1 00:00:00,000 --> 00:00:24,000 2 00:00:24,000 --> 00:00:28,062 Last time we discussed the interference pattern of two 3 00:00:28,062 --> 00:00:31,818 openings in the screen, which established the wave 4 00:00:31,818 --> 00:00:35,344 character of light. Today, I am going to expand 5 00:00:35,344 --> 00:00:39,101 that to capital N. And we will make N thousands of 6 00:00:39,101 --> 00:00:42,933 openings in a screen. And, when it comes to optical 7 00:00:42,933 --> 00:00:48,745 light, we call those gratings. Suppose here each one of those 8 00:00:48,745 --> 00:00:51,949 dots is a small opening in a screen. 9 00:00:51,949 --> 00:00:57,715 It could be a little hole or it could be a slit perpendicular to 10 00:00:57,715 --> 00:01:01,650 the blackboard. And imagine that plane waves 11 00:01:01,650 --> 00:01:05,128 are coming in. And so each one of those 12 00:01:05,128 --> 00:01:10,711 openings are going to be Huygens sources and they are going to 13 00:01:10,711 --> 00:01:15,824 radiate waves. And the question now that we 14 00:01:15,824 --> 00:01:22,078 want to answer is if we look in a direction theta away from the 15 00:01:22,078 --> 00:01:26,517 normal to that screen, what is then the light 16 00:01:26,517 --> 00:01:32,671 intensity that you will see as a result of the interference of 17 00:01:32,671 --> 00:01:38,329 all these Huygens sources? Suppose that the separation 18 00:01:38,329 --> 00:01:41,865 between two adjacent Huygens sources is d. 19 00:01:41,865 --> 00:01:45,488 In other words, here is one and here is the 20 00:01:45,488 --> 00:01:48,507 other. This is a blow-up of what you 21 00:01:48,507 --> 00:01:51,095 see here. Here you see hundred, 22 00:01:51,095 --> 00:01:54,287 maybe thousands. Here I have only two. 23 00:01:54,287 --> 00:01:59,376 And, if that separation is d, then I can calculate the phase 24 00:01:59,376 --> 00:02:04,505 difference between those two. If this is theta, 25 00:02:04,505 --> 00:02:09,286 this is no different from what we did last time for the 26 00:02:09,286 --> 00:02:14,244 double-slit, then the past difference from this spherical 27 00:02:14,244 --> 00:02:19,557 wave to a point far away at an angle theta and this spherical 28 00:02:19,557 --> 00:02:22,921 wave is this. And this is d sine theta. 29 00:02:22,921 --> 00:02:27,348 And so, we will introduce, just like we did before, 30 00:02:27,348 --> 00:02:31,776 a phase angle delta, which is the phase between the 31 00:02:31,776 --> 00:02:37,000 spherical wave from this point and that point. 32 00:02:37,000 --> 00:02:40,569 Two neighboring points. And here are a thousand. 33 00:02:40,569 --> 00:02:44,443 So this is only between the two neighboring sources. 34 00:02:44,443 --> 00:02:48,696 And so, first we want to know how many times can we fit a 35 00:02:48,696 --> 00:02:52,797 wavelength on this path? And, for each time that we can 36 00:02:52,797 --> 00:02:56,974 fit a wavelength on there, we have a phase difference of 37 00:02:56,974 --> 00:02:59,253 2pi. So this delta is the phase 38 00:02:59,253 --> 00:03:03,031 difference. And we had the same equation 39 00:03:03,031 --> 00:03:05,787 last time for double-slit interference. 40 00:03:05,787 --> 00:03:08,471 No difference. And, if this delta is a 41 00:03:08,471 --> 00:03:12,098 multiple times 2pi, then you will have constructive 42 00:03:12,098 --> 00:03:14,854 interference. My goal today is way more 43 00:03:14,854 --> 00:03:17,683 ambitious. I want to know what the light 44 00:03:17,683 --> 00:03:20,005 intensity is for any angle theta. 45 00:03:20,005 --> 00:03:23,704 It is going to be an extremely complicated function. 46 00:03:23,704 --> 00:03:29,000 I first want to revisit some of your high school geometry. 47 00:03:29,000 --> 00:03:33,461 If you have a triangle, and this side is A and this 48 00:03:33,461 --> 00:03:37,209 side is also A, the length of the triangle, 49 00:03:37,209 --> 00:03:41,849 the length of the sides, and if you want to know what 50 00:03:41,849 --> 00:03:47,292 this one is, and if this angle is delta then this side here is 51 00:03:47,292 --> 00:03:50,593 2A times the cosine of one-half delta. 52 00:03:50,593 --> 00:03:54,966 You can revisit that in your high school geometry, 53 00:03:54,966 --> 00:04:00,574 but I need that today. And now I am going to do the 54 00:04:00,574 --> 00:04:05,632 vectorial superposition of capital N E vectors that come 55 00:04:05,632 --> 00:04:10,689 from these various sources. And the neighboring ones are 56 00:04:10,689 --> 00:04:14,735 off by delta. That is the phase angle between 57 00:04:14,735 --> 00:04:17,402 them. I will raise this later, 58 00:04:17,402 --> 00:04:23,103 but I first want to work above my head so that you can see what 59 00:04:23,103 --> 00:04:25,586 I do. I start with a circle. 60 00:04:25,586 --> 00:04:30,000 You will see shortly why I do that. 61 00:04:30,000 --> 00:04:33,926 The radius of the circle is unimportant. 62 00:04:33,926 --> 00:04:37,147 You will see that it will cancel. 63 00:04:37,147 --> 00:04:40,671 This radius arbitrarily chosen is R. 64 00:04:40,671 --> 00:04:45,906 This length here is also R. I will call this point C, 65 00:04:45,906 --> 00:04:51,644 this point O and this point P. And I am going to add three 66 00:04:51,644 --> 00:04:57,583 vectors, which are all offset relative to each other over an 67 00:04:57,583 --> 00:05:02,753 angle delta. But I will make the calculation 68 00:05:02,753 --> 00:05:07,727 as if there were capital N. Capital N could be a million. 69 00:05:07,727 --> 00:05:11,636 There is no limit. But I am going to make the 70 00:05:11,636 --> 00:05:15,366 drawing only for three. This is one vector. 71 00:05:15,366 --> 00:05:20,785 And let's say it has a length A, but it is really the electric 72 00:05:20,785 --> 00:05:25,759 vector that is going to be added vectorially to the other 73 00:05:25,759 --> 00:05:31,000 electric vectors. And so here is the second one. 74 00:05:31,000 --> 00:05:36,200 And here is the third one. And so, the angle between the 75 00:05:36,200 --> 00:05:40,928 second and the third one, this angle here is delta. 76 00:05:40,928 --> 00:05:44,048 And this angle here is also delta. 77 00:05:44,048 --> 00:05:48,681 That is the delta between the neighboring sources, 78 00:05:48,681 --> 00:05:53,693 which we just derived. It follows from the geometry of 79 00:05:53,693 --> 00:05:59,650 a circle that this angle here is delta, this angle here is delta 80 00:05:59,650 --> 00:06:04,000 and this angle here is also delta. 81 00:06:04,000 --> 00:06:13,894 So that means if I have capital N of these vectors that this 82 00:06:13,894 --> 00:06:19,260 angle here is then N times delta. 83 00:06:19,260 --> 00:06:28,987 And that means that the angle here is then pi minus N times 84 00:06:28,987 --> 00:06:33,419 delta. I will draw one more line right 85 00:06:33,419 --> 00:06:35,995 through the middle of this vector. 86 00:06:35,995 --> 00:06:40,443 I think of this as length A but it is the electric vector. 87 00:06:40,443 --> 00:06:43,097 You can also think of it as E zero. 88 00:06:43,097 --> 00:06:46,219 And I draw a line straight through there. 89 00:06:46,219 --> 00:06:50,512 This angle is 90 degrees. And so, that means I know this 90 00:06:50,512 --> 00:06:54,882 angle now is one-half delta. And this is all the geometry 91 00:06:54,882 --> 00:06:59,253 that I need to calculate the incredibly complicated light 92 00:06:59,253 --> 00:07:03,000 intensity as a function of theta. 93 00:07:03,000 --> 00:07:07,363 My goal is to find the magnitude of the vector OP 94 00:07:07,363 --> 00:07:11,000 because that is the result, in this case, 95 00:07:11,000 --> 00:07:15,818 of these three vectors. But I will make my calculation 96 00:07:15,818 --> 00:07:20,000 as if they were N. First, look at this triangle 97 00:07:20,000 --> 00:07:23,909 here, which has a one-half delta angle here. 98 00:07:23,909 --> 00:07:27,909 This is 90 degrees. And so, this part here is 99 00:07:27,909 --> 00:07:32,782 one-half A. I just give it the length A. 100 00:07:32,782 --> 00:07:38,544 That means that one-half A divided by R is exactly the sine 101 00:07:38,544 --> 00:07:41,724 of delta over two. That is exact. 102 00:07:41,724 --> 00:07:45,400 It is not a small angle approximation. 103 00:07:45,400 --> 00:07:49,176 This is exactly the sine of that angle. 104 00:07:49,176 --> 00:07:53,150 In other words, my radius R is one-half A 105 00:07:53,150 --> 00:07:56,826 divided by the sine of delta over two. 106 00:07:56,826 --> 00:08:03,644 I am really interested in OP. And now I am going to use this 107 00:08:03,644 --> 00:08:06,112 knowledge. I have a triangle. 108 00:08:06,112 --> 00:08:11,224 I know two sides and I know this angle then I can calculate 109 00:08:11,224 --> 00:08:13,516 this one. I know two sides, 110 00:08:13,516 --> 00:08:18,276 this is R and this is R. I want to know the third side. 111 00:08:18,276 --> 00:08:22,419 And I know this angle which is pi minus N delta. 112 00:08:22,419 --> 00:08:26,209 And, therefore, this length here is 2R times 113 00:08:26,209 --> 00:08:32,678 the cosine of half that angle. Pi minus N delta divided by 114 00:08:32,678 --> 00:08:35,654 two. Pi over two is 90 degrees, 115 00:08:35,654 --> 00:08:41,109 and 90 degrees minus the angle is the sine of the angle. 116 00:08:41,109 --> 00:08:47,160 So I can also write for this 2R times the sine of N delta over 117 00:08:47,160 --> 00:08:50,434 two. Now, I take this R here and I 118 00:08:50,434 --> 00:08:53,806 pop that in here to eliminate my R. 119 00:08:53,806 --> 00:08:58,965 And so, now I get that the lengths of that vector OP, 120 00:08:58,965 --> 00:09:04,321 which is really my goal, that is the vectorially sum of 121 00:09:04,321 --> 00:09:09,801 N E vectors. The adjacent ones off by phase 122 00:09:09,801 --> 00:09:13,630 angle delta. That is going to be two times 123 00:09:13,630 --> 00:09:16,245 R. And so, that gives me an A 124 00:09:16,245 --> 00:09:19,700 upstairs. And then I get the sine of N 125 00:09:19,700 --> 00:09:25,210 delta divided by two divided by the sine of delta divided by 126 00:09:25,210 --> 00:09:28,291 two. And this is really the key of 127 00:09:28,291 --> 00:09:33,486 what is following. This is now the length, 128 00:09:33,486 --> 00:09:40,081 the magnitude of the E vector. If you think of this A as being 129 00:09:40,081 --> 00:09:45,378 E zero, it is the magnitude of the electric field. 130 00:09:45,378 --> 00:09:51,000 So the light intensity, the pointing vector obviously 131 00:09:51,000 --> 00:09:55,864 goes by the square. And so, now comes the very 132 00:09:55,864 --> 00:10:00,837 famous equation that I, as a function of delta, 133 00:10:00,837 --> 00:10:07,324 is I0 times the sine of N delta divided by two divided by the 134 00:10:07,324 --> 00:10:15,000 sine of delta divided by two and this whole thing squared. 135 00:10:15,000 --> 00:10:17,750 And you could call this the grading equation. 136 00:10:17,750 --> 00:10:21,062 You will see that this is a very complicated function. 137 00:10:21,062 --> 00:10:23,125 We will beat it to death together. 138 00:10:23,125 --> 00:10:26,437 This is an exact derivation. This is no approximation. 139 00:10:26,437 --> 00:10:30,000 Delta can be anything from zero to 10 million pi. 140 00:10:30,000 --> 00:10:33,322 There is no approximation made here. 141 00:10:33,322 --> 00:10:37,784 10pi, 20pi, 30pi, anything for delta is allowed. 142 00:10:37,784 --> 00:10:42,151 It is not an approximation. Now, the intensity, 143 00:10:42,151 --> 00:10:46,234 we think in terms of watts per square meter. 144 00:10:46,234 --> 00:10:51,835 And the meaning of I0 is that if there were only one opening 145 00:10:51,835 --> 00:10:57,626 in the screen instead of N then this is the intensity that you 146 00:10:57,626 --> 00:11:02,285 would see I0. Now, if the upstairs here is 147 00:11:02,285 --> 00:11:06,204 zero, you would think that the intensity is zero. 148 00:11:06,204 --> 00:11:10,693 That is not always the case. Because, if the upstairs is 149 00:11:10,693 --> 00:11:15,591 zero and the downstairs is also zero, you get zero divided by 150 00:11:15,591 --> 00:11:18,367 zero. And now you need l'Hopital to 151 00:11:18,367 --> 00:11:23,265 calculate what that ratio is. And that ratio then becomes the 152 00:11:23,265 --> 00:11:28,000 maximum value possible. I will write that here. 153 00:11:28,000 --> 00:11:31,898 The maximum value possible, if you use l'Hopital, 154 00:11:31,898 --> 00:11:35,309 you will find that that ratio is N squared. 155 00:11:35,309 --> 00:11:38,477 And so, this becomes N squared times I0. 156 00:11:38,477 --> 00:11:43,106 That is the case when you get zero divided by zero in that 157 00:11:43,106 --> 00:11:45,949 equation. Before I will show you how 158 00:11:45,949 --> 00:11:50,578 dramatic this function is, I want to remind you that for N 159 00:11:50,578 --> 00:11:54,314 equals two, which is what we covered last time, 160 00:11:54,314 --> 00:12:00,000 you can use this equation. This holds for any capital N. 161 00:12:00,000 --> 00:12:02,146 It holds for any value for delta. 162 00:12:02,146 --> 00:12:05,768 If you substitute in this equation that you have there, 163 00:12:05,768 --> 00:12:08,384 N equals two, and you do a little bit of 164 00:12:08,384 --> 00:12:11,603 massaging of the algebra, I will let you do that, 165 00:12:11,603 --> 00:12:14,554 and this is for the double-slit interference, 166 00:12:14,554 --> 00:12:18,176 you will find that I is 4I0 times the cosine squared of 167 00:12:18,176 --> 00:12:21,195 delta over two. And, for those of you who have 168 00:12:21,195 --> 00:12:24,213 a good memory, remember that I derived this in 169 00:12:24,213 --> 00:12:26,628 class. When you only have two vectors 170 00:12:26,628 --> 00:12:30,182 that you add that light intensity goes with the cosine 171 00:12:30,182 --> 00:12:35,446 square delta over two. Now, you also have this here. 172 00:12:35,446 --> 00:12:40,003 You also know that at the maxima with two slits you see 173 00:12:40,003 --> 00:12:44,812 four times more light than if there were only one opening. 174 00:12:44,812 --> 00:12:48,609 And you can do this, of course, on your own by 175 00:12:48,609 --> 00:12:52,153 substituting in this equation N equals two. 176 00:12:52,153 --> 00:12:57,046 The first thing that I want to do now is to make a drawing, 177 00:12:57,046 --> 00:13:02,703 a plot of that function. And I will do that for N equals 178 00:13:02,703 --> 00:13:05,581 four. And then we will discuss all 179 00:13:05,581 --> 00:13:11,075 the consequences also for cases that N is much larger than four. 180 00:13:11,075 --> 00:13:14,651 I am going to plot this for N equals four. 181 00:13:14,651 --> 00:13:19,709 We only have four openings now. And I always plot only sine 182 00:13:19,709 --> 00:13:22,761 theta. The reason why I like to plot 183 00:13:22,761 --> 00:13:27,296 always things in terms of sine theta, theta is a real 184 00:13:27,296 --> 00:13:31,904 geometrical angle. Here is this screen with the 185 00:13:31,904 --> 00:13:34,625 openings. And theta is an actual angle in 186 00:13:34,625 --> 00:13:37,755 the lecture hall. Theta is something that I can 187 00:13:37,755 --> 00:13:40,544 immediately relate to. This is 10 degrees. 188 00:13:40,544 --> 00:13:43,129 This is 20 degrees. This is 30 degrees. 189 00:13:43,129 --> 00:13:46,530 Delta is a phase angle. That is not a real angle in 190 00:13:46,530 --> 00:13:48,911 phase. And so, I always like to plot 191 00:13:48,911 --> 00:13:52,993 the intensity in terms of sine theta, but you can also do it, 192 00:13:52,993 --> 00:13:55,034 if you want, in terms of delta. 193 00:13:55,034 --> 00:14:00,000 If sine theta is zero then you get zero divided by zero. 194 00:14:00,000 --> 00:14:03,082 And you are going to get a maximum. 195 00:14:03,082 --> 00:14:08,431 If sine theta is lambda divided by D, you are going to get a 196 00:14:08,431 --> 00:14:11,967 maximum. If sine theta is lambda divided 197 00:14:11,967 --> 00:14:17,497 by D, you see that delta is 2pi. That means you get a maximum. 198 00:14:17,497 --> 00:14:22,030 You get a maximum here. If sine theta is two lambda 199 00:14:22,030 --> 00:14:24,840 divided by D, you get a maximum. 200 00:14:24,840 --> 00:14:28,920 And, of course, on the other side minus lambda 201 00:14:28,920 --> 00:14:35,597 over d, you also get a maximum. And, according to the equation, 202 00:14:35,597 --> 00:14:39,283 if you really believe that equation verbatim, 203 00:14:39,283 --> 00:14:43,305 then all these peaks would have the same maximum. 204 00:14:43,305 --> 00:14:47,159 It would be 16I0 because this is the N squared. 205 00:14:47,159 --> 00:14:50,929 And I will show you, but first I will plot it, 206 00:14:50,929 --> 00:14:55,956 that if there are four slits that in between the prime maxima 207 00:14:55,956 --> 00:15:00,816 there are N minus one locations whereby you have completely 208 00:15:00,816 --> 00:15:05,750 destructive interference. There is zero light. 209 00:15:05,750 --> 00:15:07,939 N minus one, in this case, 210 00:15:07,939 --> 00:15:11,178 is three. You will shortly why it is N 211 00:15:11,178 --> 00:15:14,680 minus one. There are three locations here 212 00:15:14,680 --> 00:15:19,670 whereby there is zero light. I put them in here and then I 213 00:15:19,670 --> 00:15:22,909 will draw the curve. That is the zero. 214 00:15:22,909 --> 00:15:27,986 And so, I am going to make an attempt now to draw the light 215 00:15:27,986 --> 00:15:32,700 intensity. It is prime maxima when zero 216 00:15:32,700 --> 00:15:38,392 divided by zero is N squared. Here is another one where zero 217 00:15:38,392 --> 00:15:41,575 divided by zero becomes N squared. 218 00:15:41,575 --> 00:15:47,363 And here is another one where by zero divided by zero becomes 219 00:15:47,363 --> 00:15:51,221 N squared. If you wanted to know what the 220 00:15:51,221 --> 00:15:55,369 delta is here, well, the delta here is zero, 221 00:15:55,369 --> 00:16:00,000 of course. And the delta here is 2pi. 222 00:16:00,000 --> 00:16:04,022 And the delta here is 4pi. I first want to show you, 223 00:16:04,022 --> 00:16:08,597 or at least draw your attention to the fact that there is a 224 00:16:08,597 --> 00:16:13,171 wavelength dependent lambda. And what that means is that if 225 00:16:13,171 --> 00:16:16,090 you take 650 nanometers, which is red, 226 00:16:16,090 --> 00:16:20,349 the red would have a maximum here, the red would have a 227 00:16:20,349 --> 00:16:23,583 maximum here, the red would have a maximum 228 00:16:23,583 --> 00:16:28,000 here and the red would have a maximum there. 229 00:16:28,000 --> 00:16:31,529 But if now you have 400 nanometers, which is violet 230 00:16:31,529 --> 00:16:35,270 light, it would have a maximum at different locations. 231 00:16:35,270 --> 00:16:39,011 Here at zero it would always have the same location as 232 00:16:39,011 --> 00:16:42,329 maximum, but the wavelength is shorter for blue, 233 00:16:42,329 --> 00:16:45,152 for violet. Here would be the maximum for 234 00:16:45,152 --> 00:16:49,035 violet, here would be the maximum for violet and roughly 235 00:16:49,035 --> 00:16:52,776 here would be the maximum for violet and roughly here. 236 00:16:52,776 --> 00:16:58,000 The reason why the red and the blue there almost coincide -- 237 00:16:58,000 --> 00:17:02,347 I mentioned that also last time for the double-slit 238 00:17:02,347 --> 00:17:06,086 interference. Two times 650 is roughly three 239 00:17:06,086 --> 00:17:09,391 times 400. So they live a life of their 240 00:17:09,391 --> 00:17:12,347 own. Let's now address the issue of 241 00:17:12,347 --> 00:17:16,869 the N minus one zeros. I first want to calculate what 242 00:17:16,869 --> 00:17:21,304 the location is here where you have your first zero, 243 00:17:21,304 --> 00:17:25,391 completely zero. Well, you would have your first 244 00:17:25,391 --> 00:17:32,000 zero when the upstairs is zero but the downstairs is not zero. 245 00:17:32,000 --> 00:17:36,641 Because, if they are both zero, you are at what we call a prime 246 00:17:36,641 --> 00:17:39,262 maximum. What is the first time that 247 00:17:39,262 --> 00:17:42,930 this one becomes zero and the downstairs not zero? 248 00:17:42,930 --> 00:17:46,748 It is when this is pi. When it is zero they are both 249 00:17:46,748 --> 00:17:51,016 zero, but when that is pi then, of course, I would have my 250 00:17:51,016 --> 00:17:53,262 first zero. Let's do that here. 251 00:17:53,262 --> 00:17:57,379 I call that my first zero. That is the case when N delta 252 00:17:57,379 --> 00:18:03,799 divided by two is pi. And so, that means when delta 253 00:18:03,799 --> 00:18:09,825 is 2pi divided by N. Now I go to this equation, 254 00:18:09,825 --> 00:18:15,589 and I put in here for delta 2pi divided by N. 255 00:18:15,589 --> 00:18:22,401 And what you see then, that sine theta becomes lambda 256 00:18:22,401 --> 00:18:30,000 divided by sine theta, is now lambda divided by Nd. 257 00:18:30,000 --> 00:18:32,307 In other words, this point here, 258 00:18:32,307 --> 00:18:36,774 in terms of sine theta which, of course, is the same as angle 259 00:18:36,774 --> 00:18:40,272 theta in radians because these angles are small. 260 00:18:40,272 --> 00:18:42,878 This is an angular dimensional plot. 261 00:18:42,878 --> 00:18:45,334 This here is lambda divided by Nd. 262 00:18:45,334 --> 00:18:48,908 And then, of course, the second one will be twice 263 00:18:48,908 --> 00:18:53,151 lambda divided by Nd and the third one will be three times 264 00:18:53,151 --> 00:18:56,947 lambda divided by Nd. You will again have completely 265 00:18:56,947 --> 00:19:02,592 destructive interference. You may be interested in what 266 00:19:02,592 --> 00:19:06,346 the magnitude is, what the light intensity, 267 00:19:06,346 --> 00:19:10,189 I should say, is of this little mini-maxima. 268 00:19:10,189 --> 00:19:13,407 Well, that is very easy to calculate. 269 00:19:13,407 --> 00:19:18,592 If we know that the sine of theta here is lambda divided by 270 00:19:18,592 --> 00:19:24,044 Nd and we know that here it is twice that much then all I have 271 00:19:24,044 --> 00:19:29,407 to ask that equation is what is your light intensity when the 272 00:19:29,407 --> 00:19:34,596 sine of theta -- Now we go to that mini-max, 273 00:19:34,596 --> 00:19:39,307 the first mini-max right here. I ask the equation, 274 00:19:39,307 --> 00:19:44,884 what is your intensity when the sine theta is now 1.5 times 275 00:19:44,884 --> 00:19:49,403 lambda divided by Nd? Then I am right in between 276 00:19:49,403 --> 00:19:53,634 these two zeros. Now, whether I am exactly on 277 00:19:53,634 --> 00:19:56,134 that maximum, I don't know. 278 00:19:56,134 --> 00:20:02,000 I am really not interested, but I surely am close. 279 00:20:02,000 --> 00:20:07,486 When sine theta is 1.5 lambda divided by Nd then delta is 280 00:20:07,486 --> 00:20:13,168 going to be 3pi divided by N. That is easy because you take 281 00:20:13,168 --> 00:20:16,402 this equation, 1.5 lambda over Nd, 282 00:20:16,402 --> 00:20:21,006 you put in sine theta, 1.5 lambda divided by Nd. 283 00:20:21,006 --> 00:20:26,003 The 1.5 times 2.0 becomes 3pi. You get N downstairs, 284 00:20:26,003 --> 00:20:31,000 so you get the delta of 3pi divided by N. 285 00:20:31,000 --> 00:20:35,649 And now you revisit this equation and you just put in 286 00:20:35,649 --> 00:20:39,850 there N equals four, you know what capital N is, 287 00:20:39,850 --> 00:20:45,036 you know now what delta is, so you calculate your upstairs, 288 00:20:45,036 --> 00:20:50,400 you calculate the downstairs. And you will find now that I is 289 00:20:50,400 --> 00:20:54,692 approximately 1.17I0. That is low compared to 16. 290 00:20:54,692 --> 00:20:58,000 This is only some 7%, 6.8%. 291 00:20:58,000 --> 00:21:01,505 This height here is only some 7% of that height, 292 00:21:01,505 --> 00:21:03,967 so it is very low. You have C now, 293 00:21:03,967 --> 00:21:08,517 the bizarre consequences that if you get zero divided by zero, 294 00:21:08,517 --> 00:21:12,918 you get these maxima at 16I0. Then you get N minus one point 295 00:21:12,918 --> 00:21:15,529 whereby you get complete zeros here. 296 00:21:15,529 --> 00:21:19,034 But even these mini-maxima don't mean very much. 297 00:21:19,034 --> 00:21:21,720 They are very low in light intensity. 298 00:21:21,720 --> 00:21:24,405 And, even if you go to an N of 1,000. 299 00:21:24,405 --> 00:21:27,762 And today we will even go beyond that with our 300 00:21:27,762 --> 00:21:31,418 experiments. We will have capital N. 301 00:21:31,418 --> 00:21:34,255 We go up to 2,000. If you make capital N 1,000, 302 00:21:34,255 --> 00:21:37,215 you can redo all this. And you will see that this 303 00:21:37,215 --> 00:21:39,929 mini-maximum is roughly 4.5% of this maximum. 304 00:21:39,929 --> 00:21:42,581 That maximum now becomes a million times I0. 305 00:21:42,581 --> 00:21:46,281 If you have a thousand of these openings, they will add up at 306 00:21:46,281 --> 00:21:48,625 the maxima. You get a million times the 307 00:21:48,625 --> 00:21:50,599 intensity that one alone will do. 308 00:21:50,599 --> 00:21:54,361 The reason is obviously you get a thousand times the E vector, 309 00:21:54,361 --> 00:21:58,000 and they are all in phase with each other. 310 00:21:58,000 --> 00:22:02,613 And the pointing vector is the square of the amplitude of the E 311 00:22:02,613 --> 00:22:04,623 vector. You get the million. 312 00:22:04,623 --> 00:22:07,302 All these maxima have the same width. 313 00:22:07,302 --> 00:22:11,544 And if this is lambda divided by Nd then this here on this 314 00:22:11,544 --> 00:22:15,041 side in terms of angular distance is, of course, 315 00:22:15,041 --> 00:22:17,720 the same. There is complete symmetry. 316 00:22:17,720 --> 00:22:21,367 And so the width here, if I take the width roughly 317 00:22:21,367 --> 00:22:25,609 without being very precise, the width of each one of those 318 00:22:25,609 --> 00:22:30,000 peaks must be roughly lambda divided by Nd. 319 00:22:30,000 --> 00:22:33,888 So I take half this distance, angular distance. 320 00:22:33,888 --> 00:22:38,876 All these things are angles in radians because sine theta is 321 00:22:38,876 --> 00:22:43,442 very much smaller than theta. Now, you can see that the 322 00:22:43,442 --> 00:22:47,500 larger N is, the more of these openings you have, 323 00:22:47,500 --> 00:22:52,572 the narrower these lines are going to be, if I think of these 324 00:22:52,572 --> 00:22:56,461 as being a line. And that means your ability to 325 00:22:56,461 --> 00:23:01,111 distinguish two neighboring frequencies from each other, 326 00:23:01,111 --> 00:23:05,000 two different lambdas increases. 327 00:23:05,000 --> 00:23:09,387 And that is what we call spectral resolution. 328 00:23:09,387 --> 00:23:13,974 The larger N is, the better spectral resolution 329 00:23:13,974 --> 00:23:18,860 you have, your ability to separate two neighboring 330 00:23:18,860 --> 00:23:24,643 frequencies then increases. There is a very easy way that I 331 00:23:24,643 --> 00:23:28,034 can convince you, without any math, 332 00:23:28,034 --> 00:23:32,720 well, little math, why the width of these peaks, 333 00:23:32,720 --> 00:23:40,000 the width of these maxima must be proportional to one over N. 334 00:23:40,000 --> 00:23:42,906 And that is purely an energy conservation argument. 335 00:23:42,906 --> 00:23:45,872 And follow me closely. If I have N of these openings 336 00:23:45,872 --> 00:23:48,197 in the screen, they will let N times more 337 00:23:48,197 --> 00:23:51,395 light through than one opening. That is straightforward. 338 00:23:51,395 --> 00:23:53,546 You can tell that to your kid brother. 339 00:23:53,546 --> 00:23:56,860 If you have N openings in the screen, you get N times more 340 00:23:56,860 --> 00:24:00,000 light through than if you had one opening. 341 00:24:00,000 --> 00:24:02,967 That is nonnegotiable. But if each maximum is N 342 00:24:02,967 --> 00:24:05,935 squared times higher, the only way that you can 343 00:24:05,935 --> 00:24:09,677 conserve energy is if you make the maximum N times smaller. 344 00:24:09,677 --> 00:24:12,903 Then you know that N times more light went through. 345 00:24:12,903 --> 00:24:16,322 The argument once more, you have N openings that gives 346 00:24:16,322 --> 00:24:18,903 you N times more energy than one opening. 347 00:24:18,903 --> 00:24:22,903 But, if each maximum gives you a light intensity that goes with 348 00:24:22,903 --> 00:24:26,838 N squared, the only way that you can conserve energy is if you 349 00:24:26,838 --> 00:24:30,000 make the maxima N times narrower. 350 00:24:30,000 --> 00:24:34,655 And so that is a very easy way to see that the width here must 351 00:24:34,655 --> 00:24:38,547 go down with increasing N. A very powerful argument. 352 00:24:38,547 --> 00:24:41,447 Now I will make you see in another way. 353 00:24:41,447 --> 00:24:45,034 If we have four of these openings in the screen, 354 00:24:45,034 --> 00:24:47,400 why there are only three minima. 355 00:24:47,400 --> 00:24:50,986 And all these methods that I am using, in a way, 356 00:24:50,986 --> 00:24:54,268 are complimentary. It is all the same thing, 357 00:24:54,268 --> 00:24:59,000 but I just want you to see it in different ways. 358 00:24:59,000 --> 00:25:03,259 It helps me enormously to look at it in different ways. 359 00:25:03,259 --> 00:25:07,282 We have N equals four. And I start with delta equals 360 00:25:07,282 --> 00:25:09,885 zero. There is no phase difference 361 00:25:09,885 --> 00:25:14,145 between adjacent sources. This is the case that this is 362 00:25:14,145 --> 00:25:18,641 the E vector of source number one, this is the E vector of 363 00:25:18,641 --> 00:25:22,348 source number two, number three and number four. 364 00:25:22,348 --> 00:25:26,134 All four E vectors line up in the same direction. 365 00:25:26,134 --> 00:25:30,000 Otherwise, delta could not be zero. 366 00:25:30,000 --> 00:25:35,341 Therefore, I get 16 times the light because I square 4E and I 367 00:25:35,341 --> 00:25:38,278 get 16. This is your factor of 16. 368 00:25:38,278 --> 00:25:42,729 And you get a maximum. Now we go to delta equals pi 369 00:25:42,729 --> 00:25:47,091 over two, 90 degrees. I don't have to look at that 370 00:25:47,091 --> 00:25:49,851 equation. I don't need that one. 371 00:25:49,851 --> 00:25:54,124 I know what 90 degrees is. I went to high school. 372 00:25:54,124 --> 00:25:57,062 I am educated. This is one vector. 373 00:25:57,062 --> 00:26:02,143 This is 90 degrees. That is the second vector. 374 00:26:02,143 --> 00:26:05,688 This is 90 degrees. That's the third vector. 375 00:26:05,688 --> 00:26:08,739 This is the fourth vector. 90 degrees. 376 00:26:08,739 --> 00:26:10,965 What do I end up with? Zero. 377 00:26:10,965 --> 00:26:15,500 If all four relative two neighbors 90 degree phase angle 378 00:26:15,500 --> 00:26:20,199 then clearly you have zero here. Now I go delta equals pi. 379 00:26:20,199 --> 00:26:23,002 I know what pi is. One, two, three, 380 00:26:23,002 --> 00:26:25,228 four. What do I end up with? 381 00:26:25,228 --> 00:26:27,702 I add four vector, 180 degrees, 382 00:26:27,702 --> 00:26:31,000 flip, flip, flip, flip. 383 00:26:31,000 --> 00:26:33,636 What is the net result? Zero. 384 00:26:33,636 --> 00:26:36,931 So it is dark. This one is this one. 385 00:26:36,931 --> 00:26:42,581 And the second one is this one. Now I am going to do this one 386 00:26:42,581 --> 00:26:45,876 for you. Delta equals three-half pi. 387 00:26:45,876 --> 00:26:50,019 Three-half is 270 degrees. Well, this is one. 388 00:26:50,019 --> 00:26:53,785 This is 270 degrees, this is 270 degrees, 389 00:26:53,785 --> 00:26:58,211 and that is 270 degrees. What is the net result? 390 00:26:58,211 --> 00:27:01,560 Zero. That's this one. 391 00:27:01,560 --> 00:27:06,829 Now I am going to do 2pi. When I do 2pi I am back here. 392 00:27:06,829 --> 00:27:11,121 I have another maximum, and that is this one. 393 00:27:11,121 --> 00:27:15,609 And so, you can see, purely by playing vectors, 394 00:27:15,609 --> 00:27:19,609 very simple, high school level you can see 395 00:27:19,609 --> 00:27:25,658 that there will be N minus one minima, exact minima between the 396 00:27:25,658 --> 00:27:32,592 prime maxima. The first thing that I want you 397 00:27:32,592 --> 00:27:41,925 to see, when I take the grating that you have and I use my laser 398 00:27:41,925 --> 00:27:50,074 pointer, is incredible impact of using many, many lines. 399 00:27:50,074 --> 00:27:57,185 My laser beam has about a diameter of about three 400 00:27:57,185 --> 00:28:02,136 millimeters. Here is this laser beam. 401 00:28:02,136 --> 00:28:06,765 And this I estimated to be roughly three millimeters. 402 00:28:06,765 --> 00:28:09,525 Your grating, believe it or not, 403 00:28:09,525 --> 00:28:13,798 is a super grating. That grating has 13,400 lines 404 00:28:13,798 --> 00:28:16,913 per inch. Imagine how anyone can put 405 00:28:16,913 --> 00:28:20,474 grooves in your plastic. 13,400 per inch. 406 00:28:20,474 --> 00:28:25,727 That means that the separation d between two groups is about 407 00:28:25,727 --> 00:28:30,000 1.9 times 10 to the minus 6 meters. 408 00:28:30,000 --> 00:28:34,649 That is only two microns. How anyone can do that beats 409 00:28:34,649 --> 00:28:39,385 me, but it can be done. I can calculate now how many of 410 00:28:39,385 --> 00:28:43,596 those lines I have here in the three millimeters, 411 00:28:43,596 --> 00:28:46,315 and I end up with N about 1,600. 412 00:28:46,315 --> 00:28:50,701 When I shine my laser beam through my grating I use 413 00:28:50,701 --> 00:28:53,508 effectively 1,600 of those lines. 414 00:28:53,508 --> 00:28:57,280 And I can calculate now what the angles are, 415 00:28:57,280 --> 00:29:02,456 where on the screen there on the wall, the maxima will fall, 416 00:29:02,456 --> 00:29:06,311 those maxima. And, by the way, 417 00:29:06,311 --> 00:29:10,000 there will be 1,599 of these zeros in between. 418 00:29:10,000 --> 00:29:14,754 And these minima are so small that you won't even see them. 419 00:29:14,754 --> 00:29:19,590 You will only see the maxima. Let's calculate at what angles 420 00:29:19,590 --> 00:29:24,262 we would then see the first. By the way, these things have 421 00:29:24,262 --> 00:29:26,721 names. We call this zero order. 422 00:29:26,721 --> 00:29:32,140 And we call this first order. This is first order. 423 00:29:32,140 --> 00:29:38,455 And we call this second order. And this is also called first 424 00:29:38,455 --> 00:29:43,272 order, but it is on the other side, of course. 425 00:29:43,272 --> 00:29:47,018 We call these orders of the spectra. 426 00:29:47,018 --> 00:29:53,333 You can now calculate that the sine of theta of N is N times 427 00:29:53,333 --> 00:29:57,935 lambda divided by D. And so, when N is zero, 428 00:29:57,935 --> 00:30:02,538 that is your zero order, you get, of course, 429 00:30:02,538 --> 00:30:08,369 a maximum. So the zero's order is at theta 430 00:30:08,369 --> 00:30:12,826 equals zero. Now, your first order is when 431 00:30:12,826 --> 00:30:17,608 the sine of theta one is lambda divided by D. 432 00:30:17,608 --> 00:30:24,239 And I can take my lambda which, in my case, of my laser I will 433 00:30:24,239 --> 00:30:30,000 have to tell you, my lambda is 532 nanometers. 434 00:30:30,000 --> 00:30:33,526 It is green. And so, I can calculate what 435 00:30:33,526 --> 00:30:36,787 theta one is. And I find 16.3 degrees. 436 00:30:36,787 --> 00:30:41,019 And then I can go to the second order, theta two, 437 00:30:41,019 --> 00:30:44,898 for that color. It is different for different 438 00:30:44,898 --> 00:30:47,454 colors. And I find 34 degrees. 439 00:30:47,454 --> 00:30:52,479 And I can go to the third order, and I find theta three is 440 00:30:52,479 --> 00:30:56,358 then 57 degrees. And there is no fourth order 441 00:30:56,358 --> 00:31:02,000 that would make the sine of theta larger than one. 442 00:31:02,000 --> 00:31:06,000 There are only zero order, and then there are first, 443 00:31:06,000 --> 00:31:09,843 second and third order. The grating that you have, 444 00:31:09,843 --> 00:31:13,607 I always carry that with me no matter where I go. 445 00:31:13,607 --> 00:31:16,274 It is easy to put in your calendar. 446 00:31:16,274 --> 00:31:20,823 And so, I will show you now, by simply shining through this 447 00:31:20,823 --> 00:31:25,686 grating, I will show you there on the wall the zero order would 448 00:31:25,686 --> 00:31:30,000 fall right smack in the middle so-to-speak. 449 00:31:30,000 --> 00:31:33,515 First order 16 degrees away. If you know my distance to the 450 00:31:33,515 --> 00:31:35,878 wall, you can calculate how far that is. 451 00:31:35,878 --> 00:31:38,969 It is probably 1.5 meter. And so, you will see these 452 00:31:38,969 --> 00:31:41,515 maxima, you will see them extremely narrow, 453 00:31:41,515 --> 00:31:44,000 high value for N. And, well, it speaks for 454 00:31:44,000 --> 00:31:46,303 itself. I have to rotate my gratings to 455 00:31:46,303 --> 00:31:49,212 make sure that the grooves are in this direction. 456 00:31:49,212 --> 00:31:52,666 If the grooves are in this direction, the spreading is out 457 00:31:52,666 --> 00:31:55,575 in this direction. The one that you see right now 458 00:31:55,575 --> 00:32:00,000 on the right side of the screen there is my zero order. 459 00:32:00,000 --> 00:32:03,737 Now, you cannot tell that it is zero order because they have 460 00:32:03,737 --> 00:32:05,764 only one color. And then you see, 461 00:32:05,764 --> 00:32:08,932 I will move it a little, the first order now on the 462 00:32:08,932 --> 00:32:12,352 right side of the screen. And that angle should be very 463 00:32:12,352 --> 00:32:15,520 accurately about what I calculated, the 16 degrees. 464 00:32:15,520 --> 00:32:17,990 And then you see on the blackboard here, 465 00:32:17,990 --> 00:32:20,904 we saw it right here, you see the second order. 466 00:32:20,904 --> 00:32:24,705 And then you see here the third order, if you have good eyes. 467 00:32:24,705 --> 00:32:28,000 And the fourth order does not exist. 468 00:32:28,000 --> 00:32:33,881 And imagine that between those maxima, if I really use 1,600 469 00:32:33,881 --> 00:32:38,965 lines, there would be 1,599 points with exact zeros, 470 00:32:38,965 --> 00:32:44,448 and then all these silly mini-maxima that you don't even 471 00:32:44,448 --> 00:32:47,638 see. That is the power of grating 472 00:32:47,638 --> 00:32:52,124 when you use many lines. If I use 1,600 lines, 473 00:32:52,124 --> 00:32:57,507 I use 800 times more lines than when I have double-slit 474 00:32:57,507 --> 00:33:02,438 interference. The double-slit interference 475 00:33:02,438 --> 00:33:06,356 pattern will be extremely different from this. 476 00:33:06,356 --> 00:33:10,623 That is why those locations are so narrow because, 477 00:33:10,623 --> 00:33:13,671 if it were double-slit interference, 478 00:33:13,671 --> 00:33:17,154 you would get the cosine square function. 479 00:33:17,154 --> 00:33:22,553 And so you would see the maxima would be 800 times broader than 480 00:33:22,553 --> 00:33:25,252 this. That is the power of using 481 00:33:25,252 --> 00:33:28,561 1,600 lines. Now I want you to get your 482 00:33:28,561 --> 00:33:34,484 gratings out. And I want you to look simply 483 00:33:34,484 --> 00:33:40,131 at a desktop plan. And the reason why I want you 484 00:33:40,131 --> 00:33:47,701 to do that is that I want you to be disappointed because you may 485 00:33:47,701 --> 00:33:54,910 not see what you erroneously expected, that the zero order is 486 00:33:54,910 --> 00:33:58,275 incredibly narrow. It is not. 487 00:33:58,275 --> 00:34:04,450 It is the lamp itself. Of course, if you light source 488 00:34:04,450 --> 00:34:08,943 in angular size is way larger than this angular size you 489 00:34:08,943 --> 00:34:12,946 cannot expect to see the light source get smaller. 490 00:34:12,946 --> 00:34:16,622 In other words, clearly the limiting factor of 491 00:34:16,622 --> 00:34:21,769 seeing zero order very narrow is only if the light source itself 492 00:34:21,769 --> 00:34:26,098 is small enough in size. And so, when you look at this 493 00:34:26,098 --> 00:34:31,000 light now, you will see all colors at zero order. 494 00:34:31,000 --> 00:34:33,322 That is one thing that is important. 495 00:34:33,322 --> 00:34:36,308 You just see the lamp. That is your zero order 496 00:34:36,308 --> 00:34:37,901 maximum. That is the red, 497 00:34:37,901 --> 00:34:40,755 the blue, the green, the yellow, the violet. 498 00:34:40,755 --> 00:34:43,409 That is all of it. And then you will see, 499 00:34:43,409 --> 00:34:45,201 on either side, blue appear. 500 00:34:45,201 --> 00:34:48,054 First the blue. That has the lowest shortest 501 00:34:48,054 --> 00:34:50,643 wavelength. I do not think there is much 502 00:34:50,643 --> 00:34:54,027 violet in this lamp. And then you will see the blue, 503 00:34:54,027 --> 00:34:56,482 the red. Make sure that you line it up 504 00:34:56,482 --> 00:35:00,000 so that the grooves are vertical. 505 00:35:00,000 --> 00:35:03,782 That is so that you get the spread in the horizontal plane. 506 00:35:03,782 --> 00:35:06,847 And then you see the green, and you see the red. 507 00:35:06,847 --> 00:35:10,434 And it really looks like you have a continuous spectrum. 508 00:35:10,434 --> 00:35:13,173 Even at second order, it still looks like a 509 00:35:13,173 --> 00:35:16,369 reasonable spectrum. It starts in blue and it goes 510 00:35:16,369 --> 00:35:18,717 to red. But, when you go to the third 511 00:35:18,717 --> 00:35:22,500 one, you already see that the blue and the red are going to 512 00:35:22,500 --> 00:35:26,413 interfere with each other there. Then they really do not look 513 00:35:26,413 --> 00:35:30,000 any more clearly like separate spectra. 514 00:35:30,000 --> 00:35:33,980 I want you to appreciate the fact that if your light source 515 00:35:33,980 --> 00:35:37,549 is huge that each little location of the light source 516 00:35:37,549 --> 00:35:41,598 gives you a line which is this narrow, but if you have many, 517 00:35:41,598 --> 00:35:44,343 many of those locations it smears it out, 518 00:35:44,343 --> 00:35:48,254 of course, and you see a very broad line, if I call this a 519 00:35:48,254 --> 00:35:51,000 line, this zero order maximum. 520 00:35:51,000 --> 00:35:56,000 521 00:35:56,000 --> 00:36:02,000 We have here a grating that works not in transmission. 522 00:36:02,000 --> 00:36:04,515 Your grating works in transmission. 523 00:36:04,515 --> 00:36:08,509 The light goes through it, like what we discussed here. 524 00:36:08,509 --> 00:36:12,873 But we have here one that is metal, whereby grooves I put on 525 00:36:12,873 --> 00:36:15,906 the metal. And this is called a reflection 526 00:36:15,906 --> 00:36:17,978 gradient. And I can show you, 527 00:36:17,978 --> 00:36:22,194 with this reflection grading that has a large light source 528 00:36:22,194 --> 00:36:24,191 just like this, it is large, 529 00:36:24,191 --> 00:36:29,000 the spectrum when we project it here on the screen. 530 00:36:29,000 --> 00:36:31,949 If Markos can give me a hand? Thank you, Markos. 531 00:36:31,949 --> 00:36:35,589 We have white light that goes onto this reflection grating. 532 00:36:35,589 --> 00:36:37,534 It is a big spot of white light. 533 00:36:37,534 --> 00:36:41,487 And you will see them the first order will not be narrow because 534 00:36:41,487 --> 00:36:43,558 the light source itself is so big. 535 00:36:43,558 --> 00:36:45,880 And then you will see, on either side, 536 00:36:45,880 --> 00:36:48,265 something similar to what you saw here. 537 00:36:48,265 --> 00:36:51,465 You will see the white. The zero order is always the 538 00:36:51,465 --> 00:36:54,540 light of the source itself which is, in this case, 539 00:36:54,540 --> 00:36:56,862 white. And then on both sides you will 540 00:36:56,862 --> 00:37:00,000 see first order, second order. 541 00:37:00,000 --> 00:37:02,836 And I think you see up to four or five orders. 542 00:37:02,836 --> 00:37:06,052 I do not remember how many lines there are per inch. 543 00:37:06,052 --> 00:37:09,960 It is not important because it is really the effect that I want 544 00:37:09,960 --> 00:37:12,482 to show you. I was quantitative on my own 545 00:37:12,482 --> 00:37:14,563 demonstration with my green laser. 546 00:37:14,563 --> 00:37:17,021 Let us turn this on. I think that is it. 547 00:37:17,021 --> 00:37:20,552 And we turn all the lights off so that you can enjoy this 548 00:37:20,552 --> 00:37:22,758 fully. Here you see the size of that 549 00:37:22,758 --> 00:37:25,910 zero order maximum. It is not narrow at all because 550 00:37:25,910 --> 00:37:31,228 the light source is not narrow. And then you see here nicely 551 00:37:31,228 --> 00:37:34,810 the blue, the green. And you see here the red. 552 00:37:34,810 --> 00:37:36,084 Blue, green, red. 553 00:37:36,084 --> 00:37:40,383 You begin to see spectra, and then the whole thing sort 554 00:37:40,383 --> 00:37:43,805 of fettles out. And you see the same on this 555 00:37:43,805 --> 00:37:46,114 side. So, this is a reflection 556 00:37:46,114 --> 00:37:49,139 grating. If we turn on a red laser that 557 00:37:49,139 --> 00:37:53,437 is much narrow then you get the advantage of your many, 558 00:37:53,437 --> 00:37:55,666 many lines. And then you get, 559 00:37:55,666 --> 00:38:01,000 of course, an extremely narrow zero order maximum. 560 00:38:01,000 --> 00:38:03,894 And that is this laser, I believe. 561 00:38:03,894 --> 00:38:08,368 We will know very shortly. This is the 633 nanometer 562 00:38:08,368 --> 00:38:11,614 laser. And so, now I take advantage of 563 00:38:11,614 --> 00:38:16,614 the many lines that I cover. Now my light source is narrow 564 00:38:16,614 --> 00:38:21,964 enough to let me benefit from the grating, the many slits that 565 00:38:21,964 --> 00:38:23,543 I use. And you see, 566 00:38:23,543 --> 00:38:27,403 as you expect, that the red zero order always 567 00:38:27,403 --> 00:38:33,058 falls at the same location. And then the red here, 568 00:38:33,058 --> 00:38:37,361 the 633 coincides, of course, with the red from 569 00:38:37,361 --> 00:38:42,225 the white light and so on. And you even see here 633. 570 00:38:42,225 --> 00:38:46,341 And this is first order, second, third order. 571 00:38:46,341 --> 00:38:51,112 There is no fourth order. The separation in terms of 572 00:38:51,112 --> 00:38:56,819 angle is only a function of d, the spacing between the grooves 573 00:38:56,819 --> 00:39:00,000 and, of course, lambda. 574 00:39:00,000 --> 00:39:05,654 And so, if the spacing of a double-slit interference pattern 575 00:39:05,654 --> 00:39:09,872 is the same as the spacing of a multiple one, 576 00:39:09,872 --> 00:39:13,322 the separation and angle is the same. 577 00:39:13,322 --> 00:39:17,635 But, of course, the gain is that you make them 578 00:39:17,635 --> 00:39:21,086 narrow and they go up with N squared. 579 00:39:21,086 --> 00:39:25,878 That is what you gain. Now I would like you to take 580 00:39:25,878 --> 00:39:32,267 full advantage of your grating. And we have prepared light 581 00:39:32,267 --> 00:39:37,324 sources that are narrow enough so that you come very close, 582 00:39:37,324 --> 00:39:41,160 maybe not exactly, to seeing the width of the 583 00:39:41,160 --> 00:39:44,038 light source about lambda over Nd. 584 00:39:44,038 --> 00:39:49,356 If you put the grating in front of your eye, your pupil itself 585 00:39:49,356 --> 00:39:52,931 has a diameter of about three millimeters. 586 00:39:52,931 --> 00:39:56,768 You will only use, effectively with your eye, 587 00:39:56,768 --> 00:40:02,000 about 1,600 lines, just like my laser beam did. 588 00:40:02,000 --> 00:40:04,991 Because it is the same three millimeters. 589 00:40:04,991 --> 00:40:08,207 You look through about 1,600 of these lines. 590 00:40:08,207 --> 00:40:12,470 And you know what the d is, the separation of the grooves. 591 00:40:12,470 --> 00:40:16,583 And so, you can now look at a line source, which we have 592 00:40:16,583 --> 00:40:19,425 prepared. We have prepared line sources 593 00:40:19,425 --> 00:40:22,416 which are very narrow, not like this one, 594 00:40:22,416 --> 00:40:24,959 but very narrow, which are in here. 595 00:40:24,959 --> 00:40:28,997 Now, if you look at that light source, which is helium, 596 00:40:28,997 --> 00:40:33,260 and now you use your grating, then you begin to understand 597 00:40:33,260 --> 00:40:39,665 the idea of spectral resolution. You will now begin to see the 598 00:40:39,665 --> 00:40:45,272 individual atomic lines nicely separated through your grating. 599 00:40:45,272 --> 00:40:50,602 And so we will turn this on. And we will make it completely 600 00:40:50,602 --> 00:40:53,084 dark. And then I want you to 601 00:40:53,084 --> 00:40:58,415 appreciate this and spend some time looking at these lines. 602 00:40:58,415 --> 00:41:04,040 It is really quite remarkable. And this, of course, 603 00:41:04,040 --> 00:41:09,040 you could never do with the two slit interference. 604 00:41:09,040 --> 00:41:12,204 You need these many, many lines. 605 00:41:12,204 --> 00:41:18,224 You look through about 1,600. This very strong yellow in the 606 00:41:18,224 --> 00:41:24,448 helium is a well-known helium line, and it has a wavelength of 607 00:41:24,448 --> 00:41:28,020 587 nanometers. It is the brightest, 608 00:41:28,020 --> 00:41:34,561 the strongest one in helium. And you can see them in first 609 00:41:34,561 --> 00:41:38,006 order, you can see them in second order. 610 00:41:38,006 --> 00:41:42,334 And then, gradually, when you go to higher orders, 611 00:41:42,334 --> 00:41:47,457 the various colors begin to overlap with each other because 612 00:41:47,457 --> 00:41:52,757 they each live their own lives. The angles are only dependant 613 00:41:52,757 --> 00:41:56,731 on lambda over d. And now I can show you neon, 614 00:41:56,731 --> 00:42:00,000 which has even more lines. 615 00:42:00,000 --> 00:42:05,000 616 00:42:05,000 --> 00:42:07,720 You can look at your grating. Amazing. 617 00:42:07,720 --> 00:42:11,985 I mean, the grating does not cost more than maybe a dollar. 618 00:42:11,985 --> 00:42:15,661 It is absolutely stunning. And it has an incredible 619 00:42:15,661 --> 00:42:20,000 spectral resolution already because you need a prepared line 620 00:42:20,000 --> 00:42:23,823 source to take advantage of that spectral resolution. 621 00:42:23,823 --> 00:42:26,911 And, as I said, you will probably approach, 622 00:42:26,911 --> 00:42:30,882 certainly the audience all the way in the back of 6120, 623 00:42:30,882 --> 00:42:35,000 you will approach this angular resolution. 624 00:42:35,000 --> 00:42:39,077 The ones that are closer may not approach it because they see 625 00:42:39,077 --> 00:42:41,320 the line source, of course, wider. 626 00:42:41,320 --> 00:42:45,466 The angle at which they see the line source may well be larger 627 00:42:45,466 --> 00:42:48,660 than this width, but the ones in the back of the 628 00:42:48,660 --> 00:42:51,718 audience are there for a little [better off?]. 629 00:42:51,718 --> 00:42:55,728 The angle that you see to the line source, the widths of the 630 00:42:55,728 --> 00:42:59,194 line source become smaller the farther you are away. 631 00:42:59,194 --> 00:43:03,000 This is remarkable. Absolutely incredible. 632 00:43:03,000 --> 00:43:10,000 633 00:43:10,000 --> 00:43:15,090 I think this is a great moment to rest and to digest this 634 00:43:15,090 --> 00:43:19,818 wonderful experience and to have a four-minute break. 635 00:43:19,818 --> 00:43:24,642 Thank you very much. What I want to discuss now is 636 00:43:24,642 --> 00:43:30,382 the logical consequence of this whole concept of Huygens sources 637 00:43:30,382 --> 00:43:35,667 where spherical waves come from each point in the aperture. 638 00:43:35,667 --> 00:43:40,496 And we are now going to extend it to a single opening. 639 00:43:40,496 --> 00:43:43,685 Not multiple but one single opening. 640 00:43:43,685 --> 00:43:48,423 And the opening is now d. This separation is now this 641 00:43:48,423 --> 00:43:49,881 opening. It is d. 642 00:43:49,881 --> 00:43:54,346 Think of them as being a slit which has a width d. 643 00:43:54,346 --> 00:43:59,266 It is open, single-slit. And we have plane waves coming 644 00:43:59,266 --> 00:44:04,900 in like this. And now the question is if I 645 00:44:04,900 --> 00:44:10,492 look in various directions, and that is my famous angle 646 00:44:10,492 --> 00:44:16,603 theta, what will I see now on a screen that I place very far 647 00:44:16,603 --> 00:44:19,607 away? Well, each point in this 648 00:44:19,607 --> 00:44:25,821 aperture can now be considered, according the Huygens-Fresnel 649 00:44:25,821 --> 00:44:31,000 Principle as a source of spherical waves. 650 00:44:31,000 --> 00:44:33,542 And they are going to interfere with each other. 651 00:44:33,542 --> 00:44:35,652 Strangely enough, for reasons beyond me, 652 00:44:35,652 --> 00:44:38,412 we call this diffraction, but it is exactly the same 653 00:44:38,412 --> 00:44:41,442 phenomenon as interference. We draw a strange distinction 654 00:44:41,442 --> 00:44:44,688 in physics between interference which was the grating and the 655 00:44:44,688 --> 00:44:47,069 double-slit, we say double-slit interference. 656 00:44:47,069 --> 00:44:49,503 No one would ever say double-slit diffraction, 657 00:44:49,503 --> 00:44:52,263 but it is the same thing. Somehow, when we deal with 658 00:44:52,263 --> 00:44:54,643 individual openings we call that diffraction. 659 00:44:54,643 --> 00:44:57,132 It is the same. You can use it any way you want 660 00:44:57,132 --> 00:45:00,000 to. You can call it interference. 661 00:45:00,000 --> 00:45:04,010 But the individual Huygens sources, there they are, 662 00:45:04,010 --> 00:45:07,219 they are all going to do their own thing. 663 00:45:07,219 --> 00:45:12,192 And I pick this one at the top, number one, and I pick this one 664 00:45:12,192 --> 00:45:16,283 number two right in the middle. Why don't I do that? 665 00:45:16,283 --> 00:45:20,695 You will see why I do that. Now I can calculate what the 666 00:45:20,695 --> 00:45:25,267 past difference is between the Huygens source right in the 667 00:45:25,267 --> 00:45:30,000 middle and the Huygens source right at the top. 668 00:45:30,000 --> 00:45:34,879 Well, this past difference here is clearly one-half d times the 669 00:45:34,879 --> 00:45:37,476 sine of theta. We did that before. 670 00:45:37,476 --> 00:45:41,726 We had a little d here. If I make that one-half lambda, 671 00:45:41,726 --> 00:45:45,898 I claim that in that direction there will be darkness. 672 00:45:45,898 --> 00:45:50,305 Why will there be darkness? Because, if source number one 673 00:45:50,305 --> 00:45:55,106 can kill number two because they are 180 degrees out of phase, 674 00:45:55,106 --> 00:45:59,907 then the source just below one can kill this one below two and 675 00:45:59,907 --> 00:46:04,000 the one below there can kill this one. 676 00:46:04,000 --> 00:46:08,059 So I can always identify two pairs which kill each other. 677 00:46:08,059 --> 00:46:11,539 That means darkness. This must be a criterion for 678 00:46:11,539 --> 00:46:15,309 destructive interference. Now, it may not be the only 679 00:46:15,309 --> 00:46:18,862 angle for which there is destructive interference. 680 00:46:18,862 --> 00:46:23,357 But I want to convince you that there is at least one for which 681 00:46:23,357 --> 00:46:26,619 you see no light. I will now introduce a phase 682 00:46:26,619 --> 00:46:30,897 angle beta, which is the phase difference between source one 683 00:46:30,897 --> 00:46:34,471 and source two. If this is the slit, 684 00:46:34,471 --> 00:46:37,930 one side of the slit and the middle of the slit, 685 00:46:37,930 --> 00:46:42,125 it is not the phase angle between two neighboring sources. 686 00:46:42,125 --> 00:46:45,658 Because the neighboring sources touch each other. 687 00:46:45,658 --> 00:46:49,043 There is an infinite number of Huygens sources. 688 00:46:49,043 --> 00:46:51,471 It is a continuous Huygens source. 689 00:46:51,471 --> 00:46:55,666 It is the phase angle between the edge of the slit and the 690 00:46:55,666 --> 00:46:59,272 center of the slit. That is the way I define beta. 691 00:46:59,272 --> 00:47:03,835 And so my beta now becomes 2pi divided by lambda times one-half 692 00:47:03,835 --> 00:47:09,223 d times the sine of theta. And so, you see the one-half 693 00:47:09,223 --> 00:47:14,226 eats up the two so you get pi d divided by lambda times the sine 694 00:47:14,226 --> 00:47:17,244 of theta. And now I will not derive for 695 00:47:17,244 --> 00:47:20,420 you as I did. Precisely for the gratings, 696 00:47:20,420 --> 00:47:24,788 I will not derive for you what the light intensity is at 697 00:47:24,788 --> 00:47:28,679 function of angle. Again, it is a matter of adding 698 00:47:28,679 --> 00:47:31,458 vectors. But you can look that up in 699 00:47:31,458 --> 00:47:36,050 Becefi and Barrett. It is done in Section 8.7. 700 00:47:36,050 --> 00:47:39,126 And so, I will only give you the answer. 701 00:47:39,126 --> 00:47:42,360 I did it for the grating. You do this one. 702 00:47:42,360 --> 00:47:46,461 And you can show now that the intensity of the light, 703 00:47:46,461 --> 00:47:51,115 as a function of that phase angle beta, is I0 times the sine 704 00:47:51,115 --> 00:47:54,191 of beta divided by beta. And, of course, 705 00:47:54,191 --> 00:47:58,766 no surprise that you get a square there because that has to 706 00:47:58,766 --> 00:48:04,095 do with the pointing vector. And this function is very 707 00:48:04,095 --> 00:48:06,862 different from a grating function. 708 00:48:06,862 --> 00:48:11,305 And, before I plot it, let me make a few calculations. 709 00:48:11,305 --> 00:48:16,167 I will do it here on the center board so that you can still 710 00:48:16,167 --> 00:48:20,862 compare with this equation. I am going to write down here 711 00:48:20,862 --> 00:48:25,473 what the sine of theta is. And then here my column comes 712 00:48:25,473 --> 00:48:28,491 beta. And then here comes the sine of 713 00:48:28,491 --> 00:48:32,384 beta. And here comes the intensity I. 714 00:48:32,384 --> 00:48:35,307 I first take the sine of theta as zero. 715 00:48:35,307 --> 00:48:39,923 Well, if the sine of theta is zero, it is immediately obvious 716 00:48:39,923 --> 00:48:43,692 that beta is zero. And it is also obvious that the 717 00:48:43,692 --> 00:48:47,307 sine of beta is zero. So you get zero divided by 718 00:48:47,307 --> 00:48:51,307 zero, you use l'Hopital, and this ration becomes one. 719 00:48:51,307 --> 00:48:53,846 And now the light intensity is I0. 720 00:48:53,846 --> 00:48:58,384 Well, that is not so surprising that you have there a lot of 721 00:48:58,384 --> 00:49:01,233 light. Because, of course, 722 00:49:01,233 --> 00:49:05,000 if you have an opening here and you shine light through it, 723 00:49:05,000 --> 00:49:08,376 then you expect that you see right on the wall there, 724 00:49:08,376 --> 00:49:12,142 in the middle of the slit you expect to see a lot of light. 725 00:49:12,142 --> 00:49:15,129 So that is not so surprising. This, by the way, 726 00:49:15,129 --> 00:49:18,896 is the way that we define I0. I0 is defined as that maximum 727 00:49:18,896 --> 00:49:22,662 that you will see when you so-to-speak look straightforward 728 00:49:22,662 --> 00:49:25,844 at angle theta zero. That is the way we define I0. 729 00:49:25,844 --> 00:49:30,000 Now let's take sine theta as lambda divided by D. 730 00:49:30,000 --> 00:49:35,201 Now beta is pi. We put in here sine theta is 731 00:49:35,201 --> 00:49:38,951 lambda divided by D. Here it is. 732 00:49:38,951 --> 00:49:45,725 So you see that beta is pi. The sine of beta is now zero. 733 00:49:45,725 --> 00:49:51,048 And, therefore, the intensity is zero because 734 00:49:51,048 --> 00:49:57,217 the upstairs is zero but the downstairs is not zero. 735 00:49:57,217 --> 00:50:00,914 That is this. Because, look, 736 00:50:00,914 --> 00:50:04,441 if you take this half away and you take this half away, 737 00:50:04,441 --> 00:50:07,903 you get that the sine of theta is lambda divided by D. 738 00:50:07,903 --> 00:50:11,168 I already predicted that you would have destructive 739 00:50:11,168 --> 00:50:13,716 interference. That is exactly this case, 740 00:50:13,716 --> 00:50:16,002 of course. And now I have two lambda 741 00:50:16,002 --> 00:50:18,614 divided by D. That gives me a beta now of 742 00:50:18,614 --> 00:50:22,729 2pi, that gives me again a zero here, and there is another zero. 743 00:50:22,729 --> 00:50:26,387 There are more locations in space where there is complete 744 00:50:26,387 --> 00:50:29,000 darkness. Not just one. 745 00:50:29,000 --> 00:50:32,552 In fact, there is an infinite number of them. 746 00:50:32,552 --> 00:50:36,750 You can go on like this. I will first plot the curve, 747 00:50:36,750 --> 00:50:41,190 and then we will discuss it in a little bit more detail. 748 00:50:41,190 --> 00:50:46,195 And I will only plot curves in terms of sine theta because that 749 00:50:46,195 --> 00:50:48,778 is an angle that I can relate to. 750 00:50:48,778 --> 00:50:51,361 I can tell my mother about theta. 751 00:50:51,361 --> 00:50:54,752 I cannot tell my mother about phase angles, 752 00:50:54,752 --> 00:51:00,000 but I can tell her about theta. Mom, this is theta. 753 00:51:00,000 --> 00:51:02,020 10 degrees. 20 degrees. 754 00:51:02,020 --> 00:51:05,419 30 degrees. That is a real angle in my 755 00:51:05,419 --> 00:51:08,175 laboratory. That is that theta. 756 00:51:08,175 --> 00:51:12,769 That I can relate to. I always plot things in terms 757 00:51:12,769 --> 00:51:16,259 of sine theta. And so, here is my zero. 758 00:51:16,259 --> 00:51:20,669 And let this be lambda divided by that capital D. 759 00:51:20,669 --> 00:51:25,078 That is the slit. This is two lambda divided by D 760 00:51:25,078 --> 00:51:30,774 and here is three lambda divided by D, and the other side minus 761 00:51:30,774 --> 00:51:36,620 lambda divided by D and so on. I will put one more in. 762 00:51:36,620 --> 00:51:40,989 And so, now the curve that you see, not so obvious but when you 763 00:51:40,989 --> 00:51:44,653 plot it you will see that, you get here this maximum, 764 00:51:44,653 --> 00:51:48,246 which we define to be I0. It is the central maximum. 765 00:51:48,246 --> 00:51:51,205 And then you will get here your first zero. 766 00:51:51,205 --> 00:51:54,024 You get here some kind of a mini-maximum. 767 00:51:54,024 --> 00:51:56,420 And then you get another zero here. 768 00:51:56,420 --> 00:52:01,000 And you get an infinite number of zeroes every time. 769 00:52:01,000 --> 00:52:04,308 You get three lambda over D, four lambda over D, 770 00:52:04,308 --> 00:52:07,616 five lambda over D, on this side here and so on. 771 00:52:07,616 --> 00:52:11,065 The central maximum, the width in terms of angles, 772 00:52:11,065 --> 00:52:14,655 this of this as angles. Right sine theta is close to 773 00:52:14,655 --> 00:52:18,385 theta in terms of radians, so this is an angular size. 774 00:52:18,385 --> 00:52:22,750 The linear size depends on how far you are away from the screen 775 00:52:22,750 --> 00:52:26,339 on which you show it. Then you have to multiply this 776 00:52:26,339 --> 00:52:28,381 by L. L is the distance to the 777 00:52:28,381 --> 00:52:32,688 screen. This is the angular side. 778 00:52:32,688 --> 00:52:35,806 This width here, very crudely, 779 00:52:35,806 --> 00:52:40,645 is about half this. And so that width is about 780 00:52:40,645 --> 00:52:46,129 lambda divided by D. Let's take an example which is, 781 00:52:46,129 --> 00:52:52,795 I think, the demonstration that I have lined up for you anyhow. 782 00:52:52,795 --> 00:53:00,000 We will take a laser light, which is about 600 nanometers. 783 00:53:00,000 --> 00:53:05,465 The fact that it is 633, of course, is not so important. 784 00:53:05,465 --> 00:53:10,832 I give you easy numbers. And suppose we have d which is 785 00:53:10,832 --> 00:53:15,602 about 0.1 millimeters. We have a slit which has a 786 00:53:15,602 --> 00:53:19,279 width opening of only 0.1 millimeters. 787 00:53:19,279 --> 00:53:25,341 And we will put a screen there, L at a distance of about three 788 00:53:25,341 --> 00:53:28,521 meters. We can calculate now what 789 00:53:28,521 --> 00:53:34,697 lambda divided by D is. That would give you the angle 790 00:53:34,697 --> 00:53:38,232 in radians. That is six times 10 to the 791 00:53:38,232 --> 00:53:40,000 minus 3 radians. 792 00:53:40,000 --> 00:53:45,000 793 00:53:45,000 --> 00:53:47,719 Sine theta is very close to theta. 794 00:53:47,719 --> 00:53:52,664 And so, now you can calculate the linear size of this central 795 00:53:52,664 --> 00:53:57,115 maximum, as I am going to show you there on the screen. 796 00:53:57,115 --> 00:54:02,834 And the linear size is now L. The linear dimension is L times 797 00:54:02,834 --> 00:54:06,765 lambda over D. That is how wide that central 798 00:54:06,765 --> 00:54:10,331 maximum will be. And that, in this case, 799 00:54:10,331 --> 00:54:13,440 will then be about two centimeters. 800 00:54:13,440 --> 00:54:17,645 Now, think about it. Think about the absurdity. 801 00:54:17,645 --> 00:54:22,400 We have a slit which has an opening of one-tenth of a 802 00:54:22,400 --> 00:54:25,051 millimeter. And because of Mr. 803 00:54:25,051 --> 00:54:32,000 Huygens, it will show up there with a width of two centimeters. 804 00:54:32,000 --> 00:54:34,722 Two hundred times broader than the actual opening. 805 00:54:34,722 --> 00:54:37,944 Whereas, if you would think high school, then you would say 806 00:54:37,944 --> 00:54:41,000 if you have light going through a tenth of a millimeter, 807 00:54:41,000 --> 00:54:44,055 you look, what you see on the wall would be a tenth of a 808 00:54:44,055 --> 00:54:46,222 millimeter. No, you see two centimeters. 809 00:54:46,222 --> 00:54:48,277 And that is the result of diffraction. 810 00:54:48,277 --> 00:54:51,666 That is the result of the fact that each one of those sources, 811 00:54:51,666 --> 00:54:54,277 in this aperture, are going to radiate spherical 812 00:54:54,277 --> 00:54:56,111 waves. They are going to interfere 813 00:54:56,111 --> 00:55:00,606 with each other. And they then cause this huge 814 00:55:00,606 --> 00:55:04,202 broad center. And the smaller you make D, 815 00:55:04,202 --> 00:55:09,325 the more you tighten the nuts on the slit, the wider it is 816 00:55:09,325 --> 00:55:11,662 going to be. Because, look, 817 00:55:11,662 --> 00:55:17,146 if you make this smaller d then this angle will become larger. 818 00:55:17,146 --> 00:55:21,101 Very known intuitive. Before I show you this, 819 00:55:21,101 --> 00:55:25,415 I want to know roughly what that maximum is here. 820 00:55:25,415 --> 00:55:30,000 That mini-maximum. Well, that is easy. 821 00:55:30,000 --> 00:55:36,271 You could do that now on your own, because all you have to do 822 00:55:36,271 --> 00:55:42,229 is substitute in this equation sine theta which is sort of 823 00:55:42,229 --> 00:55:45,679 half-way in between. If I do that, 824 00:55:45,679 --> 00:55:50,174 halfway in between, that is 1.5 times lambda 825 00:55:50,174 --> 00:55:54,668 divided by D. That is right in between these 826 00:55:54,668 --> 00:55:58,013 two. That gives me then a beta of 827 00:55:58,013 --> 00:56:02,848 1.5 pi. I can go to this equation and 828 00:56:02,848 --> 00:56:06,678 put in for sine theta 1.5 lambda over d. 829 00:56:06,678 --> 00:56:10,901 The lambda and the d cancel, you get 1.5 pi. 830 00:56:10,901 --> 00:56:14,732 Straightforward. Just turning the crank. 831 00:56:14,732 --> 00:56:19,544 Now you have beta. You can calculate what the sine 832 00:56:19,544 --> 00:56:25,437 of beta is, which is minus one. And so, now you know what the 833 00:56:25,437 --> 00:56:31,428 sine of beta divided by beta is. And you will find 0.045 times 834 00:56:31,428 --> 00:56:35,300 I0. This mini-maximum is 4.5% of 835 00:56:35,300 --> 00:56:39,243 the central maximum. And when you go further out, 836 00:56:39,243 --> 00:56:43,762 these maxima are even smaller. But, when I show you this 837 00:56:43,762 --> 00:56:48,198 phenomenon, which I will, you will see distinctly these 838 00:56:48,198 --> 00:56:51,237 zeros. You will see actually very nice 839 00:56:51,237 --> 00:56:54,770 dark locations. And you will see the central 840 00:56:54,770 --> 00:56:59,864 maximum and then a little bit of light, but not very much light 841 00:56:59,864 --> 00:57:03,019 on this side. And, of course, 842 00:57:03,019 --> 00:57:05,398 again, this is wavelength dependant. 843 00:57:05,398 --> 00:57:08,932 So whether you do this in red light or in blue light, 844 00:57:08,932 --> 00:57:11,446 you will see something very different. 845 00:57:11,446 --> 00:57:14,844 If you do it in red light, you will see this white. 846 00:57:14,844 --> 00:57:18,378 If you do it in blue light, you will see it narrower. 847 00:57:18,378 --> 00:57:22,184 And also those locations, of course, are then further in. 848 00:57:22,184 --> 00:57:25,922 And I have a slide which shows you the idea in different 849 00:57:25,922 --> 00:57:28,368 colors. Maybe we can make it a little 850 00:57:28,368 --> 00:57:31,963 darker. Here you see it in three 851 00:57:31,963 --> 00:57:36,707 colors, red, green and blue. Notice how very different this 852 00:57:36,707 --> 00:57:40,306 is from a grating. You see some broad central 853 00:57:40,306 --> 00:57:43,250 maximum. That is that central maximum 854 00:57:43,250 --> 00:57:46,113 that you see here on the blackboard. 855 00:57:46,113 --> 00:57:50,693 And then you see the sharp black locations where there is 856 00:57:50,693 --> 00:57:54,701 almost no light here. And you see infrared farther 857 00:57:54,701 --> 00:57:58,627 apart than for green. And for green farther apart 858 00:57:58,627 --> 00:58:03,992 than for blue. And then, when you do it with 859 00:58:03,992 --> 00:58:09,153 white light, of course, then it becomes always more 860 00:58:09,153 --> 00:58:13,074 difficult. You see the sharp dark areas 861 00:58:13,074 --> 00:58:16,893 because the colors overlap. All right. 862 00:58:16,893 --> 00:58:20,711 Now I want to demonstrate this to you. 863 00:58:20,711 --> 00:58:26,800 And the way that we are going to do this is with a slit that 864 00:58:26,800 --> 00:58:33,521 we can vary in size. Where was my calculation? 865 00:58:33,521 --> 00:58:41,408 I made a calculation here. I am going to do it with a 633 866 00:58:41,408 --> 00:58:48,732 nanometer laser light. That the beam of that laser is 867 00:58:48,732 --> 00:58:56,197 about three millimeters, and then we have a slit here. 868 00:58:56,197 --> 00:59:01,260 This is the opening. This is the d. 869 00:59:01,260 --> 00:59:06,214 And we can make the d smaller. And so, we already made a 870 00:59:06,214 --> 00:59:09,906 prediction that the linear dimension then, 871 00:59:09,906 --> 00:59:14,049 if the opening is only a tenth of a millimeter, 872 00:59:14,049 --> 00:59:19,093 you would expect that the central maximum on that screen, 873 00:59:19,093 --> 00:59:24,136 which is about three meters away, that is why I chose the 874 00:59:24,136 --> 00:59:29,000 three meters, will be two centimeters wide. 875 00:59:29,000 --> 00:59:37,313 But I can make it way wider because I can make d way smaller 876 00:59:37,313 --> 00:59:45,345 than a tenth of a millimeter. What you see now is that the 877 00:59:45,345 --> 00:59:49,431 slit is very large, very open. 878 00:59:49,431 --> 00:59:54,786 I don't know, maybe a millimeter or so. 879 00:59:54,786 --> 1:00:01,696 And I am going to tighten it. And I will stop at one point 880 1:00:01,696 --> 1:00:03,939 here where it is a nice point to stop. 881 1:00:03,939 --> 1:00:05,939 Here you see that central maximum. 882 1:00:05,939 --> 1:00:09,090 See how powerful and overwhelming that is in terms of 883 1:00:09,090 --> 1:00:11,333 its brightness? We understand now why, 884 1:00:11,333 --> 1:00:14,666 because of this crazy function sine beta divided by beta 885 1:00:14,666 --> 1:00:16,545 squared. Already now it is here, 886 1:00:16,545 --> 1:00:18,484 oh, I would say five centimeters. 887 1:00:18,484 --> 1:00:21,939 Already now this slit width must be less than a tenth of a 888 1:00:21,939 --> 1:00:25,515 millimeter, because it would be two centimeters if it were a 889 1:00:25,515 --> 1:00:29,481 tenth of a millimeter. And I hope you can see 890 1:00:29,481 --> 1:00:33,655 distinctly those dark locations. And you see there are a lot of 891 1:00:33,655 --> 1:00:35,203 them. But, keep in mind, 892 1:00:35,203 --> 1:00:39,243 that this mini-maxima next to the broad maximum is only 4.5%. 893 1:00:39,243 --> 1:00:42,879 And it gets smaller and smaller, lower and lower as you 894 1:00:42,879 --> 1:00:45,908 go further away. Now I go way beyond one-tenth 895 1:00:45,908 --> 1:00:47,794 of a millimeter, way smaller. 896 1:00:47,794 --> 1:00:51,900 Now, keep in mind that when I make the slit width smaller less 897 1:00:51,900 --> 1:00:54,661 light will go through. I cannot help that. 898 1:00:54,661 --> 1:00:57,017 The whole image will become fainter. 899 1:00:57,017 --> 1:01:02,000 That is the price I pay for letting less light through. 900 1:01:02,000 --> 1:01:06,513 But what I gain is to show you the absurdity that the center 901 1:01:06,513 --> 1:01:11,409 maximum gets wider and wider and wider as I make the slit smaller 902 1:01:11,409 --> 1:01:15,387 and smaller and smaller. The slit is now narrower and 903 1:01:15,387 --> 1:01:19,978 narrower and narrow and narrows, and the central maximum that 904 1:01:19,978 --> 1:01:24,568 you see there is almost a foot. The center, the opening of my 905 1:01:24,568 --> 1:01:30,000 slit must now be something like maybe only ten microns or so. 906 1:01:30,000 --> 1:01:33,622 This is a very highly accurate device, whereby, 907 1:01:33,622 --> 1:01:36,929 we have the option of making the slit with, 908 1:01:36,929 --> 1:01:39,370 indeed, as small as ten microns. 909 1:01:39,370 --> 1:01:42,755 And you see this is the result that you get. 910 1:01:42,755 --> 1:01:47,165 I will now make the slit open and open and open and open. 911 1:01:47,165 --> 1:01:51,338 And here we have the point. If I make the center point 912 1:01:51,338 --> 1:01:56,141 about two centimeters then the slit width, which is about now, 913 1:01:56,141 --> 1:02:00,000 is about one-tenth of a millimeter. 914 1:02:00,000 --> 1:02:05,000 915 1:02:05,000 --> 1:02:11,103 I now have to make an important confession about the grating 916 1:02:11,103 --> 1:02:14,620 equation. Some of you who were very 917 1:02:14,620 --> 1:02:21,137 observant may have noticed that the maxima of the grating that I 918 1:02:21,137 --> 1:02:26,931 showed, also when I did the experiment with my own laser, 919 1:02:26,931 --> 1:02:32,000 were not all exactly the same brightness. 920 1:02:32,000 --> 1:02:35,978 There was a difference. And no one asked me about it, 921 1:02:35,978 --> 1:02:38,885 and I was hoping that no one would ask. 922 1:02:38,885 --> 1:02:43,628 And the reason for that is that each one of those grooves has a 923 1:02:43,628 --> 1:02:47,453 finite size opening. And each one of those openings 924 1:02:47,453 --> 1:02:50,284 acts this way. They cause diffraction. 925 1:02:50,284 --> 1:02:53,803 We call that diffraction. It is just semantics. 926 1:02:53,803 --> 1:02:58,010 And so, superimposed on the grating equation this causes 927 1:02:58,010 --> 1:03:02,530 diffraction. And the net result then is that 928 1:03:02,530 --> 1:03:06,693 you get the product of the two. If I amend here now, 929 1:03:06,693 --> 1:03:11,673 and I am removing this beta now because this is for a grating, 930 1:03:11,673 --> 1:03:16,326 but the beta is defined this way, this d is the opening of 931 1:03:16,326 --> 1:03:21,061 each groove in your grating. And little d is the separation 932 1:03:21,061 --> 1:03:25,142 between the grating. Then I can write down now here 933 1:03:25,142 --> 1:03:30,204 times the sine of N delta over two divided by the sine of delta 934 1:03:30,204 --> 1:03:34,485 over two. And now I put here the square. 935 1:03:34,485 --> 1:03:37,800 And now I have the real grating equation. 936 1:03:37,800 --> 1:03:42,440 And so, what you see now is that since little d is always 937 1:03:42,440 --> 1:03:46,665 larger than capital d, you are going to see that the 938 1:03:46,665 --> 1:03:51,719 maxima, which comes from this equation, are being modulated by 939 1:03:51,719 --> 1:03:54,951 this one. And so, if this were a grating 940 1:03:54,951 --> 1:03:59,177 whereby capital d was the opening of each individual 941 1:03:59,177 --> 1:04:02,843 groove -- And if the grating wanted, 942 1:04:02,843 --> 1:04:06,248 of course, the zero or the maxima is always here, 943 1:04:06,248 --> 1:04:10,574 and if the first order maximum grating would fall here and the 944 1:04:10,574 --> 1:04:15,042 second order would fall here and the third here then this is the 945 1:04:15,042 --> 1:04:19,368 price you pay for the fact that these grooves have an opening. 946 1:04:19,368 --> 1:04:23,482 And so, you see a modulation in the strength of your first, 947 1:04:23,482 --> 1:04:26,106 second, third, fourth order and so on. 948 1:04:26,106 --> 1:04:31,000 And so, when you look carefully at the grating lights -- 949 1:04:31,000 --> 1:04:33,825 And I will demonstrate that to you. 950 1:04:33,825 --> 1:04:37,979 They are not 16 times I0 if you have N equals four. 951 1:04:37,979 --> 1:04:42,716 And if you have N equals 1,000, they are not all a million 952 1:04:42,716 --> 1:04:46,040 times pi zero, but they have this overall 953 1:04:46,040 --> 1:04:50,776 envelope which modulates it. And that is the result of the 954 1:04:50,776 --> 1:04:55,014 finite opening of the grooves. To make sure that you 955 1:04:55,014 --> 1:05:00,000 understand the difference between the two D's -- 956 1:05:00,000 --> 1:05:05,000 957 1:05:05,000 --> 1:05:09,669 If this is my grating and this is the open area, 958 1:05:09,669 --> 1:05:13,445 this is, say, where the light cannot go 959 1:05:13,445 --> 1:05:19,605 through, then the definition of D is this and the definition of 960 1:05:19,605 --> 1:05:23,679 this is d. And that d shows up in here and 961 1:05:23,679 --> 1:05:26,858 this capital D shows up in there. 962 1:05:26,858 --> 1:05:32,125 So this is the single-slit diffraction and this is the 963 1:05:32,125 --> 1:05:38,575 multiple-slit interference. There you see again we make the 964 1:05:38,575 --> 1:05:43,060 distinction in wording, but that has no meaning because 965 1:05:43,060 --> 1:05:45,718 it is all diffraction, of course. 966 1:05:45,718 --> 1:05:49,789 If somehow D were approximately d divided by five, 967 1:05:49,789 --> 1:05:54,939 and it just so happens we have a grating here for which that is 968 1:05:54,939 --> 1:06:00,006 the case, then the fifth order maximum of the grating is going 969 1:06:00,006 --> 1:06:03,745 to be killed. Because that is when this 970 1:06:03,745 --> 1:06:07,097 function becomes zero. So you would see then zero 971 1:06:07,097 --> 1:06:08,982 order one, two, three, four, 972 1:06:08,982 --> 1:06:12,892 five would be killed here. And then they would build up a 973 1:06:12,892 --> 1:06:15,266 little again. And then, ultimately, 974 1:06:15,266 --> 1:06:17,710 of course, they would all peter out. 975 1:06:17,710 --> 1:06:20,852 And so, if I show you a spectrum of a grating, 976 1:06:20,852 --> 1:06:24,693 you can actually roughly estimate what the ratio capital 977 1:06:24,693 --> 1:06:30,000 D over little d is by seeing sort of that modulation pattern. 978 1:06:30,000 --> 1:06:33,280 And so that is what I want you to see now. 979 1:06:33,280 --> 1:06:37,440 It is not so exciting. Many of you who were observant 980 1:06:37,440 --> 1:06:41,360 may have seen it anyhow, because it was every time 981 1:06:41,360 --> 1:06:44,559 there, even when I showed my own grating. 982 1:06:44,559 --> 1:06:48,000 We are going to make it quite dark for this. 983 1:06:48,000 --> 1:06:52,320 This is the wrong switch. I have so many switches here. 984 1:06:52,320 --> 1:06:55,199 There we go. This is a grating that I 985 1:06:55,199 --> 1:06:58,880 purposely offset. I purposely offset it so that 986 1:06:58,880 --> 1:07:03,440 here is the zero order. This is the zero order. 987 1:07:03,440 --> 1:07:05,444 I think it is that one, actually. 988 1:07:05,444 --> 1:07:08,012 It is easy to test where my zero order is. 989 1:07:08,012 --> 1:07:10,893 This is the zero order, so we aim exactly here. 990 1:07:10,893 --> 1:07:13,711 And so, this is the first order, second order, 991 1:07:13,711 --> 1:07:15,277 third order, fourth order. 992 1:07:15,277 --> 1:07:17,657 Look at this sucker. It is almost gone. 993 1:07:17,657 --> 1:07:21,540 That is the result of the fact of the single-slit interference. 994 1:07:21,540 --> 1:07:23,482 And then it comes up again here. 995 1:07:23,482 --> 1:07:27,240 And the reason why it comes up again is because now you enter 996 1:07:27,240 --> 1:07:32,000 this little mini-maximum in the single-slit interference. 997 1:07:32,000 --> 1:07:36,524 You see it here quite well. Sometimes with gratings you can 998 1:07:36,524 --> 1:07:40,425 see it remarkably well. Other times it is harder to 999 1:07:40,425 --> 1:07:42,453 see. It depends, of course, 1000 1:07:42,453 --> 1:07:47,056 on how many maxima you have. But here you see quite well the 1001 1:07:47,056 --> 1:07:50,333 modulation. You see it comes up here again. 1002 1:07:50,333 --> 1:07:54,780 And this little point here would then be somewhere here in 1003 1:07:54,780 --> 1:07:57,744 this maximum. This is then the complete 1004 1:07:57,744 --> 1:08:01,645 equation that combines single-slit diffraction with 1005 1:08:01,645 --> 1:08:07,482 multiple-slit interference. If we change a single opening 1006 1:08:07,482 --> 1:08:11,335 from a slit to a circle, your eye is a circle, 1007 1:08:11,335 --> 1:08:15,359 your pupil is a circle, then very little change, 1008 1:08:15,359 --> 1:08:20,496 except of course if you have a circular opening everything is 1009 1:08:20,496 --> 1:08:24,691 now axial symmetric. And so, you will get circles. 1010 1:08:24,691 --> 1:08:30,000 These things become circles, which is not so obvious. 1011 1:08:30,000 --> 1:08:34,827 And then, which is not so obvious, this minimum does not 1012 1:08:34,827 --> 1:08:39,479 fall at lambda over two in terms of angular dimension, 1013 1:08:39,479 --> 1:08:44,833 but 1.22 times lambda over two. And, if you want to use [1.2?] 1014 1:08:44,833 --> 1:08:48,344 for an approximation that is fine enough. 1015 1:08:48,344 --> 1:08:53,435 It is a little larger for a circular opening than it is for 1016 1:08:53,435 --> 1:08:56,156 a slit. I have here one of those 1017 1:08:56,156 --> 1:09:00,292 pinholes. So this is now a circular 1018 1:09:00,292 --> 1:09:04,878 opening for which this relation has to be used now. 1019 1:09:04,878 --> 1:09:08,088 This central is then a little wider. 1020 1:09:08,088 --> 1:09:12,582 And we are about four meters away from the screen. 1021 1:09:12,582 --> 1:09:16,251 It is this one, four meters away from the 1022 1:09:16,251 --> 1:09:19,644 screen. And I am going to do this with 1023 1:09:19,644 --> 1:09:23,038 a wavelength lambda of 594 nanometers. 1024 1:09:23,038 --> 1:09:29,000 It is a circular opening. We will call it a pinhole. 1025 1:09:29,000 --> 1:09:32,011 And lambda is about 594 nanometers. 1026 1:09:32,011 --> 1:09:36,174 It is also a laser. And the distance through the 1027 1:09:36,174 --> 1:09:41,399 screen L is about four meters. And what you are going to see 1028 1:09:41,399 --> 1:09:46,448 is a ring, which is this ring, and then you see this light 1029 1:09:46,448 --> 1:09:51,762 inside, which is very difficult for me, this is the very high 1030 1:09:51,762 --> 1:09:54,774 maximum. And then you will see more 1031 1:09:54,774 --> 1:09:57,608 rings outside. This ring is quite 1032 1:09:57,608 --> 1:10:01,861 well-defined. You are going to see that very 1033 1:10:01,861 --> 1:10:03,909 sharply defined. And if this ring, 1034 1:10:03,909 --> 1:10:06,702 which I measured, is about five centimeters in 1035 1:10:06,702 --> 1:10:10,611 diameter, you should be able to tell me what the diameter of the 1036 1:10:10,611 --> 1:10:13,218 opening is. Of course, because you know now 1037 1:10:13,218 --> 1:10:16,196 what the angle is. And so, you can calculate what 1038 1:10:16,196 --> 1:10:17,810 D is. And, when I did that, 1039 1:10:17,810 --> 1:10:21,099 I came up with something like, I think, an eighth of a 1040 1:10:21,099 --> 1:10:23,705 millimeter or so, but you can confirm that. 1041 1:10:23,705 --> 1:10:26,746 A very small opening of an eighth of a millimeter, 1042 1:10:26,746 --> 1:10:30,035 if I did that correctly, would then give you a central 1043 1:10:30,035 --> 1:10:35,000 maximum, which is from zero to zero, five centimeters wide. 1044 1:10:35,000 --> 1:10:37,035 And so let's take a look at that. 1045 1:10:37,035 --> 1:10:40,089 These single pinhole diffractions are always very 1046 1:10:40,089 --> 1:10:43,905 difficult because the pinholes have to be so small to see it, 1047 1:10:43,905 --> 1:10:46,959 and that means very little light will go through. 1048 1:10:46,959 --> 1:10:50,012 And so, here you see it. For those of you who are 1049 1:10:50,012 --> 1:10:52,493 sitting close, you will clearly see that 1050 1:10:52,493 --> 1:10:55,865 central circular maximum. And then you clearly see the 1051 1:10:55,865 --> 1:11:00,000 first ring, the dark ring. My pinky is right on it. 1052 1:11:00,000 --> 1:11:03,126 This is about five centimeters across. 1053 1:11:03,126 --> 1:11:07,774 I see a second dark ring, but if you are far away in the 1054 1:11:07,774 --> 1:11:10,901 audience you may not see that so well. 1055 1:11:10,901 --> 1:11:14,704 This is a nice example of circular single-slit 1056 1:11:14,704 --> 1:11:18,338 diffraction. To make you see it even better, 1057 1:11:18,338 --> 1:11:22,816 we have handed out cards. And those cards have a small 1058 1:11:22,816 --> 1:11:28,056 pinhole in one location and they have double-slits in the other 1059 1:11:28,056 --> 1:11:31,838 location. And so, I am going to aim at 1060 1:11:31,838 --> 1:11:35,711 you, very slowly I am going to scan this over the audience a 1061 1:11:35,711 --> 1:11:37,943 light-emitting diode, bright light. 1062 1:11:37,943 --> 1:11:41,225 And, as it passes you, you only get one shot at it, 1063 1:11:41,225 --> 1:11:44,638 and you look through it, maybe we can make it darker. 1064 1:11:44,638 --> 1:11:47,986 If you look through the pinhole, you will really see 1065 1:11:47,986 --> 1:11:51,794 beautifully this ring structure with the dark lines and the 1066 1:11:51,794 --> 1:11:54,617 center maximum. But, if you look through the 1067 1:11:54,617 --> 1:11:57,242 other opening, you will see the beautiful 1068 1:11:57,242 --> 1:12:02,364 double-slit interference. Notice that the widths of the 1069 1:12:02,364 --> 1:12:07,173 dark lines and the widths of the bright lines is about equal 1070 1:12:07,173 --> 1:12:12,146 because you only have two slits. Remember, you get this cosine 1071 1:12:12,146 --> 1:12:15,570 square function. It is not as dramatic as a 1072 1:12:15,570 --> 1:12:18,586 grating. Now I am going to rotate this 1073 1:12:18,586 --> 1:12:23,559 through the class so that each one of you get a chance to look 1074 1:12:23,559 --> 1:12:26,413 through both openings one at a time. 1075 1:12:26,413 --> 1:12:32,450 If I go too slowly let me know. And you can keep these cards. 1076 1:12:32,450 --> 1:12:37,183 You need a very bright light. You need a very small like, 1077 1:12:37,183 --> 1:12:41,492 too, because if the light is too large in size then, 1078 1:12:41,492 --> 1:12:46,563 of course, your dark and your bright areas are going to merge 1079 1:12:46,563 --> 1:12:49,267 with each other. You wash it out. 1080 1:12:49,267 --> 1:12:54,507 Your light source always has to be carefully thought through in 1081 1:12:54,507 --> 1:13:00,000 terms of its dimension, in terms of angular dimension. 1082 1:13:00,000 --> 1:13:04,830 If the angular dimension of your light source is too large 1083 1:13:04,830 --> 1:13:09,661 you kill all the phenomenon. I am going to rotate it back. 1084 1:13:09,661 --> 1:13:13,135 Who has not seen it? You have not seen it. 1085 1:13:13,135 --> 1:13:17,711 How could I do that to you? But now you can see it even 1086 1:13:17,711 --> 1:13:20,000 longer than others. 1087 1:13:20,000 --> 1:13:25,000 1088 1:13:25,000 --> 1:13:29,965 This single-slit diffraction or single-opening diffraction has 1089 1:13:29,965 --> 1:13:34,034 major consequences, even for our daily experiences, 1090 1:13:34,034 --> 1:13:37,779 because it ultimately determines our ability to 1091 1:13:37,779 --> 1:13:40,790 separate two light sources in the sky. 1092 1:13:40,790 --> 1:13:45,918 If you have a telescope and the telescope has a lens or it has a 1093 1:13:45,918 --> 1:13:50,639 mirror which has a diameter D then there is a limitation to 1094 1:13:50,639 --> 1:13:54,058 which it can separate two stars in the sky. 1095 1:13:54,058 --> 1:14:00,000 Let's assume there are two stars at roughly equal strength. 1096 1:14:00,000 --> 1:14:03,481 Here is star number one and here is another star which is 1097 1:14:03,481 --> 1:14:06,341 start number two. And the angle between them is 1098 1:14:06,341 --> 1:14:08,330 delta theta. Then somewhere here, 1099 1:14:08,330 --> 1:14:10,941 on the photographic plate or, in your case, 1100 1:14:10,941 --> 1:14:13,303 on your retina, there will be an image. 1101 1:14:13,303 --> 1:14:16,101 And that image will be like this for one star. 1102 1:14:16,101 --> 1:14:19,955 And there will be another image a little bit displaced from the 1103 1:14:19,955 --> 1:14:22,504 other star. And if those two blurs are too 1104 1:14:22,504 --> 1:14:25,364 close together, you don't see two stars anymore 1105 1:14:25,364 --> 1:14:28,907 but you see only one star. And so, now comes the question, 1106 1:14:28,907 --> 1:14:32,761 how small can this angle be so that you still say there are two 1107 1:14:32,761 --> 1:14:37,214 light sources? A car comes toward you with two 1108 1:14:37,214 --> 1:14:41,428 headlights, how close does the car have to be that you still 1109 1:14:41,428 --> 1:14:45,500 say there are two and not one? Well, there is a criterion, 1110 1:14:45,500 --> 1:14:49,142 which is a little bit arbitrary, called the [rally?] 1111 1:14:49,142 --> 1:14:51,428 criterion for angular resolution. 1112 1:14:51,428 --> 1:14:55,785 And that is we want the angle between the two lights larger or 1113 1:14:55,785 --> 1:14:59,928 equal to this angle so that the maximum of the second light 1114 1:14:59,928 --> 1:15:03,972 would fall here. And so, you would clearly see 1115 1:15:03,972 --> 1:15:06,217 then that this thing is broadened. 1116 1:15:06,217 --> 1:15:09,551 And you may even see a little dip in that [curl?]. 1117 1:15:09,551 --> 1:15:13,496 On your photographic plate you would really be able to say, 1118 1:15:13,496 --> 1:15:16,897 yes, there are two sources and not just one source. 1119 1:15:16,897 --> 1:15:19,891 And so, the angular resolution would then be, 1120 1:15:19,891 --> 1:15:23,156 in terms of angle, 1.2 times lambda divided by D. 1121 1:15:23,156 --> 1:15:26,965 Delta theta would have to be larger than 1.2 times lambda 1122 1:15:26,965 --> 1:15:32,000 over D for you to be able to say, yes, there are two stars. 1123 1:15:32,000 --> 1:15:37,719 Which is the ultimate limit of angular resolution for you, 1124 1:15:37,719 --> 1:15:41,632 for me, but also for optical telescopes. 1125 1:15:41,632 --> 1:15:45,846 Suppose we take the Hubble Space Telescope, 1126 1:15:45,846 --> 1:15:51,063 HST, that has a mirror D, which has a diameter of 2.4 1127 1:15:51,063 --> 1:15:54,876 meters. And it is prepared so carefully 1128 1:15:54,876 --> 1:16:02,000 that the claim is made that it is really diffraction limited. 1129 1:16:02,000 --> 1:16:07,137 And so, that means if I take an average wavelength in the 1130 1:16:07,137 --> 1:16:12,733 optical spectrum of about 500 nanometers, I realize that it is 1131 1:16:12,733 --> 1:16:17,871 all the way goes from 400 to 650, but if I take this as a 1132 1:16:17,871 --> 1:16:23,284 representative wavelength then 1.2 times lambda divided by D 1133 1:16:23,284 --> 1:16:28,055 translates into about one-twentieth of an arc second. 1134 1:16:28,055 --> 1:16:32,000 That is an incredible resolution. 1135 1:16:32,000 --> 1:16:35,821 One-twentieth of an arc second. If two stars of equal 1136 1:16:35,821 --> 1:16:39,569 brightness are one-twentieth of an arc second apart, 1137 1:16:39,569 --> 1:16:43,538 the Hubble Space Telescope can see there are two stars. 1138 1:16:43,538 --> 1:16:47,800 The same telescope on earth would do no better than half an 1139 1:16:47,800 --> 1:16:51,842 arc second, this is an arc second, to maybe even two arc 1140 1:16:51,842 --> 1:16:54,561 seconds. Why is it so much worse for a 1141 1:16:54,561 --> 1:17:00,000 telescope on the ground than it is for Hubble Space Telescope? 1142 1:17:00,000 --> 1:17:02,311 Any one of you know that? Atmosphere. 1143 1:17:02,311 --> 1:17:06,163 The earth's atmosphere is in turbulence, is always in thermal 1144 1:17:06,163 --> 1:17:08,474 motion. And it is that thermal motion 1145 1:17:08,474 --> 1:17:11,749 that is the problem. That makes your imagine on your 1146 1:17:11,749 --> 1:17:14,638 photographic plate or on your CCD move around. 1147 1:17:14,638 --> 1:17:17,912 They work like little lenses. And so it broadens it, 1148 1:17:17,912 --> 1:17:20,416 but it broadens it in an incredible way. 1149 1:17:20,416 --> 1:17:23,626 We call this the seeing. If the seeing were one arc 1150 1:17:23,626 --> 1:17:27,285 second then Hubble's Telescope, which is above the earth's 1151 1:17:27,285 --> 1:17:30,688 atmosphere, has an angular resolution that is 20 times 1152 1:17:30,688 --> 1:17:35,355 better linearly. That means over a surface. 1153 1:17:35,355 --> 1:17:39,976 It has 400 times more resolution elements because it 1154 1:17:39,976 --> 1:17:44,144 is two-dimensional, so it is highly superior in 1155 1:17:44,144 --> 1:17:49,671 terms of angular resolution than any ground based observatory. 1156 1:17:49,671 --> 1:17:53,657 And now comes your human eye. Your human eye, 1157 1:17:53,657 --> 1:18:00,000 the opening of your human eye depends on the time of the day. 1158 1:18:00,000 --> 1:18:04,271 At night, when it is dark, your pupil opens. 1159 1:18:04,271 --> 1:18:07,847 During the day it goes down a little. 1160 1:18:07,847 --> 1:18:12,417 If we take about four millimeters, a reasonable 1161 1:18:12,417 --> 1:18:18,476 number, and we take again 500 nanometers as our representative 1162 1:18:18,476 --> 1:18:24,834 wavelength then we can calculate what 1.2 times lambda over D is. 1163 1:18:24,834 --> 1:18:30,000 And that translates to half an arc minute. 1164 1:18:30,000 --> 1:18:33,517 That is 30 arc seconds. That is 600 times worse than 1165 1:18:33,517 --> 1:18:37,517 the Hubble Space Telescope. You cannot do better than this. 1166 1:18:37,517 --> 1:18:40,965 This is Mother Nature. You cannot beat diffraction. 1167 1:18:40,965 --> 1:18:43,655 On your retina, when you look at a light 1168 1:18:43,655 --> 1:18:47,241 source, the image on your retina will look like this, 1169 1:18:47,241 --> 1:18:49,724 will look exactly what you saw there. 1170 1:18:49,724 --> 1:18:52,000 That is what your retina will see. 1171 1:18:52,000 --> 1:18:55,241 And, if those two lights are too close together, 1172 1:18:55,241 --> 1:19:00,000 your brains will say, sorry, I do not see two lights. 1173 1:19:00,000 --> 1:19:03,325 Now, in practice, Mother Nature did not desire 1174 1:19:03,325 --> 1:19:07,684 our eyes, at least most of us not really down to diffraction 1175 1:19:07,684 --> 1:19:09,384 limitation. In practice, 1176 1:19:09,384 --> 1:19:12,783 I think it is more like one to two arc minutes. 1177 1:19:12,783 --> 1:19:15,221 This is my symbol for arc minutes. 1178 1:19:15,221 --> 1:19:19,729 The angular resolution of your eyes is not quite as good as it 1179 1:19:19,729 --> 1:19:22,167 could be, but it is close to that. 1180 1:19:22,167 --> 1:19:24,827 And I am going to test that with you. 1181 1:19:24,827 --> 1:19:28,448 I have here a screen, and this screen has holes in 1182 1:19:28,448 --> 1:19:33,131 it. And I will give you the code of 1183 1:19:33,131 --> 1:19:36,161 the holes. Here is that screen. 1184 1:19:36,161 --> 1:19:41,212 Two holes, two holes, two holes, and we repeat them 1185 1:19:41,212 --> 1:19:45,353 three times. This is one millimeter apart, 1186 1:19:45,353 --> 1:19:50,000 two millimeters apart, three millimeters apart, 1187 1:19:50,000 --> 1:19:55,050 four millimeters apart. Students who are two meters 1188 1:19:55,050 --> 1:20:00,000 away, very few are, but if you were -- 1189 1:20:00,000 --> 1:20:04,271 We take here a student who is at a distance of two meters. 1190 1:20:04,271 --> 1:20:08,843 If the student looks at the one millimeter separation of these 1191 1:20:08,843 --> 1:20:12,815 slides then the angular separation is 1.7 arc minutes. 1192 1:20:12,815 --> 1:20:16,113 You should be able to see them as two lights. 1193 1:20:16,113 --> 1:20:19,710 If you look at the two millimeter separation then 1194 1:20:19,710 --> 1:20:22,483 obviously it is about 3.4 arc minutes. 1195 1:20:22,483 --> 1:20:25,706 You should have no problems. In other words, 1196 1:20:25,706 --> 1:20:29,828 the students who are close should be able to see this as 1197 1:20:29,828 --> 1:20:35,000 two lights, this as two lights and this as two light. 1198 1:20:35,000 --> 1:20:38,361 But let's now go to the students who are five meters 1199 1:20:38,361 --> 1:20:40,668 away. These are the students who are 1200 1:20:40,668 --> 1:20:42,777 five meters away in the audience. 1201 1:20:42,777 --> 1:20:46,797 If we go to the one millimeter separation, there is no hope on 1202 1:20:46,797 --> 1:20:50,422 earth that you will see that when you are sitting there. 1203 1:20:50,422 --> 1:20:54,443 You will not be able to see the upper two as two light sources 1204 1:20:54,443 --> 1:20:57,475 because the separation is only 0.7 arc minutes. 1205 1:20:57,475 --> 1:21:01,495 And I do not think that any one of you can see lights that are 1206 1:21:01,495 --> 1:21:06,049 0.7 arc minutes apart. The two millimeter slots would 1207 1:21:06,049 --> 1:21:10,352 be 1.4 arc minutes and the three millimeter slots would be about 1208 1:21:10,352 --> 1:21:13,426 two arc minutes. And so, I am going to make it 1209 1:21:13,426 --> 1:21:16,909 dark now in the room. And I want each of you to just 1210 1:21:16,909 --> 1:21:20,461 look at these pinholes. And I am going to rotate this 1211 1:21:20,461 --> 1:21:24,560 so that all of you get a chance. And then I want you to raise 1212 1:21:24,560 --> 1:21:29,000 your hand if you can see the top two as separate ones. 1213 1:21:29,000 --> 1:21:33,720 And only those who will raise their hands will be very close 1214 1:21:33,720 --> 1:21:36,279 to me. And then we will slowly go 1215 1:21:36,279 --> 1:21:40,760 farther into the audience. Look closely at the upper one. 1216 1:21:40,760 --> 1:21:44,279 And then also try to see the one below there, 1217 1:21:44,279 --> 1:21:49,079 the two millimeters which are here, the three millimeters and 1218 1:21:49,079 --> 1:21:52,840 the four millimeters. Can you see the upper one? 1219 1:21:52,840 --> 1:21:57,640 Even the upper one you cannot. You're only three meters away. 1220 1:21:57,640 --> 1:22:02,498 Now I will just rotate this. The upper one, 1221 1:22:02,498 --> 1:22:06,671 again, is the one millimeter separation. 1222 1:22:06,671 --> 1:22:12,557 You see it repeat three times. And then two millimeters. 1223 1:22:12,557 --> 1:22:16,837 And then below that is three millimeters. 1224 1:22:16,837 --> 1:22:20,154 And then it is four millimeters. 1225 1:22:20,154 --> 1:22:26,147 Raise your hand if you can see the upper one at two light 1226 1:22:26,147 --> 1:22:27,752 sources. No one. 1227 1:22:27,752 --> 1:22:31,481 Oh, boy. Well, if you really can, 1228 1:22:31,481 --> 1:22:35,185 then your resolution is very close to 0.6 arc minutes because 1229 1:22:35,185 --> 1:22:37,283 the two meters was 0.7 arc minutes. 1230 1:22:37,283 --> 1:22:39,629 That is really remarkable but possible. 1231 1:22:39,629 --> 1:22:43,209 Who can see the second row clearly as two distinct sources? 1232 1:22:43,209 --> 1:22:46,481 Now people are coming in. Even the ones that are close 1233 1:22:46,481 --> 1:22:48,209 can only do it. You see that? 1234 1:22:48,209 --> 1:22:51,913 No one in the audience there in the back is raising his hand. 1235 1:22:51,913 --> 1:22:54,814 Now we are talking about a resolution of the two 1236 1:22:54,814 --> 1:22:58,148 millimeters, that 3.4 arc minutes for the ones that are 1237 1:22:58,148 --> 1:23:01,977 two meters. You are talking about 2.5 arc 1238 1:23:01,977 --> 1:23:04,227 minutes. Who can see the third row 1239 1:23:04,227 --> 1:23:06,204 separate? Now the hands go up. 1240 1:23:06,204 --> 1:23:09,068 And who can see the fourth row as separate? 1241 1:23:09,068 --> 1:23:12,545 I think the whole class. The ones in the back there, 1242 1:23:12,545 --> 1:23:14,113 can you not see? Really? 1243 1:23:14,113 --> 1:23:16,568 Boy, you have to go to an eye doctor. 1244 1:23:16,568 --> 1:23:20,386 You cannot see the bottom? I was there this morning where 1245 1:23:20,386 --> 1:23:24,068 you were and could see the bottom one distinctly as two 1246 1:23:24,068 --> 1:23:27,000 light sources. And I could kid myself that I 1247 1:23:27,000 --> 1:23:32,091 even saw this one as two. But I was really kidding myself 1248 1:23:32,091 --> 1:23:33,845 because I knew it, I think. 1249 1:23:33,845 --> 1:23:36,004 But this one I could clearly see. 1250 1:23:36,004 --> 1:23:40,120 Really, none of you can see the bottom one as two differences? 1251 1:23:40,120 --> 1:23:44,033 Well, that shows that your angular resolution is no better, 1252 1:23:44,033 --> 1:23:47,946 and you shouldn't be ashamed of that, it is not your fault, 1253 1:23:47,946 --> 1:23:51,320 than about two arc minutes. With that idea in mind, 1254 1:23:51,320 have a good weekend.