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Resolution in Microscopy

00:00:11.25 Light itself has an intrinsic grain. And that is, you can take an
00:00:16.07 image at very high resolution, but if you keep blowing
00:00:18.19 it up and magnifying it more and more, you don't actually
00:00:21.08 get more information. And I'd like to talk about why that is,
00:00:26.03 because this has limited traditional forms of microscopy
00:00:29.28 with a limit that's related to the diffraction of light.
00:00:34.07 I'll explain what that is. And this diffraction limit
00:00:36.17 ultimately leads us to an irresolvably small spot of light,
00:00:41.20 which is based on something known as the point spread function.
00:00:45.19 That limits the resolution of traditional forms of microscopy.
00:00:48.22 So to begin, I think we have to talk about why light doesn't
00:00:54.23 behave in a way that would just allow infinite magnification.
00:00:57.16 And that is, for most of us, anyone who's held a laser pointer,
00:01:01.12 you get the impression that light travels in straight lines. And
00:01:04.17 that should be very straightforward to understand where light
00:01:09.19 goes, and the best example of this for imaging is the simplest
00:01:14.02 of all imaging devices, the pinhole camera. I just want to show you
00:01:18.01 a picture of a pinhole camera here. A pinhole camera is a
00:01:22.17 device that has a box, and this box has a pinhole
00:01:29.05 in it. And the box usually has a piece of photographic
00:01:32.28 paper inside the box. The box is sealed, I've not shown the sides
00:01:37.22 of the box. And the photographic paper is looking at the world
00:01:41.14 and the world is being seen through a single pinhole.
00:01:44.10 In this case, the world consists of a single lamppost,
00:01:47.17 as shown on the other side of this image. And the idea of the
00:01:51.24 pinhole camera is very straightforward. It's that light from
00:01:54.15 each part of the lamppost goes in all directions, but because the
00:02:00.09 pinhole is only in one place, only one ray of light from each part
00:02:04.04 of the lamppost gets through the pinhole. So the very top of the
00:02:07.12 lamppost has light going in many directions, the two dotted lines
00:02:10.24 are two directions light is traveling that bumps into the
00:02:14.04 wall of the box. But one of the rays, the one that's solid
00:02:18.00 goes through the pinhole and ends up down here.
00:02:21.11 Similarly, light from the bottom of the lamppost goes in all
00:02:24.22 directions, but the only light that passes through the pinhole
00:02:27.16 is the part that ends up up here. And so it goes for every object
00:02:32.13 in space. So if you leave the photographic paper inside a
00:02:37.07 pinhole camera long enough, you get a very nice picture
00:02:40.08 of the world. And if this were the way light actually acts, then
00:02:47.05 we should be able to do very high resolution. Because light just
00:02:51.01 travels in straight lines, and we should be able to expand objects.
00:02:54.19 For example, if we move the piece of photographic paper
00:02:58.10 further away from the pinhole, that object gets bigger and
00:03:01.20 it should get bigger and bigger and we should get more and more
00:03:04.08 resolution. So that ultimately, every little fact of life in
00:03:09.16 the real world should end up on that piece of paper. It turns
00:03:12.26 out, however, that there are certain problems with the pinhole camera.
00:03:17.07 And that is which limits its resolution, and I want to
00:03:21.09 give you one obvious example. You can see that the pinhole
00:03:24.19 were bigger, this image would get fuzzy because there would be more
00:03:29.12 than one ray of light from the top of the lamppost that would get through
00:03:33.26 the pinhole. And it would blur the image down here. So
00:03:36.21 a big pinhole wouldn't work at all. And in fact, if there was no pinhole
00:03:40.04 at all, you wouldn't get an image of the world, you'd just get the
00:03:42.13 light on this piece of paper. However, you might imagine we should get
00:03:47.00 better resolution if we turn the pinhole down. It would be a sharper
00:03:50.16 image as the pinhole got smaller and smaller, restricting
00:03:53.21 only a single ray of light from each position from getting
00:03:57.17 through. But the surprising thing about a pinhole camera is
00:04:00.09 that if you make the pinhole very small, as shown here,
00:04:02.10 so now that only a very infinitesimal small amount of light gets through,
00:04:07.18 then suddenly the image gets blurry again. And this blurriness
00:04:11.04 ultimately is because light is not traveling in straight lines.
00:04:16.12 Light is bent at that pinhole, it's diffracted. And that bending of
00:04:21.09 light is ultimately what limits the light microscope. So I want to
00:04:25.04 explain why the light bends, because I think if you understand why
00:04:29.00 the light bends, you'll understand why microscopes have
00:04:32.00 limited resolution. It bends because light is actually not
00:04:36.19 traveling in straight lines, despite the fact it seems to.
00:04:39.24 It's actually that light is a wave and it's traveling in all
00:04:43.02 directions. This is not an intuitively obvious idea for most people.
00:04:46.27 And in fact, it required a great genius, a man named Christian Huygens,
00:04:51.22 to first figure out this fact. Christian Huygens was a Dutch
00:04:57.18 polymath. He did many, many things. Invented many things.
00:05:00.24 He was a genius in many different areas, and one of his
00:05:04.12 great claims to fame was wave optics. The idea that
00:05:07.26 light travels in waves, not in lines. This was his big idea
00:05:11.07 and it was counterbalanced by Newton's brilliant idea that light is
00:05:16.25 actually made of particles, the corpuscular theory of light.
00:05:19.17 And of course, we all know about photons, and that word
00:05:22.27 photon was first used by Newton. It turns out they were
00:05:27.04 both right, and in some ways, Christian Huygens was
00:05:30.01 closer to reality, I think, than Newton's view of light.
00:05:33.20 But that's a long story. I want to tell you just what a wave theory
00:05:38.00 says light actually is, instead of being a straight line,
00:05:41.09 that what light can be thought of as traveling as a wave.
00:05:44.26 And a wave might be thought of as a straight plane wave,
00:05:52.19 and by a plane wave, you would have a front of light moving
00:05:56.04 in a particular straight direction as these arrows say. And that
00:05:59.08 wavefront would be like the breaker at a beach. A long
00:06:03.13 linear wave moving forward, and right behind it, at its wavelength,
00:06:09.03 is another wave, and behind it is another wave. So each of those
00:06:12.09 vertical lines consists of one of these waves. And right
00:06:15.17 below that little sinusoid is a way to represent them in 2-dimensions,
00:06:20.16 just the direction of the wave. The wave is up and down, but the light
00:06:26.00 is moving left and right. It's called a plane wave because, in fact,
00:06:30.19 it's not a single line moving. But it's actually a plane waving
00:06:35.14 and right behind it another plane waving. That's why it's called a
00:06:38.23 plane wave. Wave optics says that in addition to plane waves,
00:06:42.16 light can travel in directions that either converge to a point
00:06:46.15 or diverge. And these are known as spherical waves, where
00:06:49.29 the wave front, rather than being a straight line, is now a
00:06:53.11 curved surface, it could even be a sphere, which light is traveling
00:06:56.20 out from a single point or converging in toward a single point.
00:07:00.07 And there you might draw it that way. So spherical waves
00:07:04.10 are the curvature of a circle, or a sphere, and plane waves
00:07:09.24 are the surface of a plane. Now, Huygens' idea was that the
00:07:16.29 elemental unit of light was not these plane waves, but the plane waves
00:07:22.03 and the spherical waves were composed of very small infitismally
00:07:27.12 small objects that now are called Huygens wavelets, which are
00:07:32.00 little points of light that distribute their light in all directions
00:07:35.26 that have no directionality at all. He called these wavelets
00:07:39.11 and to this day, we use that term. Many aspects of the way a
00:07:42.23 photon behaves in space before it interacts with matter
00:07:45.26 is very much like a Huygens wavelet, that is the light is traveling
00:07:49.23 in all directions and where it ends up has a lot to do with where
00:07:53.07 there's no interference. As opposed to where it doesn't
00:07:55.25 end up are sites where light is interfering. Now I'll explain that
00:07:59.26 in a second. So a Huygens wavelet can be thought of as a
00:08:03.04 pulsating little point of light. I'm showing here a sort of diagram
00:08:06.12 of that pulsation. Where there'd be a point of light and that pulsation
00:08:09.28 that is sort of the fluctuation of the wave, where the distance
00:08:14.01 between the crests of these waves are basically the
00:08:17.16 wavelength of light. This is the unit of light, it's an infinitesimally small
00:08:21.28 unit. And shorter wavelength light has the spacing closer
00:08:27.08 together, and the frequency of the pulsing higher.
00:08:30.01 At any instant in time, a wavelet might look as shown on
00:08:37.05 this side on the left here, with a series of rings, each of
00:08:42.10 which have the same amount of energy. But because
00:08:44.09 each ring is larger, the amplitude of the wave gets smaller
00:08:47.16 and smaller as one goes out at any one point.
00:08:51.00 The intensity one sees is the square of the amplitude of the wave, and
00:08:57.26 then one has to integrate that over at least one wavelength.
00:09:01.16 So one has a very bright spot, and then progressively dimmer
00:09:04.10 light if you integrate the effect of a wavelet over time.
00:09:08.26 So how do these wavelets explain plane waves and spherical waves?
00:09:13.18 That's easy to see by Huygens original drawings, and here's
00:09:18.21 a more modern version of these drawings. Which is that a
00:09:21.21 plane wave, as shown here in this diagram, is just a series
00:09:25.24 of wavelets that are aligned in a straight line, as shown here.
00:09:30.04 And those wavelets have a leading edge, which is the wave front
00:09:35.16 of each of them. And that a line of those leading edges
00:09:40.16 gives a straight line. And in that direction, there's nothing
00:09:43.19 to interfere with them, so that moves forward and the next
00:09:46.16 wave comes right behind it. Spherical waves are a similar
00:09:51.28 idea, but rather than being a straight line of wavelets, the
00:09:55.11 wavelets are on the surface of a sphere. And they can either be
00:09:59.29 diverging, so the light is leaving a point and spreading out
00:10:02.15 in space, or converging as when a laser is focused to a spot
00:10:07.06 in a specimen. Beyond the spot, the light diverges. So a
00:10:11.05 convergent spherical wave generates a divergent spherical wave.
00:10:14.21 So there are spherical waves and plane waves. Now,
00:10:18.01 how does this help us understand how a pinhole when it gets
00:10:21.28 small not work? That's explained here. When you have a very, very small
00:10:27.04 opening, and you have a plane wave with a bunch of wavelets
00:10:31.18 lined up underneath that pinhole, then if the hole is very small
00:10:35.27 only a few, maybe only one wavelet, gets through. And that
00:10:39.12 wavelet then spreads its light out in all directions. And
00:10:43.01 that is what is known as diffraction. And that diffraction explains
00:10:46.29 why if the pinhole gets very small, the light no longer travels
00:10:49.24 in a straight line, it bends in all these different directions.
00:10:53.08 And this is a theoretical approximation, and yet all experiments
00:10:58.20 suggest this is actually the way light behaves.
00:11:01.17 Now, one interesting thing is if you have two wavelets
00:11:05.18 or more that came from the same source, that are part of the
00:11:08.28 same original pulsating light, and therefore they are coherent.
00:11:13.15 Their wavelengths are in sync with each other. Then one can have
00:11:17.13 interference patterns made up of several wavelengths.
00:11:21.04 First in the example with one pinhole, light in a box will
00:11:26.07 be going in all directions. But the light coming out will bend,
00:11:28.26 that is diffraction. And then in the second example, if one has
00:11:33.18 more than one source, for example, a light source that
00:11:36.29 then goes through two slits, one sees out at the other end
00:11:40.25 rather than just uniform light, a funny banding pattern.
00:11:44.19 A destructive and constructive interference. Light goes through
00:11:47.26 one pinhole and then after it goes through two more pinholes,
00:11:53.00 you get this weird interaction of places where the crest are
00:11:58.01 on top of each other, where the troughs are adding together
00:12:01.23 in a negative direction, and places where one is a trough
00:12:04.19 and one is a peak, and together they nullify. So here in
00:12:08.29 instantaneous time is a complicated pattern of waves
00:12:12.28 based on the interactions of two different wavelets.
00:12:16.11 And I want to unpack that idea a little more, so you can
00:12:19.13 understand how light microscopes work. So if we have a light
00:12:23.07 source, as shown, that's sending out its light in all directions,
00:12:28.13 and then that generates a spherical wave made up of
00:12:31.13 a bunch of wavelets, one of the wavelets might go through
00:12:34.25 a slight here and generate a pattern of a wavelet on the other
00:12:39.29 side of that pinhole. There's a second slit in that box, and there's a
00:12:46.20 second wavelet that's gone through. And together those wavelets
00:12:51.02 interact in ways that either add or subtract from each other.
00:12:55.09 And there is that interaction pattern. So let's look at that pattern
00:12:59.28 in a little more detail. If we go to a position where the two
00:13:04.21 wavelet openings are exactly equidistant from that point,
00:13:09.26 then they will both be cresting or troughing at that distance.
00:13:14.13 And so one would have constructive interference. If we go to a position
00:13:19.12 where one of the distances is exactly half a wavelength
00:13:22.29 longer than the other, one would have destructive interference.
00:13:26.14 So there would be bright light where they constructively interfere and
00:13:30.26 no light where they destructively interfere. And that's why one
00:13:35.08 ends up with this pattern of bright and dark bands, because one
00:13:38.28 can be even a whole wavelength apart, one can be one
00:13:42.15 wavelength longer than the other, and still you would get
00:13:45.11 constructive interference. But if it's 1.5 wavelengths
00:13:48.22 difference in path length, then one would be destructive
00:13:52.05 relative to the other. So one has light and dark bands that
00:13:54.23 go over a wide range here. So how does this help us
00:13:59.04 understand microscopes? Well, let's review for a second the way
00:14:02.13 a microscope is designed. So a microscope takes
00:14:05.23 light from a specimen, shown over here, and in this case,
00:14:09.19 there's just a single point of light generating a spherical
00:14:12.09 wave. And what the microscope objective does is that point
00:14:17.29 source, which generates a spherical wave that diverges,
00:14:21.05 is that the objective lens then collects a sub portion of that
00:14:27.22 spherical wavelet and turns that light into a plane wave.
00:14:33.00 Because a microscope is designed such that the specimen
00:14:36.26 plane is exactly at the focal distance of that lens. So
00:14:40.13 those spherical waves turn into plane waves, and those plane
00:14:44.08 waves then travel through the microscope tube until it
00:14:49.10 reaches the tube lens, which is another positive lens, which now takes
00:14:54.00 a plane wave and turns it back, the tube lens does this, turns it
00:14:58.17 the plane wave back into a converging spherical wave.
00:15:02.04 And that spherical wave ends up as a point there. And this
00:15:07.04 allows us to ask the following question, which is for a point
00:15:12.04 object in the focal point, such as an infinitesimally small fluorescent
00:15:16.10 bead, what is the distribution of light in or near the image plane?
00:15:20.08 And how is that point spread? Is it spread, first of all, and if
00:15:24.18 it's spread, what is the function of that spread?
00:15:26.28 How does it end up being in a particular distribution?
00:15:31.10 So we're going to go through the same idea of wavelets,
00:15:33.28 but now look at it in the case of a converging spherical
00:15:39.04 wave. We're going to look at that part of the microscope,
00:15:41.19 because that is where the interference patterns generate
00:15:45.19 a complicated mix of focus to a point, plus light around
00:15:50.29 that point. So, this is now beyond the tube lens. The tube lens
00:15:55.09 is shown at the very extreme end of this image, there's a
00:15:58.08 converging spherical wave made up of a bunch of wavelets.
00:16:00.19 And each of those infinite points in the emerging wave front
00:16:04.11 is acting like its own point source of light, like a Huygens
00:16:08.00 wavelet. That is each point emits a wavelet, sending out light
00:16:11.17 in all directions. And all wavelets from the same wave
00:16:14.20 front are mutually coherent, they're oscillating in synchrony.
00:16:17.17 And because of that, they can interfere in a fixed way.
00:16:21.06 Therefore they interfere with each other in a very predictable
00:16:23.19 pattern. So let's start now with just two of those wavelets.
00:16:29.14 And arbitrarily pick a position midway between them
00:16:34.10 where that vertical line is, where that distance D1 and D2
00:16:38.11 are identical. The distance is the same, those two waves will
00:16:42.02 both be troughing and peaking or in between at exactly
00:16:46.11 the same point. And therefore, if you add those two waves
00:16:49.13 together, they're in phase and you will get constructive
00:16:52.19 interference. In distinction, if we move D1 and D2 slightly lower down on
00:16:59.19 that vertical line, one of those is now a half a wavelength
00:17:03.18 longer, the top one, D1 is a half a wavelength longer distance
00:17:08.10 than the bottom one, D2. And so when those two waves reach
00:17:11.14 that point, they're out of phase. Exactly half a wavelength out
00:17:15.17 of phase, and if you add up those two sinusoids, you're going to
00:17:18.22 end up with complete destructive interference and no light.
00:17:21.14 If one goes a little further down, now D1 is much longer
00:17:26.19 than D2. In fact, exactly 1 wavelength longer than D2.
00:17:30.14 You add up those two waves, you're going to get a full
00:17:33.25 cycle out of phase, but they are peaking at the same point
00:17:38.28 and that generates constructive interference. And so here is
00:17:44.01 the patterns one sees instantaneously on the top. And in
00:17:48.09 integrated over time, on the bottom. And you can see
00:17:52.15 that the troughs and the peaks of interference on the
00:17:56.14 top have all turned now into positives and zeroes.
00:18:01.04 There's no negative amplitude when you square the amplitudes, now
00:18:04.15 everything is positive amplitude or zero. And that interference
00:18:09.15 pattern ends up with a series of bands. So an instant
00:18:12.19 in time on the top, there's a mesh of dark and bright
00:18:15.28 lines. Detectors, including the eye, however don't see
00:18:19.21 amplitude. We collect intensity. And intensity is the addition
00:18:24.19 of a full cycle of those wavelets and squared the result.
00:18:29.04 And the detected interference pattern now has dark and
00:18:32.02 bright lines. And right in the middle, there's a white line
00:18:34.08 running horizontally, and that is the place where the two
00:18:38.12 wavelets are mutually constructively interfering all the way along
00:18:43.20 that length. The dark areas are the destructive interference, the
00:18:47.16 bright areas are the constructive interference. So we've dealt
00:18:49.15 with two wavelets. And now I just want to say a little bit about
00:18:53.15 the way microscopes work, in the sense that there's nothing
00:18:57.19 so far in what I've said that would tell you where the
00:19:00.17 image plane for those converging spherical waves are.
00:19:03.26 There's just a pattern of light and dark bands. You're going to
00:19:07.06 see that this approach, without any extra understanding
00:19:11.10 of ray optics, generates the image plane at a particular plane
00:19:15.06 largely because that is the one place where all the wavelets
00:19:20.11 can be in phase at the same time. But before talking about that, we
00:19:24.18 have to deal with the fact that microscope objectives
00:19:26.23 come in many flavors. And the particular flavor I want to talk
00:19:30.06 about now is not the difference in magnification,
00:19:32.20 which many beginning microscopists think is the most important
00:19:38.05 aspect of an object, but it isn't. What really matters for the
00:19:42.05 quality of an objective and its resolving power is its numerical
00:19:45.16 aperture, NA. And the NA has a weird description, it is the
00:19:50.19 index of refraction of the material between the microscope
00:19:53.25 objective and the slide, which in air would be very close to
00:19:57.24 1, or 1.0. But in water or glycerin or oil, that number could be
00:20:03.22 higher. Times the sine of the half angle between a vertical
00:20:08.16 and the largest deviation of light that could still be
00:20:13.25 collected by the objective. So a low numerical aperture lens
00:20:17.25 as shown in A, has a narrow cone that it can collect light
00:20:22.04 coming off of the specimen. So if you have a fluorescent
00:20:25.02 specimen that sends light in all directions, you only collect
00:20:27.29 a small amount with a low numerical aperture lens.
00:20:30.22 A medium numerical aperture lens has a slightly larger
00:20:34.14 cone that it collects. And a high numerical aperture lens
00:20:37.15 has the highest cone of all. It is a fact, and I'll try to explain
00:20:42.08 why, that the higher the numerical aperture is, the better
00:20:45.04 the resolving power of the microscope. And we have to go back to that
00:20:48.25 wavelet picture and look at the difference on the top between
00:20:52.09 a high numerical aperture objective and on the bottom, a low
00:20:56.06 numerical aperture objective. So in both cases, we have a wavelet
00:20:59.12 generating light and you see you're collecting a greater portion
00:21:03.07 of that spherical wave with the high NA lens. So a larger
00:21:10.03 amount of the light is collected, and that generates a
00:21:12.11 larger, basically a bigger plane wave, that eventually
00:21:17.14 goes to the tube lens. And then from the tube lens,
00:21:21.02 it's converted into a converging spherical wave.
00:21:24.28 So with a high numerical aperture lens, you have more
00:21:27.09 of the spherical wave collected. And so you generate a larger
00:21:31.22 spherical wave going out towards the image plane. With a
00:21:36.16 lower numerical aperture lens, you have a smaller amount
00:21:40.02 of the spherical wave collected. And the edges of that
00:21:46.00 spherical wave are very close together, relative to a high numerical
00:21:50.14 aperture lens where the most extreme parts of the spherical
00:21:55.25 wavelet that's converging are far apart from each other. And that turns out
00:22:00.06 to be the critical feature for resolution, as you'll see.
00:22:02.26 So if we compare the extreme wavelets from a curved
00:22:08.07 spherical wave in a high numerical aperture lens and a low
00:22:12.01 numerical aperture lens, you can see that the effect of that
00:22:15.21 is a different pattern of banding down at this arbitrary
00:22:20.17 position here, which will be the image plane but there's nothing yet to tell you
00:22:24.10 why. Because two wavelets don't focus on an image. So the
00:22:28.15 finest fringes determine the finest detail that can be found in an
00:22:32.08 objective. This is what limits the resolution of an image, how
00:22:36.13 fine that banding pattern is. And that period that is the distance
00:22:40.24 between the bright and dark areas is related to the numerical
00:22:45.26 aperture. If the two wavelets are far apart, then interference
00:22:50.00 allows a very high frequency set of dark and light bands. If the two
00:22:56.19 wavelets are very close together, as in a low numerical aperture
00:22:58.28 lens, then the banding is more spread out. And this limits the
00:23:03.09 finest detail in the image plane. So now let's try not to just look at two
00:23:07.17 wavelets, but add more and more wavelets to that converging
00:23:10.18 spherical wave. And look what happens to the integrated pattern
00:23:14.22 of interference down at a plane that ultimately becomes the image plane.
00:23:20.00 So first, high numerical aperture gives short fringe period and
00:23:24.02 gives narrow fringes. Low numerical aperture gives long fringe
00:23:26.26 period and wide fringes. So now, whatever numerical aperture
00:23:31.26 we have, we're going to have a converging spherical wave
00:23:34.02 that's that blue circumference one sees at the top of this
00:23:37.27 image, the tube lens is up above. And now what I'm showing you is the
00:23:42.23 two wavelets at the rim of the lens, plus one wavelet right at
00:23:49.20 the center. So three wavelets are interfering. And we're looking at the
00:23:53.20 pattern that is generated by these three wavelets after we
00:23:57.19 integrate over a wavelength and square the result.
00:24:02.17 The recorded intensity is the series of bar patterns that turns
00:24:06.15 into islands, instead of the stripes we now have islands
00:24:10.21 of bright and dark. And at the image plane, which will end up
00:24:14.26 where this line is here, one sees that there is an island
00:24:20.05 of bright right at the center, and that is because at that
00:24:23.07 point there, you are equidistant from those three wavelets
00:24:26.08 because that point is the center of a circle for which those
00:24:30.26 three wavelets are on the circumference of that circle. So
00:24:34.11 they're equidistant from that point and that is why that is
00:24:38.01 where the image is focused at that point for the specimen
00:24:42.00 that's located in the center of the image field.
00:24:45.00 And the brightest slide is there because the interference
00:24:49.17 pattern is strongest because you've got all three wavelets
00:24:52.08 in phase there, better than anywhere else.
00:24:55.01 If we look at five wavelets, just add two more besides the three
00:25:00.19 that I've already talked about, again the brightest fringe
00:25:03.27 is at the center of the image plane, because that is the one
00:25:07.16 place that all five are mutually coherent. Everywhere else
00:25:12.01 some interference in a negative way is destroying the
00:25:16.07 intensity. And all wavelengths are in phase at the center of the
00:25:19.23 image plane. And so that is the brightest area. And you can
00:25:22.20 see as you add more and more contrast and enhancement,
00:25:26.10 besides these three bright islands, there are lots of little
00:25:30.23 side lobes that are dimmer. So that's five wavelets.
00:25:34.19 If we go to nine wavelets, the image becomes more straight
00:25:39.27 forward in a way, that is the one place again where all those
00:25:43.02 wavelets are in phase because they're equidistant from each
00:25:45.24 other, is right at that point there. And all the wavelets
00:25:51.15 are in phase there, and that is the main peak. But there is
00:25:55.11 this set of banding on either side of that point. Now if we
00:26:00.16 just add all the wavelets together, this is the point spread function, this is
00:26:04.16 going from nine wavelets to infinite number. Let's just go
00:26:07.26 through the shape of this object now. All the wavelets are
00:26:12.02 in phase at the center of the image plane, light is concentrated
00:26:16.03 in the middle of the image plane. All of the fringes or
00:26:19.17 side lobes are progressively dimmer. In fact, you need enhanced intensity
00:26:24.09 to see them, as shown in this very bottom horizontal picture.
00:26:28.11 And most of the light is now seen within a cone, it seems that all
00:26:33.03 the interference has left very little light outside of this converging
00:26:39.07 cone of light that focuses on the specimen and then rediverges.
00:26:42.29 This is an hourglass shape. And that comes about not through
00:26:46.14 ray optics, but just through interference of wavelets.
00:26:49.11 The width of the main peak is set by the extreme wavelets,
00:26:55.04 the two on the two ends of that spherical cap. And that is
00:27:00.24 set by the numerical aperture. So the size of the image
00:27:04.23 for a very small point is related to how far apart those
00:27:09.16 wavelets are. The further apart they are, the more focused
00:27:12.15 this object is. And that's why the numerical aperture is
00:27:16.07 essential. So if we compare the effect of numerical aperture
00:27:19.24 on the point spread function, a high numerical aperture compared to low
00:27:23.02 numerical aperture, you see a profound difference. With high
00:27:26.07 numerical aperture, there's a wider separation between wavelets,
00:27:28.26 which is possible because they're far apart. As a result,
00:27:33.09 there's a small central peak, whereas with low numerical
00:27:36.05 aperture lenses, only a narrow separation between the
00:27:39.19 wavelets is possible, so you have a broad central peak
00:27:42.14 and less resolution. So, let's now think about this in terms
00:27:48.20 of what the image of a point looks like. So if we focus now
00:27:54.11 not in the axial direction, but looking down on the surface of an image that
00:28:01.27 point, right in the focal plane, the point in the specimen would appear
00:28:06.13 as a small dot. This dot is called an airy disk because
00:28:12.19 if you enhance the contrast, you find that the dot is surrounded by
00:28:17.01 these fainter rings of light, which is the diffraction pattern
00:28:21.26 of the point. This is the point spread function in one
00:28:25.12 single plane. Airy is a name of a researcher, I think he was
00:28:32.02 an astronomer, who saw airy patterns around stars
00:28:35.15 and understood that this was an artifact of diffraction.
00:28:38.12 Not the fact that every star had rings around it. The PSF is
00:28:43.26 a series of concentric rings, and larger rings have progressively
00:28:48.11 lower -- the further out the rings are, the progressively lower
00:28:52.15 the intensity. And if you have to remember anything in
00:28:56.06 light microscopy, in terms of equations, it would be
00:28:58.12 this. That the distance from the center of the airy disk,
00:29:02.15 that center circle, to the first dark ring, is 0.61 times the
00:29:10.01 wavelength of the light, lambda, divided by the numerical
00:29:13.16 aperture of the lens. 0.61 lambda over NA. You want to carry
00:29:17.08 around any fact about light microscopes, that's the one
00:29:20.16 you should carry around. Because as you'll see that is the
00:29:22.28 resolution limit of the light microscope, 0.61 lambda over NA
00:29:26.23 is the distance from the center to the first dark ring. One can also
00:29:30.17 look at the airy pattern in the not-lateral direction, but in the
00:29:34.05 axial direction, in XZ. X being the lateral, and Z being the
00:29:39.23 depth plane. And one sees that the area pattern is not
00:29:43.19 a circle, but more of an ellipse. And again, there are two
00:29:47.21 -- if you enhance it, there are two dark spots in that direction
00:29:51.16 and that's a longer distance. It's determined by another equation
00:29:55.12 2 times the index of refraction of the material between the
00:29:58.23 objective and the specimen, so 2 times n times the wavelength,
00:30:02.22 divided by NA squared. And for high numerical aperture
00:30:06.08 lenses, this is sometimes around three times bigger
00:30:09.17 than the XY resolution. So light microscopes intrinsically
00:30:12.26 have slightly better resolution in the lateral plane than their
00:30:16.24 ability to discriminate one plane from another in the depth
00:30:20.03 plane. So, can we use this to help us understand resolution
00:30:26.28 in a light microscope? This 0.61 lambda over NA.
00:30:30.18 The distance between the center and that first dark peak.
00:30:33.03 There are, if you think about this, these are the numbers
00:30:37.07 calculated for a high numerical aperture lens, the 1.4
00:30:42.01 numerical aperture lens with light at 480 nm. You can see
00:30:48.00 that the lateral resolution, and I'll explain why its resolution
00:30:51.04 is 225 nm, whereas the Z-axis resolution is 861 nm.
00:30:58.06 Not nearly as good. So how close is it possible for two
00:31:05.27 point sources to be and still be seen as two points?
00:31:09.25 This is basically the question of resolving. If two objects are
00:31:13.16 close, and you still believe they're separate objects, then
00:31:17.02 you can resolve them. If they blend together, you can't resolve them.
00:31:20.28 So here, for example, are two point sources. And they're moving
00:31:24.09 progressively closer. There they are sitting on top of each
00:31:27.22 other at the bottom, and they're far apart at the top.
00:31:30.02 No one would disagree that they're two points at the top. No one
00:31:33.10 would believe that this is two points at the bottom. How do we
00:31:36.16 decide when these two points are resolvable? This is sometimes
00:31:40.28 known as the Rayleigh criterion. It's an arbitrary criterion.
00:31:43.24 It's not a generally accepted criterion that everyone says
00:31:48.02 tells you the resolution, but most microscopists use the Rayleigh
00:31:51.07 criterion. Which is the criterion related to 0.61 lambda
00:31:55.00 over NA. That distance. So that distance, as I mentioned to you already,
00:32:01.02 is the distance between the center of one disk and its first
00:32:04.25 dark ring. And if we look at these traces here, these are the
00:32:08.06 two points that are just at the Rayleigh criterion apart, that is
00:32:12.13 0.61 lambda over NA, the center of this object is sitting
00:32:16.16 right on the first dark ring of that object. If you look down here
00:32:20.28 at an intensity trace, where the first one is at its lowest
00:32:24.19 value is where the other one is peaking. And at that
00:32:29.01 position, there is maybe a little more than 20% of a dip
00:32:35.00 in intensity between them when you add the two intensities.
00:32:37.22 One can also make the Rayleigh criterion in the z-axis.
00:32:41.11 Where you go from the center of one to the first dark ring,
00:32:45.22 to its first dark ring and its other object at that distance. And that again
00:32:50.12 gives you a value where they dip about 19%. And these are both
00:32:55.03 related to the other equation that I mentioned,
00:33:00.28 2 n lambda over NA squared. Whereas this is 0.61 lambda over NA.
00:33:05.02 Now the value of numerical aperture is that it makes those points smaller.
00:33:09.00 So presumably, the point spread function by being smaller
00:33:12.09 allows objects to be closer together. And here's a
00:33:15.07 classic example of this. It's a bunch of beads where
00:33:18.29 they've been imaged with a low numerical aperture lens, a
00:33:22.09 medium numerical aperture, and a high numerical aperture lens.
00:33:24.19 There's no difference in magnification, but there's no
00:33:27.06 doubt in C, that you have a bunch of individual beads with lots
00:33:31.00 of rings around them, but you still see them as separate.
00:33:34.05 In the medium NA, it's a little harder to do. And in the low NA
00:33:37.07 lens, it's just a big blob, you can't actually figure out what you're looking
00:33:40.24 at, at all. I want to just say one last thing, which is important
00:33:46.28 to realize, is if you want to get the resolution out of your light
00:33:53.04 microscope, you not only have to understand what limits it
00:33:56.25 in terms of the diffraction limit, but you have to sample the image
00:34:01.04 at enough frequency that you can see those high frequency
00:34:06.03 details. And this requires you to keep in mind Nyquist's limit.
00:34:10.11 And Nyquist, it was named after a scientist, Harry Nyquist,
00:34:14.06 who stated that sound must be sampled at least at its highest
00:34:19.15 frequency in the sound times two, so it must be sampled twice
00:34:28.05 at its highest frequency to extract all the information
00:34:31.02 from the bandwidth and accurately represent the original
00:34:34.13 acoustic energy. So for example, the human ear hears frequencies
00:34:37.14 up to 20 kHz, so a CD, if you have one, will sample at 44.1 kHz.
00:34:43.19 So that you can hear that 20 kHz clearly. Phone lines pass frequencies
00:34:48.27 up to 4 kHz, so phone companies sample at 8 kHz. So the
00:34:53.09 sampling theorem says, and this is just the Nyquist Theorem,
00:34:55.22 a continuous function, for example, some banding pattern
00:35:00.22 in an image, can be completely represented by a set
00:35:03.04 of equally spaced samples if the samples occur at more
00:35:07.14 than twice the frequency of the highest frequency component
00:35:10.13 of the function. So to capture a function with maximum
00:35:13.24 frequency F, whatever it is, whatever image you're looking at,
00:35:16.19 you must be resolving it, you must be taking samples that are
00:35:21.02 twice that highest resolution, two times that frequency.
00:35:24.16 And this is the Nyquist limit. And down here, it just shows why
00:35:28.25 you have to do that. If you have a banding pattern, and that red is
00:35:31.20 sort of an area that's going from light to dark, light to dark,
00:35:34.26 and you only sample at the resolution limits, so you get one
00:35:38.22 sample per wave of that light dark cycle, you're going to end up
00:35:43.12 with aliasing. You're going to end up with a pattern of
00:35:46.21 light and dark that's going to not be the frequency. You want to see that
00:35:50.29 high frequency, you have to have at least one sample from each
00:35:53.19 peak and each trough. And I just emphasized this because most
00:35:57.28 microscopists, they do very well and get very fancy
00:36:00.08 objectives, and then they fall down on the imaging side. They do not
00:36:04.10 do Nyquist limited imaging. So for example, if the resolution limit
00:36:08.11 is 0.61 lambda over NA, and you want to see that resolution,
00:36:12.24 you're going to have to sample enough so that there are at least
00:36:16.21 two pixels between those two. So the Nyquist limit is
00:36:23.16 0.3 lambda over NA. And one has to either zoom the
00:36:27.17 microscope image, use a higher magnification lens to
00:36:31.07 make sure one matches the charge coupled device,
00:36:34.11 or whatever digitizing device one is using for the image,
00:36:38.11 to make sure that the pixel size on the image is twice
00:36:42.04 the limit due to the point spread function. This is all great,
00:36:49.21 sometimes however, it's not enough. And I just want to end
00:36:54.04 this by telling you that there are still things that people
00:36:57.07 can do if you feel you need more resolution than your microscope
00:37:01.03 objectives can give you. One is that because it's 0.61 lambda over NA,
00:37:07.04 if you use shorter wavelength light, you will have better
00:37:10.03 resolution than longer wavelength lights. So blue light or
00:37:12.14 ultraviolet light will give you a sharper image than
00:37:15.26 red light or infrared light. If that's not enough, you can keep
00:37:21.08 going to higher index of immersion liquids, because 0.61 lambda over NA,
00:37:26.15 the NA term is related to the index of refraction times
00:37:31.17 the half angle. The higher the index of refraction, the higher
00:37:34.21 the resolution. So oil is better than glycerin, and glycerin is
00:37:38.24 better than water, and water is better than air.
00:37:41.22 And there are even some very esoteric oils that allow you
00:37:45.05 to push into very, very high numerical apertures. They're stinky
00:37:50.11 oils, not very easy to obtain, but there are things you can do.
00:37:53.23 A confocal microscope, which many people have access to,
00:37:58.27 if it's used properly can improve your resolution by the square root of 2,
00:38:02.13 by about 1.4. So that's something to think about.
00:38:05.10 And if you can study single spots, you can localize
00:38:11.23 them by looking at the center of the airy disk, and that can give you
00:38:15.22 super resolution. And that arbitrarily high resolution
00:38:19.00 has been used to track objects moving over time, and it is
00:38:22.18 also the basis of one of the many new techniques known as
00:38:27.13 super resolution techniques, optical nanoscopy, which I won't talk
00:38:30.12 about today. But someone will probably provide you with information.
00:38:34.01 So thanks very much.

This Talk
Speaker: Jeff Lichtman
Audience:
  • Researcher
Recorded: December 2012
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Talk Overview

In this talk about resolution in microscopy, Jeff Lichtman describes the diffraction of light, a key principle in image formation and a factor that limits the resolution of a conventional light microscope. The behavior of light traveling through an objective is described along with the concept of numerical aperture. The “point spread function” (or PSF) and Nyquist sampling are explained, which are critical concepts for understanding image resolution and detection of images.

Questions

  1. When light passes through a pair of slits, a series of bright and dark bands are produced. This is best described by:
    1. Light traveling as particles that collide and self-destruct to produce the dark bands.
    2. Wavelets that emanate from the slits and that produce constructive or destructive interference.
    3. Spherical waves traveling towards the slits emerge on the other side as planar waves.
    4. The dark bands are produced by light waves that travel the same distance or by waves that travel an extra distance of one or multiples of the wavelength of light.
  2. Resolution in the microscope can be best improved by:
    1. Using a high numerical aperture objective
    2. Increasing the magnification
    3. Decreasing the wavelength of the illuminating light
    4. Using a highly sensitive CCD camera
  3. The Rayleigh criterion for lateral resolution is
    1. equal to the lamba (the wavelengh of light)
    2. equal to (0.61 x lambda)/N.A.
    3. equal to 2 x N.A.
    4. equal to (2 x lambda)/N.A.
  4. True or False. When light travels through a lens, the width of the point spread function is determined by the most lateral wavelets emanating from the converging wavefront.
  5. The z axis resolution of a point spread function for a 1.4 NA lens and green light
    1. Similar to the x-y resolution
    2. Approximately two-fold better than the x-y resolution
    3. Approximately two-fold worse than the x-y resolution
    4. Approximately four-fold worse than the x-y resolution
  6. To achieve full resolution, the Nyquist sampling criterion states that image must be sampled at a minimum of ___ of the theoretical limit of resolution
    1. 1 X
    2. 2 X
    3. 5 X
    4. 10 X

Answers

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Speaker Bio

Jeff Lichtman

Jeff Lichtman

Jeff Lichtman’s interest in how specific neuronal connections are made and maintained began while he was a MD-PhD student at Washington University in Saint Louis. Lichtman remained at Washington University for nearly 30 years. In 2004, he moved to Harvard University where he is Professor of Molecular and Cellular Biology and a member of the… Continue Reading

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