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Abbe Diffraction Demonstration

00:00:11.25 I'm Kurt Thorn from UCSF, and today I'm going to tell you about
00:00:15.13 a very famous microscopy experiment, the Abbe Diffraction
00:00:19.21 Experiment. And mostly we're going to be in the lab today doing
00:00:22.27 a demonstration on a custom built optical system, but before we
00:00:26.28 go into the lab, I just wanted to give you some introduction to
00:00:29.16 this experiment and its background, and what it shows about
00:00:32.24 the microscope. So, this man here is Ernst Abbe, who is one
00:00:38.06 of the founders of modern microscopy, in a sense. He was active
00:00:42.10 in the late 1800's, working with Carl Zeiss and others in really
00:00:46.21 formalizing the performance of the optical microscope, and learning
00:00:50.13 how to build diffraction limited objective lenses. And in fact,
00:00:54.01 was one of the people who identified what limiting resolution
00:00:57.24 is in the light microscope. And his diffraction experiment, the
00:01:01.12 Abbe Diffraction Experiment here, is how he formalized that
00:01:05.05 definition of resolution. And that resolution definition is in fact
00:01:10.23 very famous. You've probably encountered in other lectures already
00:01:13.22 that the resolution limit is lambda over numerical aperture, or the
00:01:19.08 refractive index times the sine of the largest angle you can collect.
00:01:22.08 And this is so famous that it's actually written in stone on his monument
00:01:25.20 here at Jena. So what is the Abbe Diffraction Experiment?
00:01:30.27 So, the Abbe Diffraction Experiment is a way to think about
00:01:37.06 what the resolution limit of the light microscope is.
00:01:39.28 And Abbe's idea was to consider a striped sample. So the idea
00:01:45.20 was that we're going to take a sample with periodic stripes in
00:01:47.26 it, and ask how finely spaced can those stripes be. How close
00:01:52.24 together can those stripes be while we can still resolve them?
00:01:57.09 So we start with a very coarse grating, very coarse striped
00:01:59.23 sample that has stripes spaced very far apart, it'll be easy to
00:02:02.20 resolve. We can gradually make them more and more closely
00:02:05.27 spaced, until at some point we won't be able to distinguish
00:02:08.20 one stripe from the next, and we'll just see a featureless
00:02:10.16 blur. And this turns out to be really easy to analyze theoretically,
00:02:16.02 because of what a striped sample is. Which is basically a striped
00:02:20.03 sample is a diffraction grating. So here is the optical path of our
00:02:24.03 light microscope. Here is our sample in green, that's the striped
00:02:28.05 sample, that's diffraction grating. And then we've got a light source
00:02:31.25 here, a condenser lens here, so that collimates the light and
00:02:37.14 illuminates the sample. And then up here, we have an objective lens
00:02:40.13 that captures the light that's scattered off the sample and then
00:02:44.19 collects it. And up off the slide, not shown, would be a tube lens
00:02:48.11 and a camera and eyepiece. So the lens would then relay
00:02:51.13 this to a final image. But for today's discussion, all we really need to
00:02:56.15 consider is the objective lens. So this is our objective lens here,
00:03:00.16 schematically drawn as a lens. And it has some back focal plane
00:03:04.03 here, where light that leaves the sample at a constant angle will be
00:03:06.29 collected and focused to a plane. So if we consider illuminating
00:03:11.24 with a point on axis here, what's going to happen? So we come in
00:03:16.14 with this collimated beam of light from below, it interacts with
00:03:19.16 our sample here, and our striped sample, remember, is basically
00:03:23.26 a diffraction grating. So this is going to diffract that light beam.
00:03:28.00 And in fact, it's going to diffract it at some angle beta here. And the
00:03:33.02 diffracted angle beta will be described by Bragg's line, the first
00:03:37.28 order diffracted beams here will just be dsin(B)=wavelength of light.
00:03:42.17 So knowing the wavelength of light we're illuminating at and the
00:03:47.10 stripe spacing here, we can calculate that diffraction angle beta.
00:03:51.14 And provided beta is less than the largest angle the objective can
00:03:56.04 collect, provided it's smaller than the numerical aperture, what's going to
00:03:59.07 happen is this light will be collected by the objective, make it through the
00:04:03.00 back focal plane here, and then this can now interfere and be focused
00:04:07.28 to an image by our tube lens. And we'll get an image of this grating.
00:04:10.26 So, you can see that the angle here is going to depend on the
00:04:15.29 stripe spacing, d, and so if I make the stripe spacing smaller,
00:04:19.21 as I move those stripes closer together, this diffraction
00:04:22.09 angle's going to get larger and these beams are going to move
00:04:24.12 outwards. So that's shown here, and now we're looking at a case
00:04:28.03 where the diffracted angle is larger than the largest angle the objective
00:04:35.14 can collect. So this is now a diffraction angle that's larger than the
00:04:38.24 numerical aperture of the objective. And so what happens?
00:04:41.19 These diffracted beams come out and they're not collected by the
00:04:46.06 objective lens, they impact the back focal plane here and they're not
00:04:49.11 transmitted. And so, all we get through this is this undiffracted
00:04:52.21 light that's going straight up through the objective without really
00:04:55.23 being affected by the sample. And so, this light here doesn't have
00:04:59.15 any information about what the stripe spacing is here. The
00:05:03.11 information about the stripe spacing is all contained in these
00:05:05.24 diffracted beams, which we're not collecting. And so, in this case
00:05:09.19 we're going to get an image of the sample that just is a uniform gray
00:05:12.27 image. There's no information about the stripes here, we don't
00:05:15.29 see any of those stripes. We just see transmission of some light
00:05:19.13 through the sample. And so this is now a sample that is too
00:05:24.20 fine structured to be resolved by our objective lens.
00:05:27.00 So, if beta is greater than alpha, where alpha is the largest angle
00:05:32.23 the objective can collect, then we don't get an interference between
00:05:37.13 these beams and we don't get an image. And so, this is above
00:05:40.15 our resolution limit. And so if we just set beta equal to alpha, so if we
00:05:45.04 ask what's the very smallest stripe spacing we can see? What's the
00:05:49.10 largest diffraction angle that we can actually collect with that objective,
00:05:51.28 and we do the math. Then you would find that the resolution here
00:05:55.15 is just equal to the wavelength of light divided by the sine of the
00:05:59.03 largest angle that the objective can collect, and then there's this
00:06:01.27 factor here of the refractive index. And so, this gives us the
00:06:06.10 well-known resolution limit of lambda over NA. Often you'll see
00:06:10.28 lamba over 2NA, and that's because you can get a factor of 2
00:06:14.09 improvement here if you don't illuminate with light that's on axis
00:06:17.03 here, but you allow the full-range of angles to illuminate the sample,
00:06:20.17 as well. And so, this is the argument that Abbe used to work out
00:06:27.20 what the resolution limit of the objective lens is and show that
00:06:30.25 it's governed by the largest angle it can collect. And it also
00:06:34.01 shows you something else, which is that the diffraction in the sample
00:06:39.07 tells you something about the size of objects in it. And interference
00:06:42.19 between those diffracted beams is what gives rise to the image of the sample.
00:06:46.24 And Abbe summed this up in a very famous quote, where he said,
00:06:51.13 The microscope image is the interference effect of a diffraction
00:06:54.07 phenomenon. So you have diffraction from your sample, and then
00:06:57.19 interference between those diffracted beams that gives rise
00:07:00.04 to your image. And that's what we're going to go now into the lab
00:07:02.26 and show you. We're going to look at various samples, see their
00:07:06.07 diffraction patterns, and see how interference between those diffracted
00:07:09.01 beams gives rise to the image of the sample.
00:07:12.07 What I want to do here today is try to give you a real life demonstration
00:07:16.13 of how this works with real optics and a real microscope.
00:07:19.16 So the optical system in front of me is actually a microscope.
00:07:23.08 Although it may not look very much like any microscope that
00:07:25.01 you've used before. This is a system built out of simple optical
00:07:28.25 components, so that we can get access to all the planes of the
00:07:32.04 microscope for demonstrating these principles here. But in all
00:07:36.12 respects, it really is analogous to the kinds of upright and inverted
00:07:39.14 microscopes you've used. What we've got here is a transmitted
00:07:43.19 light microscope, we've got a lamp source on the left here, and then a
00:07:47.10 camera on the right. And the optics between here are all exactly analogous
00:07:50.18 to the optics you would see in a conventional research microscope.
00:07:53.24 And I'll walk through them now from left to right, illustrating what each
00:07:57.21 optic is and what it's doing. So on the left here, we've got an
00:08:01.20 LED light source. It's mounting on this assembly here with a
00:08:05.24 collector lens, so there's a lens here that's collecting this light
00:08:08.15 and focusing it out to infinity in a parallel bundle. The LED
00:08:12.04 is controlled by this knob here, so you can turn it up and down,
00:08:14.29 make it brighter and dimmer. And then the optics in here are the
00:08:19.05 condenser optics for this microscope. Starting with the object closest
00:08:23.15 to the LED, this is our field iris. So opening and closing this iris
00:08:26.25 controls the region illuminated on our sample. Then, we've got our
00:08:32.00 field lens. This is a color filter, so this is just a green filter I can
00:08:38.10 put in to get green monochromatic light to my sample.
00:08:42.05 And we'll mostly be using that to avoid issues of different
00:08:46.01 colors having different behaviors in this microscope. This here
00:08:49.16 is my aperture diaphragm, my condenser diaphragm. So opening
00:08:53.17 and closing this iris controls the range of angles that illuminate my
00:08:56.11 sample. And we'll mostly be leaving this closed so we're illuminating
00:08:59.19 just a pencil of light going right down the center of our sample
00:09:03.00 here, with parallel beams of light. And this then is our condenser
00:09:07.01 lens, and so this lens is then focusing light onto the sample.
00:09:11.15 And this whole system is set up for Kohler illumination, so
00:09:15.26 that the LED here is in the black focal plane of the condenser.
00:09:20.26 And then this field diaphragm is conjugate to this sample plane.
00:09:23.25 This is our sample holder here. And then, this is our objective. This is
00:09:30.08 just another simple lens, just another convex lens mounted in
00:09:34.25 a lens holder, so we can screw it in and out to focus it.
00:09:37.04 We've another sample holder here, and this one is located at the back
00:09:41.02 focal plane of this objective. And I'll have a lot more to say about this
00:09:44.12 later in the lecture. Then, one unique thing about this microscope
00:09:49.15 and the reason we built it this way, is we have a beam splitter
00:09:52.04 here so that we have two different detection paths. The straight
00:09:55.13 through detection path here is a conventional microscope.
00:09:57.22 This is our tube lens and it images our sample onto the camera.
00:10:01.10 So whatever is in focus in our sample plane here is in focus here on
00:10:05.02 our camera. The other path reflected off our beam splitter is a path
00:10:10.14 that allows us to image the back focal plane. And so this lens here
00:10:13.24 is acting as a simple magnifier lens to project the image from this back
00:10:16.24 focal plane onto our camera here. And so on the laptop, I can switch between
00:10:21.04 this camera and this camera, to see either the sample plane
00:10:24.18 or what's going on in the back focal plane of the microscope.
00:10:27.10 So let me start here by focusing on a simple grating slide. So,
00:10:33.26 you can see here, the sample that's in focus right now is just
00:10:39.10 a simple pattern of vertical lines. So these are just metal
00:10:43.10 lines deposited on a glass slide. And the metal lines don't
00:10:47.17 transmit light so they're dark. The glass transmits, so it's clear.
00:10:50.28 If we now switch cameras and look at the back focal plane,
00:10:55.04 and we turn this down, and I need to close this. What you
00:11:07.15 can see now is that this gives rise to a set of diffraction patterns.
00:11:11.21 So, let's just adjust the brightness. I closed that down as much as I can.
00:11:16.17 So you can first off see that the size of these disks here changes
00:11:19.28 as I open and close the aperture diaphragm. And so the central spot in this
00:11:24.28 image is the image of this aperture diaphragm. This corresponds
00:11:28.16 to the light that's making it straight through the microscope
00:11:30.26 without interacting with our sample. The two closely spaced beams
00:11:34.26 on either side of it are light that's been diffracted from this grating.
00:11:39.05 So, remember that's a striped sample here, so there's alternating
00:11:42.25 pattern of dark and light lines is going to act like a diffraction grating
00:11:46.10 and it will diffract light. And these are the first two diffraction orders.
00:11:49.17 If I make the light brighter here, you can see they're not the only
00:11:53.05 diffraction orders. And in fact, there are a whole bunch of diffraction
00:11:56.01 orders that extend out to much larger angles. And the white vertical stripe
00:12:02.21 that you may see here is actually an artifact of the camera. So,
00:12:05.13 that you can safely ignore. Okay, let me switch back to my image
00:12:10.23 camera here. So again, here's my grating image. And now I have many
00:12:21.09 different gratings printed on this slide. So I can just slide over here.
00:12:24.02 And now if you look at this grating. Let me make it a little brighter.
00:12:30.19 You can see that this is a very fine striped spacing grating.
00:12:37.17 So, you can see here that the lines are much more closely
00:12:44.11 spaced together than in the previous grating. So, if we now
00:12:48.27 flip to the back focal plane camera. You will see that these more
00:12:55.16 closely spaced lines correspond to much larger diffraction angles.
00:12:59.01 So, first thing to learn from this experiment is that position here in the back focal
00:13:06.16 plane tells you something about the size of your features in your sample.
00:13:10.10 So, things close to the center of the back focal plane correspond
00:13:13.20 to low spatial frequencies. So large scale structures in your image, things that don't
00:13:18.28 change very fast as you move across the image. Whereas, positions
00:13:23.01 further out in this image correspond to high spatial frequencies.
00:13:26.18 So, regions of the image that change very rapidly as you move
00:13:30.15 positions. So, because this grating has very closely spaced lines,
00:13:33.23 it has higher spatial frequencies, and so these first diffraction orders are
00:13:39.05 located further out in the back focal plane than the grating that
00:13:42.24 had coarser lines. Like the previous one, this also has higher order
00:13:47.05 spots. The other thing to realize here is that not only does the position
00:13:54.23 here tell you something about the size of the structures, but it also tells
00:13:57.28 you something about the direction of the structures. So, both the
00:14:03.00 gratings we've looked at so far are vertically oriented lines. And
00:14:06.19 so, a diffraction grating with lines oriented vertically is going to scatter
00:14:10.23 light perpendicular to the line orientation. And so that's why we get a line
00:14:14.29 of spots here that's in the horizontal direction. If instead we switch to a
00:14:20.12 grating with a different orientation, we'll see a different pattern
00:14:23.29 of spots. So let's switch back to the image plane again, and now
00:14:28.26 I have to look for one. So here now, we have a grating that's
00:14:34.19 oriented in the horizontal direction. And so based on what I've told
00:14:38.07 you so far, we would expect that the diffraction pattern from this
00:14:41.24 should be oriented in the vertical direction. So let's switch to the back focal
00:14:45.04 plane and find out. And yes, exactly as we expect, we now see a set of
00:14:55.24 spots located in the vertical direction. So we're getting now information about
00:15:00.08 these lines that are in the horizontal direction shows up in the vertical
00:15:04.25 diffraction pattern on the back focal plane of the microscope.
00:15:07.13 So let's switch back to the image plane again, and now ask
00:15:11.18 what do we see if we put in a two dimensional grating?
00:15:15.19 So I'm going to switch slides now and take this guy out.
00:15:19.21 And put this in, and our two dimensional grating here is just an ordinary
00:15:23.00 stage micrometer. And so this is just a grid with 10 micron lines
00:15:32.16 that you would buy to focus your microscope, or to calibrate your microscope rather.
00:15:56.20 Okay, so now we have a nice image of a 2D grid here. And based on everything
00:16:02.16 I've told you so far, we would expect that this should give rise
00:16:05.20 to diffraction in both directions because it has both horizontal
00:16:08.22 lines and vertical lines. So again, we'll switch to the back focal plane camera.
00:16:14.19 And indeed, that's what we see. So, we've got our zero order
00:16:22.08 light here. So this is the central spot, which is the undiffracted
00:16:25.13 light making it through our system, which hasn't interacted with
00:16:28.06 the grating. And then, we see these horizontal and vertical spots
00:16:32.24 diffracted by the grating. And again, this vertical stripe here is a
00:16:36.11 camera artifact from this bright central blob and some of these others.
00:16:40.05 This guy here is a back reflection. Okay, so now hopefully you've seen
00:16:45.15 that position in the back focal plane tells you both about the
00:16:49.13 spatial scale of your structure, the size of your structures and how closely
00:16:53.27 spaced they are to each other. And also their orientation.
00:16:56.29 Now what I want to show you is that in fact, this image here in the
00:17:01.10 back focal plane contains all the information here about your sample.
00:17:03.22 And everything you can see in your sample is encoded in the back focal
00:17:08.09 plane. And furthermore, if we manipulate what makes it through the back
00:17:12.14 focal plane by blocking some of this light, we will change what we see
00:17:16.14 in the image plane at our image camera here. Because we've removed that
00:17:20.15 information from our image. And so to do that, we have a set of
00:17:25.16 patterns here. And so these are just similar metal masks that we can
00:17:29.19 position in the back focal plane to prevent this information from
00:17:32.13 making it to the camera. So I'm going to start here by blocking all the
00:17:36.23 vertical light. So I will slide this grating in. And then position it height-wise.
00:17:43.28 And so, you can see now, I've got this set of two black bars here.
00:17:49.26 So that they're cutting off all the light from above and below here.
00:17:53.11 We're only letting through here now the band of horizontal
00:17:56.22 diffraction. And so now this diffraction pattern looks just like what we
00:18:00.13 saw from our vertically oriented lines before. And in fact, if we now go
00:18:06.11 looking back at our grating here. You see, in fact, we've lost all
00:18:12.15 the information about the horizontally oriented lines, and we now just see
00:18:17.04 a set of vertical stripes. So by blocking the diffraction information from
00:18:21.25 the horizontal lines, we prevent our ability to image them. Because that
00:18:25.11 information doesn't make it to the camera. And just to prove I'm not
00:18:28.11 cheating, if I slide this out, you can see the whole grid reappear.
00:18:32.15 And if I slide it back in, then we just get our vertical lines again.
00:18:36.25 There's nothing special about the vertical direction. I've got another
00:18:41.22 pattern here. So let me switch back to the back focal plane, so I can get a
00:18:44.18 position properly. So, we now go and slide this pattern in.
00:18:54.13 Now we can block all the horizontal information, and so based on
00:19:00.10 everything I've told you so far, you should expect to see just
00:19:03.13 the horizontal lines. And in fact, that's exactly what you see.
00:19:08.28 So hopefully, I've convinced you now that the back focal plane
00:19:14.20 of your microscope not only encodes all the information about
00:19:17.24 spatial structures in your sample by their spatial frequency,
00:19:20.28 but it's also a complete representation of that information
00:19:25.12 and blocking information there prevents you from retrieving
00:19:29.17 that information in your final image. So that the back focal
00:19:33.03 plane is the fourier transform of your sample, not just mathematically, but
00:19:37.04 physically. And that's how the information gets reconstructed
00:19:41.15 by your microscope. So you have essentially in a microscope, your
00:19:44.24 objective lens generates a fourier transform of your sample
00:19:47.13 to its back focal plane, and your tube lens then resynthesizes
00:19:50.20 that fourier transform into a real image on your camera.
00:19:53.21 So, thank you!

This Talk
Speaker: Kurt Thorn
Audience:
  • Researcher
Recorded: April 2013
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Talk Overview

This lecture/lab describes the famous Abbe Diffraction Experiments conducted by Ernst Abbe who showed how diffraction of light by a specimen (and interference with the illuminating light) gives rise to an image and how collection of diffracted light defines the resolution of the microscope. These concepts are demonstrated by using a diffraction grating as a specimen and visualizing and comparing the diffraction pattern in the back focal plane as well as the image in the image plane.

Speaker Bio

Kurt Thorn

Kurt Thorn

Kurt Thorn is an Assistant Professor of Biochemistry and Biophysics at UCSF and Director of the Nikon Imaging Center – a facility that provides cutting edge light microscopy equipment to UCSF researchers. Kurt can be followed on his blog at http://nic.ucsf.edu/blog/. Continue Reading

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