Top
SCIENCE. STORIES. IMPACT.

iBiology

Bringing the World's Best Biology to You

Point Spread Function

00:00:12.19 In a previous lecture,
00:00:14.17 I talked to you about
00:00:17.22 the way in which light interferes
00:00:21.25 through a process known as diffraction
00:00:24.04 and how that generates complex patterns
00:00:27.00 when light from a single source
00:00:29.18 comes at a particular place
00:00:31.13 from more than one direction.
00:00:33.15 That light interferes in patterns
00:00:36.02 that ultimately spread light out
00:00:39.01 that might otherwise be focused.
00:00:41.04 In this discussion,
00:00:44.24 I would like to derive for you,
00:00:47.28 in a very loose way,
00:00:49.28 the way light is spread out,
00:00:52.19 that is, from a single point in an image plane
00:00:56.14 of a microscope.
00:00:58.10 Ideally, if light was
00:01:00.28 traveling in straight lines,
00:01:02.19 a point object would generate a point image
00:01:05.18 and all the light would be collected
00:01:07.22 and end up in the right place,
00:01:10.06 the place it's intended to be
00:01:12.14 as a rendering of the object.
00:01:13.26 As you'll see, and you maybe already understand from diffraction,
00:01:17.02 because light goes in all directions and interferes,
00:01:20.05 that generates a different pattern.
00:01:23.03 More remarkably, though,
00:01:25.14 is that when one understands interference
00:01:27.21 in a more deep sense,
00:01:29.23 one sees that one can give up on straight lines altogether,
00:01:33.12 that all the behavior of light,
00:01:35.06 including where it goes and where it doesn't go,
00:01:37.08 is not related at all to lines,
00:01:39.03 it's just related to interference patterns.
00:01:42.10 So, I'd like to begin this discussion
00:01:46.07 by reminding you about the light path
00:01:49.04 of the light microscope,
00:01:52.19 especially the light path
00:01:54.16 between the specimen and the imaging plane,
00:01:58.15 which could be your eye,
00:02:00.06 but often is a camera faceplate
00:02:02.16 or some other imaging device.
00:02:05.04 So, let's begin by looking at
00:02:07.10 a kind of wave optic view
00:02:10.18 of the effect of the microscope lenses on light,
00:02:14.18 and we're going to begin with
00:02:17.09 a particle in the image plane,
00:02:19.02 right at the center of the image plane
00:02:21.00 -- right there, that blinking green dot,
00:02:25.13 could be thought of as
00:02:28.11 an infinitesimally small bead
00:02:31.12 that's fluorescing a particular wavelength
00:02:34.04 or a single molecule of a fluorescent protein
00:02:36.18 or a fluorophore,
00:02:38.02 and it's generating light
00:02:40.07 that is going in all directions.
00:02:41.23 It's a diverging spherical wave, undulating,
00:02:44.18 so there's one wave front after another
00:02:47.23 -- that's a point source.
00:02:49.20 And then what the microscope is going to do
00:02:51.18 is try to image that point source,
00:02:53.18 and all it has available to it is two lenses.
00:02:57.21 One lens is the microscope objective.
00:02:59.20 As you probably know,
00:03:01.23 microscope objectives are actually complicated lenses
00:03:03.21 that have many lens elements,
00:03:05.15 but we can simplify that down to say that
00:03:08.07 they all add up, basically, to a positive lens,
00:03:10.00 that is, a lens that's fat in the middle
00:03:11.20 and thin at its edges.
00:03:13.28 And some portion of the spherical wave
00:03:16.22 coming out of that point
00:03:18.09 is collected by that lens.
00:03:19.29 And because the lens
00:03:21.29 is exactly at the focal distance
00:03:24.06 from that point,
00:03:25.20 that is why that point is said to be in focus for this lens,
00:03:28.03 this lens turns that spherical, diverging wave...
00:03:34.14 that objective lens takes that diverging spherical wave
00:03:36.27 and turns it into a plane wave
00:03:38.22 moving in a particular direction.
00:03:40.20 If it's moving straight back,
00:03:43.05 it's because the object is at the center of the image plane...
00:03:47.27 sorry, the specimen plane.
00:03:49.27 If the point was offset, either up or down,
00:03:53.11 or towards me or away from me,
00:03:56.01 or sort of towards you or away from you,
00:03:58.00 then the plane wave would be tilted
00:04:00.27 in a corresponding way.
00:04:02.06 So each position generates
00:04:04.17 a particular angle plane wave,
00:04:06.21 but for an object that is exactly dead center
00:04:08.25 in the image plane,
00:04:10.06 the plane wave is moving straight back
00:04:11.28 along the optical axis.
00:04:13.18 It's a plane wave and that plane wave,
00:04:17.00 then, is intercepted by a second lens, known as the tube lens,
00:04:20.17 and the tube lens is also a positive lens,
00:04:23.00 and just like any other positive lens,
00:04:25.08 it can take a plane wave
00:04:28.19 and turn it back into a converging spherical wave,
00:04:31.15 doing sort of the inverse of what the objective is doing.
00:04:37.14 And so that converging spherical wave
00:04:39.29 is focused on that small point in the image plane.
00:04:42.03 So, we have a tube lens
00:04:44.10 and that tube lens generates
00:04:46.19 converging spherical waves
00:04:48.12 and that light then ends up at that point
00:04:50.28 and then here is the big question.
00:04:52.23 The question is, for a point object in the focal plane,
00:04:56.07 such as an infinitesimally small fluorescent bead,
00:05:00.15 what is the distribution of light
00:05:02.13 in the image plane over here
00:05:04.06 where that question mark is,
00:05:06.02 and not only in the plane
00:05:07.20 but around the plane?
00:05:11.11 And the idea is that that light is spread out in some way
00:05:13.19 and that spread is a function
00:05:15.08 -- it's called the point spread function --
00:05:16.25 and this is known with great precision,
00:05:19.07 it's a highly mathematical result,
00:05:23.20 but I think we can derive
00:05:26.20 an intuitive understanding of that distribution
00:05:28.17 by just understanding Huygens' wavelets,
00:05:31.16 that I spoke about in a previous lecture.
00:05:33.26 So, we have to really focus only on
00:05:37.27 what happens to the light beyond the tube lens,
00:05:39.11 in that range there.
00:05:41.13 What happens to those converging spherical waves
00:05:43.25 as they move forward?
00:05:49.12 And as you may remember,
00:05:52.09 the Huygens' view is that that converging spherical wave
00:05:55.08 is just a series of infinitesimally small wavelets.
00:06:00.17 Each of the infinite points of the emerging wave front
00:06:04.19 acts like a point source itself,
00:06:06.02 i.e. each point emits a wavelet.
00:06:09.05 All wavelets from the same wave front are
00:06:12.22 mutually coherent,
00:06:16.21 that is, they oscillate in synchrony
00:06:18.27 because they came from the same source.
00:06:20.18 So they can interfere in fixed patterns
00:06:22.07 and therefore, because they interfere in a predictable way,
00:06:24.09 we can actually quantify
00:06:26.18 exactly where the light will be
00:06:28.04 and where the light won't be
00:06:30.06 over periods of time longer than a wavelength.
00:06:34.27 So, let's look at pairs of wavelets to get started.
00:06:40.15 So, if we have two wavelets
00:06:43.02 that are equidistant from a particular point...
00:06:47.12 D1 and D2 are two distances
00:06:51.21 that are equidistant to that point
00:06:54.06 because that point is along a midline,
00:06:56.20 so since the spherical wave
00:07:00.11 is symmetric along the midline
00:07:02.03 those two points are equidistant from an arbitrary point
00:07:05.10 -- I could have put that point anywhere on that line --
00:07:07.11 at that position the two wavelets
00:07:09.29 will both be peaking and troughing in synchrony,
00:07:13.14 and their phases will be exactly the same,
00:07:16.19 and so if you add up those wavelets
00:07:18.15 they will add in a constructive way.
00:07:21.28 In phase... constructive interference.
00:07:24.23 Now, if we move to another position,
00:07:26.19 it could be anywhere,
00:07:28.17 but this is the particular position I'm choosing,
00:07:30.08 just down a little bit,
00:07:31.25 one of those wavefronts
00:07:34.06 now has to go a half a wavelength longer
00:07:36.18 and the other is a little bit shorter,
00:07:38.17 so one is sort of like a quarter wavelength longer for one
00:07:41.16 and a quarter wavelength shorter for the other,
00:07:44.02 so in sum total there's a half a wavelength phase difference,
00:07:47.13 now, between D1 and D2,
00:07:50.15 and those two wavelets at this point will add up...
00:07:52.20 wherever one is peaking the other is troughing and vice versa,
00:07:55.04 and so you'll end up with absolutely nothing...
00:07:57.14 half a cycle out of phase and destructive interference.
00:08:00.18 However, if we go down here,
00:08:02.27 although they're not the same lengths,
00:08:05.22 D1 is exactly one wavelength longer than D2
00:08:10.09 at this distance,
00:08:12.01 and so again one has constructive interference.
00:08:14.03 And if you think about that
00:08:16.16 and just do that for all the points
00:08:18.14 along that particular plane,
00:08:20.05 you're going to end up with a lot of places
00:08:22.06 where they interfere in a positive and negative way.
00:08:24.28 Now, as I've mentioned in a previous discussion,
00:08:28.04 the amplitudes are not what ultimately human beings
00:08:32.17 measure with our cameras or with our eyes.
00:08:33.19 We're looking at intensity,
00:08:35.12 which is the amplitude squared and integrated
00:08:40.00 over at least a wavelength,
00:08:41.18 and down below you see the integration of wavelets
00:08:46.10 over a full wavelength and then squared,
00:08:48.20 so we see amplitudes rather than wavelengths,
00:08:51.20 and it's a different pattern
00:08:54.02 than the instantaneous wave pattern.
00:08:55.15 So, at an instant, on the top,
00:08:58.05 it's a mesh of bright and dark lines.
00:09:00.08 Detectors, including the eye, collect intensity,
00:09:02.28 so the dark lines are the troughs,
00:09:05.01 but troughs are as much energy as peaks,
00:09:07.18 but if you square it all of those things become visible, basically.
00:09:11.26 Intensity is the addition of a full cycle squared.
00:09:14.08 The detected interference pattern
00:09:16.19 now has bright and dark lines,
00:09:18.24 and there's one single bright line
00:09:20.25 running right down the horizontal middle,
00:09:22.20 because that's equidistant from the two points
00:09:25.07 everywhere along that line.
00:09:27.01 The dark places are areas with destructive interference,
00:09:30.13 where there is no light,
00:09:31.20 and the light areas are places where light is,
00:09:33.28 and the bright areas are therefore the constructive interference.
00:09:37.05 And there's nothing with those two wavelets
00:09:40.08 to tell you where the image plane --
00:09:42.13 I've just arbitrarily put it
00:09:44.16 where I know the image plane is,
00:09:45.27 but there's nothing in the two wavelet data
00:09:48.00 to provide you any hint
00:09:50.01 that there's anything special about that plane.
00:09:51.20 But you'll see as we add more and more wavelets,
00:09:54.16 something very special happens,
00:09:56.10 especially in the middle of that image plane,
00:10:00.21 because that was the one and only place
00:10:02.18 where all the wavelets we're going to be adding together
00:10:05.02 constructively interfere
00:10:06.15 -- you'll see that in a moment.
00:10:08.21 Before we get to that, we have to ask the question of,
00:10:12.01 why am I focusing on these two wavelets
00:10:14.23 that are at the extreme edges
00:10:17.17 of that spherical cap,
00:10:18.27 and I am because those are the wavelets
00:10:21.26 that ultimately determine the resolution of the microscope
00:10:24.02 and those are the wavelets
00:10:25.26 that are most related to the numerical aperture of the lens.
00:10:28.26 You may have already heard about numerical aperture,
00:10:31.01 but I want to emphasize that that is the feature of a lens
00:10:33.25 that is most important when it comes to resolution.
00:10:37.21 So let's take a look at three objectives
00:10:39.17 that have different numerical apertures,
00:10:41.23 and the numerical aperture is the...
00:10:44.26 is measured as the index of refraction of the material
00:10:49.17 between the objective and the specimen
00:10:51.24 -- so if it's air, the index of refraction is 1,
00:10:54.10 but if it's water or a glycerin or oil,
00:10:58.18 that number can be substantially higher --
00:11:01.18 and it's the index of refractions
00:11:04.23 times the half-angle between a normal
00:11:06.23 -- a vertical going from the objective to the specimen --
00:11:09.13 and the most extreme light angle
00:11:12.17 that is still collected by the objective,
00:11:15.08 called α.
00:11:16.15 So, that half-angle times the...
00:11:18.24 it's the sine of the half-angle
00:11:20.27 times the index of refraction,
00:11:22.12 n sin(α),
00:11:24.15 which is the index...
00:11:26.17 is the numerical aperture of a lens.
00:11:28.02 And you can see a, b, and c are three different lenses,
00:11:30.20 one with a low numerical aperture.
00:11:33.07 Typically, low numerical aperture lenses
00:11:35.15 are a little further from the specimen
00:11:37.17 than higher numerical aperture lenses,
00:11:39.04 they collect a smaller cone of light from the specimen,
00:11:41.23 so the angle is less,
00:11:43.22 and they typically don't use an immersion medium,
00:11:46.06 so they have a low NA number
00:11:48.25 and they don't collect as much light
00:11:51.13 -- they're dimmer than high NA lenses.
00:11:53.11 The medium NA lens is shown in the middle
00:11:55.24 and the high NA lens is shown on the right,
00:11:59.27 where the angle is pretty obtuse,
00:12:03.22 relative to the other two.
00:12:05.20 That angle can go up to 70 degrees or so,
00:12:08.20 and often there's oil in there as well.
00:12:12.22 So, those are three numerical apertures,
00:12:15.11 objectives vary,
00:12:17.07 and I would like you to think about this
00:12:20.02 in terms of how that affects the converging spherical wave
00:12:23.01 coming out of the tube lens,
00:12:24.14 and this diagram kind of tries to show this,
00:12:28.16 that if you have a high NA lens,
00:12:30.11 high numerical aperture lens,
00:12:32.00 which is the one shown above,
00:12:34.03 a greater portion of the spherical wave
00:12:36.14 is collected by the objective,
00:12:38.02 meaning the plane wave
00:12:40.00 leaving the back aperture of the objective is wider,
00:12:43.15 so the distance between the furthest wavelets
00:12:47.11 are farther apart than a low numerical aperture lens, shown below,
00:12:50.01 where the wavelets come out...
00:12:52.27 the plane wave is shorter,
00:12:55.22 so the furthest apart the wavelets are
00:13:00.05 is not as far apart as in the case of
00:13:03.03 a high numerical aperture lens.
00:13:05.14 And, as I said, we're going to focus on
00:13:07.15 everything beyond the tube lens
00:13:09.02 and, basically, what those wavelets do
00:13:12.05 as they converge,
00:13:13.26 and the only difference between a high numerical aperture lens
00:13:16.10 and a low numerical aperture lens,
00:13:18.18 for our discussion here,
00:13:20.01 is just how big,
00:13:21.17 how much of the spherical cap is collected,
00:13:23.27 and therefore how much of the spherical cap exists,
00:13:26.24 converging back to that point.
00:13:29.26 So, let's start with those two wavelets
00:13:31.14 and look at the difference between a high numerical aperture lens
00:13:33.14 and a low numerical aperture lens,
00:13:35.13 and if we look at a high numerical aperture lens,
00:13:39.22 the finest fringes are
00:13:43.05 related, ultimately, in all lenses,
00:13:44.27 but in a high numerical aperture lens,
00:13:47.15 because those two wavelets are far apart,
00:13:50.15 the fringes that one sees
00:13:52.29 are very finely divided,
00:13:55.13 whereas if the furthest apart the two wavelets are
00:13:58.01 is not very far apart,
00:13:59.20 then the interference pattern
00:14:01.25 is a much more broad set of fringes of plus...
00:14:05.21 of bright and dim light.
00:14:08.09 So, the period is related to the
00:14:10.22 objective's numerical aperture.
00:14:12.13 The finer the grating here
00:14:15.01 the higher the numerical aperture is.
00:14:17.12 So, high numerical aperture
00:14:19.13 gives short fringe period,
00:14:20.23 short distance from one fringe to the next,
00:14:22.14 and the fringes themselves are narrow in size,
00:14:26.11 whereas a low numerical aperture lens
00:14:28.22 has a long fringe period,
00:14:30.17 a big distance between the fringes,
00:14:33.11 and the fringes themselves are wide,
00:14:35.04 and this turns out to be critical
00:14:36.27 because the finest detail that can be found in an image
00:14:39.00 is related to this
00:14:41.13 -- that fringe period.
00:14:43.02 Any wavelets that are closer together
00:14:45.08 can only generate bigger structures,
00:14:47.04 but can't generate finer structures,
00:14:49.18 so the limit of the quality of the image
00:14:53.06 is how fine this fringe period can be,
00:14:55.26 and for a high numerical aperture lens
00:14:58.10 it's finer than for a low numerical aperture lens.
00:15:01.06 A second thing to think about is that
00:15:03.07 these fringes are related
00:15:05.13 not simply to the distance the two wavelets are apart,
00:15:08.13 but also to the wavelength of the light.
00:15:11.04 So, for example, blue light and red light,
00:15:14.18 although they interfere the same way,
00:15:17.06 the number of wavelengths between the source of the wavelet
00:15:22.03 and the image plane
00:15:23.17 is different in both cases.
00:15:25.03 Because the finest fringes set the finest details in an image,
00:15:27.26 the period and width is related to wavelength,
00:15:32.00 that is, the quality of the image is related to wavelength.
00:15:34.00 So, for example, shorter blue light...
00:15:38.09 there are more cycles per micron of that
00:15:41.19 distance D1 to D2
00:15:44.07 because it's got a shorter wavelength,
00:15:46.05 so there is a larger phase difference
00:15:47.13 for the same absolute distance between D1 and D2,
00:15:50.24 so you get interference to give you a finer pattern
00:15:53.12 with blue light than you do with red light.
00:15:56.15 So you get a shorter fringe period with blue light
00:15:59.01 and narrower fringes,
00:16:00.19 and the converse for longer red light.
00:16:02.09 This implies that if you want
00:16:04.05 high quality imaging,
00:16:06.01 you should think not simply of the numerical aperture,
00:16:08.09 but also the wavelength of the light
00:16:09.23 -- the shorter wavelengths of light
00:16:11.18 will give you better quality images
00:16:13.07 in terms of the finest details.
00:16:16.27 So, what I'd like to do now is...
00:16:18.24 we've just talked about those last two wavelets,
00:16:21.24 one at one end
00:16:24.05 and the other end of the spherical cap...
00:16:25.19 of course, this is a whole circumference of wavelets like that
00:16:28.00 and we're just taking
00:16:30.15 any diametrically opposite wavelets,
00:16:31.25 we could have done this any pair...
00:16:33.07 I'd now like to add one more wavelet,
00:16:35.11 which is the one right in the middle
00:16:37.20 of the spherical cap
00:16:39.03 and ask,
00:16:41.09 how do three wavelets interfere
00:16:44.00 in the part of the microscope
00:16:47.26 that is beyond the tube lens.
00:16:50.18 And here is a picture of three wavelets interfering.
00:16:54.13 The two wavelets at the rim we've already talked about
00:16:57.03 and we've added one more wavelet,
00:16:58.24 right in the center here, of the lens,
00:17:01.14 and these three wavelets are in phase.
00:17:03.19 At the center of the image plane,
00:17:06.25 because this is the point that is the center
00:17:11.08 of a circle
00:17:14.03 which this sphere is subtending.
00:17:17.07 So this... you can think of this as the center of a sphere,
00:17:20.07 so all the wavelets that we're going to be talking about
00:17:24.08 are going to be in phase here because they're equidistant...
00:17:26.13 every distance from that spherical cap down here
00:17:30.11 is a radius,
00:17:32.02 and since a circle has uniform radius everywhere,
00:17:34.20 and a sphere does as well,
00:17:36.22 all of them will be in phase here.
00:17:38.01 And you can see why,
00:17:39.18 as we add more and more wavelets,
00:17:41.08 this will only get brighter,
00:17:42.23 because we're getting more and more constructive interference.
00:17:44.23 In addition, as we have more and more wavelets,
00:17:46.24 they'll be nowhere else
00:17:48.18 where most of them are in phase,
00:17:51.09 so everywhere else things will get dimmer.
00:17:53.26 And that is ultimately
00:17:56.29 how come this is where the image plane is
00:17:58.17 and that's how come this is where you get a bright spot,
00:18:01.01 simply by interference.
00:18:02.24 So, one wavelet at the center,
00:18:04.16 two at the edges,
00:18:06.01 the three wavelets are in phase at the center of the image plane,
00:18:08.00 so that's bright, you record intensity...
00:18:10.15 so, when you record intensity,
00:18:12.21 which is the square of the amplitudes
00:18:15.14 integrated over time,
00:18:16.24 you turned what was with two wavelets
00:18:18.21 that striping pattern,
00:18:21.06 now into bars, that were bars now into islands of light
00:18:24.00 -- places where the light is and where the light isn't --
00:18:26.01 and at the image plane the center fringes
00:18:28.12 are brighter.
00:18:30.03 This is brighter than the ones on the side
00:18:31.18 and they get dimmer all the way over,
00:18:33.28 and this is the one place
00:18:36.05 where they're all exactly in phase.
00:18:38.05 So, let's now add five,
00:18:40.13 add two more wavelets,
00:18:42.01 so we have not just the center
00:18:43.22 and the two ones on the end,
00:18:45.10 but we've cut the distance in half
00:18:47.26 between the two and added one more here
00:18:50.07 and one more there.
00:18:52.02 The brightest band fringe
00:18:54.00 is at the center of the image plane, right here,
00:18:56.19 not surprisingly,
00:18:58.15 because that's the center...
00:18:59.29 that's the radius for all of those wavelets,
00:19:01.13 it's equidistant from all those wavelets.
00:19:03.09 The fringes that come
00:19:05.24 over here and over on the other side,
00:19:09.06 right there,
00:19:11.18 have constructive interference
00:19:13.28 between the first two or the last two wavelets,
00:19:17.19 and because they don't have all of them in phase
00:19:20.17 they're dimmer than the first one,
00:19:22.15 and then there are other
00:19:25.07 interesting little areas of interference
00:19:27.00 that are constructive and destructive
00:19:28.08 -- you have to contrast enhance,
00:19:30.03 shown at the very bottom here,
00:19:31.27 to see them --
00:19:33.29 but it's a very complicated pattern,
00:19:35.09 but, no doubt, where B is
00:19:37.28 in that bottom image
00:19:39.07 is the brightest spot.
00:19:41.04 And we can now add more wavelets
00:19:43.11 and as we add more wavelets
00:19:45.12 we're beginning to see a trend,
00:19:46.29 which is more and more of the light
00:19:49.01 seems to be focused towards the central spot,
00:19:52.00 as if it is an hourglass-shaped beam
00:19:54.12 and then it rediverges on the other side.
00:19:56.20 All the wavelets, of course,
00:19:58.14 are in phase at the center,
00:20:00.00 so that's very bright, very visible,
00:20:01.28 and there are other, dimmer fringes visible,
00:20:04.04 if you contrast enhance, you see them,
00:20:06.17 and these are due to constructive interference,
00:20:09.00 typically between pairs of adjacent wavelets.
00:20:11.26 And most of the light now
00:20:14.07 appears to be in a cone,
00:20:16.01 so that hourglass shape,
00:20:17.21 that cone shape of focused, converging light
00:20:20.00 is simply a product of interference,
00:20:22.26 not angles of rays,
00:20:25.03 it's just interference.
00:20:27.12 So, finally, let's put them all in.
00:20:29.08 So, this is basically an infinite number of wavelets,
00:20:31.26 and this distribution is now
00:20:34.01 the point spread function,
00:20:35.15 it's how light is spread out,
00:20:36.21 and it's a nasty looking thing.
00:20:38.11 That little point that should have been
00:20:40.19 a nice symmetrical sphere, here,
00:20:42.09 is a very elliptical shape
00:20:44.18 and a lot of light is still left
00:20:47.01 around the area of the ideal object,
00:20:49.23 and that is the limitation of imaging
00:20:52.10 with a light microscope.
00:20:54.07 All the wavelets are in phase
00:20:55.29 in the center of the image plane;
00:20:57.14 the light is concentrated in the middle of the image plane;
00:21:00.10 the main peak is brighter, by far,
00:21:02.15 than anywhere else,
00:21:04.09 because that's the one place where every single wavelet
00:21:06.14 is constructively interfering;
00:21:08.04 there are side fringes,
00:21:09.20 they're sometimes called side lobes,
00:21:11.22 they're dimmer than the main peak,
00:21:13.14 and the intensity of them becomes progressively
00:21:15.23 -- you can see in this enhanced image --
00:21:18.12 become progressively smaller
00:21:20.07 as we go further and further away
00:21:22.13 from the center.
00:21:24.03 Most of the light now is clearly within a cone,
00:21:26.14 that hourglass shape mentioned earlier,
00:21:28.01 and that is why there is an hourglass shape,
00:21:30.06 just from interference.
00:21:32.06 And the width of the main peak,
00:21:34.10 this width here,
00:21:35.29 is set by how far apart
00:21:38.16 those wavelets were in the spherical cap
00:21:42.08 and hence by the numerical aperture of the lens.
00:21:46.29 So, if we look at the numerical aperture of the lens
00:21:49.10 on the point spread function,
00:21:51.17 you can see how dramatically distorted
00:21:54.22 the point spread function is
00:21:56.20 by having a smaller bit of the spherical cap.
00:22:00.24 With the high numerical aperture lens,
00:22:02.12 there's a wide separation between wavelets,
00:22:04.13 it's possible,
00:22:06.00 and that allows for interference,
00:22:07.21 and as a result there's a small central peak.
00:22:10.05 With a low numerical aperture,
00:22:12.08 only narrow separation between the wavelets
00:22:14.17 exists on the spherical cap,
00:22:16.08 so you have to go pretty far before you get your first interference,
00:22:18.22 so you get a broad central peak,
00:22:20.06 and that turns out to be limiting
00:22:22.08 the resolution of a low numerical aperture lens.
00:22:24.25 Similarly, if we look at the colored light,
00:22:27.28 the light that is short wavelength,
00:22:30.03 400 nm light,
00:22:31.27 has a much denser point spread function
00:22:34.02 than 700 nm light,
00:22:36.14 and the in between colors
00:22:40.08 of course have in between point spread functions.
00:22:42.24 So, let's be a little more quantitative now
00:22:45.06 about how big the image of that point is.
00:22:50.03 So, if you look down on the plane
00:22:52.15 where the image plane is,
00:22:53.29 where the light is in focus,
00:22:55.11 and you don't do any enhancement,
00:22:57.03 you see the point spread function as a spot,
00:22:59.16 which is reassuring.
00:23:00.20 You started with a point of light
00:23:02.23 from your specimen
00:23:03.22 and you'd like an image of it to look like a point
00:23:06.20 and, indeed, when you look down on that image
00:23:08.18 you do get a point of light.
00:23:10.10 That gives you the sense, ahh,
00:23:11.23 we have focused the light to a point.
00:23:14.14 If you enhance that image, however,
00:23:17.04 you find that, in addition to the point,
00:23:18.17 there are all these concentric rings,
00:23:20.05 and this is the...
00:23:22.10 known as the Airy disk
00:23:24.01 and it's a ring side-lobe pattern.
00:23:26.12 Airy is a person,
00:23:28.09 I think he was an astronomer,
00:23:30.02 who first saw this maybe looking at stars
00:23:31.29 and saw these rings around all these stars
00:23:34.02 and realized this was an artifact of diffraction,
00:23:36.02 not that every star is like Saturn
00:23:39.00 and had a ring around it.
00:23:41.04 And the concentric rings
00:23:44.29 have a very fixed distance from the first ring
00:23:48.13 and if there's anything you want to remember
00:23:51.03 in light microscopy,
00:23:52.23 any kind of equation that's worth keeping in mind,
00:23:56.08 it's the distance between the center of the Airy disk
00:23:59.16 and the first dark ring,
00:24:03.06 and that distance is 0.61 times the wavelength of the light,
00:24:06.25 divided by the numerical aperture.
00:24:08.10 The reason you want to remember this
00:24:10.13 is that that is a handy way
00:24:12.16 to figure out the resolution of the light microscope
00:24:14.19 and I'll explain that more in a moment...
00:24:18.10 that distance from the center to the first dark ring.
00:24:21.22 We can also make the same measurement
00:24:23.21 from the center to the first dark ring
00:24:25.12 not in the plane of imaging,
00:24:27.15 but in the depth axis,
00:24:30.27 so in an xz section
00:24:33.05 through image space,
00:24:34.20 and you'll find, of course,
00:24:36.14 as I've already told you,
00:24:38.11 it's kind of spread out in xz,
00:24:40.07 but, again, you can go from the center of that disk
00:24:43.15 to the first dark spot
00:24:46.05 above or below it in focus,
00:24:48.10 so, to the first dark spots
00:24:51.00 that one sees
00:24:53.18 either there or closer there.
00:24:57.23 And in this case
00:25:01.09 the distance is not 0.61λ over NA,
00:25:02.27 but 2 times the index of refraction
00:25:05.22 times the wavelength of light
00:25:07.26 divided by NA squared,
00:25:09.13 and the index of refraction I'm talking about
00:25:11.09 is of the medium
00:25:15.00 between the objective and the specimen.
00:25:16.24 For high numerical aperture lenses,
00:25:19.01 those two equations work out to
00:25:21.07 three- or four-fold different distances,
00:25:25.19 so if we look for a 1.4 numerical aperture lens
00:25:30.00 with 480 nm light,
00:25:32.18 the lateral distance
00:25:35.23 from the center to the first dark right is 225 nm,
00:25:40.06 which, as you'll see, is the resolution limit,
00:25:43.08 whereas in the z direction it's 861 nm,
00:25:47.00 so one has a much harder time
00:25:49.18 of disambiguating data in the z direction
00:25:53.04 because light is spread out
00:25:56.09 more that way than in the lateral direction
00:25:57.28 with a light microscope.
00:25:59.12 There's an anisotropy in the imaging;
00:26:02.04 the quality of the image is better
00:26:04.21 in the xy plane than in the depth plane.
00:26:06.24 And if you make a stack of images
00:26:08.07 and view them from the side,
00:26:09.20 you'll notice this almost immediately,
00:26:11.03 that the resolution is not as good.
00:26:14.03 So, let's look at this
00:26:16.24 in a little more detail,
00:26:18.24 about how numerical aperture
00:26:20.29 affects these point spread functions.
00:26:23.24 So, the numerical aperture's
00:26:26.28 influence on the point spread function...
00:26:28.16 the effect is actually much more dramatic, actually,
00:26:31.16 on the lateral spread of light...
00:26:33.17 on the axial spread of light,
00:26:36.01 in the depth axis,
00:26:37.24 that in the lateral spread.
00:26:39.12 So, with a numerical aperture of 1.0,
00:26:43.15 the lateral is a slightly larger spot
00:26:48.07 and the axial is a much longer spot
00:26:51.10 when you compare it to a 1.4,
00:26:52.24 where the lateral is a compact spot
00:26:55.01 and the axial is a much, much longer spot.
00:26:58.15 So, the effect of blurring
00:27:01.20 by going to lower numerical aperture
00:27:04.12 is much more a problem
00:27:07.29 for depth imaging than in the plane
00:27:09.22 -- you get a subtle effect in the plane,
00:27:12.04 but if you want to disambiguate
00:27:14.10 what's in one plane versus another,
00:27:16.01 the higher numerical aperture
00:27:18.10 helps you much more in the depth axis
00:27:20.19 than it does in the lateral axis.
00:27:24.19 So, I'd like to end this conversation now
00:27:34.28 and perhaps at this point
00:27:37.12 one can appreciate why it is
00:27:39.28 that the numerical aperture is the thing
00:27:41.28 that you should keep in mind
00:27:43.27 when you're using your objectives.
00:27:45.15 What I'd like to do in a subsequent lecture
00:27:47.24 is kind of tell you how,
00:27:49.24 once you know this information,
00:27:51.09 how to take best advantage of it
00:27:53.00 when looking at microscope images,
00:27:56.14 especially the question of,
00:27:58.08 what is actually the resolution limit of the light microscope?
00:28:01.04 I've hinted that it's 0.61λ/NA
00:28:05.06 for the lateral,
00:28:06.16 but I haven't told you why.
00:28:08.00 That's in a subsequent lecture.
00:28:09.22 Thanks.

This Talk
Speaker: Jeff Lichtman
Audience:
  • Researcher
Recorded: December 2012
More Talks in Microscopy Series
All Talks in Microscopy Series

Talk Overview

An infinitesimally small point appears in the microscope as a spot with a certain size, blurred in the z-direction and with concentric rings around it. This “point spread function” reveals many of the optical properties of your microscope. This lecture explains why and how the microscope images a point as a point spread function.

Questions

  1. In an infinity corrected microscope, the light of in-focus light of an on-axis object between the objective and the tube lens is:
    1. A plane wave parallel to the optical axis
    2. A plane wave traveling at an angle with respect to the optical axis
    3. A spherical wave parallel to the optical axis
    4. A spherical wave traveling at an angle with respect to the optical axis

     

  2. The half angle of light collection (α) for a 40x 1.2 na water immersion lens is:
    1. 72°
    2. 86°
    3. 67°
    4. 35°

     

  3. The highest resolution image will be obtained by using:
    1. High wavelength light and large numerical aperture lens
    2. Short wavelength light and large numerical aperture lens
    3. High wavelength light and low numerical aperture lens
    4. Short wavelength light and low numerical aperture lens

     

  4. The center bright area of the point spread function (psf) is called
    1. Huygens wavelet
    2. Airy disk
    3. Koehler’s ring
    4. Newton’s apple

     

  5. The distance between the bright center of the psf and the first dark ring in the xy plane equals:
    1. N x sin(α)
    2. 0.61 lambda / sin(α)
    3. 0.61 lambda / NA
    4. N / NA * lambda

     

  6. The distance between the bright center of the psf and the first dark ring in the z plane equals:
    1. 2 n lambda / NA2
    2. N2 / NA * lambda
    3. 0.61 lambda / NA2
    4. N NA2 / lambda

     

  7. Switching from a high to low na objective results in the following change in psf:
    1. Slightly larger central spot in xy plane, slightly larger cone in z plane
    2. Much larger central spot in xy plane, slight larger cone in z plane
    3. Slightly larger central spot in xy plane, much larger cone in z plane
    4. Much larger central spot in xy plane, much larger cone in z plane

Answers

View Answers

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

Playlist: Microscopy Series

Reader Interactions

Comments

  1. Mohammad Abdelwahab says

    Thank you very much for this amazing lecture!

    I am wondering if this numerical analysis and simulations for the wavelets interference are available in a published article or in a book.

    I also think Q2 has a small mistake where calculated alpha is much closer to 67 than 72.

    Sin (71.6) = 0.949 where 1.22/1.33 = 0.902 corresponding to alpha of 64.46

    Reply

Leave a Reply

Your email address will not be published. Required fields are marked *

iBiology and iBiology Courses are part of the Science Communication Lab (SCL). Our mission remains the same, to connect people to science. However, our focus has shifted to producing and evaluating cinematic films for education and the public, which you can find on the Science Communication Lab website. For more information, please see this blog post!