123 00:06:28,380 --> 00:06:30,140 I will let you work out one of these, 124 00:06:30,140 --> 00:06:33,230 a simple case for matching-- 125 00:06:33,230 --> 00:06:34,160 I forget. 126 00:06:34,160 --> 00:06:37,160 I think it is matching an ellipsoid to a sphere. 127 00:06:37,160 --> 00:06:39,590 So let you do that. 128 00:06:39,590 --> 00:06:42,390 What I really want you to live with out of this lecture, 129 00:06:42,390 --> 00:06:45,860 for now, is that the paraxial analysis remains 130 00:06:45,860 --> 00:06:48,570 valid for all three surfaces. 131 00:06:48,570 --> 00:06:49,070 OK? 132 00:06:49,070 --> 00:06:50,920 As long as we match the curvature 133 00:06:50,920 --> 00:06:53,720 of the sphere, ellipse, and hyperboloid, 134 00:06:53,720 --> 00:06:56,300 and we constrain ourselves to the paraxial regime, 135 00:06:56,300 --> 00:06:59,030 then our results remain approximately correct. 136 00:06:59,030 --> 00:07:02,180 So this is kind of another useful property 137 00:07:02,180 --> 00:07:04,760 of the paraxial approximation that it is not 138 00:07:04,760 --> 00:07:07,400 only valid for a sphere, but it is also 139 00:07:07,400 --> 00:07:11,120 valid for other surfaces of revolution, 140 00:07:11,120 --> 00:07:16,940 provided the curvature near the center of the surfaces 141 00:07:16,940 --> 00:07:21,560 matches the curvature of the corresponding sphere. 142 00:07:21,560 --> 00:07:23,070 OK. 143 00:07:23,070 --> 00:07:23,890 Yeah? 144 00:07:23,890 --> 00:07:24,890 Push the button, please. 145 00:07:30,570 --> 00:07:32,100 Yeah, so that's a good question. 146 00:07:32,100 --> 00:07:33,290 AUDIENCE: Could you repeat the question? 147 00:07:33,290 --> 00:07:34,457 We didn't hear the question. 148 00:07:36,215 --> 00:07:37,590 GEORGE BARBASTATHIS: The question 149 00:07:37,590 --> 00:07:39,630 was if there's an angle that we'll 150 00:07:39,630 --> 00:07:42,120 consider as a cutoff for the breakdown 151 00:07:42,120 --> 00:07:44,150 of the paraxial approximation. 152 00:07:44,150 --> 00:07:46,380 So the accurate answer is zero. 153 00:07:46,380 --> 00:07:48,270 Any angle other than zero violates 154 00:07:48,270 --> 00:07:50,250 the paraxial approximation. 155 00:07:50,250 --> 00:07:55,460 In actuality, you can calculate, with the help from this Taylor 156 00:07:55,460 --> 00:07:58,830 series, actually, that we saw in the previous slide, 157 00:07:58,830 --> 00:08:01,990 you can calculate the accuracy of the approximation. 158 00:08:01,990 --> 00:08:06,440 So treat it as a first order of a Taylor series expansion. 159 00:08:06,440 --> 00:08:10,810 And then you can calculate what is the next order accuracy. 160 00:08:10,810 --> 00:08:14,130 So this is actually algebraically quite complex. 161 00:08:14,130 --> 00:08:16,080 And typically, we resort to software 162 00:08:16,080 --> 00:08:18,720 to do it for us, numerical software. 163 00:08:18,720 --> 00:08:23,910 As a rule of thumb, anything in the range below 30 degrees, 164 00:08:23,910 --> 00:08:26,280 you can kind of trust the paraxial approximation. 165 00:08:26,280 --> 00:08:27,420 But that's a rule of thumb. 166 00:08:27,420 --> 00:08:29,280 In some cases, it's not true. 167 00:08:29,280 --> 00:08:31,440 In some cases, you need more accuracy. 168 00:08:31,440 --> 00:08:34,750 So the guideline is something like that. 169 00:08:34,750 --> 00:08:37,380 But, for example, for sure, if you 170 00:08:37,380 --> 00:08:39,059 have a ray propagating at 80 degrees, 171 00:08:39,059 --> 00:08:41,940 you know for sure that approximation breaks. 172 00:08:41,940 --> 00:08:45,420 If it goes to 1 to 3 degrees, then most likely, you're OK. 173 00:08:45,420 --> 00:08:48,420 And then there's a gray area that you 174 00:08:48,420 --> 00:08:51,600 have to be a little bit more careful. 175 00:08:51,600 --> 00:08:53,520 If the accuracy demanded by the application 176 00:08:53,520 --> 00:08:56,210 is much higher, than you have to be more careful. 177 00:08:59,190 --> 00:09:02,040 There's also surfaces that are not very well-described by this 178 00:09:02,040 --> 00:09:04,950 at all, for example, a cubic. 179 00:09:04,950 --> 00:09:06,690 If you make a refractive interface 180 00:09:06,690 --> 00:09:10,532 that looks like a cubic surface, would look kind of like this. 181 00:09:10,532 --> 00:09:12,240 And then this totally breaks down, right? 182 00:09:12,240 --> 00:09:21,710 So you have to go to then to non-paraxial [? methods. ?] 183 00:09:21,710 --> 00:09:29,470 Another thing to discuss is in one the previous equations, 184 00:09:29,470 --> 00:09:33,780 I wrote down ray elevations and ray angles. 185 00:09:33,780 --> 00:09:36,580 But I was not very careful about defining 186 00:09:36,580 --> 00:09:38,320 the signs of these angles, namely, 187 00:09:38,320 --> 00:09:42,660 whether they go up, or down, or left, or right, and so on. 188 00:09:42,660 --> 00:09:45,070 I was deliberately left this little bit vague. 189 00:09:47,860 --> 00:09:50,650 Again, this may appear like too much trouble for nothing. 190 00:09:50,650 --> 00:09:53,770 But you will see, as we go later on with the lecture 191 00:09:53,770 --> 00:10:03,080 today, you will see that this is a very useful and highly 192 00:10:03,080 --> 00:10:06,020 nontrivial aspect of geometrical optics, 193 00:10:06,020 --> 00:10:08,720 namely, the conventions of the signs, 194 00:10:08,720 --> 00:10:10,710 for when the quantities that we're dealing with 195 00:10:10,710 --> 00:10:12,840 are positive or negative. 196 00:10:12,840 --> 00:10:15,170 So bear with me for a second, and I 197 00:10:15,170 --> 00:10:19,680 will show examples of how this works later. 198 00:10:19,680 --> 00:10:24,270 So I want to read this because I don't know of a better way 199 00:10:24,270 --> 00:10:24,770 to do it. 200 00:10:24,770 --> 00:10:25,470 I will read it. 201 00:10:25,470 --> 00:10:28,100 And then as I read, please look down at the slide. 202 00:10:28,100 --> 00:10:30,060 And you will see on the left-hand side 203 00:10:30,060 --> 00:10:33,212 I have examples of all these qualities being positive. 204 00:10:33,212 --> 00:10:34,670 And on the right-hand side, we have 205 00:10:34,670 --> 00:10:37,860 example of all of these quantities being negative. 206 00:10:37,860 --> 00:10:40,580 OK, so the convention number one is that the light always 207 00:10:40,580 --> 00:10:42,300 travels from left to right. 208 00:10:42,300 --> 00:10:46,760 So you know immediately when someone draws an optical system 209 00:10:46,760 --> 00:10:48,590 or attempts to draw an optical system, 210 00:10:48,590 --> 00:10:51,030 you know immediately if this person knows their optics 211 00:10:51,030 --> 00:10:53,780 or not by the way they pick the light propagation. 212 00:10:53,780 --> 00:10:56,637 If they show light rays going from right to left, 213 00:10:56,637 --> 00:10:57,970 it means they don't know optics. 214 00:10:57,970 --> 00:11:00,767 So if they learned optics, they learned it very deficiently. 215 00:11:00,767 --> 00:11:01,600 AUDIENCE: [CHUCKLES] 216 00:11:01,600 --> 00:11:03,725 GEORGE BARBASTATHIS: So always light goes from left 217 00:11:03,725 --> 00:11:05,455 to right in this diagram. 218 00:11:05,455 --> 00:11:06,830 There's one exception, of course. 219 00:11:06,830 --> 00:11:07,760 Can anybody guess? 220 00:11:07,760 --> 00:11:10,730 One exception that I'm not dealing with here. 221 00:11:10,730 --> 00:11:11,540 Mirror. 222 00:11:11,540 --> 00:11:12,902 Push the button. 223 00:11:12,902 --> 00:11:13,610 AUDIENCE: Mirror. 224 00:11:13,610 --> 00:11:15,570 GEORGE BARBASTATHIS: Mirrors, yes. 225 00:11:15,570 --> 00:11:20,090 So in this diagram over here, I do not consider mirrors yet. 226 00:11:20,090 --> 00:11:23,330 We will deal with mirrors later, in about a week. 227 00:11:23,330 --> 00:11:26,600 And then we'll see a revised set of sign convention. 228 00:11:26,600 --> 00:11:29,630 But as long as there's no mirrors in your optical system, 229 00:11:29,630 --> 00:11:33,500 if you have what is called a purely dioptric optical system, 230 00:11:33,500 --> 00:11:38,700 then the light is always going from left to right. 231 00:11:38,700 --> 00:11:40,920 Taking this as a given, then we'll 232 00:11:40,920 --> 00:11:45,440 define the positive and negative radii of curvature 233 00:11:45,440 --> 00:11:47,140 as solved over here. 234 00:11:47,140 --> 00:11:50,140 So if the surface is convex towards the left, 235 00:11:50,140 --> 00:11:53,640 then we'll call it positive, OK? 236 00:11:53,640 --> 00:11:55,920 Then this sounds a little bit weird. 237 00:11:55,920 --> 00:11:57,780 It says that longitudinal distances 238 00:11:57,780 --> 00:12:00,168 are positive if pointing to the right. 239 00:12:00,168 --> 00:12:01,710 I'll show you an example a little bit 240 00:12:01,710 --> 00:12:05,310 later where distance can become negative by pointing 241 00:12:05,310 --> 00:12:06,210 to the left. 242 00:12:06,210 --> 00:12:08,910 But for now, take this definition for granted. 243 00:12:08,910 --> 00:12:10,680 The same is for lateral distances 244 00:12:10,680 --> 00:12:13,590 that this measures perpendicular to the optical axis. 245 00:12:13,590 --> 00:12:16,590 If they point up, then they're called positive. 246 00:12:16,590 --> 00:12:19,250 If they point down, they're called negative. 247 00:12:19,250 --> 00:12:23,210 And finally, angles are positive if they 248 00:12:23,210 --> 00:12:27,830 are acute in the counterclockwise sense, 249 00:12:27,830 --> 00:12:29,950 with respect to the optical axis. 250 00:12:29,950 --> 00:12:32,930 So clockwise, we have to be very careful when we define it, 251 00:12:32,930 --> 00:12:34,850 so we don't get any mirror effects. 252 00:12:34,850 --> 00:12:36,970 So hope that the cameras that we're using here 253 00:12:36,970 --> 00:12:38,810 do not make any mirrors. 254 00:12:38,810 --> 00:12:43,340 But counter-clockwise is basically like this, OK? 255 00:12:43,340 --> 00:12:46,190 If in this direction you get an acute angle, 256 00:12:46,190 --> 00:12:49,460 then the angle is positive. 257 00:12:49,460 --> 00:12:52,820 If you have to go all the way around to get your angle, 258 00:12:52,820 --> 00:12:54,980 or another way to put it is if you can make 259 00:12:54,980 --> 00:12:57,400 it acute by going clockwise, then you 260 00:12:57,400 --> 00:12:59,610 call the angle negative. 261 00:12:59,610 --> 00:13:00,110 OK. 262 00:13:00,110 --> 00:13:04,560 So this set of conventions is self-contained and consistent. 263 00:13:04,560 --> 00:13:08,900 To add to the confusion, set and optical text books, 264 00:13:08,900 --> 00:13:13,790 they use the opposite sign of conventions. 265 00:13:13,790 --> 00:13:16,250 It is also possible, by the way. 266 00:13:16,250 --> 00:13:17,900 The set of conventions I use here 267 00:13:17,900 --> 00:13:20,120 is consistent with a Hecht text book 268 00:13:20,120 --> 00:13:22,310 and the majority of text books. 269 00:13:22,310 --> 00:13:23,990 There's a minority of textbooks that use 270 00:13:23,990 --> 00:13:25,370 the exact opposite conventions. 271 00:13:25,370 --> 00:13:28,650 But we will not deal with those. 272 00:13:28,650 --> 00:13:32,900 OK, using these set of conventions, 273 00:13:32,900 --> 00:13:35,870 we can define the power of an optical system 274 00:13:35,870 --> 00:13:44,090 based on whether rays that propagate outwards-- 275 00:13:44,090 --> 00:13:48,230 by outwards, I mean a ray bundle that expands-- 276 00:13:48,230 --> 00:13:51,500 whether a surface, a spherical surface, 277 00:13:51,500 --> 00:13:56,990 causes the expansion to become slower or faster. 278 00:13:56,990 --> 00:14:01,010 If the surface causes the expansion to become slower, 279 00:14:01,010 --> 00:14:03,740 as in the top diagram over here, then 280 00:14:03,740 --> 00:14:08,420 the surface is said to have positive power. 281 00:14:08,420 --> 00:14:12,180 If the surface causes the expansion to become faster-- 282 00:14:12,180 --> 00:14:14,930 that is, it takes the expanding bundle 283 00:14:14,930 --> 00:14:18,650 and spreads it further outwards-- 284 00:14:18,650 --> 00:14:20,970 then the surface is said to have negative power. 285 00:14:23,870 --> 00:14:28,520 So now, let me proceed to define a specific optical system. 286 00:14:28,520 --> 00:14:33,430 And then we'll see more examples of this thing happening. 287 00:14:33,430 --> 00:14:35,800 So the system I would like to consider is a lens. 288 00:14:35,800 --> 00:14:38,380 And this is actually the first example of a lens 289 00:14:38,380 --> 00:14:40,510 that we see in the class. 290 00:14:40,510 --> 00:14:43,900 And I will take a very special case with a lens 291 00:14:43,900 --> 00:14:46,100 that we refer to as thin. 292 00:14:46,100 --> 00:14:47,260 What's a lens? 293 00:14:47,260 --> 00:14:52,720 A lens is simply a piece of glass 294 00:14:52,720 --> 00:14:57,850 that has been shaped to have spherical surfaces on one 295 00:14:57,850 --> 00:14:59,440 or both sides. 296 00:14:59,440 --> 00:15:01,300 Typically both, but we'll see examples 297 00:15:01,300 --> 00:15:03,880 of lenses that are actually flat on one surface 298 00:15:03,880 --> 00:15:08,980 and curved on the other surface. 299 00:15:08,980 --> 00:15:13,230 So when we call the lens thin, well, we call it thin. 300 00:15:15,890 --> 00:15:18,690 Consider an expanding spherical wave that 301 00:15:18,690 --> 00:15:20,550 propagates through the lens. 302 00:15:20,550 --> 00:15:24,300 And consider the maximum angle that can 303 00:15:24,300 --> 00:15:25,650 be subtended through the lens. 304 00:15:25,650 --> 00:15:28,950 In other words, angles larger than this one 305 00:15:28,950 --> 00:15:30,330 would actually miss the lens. 306 00:15:30,330 --> 00:15:35,610 So therefore, this alpha maximum is sort of the biggest angle 307 00:15:35,610 --> 00:15:38,010 that we have to worry about. 308 00:15:38,010 --> 00:15:40,740 Well, what does the paraxial approximation say? 309 00:15:40,740 --> 00:15:44,130 It says that this angle has to be relatively small. 310 00:15:44,130 --> 00:15:48,270 So assume that this is true, then. 311 00:15:48,270 --> 00:15:49,960 The lens also has a thickness. 312 00:15:49,960 --> 00:15:55,540 So this thickness, I denoted as T in this diagram over here. 313 00:15:55,540 --> 00:15:58,240 So provided this thickness, if we 314 00:15:58,240 --> 00:16:02,720 do a little bit of an analysis here, 315 00:16:02,720 --> 00:16:09,660 we can calculate this thickness relative to the radius 316 00:16:09,660 --> 00:16:11,670 of curvature of the sphere. 317 00:16:11,670 --> 00:16:15,720 And the angle phi that is defined as shown, 318 00:16:15,720 --> 00:16:19,560 from the center of the sphere relative to the edge 319 00:16:19,560 --> 00:16:23,820 of the ray, that subtends the maximum angle alpha maximum. 320 00:16:23,820 --> 00:16:24,570 OK? 321 00:16:24,570 --> 00:16:27,630 So we can see the definition phi for this angle 322 00:16:27,630 --> 00:16:30,660 and the radius r. 323 00:16:30,660 --> 00:16:34,000 So if, given this angle of arrival 324 00:16:34,000 --> 00:16:41,020 is alpha maximum, if it is true that the angle phi is also 325 00:16:41,020 --> 00:16:44,720 very small, then we can consider the lens to be thin. 326 00:16:44,720 --> 00:16:50,140 So we can basically pretend that the space between these two 327 00:16:50,140 --> 00:16:53,770 points on the lens, the space between the entrance 328 00:16:53,770 --> 00:16:57,580 to the optical axis and the exit of the optical axis 329 00:16:57,580 --> 00:17:00,140 through the lens, we can neglect the space. 330 00:17:00,140 --> 00:17:03,440 This is what the thin lens approximation says. 331 00:17:03,440 --> 00:17:08,290 That's very convenient because we can now analyze the lens. 332 00:17:08,290 --> 00:17:11,140 As you can see, if I neglect this space, 333 00:17:11,140 --> 00:17:16,220 then the ray elevation before and after 334 00:17:16,220 --> 00:17:18,089 the lens remains the same. 335 00:17:18,089 --> 00:17:20,180 And also can neglect a refraction 336 00:17:20,180 --> 00:17:22,286 that might happen within the lens. 337 00:17:22,286 --> 00:17:23,869 So basically, we have the system where 338 00:17:23,869 --> 00:17:29,480 the ray arrives then is bent in one step and then propagates 339 00:17:29,480 --> 00:17:35,960 outwards at a slightly different angle. 340 00:17:35,960 --> 00:17:38,440 So in order to analyze the system, now, what I will do 341 00:17:38,440 --> 00:17:41,360 is I will break it into two components. 342 00:17:41,360 --> 00:17:46,350 One of them is the curved surface on the left-hand side. 343 00:17:46,350 --> 00:17:50,610 And the second is the curved surface on the right-hand side. 344 00:17:50,610 --> 00:17:52,060 OK. 345 00:17:52,060 --> 00:17:53,650 And now, for each one of those, I 346 00:17:53,650 --> 00:18:00,190 can define an angle on the right and then angle on the left, 347 00:18:00,190 --> 00:18:03,367 as I have done in the simple [INAUDIBLE] 348 00:18:03,367 --> 00:18:04,450 of the refractive surface. 349 00:18:04,450 --> 00:18:07,340 Now, each one of those is a simple refractive surface 350 00:18:07,340 --> 00:18:09,340 interface that we know how to deal 351 00:18:09,340 --> 00:18:11,770 with from the previous lecture. 352 00:18:11,770 --> 00:18:12,880 And I can add equations. 353 00:18:12,880 --> 00:18:15,100 Remember the equation for the refraction 354 00:18:15,100 --> 00:18:16,830 from the spherical surface. 355 00:18:16,830 --> 00:18:21,940 I will add it here on my pad, just to remember. 356 00:18:21,940 --> 00:18:23,080 So it goes like this. 357 00:18:25,680 --> 00:18:28,740 Hopefully, someone will notice and they will project it. 358 00:18:28,740 --> 00:18:39,090 So n to the right, alpha to the right, x to the right equals-- 359 00:18:39,090 --> 00:18:40,280 can anybody see? 360 00:18:43,425 --> 00:18:45,300 OK, we're having technical difficulties here. 361 00:18:45,300 --> 00:18:45,967 Oh, there we go. 362 00:18:45,967 --> 00:18:48,133 AUDIENCE: Can you lower the piece of paper a little? 363 00:18:48,133 --> 00:18:48,784 Yeah, got it. 364 00:18:48,784 --> 00:18:50,230 Yeah. 365 00:18:50,230 --> 00:18:54,520 GEORGE BARBASTATHIS: So this equals 1, 0, 1. 366 00:18:54,520 --> 00:18:55,840 This is easy. 367 00:18:55,840 --> 00:19:00,530 And then we have apply our mnemonic, minus n to the right, 368 00:19:00,530 --> 00:19:06,350 minus n to the left, over the radius of curvature, 369 00:19:06,350 --> 00:19:11,060 times the same property for the left. 370 00:19:11,060 --> 00:19:16,700 And left alpha left, x left. 371 00:19:20,921 --> 00:19:23,028 OK. 372 00:19:23,028 --> 00:19:24,570 Now, what we're have to do is we have 373 00:19:24,570 --> 00:19:28,450 to apply this rule successively for the two interfaces. 374 00:19:28,450 --> 00:19:35,480 So here's the question that we get for the second interface. 375 00:19:35,480 --> 00:19:38,410 So this case, n to the right is 1 376 00:19:38,410 --> 00:19:43,730 because to the right of this interface, I have air. 377 00:19:43,730 --> 00:19:47,330 And n to the left is n because to the left of this interface, 378 00:19:47,330 --> 00:19:48,020 I have glass. 379 00:19:50,990 --> 00:19:52,660 OK. 380 00:19:52,660 --> 00:19:55,540 Now, what I will do is I will apply the same rule. 381 00:19:55,540 --> 00:20:00,790 But I will now apply it on the first interface. 382 00:20:00,790 --> 00:20:04,460 And if I do to the first interface, I get this equation. 383 00:20:04,460 --> 00:20:07,600 Now, notice that in the numerator 384 00:20:07,600 --> 00:20:11,990 of the element in the first row, second column, 385 00:20:11,990 --> 00:20:15,830 the quantities are reversed because now n to the right 386 00:20:15,830 --> 00:20:19,790 is the index of refraction of glass, whereas n to the left 387 00:20:19,790 --> 00:20:23,690 is the index of refraction in air, that is 1. 388 00:20:23,690 --> 00:20:25,730 So therefore, I get this matrix. 389 00:20:25,730 --> 00:20:28,190 Now, what it can do, I can cascade the two matrices 390 00:20:28,190 --> 00:20:33,920 because I can substitute the vector n over 1 x1 that 391 00:20:33,920 --> 00:20:35,300 is on the top equation. 392 00:20:35,300 --> 00:20:38,235 I can substitute it from the second equation. 393 00:20:38,235 --> 00:20:39,860 And if I do that, of course, I will get 394 00:20:39,860 --> 00:20:43,200 a cascade of the two matrices. 395 00:20:43,200 --> 00:20:44,990 And notice that in the cascade, it 396 00:20:44,990 --> 00:20:48,380 is very important to remember that in the cascade, 397 00:20:48,380 --> 00:20:53,063 the matrix of the last element to the right appears left. 398 00:20:53,063 --> 00:20:54,980 And that is very important to remember and not 399 00:20:54,980 --> 00:20:59,000 confuse because as you know, matrix multiplication is not 400 00:20:59,000 --> 00:21:00,230 commutative. 401 00:21:00,230 --> 00:21:02,270 If you mess up the order of these matrices, 402 00:21:02,270 --> 00:21:04,340 you will get the wrong result. So you 403 00:21:04,340 --> 00:21:06,590 have to remember that you start from the right. 404 00:21:06,590 --> 00:21:09,970 That is, you start from the end of the optical system 405 00:21:09,970 --> 00:21:12,350 where the rays depart, and you start 406 00:21:12,350 --> 00:21:15,530 cascading the matrices going backwards towards the beginning 407 00:21:15,530 --> 00:21:17,100 of the optical system. 408 00:21:17,100 --> 00:21:18,530 So this is what I've done here. 409 00:21:18,530 --> 00:21:20,090 I've put the matrix corresponding 410 00:21:20,090 --> 00:21:21,950 to the second interface. 411 00:21:21,950 --> 00:21:23,550 This matrix appears first. 412 00:21:23,550 --> 00:21:27,140 And the matrix corresponding to the first interface 413 00:21:27,140 --> 00:21:28,040 appears second. 414 00:21:28,040 --> 00:21:29,720 And then I cascade them. 415 00:21:29,720 --> 00:21:31,550 I do the matrix multiplication, which 416 00:21:31,550 --> 00:21:35,150 is a relatively straightforward task to do. 417 00:21:35,150 --> 00:21:37,010 And I get the second, the last equation 418 00:21:37,010 --> 00:21:38,860 that appears on the slide. 419 00:21:38,860 --> 00:21:40,260 OK. 420 00:21:40,260 --> 00:21:42,378 Now, let's go ahead and interpret this equation. 421 00:21:42,378 --> 00:21:43,170 It looks very nice. 422 00:21:43,170 --> 00:21:44,628 But let's see what it really means. 423 00:21:44,628 --> 00:21:47,920 Also, before I move on, let me notice one more thing. 424 00:21:47,920 --> 00:21:51,360 Notice that in the vectors of the ray, angles 425 00:21:51,360 --> 00:21:55,440 of the ray elevations, the index of refraction has disappeared. 426 00:21:55,440 --> 00:21:56,760 It has not really disappeared. 427 00:21:56,760 --> 00:22:01,140 It is simply because the angles alpha right and alpha left 428 00:22:01,140 --> 00:22:02,370 are in air. 429 00:22:02,370 --> 00:22:04,560 And therefore, the index of refraction is 1. 430 00:22:04,560 --> 00:22:08,950 That's why there's no index of refraction in that location. 431 00:22:08,950 --> 00:22:09,450 OK. 432 00:22:12,020 --> 00:22:14,560 Let's look at this equation again. 433 00:22:14,560 --> 00:22:21,700 The top quantity that appears relatively complicated, 434 00:22:21,700 --> 00:22:22,420 it has a name. 435 00:22:22,420 --> 00:22:26,290 It is called 1 over the focal length. 436 00:22:26,290 --> 00:22:29,210 Now, you might wonder, why did they 437 00:22:29,210 --> 00:22:30,680 call this the focal length? 438 00:22:30,680 --> 00:22:34,750 I will justify that in a second. 439 00:22:34,750 --> 00:22:38,140 But if you believe me that the focal length follows 440 00:22:38,140 --> 00:22:41,500 from this equation, then this equation 441 00:22:41,500 --> 00:22:43,990 is known as the lens maker's equation. 442 00:22:43,990 --> 00:22:47,260 And it is very useful because it gives you 443 00:22:47,260 --> 00:22:51,280 the focal length of the thin lens as a function of the three 444 00:22:51,280 --> 00:22:55,210 quantities that define the thin lens, that radii of curvature 445 00:22:55,210 --> 00:22:59,290 to the left and to the right and the index of refraction 446 00:22:59,290 --> 00:23:00,970 of the lens. 447 00:23:00,970 --> 00:23:03,600 And what I want to emphasize, again, 448 00:23:03,600 --> 00:23:06,280 is that this equation is valid if the lens is 449 00:23:06,280 --> 00:23:08,020 surrounded by air. 450 00:23:08,020 --> 00:23:11,910 If the lens is surrounded by another material, for example, 451 00:23:11,910 --> 00:23:14,410 water, or oil or something with index 452 00:23:14,410 --> 00:23:17,350 of refraction different than 1, then this equation 453 00:23:17,350 --> 00:23:18,980 does not hold anymore. 454 00:23:18,980 --> 00:23:22,420 And as a bonus, in the next homework, which has already 455 00:23:22,420 --> 00:23:26,410 been posted, you will actually derive the corrected equation 456 00:23:26,410 --> 00:23:29,260 if the lens is immersed in a material that 457 00:23:29,260 --> 00:23:31,360 is different than air. 458 00:23:31,360 --> 00:23:32,910 But for now, this equation is correct 459 00:23:32,910 --> 00:23:34,740 because we put our lens in air. 460 00:23:34,740 --> 00:23:36,620 So therefore, everything is fine. 461 00:23:36,620 --> 00:23:38,630 And this is a lens maker formula. 462 00:23:38,630 --> 00:23:41,490 So then let me justify that this quantity that appeared over 463 00:23:41,490 --> 00:23:47,910 there and has units 1 over distance, why on over distance? 464 00:23:47,910 --> 00:23:53,760 Because this quantity has 1 over the radius of curvature 465 00:23:53,760 --> 00:23:54,690 in the denominator. 466 00:23:54,690 --> 00:23:56,880 So therefore, it is 1 over distance. 467 00:23:56,880 --> 00:23:59,490 So let me justify why this quantity is actually 468 00:23:59,490 --> 00:24:02,030 the focal length of the lens. 469 00:24:02,030 --> 00:24:04,850 So to do that, let me consider a ray that 470 00:24:04,850 --> 00:24:09,290 is arriving from infinity horizontally at an elevation x1 471 00:24:09,290 --> 00:24:12,060 with respect to the optical axis. 472 00:24:12,060 --> 00:24:18,155 So this ray, I can retrace through the system. 473 00:24:18,155 --> 00:24:20,030 Basically to retrace through the system means 474 00:24:20,030 --> 00:24:23,770 that I have to propagate this ray by a distance z. 475 00:24:23,770 --> 00:24:28,810 And then I have to find out what is the ray angle alpha 476 00:24:28,810 --> 00:24:33,460 2 with respect to the optical axis and the ray elevation x2, 477 00:24:33,460 --> 00:24:35,590 also with respect to the optical axis, 478 00:24:35,590 --> 00:24:39,020 as function of this propagation distance z. 479 00:24:39,020 --> 00:24:41,980 So now you can see why it was convenient to define 480 00:24:41,980 --> 00:24:47,020 the matrix for propagation through free space. 481 00:24:47,020 --> 00:24:49,300 Because now, in order to figure out 482 00:24:49,300 --> 00:24:55,250 where this ray lands, what is its elevation and propagation 483 00:24:55,250 --> 00:24:58,150 angle, all I have to do is cascade 484 00:24:58,150 --> 00:25:01,330 the matrix that describes the lens 485 00:25:01,330 --> 00:25:05,520 with a matrix that describes propagation through free space. 486 00:25:05,520 --> 00:25:08,890 And we have seen this matrix of propagation through free space. 487 00:25:08,890 --> 00:25:13,390 It is simply 1, 0 in the first row. 488 00:25:13,390 --> 00:25:17,387 And then the distance and 1 in the second row. 489 00:25:17,387 --> 00:25:19,720 And there's actually, if you look back at your equation, 490 00:25:19,720 --> 00:25:23,710 it is distance divided by the index of refraction. 491 00:25:23,710 --> 00:25:26,320 But, again, because we are propagating in air here, 492 00:25:26,320 --> 00:25:27,940 the index of refraction is 1. 493 00:25:27,940 --> 00:25:30,640 So we don't have to worry about this at the moment. 494 00:25:30,640 --> 00:25:35,200 And because the free space is following the lens, 495 00:25:35,200 --> 00:25:37,960 the matrix corresponding to free space 496 00:25:37,960 --> 00:25:42,190 will actually appear first before the matrix corresponding 497 00:25:42,190 --> 00:25:43,700 to the lens. 498 00:25:43,700 --> 00:25:45,910 So this is the cascade [INAUDIBLE] correctly here. 499 00:25:45,910 --> 00:25:49,630 And we can solve this. 500 00:25:49,630 --> 00:25:53,900 And we could find that is given by this equation over here. 501 00:25:53,900 --> 00:25:57,270 So what we observe now is that if we set the propagation 502 00:25:57,270 --> 00:26:02,910 distance to-- first of all, I did this in the general case. 503 00:26:02,910 --> 00:26:11,000 So as the ray entrance angle, I left a general symbol alpha 1. 504 00:26:11,000 --> 00:26:14,510 But I said already that the ray is arriving horizontal. 505 00:26:14,510 --> 00:26:16,820 That means alpha 1 equals 0. 506 00:26:16,820 --> 00:26:20,990 So therefore, the elevation of the ray, 507 00:26:20,990 --> 00:26:22,970 with respect to the optical axis, 508 00:26:22,970 --> 00:26:24,290 is given by this equation. 509 00:26:24,290 --> 00:26:35,420 It is simply x2 equals x1, 1 minus z over f, OK? 510 00:26:35,420 --> 00:26:40,460 And we can see very easily over here that if z equals f then 511 00:26:40,460 --> 00:26:43,090 x2 will equal to 0. 512 00:26:43,090 --> 00:26:49,120 So this means that if I let the ray propagate 513 00:26:49,120 --> 00:26:53,170 by a quantity equal to this magical amount that 514 00:26:53,170 --> 00:26:56,600 appeared in the lens maker's formula, then 515 00:26:56,600 --> 00:27:00,210 the ray will hit the axis. 516 00:27:00,210 --> 00:27:02,310 So the ray will land on the optical axis. 517 00:27:02,310 --> 00:27:04,650 And moreover, we can very easily verify 518 00:27:04,650 --> 00:27:09,180 that this happens independently of x1, because as you can see, 519 00:27:09,180 --> 00:27:11,400 x1 is outside the parenthesis here. 520 00:27:11,400 --> 00:27:13,950 If I set this quantity to equal 0, then 521 00:27:13,950 --> 00:27:19,440 independent of what x1 was, we will actually 522 00:27:19,440 --> 00:27:22,590 get all of the rays to go through the same point 523 00:27:22,590 --> 00:27:24,130 of the optical axis. 524 00:27:24,130 --> 00:27:25,800 So this is a central focus. 525 00:27:25,800 --> 00:27:29,220 You can see that the ray bundle that arrived parallel 526 00:27:29,220 --> 00:27:32,400 from infinity actually now comes to focus 527 00:27:32,400 --> 00:27:36,810 at this location of the optical axis because 528 00:27:36,810 --> 00:27:41,710 of this simple equations that we just described. 529 00:27:41,710 --> 00:27:44,890 And this property over here justifies the name 530 00:27:44,890 --> 00:27:46,030 focal length. 531 00:27:46,030 --> 00:27:49,200 It is the length at which the rays come to focus. 532 00:27:49,200 --> 00:27:49,700 OK. 533 00:27:52,980 --> 00:27:56,250 The inverse of the focal length, just on its own, 534 00:27:56,250 --> 00:28:00,750 as a sort of standalone quantity, has a name, also. 535 00:28:00,750 --> 00:28:03,570 It is called the optical power. 536 00:28:03,570 --> 00:28:06,570 And that is a little bit confusing because in our minds, 537 00:28:06,570 --> 00:28:09,870 usually, power is measured in what? 538 00:28:09,870 --> 00:28:12,780 Well, this power is not the usual power 539 00:28:12,780 --> 00:28:16,650 that we measure as energy per time. 540 00:28:16,650 --> 00:28:18,570 It is a different power. 541 00:28:18,570 --> 00:28:24,320 And for historical reasons, we use the same name in optics, 542 00:28:24,320 --> 00:28:28,400 but it is measured in inverse meters. 543 00:28:28,400 --> 00:28:30,890 And the inverse meters, they're also 544 00:28:30,890 --> 00:28:33,800 known as diopters in optics. 545 00:28:33,800 --> 00:28:37,770 And they define the optical power 546 00:28:37,770 --> 00:28:39,300 as the inverse of the focal length. 547 00:28:39,300 --> 00:28:41,190 So, for example, if the focal length 548 00:28:41,190 --> 00:28:43,890 is 1 meter, which is a pretty long focal length, 549 00:28:43,890 --> 00:28:45,150 you call it one diopter. 550 00:28:45,150 --> 00:28:48,900 If the focal length is 10 centimeters, 551 00:28:48,900 --> 00:28:50,690 it is actually 10 diopters. 552 00:28:50,690 --> 00:28:53,970 If it is 1 centimeter, it is 100 diopters, 553 00:28:53,970 --> 00:28:55,770 and so on and so forth. 554 00:28:55,770 --> 00:29:01,860 Very often, people like myself who are myopic-- 555 00:29:01,860 --> 00:29:04,950 is anybody in the class, is any of your myopic? 556 00:29:04,950 --> 00:29:10,030 Anybody use corrective lenses because you have myopia? 557 00:29:10,030 --> 00:29:11,520 OK. 558 00:29:11,520 --> 00:29:14,070 Do you know what is the power of your prescription? 559 00:29:14,070 --> 00:29:15,620 AUDIENCE: 0.75. 560 00:29:15,620 --> 00:29:17,620 GEORGE BARBASTATHIS: 0.75, you're pretty likely. 561 00:29:17,620 --> 00:29:18,943 You have very small correction. 562 00:29:18,943 --> 00:29:19,770 AUDIENCE: [INAUDIBLE] 563 00:29:19,770 --> 00:29:22,020 GEORGE BARBASTATHIS: That's a pretty small correction. 564 00:29:22,020 --> 00:29:23,910 Can you say it in the microphone? 565 00:29:23,910 --> 00:29:27,740 AUDIENCE: 0.75 in the one eye and in the other, 1.25. 566 00:29:27,740 --> 00:29:29,070 GEORGE BARBASTATHIS: OK, 0.25. 567 00:29:29,070 --> 00:29:31,200 Is it 0.25 or minus 0.25. 568 00:29:31,200 --> 00:29:32,490 AUDIENCE: Minus 1.25. 569 00:29:32,490 --> 00:29:34,320 GEORGE BARBASTATHIS: Minus 1.25, OK. 570 00:29:34,320 --> 00:29:35,560 So that is also in diopters. 571 00:29:35,560 --> 00:29:35,800 AUDIENCE: Yeah. 572 00:29:35,800 --> 00:29:37,925 GEORGE BARBASTATHIS: It means that the focal length 573 00:29:37,925 --> 00:29:40,860 of your lenses is approximately 80 centimeters, right? 574 00:29:40,860 --> 00:29:41,892 AUDIENCE: Right, mhm. 575 00:29:41,892 --> 00:29:42,975 GEORGE BARBASTATHIS: Yeah. 576 00:29:42,975 --> 00:29:43,590 Do you have a--? 577 00:29:43,590 --> 00:29:43,750 Yeah. 578 00:29:43,750 --> 00:29:44,410 AUDIENCE: Mhm. 579 00:29:44,410 --> 00:29:45,410 GEORGE BARBASTATHIS: OK. 580 00:29:45,410 --> 00:29:46,322 AUDIENCE: Yes. 581 00:29:46,322 --> 00:29:48,030 GEORGE BARBASTATHIS: So we will see later 582 00:29:48,030 --> 00:29:53,030 why then correction for myopic people is negative. 583 00:29:53,030 --> 00:29:54,133 I'm also myopic. 584 00:29:54,133 --> 00:29:56,550 And my correction, I actually have pretty high correction. 585 00:29:56,550 --> 00:30:00,330 Mine minus 7 and minus 5 diopters 586 00:30:00,330 --> 00:30:02,370 in my left and right eye, respectively. 587 00:30:02,370 --> 00:30:07,050 So it means, again, that the focal length of my glasses 588 00:30:07,050 --> 00:30:10,417 is approximately minus 1 over 7, which is-- 589 00:30:10,417 --> 00:30:11,250 I don't know, what-- 590 00:30:11,250 --> 00:30:14,970 17 centimeters or something. 591 00:30:14,970 --> 00:30:17,400 OK. 592 00:30:17,400 --> 00:30:18,480 So that is the power. 593 00:30:18,480 --> 00:30:22,380 Now, why we use the term optical power? 594 00:30:22,380 --> 00:30:24,570 Actually, it is best explained if you 595 00:30:24,570 --> 00:30:29,370 look at the final animation on this slide, which 596 00:30:29,370 --> 00:30:33,810 is what happens if the ray bundle comes from infinity 597 00:30:33,810 --> 00:30:36,420 again, but it is not horizontal. 598 00:30:36,420 --> 00:30:38,430 It is coming at an angle alpha 1. 599 00:30:38,430 --> 00:30:41,220 So now, the angle alpha 1 is not 0 anymore. 600 00:30:41,220 --> 00:30:50,720 But I let it have a finite non-zero value. 601 00:30:50,720 --> 00:30:57,790 And if you do that, then you find relatively easily 602 00:30:57,790 --> 00:30:59,617 from the equation over here. 603 00:30:59,617 --> 00:31:00,700 Let me write the equation. 604 00:31:05,345 --> 00:31:20,780 We have that x2 equals alpha 1 z plus x1, 1 minus z over f. 605 00:31:20,780 --> 00:31:21,280 OK. 606 00:31:21,280 --> 00:31:24,160 So if I set z equals f to the above equation, 607 00:31:24,160 --> 00:31:28,480 then I get x2 because alpha 1 times f. 608 00:31:28,480 --> 00:31:31,150 This term will disappear, of course. 609 00:31:31,150 --> 00:31:32,890 This will go to 0. 610 00:31:32,890 --> 00:31:35,080 And z will become f. 611 00:31:35,080 --> 00:31:38,890 So get x2 equals alpha 1 times f. 612 00:31:38,890 --> 00:31:41,950 So this is the equation for the elevation 613 00:31:41,950 --> 00:31:45,910 of the focus for a parallel ray bundle 614 00:31:45,910 --> 00:31:49,500 that is arriving non-horizontal from infinity 615 00:31:49,500 --> 00:31:52,000 at some angle alpha 1. 616 00:31:52,000 --> 00:31:53,670 So you can also write this equation-- 617 00:31:53,670 --> 00:31:54,670 it is not on the slide-- 618 00:31:54,670 --> 00:31:56,560 I will just add it on the white board. 619 00:31:56,560 --> 00:32:01,320 x2 equals alpha 1 over the power, the optical power 620 00:32:01,320 --> 00:32:02,170 of the system. 621 00:32:02,170 --> 00:32:04,410 So the power is kind of like a lever. 622 00:32:04,410 --> 00:32:07,770 It tells you as you change the angle alpha 1, 623 00:32:07,770 --> 00:32:12,900 it tells you how does the elevation of the focus change, 624 00:32:12,900 --> 00:32:15,460 with respect to the optical axis. 625 00:32:15,460 --> 00:32:17,460 So it is basically, again, if your thinking 626 00:32:17,460 --> 00:32:20,580 is if you order transduction amplification or something 627 00:32:20,580 --> 00:32:23,010 like that, it tells you as you change 628 00:32:23,010 --> 00:32:26,730 the angle that the rays arrive into the optical system, 629 00:32:26,730 --> 00:32:31,580 how does the focus move with respect to the optical axis. 630 00:32:31,580 --> 00:32:34,250 OK. 631 00:32:34,250 --> 00:32:36,120 Now, basically, these equations that we just 632 00:32:36,120 --> 00:32:39,240 described, let's see some different cases of what 633 00:32:39,240 --> 00:32:41,530 would happen to the rays. 634 00:32:41,530 --> 00:32:46,270 So what I've done here is I put the different types of lenses 635 00:32:46,270 --> 00:32:50,400 together with a simple spherical refractor. 636 00:32:50,400 --> 00:32:52,470 And the different types of lenses 637 00:32:52,470 --> 00:32:55,500 are plain or convex where you have one convex surface 638 00:32:55,500 --> 00:32:59,100 followed by a planar surface, Then 639 00:32:59,100 --> 00:33:02,850 biconvex where you have two planar surfaces, 640 00:33:02,850 --> 00:33:07,100 then planar concave and biconcave. 641 00:33:07,100 --> 00:33:09,830 So you can see from these that if you have a ray bundle that 642 00:33:09,830 --> 00:33:13,460 is arriving horizontal, now, horizontal from infinity, 643 00:33:13,460 --> 00:33:17,120 you can see that what will happen at the output 644 00:33:17,120 --> 00:33:19,850 is actually different, depending on the type of refraction 645 00:33:19,850 --> 00:33:23,120 that you get in the different surfaces. 646 00:33:23,120 --> 00:33:28,340 So I will let you work out, based on the equation, the lens 647 00:33:28,340 --> 00:33:31,070 maker's equation, I will let you work out why 648 00:33:31,070 --> 00:33:33,360 these cases are all different. 649 00:33:33,360 --> 00:33:38,750 But what I would like to do is actually work out 650 00:33:38,750 --> 00:33:41,840 the location of the image, depending 651 00:33:41,840 --> 00:33:44,990 on the different cases, and also, 652 00:33:44,990 --> 00:33:49,680 see how to ask with respect to the sign convention. 653 00:33:49,680 --> 00:33:52,080 So starting, for example, with a biconvex lens, 654 00:33:52,080 --> 00:33:54,020 we can see that the biconvex will 655 00:33:54,020 --> 00:33:58,250 focus the parallel bundle to the right-hand side of the lens. 656 00:33:58,250 --> 00:34:02,910 This is the case that we have implicitly done so far. 657 00:34:02,910 --> 00:34:06,430 In my drawing so far, I have assumed that this is the case. 658 00:34:06,430 --> 00:34:09,780 However, you can see that if I have a biconcave lens, 659 00:34:09,780 --> 00:34:12,489 like the one shown at the bottom of the slide, 660 00:34:12,489 --> 00:34:17,520 you can see that each one of these rays 661 00:34:17,520 --> 00:34:20,670 will actually expand outwards after it 662 00:34:20,670 --> 00:34:22,620 passes through the lens. 663 00:34:22,620 --> 00:34:26,070 So in a sense that we described so far, 664 00:34:26,070 --> 00:34:28,889 this lens does not really focus the rays. 665 00:34:28,889 --> 00:34:31,770 It creates a divergent ray bundle. 666 00:34:31,770 --> 00:34:38,699 You can get a sense of focus if you extend the divergent rays, 667 00:34:38,699 --> 00:34:42,000 if you extend them backwards towards the optical axis. 668 00:34:42,000 --> 00:34:43,380 If you do that, you will actually 669 00:34:43,380 --> 00:34:45,270 see that the rays meet. 670 00:34:45,270 --> 00:34:46,290 They do meet. 671 00:34:46,290 --> 00:34:49,290 But they meat on the left-hand side of the lens. 672 00:34:49,290 --> 00:34:53,909 So now, this is an example of focusing that happens not 673 00:34:53,909 --> 00:34:55,639 on the right-hand side of the lens, 674 00:34:55,639 --> 00:34:59,070 as our equations have shown, but on the left-hand side. 675 00:34:59,070 --> 00:35:04,740 And we can now justify this fact if you actually 676 00:35:04,740 --> 00:35:09,550 define z to be negative and f to be negative. 677 00:35:09,550 --> 00:35:13,960 So this is what we call a negative lens, 678 00:35:13,960 --> 00:35:15,820 or a lens with negative focal length, 679 00:35:15,820 --> 00:35:18,670 or a lens with negative power. 680 00:35:18,670 --> 00:35:21,000 And if you do that now, then you will 681 00:35:21,000 --> 00:35:24,090 see that the focus will appear at the left-hand side, 682 00:35:24,090 --> 00:35:26,700 consistent with our sign conventions. 683 00:35:26,700 --> 00:35:30,780 And this is what we call a virtual image, as opposed 684 00:35:30,780 --> 00:35:36,238 to the case of the biconvex lens that we'll call a real image. 685 00:35:36,238 --> 00:35:41,730 So this is an example of a sign conventions, not convections, 686 00:35:41,730 --> 00:35:46,510 the sign conventions in action. 687 00:35:46,510 --> 00:35:49,880 Are there any questions about this? 688 00:35:49,880 --> 00:35:53,567 This is a point when I often get a lot of questions. 689 00:36:01,250 --> 00:36:03,948 Let me move on to a different case. 690 00:36:03,948 --> 00:36:05,240 And if you think of a question. 691 00:36:05,240 --> 00:36:09,012 Please interrupt me, or say something aloud. 692 00:36:09,012 --> 00:36:13,120 Push on your microphone button first. 693 00:36:13,120 --> 00:36:15,640 Let me do a slightly different case, which 694 00:36:15,640 --> 00:36:18,340 is what happens if you try to-- 695 00:36:18,340 --> 00:36:21,280 so in the previous example, we saw 696 00:36:21,280 --> 00:36:24,940 how a lens can take an object at infinity 697 00:36:24,940 --> 00:36:26,980 and focus it to a finite distance 698 00:36:26,980 --> 00:36:29,250 equal to the focal length. 699 00:36:29,250 --> 00:36:31,110 What I will do now is I will argue 700 00:36:31,110 --> 00:36:39,610 that the positive lens actually can take a point object located 701 00:36:39,610 --> 00:36:45,040 at one focal distance to the left and image it at infinity. 702 00:36:45,040 --> 00:36:46,380 Now, why is that true? 703 00:36:50,247 --> 00:36:52,870 I will do this in a second. 704 00:36:52,870 --> 00:36:56,890 But you can very easily justify it to yourself 705 00:36:56,890 --> 00:36:59,730 if you simply reverse the orientation of the rays 706 00:36:59,730 --> 00:37:01,190 in the previous case. 707 00:37:01,190 --> 00:37:09,220 What we did before is we had the ray bundle that 708 00:37:09,220 --> 00:37:10,710 was starting from infinity. 709 00:37:16,470 --> 00:37:18,470 And the lens was focusing it. 710 00:37:33,876 --> 00:37:37,370 We can very easily reverse the radiation of the rays. 711 00:37:37,370 --> 00:37:39,590 And if we do that, then we can see 712 00:37:39,590 --> 00:37:41,930 that if I have a point source over here, 713 00:37:41,930 --> 00:37:44,980 it will actually get imaged at infinity. 714 00:37:44,980 --> 00:37:47,360 Now, of course, according to my sign conventions, 715 00:37:47,360 --> 00:37:50,063 this is not the proper way to write it. 716 00:37:50,063 --> 00:37:51,980 We have to make sure that the light propagates 717 00:37:51,980 --> 00:37:53,152 from left to right. 718 00:37:53,152 --> 00:37:54,610 So therefore, the correct to write, 719 00:37:54,610 --> 00:37:58,040 it is not what is showing over here, simply by convention, 720 00:37:58,040 --> 00:38:00,080 not because this is physically wrong. 721 00:38:00,080 --> 00:38:02,160 But it does not obey the convention. 722 00:38:02,160 --> 00:38:06,220 So we never actually draw something like this. 723 00:38:06,220 --> 00:38:10,040 What we draw is what is shown on the slide where the rays are 724 00:38:10,040 --> 00:38:13,870 propagating from left to right. 725 00:38:13,870 --> 00:38:17,200 They have a focus that is a point 726 00:38:17,200 --> 00:38:19,420 object on the left-hand side of the lens. 727 00:38:19,420 --> 00:38:22,510 And then the lens, of course, will collimate these rays. 728 00:38:22,510 --> 00:38:29,170 They will convert them from a divergence spherical wave 729 00:38:29,170 --> 00:38:31,710 to a parallel plane wave. 730 00:38:31,710 --> 00:38:34,490 And therefore, we limit at infinity. 731 00:38:34,490 --> 00:38:37,000 The same thing can be said about the negative lens, 732 00:38:37,000 --> 00:38:40,810 except in this case, the ray bundle arriving at the lens 733 00:38:40,810 --> 00:38:44,740 has to be convergent and has to come 734 00:38:44,740 --> 00:38:47,380 to a focus on the right-hand side of the lens. 735 00:38:47,380 --> 00:38:50,140 So therefore, in the same way that this type of lens 736 00:38:50,140 --> 00:38:54,400 formed the virtual image in the previous case, in this case, 737 00:38:54,400 --> 00:38:58,220 we talk about the virtual object for this type of lens. 738 00:38:58,220 --> 00:38:58,720 OK. 739 00:38:58,720 --> 00:39:02,650 So let me now try to put this all together so it 740 00:39:02,650 --> 00:39:03,850 can make some sort of sense. 741 00:39:08,430 --> 00:39:11,610 Again, recall the first sign convention, 742 00:39:11,610 --> 00:39:15,043 which sees that the light propagates from left to right. 743 00:39:15,043 --> 00:39:16,710 If it propagates from the left to right, 744 00:39:16,710 --> 00:39:21,000 it means that the object should be to the left of anything that 745 00:39:21,000 --> 00:39:21,720 is of interest. 746 00:39:21,720 --> 00:39:25,050 And the image should be to the right of anything 747 00:39:25,050 --> 00:39:27,700 that is of interest to the optical system. 748 00:39:27,700 --> 00:39:30,300 So that is why if an object is to the left 749 00:39:30,300 --> 00:39:34,440 of the optical element, as it is in this case, then 750 00:39:34,440 --> 00:39:37,260 we say that the distance from the object to the element 751 00:39:37,260 --> 00:39:39,880 is positive. 752 00:39:39,880 --> 00:39:42,370 If, on the other hand, an object happens 753 00:39:42,370 --> 00:39:44,770 to be on the right of an optical element, 754 00:39:44,770 --> 00:39:47,980 as it happens in this case with a negative lens, 755 00:39:47,980 --> 00:39:51,100 then we say that the object is virtual. 756 00:39:51,100 --> 00:39:53,020 And the image from the object to the element 757 00:39:53,020 --> 00:39:57,670 is negative, because it has to be on the right-hand side, 758 00:39:57,670 --> 00:40:02,750 whereas in the proper case, it had to be on the left. 759 00:40:02,750 --> 00:40:06,150 That's why we put a negative sign. 760 00:40:06,150 --> 00:40:09,230 Now, similar things can be can be said about an image. 761 00:40:09,230 --> 00:40:12,200 Except, now, the image is properly positioned 762 00:40:12,200 --> 00:40:14,690 or the right-hand side of the optical element. 763 00:40:14,690 --> 00:40:17,690 So everything is really on the right-hand side, 764 00:40:17,690 --> 00:40:21,380 as it happens with a positive biconvex lens. 765 00:40:24,620 --> 00:40:27,500 Then the distance from the lens to the image 766 00:40:27,500 --> 00:40:31,930 will be referred to as positive, whereas on the other hand, 767 00:40:31,930 --> 00:40:34,100 if it happens to be on the left, as it 768 00:40:34,100 --> 00:40:36,560 is for the case of the negative lens, 769 00:40:36,560 --> 00:40:43,370 then this image will be referred to as virtual, 770 00:40:43,370 --> 00:40:47,438 and the distance will be referred to as negative. 771 00:40:50,510 --> 00:40:53,770 Finally, and I will conclude with this, 772 00:40:53,770 --> 00:40:58,710 and I will let Pepe do his demo, is how this 773 00:40:58,710 --> 00:41:00,740 applies to off-axis objects. 774 00:41:00,740 --> 00:41:03,450 So remember we derived this equation, 775 00:41:03,450 --> 00:41:09,900 x2 equals alpha 1 times f, for an object that is at infinity. 776 00:41:09,900 --> 00:41:13,020 Now, for a positive lens, because the focal length 777 00:41:13,020 --> 00:41:16,450 is positive, if the angle alpha 1 778 00:41:16,450 --> 00:41:19,060 is also positive, as shown here, it 779 00:41:19,060 --> 00:41:21,810 means that x2 must also be positive. 780 00:41:21,810 --> 00:41:23,950 And this is indeed what we can verify. 781 00:41:23,950 --> 00:41:25,715 If we apply Snell's law repeatedly 782 00:41:25,715 --> 00:41:27,340 for each one of these rays, we will see 783 00:41:27,340 --> 00:41:28,910 that indeed this is the case. 784 00:41:28,910 --> 00:41:31,810 On the other hand, if I use a negative lens, 785 00:41:31,810 --> 00:41:39,100 where the quantity f, the focal distance, is negative, 786 00:41:39,100 --> 00:41:43,670 but the angle of arrival is still positive, 787 00:41:43,670 --> 00:41:45,850 then I will get x2 is negative. 788 00:41:45,850 --> 00:41:50,950 That is, x2 will also appear below the optical axis, OK? 789 00:41:50,950 --> 00:41:54,130 So this virtual image not only appears 790 00:41:54,130 --> 00:41:56,440 on the left of the negative lens, 791 00:41:56,440 --> 00:41:59,480 but it is also below the optical axis. 792 00:41:59,480 --> 00:42:01,180 OK? 793 00:42:01,180 --> 00:42:09,290 Now, what about an off-axis image at infinity? 794 00:42:09,290 --> 00:42:10,700 I will skip this derivation. 795 00:42:10,700 --> 00:42:12,470 It is very similar to the derivation 796 00:42:12,470 --> 00:42:18,230 that we did before for the case of an object at infinity. 797 00:42:18,230 --> 00:42:21,390 Basically, what I've done here is I have reversed it. 798 00:42:21,390 --> 00:42:23,330 I have used an object at a distance 799 00:42:23,330 --> 00:42:26,430 f to the left of the lens. 800 00:42:26,430 --> 00:42:32,040 I have cascaded the propagation matrix for this free space 801 00:42:32,040 --> 00:42:35,160 propagation to the propagation method corresponding 802 00:42:35,160 --> 00:42:36,280 to the lens. 803 00:42:36,280 --> 00:42:38,220 And I have proceeded to solve. 804 00:42:38,220 --> 00:42:40,500 If I do this, I will let you do it at home 805 00:42:40,500 --> 00:42:43,320 and then come back tomorrow if something 806 00:42:43,320 --> 00:42:44,940 is unclear about this. 807 00:42:44,940 --> 00:42:49,078 But what I've done is I have solved it. 808 00:42:49,078 --> 00:42:51,120 And when I find an equation that looks like this, 809 00:42:51,120 --> 00:42:54,030 I find that the angle of propagation 810 00:42:54,030 --> 00:42:58,050 of the ray bundle to the right-hand side of the lens 811 00:42:58,050 --> 00:43:07,077 actually equals minus the elevation of the object divided 812 00:43:07,077 --> 00:43:07,910 by the focal length. 813 00:43:07,910 --> 00:43:11,000 Or another way to write this equation is 814 00:43:11,000 --> 00:43:16,160 alpha 2 equals minus x1 times the power 815 00:43:16,160 --> 00:43:17,840 of the optical element, according 816 00:43:17,840 --> 00:43:21,800 to the definition of power that I remind you by definition 817 00:43:21,800 --> 00:43:24,030 equals 1 over the focal length. 818 00:43:24,030 --> 00:43:24,530 OK. 819 00:43:24,530 --> 00:43:28,920 So there's a minus sign now, which 820 00:43:28,920 --> 00:43:33,390 means that if we apply this equation to our positive lens, 821 00:43:33,390 --> 00:43:36,030 the ray bundle that will emerge from the lens 822 00:43:36,030 --> 00:43:39,030 will actually now propagate downwards, OK? 823 00:43:39,030 --> 00:43:42,450 Because in this case, x1 is positive 824 00:43:42,450 --> 00:43:44,370 and the focal length is positive, 825 00:43:44,370 --> 00:43:46,440 alpha 2 has to be negative. 826 00:43:46,440 --> 00:43:48,540 And negative alpha 2 means that ray bundle 827 00:43:48,540 --> 00:43:51,040 has to propagate downwards. 828 00:43:51,040 --> 00:43:51,930 OK? 829 00:43:51,930 --> 00:43:55,440 If I do the same exercise for the negative lens, 830 00:43:55,440 --> 00:44:03,250 then we can see, again, very easily that because x1, 831 00:44:03,250 --> 00:44:06,950 If I pick it to be positive, f is negative 832 00:44:06,950 --> 00:44:10,490 and I have a spare negative sign from the equation. 833 00:44:10,490 --> 00:44:15,560 Then I can see that the propagation angle, alpha 2, 834 00:44:15,560 --> 00:44:16,640 will be positive. 835 00:44:16,640 --> 00:44:22,220 That is, the image at infinity will 836 00:44:22,220 --> 00:44:28,580 be on the positive side, above the optical axis, OK? 837 00:44:28,580 --> 00:44:32,520 I hope the positive and negative signs are not confusing. 838 00:44:32,520 --> 00:44:33,470 Let me restate. 839 00:44:33,470 --> 00:44:35,450 When I learned optics, I found this 840 00:44:35,450 --> 00:44:38,080 to be also horribly confusing. 841 00:44:38,080 --> 00:44:42,440 So I realize and I sympathize how at the beginning, 842 00:44:42,440 --> 00:44:46,040 this can be a little bit confusing. 843 00:44:46,040 --> 00:44:48,980 I've done my best here to present them 844 00:44:48,980 --> 00:44:54,200 better than the professional who taught them to me solve them. 845 00:44:54,200 --> 00:44:57,770 So I hope, because I sort of used my own confusion 846 00:44:57,770 --> 00:45:00,050 as a guide, I hope that I managed 847 00:45:00,050 --> 00:45:02,510 to make it a little bit clearer for you 848 00:45:02,510 --> 00:45:04,173 if it is the first time you see it. 849 00:45:04,173 --> 00:45:05,840 But, of course, it will become even more 850 00:45:05,840 --> 00:45:07,940 clearer as you practice, OK? 851 00:45:07,940 --> 00:45:11,420 So the homeworks and the examples that we see later, 852 00:45:11,420 --> 00:45:15,020 they will make it progressively clearer. 853 00:45:15,020 --> 00:45:17,420 For now, what I want you to do between now 854 00:45:17,420 --> 00:45:21,530 and tomorrow's lecture is to go back and make sure 855 00:45:21,530 --> 00:45:25,790 that these pictures that I've drawn, that they make sense 856 00:45:25,790 --> 00:45:26,653 with Snell's law. 857 00:45:26,653 --> 00:45:28,070 In other words, what you do is you 858 00:45:28,070 --> 00:45:31,250 go to the spherical surface. 859 00:45:31,250 --> 00:45:34,400 You imagine a normal to the surface at each point. 860 00:45:34,400 --> 00:45:35,960 You apply Snell's law. 861 00:45:35,960 --> 00:45:39,890 And you convince yourselves that the rays bend the correct way. 862 00:45:39,890 --> 00:45:43,580 And this is consistent with the way they image is formed, OK? 863 00:45:43,580 --> 00:45:46,140 And then you apply all the methods, all the equations, 864 00:45:46,140 --> 00:45:48,450 and so on and so forth. 865 00:45:48,450 --> 00:45:51,325 I broke my promise and I did not leave Pepe with his 5 minutes. 866 00:45:51,325 --> 00:45:52,950 But that's because it took us 5 minutes 867 00:45:52,950 --> 00:45:55,740 to fix the figures, the display, or whatever 868 00:45:55,740 --> 00:45:57,280 was happening at the beginning. 869 00:45:57,280 --> 00:45:59,923 So hopefully, you guys can stay for a little bit longer, 870 00:45:59,923 --> 00:46:00,840 so we can do the demo. 871 00:46:00,840 --> 00:46:01,620 Is that OK? 872 00:46:01,620 --> 00:46:03,030 PROFESSOR: Sure. 873 00:46:03,030 --> 00:46:04,050 I'll try to do it. 874 00:46:04,050 --> 00:46:05,520 We'll try to do it fast. 875 00:46:05,520 --> 00:46:07,658 GEORGE BARBASTATHIS: Also, and look at the-- 876 00:46:07,658 --> 00:46:08,450 PROFESSOR: So now-- 877 00:46:08,450 --> 00:46:09,040 GEORGE BARBASTATHIS: Yeah. 878 00:46:09,040 --> 00:46:11,123 PROFESSOR: We can switch to their cameras, please. 879 00:46:13,540 --> 00:46:14,040 OK. 880 00:46:14,040 --> 00:46:18,000 So here we go. 881 00:46:18,000 --> 00:46:19,900 OK so we tried to set up these webcams 882 00:46:19,900 --> 00:46:23,340 because last time it was a bit hard to see the demo. 883 00:46:25,900 --> 00:46:27,610 So this demo has actually two parts. 884 00:46:27,610 --> 00:46:31,650 But today, we're just going to show one part, namely, 885 00:46:31,650 --> 00:46:32,502 the refraction. 886 00:46:32,502 --> 00:46:34,960 And we're going to basically witness how Fermat's principle 887 00:46:34,960 --> 00:46:37,252 that we've been learning in the past couple of classes, 888 00:46:37,252 --> 00:46:39,970 the Snell's law, applies in these systems, 889 00:46:39,970 --> 00:46:43,310 like lenses or prisms that we've been learning so far. 890 00:46:43,310 --> 00:46:46,930 So before actually changing the exposure, 891 00:46:46,930 --> 00:46:50,170 let me just show the set up. 892 00:46:50,170 --> 00:46:54,040 So we are going to be focusing on this half of the set up. 893 00:46:54,040 --> 00:46:56,620 So hopefully, you can see it fine here. 894 00:46:56,620 --> 00:46:58,810 Let me just move this window. 895 00:46:58,810 --> 00:47:02,740 So to avoid any problems about chromatic dispersion 896 00:47:02,740 --> 00:47:04,690 or wavelength-dependent behavior, 897 00:47:04,690 --> 00:47:06,970 we're using a laser, a green laser. 898 00:47:06,970 --> 00:47:09,820 So this component here is the laser. 899 00:47:09,820 --> 00:47:11,590 Then this is a new component that we're 900 00:47:11,590 --> 00:47:14,470 going to call-- it's called spatial filter. 901 00:47:14,470 --> 00:47:17,618 This, for now, it has a very interesting property 902 00:47:17,618 --> 00:47:19,660 that we'll learn in the second half of the course 903 00:47:19,660 --> 00:47:21,850 when we learn about Fourier optics. 904 00:47:21,850 --> 00:47:25,780 For this first part, just think about this produces a very nice 905 00:47:25,780 --> 00:47:27,300 point source object. 906 00:47:27,300 --> 00:47:28,890 So therefore, after this lens, there's 907 00:47:28,890 --> 00:47:32,440 going to be a spherical wave emanating, propagating. 908 00:47:32,440 --> 00:47:35,260 So what we want to do after that is 909 00:47:35,260 --> 00:47:38,230 to convert that spherical wave into a plane wave, right? 910 00:47:38,230 --> 00:47:41,120 So this is what this lens here is doing. 911 00:47:41,120 --> 00:47:42,880 And that's what we call collimation. 912 00:47:42,880 --> 00:47:45,400 So after this lens, as you can see-- well, 913 00:47:45,400 --> 00:47:48,770 you'll see it in a second. 914 00:47:48,770 --> 00:47:49,270 OK. 915 00:47:49,270 --> 00:47:51,370 So now, I reduce the exposure of the camera, 916 00:47:51,370 --> 00:47:52,600 and I can see the rays. 917 00:47:52,600 --> 00:47:54,540 Now, I see parallel rays. 918 00:47:54,540 --> 00:47:56,040 Hopefully, you can see them clearly. 919 00:47:56,040 --> 00:47:58,540 Could you zoom in a little bit? 920 00:47:58,540 --> 00:47:59,540 OK. 921 00:47:59,540 --> 00:48:02,700 And we have basically parallel rays coming out 922 00:48:02,700 --> 00:48:04,880 of the collimated lens. 923 00:48:04,880 --> 00:48:06,722 So now, this brings another interesting. 924 00:48:06,722 --> 00:48:08,180 As you can see, our eyes, and we'll 925 00:48:08,180 --> 00:48:10,700 see it next lecture, how they are really 926 00:48:10,700 --> 00:48:13,070 robust to focusing in different conditions, 927 00:48:13,070 --> 00:48:14,360 to change illumination. 928 00:48:14,360 --> 00:48:17,270 And more particularly, they're adapted 929 00:48:17,270 --> 00:48:22,610 to high-contrast in light, like very high intensity 930 00:48:22,610 --> 00:48:24,260 in the light and low intensity. 931 00:48:24,260 --> 00:48:26,360 Your eyes can see it fine, whereas this camera, 932 00:48:26,360 --> 00:48:27,840 we just need it to change exposure, 933 00:48:27,840 --> 00:48:29,820 so you could be able to see that. 934 00:48:29,820 --> 00:48:31,850 So this is a problem of dynamic range 935 00:48:31,850 --> 00:48:34,420 that occurs in optical systems. 936 00:48:34,420 --> 00:48:37,130 So, all right, let's look at the first component. 937 00:48:37,130 --> 00:48:39,050 It's this lens here. 938 00:48:39,050 --> 00:48:42,930 I'm going to put cylindrical lens. 939 00:48:42,930 --> 00:48:47,030 It's a plano curve lens here that it's cylindrical. 940 00:48:47,030 --> 00:48:49,010 Basically, it means that the surface only 941 00:48:49,010 --> 00:48:53,715 depends on the x-coordinate. 942 00:48:53,715 --> 00:48:54,590 So we'll put it here. 943 00:48:54,590 --> 00:48:55,430 And [INAUDIBLE] wavelength. 944 00:48:55,430 --> 00:48:57,030 GEORGE BARBASTATHIS: If I may interject here. 945 00:48:57,030 --> 00:48:58,430 Pepe, if I may interject something? 946 00:48:58,430 --> 00:48:59,340 PROFESSOR: Sure, sure, sure. 947 00:48:59,340 --> 00:49:00,140 GEORGE BARBASTATHIS: That equations 948 00:49:00,140 --> 00:49:01,770 that we derived, strictly speaking, 949 00:49:01,770 --> 00:49:03,410 they're for cylindrical lenses, right, 950 00:49:03,410 --> 00:49:06,130 because we only use x in our equations. 951 00:49:06,130 --> 00:49:06,800 PROFESSOR: Yeah. 952 00:49:06,800 --> 00:49:09,350 GEORGE BARBASTATHIS: If you have an x and y, 953 00:49:09,350 --> 00:49:12,060 then you get the proper spherical lens. 954 00:49:12,060 --> 00:49:13,775 So strictly speaking, this is-- 955 00:49:13,775 --> 00:49:14,442 PROFESSOR: Yeah. 956 00:49:14,442 --> 00:49:16,430 GEORGE BARBASTATHIS: --what that equations-- 957 00:49:16,430 --> 00:49:16,830 PROFESSOR: Exactly. 958 00:49:16,830 --> 00:49:17,830 GEORGE BARBASTATHIS: OK. 959 00:49:17,830 --> 00:49:21,110 PROFESSOR: So now, you can see now the ray tracing, but now 960 00:49:21,110 --> 00:49:21,890 done optically. 961 00:49:21,890 --> 00:49:27,260 You can see how the rays started focusing in to this point here. 962 00:49:27,260 --> 00:49:30,690 Well, in this case, now, since this is cylindrical lens, 963 00:49:30,690 --> 00:49:31,910 we see that it's a line. 964 00:49:31,910 --> 00:49:34,580 And I can put this card here and we see that we actually 965 00:49:34,580 --> 00:49:38,170 form a line instead of a point. 966 00:49:38,170 --> 00:49:40,740 Other nice things to see about this lens 967 00:49:40,740 --> 00:49:43,130 is that so far, we've been ignoring reflections 968 00:49:43,130 --> 00:49:44,360 that occur with this lens. 969 00:49:44,360 --> 00:49:47,030 But let me just show you some of the reflections 970 00:49:47,030 --> 00:49:48,870 in this cardboard here. 971 00:49:48,870 --> 00:49:52,550 So we'll see in the second part of the course 972 00:49:52,550 --> 00:49:54,560 that these reflections are occurring, 973 00:49:54,560 --> 00:49:58,800 transitioning from the air and glass interface and back again. 974 00:49:58,800 --> 00:50:02,690 So in these interfaces are very sensitive to the angle 975 00:50:02,690 --> 00:50:05,210 of incidence and to the refractive indexes 976 00:50:05,210 --> 00:50:08,817 of both the lens and the surrounding media. 977 00:50:08,817 --> 00:50:11,150 So in this case, you can see that quite a lot of power-- 978 00:50:11,150 --> 00:50:12,890 so this is the incidence wave. 979 00:50:12,890 --> 00:50:14,390 And you can see quite a lot of power 980 00:50:14,390 --> 00:50:17,900 is actually going to this side because this lens might not 981 00:50:17,900 --> 00:50:21,710 be AR-coated, or coated with an anti-reflection coating 982 00:50:21,710 --> 00:50:25,720 as a better lens would be. 983 00:50:25,720 --> 00:50:29,690 So now, let's witness Snell's law with this prism here. 984 00:50:29,690 --> 00:50:32,830 So you see the little rays here that are going straight. 985 00:50:32,830 --> 00:50:36,220 So now, I introduce this prism shown here. 986 00:50:36,220 --> 00:50:37,090 It looks very dark. 987 00:50:37,090 --> 00:50:39,740 But believe me, there's a prism here. 988 00:50:39,740 --> 00:50:42,860 And then as soon as I started putting it into the system, 989 00:50:42,860 --> 00:50:47,950 you see how the light starts bending to one side more. 990 00:50:52,340 --> 00:50:56,710 And then you can see how basically, the tilted surface 991 00:50:56,710 --> 00:50:59,360 is bending the light, and I can just change the angle. 992 00:50:59,360 --> 00:51:01,460 And let's now see the other phenomena 993 00:51:01,460 --> 00:51:04,220 that we learned in the previous class, which is TIR. 994 00:51:04,220 --> 00:51:06,320 So I still rotate this more. 995 00:51:06,320 --> 00:51:09,620 I make the angle to be larger than the critical angle, 996 00:51:09,620 --> 00:51:10,370 and boom. 997 00:51:10,370 --> 00:51:13,880 All the light gets TIRed and we can actually see, now 998 00:51:13,880 --> 00:51:15,540 that it's getting to this side. 999 00:51:15,540 --> 00:51:18,803 But in this side of the prism, there is a diffuser. 1000 00:51:18,803 --> 00:51:20,720 And that's why you see the diffused light here 1001 00:51:20,720 --> 00:51:21,540 very bright. 1002 00:51:21,540 --> 00:51:23,390 So I'm going to do it one more time. 1003 00:51:23,390 --> 00:51:24,320 Without TIR. 1004 00:51:24,320 --> 00:51:27,496 All the light goes, escapes out straight through. 1005 00:51:27,496 --> 00:51:31,740 And now, I start rotating this more. 1006 00:51:31,740 --> 00:51:33,100 And then you see TIR here. 1007 00:51:36,150 --> 00:51:37,512 There are any questions? 1008 00:51:43,910 --> 00:51:47,840 So I also brought a parabolic reflector for you guys, 1009 00:51:47,840 --> 00:51:50,420 like the people here in MIT if you guys 1010 00:51:50,420 --> 00:51:54,950 want to see the reflection, how you see your face. 1011 00:51:54,950 --> 00:51:57,150 Look at your face with this parabolic reflector. 1012 00:51:57,150 --> 00:51:59,600 It's actually quite funny to see the virtual 1013 00:51:59,600 --> 00:52:00,630 and the real images. 1014 00:52:00,630 --> 00:52:02,255 And in this case, you'll see, and we'll 1015 00:52:02,255 --> 00:52:04,970 learn in next class, how the virtual image generated 1016 00:52:04,970 --> 00:52:09,680 by such a reflector, it's actually not inverted. 1017 00:52:09,680 --> 00:52:11,720 It's erected, as it's called. 1018 00:52:11,720 --> 00:52:16,060 So I'll leave it here, so you can come after class.