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