How a Lens creates an Image.
Summary
TLDR本视频通过生动的动画和实验,探讨了透镜如何通过波动而非射线来形成图像。视频首先解释了传统射线模型的局限性,然后通过波的视角,展示了透镜如何通过改变波前的形状和方向来聚焦光波。此外,视频还讨论了数值孔径的概念,并通过显微镜实验展示了数值孔径对图像清晰度的影响。最后,通过双缝实验和菲涅耳衍射图案,视频展示了如何利用衍射模式来提高图像分辨率,以及波长对分辨率的影响。
Takeaways
- 🌟 透镜通过折射光线来形成图像,但光线的实际行为更接近波动而非射线。
- 🔍 射线模型无法完全解释光线在不同介质中折射方向变化的原因。
- 💡 波模型可以更准确地描述光线如何在透镜中传播和聚焦。
- 📏 透镜的数值孔径(NA)决定了其最大锐度,与透镜的开口角度成正比。
- 📊 实验表明,减小数值孔径会降低图像的清晰度,因为丢失了高角度衍射的信息。
- 🔬 显微镜实验中,通过改变数值孔径,可以直接观察到图像分辨率的变化。
- 🌐 衍射光栅可以模拟透镜的作用,通过不同角度的衍射来聚焦光线。
- 🔄 通过增加衍射环的数量,可以提高图像的分辨率,类似于傅里叶级数近似。
- 🎨 缺少中心环的衍射图案会导致图像对比度降低,类似于JPEG压缩过度的图像。
- 🌈 使用不同波长的光进行成像,短波长的光能提供更高的图像分辨率。
- 🚀 ASML公司利用衍射和折射原理制造现代微芯片,展示了这些原理的高级应用。
Q & A
透镜是如何形成图像的?
-透镜通过收集物体发出或散射的光线,改变光线的方向并将其聚焦到图像平面上,从而形成物体的图像。这是通过透镜的曲面设计实现的,它能够将来自物体每个点的光线精确地投影到图像平面上的对应点。
为什么光线在进入不同折射率的介质时会发生折射?
-光线在进入折射率不同的介质时,其传播速度会发生变化,导致光线方向的改变。这是因为光波在不同介质中的传播速度不同,当光波从一种介质进入另一种介质时,波前会重新弯曲以适应新的传播速度,从而发生折射。
透镜的最大锐度为什么依赖于光的波长?
-透镜的最大锐度依赖于光的波长,因为不同波长的光在介质中的传播速度不同,这会影响光波在透镜内部的弯曲程度和聚焦效果。较短的波长允许产生更高频率的干涉,从而能够再现更小的特征。
为什么直径较小的透镜在相同焦距下不够锐利?
-直径较小的透镜开口角较小,这意味着到达焦点平面的光线角度受限,从而限制了能够产生的最高空间频率的干涉模式。因此,较小的数值孔径导致在焦点平面上的波能量分布不够明确,图像的细节就会丢失。
数值孔径(Numerical Aperture, NA)对成像有什么影响?
-数值孔径决定了透镜系统能够接受光线的最大角度,从而影响成像的锐度和分辨率。较大的数值孔径允许收集更多的光线信息,包括更高空间频率的干涉模式,从而提高成像的清晰度和细节表现。
实验中使用的光栅是如何通过干涉模式重建图像的?
-实验中的光栅通过一系列具有衍射边缘的缝隙来重建图像。这些缝隙的位置设计得当,使得它们在期望的焦点处产生1波长的路径差,从而通过1阶衍射在该点重建焦点。通过增加更多的缝隙,我们实际上是在增加更高频率的正弦波衍射模式,这有助于更准确地重建图像。
为什么使用不同波长的光进行成像时,图像的分辨率会有所不同?
-使用不同波长的光进行成像时,较短的波长能够产生更高频率的干涉模式,从而提供更好的对比度和定义。这是因为较短的波长允许更精细的光波干涉,使得图像的细节更加清晰。
为什么减少数值孔径会类似于JPEG压缩的效果?
-减少数值孔径相当于丢失了更高空间频率的衍射信息,这与JPEG压缩中丢失高频信息类似。JPEG压缩通过编码对图像质量影响较小的频率来减小文件大小,但如果压缩过度,就会丢失重要频率,导致图像质量下降,出现压缩伪影。同样,透镜如果缺少高空间频率的衍射信息,图像也会失去清晰度和对比度。
光的波动性如何帮助我们更好地理解透镜成像?
-光的波动性让我们能够从波的角度来理解光的行为,包括衍射、干涉等现象。这些波动性质使得我们能够更全面地理解透镜如何通过收集和聚焦光波来形成图像,尤其是在解释像差、分辨率和成像质量方面。
实验中提到的Fresnel区板是什么?
-Fresnel区板是一种基于菲涅耳衍射理论的光学元件,它由一系列同心的环形透明和不透明区域组成。这些区域的边缘被设计为在特定焦点处产生1波长的路径差,从而通过1阶衍射重建焦点。Fresnel区板可以用来聚焦光波,类似于透镜的作用。
如何通过增加透镜的数值孔径来提高成像质量?
-通过增加透镜的数值孔径,我们可以收集更广泛角度的光线,这意味着能够捕获更高空间频率的干涉模式。这有助于提高成像的锐度和分辨率,使得图像的细节更加清晰。
Outlines
🌟 透镜成像原理
本段介绍了透镜如何通过光线的折射来形成图像。最初,作者以高中物理知识为基础,解释了光线从物体发出或散射,并被透镜收集、聚焦成图像。提到了光线在不同介质交界面上因折射率差异而改变方向,以及透镜的曲面如何巧妙地将来自物体上一点的光线投影到图像平面上的对应点。然而,作者指出用光线来描述光与透镜的相互作用虽然方便,但并不完全准确。通过举例说明光线模型无法解释的一些现象,如光线为何在进入不同折射率的介质时改变方向,以及透镜的最大锐度为何依赖于光的波长。
🌊 光的波动性
在这一段中,作者通过波动的角度来解释光的行为,强调光实际上是电磁波能量,而在可见波长范围内,光表现得像波一样。通过展示一个波动动画,说明了两个点光源通过透镜聚焦到图像平面上的过程。动画中,波前在透镜界面处整体改变形状和方向,这是由于透镜界面的球形形状和透镜内部波速降低造成的。作者还提到了Nils Berglund制作的动画,并感谢他的贡献。
🔍 数值孔径的影响
本段讨论了数值孔径(NA)对图像清晰度的影响。数值孔径相当于透镜最大开口角度的正弦值。作者通过比较不同直径的透镜在相同焦距下的模拟结果,说明了较小数值孔径的透镜在焦平面上的波能量分布不够明确。通过解释波前的形成和干涉现象,作者阐述了限制波到达焦平面的角度会限制最大空间频率,从而影响图像细节。此外,还介绍了透镜系统最大清晰度的一般公式,以及短波长和增大光学系统接受光的最大角度如何帮助重现更小的特征。
🧪 显微镜下的实验
作者通过显微镜下的实验来展示数值孔径对图像清晰度的影响。通过在显微镜系统中改变数值孔径,作者展示了插入小孔限值透镜开口角度后,图像中小特征的定义如何显著降低。实验使用了具有不同尺寸特征的测试图案,并使用10倍放大目标镜头观察。实验结果表明,减少NA会导致更高阶的衍射信息丢失,从而影响图像的清晰度。
🎨 衍射与成像
在这段视频中,作者通过一系列有趣的实验来探讨衍射和成像。首先,通过两个线性缝隙的实验,展示了在显微镜下观察到的干涉模式。随着缝隙间距的增加,干涉模式变得更加密集,空间频率增加。接着,作者通过在光束中添加更多具有衍射边缘的缝隙,展示了如何创建更清晰的焦点。实验使用了基于菲涅耳区域板配置的缝隙模式,通过增加缝隙数量来提高数值孔径和收集更多相位信息。最后,作者讨论了不同波长对图像分辨率的影响,指出使用更短波长的光可以得到更好的图像对比度和定义。
Mindmap
Keywords
💡透镜
💡折射率
💡光线
💡波前
💡数值孔径
💡衍射
💡干涉
💡光波
💡傅里叶变换
💡分辨率
Highlights
透镜成像的基本原理是通过收集物体发出或散射的光,并将其聚焦成图像。
光的折射是由于光在不同介质之间传播时速度的变化,导致光线方向的改变。
透镜的曲率表面能够巧妙地将来自物体上一点的光投射到图像平面上相应的点。
光线模型虽然方便,但并不能准确描述光的物理行为,例如无法解释光在不同介质中传播方向为何改变。
光实际上是电磁波能量,而非简单的光线。
通过观察波的行为,我们能更深刻地理解光的真实行为。
波前通过透镜时,整体形状和方向会发生变化,这是由于透镜界面的球形和透镜内部波速降低造成的。
即使点光源在焦平面上没有被完美再现为尖点,我们仍可以在波强度分布中找到两个明显的极大值。
数值孔径(Numerical Aperture, NA)是透镜最大锐度的关键因素,它与透镜的开口角度的正弦值成正比。
通过限制到达焦平面的波的入射角度,我们限制了可以在此创建的最大空间频率的强度变化。
透镜系统的最锐度可以通过公式CD = k * λ / NA来描述,其中CD代表可分辨的最小特征尺寸。
通过显微镜实验,我们可以直观地观察数值孔径对图像锐度的影响。
通过增加透镜中的缝隙数量,我们可以提高图像的分辨率,这是因为我们收集了更多的相位信息和更高频的衍射模式。
使用衍射环的模式可以创建一个紧密的焦点,这类似于傅里叶级数近似,通过添加更高频率的正弦衍射模式,可以更准确地重建图像。
减少波长可以提高图像的分辨率,这是因为较短的波长允许创建更高频率的干涉模式。
通过移除中心环,我们可以看到为了准确创建包含小特征和大特征的图像,使用高低空间频率的衍射是至关重要的。
这个视频展示了如何通过衍射和干涉的原理来创建图像,而ASML公司将这些原理应用于现代微芯片的纳米级特征制造。
Transcripts
Hey everyone, Can you tell me
how a lens creates an image? If you would have asked me this question a couple of years ago,
I would probably have explained it to you as it was explained to me in a high school.
I would have told you something similar to what is described in these pictures:
that light is emitted by or scattered of an object and spreads out in space in different directions.
A part of the light is collected by the lens, where it changes direction and is then focused
into an image. I would have explained the focusing by saying that the light rays change direction at
the boundary between different materials due to a difference in refractive index. And that because
a lens has curves surfaces, it can cleverly project the light originating from a point
in an object to a corresponding point in the image plane. And by doing this for every point,
this then yields an image of the object. So that is how a lens creates an image, right?
As the images in this physics book illustrate, the interaction between light and lenses,
is generally described by using “rays” of light. Drawing rays is very convenient because they show
you where the light goes and they can quickly make you understand the basic principle of for
example a focal point. But they also have a downside in the sense that they are in fact a
very poor representation of the physics that is going on. Let me give you some examples
where rays completely miss this point. For example, why in the first place would
a ray change direction when entering a medium with a different refractive index? I mean, you
could think of reasons why a denser medium would slow down light, but why change its direction?
And how can a ray model explain that the maximum sharpness of a lens is dependent on the wavelength
of the light? Here is another example where rays fail to explain what we observe in real life:
say we have 2 perfect lenses with the same focal length. Why is it that the one with the smallest
diameter or opening angle is fundamentally less sharp? All these questions are in fact
quite hard to explain at the fundamental level if we view light from a ray perspective.
Now, most of you will of course know that light isn’t actually rays, but electromagnetic wave
energy. And light in the visible wavelength range very much behaves like a wave. What rays
are trying to depict is the local direction of wave propagation. And so, to understand why light
really behaves the way it does, we should actually be looking at how waves behave.
Let me just show you a wave animation. It features 2 point sources that emit waves at
a fixed wavelength. Furthermore, we have a lens, which is an area where the waves propagate much
slower and which has 2 curves surfaces. And on the right side, we have a plane where this lens
focusses the wave energy. Now these lines show the ray representation of how these two sources are
imaged onto the image plane. But this is how it looks from the wave perspective. As you can see,
the waves spread out in space. Because the sources are emitting the wave energy coherently, they
create a nice interference pattern. When the waves pass the lens interface, the wavefront as a whole
changes shape and direction. And this is because of the spherical shape of the lens interface and
because the velocity of wave propagation is lower inside the lens. And the same sort of
phenomenon happens on the other side of the lens where the waves bend again due to the difference
in propagation speed between the media and then reach the focal plane. What you observe is that
even though the point sources are not perfectly reproduced in the focal plane as sharp points,
we find two very distinct maxima for the wave intensity. So, when this lens creates
an image in the focal plane using waves, it can easily resolve the two individual wave sources.
Now before I continue, I want to mention that this animation was created by Nils Berglund who many of
you will know from his YouTube Channel where he presents all kinds of cool physics animations.
I’m a big fan of Nils and I asked whether he could maybe create this type of lens simulation
and he gladly accepted the challenge. By the way, notice that the number of videos he published has
recently exceeded one thousand. I mean, how? Well, basically by publishing a video every single day
for the last 3 years. So yeah, Nils has really been pretty busy. Anyway, I want to thank Nils
for his efforts and for those interested I’ll post a link to his channel in the description.
To demonstrate the effect of Numerical Aperture, which is basically equivalent to the sin of the
maximum opening angle of a lens, I asked Nils to do the same simulation for a smaller diameter
lens with the same focal length. Here you see how that works out. I’m showing the previous and the
new simulation together here so you can compare the difference in the outcome more easily. And
what you observe is that the distribution of the wave energy in the focal plane with the smaller
Numerical Aperture is much less well-defined. So, can you from the simulation spot why that
is exactly? It’s pretty hard to see right because what we observe isn’t even close to what we would
expect based on the ray representation. The simplest way that I know how to roughly
explain it is the following: say that we have an array of very small individual wave sources that
emit waves coherently and that we want to resolve in an image. If we look at the wavefront created
some distance away from the sources that reaches the image plane heads on, we observe that it has
become almost flat because of spatial coherence. And if you were to place and image detector here,
then there would be very little to no variation in the wave intensity. So then the question arises:
how can we introduce the intensity variations needed to resolve the sources using just
waves? Well, the only way that we can do this is is by means of introducing
wave interference. And in order to create this, we need additional waves that arrive
at the focal plane under a different angle. In fact, the higher the spatial frequency that
we want to reproduce in the image plane, the larger the angle needs to be between
the incoming waves. So, by limiting the angle at which waves can arrive in the focal plane,
we also limit the maximum spatial frequency of the intensity variations that can be created
here. This basically means that by limiting the opening angle of a lens, we lose important
information contained in the diffraction pattern, and therefore will lose detail in the image.
If you keep this view in mind, it is very easy to understand the general formula that
describes the maximum sharpness of a lens system. In this formula CD stands for critical dimension,
which is basically equivalent to the dimension of smallest features that can be resolved. The
critical dimension is equal to a constant, times the wavelength, divided by the numerical aperture.
And the numerical aperture is in this case proportional to the sin of the opening angle
of the lens. From the view point of creating interference, having a shorter wavelength allows
us to create higher frequency interference and eventually to reproduce smaller features with
the same angle. The same is true for increasing the maximum angle at which an optical system
can accept light. This will also allow us to create a higher-density interference pattern
and eventually to resolve smaller features. Okay let me show you an experiment that
illustrates the effect of Numerical Aperture in a microscope. Say this is a schematic of our
microscope with the objective depicted here, a tube lens and a focal plane. We can examine
the image that the microscope produces by either placing a CMOS or CCD sensor in the focal plane
or look at the aerial image using an eye piece. If we want to change the numerical aperture of this
system. Then the easiest way to do this is by inserting a small aperture in the optical path,
for example here. This area is called infinity space, and it allows you to insert filters or
beam splitters, into the optical path of the microscope. And as long as these have flat and
parallel optical surfaces, they don’t introduce optical aberrations into the system. Now by
placing a pinhole here, we effectively limit the opening angle of the objective, in other words,
we limit the numerical aperture. So let’s have a look at the effect in
practice. Here is an optical microscope and this one actually gives us easy access to the
infinity space inside the microscope. Here you can insert filters but we can also insert an aperture
to effectively limit Numerical Aperture. Under the microscope is a small glass disk with
a chromium surface layer. The chromium contains a test pattern etched in it. So in the areas
where the chromium was etched away, the sample has become transparent. I actually made this pattern
using photolithography with my maskless wafer stepper. And if you want to know more about that,
please look in the description of this video. The test pattern is illuminated from the back
with white light. The pattern itself is pretty small and contains features of various sizes.
The total diameter of this particular round pattern is 0.5mm, meaning that the smallest
features in the pattern are only a few microns in size. And currently the pattern is viewed with a
10x magnification objective at full aperture. Now let me show you what happens when we insert
an aperture into the optical path and thereby reducing the NA. Here is a comparison: we
observe is that the definition of the smaller features suffers significantly due to the absence
of higher order diffraction from the object. In other words: by throwing away the information
contained in the light diffracted under larger angles by the test pattern. Now I think it is
pretty cool that we can directly observe the effect of NA in a microscope in this way.
Okay, so the previous was basically the main message of this video and I think it
explains intuitively why numerical aperture is so important to create sharp images. Now,
in the rest of this video I’m going to goof around a bit with diffraction and image formation and do
a few experiments. But at the same time, I’ll also dive in really deep. Now I’m not going
to explain every single aspect that you are about to see. But, I can assure you,
if you stick around you will not be disappointed. The first experiment, that I want to show you is
very simple and you have probably seen it presented quite often. It involves just 2
linear slits and because they are very small, we view them under a microscope. They are illuminated
from the other side using the coherent light of a HeNe laser. The width of these slits is
around 3 microns and they are spaced about the same distance apart. Here we view them
in a bit more detail and can measure the light intensity in the horizontal direction in a graph.
If we now move away from the slits, we observe an interference pattern, which is caused by the
diffraction of the light from both slits. This diffraction pattern is actually quite similar
to the diffraction pattern that we just observed previously in the simulations. If we now place
the slits further apart, we observe that the interference pattern that appear behind the slits
becomes denser and so the maxima and minima are spaced closer together. In other words,
the interference has a higher spatial frequency. Here I’ve schematically drawn the configuration:
this is the mask containing the slits, with the coherent light source behind it and here some
distance (l) away we observe the interference. If you do a little math, it turns out that you
can formulate the relationship between the spacing in the diffraction pattern (delta x)
to the wavelength (lambda), the distance between the slits (d) and the distance (l) from the slits
to where we observe the interference. Now this formula is an approximation, but illustrates the
fact that, when the distance between the slits gets larger, delta x gets smaller,
so the spatial frequency in the interference pattern increases. And this is basically due
to the angles under which the slits interfere. Now of course a double slit isn’t a lens, because
normal lenses are generally round. But what if we were to bend these two lines into a single
circular slit, would that be a lens? Take example, this little fellah which has just 2 circular
diffractive edges and only 70um in diameter? No, that cannot possibly be a lens. But just
to be sure, let’s place it in a coherent beam of light of laser light and look at the diffraction
pattern. At a distance of 2mm away from the slit, we observe a circular diffraction pattern with,
what appears to be a focal point. Here you can see it in a bit more detail together with
a plot of the intensity distribution based on the diffraction pattern. So the circular slit
seems to be lens after all. It is not really impressive, because let’s face it, the focal
point is almost as large as the lens. But I think we are on to something. Now let’s place a few more
slits with diffractive edges in strategic places and see what happens. Here you see the result
with 6 slits and here with thirty. Now, this is starting to look like a real focal point!
I want to emphasize that these images are not simulations. They are real images that
were collected using a microscope from real slit patterns ranging in diameter from 70 to
500um. And these patterns were also created using photolithography. Now of course, the slit patterns
aren’t just random circular patterns. They are actually based on the configuration of a
Fresnel zone plate, named after Augustin Fresnel, a French scientist. The edges in the patterns are
placed such that each creates 1 wavelength of path difference to the desired focal point. Basically,
these recreate a focus from 1st order diffraction at this point. It’s definitely not the same thing
as a refractive lens but it is quite similar. And the fun thing is that using these, we can build up
numerical aperture in discrete little steps. If you look at the focal point in the last
pattern, you might get the idea that a tight focus is mainly achieved by the outer rings.
But look what happens if we take away the center rings: the total size of focal point
increases again because of what appears to be high-frequency diffraction. I’ll get back to
this lens pattern later in the video. Okay, so these patterns can create a tight
focal point from laser light by adding diffraction patterns. But are they in fact also real lenses
when we use incoherent light, like the light from a standard candescent lamp? Again, let’s just find
out. Here is the schematic of the setup: light from the candescent lamp is filtered with a red
color filter to make it a bit more monochromatic. The test pattern used previously is placed in the
beam, then, at some distance, the circular slit pattern. And the slit pattern will hopefully
create an aerial image of the test pattern, which we can then observe using a microscope. So here
you see how that works in practice. This is the plate containing the test patterns which is 15mm
below the x-y table. The lens patterns are placed on the x-y table and so we can easily choose which
lens to use. And with the microscope, we will take a look at the aerial image of each lens
Here I’ve got the simplest pattern containing only 1 circular slit in focus with the microscope and
if I now move the focus away from the pattern itself, we observe how light is diffracted of
the edges and eventually creates an image of the original pattern. Hmm, I admit it is not
very sharp is it? You can see for example that there is a line, but you cannot see the central
spacing at all. So this lens is probably missing out on a lot of the light that is diffracted under
larger angles by the test pattern. So let’s add a few more ring-shaped slits and see what happens.
Here is another one, and another one [ let’s go] This is the resulting image of 12 diffractive
rings. Now, who would have thought that what is basically a simple pattern containing a very
limited number of slits could reproduce such small features. But of course, we are not done yet,
we can do better. Let’s go straight to the maximum number of 30 rings and see what that this pattern
can do. Here you see the slit pattern in focus and is we now slowly move the focus
of the microscope towards the focal plane of this lens we can see how the image is created. I mean,
look at that. Isn’t that just amazing. That adding what is basically a limited set of
interference patterns create by a bunch of slits can recreate a pattern with such amazing
resolution. The round feature here is only 76um in diameter in the image plane, making the smallest
features imaged in the order of 1 um. So why does the resolution improve with
the increasing number of rings? It is actually two-fold: by adding more slits in the lens pattern
we are of course also increasing the numerical aperture of our lens and collecting more phase
information from the light diffracted by the test pattern. But at the same time,
we are also getting a larger set of high frequency diffraction patterns available to reconstruct the
image with. With just 2 diffractive edges in the center, the reproduction of the pattern
is very poor, because the lens can only create low spatial frequency diffraction
patterns. But as we add more and especially wider rings, smaller features in the pattern
can be resolved because the wider rings are able to create higher frequency spatial diffraction.
If you are familiar to the Fourier transform, you may have noticed that what you just witnessed
was real-life version of the Fourier series approximation. With this method, basically any
function can be approximated using a specific set of sinusoidal functions with specific frequency,
amplitude and phase. By adding more and especially higher frequency harmonics, we can more accurately
approximate the function. And that is exactly what we did here. By adding diffraction under
increasing angles, we basically added higher and higher frequency sinusoidal diffraction patterns,
which eventually resulted in a fairly high-quality reproduction of the image.
Now I mentioned a few minutes ago that I was going to return to this particular pattern where the
center rings are missing. With the Fourier series approximation in mind, it is interesting to look
at the image that this pattern creates. Here it is and what you can observe is that the image looks
somewhat weird: it has lost contrast compared to the full pattern. It has lost uniformity in the
larger areas and it has these faint borders around the intensity transients. The image looks a bit
like a very heavily compressed JPEG image. And that is because what you observe is very similar
to bad JPEG compression. JPEG compression and decompression is also based on the principles
of the Fourier. The compression works by only encoding the frequencies that are essential for
creating an acceptable reproduction of an image. But if you compress the information in an image to
the extend that essential frequencies are omitted, this then leads to artefacts. And these are very
similar to the ones observed here. So basically what this demonstrates that in order to accurately
create an image that contains both small and large features, it is essential that you use
both high and low spatial frequency diffraction. The last thing I want to show you is the effect of
wavelength. These 3 images are all created using a lens of 30 diffractive rings, but in each case,
I’ve used a filter of a different wavelength. And if you look carefully you can see that reducing
wavelength results in better image resolution. I admit that it is hard to see, so I tried to
quantify the effect here in the line and space pattern. Here I’ve plotted the intensity profile
over these lines and it is very clear that when using blue light, the picture has better contrast
and definition, compared to the one in red. So that is how lenses create images using waves:
by adding up a whole lot of diffraction. And in the upcoming video, which will also
be about image formation, I’ll tell you about my visit to a company that takes the principles of
diffraction and refraction to a whole new level. The name of this company is Advanced Semiconductor
Materials Lithography, ASML for short. In order to create the nanometer features that
populate modern micro-chips, they literally have to use every trick in the book of diffraction.
So, I hope that this video gave you some new insights and who knows, maybe we’ll meet again.
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