How a Lens creates an Image. — Transcript

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1. 0:01Hey everyone, Can you tell me
2. 0:03how a lens creates an image? If you would have  asked me this question a couple of years ago,
3. 0:08I would probably have explained it to you  as it was explained to me in a high school.
4. 0:13I would have told you something similar  to what is described in these pictures:
5. 0:17that light is emitted by or scattered of an object  and spreads out in space in different directions.
6. 0:24A part of the light is collected by the lens,  where it changes direction and is then focused
7. 0:28into an image. I would have explained the focusing  by saying that the light rays change direction at
8. 0:34the boundary between different materials due to a  difference in refractive index. And that because
9. 0:40a lens has curves surfaces, it can cleverly  project the light originating from a point
10. 0:45in an object to a corresponding point in the  image plane. And by doing this for every point,
11. 0:51this then yields an image of the object. So  that is how a lens creates an image, right?
12. 1:00As the images in this physics book illustrate,  the interaction between light and lenses,
13. 1:05is generally described by using “rays” of light.  Drawing rays is very convenient because they show
14. 1:11you where the light goes and they can quickly  make you understand the basic principle of for
15. 1:16example a focal point. But they also have a  downside in the sense that they are in fact a
16. 1:21very poor representation of the physics that  is going on. Let me give you some examples
17. 1:27where rays completely miss this point. For example, why in the first place would
18. 1:32a ray change direction when entering a medium  with a different refractive index? I mean, you
19. 1:38could think of reasons why a denser medium would  slow down light, but why change its direction?
20. 1:44And how can a ray model explain that the maximum  sharpness of a lens is dependent on the wavelength
21. 1:50of the light? Here is another example where rays  fail to explain what we observe in real life:
22. 1:56say we have 2 perfect lenses with the same focal  length. Why is it that the one with the smallest
23. 2:02diameter or opening angle is fundamentally  less sharp? All these questions are in fact
24. 2:08quite hard to explain at the fundamental level  if we view light from a ray perspective.
25. 2:14Now, most of you will of course know that light  isn’t actually rays, but electromagnetic wave
26. 2:19energy. And light in the visible wavelength  range very much behaves like a wave. What rays
27. 2:25are trying to depict is the local direction of  wave propagation. And so, to understand why light
28. 2:31really behaves the way it does, we should  actually be looking at how waves behave.
29. 2:37Let me just show you a wave animation. It  features 2 point sources that emit waves at
30. 2:43a fixed wavelength. Furthermore, we have a lens,  which is an area where the waves propagate much
31. 2:49slower and which has 2 curves surfaces. And on  the right side, we have a plane where this lens
32. 2:54focusses the wave energy. Now these lines show the  ray representation of how these two sources are
33. 3:01imaged onto the image plane. But this is how it  looks from the wave perspective. As you can see,
34. 3:07the waves spread out in space. Because the sources  are emitting the wave energy coherently, they
35. 3:13create a nice interference pattern. When the waves  pass the lens interface, the wavefront as a whole
36. 3:19changes shape and direction. And this is because  of the spherical shape of the lens interface and
37. 3:24because the velocity of wave propagation is  lower inside the lens. And the same sort of
38. 3:30phenomenon happens on the other side of the lens  where the waves bend again due to the difference
39. 3:35in propagation speed between the media and then  reach the focal plane. What you observe is that
40. 3:42even though the point sources are not perfectly  reproduced in the focal plane as sharp points,
41. 3:48we find two very distinct maxima for the  wave intensity. So, when this lens creates
42. 3:53an image in the focal plane using waves, it can  easily resolve the two individual wave sources.
43. 4:01Now before I continue, I want to mention that this  animation was created by Nils Berglund who many of
44. 4:06you will know from his YouTube Channel where he  presents all kinds of cool physics animations.
45. 4:11I’m a big fan of Nils and I asked whether he  could maybe create this type of lens simulation
46. 4:16and he gladly accepted the challenge. By the way,  notice that the number of videos he published has
47. 4:23recently exceeded one thousand. I mean, how? Well,  basically by publishing a video every single day
48. 4:31for the last 3 years. So yeah, Nils has really  been pretty busy. Anyway, I want to thank Nils
49. 4:37for his efforts and for those interested I’ll  post a link to his channel in the description.
50. 4:44To demonstrate the effect of Numerical Aperture,  which is basically equivalent to the sin of the
51. 4:49maximum opening angle of a lens, I asked Nils  to do the same simulation for a smaller diameter
52. 4:55lens with the same focal length. Here you see how  that works out. I’m showing the previous and the
53. 5:01new simulation together here so you can compare  the difference in the outcome more easily. And
54. 5:06what you observe is that the distribution of the  wave energy in the focal plane with the smaller
55. 5:11Numerical Aperture is much less well-defined.  So, can you from the simulation spot why that
56. 5:17is exactly? It’s pretty hard to see right because  what we observe isn’t even close to what we would
57. 5:21expect based on the ray representation. The simplest way that I know how to roughly
58. 5:23explain it is the following: say that we have an  array of very small individual wave sources that
59. 5:29emit waves coherently and that we want to resolve  in an image. If we look at the wavefront created
60. 5:35some distance away from the sources that reaches  the image plane heads on, we observe that it has
61. 5:41become almost flat because of spatial coherence.  And if you were to place and image detector here,
62. 5:47then there would be very little to no variation in  the wave intensity. So then the question arises:
63. 5:53how can we introduce the intensity variations  needed to resolve the sources using just
64. 5:59waves? Well, the only way that we can  do this is is by means of introducing
65. 6:04wave interference. And in order to create  this, we need additional waves that arrive
66. 6:09at the focal plane under a different angle. In fact, the higher the spatial frequency that
67. 6:16we want to reproduce in the image plane,  the larger the angle needs to be between
68. 6:20the incoming waves. So, by limiting the angle  at which waves can arrive in the focal plane,
69. 6:25we also limit the maximum spatial frequency of  the intensity variations that can be created
70. 6:31here. This basically means that by limiting  the opening angle of a lens, we lose important
71. 6:37information contained in the diffraction pattern,  and therefore will lose detail in the image.
72. 6:44If you keep this view in mind, it is very  easy to understand the general formula that
73. 6:49describes the maximum sharpness of a lens system.  In this formula CD stands for critical dimension,
74. 6:55which is basically equivalent to the dimension  of smallest features that can be resolved. The
75. 7:00critical dimension is equal to a constant, times  the wavelength, divided by the numerical aperture.
76. 7:07And the numerical aperture is in this case  proportional to the sin of the opening angle
77. 7:12of the lens. From the view point of creating  interference, having a shorter wavelength allows
78. 7:19us to create higher frequency interference and  eventually to reproduce smaller features with
79. 7:25the same angle. The same is true for increasing  the maximum angle at which an optical system
80. 7:30can accept light. This will also allow us to  create a higher-density interference pattern
81. 7:35and eventually to resolve smaller features. Okay let me show you an experiment that
82. 7:42illustrates the effect of Numerical Aperture  in a microscope. Say this is a schematic of our
83. 7:47microscope with the objective depicted here,  a tube lens and a focal plane. We can examine
84. 7:53the image that the microscope produces by either  placing a CMOS or CCD sensor in the focal plane
85. 8:00or look at the aerial image using an eye piece. If  we want to change the numerical aperture of this
86. 8:05system. Then the easiest way to do this is by  inserting a small aperture in the optical path,
87. 8:10for example here. This area is called infinity  space, and it allows you to insert filters or
88. 8:16beam splitters, into the optical path of the  microscope. And as long as these have flat and
89. 8:21parallel optical surfaces, they don’t introduce  optical aberrations into the system. Now by
90. 8:27placing a pinhole here, we effectively limit the  opening angle of the objective, in other words,
91. 8:32we limit the numerical aperture. So let’s have a look at the effect in
92. 8:37practice. Here is an optical microscope and  this one actually gives us easy access to the
93. 8:43infinity space inside the microscope. Here you can  insert filters but we can also insert an aperture
94. 8:50to effectively limit Numerical Aperture. Under the microscope is a small glass disk with
95. 8:57a chromium surface layer. The chromium contains  a test pattern etched in it. So in the areas
96. 9:03where the chromium was etched away, the sample has  become transparent. I actually made this pattern
97. 9:08using photolithography with my maskless wafer  stepper. And if you want to know more about that,
98. 9:13please look in the description of this video. The test pattern is illuminated from the back
99. 9:21with white light. The pattern itself is pretty  small and contains features of various sizes.
100. 9:27The total diameter of this particular round  pattern is 0.5mm, meaning that the smallest
101. 9:33features in the pattern are only a few microns in  size. And currently the pattern is viewed with a
102. 9:3810x magnification objective at full aperture. Now let me show you what happens when we insert
103. 9:44an aperture into the optical path and thereby  reducing the NA. Here is a comparison: we
104. 9:49observe is that the definition of the smaller  features suffers significantly due to the absence
105. 9:54of higher order diffraction from the object. In  other words: by throwing away the information
106. 10:00contained in the light diffracted under larger  angles by the test pattern. Now I think it is
107. 10:05pretty cool that we can directly observe the  effect of NA in a microscope in this way.
108. 10:12Okay, so the previous was basically the  main message of this video and I think it
109. 10:17explains intuitively why numerical aperture  is so important to create sharp images. Now,
110. 10:22in the rest of this video I’m going to goof around  a bit with diffraction and image formation and do
111. 10:28a few experiments. But at the same time, I’ll  also dive in really deep. Now I’m not going
112. 10:34to explain every single aspect that you  are about to see. But, I can assure you,
113. 10:39if you stick around you will not be disappointed. The first experiment, that I want to show you is
114. 10:46very simple and you have probably seen it  presented quite often. It involves just 2
115. 10:53linear slits and because they are very small, we  view them under a microscope. They are illuminated
116. 10:58from the other side using the coherent light  of a HeNe laser. The width of these slits is
117. 11:04around 3 microns and they are spaced about  the same distance apart. Here we view them
118. 11:09in a bit more detail and can measure the light  intensity in the horizontal direction in a graph.
119. 11:16If we now move away from the slits, we observe  an interference pattern, which is caused by the
120. 11:21diffraction of the light from both slits. This  diffraction pattern is actually quite similar
121. 11:26to the diffraction pattern that we just observed  previously in the simulations. If we now place
122. 11:31the slits further apart, we observe that the  interference pattern that appear behind the slits
123. 11:35becomes denser and so the maxima and minima  are spaced closer together. In other words,
124. 11:41the interference has a higher spatial frequency. Here I’ve schematically drawn the configuration:
125. 11:48this is the mask containing the slits, with the  coherent light source behind it and here some
126. 11:53distance (l) away we observe the interference.  If you do a little math, it turns out that you
127. 11:59can formulate the relationship between the  spacing in the diffraction pattern (delta x)
128. 12:04to the wavelength (lambda), the distance between  the slits (d) and the distance (l) from the slits
129. 12:10to where we observe the interference. Now this  formula is an approximation, but illustrates the
130. 12:15fact that, when the distance between the  slits gets larger, delta x gets smaller,
131. 12:20so the spatial frequency in the interference  pattern increases. And this is basically due
132. 12:26to the angles under which the slits interfere. Now of course a double slit isn’t a lens, because
133. 12:34normal lenses are generally round. But what if  we were to bend these two lines into a single
134. 12:41circular slit, would that be a lens? Take example,  this little fellah which has just 2 circular
135. 12:47diffractive edges and only 70um in diameter?  No, that cannot possibly be a lens. But just
136. 12:55to be sure, let’s place it in a coherent beam of  light of laser light and look at the diffraction
137. 13:00pattern. At a distance of 2mm away from the slit,  we observe a circular diffraction pattern with,
138. 13:07what appears to be a focal point. Here you  can see it in a bit more detail together with
139. 13:12a plot of the intensity distribution based on  the diffraction pattern. So the circular slit
140. 13:18seems to be lens after all. It is not really  impressive, because let’s face it, the focal
141. 13:24point is almost as large as the lens. But I think  we are on to something. Now let’s place a few more
142. 13:30slits with diffractive edges in strategic places  and see what happens. Here you see the result
143. 13:36with 6 slits and here with thirty. Now, this  is starting to look like a real focal point!
144. 13:45I want to emphasize that these images are  not simulations. They are real images that
145. 13:50were collected using a microscope from real  slit patterns ranging in diameter from 70 to
146. 13:56500um. And these patterns were also created using  photolithography. Now of course, the slit patterns
147. 14:03aren’t just random circular patterns. They  are actually based on the configuration of a
148. 14:08Fresnel zone plate, named after Augustin Fresnel,  a French scientist. The edges in the patterns are
149. 14:15placed such that each creates 1 wavelength of path  difference to the desired focal point. Basically,
150. 14:22these recreate a focus from 1st order diffraction  at this point. It’s definitely not the same thing
151. 14:29as a refractive lens but it is quite similar. And  the fun thing is that using these, we can build up
152. 14:35numerical aperture in discrete little steps. If you look at the focal point in the last
153. 14:44pattern, you might get the idea that a tight  focus is mainly achieved by the outer rings.
154. 14:50But look what happens if we take away the  center rings: the total size of focal point
155. 14:55increases again because of what appears to be  high-frequency diffraction. I’ll get back to
156. 15:01this lens pattern later in the video. Okay, so these patterns can create a tight
157. 15:08focal point from laser light by adding diffraction  patterns. But are they in fact also real lenses
158. 15:15when we use incoherent light, like the light from  a standard candescent lamp? Again, let’s just find
159. 15:18out. Here is the schematic of the setup: light  from the candescent lamp is filtered with a red
160. 15:24color filter to make it a bit more monochromatic.  The test pattern used previously is placed in the
161. 15:30beam, then, at some distance, the circular slit  pattern. And the slit pattern will hopefully
162. 15:35create an aerial image of the test pattern, which  we can then observe using a microscope. So here
163. 15:42you see how that works in practice. This is the  plate containing the test patterns which is 15mm
164. 15:48below the x-y table. The lens patterns are placed  on the x-y table and so we can easily choose which
165. 15:56lens to use. And with the microscope, we will  take a look at the aerial image of each lens
166. 16:04Here I’ve got the simplest pattern containing only  1 circular slit in focus with the microscope and
167. 16:10if I now move the focus away from the pattern  itself, we observe how light is diffracted of
168. 16:15the edges and eventually creates an image of  the original pattern. Hmm, I admit it is not
169. 16:22very sharp is it? You can see for example that  there is a line, but you cannot see the central
170. 16:28spacing at all. So this lens is probably missing  out on a lot of the light that is diffracted under
171. 16:34larger angles by the test pattern. So let’s add a  few more ring-shaped slits and see what happens.
172. 16:41Here is another one, and another one [ let’s go] This is the resulting image of 12 diffractive
173. 17:11rings. Now, who would have thought that what  is basically a simple pattern containing a very
174. 17:16limited number of slits could reproduce such small  features. But of course, we are not done yet,
175. 17:22we can do better. Let’s go straight to the maximum  number of 30 rings and see what that this pattern
176. 17:29can do. Here you see the slit pattern in  focus and is we now slowly move the focus
177. 17:35of the microscope towards the focal plane of this  lens we can see how the image is created. I mean,
178. 17:42look at that. Isn’t that just amazing. That  adding what is basically a limited set of
179. 17:48interference patterns create by a bunch of  slits can recreate a pattern with such amazing
180. 17:54resolution. The round feature here is only 76um in  diameter in the image plane, making the smallest
181. 18:02features imaged in the order of 1 um. So why does the resolution improve with
182. 18:10the increasing number of rings? It is actually  two-fold: by adding more slits in the lens pattern
183. 18:17we are of course also increasing the numerical  aperture of our lens and collecting more phase
184. 18:23information from the light diffracted by  the test pattern. But at the same time,
185. 18:28we are also getting a larger set of high frequency  diffraction patterns available to reconstruct the
186. 18:34image with. With just 2 diffractive edges in  the center, the reproduction of the pattern
187. 18:39is very poor, because the lens can only  create low spatial frequency diffraction
188. 18:44patterns. But as we add more and especially  wider rings, smaller features in the pattern
189. 18:51can be resolved because the wider rings are able  to create higher frequency spatial diffraction.
190. 19:00If you are familiar to the Fourier transform,  you may have noticed that what you just witnessed
191. 19:05was real-life version of the Fourier series  approximation. With this method, basically any
192. 19:11function can be approximated using a specific set  of sinusoidal functions with specific frequency,
193. 19:17amplitude and phase. By adding more and especially  higher frequency harmonics, we can more accurately
194. 19:25approximate the function. And that is exactly  what we did here. By adding diffraction under
195. 19:30increasing angles, we basically added higher and  higher frequency sinusoidal diffraction patterns,
196. 19:36which eventually resulted in a fairly  high-quality reproduction of the image.
197. 19:44Now I mentioned a few minutes ago that I was going  to return to this particular pattern where the
198. 19:49center rings are missing. With the Fourier series  approximation in mind, it is interesting to look
199. 19:55at the image that this pattern creates. Here it is  and what you can observe is that the image looks
200. 20:01somewhat weird: it has lost contrast compared to  the full pattern. It has lost uniformity in the
201. 20:08larger areas and it has these faint borders around  the intensity transients. The image looks a bit
202. 20:14like a very heavily compressed JPEG image. And  that is because what you observe is very similar
203. 20:19to bad JPEG compression. JPEG compression and  decompression is also based on the principles
204. 20:26of the Fourier. The compression works by only  encoding the frequencies that are essential for
205. 20:31creating an acceptable reproduction of an image.  But if you compress the information in an image to
206. 20:37the extend that essential frequencies are omitted,  this then leads to artefacts. And these are very
207. 20:43similar to the ones observed here. So basically  what this demonstrates that in order to accurately
208. 20:49create an image that contains both small and  large features, it is essential that you use
209. 20:54both high and low spatial frequency diffraction. The last thing I want to show you is the effect of
210. 21:02wavelength. These 3 images are all created using  a lens of 30 diffractive rings, but in each case,
211. 21:09I’ve used a filter of a different wavelength. And  if you look carefully you can see that reducing
212. 21:14wavelength results in better image resolution.  I admit that it is hard to see, so I tried to
213. 21:20quantify the effect here in the line and space  pattern. Here I’ve plotted the intensity profile
214. 21:26over these lines and it is very clear that when  using blue light, the picture has better contrast
215. 21:31and definition, compared to the one in red. So that is how lenses create images using waves:
216. 21:40by adding up a whole lot of diffraction.  And in the upcoming video, which will also
217. 21:45be about image formation, I’ll tell you about my  visit to a company that takes the principles of
218. 21:51diffraction and refraction to a whole new level.  The name of this company is Advanced Semiconductor
219. 21:58Materials Lithography, ASML for short. In  order to create the nanometer features that
220. 22:04populate modern micro-chips, they literally have  to use every trick in the book of diffraction.
221. 22:10So, I hope that this video gave you some new  insights and who knows, maybe we’ll meet again.