How a Lens creates an Image.

Summary

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

00:00

🌟 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。

05:01

🌊 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。

10:05

🔍 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。

15:08

🧪 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。

20:15

🎨 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。

Keywords

💡透镜

透镜是一种光学元件，能够通过折射光线来聚焦或散射光波。在视频中，透镜被用来解释如何通过改变光线的方向来形成图像。透镜的曲率表面能够使来自物体的光线汇聚于一点，从而在成像平面上形成清晰的图像。

💡折射率

折射率是描述光在不同介质中传播速度变化的物理量。当光从一个折射率较低的介质进入一个折射率较高的介质时，光线会向折射率较高的介质的法线方向偏折。

💡光线

光线是光学中用来描述光传播路径的直线。在视频中，光线被用来形象地表示光的传播方向和成像原理，但同时也指出了光线模型在解释光的波动性质时的局限性。

💡波前

波前是指在波动传播过程中，相位相同的点连成的面。在视频中，波前的变化和传播速度的变化是解释透镜如何聚焦光波的关键。

💡数值孔径

数值孔径是描述光学系统聚焦能力的参数，与透镜的最大开口角度的正弦值成正比。数值孔径越大，系统能够接受的光线角度越宽，成像的分辨率和清晰度越高。

💡衍射

衍射是指波在遇到障碍物或通过狭缝时发生的弯曲和扩散现象。在视频中，衍射现象被用来解释如何通过一系列狭缝（或透镜上的环）来形成聚焦的光波。

💡干涉

干涉是指两个或多个波在空间中相遇时，由于相位差引起的波的强度变化现象。在视频中，干涉是形成清晰图像的关键机制之一，通过不同角度到达的波的干涉来创建图像的强度变化。

💡光波

光波是电磁波的一种，它在可见光谱范围内表现出波动性质。视频中强调了光波模型对于理解透镜如何形成图像的重要性，特别是在解释光的衍射和干涉现象时。

💡傅里叶变换

傅里叶变换是一种数学工具，用于将信号或图像从空间域转换到频率域。在视频中，通过添加不同频率的衍射模式，实际上是在进行一种傅里叶级数近似，以更准确地重建图像。

💡分辨率

分辨率是指成像系统能够区分两个接近点的最小距离的能力。在视频中，分辨率受到数值孔径、波长和衍射模式的影响，这些因素共同决定了成像的清晰度和细节程度。

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.