Light & Coherence part 2: Spatial Coherence (and the Double Slit Experiment)

Summary

本视频深入探讨了空间相干性及其对光性质的影响。通过荷兰希尔弗瑟姆的洛伦兹池塘实例，阐释了波的空间相干性如何随距离源点增加而增强。视频还通过数值模拟展示了波的空间相干性发展过程，并定量分析了相干区域与波源几何尺寸的关系。此外，视频对托马斯·杨的双缝实验进行了讨论，指出了科学传播中对实验配置的简化和误解。最后，通过双缝实验和光栅实验，展示了空间相干性在光的干涉和频谱分析中的作用，引发了对光的波动性和量子性质的深入思考。

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Q & A

什么是空间相干性？

空间相干性是指光波在空间分布上的有序性和一致性。当光波的相位在一定区域内保持一致时，我们可以说这些光波在该区域内具有空间相干性。

视频中提到的Hilversum的池塘是如何展示空间相干性的？

视频中的Hilversum池塘通过喷泉产生的水波来形象地展示空间相干性。喷泉附近的水面波动较为混乱，但随着距离的增加，水波逐渐变得有序并朝同一方向移动，展示了空间相干性随距离增加而增强的现象。

视频中提到的Nils Berglund的模拟实验是如何帮助理解波的空间相干性的？

Nils Berglund的模拟实验通过展示15个随机位置的波源发出的波如何在时间和空间上发展，帮助我们更详细地理解波是如何逐渐获得空间相干性的。实验中波源发出的波最初是混乱的，但随着距离的增加，波的模式变得更加有序，这与实际池塘中观察到的现象相似。

波的空间相干性与波的几何特性是如何关联的？

波的空间相干性可以通过相干面积来表征，该面积与波源距离的平方、波长的平方成正比，与波源直径的平方成反比。这个关系简化了实际情况，但提供了一个有用的公式来估算相干区域的大小。

Thomas Young的双缝实验是如何利用空间相干性的？

Thomas Young的双缝实验通过使用单色光和两个非常接近的缝隙来展示空间相干性的效果。只有当光源产生的相干面积足够大，以至于覆盖了两个缝隙的整体区域时，才能在屏幕上观察到明显的干涉条纹模式。

视频中提到的关于Antares星的空间相干性计算结果是什么？

根据视频中的计算，Antares星的光到达地球后，其空间相干区域大约是2.3平方米。这意味着尽管Antares是一个巨大的恒星，其发出的光在地球上表现出几乎均匀的相干性。

为什么我们需要空间相干性来进行双缝实验？

进行双缝实验需要空间相干性，因为只有当来自两个缝隙的光波在空间上具有固定的相位关系时，它们才能在重叠区域形成干涉条纹。如果光波在缝隙区域内没有相位关系，就无法形成明显的干涉模式。

视频中提到的关于光的经典视角和量子力学视角有何不同？

经典视角主要关注光作为电磁波的传播、散射、衍射等现象，而量子力学视角更侧重于光的发射和吸收过程中的能量包转移，以及这些过程中的量子化特性。

视频中提到的关于光的波粒二象性是如何理解的？

光的波粒二象性是指光既表现出波动特性，如干涉和衍射，也表现出粒子特性，如光电效应和康普顿散射。视频中提到，尽管光的相互作用过程是量子化的，但这并不一定意味着光本身就必须被看作是能量的离散包。

视频中提到的关于光的相干面积公式是如何推导的？

相干面积公式是基于波源几何形状和波的干涉条件推导出来的。考虑一个扁平的圆形波源，只有来自波源最外侧的波在远离中心轴的某一点上保持同相。随着距离中心轴的增加，这些波逐渐失去同相性。定义一个距离Y，使得两点间的相位差小于一个可接受的值，这个距离Y与波长成正比，与波源距离和波源大小成反比。

视频中提到的关于光的传播方向对空间相干性的影响是什么？

光的传播方向对空间相干性有重要影响。当光波从不同的波源发射出来时，它们的传播方向会随着距离的增加而逐渐对齐。这种对齐导致了波的相位关系变得一致，从而在一定距离外形成了空间相干性。

视频中提到的关于光的干涉和衍射现象有什么区别？

干涉现象是指两个或多个相干光波在空间某点相遇时，由于相位差引起的光强增强或减弱的现象。而衍射是指光波遇到障碍物或通过狭缝时，波前发生弯曲和扩散的现象。在双缝实验中，衍射和干涉同时发生，共同作用形成了复杂的干涉条纹模式。

Outlines

00:00

🌟 视频简介与空间相干性概述

本视频是关于相干性的第二部分，继上一个关于时间相干性的视频之后，这一集主要讨论空间相干性，以及它是如何影响光的特性的。视频提到了一个位于荷兰Hilversum的名为Hendrik Antoon Lorentz的长矩形池塘，通过池塘中的喷泉产生的水波来形象地展示空间相干性。随着远离喷泉，波浪逐渐变得有序，展示了空间相干性的形成和特性。视频还提到了即将推出的第三部分，以及通过数值模拟来更详细地了解波是如何随时间发展的。

05:06

📏 空间相干性的量化与公式解释

这一部分详细讨论了如何量化三维空间中波（例如光）的空间相干性，并通过一个简化的公式来解释其背后的原理。公式表明，相干面积与距离的平方、波长的平方成正比，与源的直径的平方成反比。视频通过一个假设的圆形波源来形象解释这个公式，并以太阳为例，计算了其相干面积。此外，还提到了Thomas Young在1801年的光和颜色理论实验，以及他是如何使用太阳光来进行实验的。

10:08

🌈 托马斯·杨的双缝实验与误解

在这一部分中，视频讨论了托马斯·杨的双缝实验，指出历史上对该实验的描述并不完全准确。实际上，杨并没有使用双缝，而是使用了一张薄片来产生衍射图案，并观察光的分离成不同颜色。视频还讨论了科学传播中的问题，即我们通常传播的是代表实验本质的隐喻，而不是实际配置的细节。此外，视频还展示了如何使用真实的双缝来演示空间相干性的效果，并通过白光来进行实验，展示了光通过双缝时的干涉模式。

15:09

🌠 星光照明白空间相干性与傅里叶变换

这一部分探讨了如何通过星光照明白光的空间相干性，并通过一个类似光栅的结构来执行傅里叶变换。视频以LED的光谱为例，展示了如何通过小孔衍射图案来实现光谱分析。然后，视频以天蝎座的红色巨星Antares为例，计算了其光在地球上的相干面积，并讨论了这一结果对于理解光的本质的意义。视频最后提出了关于光的波动行为和量子力学视角之间的问题，为第三部分的讨论留下了悬念。

20:15

🤔 光的本性与量子力学的探讨

视频的最后一部分提出了关于光的本性的问题，探讨了经典电磁观点和量子力学观点之间的差异。视频作者表达了对量子力学的不理解，并提出了关于光的相互作用是否本质上是概率性的疑问。这部分为视频的续集埋下了伏笔，预示着在未来的视频中可能会探讨更多关于光的本质和量子力学的内容。

Mindmap

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Keywords

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Highlights

Introduction to spatial coherence and its importance in understanding light properties.

Real-life demonstration of spatial coherence using a fountain in a pond named after Hendrik Antoon Lorentz.

Transition from chaotic wave movements to organized, spatially coherent waves demonstrated with water jets.

Numerical simulation of wave sources by Nils Berglund to further illustrate spatial coherence.

Explanation of wave propagation and the development of spatial coherence from chaotic sources.

Introduction to the concept of spatial coherence in waves and its mathematical foundation.

Derivation and explanation of the formula relating coherence area to distance, wavelength, and source diameter.

Application of spatial coherence principles to the sun's light and the calculation of its coherence area.

Historical context of Thomas Young's experiments on light and the evolution of the double slit experiment.

Explanation of spatial coherence requirement for conducting a proper double slit experiment.

Demonstration of a double slit experiment with white light and the effect of spatial coherence on interference patterns.

Introduction to the 'bunch-a-slits' experiment and its application in performing Fourier transforms.

Calculation of the area of coherence for light from Antares and its implications for understanding stellar light properties.

Discussion on the differing perspectives of light in classical physics versus quantum mechanics.

Reflection on the challenges of reconciling quantum and classical views of light and the anticipation of further discussion in part 3.

Transcripts

Hey Everyone, This is my second video on  

coherence. The previous episode was mostly about  temporal coherence and how it is directly related  

to the spectral properties of light. If you haven’  t seen it, it might be a good idea to watch that  

one first, because the current video it really  a sequel of part 1. Today, I will mainly talk  

about spatial coherence, how it comes about and  what it means for the properties of light. And,  

I will also take a few side tracks. I found that I  could not fit all the side tracks into this video,  

so good news for those of you who  like sequels: there will be a part 3.  

I live in Hilversum, the Netherlands and close  to my home there is this a long rectangular pond  

that is named after Hendrik Antoon Lorentz,  in my opinion one of the greatest Dutch  

scientists ever. And this pond is just ideal  to demonstrate real life spatial coherence.  

The pond has a fountain in it, which consists of  3 water jets that generate a lot of splashes. And  

so, in the vicinity of the fountain, every point  on the surface moves rather chaotically. However,  

if we move away from the actual splash zone, we  observe that the waves gradually all start moving  

in this same direction, away from the source and  become ever more organized. Shh, go away.If we  

consider a small enough part of the wavefront, we  observe that the waves are almost linear. Which  

means that all points in the direction transverse  to wave propagation move in unison or in sync.  

And so, if we consider a small enough area of  the wavefront, we can say that the waves have  

gradually become spatially coherent. If you look  at the spacing between the waves you will still  

observe that the frequency of the waves changes  with time in the direction of wave propagation.  

And this is related to temporal coherence, a  phenomenon that was covered in the previous video.  

What you observe in a general sense is that  the region of spatial coherence gradually  

expands when the waves move further away from  the sources. Now, although, quite illustrative,  

this pond is not ideal as a model, especially  with birds and the wind contributing to the  

total wave pattern. So let me take you the more  controlled environment of numerical simulation.  

The simulation that I’m about to show you here is  made by Nils Berglund who runs a YouTube channel  

in his name. On his channel, he regularly posts  visualizations of all kinds of physical phenomena,  

from chemical reactions to lasers solving mazes.  And he very kindly accepted the challenge to do  

this wave simulation, which allows us to look  in more detail how waves develop in time.  

Let’s go to the start of the simulation. It  involves 15 randomly positioned wave sources that  

each emit a longitudinal wave in one long burst.  The sources have a slightly different emission  

frequency and start at a random phase. As you can  see, they emit waves in all directions and what we  

observe is how, relatively close to the sources,  the wave patterns are quite chaotic, a bit like  

in the pond. But as the waves move away from the  disorder, we see that things settle down pretty  

quickly. And again, if we consider only a part  of the wavefront, we can actually see how spatial  

coherence grows with distance. But there is more:  we can identify these areas where the waves are  

relatively strong. These are separated by what  appear to be boundaries where the amplitude is  

low due to destructive interference. Now for the  casual observer it may seem that the waves and the  

wave energy is non-uniformly distributed in space.  But that is actually not the case, because the  

simulation basically only highlights potential  energy and not so much the kinetic energy in  

the waves. But what we do observe is that the  waves from apparently randomly emitting sources  

result in a wave pattern that gradually becomes  spatially coherent some distance further away.  

It also illustrates nicely the most important  aspects of temporal and spatial decoherence.  

But why do waves seem to add up to a  fairly regular pattern when they move  

away from the sources? Well, one important  aspect is that as they move further away,  

the wave propagation directions of different  sources are lining up. And what you should realize  

is that when waves of the same frequency  but with a different phase are lined up,  

they will result in one single regular wave  of the same frequency. And so, all the initial  

chaos related to just phase difference from  the various wave sources gradually disappears  

when we move further away. Now in this case,  the linear addition of individual waves will  

never show a 100% regular pattern, because the  sources all emit at a different frequency and so  

phase relationships are not constant in time and  space. And that is why we observe these areas of  

decoherence in both the direction of propagation  as well as in the transverse direction.Let’s  

see if we can understand quantitatively how  spatial coherence in waves is related to geometry.  

Spatial coherence of waves in 3D space, like  for example light, can be characterized by  

an area of coherence and it is interesting to  see what defines this area. I’ll just start by  

giving you a formula. You see that the coherence  area is proportional to the distance from the  

source squared, and the wavelength squared  and inversely proportional to the diameter  

of the source squared. I should add that this  is a simplification which only applies if the  

distance R is much larger than the size of the  emitter. Now there is actually quite a lot to it  

if you want to derive this formula property in the  case of light, which I don’t want to do. However,  

I think it would be good to give you some  very general insight into this formula.  

Imagine we have a source in the shape of a  flat round disk emitting waves. And let’s  

for now consider only those waves arriving from  the extreme outside of this source which arrive  

on a surface located somewhat further away. This  surface is perpendicular to the axis pointing in  

the direction of the source and we observe  that these waves are in phase on this axis.  

If we move some distance in the y-direction  away from the central axis, the waves will  

gradually get out of phase more and more  until they are completely out of phase.  

Now if we define Y as the distance over which  the phase shift between these 2 waves is lower  

than an acceptable value, say for example  60 degrees phase shift, we find that this  

distance Y is proportional to the wavelength:  so, the shorter the wavelength, the smaller  

this distance Y. And this relationship between  Y and wavelength is actually a linear one.The  

same kind of relationship holds for the distance  from the source: the longer the distance between  

source and area, the larger the value of Y  within which waves will still be considered  

to be in phase. However, for the size of the  emitter, it is exactly the opposite: the smaller  

the distance between the outer boundaries  of the emitter, the larger the value of Y.  

And so, we can summarize that the distance Y  is linearly proportional to the wavelength and  

the distance from the source, and is inversely  proportional to the size of the emitter. And that  

means that the Area of coherence which is defined  by Y as the radius of a circle, is proportional to  

Y squared. Now if you compare this result with  the formula I showed you, you may notice there  

is one detail which is still a bit out of place  and that is the one over pi. This is actually a  

proportionality factor that arises if we derive  this formula in a mathematically more rigorous  

way, based on the exact geometric configuration  of a round source and our coherence requirement.  

Now as stated, this is a somewhat simplified  representation, but nevertheless the resulting  

formula is actually quite useful. So, let’s  calculate the area of coherence for the sun using  

this formula. I must remark that, technically  speaking, on earth the constraint between  

distance and size is not truly satisfied.  You should also keep in mind that the sun  

is spectrally very broad, so we have to choose  a wavelength for which to do the calculation.  

But when we fill in the values for the diameter  of the sun, the distance between the sun and  

earth and choose a value for Lamba in the center  of the visible spectrum, say 500nm, we arrive at  

an area of coherence of almost 900 square microns  which is equivalent to an area with a diameter of  

34 um. This area is comparable to the cross  section of a human hair. Of course, this only  

works if there are no clouds in the sky because in  that case, the area of coherence will definitely  

be smaller. And since wavelength is a parameter,  for blue light with a shorter wavelength the area  

of coherence it will be somewhat smaller  and for red light it somewhat larger.It  

is interesting to note that when Thomas Young  wrote his paper about the theory of light and  

colors in 1801, he did of course not have a laser  or other really bright light sources available,  

other than the sun. And so Young was more or less  forced to use sunlight for his experiment. From  

what I’ve read on the internet, he used a mirror  to direct the sunlight through a tiny pinhole into  

a darkened room, to create a bright beam of light.  And with the knowledge you have just aquired about  

the area of coherence of the sun, I think it  should be fairly obvious to you what the function  

of the pinhole is: it’s intended to make all the  light entering the room spatially coherent.  

When I tried to confirm that this is how he  actually did it in the original paper, I could not  

find any reference to a mirror or an aperture in  the window. Or a double slit for that matter. Yes,  

that is correct: Thomas Youngs’ famous double  slit experiment actually did not involve a  

double slit configuration. Instead, he placed  a thin card in the beam that split it in half,  

which caused the diffraction pattern. And this  is how he could observe the separation of light  

into different colors. Even a few years later  in his 1803-1804 paper on the same subject,  

there is no reference to a double slit  configuration anywhere. However, six years  

after the original experiment, Young published  2 books that contained a large collection  

of lectures. And in volume 1, he did actually  mention a configuration resembling a double slit.  

The reason why I think this is interesting  is because it illustrates how we communicate  

about science. We don’t necessarily communicate  facts. Most of the time we communicate metaphors  

that represent the essence of an experiment  and not necessarily the details of the actual  

configuration. So somewhere in history, someone  decided to name it the Thomas Young’s double slit  

experiment and this then became synonymous with  the phenomenon observed. Now, don’t get me wrong:  

There is nothing like a good metaphor to explain  difficult concepts to an audience of non-experts.  

However, extending the use of metaphors beyond  what they were intended for can become pretty  

confusing. Take the double slit experiment for  single electrons instead of photons. Here are a  

few examples of how that experiment is depicted:  see if you can spot the confusing aspect.  

It’s not this cougar here, which might very well  be a metaphor for something else too, who knows.  

No, the confusing aspect is that all these visuals  show narrow slits with a very substantial distance  

between then. Which makes the experiment  intrinsically mysterious: because how can a  

particle small enough to pass through a narrow  slit, pass through two widely spaced slits at  

the same time?However, if we refer to the original  experiment performed at Hitachi, the interference  

was actually created by using an electron  biprism. This device basically consists of  

an extremely thin charged wire, with a thickness  much smaller than a micron. When you look at this  

configuration, the interference effect suddenly  becomes much less mysterious because it is  

basically about how the electron interacts with  an object small enough as to demonstrate that the  

electron has wave properties. And so, the metaphor  of the double slit and the way it is presented  

doesn’t necessarily help people understand.  If anything it just adds to the confusion.Ah  

well, all of this does not change the fact that  actual double slits do a good job in demonstrating  

the effects that Thomas Young observed. So, let’s  do a true double slit experiment that illustrates  

the effect of spatial coherence in light. And for  this demonstration we will use white light. So  

here are 2 slits etched in a layer of chromium on  a glass slide and we are looking at them under a  

microscope. They are quite small about 300 microns  long and 5 microns wide. Why so small? Well,  

because we really want to bring out the colors  and the interference here in a lot of detail. I’ll  

place a link to another video in the description  about how you can record this type of images.  

We will start out by looking at how the light  through the slits develops when the area of  

coherence is much smaller than the dimensions of  the current double slit configuration. Basically,  

this can be achieved by placing a white  light LED very near to the slits. If we  

now move away from the slits, we’ll observe  how the light passing the slits is diffracted  

and how this results in 2 blurry blobs that merge  without any noticeable interference pattern. And  

this is because the light does not contain any  phase relationships on points in the slit area.  

Not particularly fascinating right? Now, let’s  replace the close-by positioned LED with a light  

source that produces spatially coherent light.  And this means that the area of coherence is much  

larger than the total area of the 2 slits. So in  this case there is a fixed phase relationship for  

every wavelength at every point on the slits.  And look what happens now if we do the same  

experiment. Once light from the 2 beams starts to  overlap, we observe this fascinating pattern of  

colored lines developing, because of different  wavelengths going in and out of phase.What  

we observe in this pattern is actually temporal  decoherence. In the center of the pattern,  

all wavelengths are in phase with each other,  because the distance to the two slits is equal  

for every point on that line. But as we move  away from the center, every wavelength has its  

own interference periodicity and so the phase  differences between the light originating from  

the 2 different slits will be different for  every wavelength. So this is the reason that  

you need spatial coherence to do a proper  double slit experiment. Pretty cool huh?  

Want to see something even cooler? Here I’ve got  a bunch of slits on which we can do the famous  

bunch-a-slits experiment. Together these can  actually perform a little math operation for us.  

They can do a Fourier transforms on the temporal  signal of the light wave passing the area of the  

slits. Here you see how the transformation  develops in space as we move away from the  

grating. It results in a spectrum showing all  frequencies present in the LED used for the  

experiment. In other words, it’s a microscopically  small spectrometer in one pattern. Previously we  

calculated the area of coherence for light from  the sun and found that the area is pretty small in  

the visible range. But of course, for stars a bit  further away, we can do the same thing. That is,  

if we know their size and distance and that is for  example the case for this particular star here in  

the Scorpio constellation: The star is called  Antares after the Greek god of war. And indeed,  

Antares is not a friendly little neighborhood  star. Depending on which source you consult,  

Antares is about 600- 800 times larger than our  sun. And I mean larger in diameter. If we were  

to place this big boy in our own solar system,  it’s physical boundaries would extend beyond  

the orbit of the planet mars. It’s a monster, but  fortunately for us, it’s also 550 light years away  

from earth. Now, if we do the same calculation for  the area of coherence of light Antares on earth,  

we find a value of about 2.3 square meters. Now  I want you to let this sink in for a second.  

Here we have an object so massive it’s beyond  human comprehension. Which emits unimaginable  

amounts of random emissions every second. Yet  by the time the radiation arrives on earth,  

the sum of these individual contributions yields  a field that is as good as uniform within an area  

of several square meters. And you can measure  this with methods similar to the double slit  

experiment. Of course, as for the temporal field  properties: it still contains a lot of frequencies  

since the light emitted by Antares is spectrally  also very broad. But just think about how you  

could explain for example the phase relationship  from a corpuscular perspective on light.  

Also, keep in mind Antares is a huge  star, relatively close by. Smaller,  

further away stars can easily have an area  of coherence the size of a square kilometer.  

As you may have noticed, I’ve focused only on the  wave behavior of light so far, in this video but  

also in the previous one. And if you want, you  can consider this a “classical” perspective. In  

the previous video I made friends for life with  the quantum mechanics community by stating that  

the properties of light arise purely from wave  behavior. And as much as I would elaborate on  

this a bit further now, I don’t think it would be  wise to do that near the end of a video that is  

already too long. By the way, apart from the use  of the word photon in video part 1, I did not make  

a single reference to quantum mechanics. Because I  think it is safe to say I don’t understand Quantum  

Mechanics. I did take courses in the subject at  university and I did pass the exams without a  

problem. But somehow it never made sense to  me at. I guess I just wasn’t smart enough.  

There was one thing that struck me  during the discussion. And that is  

that at some fundamental level, we do not seem  to agree on what exactly we mean with the word  

“light”.If you’d ask someone like me with  a classical mindset to describe light,  

that person would probably mainly refer to the  electromagnetic phenomenon. So, I would say  

something like: we have a source that emits light.  This light propagates in space, is scattered,  

diffracted, reflected whatever and is then  detected or absorbed, for example in your eye.  

From this perspective, I’m referring to  the electromagnetic radiation as light.  

However, if you’d ask the same question to  someone deeply involved in quantum mechanics,  

the emphasis would not be on the radiation  aspect. It would likely be more on the emission  

and absorption processes and how these involve the  transfer of discrete energy packages. So I guess,  

from the viewpoint of someone in quantum  mechanics, double slit experiments are only  

marginally interesting at best. They would argue  that the quantized interaction between radiation  

and matter is proof that the radiation or the  field itself must also quantized. And that this  

proof is in many experiments, including the  photoelectric effect and Compton scattering.  

Now just to be clear, I’m not questioning the  quantized nature of the interaction between  

radiation and matter. But do these experiments  truly show that the radiation which I refer to  

as light, is quantized into discrete packages of  energy? Or do they merely demonstrate that the  

interaction of electromagnetic radiation with  all matter is fundamentally probabilistic? 

I have a feeling that we’ll have plenty  to talk about in part 3. And maybe we  

can even discover a suitable common metaphor.