1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,850 Commons license. 3 00:00:03,850 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 00:00:10,150 continue to offer high quality educational resources for free. 5 00:00:10,150 --> 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 --> 00:00:18,012 at ocw.mit.edu. 8 00:00:21,545 --> 00:00:24,500 PROFESSOR: So today, in some ways, that word for what 9 00:00:24,500 --> 00:00:28,110 we've been suffering through for the last few lectures 10 00:00:28,110 --> 00:00:31,680 because we'll actually get to see some optical instruments 11 00:00:31,680 --> 00:00:33,790 and how they work. 12 00:00:33,790 --> 00:00:36,840 So we'll basically put all of these tools 13 00:00:36,840 --> 00:00:39,870 and all of the terminology and the matrices 14 00:00:39,870 --> 00:00:42,750 and the formulas and all that, we'll 15 00:00:42,750 --> 00:00:47,190 put it to work for the basic optical systems, namely 16 00:00:47,190 --> 00:00:49,740 microscope and telescope. 17 00:00:49,740 --> 00:00:53,130 And we'll also look at sort of the building 18 00:00:53,130 --> 00:00:58,130 block of a microscope which is a magnifier lens. 19 00:00:58,130 --> 00:01:00,530 But before we do that, I would like 20 00:01:00,530 --> 00:01:05,600 to spend some time just reviewing basically 21 00:01:05,600 --> 00:01:08,720 all the fundamental material before it will go 22 00:01:08,720 --> 00:01:12,120 into the application systems. 23 00:01:12,120 --> 00:01:15,290 I would like to do review the fundamental material. 24 00:01:15,290 --> 00:01:18,362 So the first four or five slides-- 25 00:01:18,362 --> 00:01:19,070 I don't remember. 26 00:01:19,070 --> 00:01:20,780 I think five. 27 00:01:20,780 --> 00:01:21,840 They're not really new. 28 00:01:21,840 --> 00:01:24,050 I just brought them-- 29 00:01:24,050 --> 00:01:26,030 just copied them from the previous lectures 30 00:01:26,030 --> 00:01:28,280 as a kind of overview. 31 00:01:28,280 --> 00:01:32,000 And I would like, instead of just describing it again, 32 00:01:32,000 --> 00:01:33,510 I would like to encourage you-- 33 00:01:33,510 --> 00:01:35,237 I will pass on it and I would like 34 00:01:35,237 --> 00:01:37,070 to encourage you to ask any questions if you 35 00:01:37,070 --> 00:01:42,290 have on this particular topic or something related. 36 00:01:42,290 --> 00:01:45,200 So the first topic is the ray transfer matrices. 37 00:01:47,750 --> 00:01:49,540 And again, very briefly to remind you, 38 00:01:49,540 --> 00:01:55,180 this comes about because at a fraction at the spherical 39 00:01:55,180 --> 00:01:58,880 dielectric interface is very complex to deal with, 40 00:01:58,880 --> 00:02:00,050 not the very complex. 41 00:02:00,050 --> 00:02:01,400 It is actually pretty simple. 42 00:02:01,400 --> 00:02:03,920 But algebraically complicated, or I 43 00:02:03,920 --> 00:02:07,070 should say trigonometrically, very complicated. 44 00:02:07,070 --> 00:02:09,860 So because we don't want to deal with that stuff. 45 00:02:09,860 --> 00:02:13,220 And it also hides a lot of the intuition, 46 00:02:13,220 --> 00:02:16,640 we derived this paraxial approximation which 47 00:02:16,640 --> 00:02:19,820 is true for a pretty limited-- 48 00:02:19,820 --> 00:02:23,900 typically 10 to 30 degrees. 49 00:02:23,900 --> 00:02:27,440 Actually 30 is too optimistic, maybe 10 to 20 degrees 50 00:02:27,440 --> 00:02:29,870 from the optical axis. 51 00:02:29,870 --> 00:02:32,540 We discovered that there is a linear relationship 52 00:02:32,540 --> 00:02:40,900 between them, elevation and angle of a ray that is arriving 53 00:02:40,900 --> 00:02:43,540 from the left to the interface. 54 00:02:43,540 --> 00:02:48,010 And the elevation and angle of propagation of array 55 00:02:48,010 --> 00:02:52,430 that is departing to the right over the interface, 56 00:02:52,430 --> 00:02:54,130 the same rate, that is [INAUDIBLE].. 57 00:02:54,130 --> 00:02:57,370 And of course, the elevation is the same because 58 00:02:57,370 --> 00:02:59,590 of the paraxial approximation, we 59 00:02:59,590 --> 00:03:02,650 neglect the thickness of the instrument. 60 00:03:02,650 --> 00:03:05,890 So basically, no matter where the ray hits, here or here 61 00:03:05,890 --> 00:03:08,020 or here, it is all the same. 62 00:03:08,020 --> 00:03:09,850 So that helps. 63 00:03:09,850 --> 00:03:15,070 And the second is that the angle, basically the angle 64 00:03:15,070 --> 00:03:18,160 satisfies Snell's law with respect 65 00:03:18,160 --> 00:03:22,180 to the normal at this sphere at this location over here. 66 00:03:22,180 --> 00:03:24,070 But of course, because this orientation 67 00:03:24,070 --> 00:03:29,640 depends on the relative position of the arrival of the ray 68 00:03:29,640 --> 00:03:32,650 and the center of the sphere, then we 69 00:03:32,650 --> 00:03:37,940 simplified it using the paraxial approximations. 70 00:03:37,940 --> 00:03:40,970 And we discovered this expression over here. 71 00:03:40,970 --> 00:03:44,680 So the relationship between the right and the left angle, 72 00:03:44,680 --> 00:03:48,370 it has to do with the elevation. 73 00:03:48,370 --> 00:03:50,440 So if you look at this equation, it 74 00:03:50,440 --> 00:03:55,360 says that n-right, alpha-right equals n-left alpha-left. 75 00:03:55,360 --> 00:03:56,800 That's the first term. 76 00:03:56,800 --> 00:03:59,350 And then the interesting part is the second one that says 77 00:03:59,350 --> 00:04:04,660 equals minus this quantity times the elevation of the ray. 78 00:04:04,660 --> 00:04:06,940 So that tells you that the more-- 79 00:04:06,940 --> 00:04:12,830 the higher the arrival elevation of the ray, the more it bends. 80 00:04:12,830 --> 00:04:15,460 And that makes sense because the higher the arrival, 81 00:04:15,460 --> 00:04:18,850 the larger the angle that it makes with respect 82 00:04:18,850 --> 00:04:23,920 to the normal to the sphere and, therefore, if you wish, 83 00:04:23,920 --> 00:04:26,890 the more violent the Snell's law will be. 84 00:04:26,890 --> 00:04:29,890 And of course, you can imagine, as you increase the angle, 85 00:04:29,890 --> 00:04:33,220 eventually the angle becomes so large 86 00:04:33,220 --> 00:04:35,250 that the paraxial approximation breaks down. 87 00:04:35,250 --> 00:04:37,570 But this is all true. 88 00:04:37,570 --> 00:04:39,535 Before the approximation breaks down, 89 00:04:39,535 --> 00:04:41,950 the progression is linear. 90 00:04:41,950 --> 00:04:44,730 It is linear in the position. 91 00:04:44,730 --> 00:04:48,090 And the quantity that appears here, does anybody remember? 92 00:04:48,090 --> 00:04:49,340 How do you call this quantity? 93 00:04:55,580 --> 00:04:56,200 Optical power. 94 00:04:56,200 --> 00:04:56,630 Correct. 95 00:04:56,630 --> 00:04:58,005 Could you please push the button? 96 00:05:01,160 --> 00:05:01,660 Correct. 97 00:05:01,660 --> 00:05:03,540 It's called the optical power. 98 00:05:03,540 --> 00:05:06,570 And it is actually the first instance 99 00:05:06,570 --> 00:05:08,970 that we saw this quantity. 100 00:05:08,970 --> 00:05:12,620 It is the inverse distance units. 101 00:05:12,620 --> 00:05:14,760 But we don't call it inverse meters. 102 00:05:14,760 --> 00:05:18,450 We call it-- how do we call the unit of this quantity? 103 00:05:18,450 --> 00:05:20,835 Diopters, right? 104 00:05:20,835 --> 00:05:24,020 AUDIENCE: So professor, I have a quick question. 105 00:05:24,020 --> 00:05:27,850 Here we have quantity over radius r. 106 00:05:27,850 --> 00:05:31,540 Does this describe a hyperboloid or elliptical shaped 107 00:05:31,540 --> 00:05:34,203 or a spherical lens? 108 00:05:34,203 --> 00:05:35,120 And if so, [INAUDIBLE] 109 00:05:35,120 --> 00:05:36,537 PROFESSOR: That's a good question. 110 00:05:36,537 --> 00:05:39,420 AUDIENCE: For example, with the spherical or an elliptical lens 111 00:05:39,420 --> 00:05:40,970 that has two different radiuses. 112 00:05:40,970 --> 00:05:42,137 How do you account for that? 113 00:05:46,800 --> 00:05:50,100 PROFESSOR: So this derivation, we actually did for a sphere. 114 00:05:50,100 --> 00:05:55,650 So the radius here is the radius of the spherical interface. 115 00:05:55,650 --> 00:06:01,530 And a very similar relationship would 116 00:06:01,530 --> 00:06:04,150 hold if instead of a sphere you had 117 00:06:04,150 --> 00:06:10,530 another ovoid, another surface of revolution which 118 00:06:10,530 --> 00:06:14,220 would be symmetric around the axis over here. 119 00:06:14,220 --> 00:06:17,280 So for example, if I put a parabola. 120 00:06:17,280 --> 00:06:21,540 Imagine a parabolic bowl where this is the cross section 121 00:06:21,540 --> 00:06:23,670 of the parabola. 122 00:06:23,670 --> 00:06:28,330 But then if you spin it around this axis, it remains the same. 123 00:06:28,330 --> 00:06:31,810 This is called a paraboloid of a revolution. 124 00:06:31,810 --> 00:06:33,810 This is actually-- I believe this was a homework 125 00:06:33,810 --> 00:06:36,600 or actually we'll do it a little bit later today. 126 00:06:36,600 --> 00:06:38,790 We'll discover that a very similar-- 127 00:06:38,790 --> 00:06:41,800 actually in the same equation holds, except in the case 128 00:06:41,800 --> 00:06:43,300 r is not that radius of the parabola 129 00:06:43,300 --> 00:06:45,390 if the parabola does not have a radius, 130 00:06:45,390 --> 00:06:49,680 but it does have a curvature near the axis. 131 00:06:49,680 --> 00:06:52,200 So the curvature of the parabola, actually 132 00:06:52,200 --> 00:06:55,050 the inverse of the curvature is-- the radius of curvature 133 00:06:55,050 --> 00:06:57,720 is what enters this equation. 134 00:06:57,720 --> 00:07:00,030 Now we discussed. 135 00:07:00,030 --> 00:07:00,950 an ellipsoid. 136 00:07:00,950 --> 00:07:04,230 And you are right that the ellipsoid has two axes. 137 00:07:04,230 --> 00:07:06,100 But the ellipsoid that we saw, the two axes 138 00:07:06,100 --> 00:07:08,440 are oriented a different way. 139 00:07:08,440 --> 00:07:11,950 So in the ellipsoid that we saw, when 140 00:07:11,950 --> 00:07:14,830 the actual propagation is z, then we have 141 00:07:14,830 --> 00:07:17,640 one of the lateral axis is x. 142 00:07:17,640 --> 00:07:20,440 And we show that the perfect focusing element, 143 00:07:20,440 --> 00:07:23,950 the perfect focusing refractive element 144 00:07:23,950 --> 00:07:27,170 is an ellipsoid looks like this. 145 00:07:27,170 --> 00:07:29,090 So of course, this has a major and minor axis. 146 00:07:29,090 --> 00:07:33,320 But you can see that the ellipsoid is still 147 00:07:33,320 --> 00:07:36,300 symmetric with respect to the optical axis. 148 00:07:36,300 --> 00:07:39,680 So in other words, it is still an ellipsoid of revolution 149 00:07:39,680 --> 00:07:43,320 around the z-axis. 150 00:07:43,320 --> 00:07:46,650 So I don't have to worry about 2 radii of curvature. 151 00:07:46,650 --> 00:07:48,900 But it is still a valid question, what you are asking. 152 00:07:48,900 --> 00:07:52,010 What if I did have-- what if I do put the surface there 153 00:07:52,010 --> 00:07:54,140 with [INAUDIBLE] of curvature? 154 00:07:54,140 --> 00:07:56,370 Well, it turns out, you will seldom 155 00:07:56,370 --> 00:07:59,000 see this in books except basic-- 156 00:07:59,000 --> 00:08:00,770 I don't think I've ever seen it in a book. 157 00:08:00,770 --> 00:08:06,180 But we can reconstruct what you should do. 158 00:08:06,180 --> 00:08:08,700 This is slightly incomplete because I'm only 159 00:08:08,700 --> 00:08:11,460 dealing with elevation of the ray and the angle 160 00:08:11,460 --> 00:08:13,490 in a cross section of the plane. 161 00:08:13,490 --> 00:08:18,780 And that is fine here because it is a surface of revolution. 162 00:08:18,780 --> 00:08:20,670 But if it is not a surface of revolution, 163 00:08:20,670 --> 00:08:23,340 then I have to grow this to a 4 vector 164 00:08:23,340 --> 00:08:29,130 with two angles and two elevations. 165 00:08:29,130 --> 00:08:32,299 And then this would become a 4 by 4 matrix. 166 00:08:32,299 --> 00:08:36,610 And then it would have a different power in each axis, 167 00:08:36,610 --> 00:08:38,909 depending on the Gaussian curvature 168 00:08:38,909 --> 00:08:41,659 of the surface in the corresponding location where 169 00:08:41,659 --> 00:08:43,049 the ray hits the axis. 170 00:08:43,049 --> 00:08:45,840 So you can imagine it will become a very complicated kind 171 00:08:45,840 --> 00:08:47,020 of situation. 172 00:08:47,020 --> 00:08:51,180 So typically we don't deal with this. 173 00:08:51,180 --> 00:08:56,456 I don't even know if optical elements are designed this way. 174 00:08:56,456 --> 00:09:12,892 AUDIENCE: [INAUDIBLE] 175 00:09:12,892 --> 00:09:16,080 CREW: Can you press the button, please? 176 00:09:16,080 --> 00:09:16,580 Oh. 177 00:09:16,580 --> 00:09:18,030 You cannot hear? 178 00:09:18,030 --> 00:09:19,130 AUDIENCE: Sorry. 179 00:09:19,130 --> 00:09:22,800 I pressed it but it didn't seem to do anything. 180 00:09:22,800 --> 00:09:23,760 You can hear now? 181 00:09:23,760 --> 00:09:24,260 CREW: Yes. 182 00:09:24,260 --> 00:09:24,760 Yes. 183 00:09:24,760 --> 00:09:28,610 AUDIENCE: I was just saying that I haven't come across this 184 00:09:28,610 --> 00:09:31,010 for geometrical objects either. 185 00:09:31,010 --> 00:09:35,060 But I have seen it for diffraction optics, 186 00:09:35,060 --> 00:09:37,143 in terms of the Wigner distribution function. 187 00:09:37,143 --> 00:09:38,810 And what I was going to carry on to say, 188 00:09:38,810 --> 00:09:41,600 though, was for the geometric optics case, 189 00:09:41,600 --> 00:09:43,940 it might be quite interesting because it 190 00:09:43,940 --> 00:09:48,620 would allow you to look at the behavior of skew rays. 191 00:09:48,620 --> 00:09:51,690 So it might actually give some quite interesting information, 192 00:09:51,690 --> 00:09:52,190 I think. 193 00:09:52,190 --> 00:09:52,640 PROFESSOR: Yeah. 194 00:09:52,640 --> 00:09:53,320 That's true, yeah. 195 00:09:53,320 --> 00:09:55,612 And you could also look at things like caustics, right? 196 00:09:55,612 --> 00:09:59,810 You know, if you make the surface non-rotationally 197 00:09:59,810 --> 00:10:03,170 symmetric, you get new different caustics. 198 00:10:03,170 --> 00:10:03,670 Forgive us. 199 00:10:03,670 --> 00:10:06,950 We got into a slightly esoteric conversation here. 200 00:10:06,950 --> 00:10:08,770 But that's a really good question. 201 00:10:08,770 --> 00:10:11,330 So we may end up writing a paper about this in a few months 202 00:10:11,330 --> 00:10:14,710 or something. 203 00:10:14,710 --> 00:10:15,210 Great. 204 00:10:15,210 --> 00:10:17,001 Any other questions? 205 00:10:24,540 --> 00:10:29,375 AUDIENCE: Isn't this problem separable in x and y? 206 00:10:29,375 --> 00:10:31,350 PROFESSOR: Actually, before I answer that. 207 00:10:31,350 --> 00:10:33,390 Let me bring up a trivial case where 208 00:10:33,390 --> 00:10:34,980 you might have this situation. 209 00:10:34,980 --> 00:10:37,830 That's a cylindrical lens. 210 00:10:37,830 --> 00:10:41,250 The analysis with it is actually valid for two cases. 211 00:10:41,250 --> 00:10:44,340 One is when you have a surface of revolution, like a sphere. 212 00:10:44,340 --> 00:10:45,840 The other is if you have a cylinder, 213 00:10:45,840 --> 00:10:48,060 in which case, everything is invariant 214 00:10:48,060 --> 00:10:50,110 in the out-of-place direction. 215 00:10:50,110 --> 00:10:51,940 So that's just a trivial case where 216 00:10:51,940 --> 00:10:55,170 you have a radius of curvature in one plane. 217 00:10:55,170 --> 00:10:56,440 And the curvature is-- 218 00:10:56,440 --> 00:10:58,240 the register is infinite in other plane. 219 00:10:58,240 --> 00:11:00,520 That is, we a plane, a planar surface, right? 220 00:11:00,520 --> 00:11:02,250 That's also called a cylindrical lens. 221 00:11:02,250 --> 00:11:05,040 And of course the cylindrical lens does not focus to a point. 222 00:11:05,040 --> 00:11:08,658 It focuses on a line. 223 00:11:08,658 --> 00:11:09,700 There's no focus in here. 224 00:11:09,700 --> 00:11:12,670 But you can imagine if you focus on the line out. 225 00:11:12,670 --> 00:11:13,170 I'm sorry. 226 00:11:13,170 --> 00:11:14,360 So what was the question? 227 00:11:14,360 --> 00:11:16,660 AUDIENCE: Yeah, the question was, 228 00:11:16,660 --> 00:11:18,960 isn't the problem separable in x and y. 229 00:11:18,960 --> 00:11:25,230 So we could just extend the matrix in y direction, 230 00:11:25,230 --> 00:11:25,945 and [INAUDIBLE] 231 00:11:25,945 --> 00:11:26,940 PROFESSOR: Yes. 232 00:11:26,940 --> 00:11:27,870 I think that's true. 233 00:11:27,870 --> 00:11:32,360 The paraxial approximation, it is separable, yeah. 234 00:11:32,360 --> 00:11:34,700 But of course, if you go off the paraxial approximation, 235 00:11:34,700 --> 00:11:37,740 things like [INAUDIBLE] and so on, then all bets are off. 236 00:11:37,740 --> 00:11:40,040 They become highly coupled. 237 00:11:40,040 --> 00:11:45,030 So for example, astigmatism is highly coupled. 238 00:11:45,030 --> 00:11:48,440 You cannot [INAUDIBLE] with-- 239 00:11:48,440 --> 00:11:53,000 astigmatism for those of you who are not unfortunate enough 240 00:11:53,000 --> 00:11:56,240 to have it in your own eyeglass prescription, 241 00:11:56,240 --> 00:12:00,222 astigmatism is a situation where an optical system focuses-- 242 00:12:00,222 --> 00:12:04,850 it's kind of difficult. I'll go into it in next week. 243 00:12:04,850 --> 00:12:07,880 It's a form of aberration where an optical system has 244 00:12:07,880 --> 00:12:11,960 different focusing properties for tangential rays as opposed 245 00:12:11,960 --> 00:12:13,677 to the sagittal rays. 246 00:12:13,677 --> 00:12:15,260 I don't want to confuse you right now. 247 00:12:15,260 --> 00:12:16,880 But it is a form of operation that 248 00:12:16,880 --> 00:12:20,090 has to do with skew rays that arrive in sort 249 00:12:20,090 --> 00:12:25,120 of off-axis and off-plane. 250 00:12:25,120 --> 00:12:28,810 And it is actually very common in people. 251 00:12:28,810 --> 00:12:30,790 Very often those of us who wear-- 252 00:12:30,790 --> 00:12:33,005 does anybody know you if you have astigmatism 253 00:12:33,005 --> 00:12:33,880 in your prescription? 254 00:12:33,880 --> 00:12:34,380 Yeah. 255 00:12:37,048 --> 00:12:37,590 That's right. 256 00:12:37,590 --> 00:12:38,090 Yes. 257 00:12:38,090 --> 00:12:41,309 So it means that his glasses are not exactly spherical. 258 00:12:47,060 --> 00:12:47,990 Other questions? 259 00:13:01,760 --> 00:13:03,560 The next step-- so this we did, I 260 00:13:03,560 --> 00:13:06,980 believe it was two weeks ago or so. 261 00:13:06,980 --> 00:13:09,440 Then the next thing that we did is 262 00:13:09,440 --> 00:13:12,320 we took this formulation basically-- oh, 263 00:13:12,320 --> 00:13:14,540 and before I move on let me just remind you 264 00:13:14,540 --> 00:13:16,730 very briefly why we did this. 265 00:13:16,730 --> 00:13:18,530 This looks kind of silly. 266 00:13:18,530 --> 00:13:22,910 It tells you that if you have a ray propagating in free space, 267 00:13:22,910 --> 00:13:26,240 again you can relate the elevation and angle 268 00:13:26,240 --> 00:13:31,180 to the left of some chunk of free space 269 00:13:31,180 --> 00:13:34,060 to the ray elevation and angle to the right 270 00:13:34,060 --> 00:13:35,847 of that chunk of free space. 271 00:13:35,847 --> 00:13:37,180 That's, again, that looks silly. 272 00:13:37,180 --> 00:13:39,220 Because we know the ray propagates 273 00:13:39,220 --> 00:13:43,120 in a straight line in free space or in uniform space because 274 00:13:43,120 --> 00:13:45,033 of Fermat principle. 275 00:13:45,033 --> 00:13:46,450 And of course, what will happen is 276 00:13:46,450 --> 00:13:51,920 that the angle of propagation will remain the same. 277 00:13:51,920 --> 00:13:53,890 That's what this law says. 278 00:13:53,890 --> 00:13:57,940 And also the elevation will increase 279 00:13:57,940 --> 00:14:01,270 by the distance times the tangent 280 00:14:01,270 --> 00:14:04,030 of the angle of propagation. 281 00:14:04,030 --> 00:14:06,450 But of course, with the paraxial approximations, 282 00:14:06,450 --> 00:14:09,533 instead of the tangent we put the angle itself. 283 00:14:09,533 --> 00:14:11,200 So this is what this equation says here. 284 00:14:11,200 --> 00:14:15,280 If you multiply this by this, the indices of refraction 285 00:14:15,280 --> 00:14:16,430 will drop out. 286 00:14:16,430 --> 00:14:20,680 And you're left with d, the distance times alpha. 287 00:14:20,680 --> 00:14:24,880 So it is really the tangent. 288 00:14:24,880 --> 00:14:25,750 So this seems silly. 289 00:14:25,750 --> 00:14:28,030 But the reason we did it is so that when 290 00:14:28,030 --> 00:14:32,650 we have optical systems with a cascade of uniform space 291 00:14:32,650 --> 00:14:36,010 then add the refractive interface and then more uniform 292 00:14:36,010 --> 00:14:38,620 space and so on, then you can basically 293 00:14:38,620 --> 00:14:42,260 ray trace those systems simply by cascading these matrices. 294 00:14:42,260 --> 00:14:44,950 So this is why we did this particular pattern. 295 00:14:44,950 --> 00:14:46,990 But this really doesn't say anything other 296 00:14:46,990 --> 00:14:50,560 than rays must propagate in a straight line 297 00:14:50,560 --> 00:14:54,100 in uniform refractive index because that is the shortest 298 00:14:54,100 --> 00:14:58,200 path for them to follow. 299 00:14:58,200 --> 00:15:01,740 And then what we did, after we developed this formulation, 300 00:15:01,740 --> 00:15:05,670 is we actually put into practice in the special case of two 301 00:15:05,670 --> 00:15:10,650 adjacent spherical surfaces. 302 00:15:10,650 --> 00:15:14,970 So typically, you have air outside. 303 00:15:14,970 --> 00:15:17,400 And between the two surfaces, you have glass. 304 00:15:17,400 --> 00:15:19,170 And this kind of element is what we call 305 00:15:19,170 --> 00:15:21,060 in everyday language a lens. 306 00:15:21,060 --> 00:15:22,290 It does not have to be glass. 307 00:15:22,290 --> 00:15:24,630 For example, in the case of our eyes, it's not glass. 308 00:15:24,630 --> 00:15:30,210 It is actually tissue, mostly water in fact. 309 00:15:30,210 --> 00:15:33,070 But anyway, it is higher index of refraction 310 00:15:33,070 --> 00:15:34,500 than the surrounding air. 311 00:15:34,500 --> 00:15:35,785 So it is acting-- 312 00:15:35,785 --> 00:15:39,650 it is doing the same job as a lens. 313 00:15:39,650 --> 00:15:42,290 And what we did then, if you recall, I will not do all of it 314 00:15:42,290 --> 00:15:45,860 again is but what we did is we did a cascade of the matrices 315 00:15:45,860 --> 00:15:47,510 for the two interfaces. 316 00:15:47,510 --> 00:15:51,170 And based on this, we derived the properties of this element. 317 00:15:51,170 --> 00:15:53,450 And what we discovered is the basic properties 318 00:15:53,450 --> 00:15:56,450 that if you have had a ray bundle that is arriving 319 00:15:56,450 --> 00:15:59,630 from infinity, therefore it is razor parallel, 320 00:15:59,630 --> 00:16:01,670 this type of optical element will actually 321 00:16:01,670 --> 00:16:03,050 bring them to a focus. 322 00:16:03,050 --> 00:16:06,710 So the rays after, they will propagate a certain distance 323 00:16:06,710 --> 00:16:08,160 after the element. 324 00:16:08,160 --> 00:16:10,350 And then they will focus to a point. 325 00:16:10,350 --> 00:16:12,440 And we also derive the other useful quantities, 326 00:16:12,440 --> 00:16:14,970 we derive the elevation of this point. 327 00:16:14,970 --> 00:16:16,910 We discover that it is proportional 328 00:16:16,910 --> 00:16:21,110 to this focal distance times the annual propagation. 329 00:16:21,110 --> 00:16:23,210 And then we saw that there's different cases 330 00:16:23,210 --> 00:16:26,480 of focusing depending on the nature of the lens. 331 00:16:26,480 --> 00:16:30,330 Namely its optical power. 332 00:16:30,330 --> 00:16:34,610 So positive lens will focus the rays at a finite distance. 333 00:16:34,610 --> 00:16:37,410 A negative lens will actually cause the rays to diverge. 334 00:16:37,410 --> 00:16:40,170 So they, strictly speaking, do not focus. 335 00:16:40,170 --> 00:16:42,140 But if you extend them backwards, 336 00:16:42,140 --> 00:16:44,690 then that create what is called a virtual image. 337 00:16:44,690 --> 00:16:48,830 And we will see later today what exactly this virtual image 338 00:16:48,830 --> 00:16:52,190 means and why it is an extremely useful concept 339 00:16:52,190 --> 00:16:53,600 in optical design. 340 00:16:53,600 --> 00:16:55,220 For now, it appears as an oddity. 341 00:16:55,220 --> 00:16:56,210 What exactly is that? 342 00:16:56,210 --> 00:16:59,560 Why did it move-- why did it trace the rays backwards. 343 00:16:59,560 --> 00:17:04,963 But today, actually, you will see why this is useful. 344 00:17:04,963 --> 00:17:06,380 The main point to take out of here 345 00:17:06,380 --> 00:17:09,619 is that the lens is acting kind of like a lever. 346 00:17:09,619 --> 00:17:11,906 So if you have-- 347 00:17:11,906 --> 00:17:16,910 if you imagine that you are rotating the angle 348 00:17:16,910 --> 00:17:19,910 that the rays are arriving to the left of the lens, 349 00:17:19,910 --> 00:17:23,480 then this point is moving linearly. 350 00:17:23,480 --> 00:17:26,150 It is staying on this plane, ideally at least, 351 00:17:26,150 --> 00:17:27,920 within the paraxial approximation. 352 00:17:27,920 --> 00:17:31,880 It is staying on this plane but it is moving. 353 00:17:31,880 --> 00:17:36,970 So as you rotate the angle, the elevation changes. 354 00:17:36,970 --> 00:17:39,760 And the constant of proportionality in this change 355 00:17:39,760 --> 00:17:41,570 is the focal length of the lens. 356 00:17:41,570 --> 00:17:45,190 This is a very key point to remember about lenses 357 00:17:45,190 --> 00:17:48,160 and how they function. 358 00:17:48,160 --> 00:17:51,400 And the other thing that we said is that it can also 359 00:17:51,400 --> 00:17:53,140 work the other way around. 360 00:17:53,140 --> 00:17:57,000 If you take this picture, the rule 361 00:17:57,000 --> 00:18:00,720 is that the rays always go from the left to the right. 362 00:18:00,720 --> 00:18:02,285 It's not a rule, it is a convention. 363 00:18:02,285 --> 00:18:04,410 There's no physical law, obviously, that says that. 364 00:18:04,410 --> 00:18:06,720 But it's a convention. 365 00:18:06,720 --> 00:18:08,580 But nevertheless, there is a physical law 366 00:18:08,580 --> 00:18:15,030 that says that if a light propagates in one direction 367 00:18:15,030 --> 00:18:17,410 and then you flip the time backwards, 368 00:18:17,410 --> 00:18:20,130 that is you reverse the propagation of the light, 369 00:18:20,130 --> 00:18:23,060 then the physical law actually says that the propagation 370 00:18:23,060 --> 00:18:24,570 should remain invariant. 371 00:18:24,570 --> 00:18:28,930 That is if I knew that the light propagated this way focuses, 372 00:18:28,930 --> 00:18:31,800 if I turn time backwards like these old movies 373 00:18:31,800 --> 00:18:34,020 that you see where they play the movie backwards 374 00:18:34,020 --> 00:18:37,110 and you see it pass, you know, moving towards the back. 375 00:18:37,110 --> 00:18:40,710 If I did that, I would have the rays going like this. 376 00:18:40,710 --> 00:18:42,240 So this against our convention. 377 00:18:42,240 --> 00:18:44,560 So that's why I flipped it here. 378 00:18:44,560 --> 00:18:47,100 Now again, the rays are going from left to right 379 00:18:47,100 --> 00:18:49,950 but in this case they're going through the focus. 380 00:18:49,950 --> 00:18:54,360 What this picture is telling us is that if you have rays that 381 00:18:54,360 --> 00:18:58,980 originate at the point one focal distance behind the lens, 382 00:18:58,980 --> 00:19:01,510 then this will form an image at infinity. 383 00:19:01,510 --> 00:19:04,500 That is, the lens will collimate this ray bundle. 384 00:19:04,500 --> 00:19:07,405 And it will send the rays out parallel at a 90 385 00:19:07,405 --> 00:19:09,030 that obeys, actually if you look at it, 386 00:19:09,030 --> 00:19:12,240 is the same relationship that was obeyed before. 387 00:19:12,240 --> 00:19:14,230 With the next reminder sign. 388 00:19:14,230 --> 00:19:15,930 And why did the minus sign come about? 389 00:19:25,573 --> 00:19:27,740 Everything's symmetric here except for this annoying 390 00:19:27,740 --> 00:19:28,560 minus sign. 391 00:19:28,560 --> 00:19:29,600 Why has this come about? 392 00:19:45,344 --> 00:19:47,840 Button. 393 00:19:47,840 --> 00:19:50,220 Did you do a button? 394 00:19:50,220 --> 00:19:51,860 Go to some desk, so you have a button. 395 00:19:51,860 --> 00:19:54,240 [LAUGHTER] 396 00:19:54,240 --> 00:19:56,580 AUDIENCE: The object distance, is it? 397 00:19:59,412 --> 00:20:01,000 PROFESSOR: Maybe not the distance. 398 00:20:01,000 --> 00:20:03,430 We have distances and what else here? 399 00:20:03,430 --> 00:20:04,840 AUDIENCE: The angle is negative. 400 00:20:04,840 --> 00:20:06,490 PROFESSOR: The angle, that's right. 401 00:20:06,490 --> 00:20:08,050 The only reason this negative sign 402 00:20:08,050 --> 00:20:10,720 appears there is the sign convention. 403 00:20:10,720 --> 00:20:15,320 This angle here, we define it as positive. 404 00:20:15,320 --> 00:20:16,660 This angle here is the same. 405 00:20:16,660 --> 00:20:18,290 But that's not how we measure angles. 406 00:20:18,290 --> 00:20:21,170 We always measure angles this way. 407 00:20:21,170 --> 00:20:23,100 So therefore, this angle is negative. 408 00:20:23,100 --> 00:20:24,850 So to account for this negative angle, 409 00:20:24,850 --> 00:20:27,230 a negative sign appears over there. 410 00:20:27,230 --> 00:20:29,630 AUDIENCE: I have another quick question. 411 00:20:29,630 --> 00:20:32,510 Let's say we have a lens where our left and our right 412 00:20:32,510 --> 00:20:35,390 are the same, which I believe is pretty common in optics. 413 00:20:35,390 --> 00:20:37,130 So from the lens makers equation you 414 00:20:37,130 --> 00:20:40,053 would get that 1/f is equal to 0. 415 00:20:40,053 --> 00:20:40,720 PROFESSOR: Nope. 416 00:20:40,720 --> 00:20:41,960 AUDIENCE: Or is that wrong? 417 00:20:41,960 --> 00:20:44,630 Because you get 1 minus a quantity-- 418 00:20:44,630 --> 00:20:48,510 PROFESSOR: [INAUDIBLE] That's not true, actually. 419 00:20:48,510 --> 00:20:51,360 If the red is to the left is equal to the-- well, 420 00:20:51,360 --> 00:20:52,450 depends how you are. 421 00:20:52,450 --> 00:20:54,600 Is it like this or like this? 422 00:20:54,600 --> 00:20:56,310 AUDIENCE: It's like this, 423 00:20:56,310 --> 00:20:57,350 PROFESSOR: Like this? 424 00:20:57,350 --> 00:20:58,142 AUDIENCE: Yes, sir. 425 00:20:58,142 --> 00:20:58,770 PROFESSOR: OK. 426 00:20:58,770 --> 00:21:00,030 Let me give you a hint. 427 00:21:00,030 --> 00:21:07,340 If it was really like this, we have 1 and 1. 428 00:21:10,080 --> 00:21:12,600 Now it is the same radius of curvature, 429 00:21:12,600 --> 00:21:17,130 and that would indeed have no power, no optical power. 430 00:21:17,130 --> 00:21:19,410 Their focal length would be infinity 431 00:21:19,410 --> 00:21:22,830 or the optical power would be 0. 432 00:21:22,830 --> 00:21:26,332 In this case, are the radii of curvature the same or not? 433 00:21:26,332 --> 00:21:28,290 AUDIENCE: So one's negative and one's positive. 434 00:21:28,290 --> 00:21:29,000 Is that what you're saying? 435 00:21:29,000 --> 00:21:29,830 PROFESSOR: Exactly. 436 00:21:29,830 --> 00:21:30,430 They're not the same. 437 00:21:30,430 --> 00:21:31,258 They're opposite. 438 00:21:31,258 --> 00:21:31,800 AUDIENCE: OK. 439 00:21:31,800 --> 00:21:32,300 Thanks. 440 00:21:32,300 --> 00:21:35,372 PROFESSOR: So they may be the same in absolute value. 441 00:21:35,372 --> 00:21:36,580 But they have opposite signs. 442 00:21:36,580 --> 00:21:38,530 Therefore in this case, you actually 443 00:21:38,530 --> 00:21:40,000 get a finite focal length. 444 00:21:42,508 --> 00:21:44,550 I'm sorry, I don't think you could see what I do. 445 00:21:56,360 --> 00:21:57,260 Other questions? 446 00:22:01,033 --> 00:22:02,200 It was a very good question. 447 00:22:02,200 --> 00:22:03,990 I'm very glad you asked that. 448 00:22:03,990 --> 00:22:09,570 We have to be very careful when we apply these formulas to use 449 00:22:09,570 --> 00:22:12,010 the proper sign. 450 00:22:12,010 --> 00:22:17,500 So remember the curvatures by the sign convention, which 451 00:22:17,500 --> 00:22:19,180 I really recommend that you-- 452 00:22:19,180 --> 00:22:21,100 I don't have it available here. 453 00:22:21,100 --> 00:22:23,580 But I really recommend you go back and study it. 454 00:22:23,580 --> 00:22:28,990 The convention is that if the curvature points like this, 455 00:22:28,990 --> 00:22:31,750 it is convex towards the left, then 456 00:22:31,750 --> 00:22:33,640 it is called a positive curvature. 457 00:22:33,640 --> 00:22:36,100 So in this case, you would write, for example, 458 00:22:36,100 --> 00:22:42,390 you would write r equals maybe it is 10 centimeters. 459 00:22:42,390 --> 00:22:45,870 In this case, because the surface now is concave, 460 00:22:45,870 --> 00:22:49,780 then you would say that R equals minus 10 centimeters. 461 00:22:54,780 --> 00:22:57,900 So the lens make an equation in this case, 462 00:22:57,900 --> 00:23:02,180 we'll say-- let's say n equals 1.5 for example. 463 00:23:02,180 --> 00:23:08,650 Let's make it, I would say, 1/f equals 1.5 minus 1 times 464 00:23:08,650 --> 00:23:12,140 1 over 10 minus 1 over minus 10. 465 00:23:12,140 --> 00:23:13,015 And this is whatever. 466 00:23:20,390 --> 00:23:22,460 I believe it is 3 or something. 467 00:23:25,460 --> 00:23:26,030 Wait. 468 00:23:26,030 --> 00:23:32,510 This is 1/5, or there's 0.5 times 1/5. 469 00:23:32,510 --> 00:23:34,880 So it is 1 over 10 so it is 10. 470 00:23:44,180 --> 00:23:45,520 Other questions? 471 00:24:00,950 --> 00:24:04,040 So this was true for what we call the thin lens, where 472 00:24:04,040 --> 00:24:08,330 it is thin because we assume that the distance 473 00:24:08,330 --> 00:24:12,500 between the surfaces within the paraxial approximation 474 00:24:12,500 --> 00:24:14,970 can be neglected. 475 00:24:14,970 --> 00:24:20,070 If we cannot do that, if we cannot neglect the distance 476 00:24:20,070 --> 00:24:24,990 between the two surfaces, our formulation does not abandon 477 00:24:24,990 --> 00:24:28,050 us, we just get a more complicated formula. 478 00:24:28,050 --> 00:24:32,610 And the reason the formula becomes more complicated 479 00:24:32,610 --> 00:24:37,410 is because if we drop the approximation 480 00:24:37,410 --> 00:24:40,740 of negligible spacing then the ray 481 00:24:40,740 --> 00:24:43,480 refracts twice before exiting. 482 00:24:43,480 --> 00:24:45,620 It will diffract at the first interface 483 00:24:45,620 --> 00:24:48,480 and the second interface. 484 00:24:48,480 --> 00:24:49,800 We'll get the formula for this. 485 00:24:49,800 --> 00:24:52,950 The thick lens, I don't want to go back into the formula. 486 00:24:52,950 --> 00:24:54,900 It just comes out of the algebra. 487 00:24:54,900 --> 00:24:56,970 What I really want to emphasize in this case 488 00:24:56,970 --> 00:25:01,020 is that even though the refracts doubly, 489 00:25:01,020 --> 00:25:04,740 we can pretend that it only refracted once. 490 00:25:04,740 --> 00:25:09,240 And we can do that by extending the incoming ray, 491 00:25:09,240 --> 00:25:11,460 extending the outgoing ray. 492 00:25:11,460 --> 00:25:15,780 And then what the plane, where they intersect, 493 00:25:15,780 --> 00:25:18,210 that is called the principal plane. 494 00:25:18,210 --> 00:25:21,460 Actually that is called the second principal plane. 495 00:25:21,460 --> 00:25:25,100 If I do it for an incoming ray bundle that 496 00:25:25,100 --> 00:25:28,360 is imaged at infinity then it is the other way around. 497 00:25:28,360 --> 00:25:32,110 Again, it can extend the incoming ray, the outgoing ray, 498 00:25:32,110 --> 00:25:34,930 and again get the first principal plane. 499 00:25:34,930 --> 00:25:37,270 And the reason we did all this exercise were 500 00:25:37,270 --> 00:25:40,480 to discover principal planes and so on 501 00:25:40,480 --> 00:25:42,740 is because this allows us to do ray 502 00:25:42,740 --> 00:25:44,350 tracing as shown over there. 503 00:25:49,010 --> 00:25:50,460 Can you still hear me? 504 00:25:54,095 --> 00:25:54,595 Sorry. 505 00:25:54,595 --> 00:25:56,410 This microphone has a tendency to fall. 506 00:25:59,130 --> 00:25:59,630 OK. 507 00:25:59,630 --> 00:26:01,186 You can still hear me? 508 00:26:01,186 --> 00:26:02,640 AUDIENCE: Yes. 509 00:26:02,640 --> 00:26:03,890 PROFESSOR: Thanks. 510 00:26:03,890 --> 00:26:04,390 OK. 511 00:26:07,110 --> 00:26:09,090 So the reason we did this exercise 512 00:26:09,090 --> 00:26:11,290 of effective focal length and so on 513 00:26:11,290 --> 00:26:13,830 is because we can play the following take. 514 00:26:13,830 --> 00:26:16,545 I can give you an optical system with a bunch of lenses. 515 00:26:20,080 --> 00:26:22,290 I don't know, whatever. 516 00:26:22,290 --> 00:26:23,160 Very complicated. 517 00:26:23,160 --> 00:26:25,370 And I'm asking you to find the imaging condition. 518 00:26:25,370 --> 00:26:26,640 Here's an object. 519 00:26:26,640 --> 00:26:28,725 And I'm asking you where is the image. 520 00:26:33,160 --> 00:26:35,650 What is the magnification? 521 00:26:35,650 --> 00:26:39,660 Lateral, angular, and so on. 522 00:26:39,660 --> 00:26:41,550 This [INAUDIBLE] of principal planes, 523 00:26:41,550 --> 00:26:47,910 it applies as we discussed not only on a thick lens 524 00:26:47,910 --> 00:26:51,330 but in general in any compound optical system that 525 00:26:51,330 --> 00:26:54,210 contains multiple elements like this one. 526 00:26:54,210 --> 00:26:58,180 And in order to utilize it, what you do is you throw out, 527 00:26:58,180 --> 00:27:01,350 so to speak, all of this entire system and you 528 00:27:01,350 --> 00:27:04,230 replace it with its principal planes. 529 00:27:04,230 --> 00:27:07,670 So here is, let's say that the principal planes are like this. 530 00:27:07,670 --> 00:27:10,000 Here's the first principal plane, 531 00:27:10,000 --> 00:27:12,960 the second principal plane. 532 00:27:12,960 --> 00:27:19,490 And now I can do the imaging business very simply. 533 00:27:19,490 --> 00:27:22,670 Here's my object. 534 00:27:22,670 --> 00:27:28,690 If I take a ray that goes horizontal from the object, 535 00:27:28,690 --> 00:27:36,660 this ray, it is as if it came from infinity. 536 00:27:36,660 --> 00:27:41,400 So therefore this ray will go to the second principal plane, 537 00:27:41,400 --> 00:27:43,960 it will bend. 538 00:27:43,960 --> 00:27:47,440 And it will meet the axis one effective focal length 539 00:27:47,440 --> 00:27:51,160 to the right of the second principal plane. 540 00:27:55,390 --> 00:27:57,730 Then I can take another ray which goes 541 00:27:57,730 --> 00:28:00,190 through the first focal point. 542 00:28:00,190 --> 00:28:04,130 That goes one focal length to the-- 543 00:28:04,130 --> 00:28:05,917 I'm sorry, did I say to the left before? 544 00:28:05,917 --> 00:28:07,250 I should have said to the right. 545 00:28:07,250 --> 00:28:08,100 Let me repeat. 546 00:28:08,100 --> 00:28:09,540 So this ray will go-- 547 00:28:09,540 --> 00:28:12,410 will hit the optical access one focal distance 548 00:28:12,410 --> 00:28:15,410 to the right of the second principal plane. 549 00:28:15,410 --> 00:28:18,260 I think I said the wrong thing before. 550 00:28:18,260 --> 00:28:21,650 Anyway, to the right. 551 00:28:21,650 --> 00:28:25,970 Now let me take a ray that goes through the focal point one 552 00:28:25,970 --> 00:28:31,182 focal distance to the left of the first principal plane. 553 00:28:31,182 --> 00:28:32,390 I will run out of space here. 554 00:28:32,390 --> 00:28:33,820 But that's OK. 555 00:28:33,820 --> 00:28:36,950 This ray will have to hit the first principal plane 556 00:28:36,950 --> 00:28:38,120 and then turn horizontal. 557 00:28:38,120 --> 00:28:43,460 Because this is equivalent to a point source one focal distance 558 00:28:43,460 --> 00:28:46,620 to the left of the lens that would be [INAUDIBLE] infinity. 559 00:28:46,620 --> 00:28:49,620 So therefore, this ray must come out horizontal. 560 00:28:49,620 --> 00:28:51,980 And where these rays meet is actually 561 00:28:51,980 --> 00:28:53,400 where my image is located. 562 00:28:58,100 --> 00:29:01,850 This construction is very useful and very basic and fundamental. 563 00:29:01,850 --> 00:29:08,920 And that's why the principal planes are important. 564 00:29:08,920 --> 00:29:10,660 Now let's try this. 565 00:29:10,660 --> 00:29:11,740 Any questions about this? 566 00:29:15,870 --> 00:29:18,660 AUDIENCE: Since we have principal planes 567 00:29:18,660 --> 00:29:25,590 for the system, are we going to substitute the system 568 00:29:25,590 --> 00:29:29,580 by a matrix which is for single lens? 569 00:29:29,580 --> 00:29:33,270 Like 1, minus 1 over f, 0. 570 00:29:33,270 --> 00:29:34,950 But that's not going to be possible 571 00:29:34,950 --> 00:29:37,960 because the single lens doesn't change the angle 572 00:29:37,960 --> 00:29:41,473 but the system can change the position and angle both. 573 00:29:41,473 --> 00:29:42,140 PROFESSOR: Yeah. 574 00:29:42,140 --> 00:29:44,280 The system is not equivalent to a thin lens. 575 00:29:44,280 --> 00:29:45,280 It's a little bit weird. 576 00:29:45,280 --> 00:29:46,950 This equivalent, you can think of it 577 00:29:46,950 --> 00:29:49,200 as having two thick lenses. 578 00:29:49,200 --> 00:29:52,380 One of them is operating for this ray. 579 00:29:52,380 --> 00:29:55,290 This is like a thin lens but for this ray. 580 00:29:55,290 --> 00:29:57,600 This is like a thin lens but for this ray. 581 00:30:00,550 --> 00:30:02,550 That's a bit confusing to think of it like that. 582 00:30:02,550 --> 00:30:03,050 Right? 583 00:30:03,050 --> 00:30:07,527 So I so would rather not go there. 584 00:30:07,527 --> 00:30:09,610 But yeah, if you look at the matrix of the system, 585 00:30:09,610 --> 00:30:11,080 it is not a matrix of a thin lens. 586 00:30:11,080 --> 00:30:11,570 Of course not. 587 00:30:11,570 --> 00:30:12,070 Yeah. 588 00:30:12,070 --> 00:30:13,525 Yeah? 589 00:30:13,525 --> 00:30:20,650 AUDIENCE: [INAUDIBLE] 590 00:30:20,650 --> 00:30:21,390 PROFESSOR: OK. 591 00:30:21,390 --> 00:30:22,670 That was actually my question. 592 00:30:22,670 --> 00:30:25,912 So he's asking, what if the focal length is negative. 593 00:30:25,912 --> 00:30:26,870 How would you the ray-- 594 00:30:26,870 --> 00:30:28,700 this is what you're asking, right? 595 00:30:28,700 --> 00:30:29,390 Yeah. 596 00:30:29,390 --> 00:30:31,280 So that was actually my-- the next thing 597 00:30:31,280 --> 00:30:32,660 I was going to do also here. 598 00:30:32,660 --> 00:30:34,620 So let's take this out of the way. 599 00:30:34,620 --> 00:30:37,760 Take this out all the way too because the marker leaked. 600 00:30:37,760 --> 00:30:38,855 The marker leaked here. 601 00:30:38,855 --> 00:30:39,832 Oh, OK. 602 00:30:39,832 --> 00:30:41,040 We're wasting too much paper. 603 00:30:43,790 --> 00:30:45,140 So let's do the same game now. 604 00:30:45,140 --> 00:30:47,950 But now let's say that the focal length is negative. 605 00:30:51,030 --> 00:30:59,300 So here is an object. 606 00:31:04,320 --> 00:31:06,260 And here are the two principal planes. 607 00:31:06,260 --> 00:31:09,180 Let's say this is the first principal plane, 608 00:31:09,180 --> 00:31:11,668 the second principal plane. 609 00:31:11,668 --> 00:31:13,210 How would the ray trace in this case? 610 00:31:13,210 --> 00:31:15,000 Would I have a negative focal length? 611 00:31:20,220 --> 00:31:22,030 Does someone want to help me trace it? 612 00:31:27,290 --> 00:31:28,460 Button. 613 00:31:28,460 --> 00:31:31,380 AUDIENCE: You first propagate to second principal plane 614 00:31:31,380 --> 00:31:31,880 and then-- 615 00:31:31,880 --> 00:31:33,713 PROFESSOR: So [INAUDIBLE] the horizontal ray 616 00:31:33,713 --> 00:31:37,380 to the second principal plane. 617 00:31:37,380 --> 00:31:37,980 Now what? 618 00:31:37,980 --> 00:31:39,228 AUDIENCE: Come back. 619 00:31:39,228 --> 00:31:40,116 PROFESSOR: Come back? 620 00:31:40,116 --> 00:31:45,330 AUDIENCE: In the sense that the ray diverges 621 00:31:45,330 --> 00:31:47,166 from the first plane. 622 00:31:47,166 --> 00:31:50,293 PROFESSOR: But we'll have to go-- 623 00:31:50,293 --> 00:31:51,710 well, why drawing's not very good. 624 00:31:51,710 --> 00:31:55,785 Let's say that this is the focal point. 625 00:31:55,785 --> 00:31:57,410 So [INAUDIBLE] diverge in this fashion. 626 00:31:57,410 --> 00:31:57,800 Right? 627 00:31:57,800 --> 00:31:58,150 AUDIENCE: Yeah. 628 00:31:58,150 --> 00:31:58,690 Yeah. 629 00:31:58,690 --> 00:32:00,190 PROFESSOR: What about the other one? 630 00:32:17,040 --> 00:32:20,937 AUDIENCE: [INAUDIBLE] 631 00:32:20,937 --> 00:32:23,270 PROFESSOR: First of all, where is the first focal point? 632 00:32:23,270 --> 00:32:25,377 AUDIENCE: On the right panel [INAUDIBLE] 633 00:32:25,377 --> 00:32:26,710 PROFESSOR: They cannot hear you. 634 00:32:32,662 --> 00:32:34,750 AUDIENCE: Probably should take rays 635 00:32:34,750 --> 00:32:37,975 from the object to the first focal point, 636 00:32:37,975 --> 00:32:39,100 there's a back focal point. 637 00:32:39,100 --> 00:32:39,970 PROFESSOR: So where is that? 638 00:32:39,970 --> 00:32:41,387 AUDIENCE: That's on the right side 639 00:32:41,387 --> 00:32:43,240 of the first principal plane. 640 00:32:43,240 --> 00:32:45,616 PROFESSOR: Should be somewhere around here, right? 641 00:32:45,616 --> 00:32:47,995 AUDIENCE: And at the point where they intersect, 642 00:32:47,995 --> 00:32:48,870 you make it parallel. 643 00:32:48,870 --> 00:32:52,660 PROFESSOR: So this is minus f. 644 00:32:52,660 --> 00:32:54,000 This is minus f. 645 00:32:54,000 --> 00:32:54,500 Right? 646 00:32:54,500 --> 00:32:55,125 AUDIENCE: Yeah. 647 00:32:55,125 --> 00:32:56,720 PROFESSOR: With the distances. 648 00:32:56,720 --> 00:32:57,220 OK. 649 00:32:57,220 --> 00:32:58,095 So then what do I do? 650 00:33:01,970 --> 00:33:03,900 AUDIENCE: The point where the ray 651 00:33:03,900 --> 00:33:06,270 meets the first principal plane we make it 652 00:33:06,270 --> 00:33:09,710 parallel to the optical axis? 653 00:33:09,710 --> 00:33:10,710 PROFESSOR: That's right. 654 00:33:10,710 --> 00:33:12,430 So what I have to do is first of all 655 00:33:12,430 --> 00:33:15,510 I draw a dashed line that goes to this point. 656 00:33:21,850 --> 00:33:24,640 This tells me which way this would be pointing. 657 00:33:24,640 --> 00:33:26,320 This will have to go all the way here. 658 00:33:33,070 --> 00:33:33,990 Yeah, that's right. 659 00:33:33,990 --> 00:33:40,688 And then-- So where is the image? 660 00:33:52,978 --> 00:33:53,520 That's right. 661 00:33:53,520 --> 00:33:55,610 The image is actually where these rays 662 00:33:55,610 --> 00:33:56,510 are supposed to meet. 663 00:33:56,510 --> 00:33:57,170 Right? 664 00:33:57,170 --> 00:33:58,430 They never quite meet. 665 00:33:58,430 --> 00:34:00,590 But I have to extend one of them backwards so 666 00:34:00,590 --> 00:34:01,870 they would actually meet here. 667 00:34:05,220 --> 00:34:07,000 And notice the significant difference 668 00:34:07,000 --> 00:34:09,380 that here the image erect. 669 00:34:09,380 --> 00:34:15,540 It is upside, which is customary for virtual images. 670 00:34:15,540 --> 00:34:18,780 Whereas in this case, the image if you recall the image quite 671 00:34:18,780 --> 00:34:22,025 clearly, it was inverted. 672 00:34:26,469 --> 00:34:28,989 So it is really the same approach. 673 00:34:28,989 --> 00:34:30,730 I haven't changed anything. 674 00:34:30,730 --> 00:34:36,429 But the only thing I have to do, really, 675 00:34:36,429 --> 00:34:38,520 is be careful with the signs. 676 00:34:38,520 --> 00:34:41,105 Who is to the left of whom. 677 00:34:41,105 --> 00:34:44,639 And the way we keep track of that and we don't lose our head 678 00:34:44,639 --> 00:34:50,260 is by faithfully following the sign conventions. 679 00:34:50,260 --> 00:34:53,520 So the definition is that the positive focal length 680 00:34:53,520 --> 00:34:56,980 would be to the right of the second principal plane. 681 00:34:56,980 --> 00:34:57,480 OK. 682 00:34:57,480 --> 00:34:58,110 Fine. 683 00:34:58,110 --> 00:35:00,240 If I'm given a negative focal length, 684 00:35:00,240 --> 00:35:04,110 that means that I should go to the left 685 00:35:04,110 --> 00:35:05,550 of the second principal plane. 686 00:35:08,230 --> 00:35:09,480 And so on and so forth. 687 00:35:09,480 --> 00:35:11,103 That's the principle. 688 00:35:15,745 --> 00:35:16,620 What about this case? 689 00:35:16,620 --> 00:35:17,287 Can I have this? 690 00:35:31,750 --> 00:35:34,110 Of course I can. 691 00:35:34,110 --> 00:35:36,240 Interactive homework, you'll see a system 692 00:35:36,240 --> 00:35:38,440 that has that property. 693 00:35:38,440 --> 00:35:41,130 The second principal plane is to the left 694 00:35:41,130 --> 00:35:42,600 of the first principal plane. 695 00:35:42,600 --> 00:35:45,800 There's no rule or law against that. 696 00:35:45,800 --> 00:35:47,600 What we do? 697 00:35:47,600 --> 00:35:48,600 Nothing changed, really. 698 00:35:48,600 --> 00:35:51,330 I can still do my ray tracing as before. 699 00:35:51,330 --> 00:35:53,480 If I were to do my rotation here, 700 00:35:53,480 --> 00:35:57,000 well, it would go up to the second principal plane. 701 00:35:57,000 --> 00:35:59,468 I would shift the ray down. 702 00:35:59,468 --> 00:36:01,010 Let's say this is one focal distance. 703 00:36:04,000 --> 00:36:06,520 And then now let me do the other one correctly. 704 00:36:06,520 --> 00:36:09,675 So this is the first principal plane so to go from here-- 705 00:36:12,950 --> 00:36:13,850 something like that. 706 00:36:13,850 --> 00:36:17,150 I mean, that produced a ridiculously-- 707 00:36:17,150 --> 00:36:17,970 Well, we're here. 708 00:36:20,796 --> 00:36:23,912 I did something wrong here so the image ended up very close 709 00:36:23,912 --> 00:36:24,870 to the principal plane. 710 00:36:24,870 --> 00:36:27,270 But this is how you would ray trace the system. 711 00:36:35,000 --> 00:36:37,330 Any questions about the stuff I'm doing here? 712 00:36:48,230 --> 00:36:49,930 So when you're given an optical system 713 00:36:49,930 --> 00:36:52,140 it's a little bit of a judgment call. 714 00:36:52,140 --> 00:36:56,580 You have-- what happened. 715 00:36:56,580 --> 00:36:57,570 Some message flashed. 716 00:36:57,570 --> 00:36:58,920 Something got disconnected. 717 00:36:58,920 --> 00:37:01,263 Are we still there? 718 00:37:01,263 --> 00:37:04,560 CREW: Yep. 719 00:37:04,560 --> 00:37:08,710 PROFESSOR: So when you have an optical system 720 00:37:08,710 --> 00:37:11,330 there is three ways you can analyze it. 721 00:37:11,330 --> 00:37:15,320 One is you can just blindly write down 722 00:37:15,320 --> 00:37:19,660 the cascade of matrices, figure out the incoming 723 00:37:19,660 --> 00:37:21,750 and an outgoing angles. 724 00:37:21,750 --> 00:37:23,730 And then try to solve it this way. 725 00:37:23,730 --> 00:37:26,790 The second way is you find the principal planes. 726 00:37:26,790 --> 00:37:28,560 And then you apply one of these techniques 727 00:37:28,560 --> 00:37:30,120 that I just did on paper. 728 00:37:30,120 --> 00:37:32,040 And this is another way to analyze it. 729 00:37:32,040 --> 00:37:35,250 The third way, which is what I think [INAUDIBLE] 730 00:37:35,250 --> 00:37:38,680 did in the class and was also the supplement of your nodes 731 00:37:38,680 --> 00:37:42,690 is where you take each length. 732 00:37:42,690 --> 00:37:46,622 You apply the imaging condition individually and then move on. 733 00:37:46,622 --> 00:37:47,580 I will not do it again. 734 00:37:47,580 --> 00:37:49,500 It's in the example of your notes. 735 00:37:49,500 --> 00:37:51,450 But it's a little bit of a judgment call. 736 00:37:51,450 --> 00:37:53,730 And sometimes it's a different matter of preference. 737 00:37:53,730 --> 00:37:56,340 Some people are very good with matrices. 738 00:37:56,340 --> 00:37:57,990 So it is just better in that case 739 00:37:57,990 --> 00:38:01,470 to just multiply the matrices and get done with it. 740 00:38:01,470 --> 00:38:05,070 The principal planes that give you more intuition. 741 00:38:05,070 --> 00:38:09,920 So I'm personally not in favor of just blindly multiplying 742 00:38:09,920 --> 00:38:10,480 matrices. 743 00:38:10,480 --> 00:38:12,120 That's a little bit mind numbing. 744 00:38:12,120 --> 00:38:16,080 And also very prone to algebraic errors because we're all human. 745 00:38:16,080 --> 00:38:18,110 But this method of principal planes, 746 00:38:18,110 --> 00:38:22,830 they're a little bit less prone to algebraic errors because 747 00:38:22,830 --> 00:38:25,690 with these methods, you can actually test your intuition. 748 00:38:25,690 --> 00:38:29,130 You can see easily, whether what you are doing is correct 749 00:38:29,130 --> 00:38:29,820 or not. 750 00:38:29,820 --> 00:38:32,610 Whereas you make an algebraic mistake doing the matrix 751 00:38:32,610 --> 00:38:35,340 multiplications, well, it's very difficult 752 00:38:35,340 --> 00:38:37,850 to know where it happened. 753 00:38:37,850 --> 00:38:40,860 And in fact, I confess when I wrote the supplement 754 00:38:40,860 --> 00:38:42,805 to the notes which is-- 755 00:38:42,805 --> 00:38:44,680 I don't know if you had the chance to see it, 756 00:38:44,680 --> 00:38:47,270 but it is the same optical system solved 757 00:38:47,270 --> 00:38:48,830 with the three different ways. 758 00:38:48,830 --> 00:38:51,450 You know, with first of all with the cascade of lenses 759 00:38:51,450 --> 00:38:54,150 then with matrices and then with principal planes. 760 00:38:54,150 --> 00:38:59,100 So I confess that I first did it with a cascade of lenses. 761 00:38:59,100 --> 00:39:00,330 So I took the first lens. 762 00:39:00,330 --> 00:39:01,170 I imaged it. 763 00:39:01,170 --> 00:39:04,320 Then I used the image of the first lens 764 00:39:04,320 --> 00:39:07,170 as objects to the second lens and I imaged again 765 00:39:07,170 --> 00:39:09,310 and I found the answer. 766 00:39:09,310 --> 00:39:13,670 Then I took a plan, I got a big cup of coffee, 767 00:39:13,670 --> 00:39:15,328 I did all the algebra of the matrices, 768 00:39:15,328 --> 00:39:17,120 I actually got the wrong result. It was not 769 00:39:17,120 --> 00:39:20,580 a green with the first result. So I went back and made it 770 00:39:20,580 --> 00:39:23,160 again and finally I got the algebra correct 771 00:39:23,160 --> 00:39:24,840 and I got the two results to agree. 772 00:39:24,840 --> 00:39:26,620 But for sure, the first thing I did it 773 00:39:26,620 --> 00:39:28,950 with the cascade of lenses was correct. 774 00:39:28,950 --> 00:39:31,020 Then every time I got a different result 775 00:39:31,020 --> 00:39:32,520 with the matrices, I knew that I was 776 00:39:32,520 --> 00:39:34,380 making an algebraic mistake. 777 00:39:34,380 --> 00:39:36,210 Until I finally got the matrices to agree 778 00:39:36,210 --> 00:39:39,300 with the actual result. And I got it right. 779 00:39:39,300 --> 00:39:40,800 So there reason I'm telling you this 780 00:39:40,800 --> 00:39:43,980 is because I've taught this class for 10 years, 781 00:39:43,980 --> 00:39:45,330 believe it or not. 782 00:39:45,330 --> 00:39:48,930 So every time when we do the quiz number one, 783 00:39:48,930 --> 00:39:51,120 your colleagues in the past they all 784 00:39:51,120 --> 00:39:53,730 try to solve the problem with matrices. 785 00:39:53,730 --> 00:39:55,830 Because it is easier. 786 00:39:55,830 --> 00:39:56,950 Matrices is a no-brainer. 787 00:39:56,950 --> 00:40:00,000 You just multiply and get the result. 788 00:40:00,000 --> 00:40:03,260 About 70%-- over all these years, 789 00:40:03,260 --> 00:40:04,590 I have accurate statistics. 790 00:40:04,590 --> 00:40:08,700 About 70% they get stuck into some algebraic error. 791 00:40:08,700 --> 00:40:12,120 And then they cannot solve the problem because the measure 792 00:40:12,120 --> 00:40:14,732 of this is if you get the ray tracing wrong, 793 00:40:14,732 --> 00:40:15,690 you cannot do anything. 794 00:40:15,690 --> 00:40:17,350 You can not do magnification. 795 00:40:17,350 --> 00:40:20,990 You can not do whatever. 796 00:40:20,990 --> 00:40:23,950 So I would only recommend the matrices 797 00:40:23,950 --> 00:40:26,830 if you are really, really sharp with algebra. 798 00:40:26,830 --> 00:40:28,540 Otherwise, you're better off-- 799 00:40:28,540 --> 00:40:30,040 and actually overall, I think you're 800 00:40:30,040 --> 00:40:32,380 better off if you land in the physics, which is not 801 00:40:32,380 --> 00:40:33,790 what the matrices give you. 802 00:40:33,790 --> 00:40:35,620 The matrices are convenient. 803 00:40:35,620 --> 00:40:38,680 For example, we derived all of these based on matrices. 804 00:40:38,680 --> 00:40:42,850 But after that we should let our intuition take over. 805 00:40:42,850 --> 00:40:45,005 And not rely just on the math, which is actually 806 00:40:45,005 --> 00:40:47,380 a pretty good principle in anything you do in engineering 807 00:40:47,380 --> 00:40:49,570 but it is especially true here. 808 00:40:53,880 --> 00:40:55,120 Any questions? 809 00:40:57,940 --> 00:41:00,030 AUDIENCE: I have a question. 810 00:41:00,030 --> 00:41:05,720 So for the focal length that is negative, 811 00:41:05,720 --> 00:41:10,190 if you retrace the rate coming from the object 812 00:41:10,190 --> 00:41:15,760 say to the center of the first principle plane. 813 00:41:15,760 --> 00:41:16,915 So this-- 814 00:41:16,915 --> 00:41:18,790 PROFESSOR: No, you don't trace through center 815 00:41:18,790 --> 00:41:21,207 with the principal plane, you trace it to the focal point. 816 00:41:21,207 --> 00:41:22,560 AUDIENCE: No, no, no. 817 00:41:22,560 --> 00:41:27,640 I mean, it means that, so in this case for the focal lens 818 00:41:27,640 --> 00:41:28,580 is a positive. 819 00:41:28,580 --> 00:41:36,560 So we have three [INAUDIBLE] characteristics arrays 820 00:41:36,560 --> 00:41:37,350 for the-- 821 00:41:37,350 --> 00:41:38,725 PROFESSOR: Oh, you mean this ray? 822 00:41:38,725 --> 00:41:41,596 AUDIENCE: So OC, ray OC. 823 00:41:41,596 --> 00:41:43,095 PROFESSOR: This ray? 824 00:41:43,095 --> 00:41:43,910 AUDIENCE: Yeah. 825 00:41:43,910 --> 00:41:44,410 Yeah. 826 00:41:44,410 --> 00:41:48,573 So for the focal length that is negative, 827 00:41:48,573 --> 00:41:49,490 can we draw that line? 828 00:41:49,490 --> 00:41:56,280 So you will find that the intersection of those rays 829 00:41:56,280 --> 00:41:59,568 would not meet the objects you found. 830 00:41:59,568 --> 00:42:00,610 PROFESSOR: Oh, it should. 831 00:42:00,610 --> 00:42:01,200 It should. 832 00:42:01,200 --> 00:42:03,660 If I had drawn it properly, it should. 833 00:42:03,660 --> 00:42:05,880 If you drew this ray, if you drew this ray 834 00:42:05,880 --> 00:42:09,210 and then you extend this ray backwards, 835 00:42:09,210 --> 00:42:13,230 it should still go through my image point. 836 00:42:13,230 --> 00:42:14,130 AUDIENCE: No. 837 00:42:14,130 --> 00:42:15,220 It shouldn't. 838 00:42:15,220 --> 00:42:17,620 PROFESSOR: You're right that they don't meet because this 839 00:42:17,620 --> 00:42:19,300 is a virtual image. 840 00:42:19,300 --> 00:42:23,020 What does meet is their extensions backwards. 841 00:42:23,020 --> 00:42:24,100 AUDIENCE: Yeah, I know. 842 00:42:24,100 --> 00:42:27,220 But when you draw it, but-- 843 00:42:27,220 --> 00:42:32,030 so you see those angle relationship. 844 00:42:32,030 --> 00:42:32,650 So there-- 845 00:42:32,650 --> 00:42:36,703 PROFESSOR: Well, that's because my drawing is not very good. 846 00:42:36,703 --> 00:42:38,120 If I had done the drawing properly 847 00:42:38,120 --> 00:42:39,245 it shouldn't have happened. 848 00:42:43,700 --> 00:42:47,360 AUDIENCE: The artwork angle is larger than the-- 849 00:42:50,680 --> 00:42:54,190 larger than the angle coming-- 850 00:42:54,190 --> 00:42:57,670 going from the say-- 851 00:43:04,340 --> 00:43:08,420 I mean if you take like a z, the angle relationship 852 00:43:08,420 --> 00:43:14,495 between this one and the one-- 853 00:43:14,495 --> 00:43:16,080 PROFESSOR: This one and this one. 854 00:43:16,080 --> 00:43:19,470 AUDIENCE: --and the one coming from the object 855 00:43:19,470 --> 00:43:23,370 to the first focal point. 856 00:43:23,370 --> 00:43:27,420 So this angle is a larger than the array coming-- 857 00:43:30,210 --> 00:43:32,640 larger than the one from the goes 858 00:43:32,640 --> 00:43:36,600 to the first focal point, which means that if you backtrack. 859 00:43:36,600 --> 00:43:38,310 PROFESSOR: This is my handwriting. 860 00:43:38,310 --> 00:43:40,450 This is my handwriting. 861 00:43:40,450 --> 00:43:41,820 AUDIENCE: Oh, I see. 862 00:43:41,820 --> 00:43:42,320 I see. 863 00:43:42,320 --> 00:43:43,460 OK. 864 00:43:43,460 --> 00:43:46,100 PROFESSOR: I forced it in order to make them eat. 865 00:43:46,100 --> 00:43:47,390 I did not make a good diagram. 866 00:43:47,390 --> 00:43:47,890 I'm sorry. 867 00:43:47,890 --> 00:43:48,650 AUDIENCE: OK. 868 00:43:48,650 --> 00:43:49,150 I see. 869 00:43:56,275 --> 00:43:57,217 PROFESSOR: Yeah. 870 00:43:57,217 --> 00:43:58,300 I'm not very good drafter. 871 00:43:58,300 --> 00:43:58,800 I'm sorry. 872 00:44:02,428 --> 00:44:03,220 How do you call it? 873 00:44:03,220 --> 00:44:04,170 Draftsman, right? 874 00:44:04,170 --> 00:44:05,765 I'm not a very good draftsman. 875 00:44:12,890 --> 00:44:14,675 OK. 876 00:44:14,675 --> 00:44:17,700 So I don't want to dwell on this too much. 877 00:44:17,700 --> 00:44:19,360 But this is basically-- 878 00:44:19,360 --> 00:44:26,650 we already discussed this idea that pretty much the only case 879 00:44:26,650 --> 00:44:31,190 where you form a real image that is the rays actually 880 00:44:31,190 --> 00:44:37,130 do meet at the image point, is when you have a positive lens 881 00:44:37,130 --> 00:44:42,600 and you place the object to the left of the front focal point 882 00:44:42,600 --> 00:44:43,200 of that lens. 883 00:44:43,200 --> 00:44:45,330 That will form a real element. 884 00:44:45,330 --> 00:44:47,760 All the other combinations, they actually form 885 00:44:47,760 --> 00:44:50,940 virtual images, which means that what is coming out 886 00:44:50,940 --> 00:44:55,640 of the system is a divergent ray, a divergent ray bundle. 887 00:44:55,640 --> 00:44:57,630 Therefore they don't really meet. 888 00:44:57,630 --> 00:45:01,720 But they do meet if you extend them backwards. 889 00:45:01,720 --> 00:45:03,750 And this backwards intersection is 890 00:45:03,750 --> 00:45:07,313 what we call the virtual image. 891 00:45:07,313 --> 00:45:09,230 And you can see there is different cases here. 892 00:45:09,230 --> 00:45:12,050 Sometimes the magnification is bigger than one. 893 00:45:12,050 --> 00:45:14,060 Sometimes it is less than one. 894 00:45:16,892 --> 00:45:18,350 These actually, can never remember. 895 00:45:18,350 --> 00:45:20,630 Every time I have to derive which is the case. 896 00:45:20,630 --> 00:45:22,310 But anyway, all the possibilities 897 00:45:22,310 --> 00:45:25,070 can happen except, of course, for this one 898 00:45:25,070 --> 00:45:27,810 where the magnification is always negative. 899 00:45:27,810 --> 00:45:29,110 It is always inverted. 900 00:45:29,110 --> 00:45:31,580 In the virtual image, the magnification [INAUDIBLE] 901 00:45:31,580 --> 00:45:32,930 it is always positive. 902 00:45:32,930 --> 00:45:37,220 It is very much is erect. 903 00:45:37,220 --> 00:45:37,720 OK? 904 00:45:47,450 --> 00:45:50,620 Any questions about this? 905 00:45:50,620 --> 00:45:51,890 Real and virtual images. 906 00:46:03,287 --> 00:46:05,120 And the last thing I wanted to remind you of 907 00:46:05,120 --> 00:46:08,200 is what we did on Monday actually. 908 00:46:08,200 --> 00:46:13,770 The definitions of aperture stops pupils, windows, 909 00:46:13,770 --> 00:46:16,500 [INAUDIBLE] and margin inlays. 910 00:46:16,500 --> 00:46:19,050 I really don't feel like going through all this over again. 911 00:46:23,760 --> 00:46:25,792 This is probably very fresh in your minds. 912 00:46:25,792 --> 00:46:27,750 So are there any questions for Monday's lecture 913 00:46:27,750 --> 00:46:30,060 that I can answer about this? 914 00:46:38,098 --> 00:46:41,170 AUDIENCE: The term vignetting applies to the case 915 00:46:41,170 --> 00:46:44,250 when the numerical aperture changes over the field of view. 916 00:46:44,250 --> 00:46:45,250 PROFESSOR: That's right. 917 00:46:45,250 --> 00:46:49,420 So to exaggerate it, suppose that you 918 00:46:49,420 --> 00:46:52,690 have an optical system where on access you 919 00:46:52,690 --> 00:46:54,820 get a large acceptance angle. 920 00:46:54,820 --> 00:46:57,880 But then somehow when you go off axis, 921 00:46:57,880 --> 00:46:59,940 the acceptance angle becomes very small. 922 00:47:03,260 --> 00:47:05,280 So now you compare this with this. 923 00:47:05,280 --> 00:47:06,550 Clearly, this is smaller. 924 00:47:06,550 --> 00:47:07,678 That is vignetting. 925 00:47:10,670 --> 00:47:12,450 AUDIENCE: And that would mean-- 926 00:47:12,450 --> 00:47:15,830 would that be independent of placing an aperture stop 927 00:47:15,830 --> 00:47:18,545 at the focal plane of the lens? 928 00:47:22,184 --> 00:47:24,145 PROFESSOR: What you can say about vignetting 929 00:47:24,145 --> 00:47:25,520 is that it generally happens when 930 00:47:25,520 --> 00:47:27,980 you place the aperture stop in the wrong place. 931 00:47:27,980 --> 00:47:29,400 AUDIENCE: In the wrong place, OK. 932 00:47:29,400 --> 00:47:29,900 OK. 933 00:47:29,900 --> 00:47:32,310 PROFESSOR: When a properly designed optical system, 934 00:47:32,310 --> 00:47:35,300 it should not happen, yeah, at least if there's 935 00:47:35,300 --> 00:47:37,580 a way to get around it. 936 00:47:37,580 --> 00:47:43,720 If you construct the optical system in a way that you cannot 937 00:47:43,720 --> 00:47:46,510 place the aperture stop then that's a badly designed optical 938 00:47:46,510 --> 00:47:47,810 system. 939 00:47:47,810 --> 00:47:49,540 So you generally you try to avoid it 940 00:47:49,540 --> 00:47:53,185 by placing your stops in a strategic location. 941 00:48:00,170 --> 00:48:02,150 There's no universal rule that says 942 00:48:02,150 --> 00:48:04,850 you have to place the aperture stop there in order 943 00:48:04,850 --> 00:48:07,580 to avoid the vignette It kind of depends if you 944 00:48:07,580 --> 00:48:09,260 have a multi-element system. 945 00:48:15,565 --> 00:48:24,680 AUDIENCE: [INAUDIBLE] If you always make sure 946 00:48:24,680 --> 00:48:28,850 that we place the aperture stop at the effective focal plane, 947 00:48:28,850 --> 00:48:31,310 then the vignetting doesn't happen? 948 00:48:31,310 --> 00:48:35,268 Because the rays coming at any angle will-- 949 00:48:35,268 --> 00:48:38,100 PROFESSOR: That is true for the case of a telescope. 950 00:48:38,100 --> 00:48:42,080 I'm not sure if it is true for a general optical system. 951 00:48:42,080 --> 00:48:44,720 Yeah, for a telescope where if you place it 952 00:48:44,720 --> 00:48:47,790 at the common focal point, then yes, you have-- 953 00:48:47,790 --> 00:48:49,040 you minimize vignetting, yeah. 954 00:48:59,450 --> 00:48:59,950 OK. 955 00:48:59,950 --> 00:49:07,492 Before we break, let me go over what we will do today. 956 00:49:07,492 --> 00:49:10,970 So today we have two items in the agenda, basically. 957 00:49:10,970 --> 00:49:14,000 One of them is mirrors. 958 00:49:14,000 --> 00:49:17,390 So far with that with spherical refractive 959 00:49:17,390 --> 00:49:20,480 dielectric interfaces whenever I said anything 960 00:49:20,480 --> 00:49:26,560 about kind of reflective surfaces, like mirrors. 961 00:49:26,560 --> 00:49:27,920 So we'll do that very briefly. 962 00:49:27,920 --> 00:49:31,160 It is simply all we need is a modification 963 00:49:31,160 --> 00:49:35,040 of the sine conventions and we're done. 964 00:49:35,040 --> 00:49:38,363 So that's a fairly painless topic. 965 00:49:38,363 --> 00:49:40,030 There's a little bit of terminology here 966 00:49:40,030 --> 00:49:41,668 which being Greek-- 967 00:49:41,668 --> 00:49:43,710 actually I don't know if being Greek makes sense. 968 00:49:43,710 --> 00:49:46,060 These are archaic words. 969 00:49:46,060 --> 00:49:47,830 Even in Greek, we don't use them anymore. 970 00:49:47,830 --> 00:49:53,710 But a system that uses mirrors is called catoptric, 971 00:49:53,710 --> 00:49:54,520 from the Greek-- 972 00:49:54,520 --> 00:49:57,590 from the ancient Greek word for mirror. 973 00:49:57,590 --> 00:49:59,890 And then if it uses only refractive elements, 974 00:49:59,890 --> 00:50:01,090 you call it dioptic. 975 00:50:01,090 --> 00:50:04,550 That's also where the term diopter comes from. 976 00:50:04,550 --> 00:50:08,680 And finally, there is a class of optical systems 977 00:50:08,680 --> 00:50:12,940 that use both mirrors and refractive lenses that 978 00:50:12,940 --> 00:50:16,780 are very popular in two areas in astronomy and lithography. 979 00:50:16,780 --> 00:50:18,940 In particular, lithography. 980 00:50:18,940 --> 00:50:22,480 If any of you have dealt with semiconductor lithography, 981 00:50:22,480 --> 00:50:25,580 semiconductor industry where they make these huge machines 982 00:50:25,580 --> 00:50:30,710 that write on-- 983 00:50:30,710 --> 00:50:33,580 write patterns on silicon to make chips and such. 984 00:50:33,580 --> 00:50:37,600 For example, [INAUDIBLE] for his project when he does 985 00:50:37,600 --> 00:50:38,440 lithography. 986 00:50:38,440 --> 00:50:40,600 Typically, these machines, they contain-- 987 00:50:40,600 --> 00:50:43,463 the optics are probably taller than me. 988 00:50:43,463 --> 00:50:44,380 OK, I'm not that tall. 989 00:50:44,380 --> 00:50:47,530 But anyway, they are taller than a basketball player. 990 00:50:47,530 --> 00:50:52,180 And they contain typically between 20 and 30 elements. 991 00:50:52,180 --> 00:50:57,432 Some of those are mirrors some of those are refractive lenses. 992 00:50:57,432 --> 00:50:59,140 So this type of system that contains both 993 00:50:59,140 --> 00:51:00,820 is called catadioptric. 994 00:51:00,820 --> 00:51:01,843 That's a mouthful. 995 00:51:01,843 --> 00:51:03,010 But that's what it's called. 996 00:51:03,010 --> 00:51:06,330 So there will see some examples of such systems. 997 00:51:06,330 --> 00:51:10,600 And we'll see them in the context of the basic imaging 998 00:51:10,600 --> 00:51:16,330 systems that is the magnifier lens, the eyepiece which 999 00:51:16,330 --> 00:51:19,480 is basically a magnifier, then the microscope 1000 00:51:19,480 --> 00:51:21,850 uses actually two magnifiers-- 1001 00:51:21,850 --> 00:51:26,080 the telescope which is a slightly strange magnifier. 1002 00:51:26,080 --> 00:51:28,980 And finally, we'll see different types of telescope. 1003 00:51:34,950 --> 00:51:35,785 Where is Le? 1004 00:51:35,785 --> 00:51:36,285 Le? 1005 00:51:36,285 --> 00:51:38,212 AUDIENCE: Yeah. 1006 00:51:38,212 --> 00:51:40,170 PROFESSOR: This is the answer to your question. 1007 00:51:40,170 --> 00:51:44,910 While you guys took a break, I drew again a careful diagram 1008 00:51:44,910 --> 00:51:46,680 with the ray that you asked. 1009 00:51:46,680 --> 00:51:50,390 So now you see so the red rays are 1010 00:51:50,390 --> 00:51:54,060 the ray traces for the case where the lens is positive. 1011 00:51:54,060 --> 00:51:59,100 But the second principal plane is to the left. 1012 00:51:59,100 --> 00:52:03,990 And the black lines are the rays that go to the-- 1013 00:52:03,990 --> 00:52:07,438 that intersect the principal planes on axis. 1014 00:52:07,438 --> 00:52:08,980 So you have to be a bit careful here. 1015 00:52:08,980 --> 00:52:11,105 And I've got to take myself actually the first time 1016 00:52:11,105 --> 00:52:11,890 I did it. 1017 00:52:11,890 --> 00:52:14,730 That they-- when you depart from here, 1018 00:52:14,730 --> 00:52:17,160 you have to meet the first principal plane 1019 00:52:17,160 --> 00:52:18,890 at this center of the axis. 1020 00:52:18,890 --> 00:52:21,810 And then you start from the second principal plane. 1021 00:52:21,810 --> 00:52:23,740 And you're drawing to the image point. 1022 00:52:23,740 --> 00:52:26,940 And you can see there the two black lines, 1023 00:52:26,940 --> 00:52:29,184 they do look sort of parallel, don't they? 1024 00:52:29,184 --> 00:52:30,108 AUDIENCE: Mm-hm. 1025 00:52:30,108 --> 00:52:32,237 Yeah. 1026 00:52:32,237 --> 00:52:34,070 PROFESSOR: And of course, the image is here. 1027 00:52:42,656 --> 00:52:44,891 AUDIENCE: Actually it's a similar thing 1028 00:52:44,891 --> 00:52:47,346 but I think it's wrong. 1029 00:52:51,770 --> 00:52:53,200 PROFESSOR: Yes, they are similar. 1030 00:52:53,200 --> 00:52:54,420 It's a little bit difficult to see here. 1031 00:52:54,420 --> 00:52:55,830 It is obvious that they're similar if you 1032 00:52:55,830 --> 00:52:57,300 do the case of the single lens. 1033 00:53:01,100 --> 00:53:02,090 Remember this diagram? 1034 00:53:08,063 --> 00:53:09,230 That's a really bad drawing. 1035 00:53:09,230 --> 00:53:11,710 But from here you can see that they're similar triangles. 1036 00:53:11,710 --> 00:53:12,210 Right? 1037 00:53:12,210 --> 00:53:13,202 AUDIENCE: Yeah. 1038 00:53:16,420 --> 00:53:17,920 PROFESSOR: And then, of course, when 1039 00:53:17,920 --> 00:53:20,410 you put the principal planes, there's 1040 00:53:20,410 --> 00:53:23,200 just dead space between the two principal planes. 1041 00:53:23,200 --> 00:53:24,882 But the geometrical relationships 1042 00:53:24,882 --> 00:53:26,590 are similar triangles and all that stuff, 1043 00:53:26,590 --> 00:53:29,620 they get presented except you are dead space. 1044 00:53:29,620 --> 00:53:31,680 And in this example, it is even worse. 1045 00:53:31,680 --> 00:53:33,550 The dead space is kind of reversed. 1046 00:53:33,550 --> 00:53:36,940 Because they moved one principal plane on top of the other. 1047 00:53:36,940 --> 00:53:38,258 You know what I mean. 1048 00:53:38,258 --> 00:53:40,800 I think if I attempt to express it, I will confuse the issue. 1049 00:53:50,010 --> 00:53:53,680 Let me talk a little bit about mirrors now. 1050 00:53:53,680 --> 00:53:58,600 So for mirrors, wherever set of the same conventions 1051 00:53:58,600 --> 00:54:00,640 that is a little bit modified with respect 1052 00:54:00,640 --> 00:54:04,260 to the same conventions for refraction. 1053 00:54:04,260 --> 00:54:07,780 And the reason, of course, is that when you have a reflective 1054 00:54:07,780 --> 00:54:10,570 surface then the rays will indeed fall 1055 00:54:10,570 --> 00:54:15,460 and they will start going from the right to the left. 1056 00:54:15,460 --> 00:54:17,320 So because of that fact, then we really 1057 00:54:17,320 --> 00:54:22,570 have to modify our sign conventions. 1058 00:54:22,570 --> 00:54:27,350 And here I put all the cases of positive quantities. 1059 00:54:27,350 --> 00:54:29,220 So most of these are familiar. 1060 00:54:29,220 --> 00:54:31,550 The positive curvature is the same as before. 1061 00:54:31,550 --> 00:54:35,820 These angles and directions here are the same as before. 1062 00:54:35,820 --> 00:54:38,330 The really interesting ones, the ones 1063 00:54:38,330 --> 00:54:40,790 that happen upon the reflection. 1064 00:54:40,790 --> 00:54:43,740 Because upon reflection, the distance 1065 00:54:43,740 --> 00:54:46,820 has now become positive if they go to their right 1066 00:54:46,820 --> 00:54:48,220 not to the left. 1067 00:54:48,220 --> 00:54:50,600 And that kind of makes intuitive sense. 1068 00:54:50,600 --> 00:54:53,780 Because now it is as if actually the best 1069 00:54:53,780 --> 00:54:57,380 way to make sense of this is to simply unfold it. 1070 00:54:57,380 --> 00:54:59,380 So if you want to take this-- 1071 00:54:59,380 --> 00:55:01,580 the part of what happened after reflection, 1072 00:55:01,580 --> 00:55:05,840 and then found it so it could go back for one direction. 1073 00:55:05,840 --> 00:55:10,280 Then these quantities would then would indeed remain positive. 1074 00:55:10,280 --> 00:55:12,770 So this is the origin of the sine conventions 1075 00:55:12,770 --> 00:55:15,110 for the mirrors. 1076 00:55:15,110 --> 00:55:17,180 So for example, why is this angle positive? 1077 00:55:17,180 --> 00:55:19,560 Well, because if I flip it the other value 1078 00:55:19,560 --> 00:55:24,650 down it would indeed be the same as a positive angle over here. 1079 00:55:24,650 --> 00:55:27,170 Of course, this angle, if it had happened 1080 00:55:27,170 --> 00:55:30,350 in a refractive system, that would have been-- 1081 00:55:30,350 --> 00:55:32,030 well, it would have been forbidden 1082 00:55:32,030 --> 00:55:35,380 to begin with because the ray is going from right to left. 1083 00:55:35,380 --> 00:55:37,760 And that's anathema. 1084 00:55:41,510 --> 00:55:44,360 So having said that, I'll come back to this. 1085 00:55:44,360 --> 00:55:47,780 But having said that, the next thing we would like to do 1086 00:55:47,780 --> 00:55:53,580 is derive a matrix relationship for mirrors. 1087 00:55:53,580 --> 00:55:55,580 Now long time ago, I think it was 1088 00:55:55,580 --> 00:55:58,120 in the first or second lecture or so, 1089 00:55:58,120 --> 00:55:59,690 we derived the ideal mirror. 1090 00:55:59,690 --> 00:56:06,260 We said that if you have rays coming from infinity 1091 00:56:06,260 --> 00:56:08,630 and you want to focus them onto a single point 1092 00:56:08,630 --> 00:56:19,890 out here, like so, we said that the way you 1093 00:56:19,890 --> 00:56:23,970 do that is with a surface, reflective surface. 1094 00:56:23,970 --> 00:56:27,480 And you remember, probably, that what the surface was. 1095 00:56:27,480 --> 00:56:28,940 It was a parabola. 1096 00:56:28,940 --> 00:56:33,360 So we wrote it in equations here. 1097 00:56:33,360 --> 00:56:37,980 If you call this elevation s, you call this axis x. 1098 00:56:37,980 --> 00:56:39,720 So we found the equation of the parabola 1099 00:56:39,720 --> 00:56:45,630 was s equals x squared over 4F, where F is 1100 00:56:45,630 --> 00:56:46,920 the focal length of the lens. 1101 00:56:51,730 --> 00:56:54,010 So the parabola is ideal. 1102 00:56:54,010 --> 00:56:57,940 But in some cases, people make spherical mirrors. 1103 00:56:57,940 --> 00:57:00,490 Actually, it is not as common as lenses. 1104 00:57:00,490 --> 00:57:02,470 Lenses are very commonly spherical. 1105 00:57:02,470 --> 00:57:05,020 Mirrors are very often actually paraboloidal, very close 1106 00:57:05,020 --> 00:57:11,830 to ideal, especially mirrors used in concentrators, things 1107 00:57:11,830 --> 00:57:17,050 like satellite antennas, solar concentrations that 1108 00:57:17,050 --> 00:57:21,910 are used in solar energy systems and so on. 1109 00:57:21,910 --> 00:57:24,850 Then, because they would want to focus 1110 00:57:24,850 --> 00:57:28,030 the light on a single point then you actually 1111 00:57:28,030 --> 00:57:29,740 go for the parabola. 1112 00:57:29,740 --> 00:57:32,530 It turns out, if you want to form a sort of a bigger image 1113 00:57:32,530 --> 00:57:34,450 here than the parabola is not very good. 1114 00:57:34,450 --> 00:57:37,120 So people actually make spherical mirrors. 1115 00:57:37,120 --> 00:57:39,070 So for a spherical mirror, I did a little bit 1116 00:57:39,070 --> 00:57:46,160 of a derivation here which basically follows 1117 00:57:46,160 --> 00:57:48,920 the paraxial approximation to connect 1118 00:57:48,920 --> 00:57:52,490 the ideal parabolic shape and the focal length 1119 00:57:52,490 --> 00:57:57,840 that we get from the parabola to the equivalent sphere. 1120 00:57:57,840 --> 00:58:00,230 And again, the parabola and the sphere, they 1121 00:58:00,230 --> 00:58:01,550 kind of look the same. 1122 00:58:01,550 --> 00:58:04,950 If you look at the parabola, maybe it goes like this. 1123 00:58:04,950 --> 00:58:07,460 If you look at the sphere, and you 1124 00:58:07,460 --> 00:58:10,430 match the curvatures near the center, it might go like this. 1125 00:58:15,900 --> 00:58:18,360 So what we're trying to find here, 1126 00:58:18,360 --> 00:58:21,690 what is the radius of this sphere here 1127 00:58:21,690 --> 00:58:25,860 that matches best the parabola, the ideal parabola. 1128 00:58:25,860 --> 00:58:30,510 So this is the ideal parabola that is given by this equation. 1129 00:58:30,510 --> 00:58:33,677 And then here's the sphere. 1130 00:58:33,677 --> 00:58:35,260 That's not a very good looking sphere. 1131 00:58:35,260 --> 00:58:37,600 But anyway, let's call it a sphere. 1132 00:58:37,600 --> 00:58:39,820 And the question is, what is the radius 1133 00:58:39,820 --> 00:58:42,640 of curvature of this sphere that will 1134 00:58:42,640 --> 00:58:45,880 match the parabola ideally. 1135 00:58:45,880 --> 00:58:47,500 So this is what I did here. 1136 00:58:47,500 --> 00:58:50,760 If you write the equation for the sphere, 1137 00:58:50,760 --> 00:58:53,180 well, we know-- actually, let me leave this up. 1138 00:58:53,180 --> 00:58:55,180 We know how to read the question for the sphere. 1139 00:58:55,180 --> 00:58:59,000 In my chosen system of coordinates, this is x. 1140 00:58:59,000 --> 00:59:00,910 And this is the positive axis z. 1141 00:59:00,910 --> 00:59:04,750 So the equation for the sphere is x plus the radius, 1142 00:59:04,750 --> 00:59:11,753 squared plus s squared equals the radius squared. 1143 00:59:11,753 --> 00:59:13,420 So this is the question for this sphere. 1144 00:59:13,420 --> 00:59:17,930 This is this displacement here, with respect to the center 1145 00:59:17,930 --> 00:59:19,300 of the coordinates. 1146 00:59:19,300 --> 00:59:21,400 And then what do you do, this is [INAUDIBLE] 1147 00:59:21,400 --> 00:59:23,410 slide but I will actually derive it 1148 00:59:23,410 --> 00:59:27,070 for you here so you can see it in this sort of animated form. 1149 00:59:27,070 --> 00:59:28,690 And then what you do is you actually 1150 00:59:28,690 --> 00:59:37,555 solve, so you have an s equals plus/minus R minus-- 1151 00:59:40,493 --> 00:59:41,160 what am I doing? 1152 00:59:41,160 --> 00:59:45,090 AUDIENCE: So you've got your axis-- your [INAUDIBLE] 1153 00:59:45,090 --> 00:59:46,160 PROFESSOR: Oh, I'm sorry. 1154 00:59:46,160 --> 00:59:46,490 Yes, I did. 1155 00:59:46,490 --> 00:59:46,990 Didn't I? 1156 00:59:46,990 --> 00:59:47,380 Yay. 1157 00:59:47,380 --> 00:59:48,255 I'm sorry about that. 1158 00:59:51,492 --> 00:59:53,440 I think I better do this over. 1159 01:00:10,170 --> 01:00:12,010 So this is s for the parabola. 1160 01:00:12,010 --> 01:00:13,670 This is s for the sphere. 1161 01:00:17,120 --> 01:00:29,050 So of course it would be s plus R squared plus x squared 1162 01:00:29,050 --> 01:00:31,240 equals R squared. 1163 01:00:31,240 --> 01:00:37,760 So that means that s equals minus R plus minus square root 1164 01:00:37,760 --> 01:00:41,440 R squared minus x squared. 1165 01:00:41,440 --> 01:00:44,860 Now which signs would I keep, the plus or the minus? 1166 01:00:44,860 --> 01:00:49,610 Well, the minus sign, you can see a little bit by inspection, 1167 01:00:49,610 --> 01:00:53,060 the minus sign is this part of the sphere which 1168 01:00:53,060 --> 01:00:57,260 is pretty far from the paraxial approximation, clearly, right? 1169 01:00:57,260 --> 01:00:59,660 So the only way to deal with a paraxial approximation 1170 01:00:59,660 --> 01:01:01,400 is to keep the plus sign. 1171 01:01:01,400 --> 01:01:06,440 So we'll write, then, s equals minus R plus the next step you 1172 01:01:06,440 --> 01:01:09,300 do to derive paraxial approximations 1173 01:01:09,300 --> 01:01:13,500 is you pull a large quantity out of the square root. 1174 01:01:13,500 --> 01:01:15,365 So the large quantities, the radius here, 1175 01:01:15,365 --> 01:01:16,490 and you write it like this. 1176 01:01:16,490 --> 01:01:20,330 1 minus x squared over R squared. 1177 01:01:20,330 --> 01:01:23,880 Then you apply a Taylor formula that says that square root of 1 1178 01:01:23,880 --> 01:01:31,250 plus small approximately equals 1 plus the small divided by 2. 1179 01:01:31,250 --> 01:01:32,870 And note, of course, that applies also 1180 01:01:32,870 --> 01:01:34,790 for a negative sign. 1181 01:01:34,790 --> 01:01:38,210 So I can do like this. 1182 01:01:38,210 --> 01:01:44,600 And this means that s equals minus R plus R 1 minus x 1183 01:01:44,600 --> 01:01:47,330 square over 2R squared. 1184 01:01:47,330 --> 01:01:49,550 And if you do it carefully here you 1185 01:01:49,550 --> 01:01:51,710 see that this kills the square. 1186 01:01:51,710 --> 01:01:53,420 This kills this. 1187 01:01:53,420 --> 01:02:05,630 And you end up with s equals minus x squared over 2R. 1188 01:02:05,630 --> 01:02:08,340 And what this means now is if you compare with equation 1189 01:02:08,340 --> 01:02:12,770 of the ideal parabola, which was x squared over 4F, 1190 01:02:12,770 --> 01:02:17,000 it means that F equals minus R over 2. 1191 01:02:21,300 --> 01:02:23,220 The next question now is what does this mean. 1192 01:02:23,220 --> 01:02:25,610 Is the focal length negative or positive? 1193 01:02:28,570 --> 01:02:31,090 To answer this question, we have to first remember 1194 01:02:31,090 --> 01:02:32,830 our sign conventions. 1195 01:02:32,830 --> 01:02:35,350 Is our surface positive or negative? 1196 01:02:35,350 --> 01:02:40,210 That is, is R a negative or positive quantity? 1197 01:02:40,210 --> 01:02:45,090 So if you recall, I'll go back one slide to the previous one. 1198 01:02:45,090 --> 01:02:47,640 This is a positive curvature. 1199 01:02:47,640 --> 01:02:51,180 if If it is pointing that way, I will not use any formal term 1200 01:02:51,180 --> 01:02:52,560 because that may be confusing. 1201 01:02:52,560 --> 01:02:56,890 If it's like that, it's positive. 1202 01:02:56,890 --> 01:02:58,000 This is not like that. 1203 01:02:58,000 --> 01:02:58,850 It's the opposite. 1204 01:02:58,850 --> 01:03:02,530 And so therefore here, R is negative. 1205 01:03:02,530 --> 01:03:05,770 So you have a negative sign that came out of the math. 1206 01:03:05,770 --> 01:03:08,140 But R is a negative quantity. 1207 01:03:08,140 --> 01:03:10,750 So therefore F is still positive. 1208 01:03:10,750 --> 01:03:12,430 So this is still a positive lens. 1209 01:03:21,400 --> 01:03:23,800 And it is clear that it is a positive lens, 1210 01:03:23,800 --> 01:03:27,010 because it forms a real image for an object at infinity. 1211 01:03:27,010 --> 01:03:30,940 By construction, we demanded that if I have parallel rays 1212 01:03:30,940 --> 01:03:33,780 from infinity, these things should focus them 1213 01:03:33,780 --> 01:03:35,350 to a real point on the axis. 1214 01:03:35,350 --> 01:03:37,450 So therefore, clearly, this is what 1215 01:03:37,450 --> 01:03:41,230 you would call a positive lens. 1216 01:03:41,230 --> 01:03:43,690 And since the lens, it actually has 1217 01:03:43,690 --> 01:03:47,230 a matrix that is very similar to the matrix of a lens, 1218 01:03:47,230 --> 01:03:50,560 all you have to do is substitute the focal length 1219 01:03:50,560 --> 01:03:52,540 with the equation that we just derived 1220 01:03:52,540 --> 01:03:55,180 for the radius of curvature. 1221 01:03:55,180 --> 01:04:00,640 And this is the matrix that describes a mirror, a mirror 1222 01:04:00,640 --> 01:04:02,510 concentrator. 1223 01:04:02,510 --> 01:04:06,500 And of course, since it is a lens, it also forms images. 1224 01:04:09,080 --> 01:04:12,920 So notice here, the way I drew it, 1225 01:04:12,920 --> 01:04:15,200 these quantities are still all positive. 1226 01:04:15,200 --> 01:04:19,290 Of course, it's not as positive because I go from the object 1227 01:04:19,290 --> 01:04:23,620 to the right, towards the instrument. 1228 01:04:23,620 --> 01:04:24,700 Now notice what happens. 1229 01:04:24,700 --> 01:04:28,930 From the instrument, I go to the left towards the image. 1230 01:04:28,930 --> 01:04:30,640 But that is still a positive quantity 1231 01:04:30,640 --> 01:04:33,520 because there was a reflection that occurred there. 1232 01:04:33,520 --> 01:04:37,780 So therefore s sub I is still a positive quantity. 1233 01:04:37,780 --> 01:04:40,570 So I can write this equation with a clear conscience, 1234 01:04:40,570 --> 01:04:48,350 so to speak, that s0 and si are all positive qualities. 1235 01:04:48,350 --> 01:04:53,150 Now can you imagine when a lens might form a virtual image? 1236 01:04:53,150 --> 01:04:56,335 AUDIENCE: [INAUDIBLE] 1237 01:04:56,335 --> 01:04:57,320 PROFESSOR: I'm sorry. 1238 01:04:57,320 --> 01:04:58,520 Yes, a mirror. 1239 01:04:58,520 --> 01:05:02,870 So clearly the mirror here is forming a real image. 1240 01:05:02,870 --> 01:05:06,150 Can you imagine how I could construct a mirror forming 1241 01:05:06,150 --> 01:05:08,360 a virtual image? 1242 01:05:16,724 --> 01:05:19,340 AUDIENCE: If the mirror diverges and you are looking 1243 01:05:19,340 --> 01:05:21,010 from the same side, then. 1244 01:05:21,010 --> 01:05:22,010 PROFESSOR: That's right. 1245 01:05:22,010 --> 01:05:22,850 That's one way. 1246 01:05:22,850 --> 01:05:25,640 If you make the mirror like this, 1247 01:05:25,640 --> 01:05:28,490 now the mirror has a positive R therefore 1248 01:05:28,490 --> 01:05:31,430 it has a negative F. Now this mirror 1249 01:05:31,430 --> 01:05:33,980 has become the equivalent of a negative lens. 1250 01:05:33,980 --> 01:05:36,470 Therefore it will form a virtual image. 1251 01:05:36,470 --> 01:05:39,420 What's another possibility for forming a virtual image 1252 01:05:39,420 --> 01:05:40,170 with a mirror? 1253 01:05:44,710 --> 01:05:46,150 What about a positive lens? 1254 01:05:46,150 --> 01:05:49,546 Could a positive lens ever form a virtual image? 1255 01:05:49,546 --> 01:05:50,482 AUDIENCE: [INAUDIBLE] 1256 01:05:50,482 --> 01:05:51,820 PROFESSOR: That's right. 1257 01:05:51,820 --> 01:05:55,570 If you put the object in between the focal point 1258 01:05:55,570 --> 01:05:59,643 and the mirror itself, again, it will form a virtual image. 1259 01:05:59,643 --> 01:06:01,810 That's actually a very funny to observe by yourself. 1260 01:06:04,997 --> 01:06:07,330 You cannot quite do it with a spoon because the spoon is 1261 01:06:07,330 --> 01:06:07,960 really too small. 1262 01:06:07,960 --> 01:06:09,127 You have to go really close. 1263 01:06:09,127 --> 01:06:11,230 And you cannot focus your eyes in there. 1264 01:06:11,230 --> 01:06:14,590 But those of you who've been at hotels-- 1265 01:06:14,590 --> 01:06:16,010 I've never seen this in a home. 1266 01:06:16,010 --> 01:06:20,350 But most hotels, in the bathroom they have a curved mirror. 1267 01:06:20,350 --> 01:06:21,880 And I don't want to speculate why 1268 01:06:21,880 --> 01:06:24,350 they put that curved mirror, there but anyway they have it. 1269 01:06:24,350 --> 01:06:26,280 And this mirror is very lightly curved. 1270 01:06:26,280 --> 01:06:29,990 It has a very long radius of curvature. 1271 01:06:29,990 --> 01:06:32,530 So it is very easy, actually, if you 1272 01:06:32,530 --> 01:06:35,530 stand in front of the mirror, you can move back and forth. 1273 01:06:35,530 --> 01:06:38,880 And you can switch your image from real to virtual. 1274 01:06:38,880 --> 01:06:40,040 It is really spectacular. 1275 01:06:40,040 --> 01:06:43,450 It is worth checking yourself in the Ritz Carlton in Boston 1276 01:06:43,450 --> 01:06:46,270 spend $400 for a night just to see that, actually. 1277 01:06:46,270 --> 01:06:49,300 Because it's very, very instructive from the optics 1278 01:06:49,300 --> 01:06:50,130 point of view. 1279 01:06:50,130 --> 01:06:53,530 But real, it's nice, because as you go near the mirror, 1280 01:06:53,530 --> 01:06:56,200 basically when you get between the focal length 1281 01:06:56,200 --> 01:06:59,680 and the mirror, then you see an erect image of yourself that 1282 01:06:59,680 --> 01:07:01,990 is virtual because you see yourself behind the mirror 1283 01:07:01,990 --> 01:07:03,060 it is magnified. 1284 01:07:03,060 --> 01:07:05,500 Actually, the magnification is so high 1285 01:07:05,500 --> 01:07:07,210 that you typically only see your eye. 1286 01:07:07,210 --> 01:07:08,710 But anyway, something like that. 1287 01:07:08,710 --> 01:07:13,240 If you move back, then you see yourself inverted, 1288 01:07:13,240 --> 01:07:15,360 as you see yourself in a spoon, by the way. 1289 01:07:15,360 --> 01:07:16,990 If you look at yourself in a spoon 1290 01:07:16,990 --> 01:07:20,020 you almost always see yourself inverted because the spoon has 1291 01:07:20,020 --> 01:07:21,410 a very short focal length. 1292 01:07:21,410 --> 01:07:23,530 So you are always in this scenario 1293 01:07:23,530 --> 01:07:24,590 when you look at a spoon. 1294 01:07:32,320 --> 01:07:34,350 Any questions? 1295 01:07:43,080 --> 01:07:45,190 There's not much else to say about mirrors really. 1296 01:07:45,190 --> 01:07:48,580 The same optics applies, with this little twist 1297 01:07:48,580 --> 01:07:52,160 of the flipping of the signs upon inversion. 1298 01:07:52,160 --> 01:07:56,800 So if you face a situation, all you need to do is keep cool. 1299 01:07:56,800 --> 01:07:59,110 Don't let yourself panic or anything. 1300 01:07:59,110 --> 01:08:02,560 It's just a matter of keeping track 1301 01:08:02,560 --> 01:08:06,660 of the positive things that happen to the right or to the-- 1302 01:08:06,660 --> 01:08:08,770 I'm sorry-- to the [INAUDIBLE] things 1303 01:08:08,770 --> 01:08:11,020 that happen after reflection. 1304 01:08:11,020 --> 01:08:12,880 And typically with the kind of problems 1305 01:08:12,880 --> 01:08:15,850 that we have to solve by hand, these are pretty simple. 1306 01:08:15,850 --> 01:08:18,130 I mean at most, you might encounter one reflection. 1307 01:08:18,130 --> 01:08:20,920 Now people who make optical design software, 1308 01:08:20,920 --> 01:08:23,272 they have to deal with arbitrary numbers of reflections. 1309 01:08:23,272 --> 01:08:24,939 I wouldn't want to be the programmer who 1310 01:08:24,939 --> 01:08:27,939 has to solve that problem. 1311 01:08:27,939 --> 01:08:30,160 Because you know, if you have a second reflection, 1312 01:08:30,160 --> 01:08:31,630 then the signs flip again. 1313 01:08:31,630 --> 01:08:34,170 As you can imagine, if I stick another mirror here. 1314 01:08:34,170 --> 01:08:35,950 So the light starts sort of zig zagging, 1315 01:08:35,950 --> 01:08:41,520 every reflection flips the signs in the sign convention again. 1316 01:08:41,520 --> 01:08:43,810 So it can get a bit tricky to keep track 1317 01:08:43,810 --> 01:08:45,609 of all of these flippings. 1318 01:08:45,609 --> 01:08:48,670 But we don't have to worry about that 1319 01:08:48,670 --> 01:08:52,096 at this introductory level. 1320 01:08:58,189 --> 01:09:02,640 So let me move onto the next topic, which 1321 01:09:02,640 --> 01:09:06,725 is a few examples of optical systems. 1322 01:09:06,725 --> 01:09:08,314 AUDIENCE: Hey, George. 1323 01:09:08,314 --> 01:09:08,939 PROFESSOR: Yes? 1324 01:09:08,939 --> 01:09:10,147 AUDIENCE: Sorry to interrupt. 1325 01:09:10,147 --> 01:09:12,920 I'm going to pass around these parabolic reflecters 1326 01:09:12,920 --> 01:09:14,260 so you can try that. 1327 01:09:14,260 --> 01:09:15,279 PROFESSOR: Good idea. 1328 01:09:17,564 --> 01:09:19,689 So maybe you don't have to go to a hotel after all. 1329 01:09:19,689 --> 01:09:22,540 [INAUDIBLE] What is the focal length of this, you know? 1330 01:09:22,540 --> 01:09:25,300 AUDIENCE: It's like 20 centimeters or so. 1331 01:09:25,300 --> 01:09:26,220 You can check it out. 1332 01:09:26,220 --> 01:09:27,000 PROFESSOR: Great. 1333 01:09:27,000 --> 01:09:31,646 So yeah, you try to go at 40 and 10 or whatever, 1334 01:09:31,646 --> 01:09:32,979 and you will see the difference. 1335 01:09:32,979 --> 01:09:35,083 It's actually quite dramatic and very interesting. 1336 01:09:41,859 --> 01:09:47,819 So the magnifier, as the name suggests, 1337 01:09:47,819 --> 01:09:52,300 it is a device that we use to aid our eyes when we're 1338 01:09:52,300 --> 01:09:56,330 looking at a very small object. 1339 01:09:56,330 --> 01:09:59,147 And this is a-- 1340 01:09:59,147 --> 01:10:01,230 dog it took me a lot of time to make this cartoon. 1341 01:10:01,230 --> 01:10:02,230 So I'm very proud of it. 1342 01:10:02,230 --> 01:10:04,700 So let me start the animation here. 1343 01:10:04,700 --> 01:10:08,980 So if you have a really small object, of course 1344 01:10:08,980 --> 01:10:12,580 it will form a very small image on your retina. 1345 01:10:12,580 --> 01:10:16,030 And very often, this image is not 1346 01:10:16,030 --> 01:10:19,582 big enough to observe with sufficient detail. 1347 01:10:19,582 --> 01:10:21,040 So the obvious thing that you would 1348 01:10:21,040 --> 01:10:22,780 do if something looks small to you 1349 01:10:22,780 --> 01:10:24,843 and you'd like to make it bigger, 1350 01:10:24,843 --> 01:10:26,260 you don't have to know any optics. 1351 01:10:26,260 --> 01:10:28,295 What you do is you bring it near your eyes. 1352 01:10:28,295 --> 01:10:29,920 But from experience, you know that this 1353 01:10:29,920 --> 01:10:30,940 doesn't work very well. 1354 01:10:30,940 --> 01:10:32,680 Because if you exceed the distance 1355 01:10:32,680 --> 01:10:35,980 of approximately 25 centimeters, then you 1356 01:10:35,980 --> 01:10:37,780 can really not focus anymore. 1357 01:10:37,780 --> 01:10:40,320 And actually for you guys who are young 1358 01:10:40,320 --> 01:10:42,730 it's about 25 centimeters. 1359 01:10:42,730 --> 01:10:46,090 As we grow older, this distance becomes longer and longer. 1360 01:10:46,090 --> 01:10:47,530 And we basically lose our ability 1361 01:10:47,530 --> 01:10:49,840 to focus, which is a very annoying, annoying thing 1362 01:10:49,840 --> 01:10:50,440 actually. 1363 01:10:50,440 --> 01:10:52,130 It has just started to happen to me. 1364 01:10:52,130 --> 01:10:55,840 As I approach age 40, I've started noticing that I cannot 1365 01:10:55,840 --> 01:10:57,920 focus as well as I used to. 1366 01:10:57,920 --> 01:11:00,190 So it's a little bit annoying. 1367 01:11:00,190 --> 01:11:02,800 But anyway it beats the alternative I guess. 1368 01:11:02,800 --> 01:11:04,550 So I'm very happy that I reached that age. 1369 01:11:04,550 --> 01:11:09,160 But anyway, the point is that there comes a point-- 1370 01:11:09,160 --> 01:11:11,140 actually for me now, the only way to focus 1371 01:11:11,140 --> 01:11:13,600 is to take off my glasses. 1372 01:11:13,600 --> 01:11:15,847 But even so, there comes a point where 1373 01:11:15,847 --> 01:11:16,930 you can not focus anymore. 1374 01:11:16,930 --> 01:11:17,980 Even you cannot focus. 1375 01:11:17,980 --> 01:11:21,790 I bring something to 5 or 10 centimeters, you cannot focus. 1376 01:11:21,790 --> 01:11:23,900 The question is, what could you do. 1377 01:11:23,900 --> 01:11:25,905 So what you do is you use a magnifier. 1378 01:11:25,905 --> 01:11:27,610 And the magnifier is something that you 1379 01:11:27,610 --> 01:11:30,820 put between the object and your eye. 1380 01:11:30,820 --> 01:11:33,340 And the way it works is like this. 1381 01:11:33,340 --> 01:11:35,350 We can do some simple ray tracing here. 1382 01:11:38,170 --> 01:11:40,500 This is where the parameters of the magnifier, 1383 01:11:40,500 --> 01:11:45,760 let's say it is a positive lens, the magnifier. 1384 01:11:45,760 --> 01:11:47,750 Let's say the focal length is F and the power 1385 01:11:47,750 --> 01:11:49,560 is the inverse of that. 1386 01:11:49,560 --> 01:11:54,530 So this is some ray tracing that shows how the image is 1387 01:11:54,530 --> 01:11:56,120 formed by this lens. 1388 01:11:56,120 --> 01:11:58,730 So all you do is you take a horizontal ray. 1389 01:11:58,730 --> 01:12:00,450 This will go somehow through-- 1390 01:12:00,450 --> 01:12:02,330 this guy will bend it. 1391 01:12:02,330 --> 01:12:06,770 And let's pretend now that this ray, somehow I tune things 1392 01:12:06,770 --> 01:12:09,740 so that this ray went through the center of the eye. 1393 01:12:09,740 --> 01:12:11,770 So this ray went and bend, right? 1394 01:12:11,770 --> 01:12:15,620 But you can see very clearly now because what these lines did 1395 01:12:15,620 --> 01:12:18,290 is it bent this ray bundle so that it 1396 01:12:18,290 --> 01:12:20,030 meets the object over here. 1397 01:12:20,030 --> 01:12:25,160 The result is that I got a much magnified image on the retina. 1398 01:12:25,160 --> 01:12:25,980 That's about it. 1399 01:12:25,980 --> 01:12:27,460 That's all there is to. 1400 01:12:27,460 --> 01:12:30,860 By using this element, you can get bigger images. 1401 01:12:30,860 --> 01:12:34,540 Now if I didn't have the eye here, 1402 01:12:34,540 --> 01:12:37,410 this is actually the most instructive thing 1403 01:12:37,410 --> 01:12:39,790 I'm going to say today. 1404 01:12:39,790 --> 01:12:42,880 This is really the meaning of the virtual image. 1405 01:12:42,880 --> 01:12:46,870 If I give you this system by itself, obviously 1406 01:12:46,870 --> 01:12:49,040 because this lens, if you look carefully, 1407 01:12:49,040 --> 01:12:51,740 it created a divergent ray bundle. 1408 01:12:51,740 --> 01:12:56,830 Obviously this object is between the focal point of the lens. 1409 01:12:56,830 --> 01:12:58,330 So therefore, this is the situation 1410 01:12:58,330 --> 01:13:00,070 that we encountered before. 1411 01:13:00,070 --> 01:13:03,370 And we've said that the system forms a virtual image. 1412 01:13:03,370 --> 01:13:05,230 Well what is the virtual image? 1413 01:13:05,230 --> 01:13:07,450 I have to take these rays that the lens produced 1414 01:13:07,450 --> 01:13:09,880 and I have to extend them backwards. 1415 01:13:09,880 --> 01:13:11,590 And with a little bit of manipulation 1416 01:13:11,590 --> 01:13:15,210 here, because this is a cartoon, it is not a real calculation. 1417 01:13:15,210 --> 01:13:16,840 But with some manipulation, I made sure 1418 01:13:16,840 --> 01:13:18,220 that the rays do meet. 1419 01:13:18,220 --> 01:13:19,460 So the rays meet. 1420 01:13:19,460 --> 01:13:23,020 And the point where the rays meet is the virtual image. 1421 01:13:23,020 --> 01:13:24,880 There's nothing there. 1422 01:13:24,880 --> 01:13:25,540 Nothing. 1423 01:13:25,540 --> 01:13:26,530 The object is here. 1424 01:13:26,530 --> 01:13:29,410 The only physical entity is here and your eye, 1425 01:13:29,410 --> 01:13:30,760 and the lens of course. 1426 01:13:30,760 --> 01:13:31,900 There's nothing here. 1427 01:13:31,900 --> 01:13:37,280 But when your eye forms an image it 1428 01:13:37,280 --> 01:13:40,730 is as if you could take this away-- 1429 01:13:40,730 --> 01:13:43,070 you could take away the combination of the object 1430 01:13:43,070 --> 01:13:48,410 and the magnifier and replace it with a virtual image 1431 01:13:48,410 --> 01:13:50,660 that this system formed. 1432 01:13:50,660 --> 01:13:57,550 This virtual element is acting as an object for your eye. 1433 01:13:57,550 --> 01:13:59,380 The eye images this object. 1434 01:13:59,380 --> 01:14:03,010 And because this object has been magnified relative 1435 01:14:03,010 --> 01:14:08,930 to the actual object, you end up observing a magnified image. 1436 01:14:08,930 --> 01:14:11,080 So this is the significance of virtual images. 1437 01:14:11,080 --> 01:14:13,570 By themselves, that don't mean anything. 1438 01:14:13,570 --> 01:14:17,080 But because at the end of an optical system you always have 1439 01:14:17,080 --> 01:14:19,830 the observer's-- 1440 01:14:19,830 --> 01:14:21,700 I mean, the observer's eye length 1441 01:14:21,700 --> 01:14:24,570 and the retina, the virtual image 1442 01:14:24,570 --> 01:14:28,390 ends up becoming a real image, hopefully on the retina. 1443 01:14:28,390 --> 01:14:30,698 If it becomes a real image elsewhere, 1444 01:14:30,698 --> 01:14:31,990 you will not be able to see it. 1445 01:14:31,990 --> 01:14:33,490 It will be out to focus. 1446 01:14:33,490 --> 01:14:35,440 Or even worse, if you create a virtual image 1447 01:14:35,440 --> 01:14:37,570 and you send it to an observer so it becomes 1448 01:14:37,570 --> 01:14:39,430 a virtual image on the retina, again you 1449 01:14:39,430 --> 01:14:40,472 don't see anything there. 1450 01:14:40,472 --> 01:14:43,750 Because you can never observe a virtual image 1451 01:14:43,750 --> 01:14:47,170 unless you can image it using the optics of your eye. 1452 01:14:47,170 --> 01:14:49,790 So this is the significance then. 1453 01:14:49,790 --> 01:14:52,870 And we'll see in the second that everything-- microscopes, 1454 01:14:52,870 --> 01:14:54,790 telescopes, magnifiers-- 1455 01:14:54,790 --> 01:14:59,710 they all form of virtual images which eventually the observer's 1456 01:14:59,710 --> 01:15:02,640 eye lens converts to a real image on the retina. 1457 01:15:07,240 --> 01:15:08,770 The next thing that we'll do now is 1458 01:15:08,770 --> 01:15:13,120 we'll put some math around this topic. 1459 01:15:13,120 --> 01:15:14,620 And this is a little bit unfortunate 1460 01:15:14,620 --> 01:15:18,180 because the book is generally very good, the textbook 1461 01:15:18,180 --> 01:15:20,050 that we have, Hecht. 1462 01:15:20,050 --> 01:15:22,180 I do think that in this particular topic, 1463 01:15:22,180 --> 01:15:24,590 the textbook is a little bit confusing. 1464 01:15:24,590 --> 01:15:27,610 So I'll try to walk you through the derivation in the textbook 1465 01:15:27,610 --> 01:15:32,190 and hopefully I will make it less confusing. 1466 01:15:32,190 --> 01:15:38,310 But if, when you go back home and you read the textbook, 1467 01:15:38,310 --> 01:15:41,460 you confused again, please bring your questions on Monday 1468 01:15:41,460 --> 01:15:44,730 so we can talk about this again. 1469 01:15:44,730 --> 01:15:47,310 So I tried to make it less confusing 1470 01:15:47,310 --> 01:15:50,070 but still I'm using the notation in the book. 1471 01:15:50,070 --> 01:15:53,520 So you don't-- in addition to the confusion in the book, 1472 01:15:53,520 --> 01:15:57,200 you don't have to overcome notational confusion. 1473 01:15:59,800 --> 01:16:04,720 So the definition of the magnifying power which cannot 1474 01:16:04,720 --> 01:16:08,810 be argued against because it is a definition is it is the ratio 1475 01:16:08,810 --> 01:16:13,870 of the angle subtended by the tip of the object before 1476 01:16:13,870 --> 01:16:15,610 and after magnification. 1477 01:16:15,610 --> 01:16:19,570 Actually the subscript here, U and A, U 1478 01:16:19,570 --> 01:16:23,800 stands for Unaided, that is the eye without the magnifier. 1479 01:16:23,800 --> 01:16:25,930 And A stands for Aided. 1480 01:16:25,930 --> 01:16:32,140 That is the eye with the aid of the magnifier lens. 1481 01:16:32,140 --> 01:16:40,280 And now we need to place symbols to all of these distances here. 1482 01:16:40,280 --> 01:16:42,530 And I think this is where the book is a bit confusing. 1483 01:16:42,530 --> 01:16:44,780 Because it tries to do everything at once. 1484 01:16:44,780 --> 01:16:47,030 But anyway, so I need this symbol 1485 01:16:47,030 --> 01:16:50,290 for the distance between the object of the lens. 1486 01:16:50,290 --> 01:16:53,760 Then I need the symbol from the lens to the eye. 1487 01:16:53,760 --> 01:16:57,230 And in the symbol from the lens to the virtual image, which 1488 01:16:57,230 --> 01:16:59,720 is of course si, and in this case 1489 01:16:59,720 --> 01:17:01,400 you would expect the side to turn out 1490 01:17:01,400 --> 01:17:04,340 to be a negative number, because it is a virtual image. 1491 01:17:04,340 --> 01:17:07,490 And finally you use the symbol L for the distance 1492 01:17:07,490 --> 01:17:12,920 between the virtual image and the eye. 1493 01:17:12,920 --> 01:17:15,710 So L is, of course, that simply the sum of these two distances 1494 01:17:15,710 --> 01:17:18,300 with a proper sign convention. 1495 01:17:24,130 --> 01:17:26,130 The first thing to note is that you don't really 1496 01:17:26,130 --> 01:17:30,150 need all these notations because the magnifying power is 1497 01:17:30,150 --> 01:17:32,790 very simple to calculate. 1498 01:17:32,790 --> 01:17:36,300 As we did in the past, all you have to do 1499 01:17:36,300 --> 01:17:39,000 is use similar triangles. 1500 01:17:39,000 --> 01:17:44,820 Because you can see that the magnifying power-- 1501 01:17:47,550 --> 01:17:51,390 the angle a sub a, the [INAUDIBLE] angle, 1502 01:17:51,390 --> 01:17:57,490 equals the ratio of the size of the virtual image over what? 1503 01:17:57,490 --> 01:18:00,240 Look at the red line. 1504 01:18:00,240 --> 01:18:04,317 From this ray angle, over the big distance L. 1505 01:18:04,317 --> 01:18:05,650 And of course there's a tangent. 1506 01:18:05,650 --> 01:18:06,700 I should have a tangent there. 1507 01:18:06,700 --> 01:18:08,200 But I dropped the tangent because 1508 01:18:08,200 --> 01:18:10,330 of the paraxial approximation. 1509 01:18:10,330 --> 01:18:16,210 Similarly for the unaided eye, it 1510 01:18:16,210 --> 01:18:19,990 equals the ratio of the natural height of the object 1511 01:18:19,990 --> 01:18:23,140 to the distance between the object and the lens. 1512 01:18:23,140 --> 01:18:28,150 Now for the unaided case, the convention is to use the near 1513 01:18:28,150 --> 01:18:32,230 point of the eye, that is the 25 centimeter distance beyond 1514 01:18:32,230 --> 01:18:33,550 which the eye cannot focus. 1515 01:18:33,550 --> 01:18:35,770 Because presumably, if you wanted 1516 01:18:35,770 --> 01:18:39,232 to get the maximum possible unaided magnification, 1517 01:18:39,232 --> 01:18:40,690 you really have to bring the object 1518 01:18:40,690 --> 01:18:43,780 within this 25 centimeters. 1519 01:18:43,780 --> 01:18:45,880 So that's why we use d0. 1520 01:18:45,880 --> 01:18:47,200 So that's a standard actually. 1521 01:18:47,200 --> 01:18:49,600 It is 25 centimeters. 1522 01:18:49,600 --> 01:18:51,350 And when you play around with this, 1523 01:18:51,350 --> 01:18:53,830 you discover that it equals the ratio of these two 1524 01:18:53,830 --> 01:18:57,140 distances, which is easy to do. 1525 01:18:57,140 --> 01:18:59,150 And this is what we used to call-- 1526 01:18:59,150 --> 01:19:01,250 what did we call this quantity when 1527 01:19:01,250 --> 01:19:04,730 we did the imaging condition? 1528 01:19:07,610 --> 01:19:08,570 Button. 1529 01:19:08,570 --> 01:19:10,060 AUDIENCE: Lateral magnification. 1530 01:19:10,060 --> 01:19:11,060 PROFESSOR: That's right. 1531 01:19:11,060 --> 01:19:13,730 This is what we used to call the lateral magnification. 1532 01:19:13,730 --> 01:19:16,360 It is the ratio of the size of the image 1533 01:19:16,360 --> 01:19:19,970 over the size of the object. 1534 01:19:19,970 --> 01:19:24,250 So this is what they say here. 1535 01:19:24,250 --> 01:19:26,820 And of course this is the lateral magnification. 1536 01:19:26,820 --> 01:19:28,510 We have the questions for it in terms 1537 01:19:28,510 --> 01:19:30,760 of the image and distances. 1538 01:19:30,760 --> 01:19:35,920 And we can substitute these equations 1539 01:19:35,920 --> 01:19:37,450 and we can actually derive a number 1540 01:19:37,450 --> 01:19:40,970 of equivalent expressions for the magnifying power. 1541 01:19:40,970 --> 01:19:43,870 So now, none of this is particularly intuitive. 1542 01:19:43,870 --> 01:19:46,210 And then the book goes ahead and gives two 1543 01:19:46,210 --> 01:19:49,450 possible uses of the magnifying lens 1544 01:19:49,450 --> 01:19:52,750 that are not particularly intuitive either. 1545 01:19:52,750 --> 01:19:55,130 So I will not even bother to deal with them. 1546 01:19:55,130 --> 01:19:57,800 One of them is when you put the magnifier 1547 01:19:57,800 --> 01:19:59,380 one focal distance away. 1548 01:20:03,230 --> 01:20:05,210 I don't know if it's necessary to do that. 1549 01:20:05,210 --> 01:20:08,100 The other is when you put the magnifier-- 1550 01:20:08,100 --> 01:20:10,463 you stick it to your eye, which is not a very good idea 1551 01:20:10,463 --> 01:20:11,880 because there's all kinds of germs 1552 01:20:11,880 --> 01:20:14,710 around so you don't want to stick magnifiers 1553 01:20:14,710 --> 01:20:17,200 onto your body. 1554 01:20:17,200 --> 01:20:20,760 The way we use magnifiers in general 1555 01:20:20,760 --> 01:20:24,960 is the number three description of the book, which 1556 01:20:24,960 --> 01:20:27,690 is why we place the object at one focal point 1557 01:20:27,690 --> 01:20:29,640 behind the magnifier. 1558 01:20:29,640 --> 01:20:34,228 So if the magnifying focal length is 5 centimeters-- 1559 01:20:34,228 --> 01:20:35,520 actually, that's a pretty long. 1560 01:20:35,520 --> 01:20:39,240 But anyway, you place it 5 centimeters from the object. 1561 01:20:39,240 --> 01:20:42,120 What do you accomplish this way? 1562 01:20:42,120 --> 01:20:45,260 What happens if you place an object at one focal distance 1563 01:20:45,260 --> 01:20:48,370 in front of the lens. 1564 01:20:48,370 --> 01:20:49,210 That's right. 1565 01:20:49,210 --> 01:20:50,680 You form an image at infinity. 1566 01:20:50,680 --> 01:20:53,950 Actually, in that case, you form a real image with infinity. 1567 01:20:53,950 --> 01:20:54,950 Now is that good or bad? 1568 01:20:59,490 --> 01:21:02,640 In the context here, you form the image with infinity 1569 01:21:02,640 --> 01:21:06,810 and you still have an observer with his or her eye 1570 01:21:06,810 --> 01:21:07,680 lens and the retina. 1571 01:21:10,810 --> 01:21:11,980 Let me draw this. 1572 01:21:16,720 --> 01:21:20,190 Now we draw the specific case, according to the book lingo, 1573 01:21:20,190 --> 01:21:25,140 this is case number three, si equals F. 1574 01:21:25,140 --> 01:21:26,380 So what does it look like? 1575 01:21:26,380 --> 01:21:27,600 Here's my magnifier. 1576 01:21:31,160 --> 01:21:36,110 Here is my eye lens. 1577 01:21:36,110 --> 01:21:38,982 And of course the eye kind of looks like this. 1578 01:21:38,982 --> 01:21:39,940 And this is the retina. 1579 01:21:44,410 --> 01:21:48,490 And here is the optical axis. 1580 01:21:48,490 --> 01:21:51,825 And here is the object. 1581 01:21:51,825 --> 01:21:53,450 So let's look at the tip of the object. 1582 01:21:53,450 --> 01:21:59,000 Since it is one focal distance away from the lens 1583 01:21:59,000 --> 01:22:01,225 I will get a parallel ray bundle. 1584 01:22:05,530 --> 01:22:07,240 Is that good or bad? 1585 01:22:07,240 --> 01:22:08,980 AUDIENCE: It's s0 equals F, right? 1586 01:22:12,144 --> 01:22:12,890 PROFESSOR: Yes. 1587 01:22:12,890 --> 01:22:13,390 Thank you. 1588 01:22:19,792 --> 01:22:20,750 So is that good or bad? 1589 01:22:23,516 --> 01:22:26,995 AUDIENCE: If we talk of aberrations, it's good. 1590 01:22:26,995 --> 01:22:27,810 PROFESSOR: Yeah. 1591 01:22:27,810 --> 01:22:30,100 Let's not go into aberrations. 1592 01:22:30,100 --> 01:22:32,040 In our simple geometrical optics, 1593 01:22:32,040 --> 01:22:34,140 is that situation happy? 1594 01:22:34,140 --> 01:22:37,012 Are we happy that we did this? 1595 01:22:37,012 --> 01:22:39,115 We're happy for a number of reasons. 1596 01:22:39,115 --> 01:22:40,000 AUDIENCE: Yes. 1597 01:22:40,000 --> 01:22:43,380 Because it will perfectly focus at the point. 1598 01:22:43,380 --> 01:22:44,380 PROFESSOR: That's right. 1599 01:22:44,380 --> 01:22:49,340 First of all, this is assumed to be a corrected eye. 1600 01:22:49,340 --> 01:22:55,710 So if it was me, that would be me wearing my eyeglasses. 1601 01:22:55,710 --> 01:22:58,020 If it is me without my eyeglasses, I have a problem. 1602 01:22:58,020 --> 01:22:59,980 You'll see in a second what problem. 1603 01:22:59,980 --> 01:23:06,135 But if it is a healthy eyesight where 1604 01:23:06,135 --> 01:23:08,010 the eye can focus perfectly, then this indeed 1605 01:23:08,010 --> 01:23:09,420 will form an image on the retina. 1606 01:23:12,930 --> 01:23:17,130 What is the other good news about this situation? 1607 01:23:17,130 --> 01:23:19,740 The fact that the image is at infinity, 1608 01:23:19,740 --> 01:23:20,990 is this good news or bad news? 1609 01:23:26,138 --> 01:23:27,990 AUDIENCE: Your eyes remain-- 1610 01:23:27,990 --> 01:23:29,925 your eye remains unaccommodated. 1611 01:23:29,925 --> 01:23:31,920 PROFESSOR: That's right. 1612 01:23:31,920 --> 01:23:37,150 We learned that the eye is most relaxed when it observes 1613 01:23:37,150 --> 01:23:39,790 remote objects at infinity. 1614 01:23:39,790 --> 01:23:42,330 The reason it cannot focus closer is because I have 1615 01:23:42,330 --> 01:23:44,690 to strain the lens of my eye. 1616 01:23:44,690 --> 01:23:49,140 And as I get older, my ability to strain it diminishes. 1617 01:23:49,140 --> 01:23:51,888 But this is true for everybody, young or old. 1618 01:23:51,888 --> 01:23:53,680 When you try to look at something close by, 1619 01:23:53,680 --> 01:23:55,260 you get tired. 1620 01:23:55,260 --> 01:23:58,080 If you read a book like this, your parents probably 1621 01:23:58,080 --> 01:23:59,080 told you-- they told me. 1622 01:23:59,080 --> 01:24:01,247 Don't read the book like this, you get the headache. 1623 01:24:01,247 --> 01:24:02,080 And indeed you do. 1624 01:24:02,080 --> 01:24:06,300 If you read for a long time you get a headache from very close. 1625 01:24:06,300 --> 01:24:08,970 So the reason this is a happy situation 1626 01:24:08,970 --> 01:24:12,180 is because the eye can be totally relaxed 1627 01:24:12,180 --> 01:24:15,650 as if it was observing something very far away 1628 01:24:15,650 --> 01:24:19,940 and still see the object with magnification. 1629 01:24:19,940 --> 01:24:21,800 And how much is the magnification? 1630 01:24:21,800 --> 01:24:24,240 Well, we'll skip the math. 1631 01:24:24,240 --> 01:24:25,850 The math is fairly simple here. 1632 01:24:25,850 --> 01:24:29,960 But it turns out to be simply the product of the near point, 1633 01:24:29,960 --> 01:24:34,670 the 25 centimeters, times their power of the lens. 1634 01:24:34,670 --> 01:24:37,010 And the only penalty you pay, the reason, 1635 01:24:37,010 --> 01:24:39,670 actually, the book covers this case over here 1636 01:24:39,670 --> 01:24:42,440 is because this is not the maximum magnification you 1637 01:24:42,440 --> 01:24:44,090 can get with a magnifier. 1638 01:24:44,090 --> 01:24:47,550 You can get slightly higher if you stick the magnifier 1639 01:24:47,550 --> 01:24:48,270 next to your eye. 1640 01:24:51,253 --> 01:24:53,170 But you never stick the magnifier to your eye, 1641 01:24:53,170 --> 01:24:53,740 actually. 1642 01:24:53,740 --> 01:24:56,524 Unless you're adventurous. 1643 01:24:59,710 --> 01:25:01,960 If I did this, I would damage my glasses. 1644 01:25:06,480 --> 01:25:08,760 So the message, perhaps, important. 1645 01:25:08,760 --> 01:25:13,800 But what I really hope that you get out of this 1646 01:25:13,800 --> 01:25:17,280 is the significance of the virtual image 1647 01:25:17,280 --> 01:25:21,600 and why the magnifier functions by forming a virtual image that 1648 01:25:21,600 --> 01:25:23,940 is erect and magnified. 1649 01:25:23,940 --> 01:25:28,880 And this also tells you why you will use a positive lens 1650 01:25:28,880 --> 01:25:30,410 as a magnifier. 1651 01:25:30,410 --> 01:25:33,610 If you recall, it is actually one of the previous slides. 1652 01:25:33,610 --> 01:25:35,870 A negative lens also forms a virtual image. 1653 01:25:35,870 --> 01:25:38,740 But unfortunately, it is not magnified. 1654 01:25:38,740 --> 01:25:41,570 The magnification, it is de-magnified. 1655 01:25:41,570 --> 01:25:44,110 The lateral magnification is less than 1. 1656 01:25:44,110 --> 01:25:46,280 So we would be out of luck in this case. 1657 01:25:58,040 --> 01:26:03,000 The next element is actually the same. 1658 01:26:03,000 --> 01:26:05,270 It is no different than the magnifier. 1659 01:26:05,270 --> 01:26:08,060 But it comes under the name eyepiece 1660 01:26:08,060 --> 01:26:13,640 because in optical instruments such as binoculars, telescopes, 1661 01:26:13,640 --> 01:26:18,680 microscopes, and the like this is the last lens 1662 01:26:18,680 --> 01:26:23,840 that you look into the instrument with your naked eye. 1663 01:26:23,840 --> 01:26:27,410 Now I realize that this is old fashioned talk. 1664 01:26:27,410 --> 01:26:31,640 Modern instruments, you seldom look at with a naked eye. 1665 01:26:31,640 --> 01:26:33,500 You actually have a camera. 1666 01:26:33,500 --> 01:26:36,050 So you put a digital camera and you register the image 1667 01:26:36,050 --> 01:26:37,070 digitally. 1668 01:26:37,070 --> 01:26:39,670 However, even a digital camera functions like a retina. 1669 01:26:39,670 --> 01:26:41,810 So in other words, even a digital camera 1670 01:26:41,810 --> 01:26:45,140 would have an eyepiece and then a condenser. 1671 01:26:45,140 --> 01:26:47,760 And then it would go onto the chip. 1672 01:26:47,760 --> 01:26:51,950 So really the situation here applies to digital imaging 1673 01:26:51,950 --> 01:26:56,540 as well as human, as well as instruments 1674 01:26:56,540 --> 01:26:59,820 meant to be used by humans. 1675 01:26:59,820 --> 01:27:06,150 So the eyepiece is basically the last element in the instrument, 1676 01:27:06,150 --> 01:27:12,710 just before you get to the actual observation stage. 1677 01:27:12,710 --> 01:27:16,580 And because the eyepieces were traditionally 1678 01:27:16,580 --> 01:27:20,180 designed for the humans, they are typically 1679 01:27:20,180 --> 01:27:23,100 designed like a magnifier with the same principle 1680 01:27:23,100 --> 01:27:27,300 that I described before that forms an image of infinity. 1681 01:27:27,300 --> 01:27:29,900 So therefore the observer, when he or she 1682 01:27:29,900 --> 01:27:35,510 looks into the eyepiece, they can use an unaccommodated eye 1683 01:27:35,510 --> 01:27:40,010 and observe the magnified image without having 1684 01:27:40,010 --> 01:27:42,110 to strain their eyes. 1685 01:27:42,110 --> 01:27:48,980 So this is the sort of standard use of an eyepiece. 1686 01:27:48,980 --> 01:27:54,350 And there's a second condition for the eyepiece which 1687 01:27:54,350 --> 01:28:02,750 will become a little bit more apparent in the next slide 1688 01:28:02,750 --> 01:28:05,000 when I talk about the microscope. 1689 01:28:05,000 --> 01:28:06,290 It is this one over here. 1690 01:28:08,997 --> 01:28:10,330 This looks a little bit cryptic. 1691 01:28:10,330 --> 01:28:13,810 It says the center of the exit pupil eyepoint 1692 01:28:13,810 --> 01:28:17,957 where the observer's eye is placed at 10 millimeters 1693 01:28:17,957 --> 01:28:18,790 from the instrument. 1694 01:28:18,790 --> 01:28:19,748 This doesn't mean much. 1695 01:28:19,748 --> 01:28:23,260 But I will describe what this means in a second, when 1696 01:28:23,260 --> 01:28:24,520 I talk about the microscope. 1697 01:28:24,520 --> 01:28:28,600 Not withstanding this last point over here, any questions 1698 01:28:28,600 --> 01:28:31,328 about the eyepiece or the magnifier or anything 1699 01:28:31,328 --> 01:28:31,870 of that sort? 1700 01:28:41,156 --> 01:28:44,780 AUDIENCE: Why is the eye [INAUDIBLE] 10 millimeter when 1701 01:28:44,780 --> 01:28:45,620 we cannot see? 1702 01:28:47,877 --> 01:28:48,460 PROFESSOR: OK. 1703 01:28:48,460 --> 01:28:52,290 What this really means is that 10 millimeters is the design 1704 01:28:52,290 --> 01:28:55,420 distance that you're supposed to place your eyes with respect 1705 01:28:55,420 --> 01:28:57,340 to the instrument. 1706 01:28:57,340 --> 01:29:00,087 When you look at the microscope, for example, most people 1707 01:29:00,087 --> 01:29:00,670 don't do that. 1708 01:29:00,670 --> 01:29:02,950 But you're supposed to be at about 10 centimeters 1709 01:29:02,950 --> 01:29:06,430 from the last surface, right? 1710 01:29:06,430 --> 01:29:10,030 So that is called the eye relief. 1711 01:29:10,030 --> 01:29:13,660 And what this is really saying is that when you look at this, 1712 01:29:13,660 --> 01:29:19,540 you're not supposed to see any blockage of the image. 1713 01:29:19,540 --> 01:29:22,452 And you're not supposed to see vignetting. 1714 01:29:22,452 --> 01:29:23,910 There's two things that can happen. 1715 01:29:23,910 --> 01:29:26,202 If you place your eye at the wrong point-- as you know, 1716 01:29:26,202 --> 01:29:28,340 if any of you have looked at a microscope, 1717 01:29:28,340 --> 01:29:31,380 if you are not positioned properly you cannot see 1718 01:29:31,380 --> 01:29:32,460 the entire field of view. 1719 01:29:32,460 --> 01:29:33,360 That's one problem. 1720 01:29:33,360 --> 01:29:35,412 You see a black hole. 1721 01:29:35,412 --> 01:29:36,870 And you kind of have to move around 1722 01:29:36,870 --> 01:29:38,550 and you can never see anything until you 1723 01:29:38,550 --> 01:29:41,847 go to the proper place and then finally you see everything. 1724 01:29:41,847 --> 01:29:42,930 So this is the eye relief. 1725 01:29:42,930 --> 01:29:45,927 This is by design where you should place your eye. 1726 01:29:45,927 --> 01:29:47,760 The other thing that can happen if you place 1727 01:29:47,760 --> 01:29:50,220 your eye at the wrong place or if the microscope is not 1728 01:29:50,220 --> 01:29:52,200 properly designed is vignetting. 1729 01:29:52,200 --> 01:29:55,080 So this requirement clearly assures 1730 01:29:55,080 --> 01:29:59,210 that there's no vignetting when your eye is placed 1731 01:29:59,210 --> 01:30:01,950 at eye relief distance, that is at 10 millimeters 1732 01:30:01,950 --> 01:30:03,090 from the instrument. 1733 01:30:03,090 --> 01:30:04,020 And of course, you don't have to worry 1734 01:30:04,020 --> 01:30:06,660 about focusing because the microscope is supposed to have 1735 01:30:06,660 --> 01:30:08,130 infinite conjugate, right? 1736 01:30:08,130 --> 01:30:11,790 So you just use with an accommodated eye. 1737 01:30:11,790 --> 01:30:13,857 And I suspect, I don't know if it's true, 1738 01:30:13,857 --> 01:30:16,190 but I suspect-- there are many reasons why they put over 1739 01:30:16,190 --> 01:30:16,670 there. 1740 01:30:16,670 --> 01:30:18,837 But also, it's kind of convenient that when you look 1741 01:30:18,837 --> 01:30:21,830 at the microscope you're so close to the microscope tube 1742 01:30:21,830 --> 01:30:24,360 itself that you cannot focus and you cannot see that. 1743 01:30:24,360 --> 01:30:25,860 So you can basically see what you're 1744 01:30:25,860 --> 01:30:28,318 supposed to observe but you don't see the actual instrument 1745 01:30:28,318 --> 01:30:29,730 because it is out of focus. 1746 01:30:29,730 --> 01:30:30,965 So that's kind of convenient. 1747 01:30:33,580 --> 01:30:35,480 So [INAUDIBLE] this about the microscope. 1748 01:30:35,480 --> 01:30:36,730 Here is a microscope. 1749 01:30:36,730 --> 01:30:39,340 Here's what it looks like. 1750 01:30:39,340 --> 01:30:40,440 It is deceptively simple. 1751 01:30:40,440 --> 01:30:43,620 Because there's, of course, a lot of engineering design 1752 01:30:43,620 --> 01:30:46,410 that goes into making all of these elements. 1753 01:30:46,410 --> 01:30:48,420 But the principle is surprisingly simple. 1754 01:30:48,420 --> 01:30:51,180 It consists of basically two magnifiers. 1755 01:30:51,180 --> 01:30:53,840 One of them is called objective. 1756 01:30:53,840 --> 01:30:55,230 The other is called eyepiece. 1757 01:30:55,230 --> 01:30:58,650 But they both function as magnifiers. 1758 01:30:58,650 --> 01:31:02,710 So the job of the objective, this is the first lens. 1759 01:31:02,710 --> 01:31:06,390 It is called objective because it is placed near the object. 1760 01:31:06,390 --> 01:31:12,120 And its job is basically to form an intermediate image that 1761 01:31:12,120 --> 01:31:14,550 is magnified to begin with. 1762 01:31:14,550 --> 01:31:15,990 What is it magnified by? 1763 01:31:15,990 --> 01:31:19,020 Well, here we simply have-- 1764 01:31:19,020 --> 01:31:20,670 actually this is a real image. 1765 01:31:20,670 --> 01:31:23,980 So we simply have a positive lens forming a real image. 1766 01:31:23,980 --> 01:31:26,100 We can compute the magnification. 1767 01:31:26,100 --> 01:31:34,340 It is equal to minus si over s0, just like the regular formula. 1768 01:31:34,340 --> 01:31:39,170 And then we have the eyepiece for which 1769 01:31:39,170 --> 01:31:44,000 the image that the objective produced actually now 1770 01:31:44,000 --> 01:31:45,980 acts as the object. 1771 01:31:45,980 --> 01:31:50,180 So this intermediate image is re-imaged by the eyepiece. 1772 01:31:50,180 --> 01:31:52,980 But the eyepiece is placed one focal distance. 1773 01:31:52,980 --> 01:31:56,960 F sub e here is the focal length of the eyepiece. 1774 01:31:56,960 --> 01:32:00,530 It is one focal distance from that intermediate image. 1775 01:32:00,530 --> 01:32:06,310 And therefore the final image is formed at infinity. 1776 01:32:06,310 --> 01:32:12,370 Now the way this is drawn for this one, what 1777 01:32:12,370 --> 01:32:16,480 is prominent in this picture is, of course, the on-axis ray. 1778 01:32:16,480 --> 01:32:19,330 And there's no magnification to talk about 1779 01:32:19,330 --> 01:32:20,830 because this is just an axis. 1780 01:32:20,830 --> 01:32:22,270 Nothing interesting is happening. 1781 01:32:22,270 --> 01:32:23,770 What is really interesting is if you 1782 01:32:23,770 --> 01:32:26,680 look at an off-axis point, which is at the bottom here. 1783 01:32:26,680 --> 01:32:28,650 It's a little bit difficult to see. 1784 01:32:28,650 --> 01:32:30,882 It looks better in the book actually. 1785 01:32:30,882 --> 01:32:34,200 So if you trace this off-axis point, 1786 01:32:34,200 --> 01:32:36,330 then you find that its final image 1787 01:32:36,330 --> 01:32:40,860 is this ray bundle that is coming off-axis 1788 01:32:40,860 --> 01:32:44,820 into your eye, this parallel ray. 1789 01:32:44,820 --> 01:32:48,020 That is the final image of the microscope produced 1790 01:32:48,020 --> 01:32:50,580 of this off-axis objective point. 1791 01:32:50,580 --> 01:32:53,700 You can see it has entered the eye at a very large angle. 1792 01:32:53,700 --> 01:32:56,120 Therefore it will form a big image. 1793 01:32:56,120 --> 01:32:59,632 So that's how the microscope magnifies small objects 1794 01:32:59,632 --> 01:33:00,840 so that you can observe them. 1795 01:33:05,060 --> 01:33:09,490 So then the magnification of the microscope, 1796 01:33:09,490 --> 01:33:11,860 very similar to the case of the objective, 1797 01:33:11,860 --> 01:33:15,610 is defined as the ratio of this angle, the angle 1798 01:33:15,610 --> 01:33:20,350 that the microscope produced for the off-axis object, 1799 01:33:20,350 --> 01:33:28,820 for the same angle that the eye would observe unaided, 1800 01:33:28,820 --> 01:33:32,210 in a very similar case as the magnifier. 1801 01:33:32,210 --> 01:33:36,450 So in order to compute this now, it's not very difficult to do. 1802 01:33:36,450 --> 01:33:37,980 We need two elements. 1803 01:33:37,980 --> 01:33:40,250 One is the lateral magnification of the objective. 1804 01:33:40,250 --> 01:33:43,850 Because recall, the objective actually forms a real image. 1805 01:33:43,850 --> 01:33:45,860 So we have one. 1806 01:33:45,860 --> 01:33:47,240 Here is the object. 1807 01:33:47,240 --> 01:33:48,950 And here is its image. 1808 01:33:48,950 --> 01:33:50,600 So we need, first of all, to find out 1809 01:33:50,600 --> 01:33:54,710 where the intermediate image appears with respect 1810 01:33:54,710 --> 01:33:55,880 to the optical axis. 1811 01:33:55,880 --> 01:33:58,920 That's the lateral magnification of the objective. 1812 01:33:58,920 --> 01:34:02,630 That is whatever. 1813 01:34:02,630 --> 01:34:06,660 Well, I will say in the second why this 160 appears. 1814 01:34:06,660 --> 01:34:12,260 But the second element that we need 1815 01:34:12,260 --> 01:34:15,380 is the angular magnification of the eyepiece. 1816 01:34:15,380 --> 01:34:20,180 Because this will allow us to say how this distance over here 1817 01:34:20,180 --> 01:34:26,150 translates into angle that goes into the eye finally. 1818 01:34:26,150 --> 01:34:30,330 So this one is actually a familiar formula. 1819 01:34:30,330 --> 01:34:33,300 This is the same formula you had for the magnifier. 1820 01:34:33,300 --> 01:34:35,570 So go back one. 1821 01:34:35,570 --> 01:34:39,460 We said that-- actually, it is over here already. 1822 01:34:39,460 --> 01:34:40,940 We said that the magnifying power 1823 01:34:40,940 --> 01:34:45,950 of an eyepiece or a magnifier equals the near distance 1824 01:34:45,950 --> 01:34:51,950 for the eye, which is about 25 centimeters, times the power 1825 01:34:51,950 --> 01:34:54,690 of the lens, of the positive lens that they use. 1826 01:34:54,690 --> 01:34:56,140 So this is the same formula. 1827 01:34:56,140 --> 01:34:58,880 In microscopes, they don't use exactly 25 centimeters. 1828 01:34:58,880 --> 01:35:02,120 They use 25.4 centimeters. 1829 01:35:02,120 --> 01:35:03,318 So this is the standard. 1830 01:35:03,318 --> 01:35:04,610 And how much is that in inches? 1831 01:35:04,610 --> 01:35:05,660 Is it 10 inches? 1832 01:35:05,660 --> 01:35:07,090 No. 1833 01:35:07,090 --> 01:35:09,220 11, 12 inches. 1834 01:35:09,220 --> 01:35:10,510 10 inches. 1835 01:35:10,510 --> 01:35:13,070 I thought it would be 22-- oh, that's pounds. 1836 01:35:13,070 --> 01:35:13,570 I'm sorry. 1837 01:35:13,570 --> 01:35:16,440 That's pounds. 1838 01:35:16,440 --> 01:35:19,400 There's 10 inches then. 1839 01:35:19,400 --> 01:35:21,350 So this is, then, the same formula 1840 01:35:21,350 --> 01:35:23,700 that we had for the magnifier. 1841 01:35:23,700 --> 01:35:25,420 Where did this come from? 1842 01:35:25,420 --> 01:35:27,600 Well, if you remember one of the ways 1843 01:35:27,600 --> 01:35:34,010 to add the lateral magnification is like this-- 1844 01:35:34,010 --> 01:35:42,340 1 minus si over F. So you can write this as 1 minus si 1845 01:35:42,340 --> 01:35:50,320 over F equals F minus si over F. And now I have to-- 1846 01:35:55,120 --> 01:35:56,230 Ah, that worked. 1847 01:35:59,020 --> 01:36:00,210 So minus si-- 1848 01:36:00,210 --> 01:36:01,600 I am sorry. 1849 01:36:01,600 --> 01:36:10,650 It is F minus si over F. So F minus si 1850 01:36:10,650 --> 01:36:23,066 is actually the distance between the objective lens and the-- 1851 01:36:25,732 --> 01:36:27,860 it's basically this distance L. That's 1852 01:36:27,860 --> 01:36:29,050 what I'm trying to get to. 1853 01:36:29,050 --> 01:36:31,700 So this is known as the tube length of the microscope. 1854 01:36:31,700 --> 01:36:36,290 It is the distance between the objective lens and the location 1855 01:36:36,290 --> 01:36:38,977 where you put the eyepiece. 1856 01:36:38,977 --> 01:36:40,810 Of course, the eyepiece is this whole thing. 1857 01:36:40,810 --> 01:36:42,580 So that's where you stick the eyepiece. 1858 01:36:42,580 --> 01:36:43,460 So this is standard. 1859 01:36:43,460 --> 01:36:46,370 In microscopes, it is 160 millimeters 1860 01:36:46,370 --> 01:36:48,100 and is known as the tube length. 1861 01:36:48,100 --> 01:36:51,990 And this is what appears in this formula over here. 1862 01:36:51,990 --> 01:36:55,800 So basically, you can see that if you pick the focal length 1863 01:36:55,800 --> 01:36:58,050 to be sufficiently small, then you 1864 01:36:58,050 --> 01:37:01,590 can get very high magnification in the order of hundreds, maybe 1865 01:37:01,590 --> 01:37:06,640 even 1,000. 1866 01:37:06,640 --> 01:37:08,720 Let me ask two questions now. 1867 01:37:08,720 --> 01:37:15,710 First of all, what I just said. 1868 01:37:15,710 --> 01:37:19,370 What do you think might stop me from getting a focal length 1869 01:37:19,370 --> 01:37:23,050 that is, let's say, 1 micron? 1870 01:37:23,050 --> 01:37:25,650 I pick a focal length to be 1 micron, and then I have-- 1871 01:37:25,650 --> 01:37:27,350 these are all in millimeters. 1872 01:37:27,350 --> 01:37:30,380 So 160 millimeters over 1 micron, that would be what? 1873 01:37:30,380 --> 01:37:33,980 That would be about 160,000. 1874 01:37:33,980 --> 01:37:36,920 And then another micron here, I would get a magnification 1875 01:37:36,920 --> 01:37:38,720 in the order of 1 billion. 1876 01:37:38,720 --> 01:37:39,970 What stops me from doing that? 1877 01:37:43,140 --> 01:37:44,570 First of all, just to get back to, 1878 01:37:44,570 --> 01:37:46,910 what is it from your experience? 1879 01:37:46,910 --> 01:37:50,825 What is a typical magnification that you get in microscopes? 1880 01:37:50,825 --> 01:37:53,880 How about 1,000 orders of magnitude? 1881 01:37:53,880 --> 01:37:56,060 But if you apply this formula here, 1882 01:37:56,060 --> 01:37:58,860 you could easily get a magnification of 1 billion 1883 01:37:58,860 --> 01:38:01,760 by picking the focal length to be 1 micron. 1884 01:38:01,760 --> 01:38:04,132 What is wrong with my statement? 1885 01:38:04,132 --> 01:38:05,090 Clearly, there's wrong. 1886 01:38:05,090 --> 01:38:07,605 Otherwise, people would be doing it. 1887 01:38:07,605 --> 01:38:09,740 AUDIENCE: Is it because you can't see 1888 01:38:09,740 --> 01:38:11,800 past the wave length of light? 1889 01:38:11,800 --> 01:38:13,307 That 1 micron is the most you could 1890 01:38:13,307 --> 01:38:14,390 focus because that's the-- 1891 01:38:19,010 --> 01:38:21,170 PROFESSOR: That is in the correct direction, yes. 1892 01:38:28,410 --> 01:38:30,380 The other possible reason is the failure 1893 01:38:30,380 --> 01:38:32,320 of the paraxial approximation. 1894 01:38:32,320 --> 01:38:36,810 But the paraxial approximation may fail all at once. 1895 01:38:36,810 --> 01:38:40,630 But I can always do something more sophisticated to do 1896 01:38:40,630 --> 01:38:42,848 non-paraxial calculations. 1897 01:38:42,848 --> 01:38:45,390 Maybe you're saying that if the paraxial approximation fails, 1898 01:38:45,390 --> 01:38:47,750 then this formula fails as well. 1899 01:38:47,750 --> 01:38:50,400 So then you don't get as good magnification. 1900 01:38:50,400 --> 01:38:53,740 I think you guys both captured-- the two of you 1901 01:38:53,740 --> 01:38:56,790 captured the answer in its entirety You have another one? 1902 01:38:56,790 --> 01:38:59,080 AUDIENCE: I don't know if it's another. 1903 01:38:59,080 --> 01:39:01,960 To make such a high-- 1904 01:39:01,960 --> 01:39:04,870 such a short focal length lens we'll 1905 01:39:04,870 --> 01:39:07,170 have to make a very big lens. 1906 01:39:07,170 --> 01:39:08,690 PROFESSOR: That's right. 1907 01:39:08,690 --> 01:39:10,300 So you have to be very highly curved. 1908 01:39:10,300 --> 01:39:12,640 Remember the focal length of the lens 1909 01:39:12,640 --> 01:39:18,820 is proportional to the radius of curvature. 1910 01:39:18,820 --> 01:39:21,030 So you would have to make a lens with a radius 1911 01:39:21,030 --> 01:39:23,460 of curvature equal to 1 micron. 1912 01:39:23,460 --> 01:39:26,140 And that's when you get into problems like the-- 1913 01:39:26,140 --> 01:39:28,510 what's your name again? 1914 01:39:28,510 --> 01:39:31,825 [INAUDIBLE] answered the question before. 1915 01:39:31,825 --> 01:39:32,980 AUDIENCE: Dean. 1916 01:39:32,980 --> 01:39:33,920 PROFESSOR: Yeah. 1917 01:39:33,920 --> 01:39:37,730 This is what Dean answered. 1918 01:39:37,730 --> 01:39:43,360 If you try to make a curvature that is as sharp as 1 micron, 1919 01:39:43,360 --> 01:39:46,960 then basically you get into the range 1920 01:39:46,960 --> 01:39:49,413 of very strong diffraction of light. 1921 01:39:49,413 --> 01:39:50,830 So basically all these geometrical 1922 01:39:50,830 --> 01:39:52,390 optics that we do here, they fail. 1923 01:39:52,390 --> 01:39:55,870 In fact, you discover if you did make such a very small element 1924 01:39:55,870 --> 01:39:58,630 that the light does not really focus. 1925 01:39:58,630 --> 01:39:59,740 We'll see what it does. 1926 01:39:59,740 --> 01:40:01,850 But it does not propagate. 1927 01:40:01,850 --> 01:40:07,510 So reasonable focal lengths are in the range of 1 centimeter, 1928 01:40:07,510 --> 01:40:08,410 a few millimeters. 1929 01:40:08,410 --> 01:40:11,660 These are typical microscope objective focal length. 1930 01:40:11,660 --> 01:40:12,910 And the eyepieces are similar. 1931 01:40:12,910 --> 01:40:14,660 And that's why you get magnification 1932 01:40:14,660 --> 01:40:19,047 on the order of 1,000. 1933 01:40:19,047 --> 01:40:20,380 The next topic is the telescope. 1934 01:40:20,380 --> 01:40:24,220 But I think I've exhausted your endurance.