And don't be afraid, it doesn't involve any integrals, it's really just very simple vectors and 1 by 2 vectors and 2 by 2 matrices.

別害怕，它不涉及任何積分，實際上只是非常簡單的向量、1 乘 2 向量和 2 乘 2 矩陣。

So let's say that I have a plane wave, so I'm just going to it's moving in, let's call this the z-direction, that would make this the x-direction and this the y-direction.

比方說，我有一個平面波，所以我把它的運動方向稱為 z 方向，這樣這裡就是 x 方向，這裡就是 y 方向。

If we know the plane wave is moving in the z-direction, there's a bunch of ways the electric field could be pointing.

如果我們知道平面波是沿 Z 方向運動的，那麼電場就有多種指向方式。

It could be pointing upwards in the x-direction, and in that case the h-field would be pointing this way in the in the y-direction, well let me label that x as y.

它可能在 x 方向上指向上方，在這種情況下，h 場將在 y 方向上指向這邊，好吧，讓我把 x 標為 y。

It could also be pointing that way in the y-direction, and then the h-field would be pointing in the minus x-direction, or it could be pointing somewhere in between.

它也可能在 y 方向上指向那個方向，然後 h 場將指向負 x 方向，或者它可能指向兩者之間的某處。

So it could be pointing in the xy, anywhere really in the xy-plane, so anywhere in this plane, and then the h-field is just 90 degrees away from that.

是以，它可以指向 xy 平面上的任何位置，也就是 xy 平面上的任何位置，而 h 場則與它成 90 度角。

Now one of the many beautiful things about plane waves is that the electric field and the magnetic field are just related by a constant eta, and this was the wave impedance of free space.

平面波有許多美妙之處，其中之一就是電場和磁場之間有一個常數 eta，這就是自由空間的波阻抗。

This only depends on the material that we're propagating through, so if we know what the e-field is, we automatically know what the h-field is, and similarly we know its direction because it has to be orthogonal to the electric field, and e cross h is pointing in the direction of propagation, so it's our pointing vector.

這隻取決於我們所傳播的材料，所以如果我們知道什麼是 e 場，我們就會自動知道什麼是 h 場，同樣，我們也知道它的方向，因為它必須與電場正交，而 e 交叉 h 指向傳播方向，所以它就是我們的指向矢量。

So knowing one of the fields of a plane wave fully determines the other, and so we're only going to worry about the electric field.

是以，知道了平面波的其中一個場，就能完全決定另一個場，所以我們只關心電場。

We're just going to study the electric field.

我們只研究電場。

This is all we need to worry about in order to fully understand polarization.

要充分理解兩極分化，我們只需關注這一點。

So how do we go about representing that polarization?

那麼，我們該如何體現這種兩極分化呢？

So let's say we know the direction of propagation of the plane wave, and let's say it's along the z-axis.

假設我們知道平面波的傳播方向，並假設它是沿著 Z 軸傳播的。

How many degrees of freedom do we have to deal with?

我們需要處理多少個自由度？

Well, the electric field, like we just said, can be anywhere in this plane, so let's pretend now that it's pointing just straight up.

好吧，電場就像我們剛才說的，可以在這個平面的任何地方，所以我們現在就假裝它只是直直地指向上方。

It's pointing in the in the x-direction.

它指向 X 方向。

How would we represent that in just mathematically?

我們如何用數學來表示？

And let's say it's got the amplitude e-naught.

比方說，它的振幅是 "E-Naught"。

Well, the electric field is a vector, and it's pointing in the x-hat direction.

那麼，電場是一個矢量，它指向 x-帽子方向。

It's got amplitude e-naught, and then it's a traveling wave in the z-direction, so we're gonna write that in complex notation e to the j omega t minus kz.

它有振幅 e-naught，然後它是 z 方向上的行波，所以我們要用複數符號 e 寫成 j 歐米茄 t 減 kz。

And how would we represent it if it were pointing in the y-direction?

如果它指向 Y 方向，我們該如何表示？

So maybe it has amplitude e-naught in the y-direction.

是以，也許它在 y 方向上有振幅 e-naught。

Well, then the electric field is just equal to y-hat times e-naught e to the j omega t minus kz.

那麼，電場就等於 y-hat 乘以 e-naught e 到 j omega t 減去 kz。

And in general, if it's pointing in some other direction, so it's got some component along x, so let's call that e-naught x, and some component along y, e-naught y, then we can write the total electric field as the x component, e-naught x, e to the j omega t minus kz, plus the y component, e-naught y, e to the j omega t minus kz.

一般來說，如果它指向其他方向，那麼它有一些沿 x 的分量，我們稱之為 e-naught x，還有一些沿 y 的分量，e-naught y，那麼我們可以把總電場寫成 x 分量，e-naught x，e 到 j omega t 減去 kz，加上 y 分量，e-naught y，e 到 j omega t 減去 kz。

Now notice that this e to the j omega t minus kz, this traveling wave component of the of the plane wave, is getting sort of redundant.

現在請注意，平面波的行波分量 j omega t 減去 kz 後的這個 e 變得有點多餘了。

So the only real information that we're representing with our polarization is the x-hat part and the y-hat part.

是以，我們用偏振表示的唯一真實資訊就是 x-帽子部分和 y-帽子部分。

So how much is pointing in x, how much is pointing in y?

那麼，有多少是指向 x 的，有多少是指向 y 的？

We don't really care that the wave is traveling because we already knew that.

我們並不關心波浪是否在行進，因為我們已經知道了這一點。

So we might just be lazy and say that the electric field is just equal to x-hat e-naught x plus y-hat e-naught y.

所以我們可以偷個懶，說電場就等於 x-hat e-naught x 加上 y-hat e-naught y。

And if you were a lazy person that decided to do this, you'd be doing exactly what Jones did when he invented his calculus a long time ago, I think in the late 1900s.

如果你是一個懶惰的人，決定這樣做，你就會像瓊斯在很久以前（我想是在 20 世紀末）發明微積分時所做的那樣。

And so this is a vector.

是以，這是一個矢量。

We can also write this as a column vector with just e-naught x and e-naught y.

我們也可以將其寫成列向量，其中只有 e-naught x 和 e-naught y。

And this is kind of cute because we have a three-dimensional problem, but because we assumed the direction of propagation, we only really have two degrees of freedom.

這有點可愛，因為我們有一個三維問題，但由於我們假定了傳播方向，我們實際上只有兩個自由度。

We only, our electric field can only be x and y.

我們的電場只能是 x 和 y。

So if we know that it's a traveling wave, we know its frequency, we can fully capture all of the information just in this vector, which is really cool.

是以，如果我們知道這是一個行波，我們知道它的頻率，我們就能完全捕捉到這個矢量中的所有資訊，這真的很酷。

You might also be wondering, well why is the electric field amplitude even important here?

你可能還會問，為什麼電場振幅在這裡如此重要？

Because all we care about is how much is contributed, or how much of the electric field is in the x direction, and how much of the electric field is in the y direction.

因為我們關心的只是貢獻了多少，或者說有多少電場在 x 方向，有多少電場在 y 方向。

So this is sort of the super compactness of the electric field.

這就是電場的超緊湊性。

Say it's pointing only in the x direction, so its equation, its mathematical description would be x-hat e-naught e to the j omega t minus kz.

假設它只指向 x 方向，那麼它的方程，它的數學描述就是 x-hat e-naught e 到 j omega t 減 kz。

The Jones vector for that, or the vector that we would we would use if we weren't being super lazy, was just e-naught zero.

瓊斯的矢量，或者說，如果我們不是超級懶惰的話，我們會使用的矢量就是 e-naught zero。

So it's just got some component in the x direction.

是以，它只是在 X 方向上有一些分量。

But if we were to divide this by the total amplitude, so let's divide this by 1 over e-naught, we would get the Jones vector, which is just 1, 0.

但如果我們將其除以總振幅，讓我們將其除以 1，就會得到瓊斯矢量，即 1，0。

And this is a normalized vector, so it's got length 1, and it just tells us in what direction the electric field is pointing.

這是一個歸一化矢量，所以長度為 1，它只是告訴我們電場指向的方向。

So if we had an electric field pointing in the y direction, or a traveling wave in the y direction, our Jones vector would be 0, 1.

是以，如果我們有一個指向 y 方向的電場，或者有一個 y 方向的行波，我們的瓊斯矢量就是 0，1。

If it were pointing half in the x direction, half in the y direction, 45 degrees, then this would be 1 over root 2, 1, 1.

如果它一半指向 x 方向，一半指向 y 方向，即 45 度，那麼這將是 1 超過根 2、1、1。

And that's just, this is just a normalized vector pointing at 45 degrees in x and y.

這只是一個歸一化矢量，在 x 和 y 方向上指向 45 度。

Now all of these vectors are what's known as linearly polarized, and that just means that the electric field is pointing in some direction.

現在，所有這些矢量都被稱為線性極化，這只是意味著電場指向某個方向。

And if you were to advance the plane wave, it would still be pointing in the same direction.

如果你將平面波向前推進，它仍然會指向同一個方向。

So take our x example from before.

就拿我們之前的 X 例子來說吧。

If we take one snapshot of the plane wave, which is pointing in, going in this direction, our electric field is pointing in the x-hat direction, so x, y, z.

如果我們對指向這個方向的平面波進行一次快照，我們的電場就會指向 x-hat 方向，即 x、y、z。

And if we advance the plane wave, so we take it some time further, the electric field is in general going to have a different amplitude, because maybe this was at the maximum of the plane wave, so maybe this is slightly lower.

如果我們將平面波向前推進一步，電場的振幅一般會有所不同，因為這可能是平面波的最大值，所以可能會略低一些。

So let's say, where's, let's draw our E0 vector pointing downward now.

是以，讓我們畫出指向下方的 E0 向量。

So this is E0, but it's still pointing in the x direction.

所以這是 E0，但它仍然指向 X 方向。

This is y, this is x.

這是 y，這是 x。

It's still pointing in the x direction.

它仍然指向 X 方向。

And so as the field, as we go in the z direction, the electric field stays pointing in the same direction.

是以，當電場沿 Z 軸方向移動時，電場的方向保持不變。

So if we just draw what it looks like over all of the space between one plane that I've outlined and the other, this is what the electric field will trace out on its path in the z direction.

是以，如果我們在我所勾畫的一個平面和另一個平面之間的所有空間內畫出它的樣子，這就是電場在 Z 方向上的軌跡。

This is what it'll look like if we freeze it in time, so at t equals 0.

如果我們把它定格在 t 等於 0 的時間點上，它就會是這個樣子。

And if we let it go in time, then it'll move forward, so the wave, the phase front will advance and the wave will go, all the electric fields will shift in space.

如果我們讓它在時間上前進，那麼它就會向前移動，所以波、相位前沿會前進，波就會前進，所有的電場都會在空間中移動。

And so this is linearly polarized light.

這就是線性偏振光。

And it doesn't have to be pointing just in the x direction to be linearly polarized.

而且，線性偏振不一定只指向 X 方向。

It could be pointing in the y direction, it could be pointing somewhere in between, it could be pointing anywhere in this two-dimensional plane.

它可能指向 Y 軸方向，也可能指向兩者之間的某處，還可能指向這個二維平面內的任何地方。

The thing that makes it linearly polarized is that it stays the same direction as it advances.

線性偏振的特點是，它在前進時保持同一方向。

And you might be asking, that seems really weird.

你可能會問，這似乎真的很奇怪。

Why would it be able to change directions as it goes forward?

為什麼它能在前進過程中改變方向？

And that's going to be the subject of the next video, and circularly polarized light, which is where things get really interesting and where Jones vectors start to become really, really valuable.

這將是下一個視頻的主題，還有圓偏振光，這才是真正有趣的地方，也是瓊斯矢量開始變得非常非常有價值的地方。

So I hope you enjoyed the video.

希望你們喜歡這段視頻。

If you did, please give it a like down below and subscribe to my channel.

如果你看了，請在下面點個贊，並訂閱我的頻道。

Also if you have any questions or comments, please feel free to post those down below and I'll try to get back to you as soon as I can.

此外，如果您有任何問題或意見，請隨時在下面留言，我會盡快給您回覆。

And thanks for watching.

感謝您的收看。

I'll see you next time.

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