Name: 円偏光の説明 (Circularly Polarized Light Explained)
Uploaded: 2024-09-21T10:25:09.000Z
Duration: 10 min 50 s

In this video we're going to talk about circularly polarized light and its Jones vector and the Jones vector that accompanies it.

このビデオでは、円偏光とそのジョーンズ・ベクトル、それに付随するジョーンズ・ベクトルについてお話しします。

基礎受動態

様態の副詞

Well, in the last video we talked about linearly polarized light.

さて、前回のビデオでは直線偏光についてお話しました。

In other words, the direction of the electric field doesn't change as you go throughout space.

言い換えれば、電場の方向は宇宙空間を移動しても変わらない。

So let's say that this is the propagation direction of our plane wave and then we've got our coordinate axis.

つまり、これが平面波の伝播方向で、座標軸があるとしよう。

A linearly polarized light, if it's polarized initially in X, so it's got some X component, that X component will change throughout space.

直線偏光された光は、最初はXに偏光しているので、あるX成分を持っているが、そのX成分は空間全体で変化する。

So it'll get bigger and smaller and bigger and smaller because this is a sinusoidal wave, this is a plane wave.

これは正弦波で、これは平面波だから、どんどん大きく小さくなる。

So we expect the magnitude to go up and down as we go throughout space.

だから、宇宙空間を移動するにつれて、マグニチュードが上がったり下がったりすることが予想される。

しかし、方向性は変わらない。

Now, circularly polarized light doesn't behave this way.

さて、円偏光はこのような振る舞いをしない。

It actually changes the direction that this arrow is pointing as you go throughout space.

宇宙空間を進むにつれて、この矢印が指し示す方向が実際に変わる。

Now, you might ask, how on earth is that possible?

一体どうすればそんなことが可能なのか？

And the answer has to do with the wave nature of light and the fact that we can... add two polarized waves together.

そしてその答えは、光の波動性と、2つの偏波を足し合わせることができるという事実に関係している。

So let's say we have this polarized wave, which is polarized in the X direction.

そこで、X方向に偏光した偏波があるとしよう。

Let's say we also want to add a wave that's polarized in the Y direction.

Y方向に偏光した波も加えたいとしよう。

But let's say that instead of having the maximum of the Y polarization match up with the maximum of the X polarization, so that would look something like this, instead of doing that, why don't we offset it?

しかし、Y偏光の最大値とX偏光の最大値を一致させる代わりに、このようにするとしましょう。

So let's say that the maximum of our Y polarization happens when our X polarization is equal to zero.

つまり、Y偏光の最大値は、X偏光がゼロに等しいときに起こるとしよう。

So this is the maximum of our Y polarization.

And similarly, it decays to zero, or the sine wave goes to zero when the X polarization is at a maximum.

そして同様に、X偏波が最大になるとゼロに減衰する、つまり正弦波がゼロになる。

So if we were to trace that out in the Y axis, it would look something like this.

つまり、Y軸でなぞると次のようになる。

So we've got arrows pointing along the Y axis, and then our arrows start pointing backwards.

つまり、Y軸に沿って矢印が指し示され、そして矢印は後方を指し示し始める。

And most importantly, at this point, so let's call this Z equals zero, we have an entirely X polarized wave, or what looks like purely X polarized light.

そして最も重要なことは、この時点で、つまりZはゼロに等しいとすると、完全にX偏光した波、つまり純粋にX偏光した光のように見えるものがある。

So if we were to add up these two polarizations, this is X and this is Y, then initially, our wave is just polarized in X.

つまり、この2つの偏光を足し合わせると、これがXで、これがYである。

And I'm going to draw this total in a different color, let's do blue.

そして、この合計を別の色で描こうと思う。

So initially, this is our total electric field, it's polarized in X.

つまり、最初はこれが全電界で、Xに分極している。

But as we go along some distance, our total field ends up being polarized in Y.

しかし、ある距離を進むにつれて、私たちの全フィールドはYに偏ってしまう。

And then if we go some distance more, it's polarized in negative X, and then negative Y, and then X, and so on and so on.

そして、さらに少し進むと、マイナスXに偏り、さらにマイナスYに偏り、さらにXに偏り......といった具合だ。

And if we were to trace out what this looks like, this would actually be, if they have the same magnitude, a circle.

そして、これがどのように見えるかをトレースしてみると、同じ大きさであれば、これは実際に円になる。

So I've done my best to draw the, this is what it would look like, where we've got, initially we're in front of the axis, then we go behind the axis, and we sort of curve around, we continually curve around this axis.

最初は軸の前にいて、次に軸の後ろに回り、この軸の周りをカーブし続ける。

And this is what's known as right-hand circularly polarized light.

これが右旋円偏光と呼ばれるものだ。

So if you stuck your thumb out in this direction, and you were to curve your fingers of your right hand, you'd get that they travel in the same direction as this wave curves around the axis.

親指をこの方向に突き出し、右手の指をカーブさせると、この波が軸を中心にカーブするのと同じ方向に進むことがわかるだろう。

And so here's a visualization more of what this looks like, this curving light, or this curving polarization.

そして、これがどのように見えるか、この湾曲した光、あるいは湾曲した偏光がどのように見えるか、さらに視覚化したものがここにある。

Notice it doesn't change its magnitude, so it stays constant in magnitude, it doesn't go from zero to one, it just changes its direction.

つまり、大きさは一定で、ゼロから1になることはなく、方向が変わるだけなのだ。

So you can kind of see how it curves to the right, or sort of a right-handed helix.

右にカーブしているのがわかるだろう。

And if we advance the time, we can watch this wave propagate as you increase the time.

時間を進めれば、この波が伝播していくのを見ることができる。

So we can see it's propagating down the Z-axis, and the electric field is sort of curving out this circle as we propagate it.

つまり、Z軸を伝搬し、電場がこの円を描くように伝搬しているのがわかる。

And if we freeze the helix in time, we can see that it's a right-handed helix, so it follows the curve of our right-handed fingers as we point our thumb in the direction of propagation.

そして、らせんを時間的に凍結させると、右巻きのらせんであることがわかる。つまり、親指を伝搬方向に向けると、右利きの指のカーブに沿う。

So that's fine and good, but how do we mathematically model this?

それはそれでいいのだが、これを数学的にモデル化するにはどうしたらいいのだろう？

And the answer is going to be with our Jones vectors.

そして、その答えはジョーンズ・ベクトルにある。

So how do we represent a wave that's shifted, that's delayed with respect to the other wave by a quarter of the wavelength?

では、波長の4分の1だけ、もう一方の波に対して遅れた、シフトした波をどう表現すればいいのだろうか？

So this distance would be lambda over four, this distance here, whoop, what the fuck?

つまり、この距離はラムダの4倍、この距離はフープ、なんだこりゃ？

So this distance is lambda over four, our wavelength over four.

つまり、この距離はラムダの4倍、つまり我々の波長の4倍ということになる。

This total distance, a full half period, would be lambda over two, and this would be our full wavelength, so we've completed a full cycle.

この合計距離、つまり半周期はλが2倍となり、これが全波長となる。

So when we write down our X-polarized wave, we can write out a formula for that, we know what it is, and then we write down our Y-polarized wave, we need to delay this wave by lambda over four.

X偏波を書き出すと、その式が書けるので、それが何であるかはわかっている。

So we need to set Z to Z minus lambda over four.

そこで、Zから4以上のラムダを引いた値をZとする必要がある。

And let's see what happens when we do that.

そうしたらどうなるか見てみよう。

So our X-polarized wave, we can just write as, let's say, some magnitude E-naught in the X direction, e to the j omega t minus kz, and now our Y-polarized wave, wow, that looks like an X, that's actually a Y, our Y-polarized wave has the same magnitude, and we're assuming that, so this is what you need to assume for circularly polarized light, e to the j omega t minus k, and now we said we need to replace Z with Z minus lambda over four, so Z minus lambda over four, and we can rewrite that, so E-naught Y e to the j omega t minus kz, plus, so what's k times lambda over four?

つまり、X偏光波は、X方向にある大きさE-naughtとして書くことができます、これは円偏光に対して仮定する必要があるもので、eからjオメガtマイナスk、そして今、ZをZから4倍以上のラムダを引いたものに置き換える必要があると言いました、Zから4倍以上のラムダを引いたものです、そして、これを書き直すことができます、E-naught Y eからjオメガtマイナスkz、プラス、では、4倍以上のラムダのk倍は？

Well, k is two pi over lambda times lambda over four, lambdas cancel, and we get pi over two, so plus pi over two.

さて、kは2πオーバーラムダ×4πオーバーラムダで、ラムダはキャンセルされ、2πオーバーラムダとなり、プラス2πオーバーラムダとなる。

 And now the really clever part, you might notice we can just take this out of our wave, so we can factor this out front, and we'll have E-naught e to the j pi over two times our traveling wave that we had before, omega t minus kz, and this remembers the part that we're not interested in, so at least not after this point, at least not after this point, because we know that it's a traveling wave, we know that it's traveling in the Z direction, we only care about the components, so if we just write down the X and the Y components into our lazy man's vector, which is almost a Jones vector, it's just E-naught X plus E-naught e to the j pi over two Y, now we could write this in column vector form, and factor out the E-naught, and in that case we'll just have one, and e to the j pi over two, which is the imaginary number I, and so for the Jones vector, we just drop the amplitude, and we normalize the vector, so you need to divide by the square root of two, and this is our Jones vector for right-hand polarized light, and this contains all the information that we had before, but it's super, super simple, so rather than writing down this plus this, we just write down this super simple column vector, and this will actually allow us to do some really cool things, now I should make one note on conventions, so I've been using the convention that a traveling wave is represented as e to the j omega t minus kz, there's some people that do e to the j kz minus omega t, and in this case right-hand polarized light would be one minus I, there are also different conventions on whether this is right-hand polarized, or left-hand polarized, and the convention that I follow, and what seems to be the most common one, is the handedness convention, so stick your right hand out, curve your fingers, if your fingers follow the path of the light, then the light is right-hand polarized, if it follows the path of your left hand, it's left-hand polarized, so if instead we had shifted the wave, instead of shifting the y polarized wave in this direction, we could have also shifted it in this direction, and that would have given us left-hand polarized light, which the Jones vector, we could just represent as one minus I, and so these are two circularly polarized forms of light, and they correspond to helices, that are either right or left-handed, and are propagating along the z direction, but you might wonder, what if we hadn't shifted by a perfect lambda over four, what if we had shifted by a little bit less, or a little bit more, and that's a perfectly reasonable question, and we'll answer that in the next video, on elliptically polarized light, and you'll get the same exact thing, if you instead had different amplitudes, in front of this x and y, but we'll go over that in the future video, so I hope you enjoyed this one, if you did please give it a like down below, and subscribe to my channel, 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, and thanks for watching, I'll see you next time.

そして今、本当に賢い部分は、波からこれを取り除くことができることにお気づきかもしれません。つまり、これを因数分解して、E-naught eを、前にあった進行波の2倍以上のjπにすることができます、Z方向に進行していることは分かっている。私たちが気にするのは成分だけだ。だから、X成分とY成分を、ほとんどジョーンズ・ベクトルである怠け者のベクトルに書き下せば、E-naught XとE-naught e to the j pi over two Yだけだ。さて、これを列ベクトル形式で書いて、E-naughtを因