We're going to be talking about the reflection and The refraction of waves when they hit an interface otherwise known as Fresnel equations if you're following along in my book What we're covering is in chapter 8 and since we're going to cover a whole chapter I'm just going to give you the highlights and not give you the derivation Although the derivations of Fresnel's equations is a really good way to understand if you you really know waves or not So if you remember in our picture We essentially had an incoming electric field and that electric field would turn on and off and it would essentially Jiggle the clouds of the atom the electron clouds of the atoms up and down and we saw that this can lead both to change in the phase constant with distance the index of refraction or the relative permittivity of the material and also some absorption or some loss and If you have a material where you have some absorption and I'm going to say it's three here and a phase constant Which is three and this is the equation that describes our plane wave going into the material and I'll note that the plane wave is Here at normal incidence that essentially in the general case You're going to get a change of the wavelength of the material that arises from a change in the velocity phase velocity and you're also going to get some kind of exponential type of attenuation as the wave goes down into the material and You can calculate the absorption coefficient alpha Which is the attenuation of the wave with distance as well as the spatial phase change of the wave Using some expressions as we saw before and these are the most complicated expressions.

我們要討論的是波碰到界面時的反射和折射 也就是所謂的菲涅爾方程 如果你在看我的書的話 我們要討論的是第8章的內容如果你還記得我們的圖片 我們基本上有一個進入的電場 電場的開啟和關閉 基本上會使原子雲 原子的電子雲上下襬動如果你有一種材料，它有一定的吸收，我假設它的吸收率是3，相位常數是3，這就是描述我們的平面波進入材料的方程式。你可以計算吸收係數α，即波隨距離的衰減以及波的空間相位變化。

You can simplify them in particular cases So that's where we were.

你可以在特殊情況下簡化它們。

Where are we going?

我們要去哪裡？

What we want to do today is talk about essentially a more general case where essentially you have the incoming Wave with an incident K vector.

我們今天要討論的是一種更普遍的情況，即入射波帶有入射 K 向量。

We'll call it case of I for instance for incident coming in some angle And as you will see we're going to measure the angles normal to the surface because that's the only way you can accurately measure things Related to a surface is to the normal and that this incident wave is going to give rise to both the reflected wave That's going back out in that way and a transmitted wave that's going to go into the medium this way So we're we're really expanding our picture here To think about how these oscillations of the atom inside the material Aren't just caused by the incident wave but affect the transmitted wave as well as give rise to a wave reflected from the surface So rather than go into a full derivation.

我們把它稱作 "I "的情況 例如，入射波從某個角度射入 你會看到，我們要測量的是表面的法線角度 因為只有法線角度才能準確測量與表面相關的東西是以，我們在這裡真正擴展了我們的圖景 去思考材料內部原子的振盪不僅僅是由入射波引起的 而是會影響透射波以及產生從表面反射回來的波

Let me just cover really the basics before we get into some cases So we're interested in waves incident on surfaces and essentially what we're going to do is we're going to assume that there's some incoming wave With a K vector case of I that's the direction It's coming in and the incident angle is defined relative to the surface normal and that incident angle is theta sub I right here And so that angle is right here the angle between the dashed line Which is normal to the surface and the direction of the incoming K vector?

是以，我們對入射到表面的波浪很感興趣，我們要做的就是假設有一個入射波，它的K矢量為I，即入射方向，入射角相對於表面法線，入射角為這裡的θ sub I，所以這個角度就是這裡的虛線與入射K矢量方向之間的夾角，也就是表面法線與入射K矢量方向之間的夾角。

There's also something called the law of reflection which says the reflected angle Theta sub R is also equal To theta sub I so here we have theta sub R the angle of reflection and this angle theta sub R And this angle theta sub I are equal to one another in other words for most flat surfaces The angle of reflection doesn't go off in a different direction than the angle of incidence Another thing that's going to make our lives a lot easier is to characterize the material by the index of refraction in so essentially we Have some incident index of refraction That's the index of refraction of the medium the incident wave is going through over on this side And we also we are also going to talk about the transmitted index of refraction that's the index of refraction of this gray medium over here that the transmitted wave goes into and The index of refraction essentially in a general case is given by the square root of epsilon R and mu sub R of course most Materials are non-magnetic so mu sub R is equal to 1 so in most cases you can simply say the index of refraction of the square root of the relative permittivity and Of course since the relative permittivity in free space is 1 the index of refraction of free space is 1 as well However, we've also seen that in lossy materials you have complex permittivity and in the general case we can essentially represent the index of refraction this way is a square root of The complex epsilon divided by epsilon naught and essentially then we can basically tie that back to an alpha and J beta term and To be perfectly correct.

還有一個叫做反射定律的東西，說的是反射角 Theta sub R 也等於 Theta sub I，所以這裡的 Theta sub R 是反射角，這個角度 Theta sub R 和這個角度 Theta sub I 是相等的，換句話說，對於大多數平面來說，反射角的方向不會與入射角的方向不同。這就是入射波經過的介質的折射率，我們還要討論透射折射率，即透射波進入的灰色介質的折射率。當然，由於自由空間的相對介電常數是 1，是以自由空間的折射率也是 1、在一般情況下，我們可以用複數ε除以εnaught的平方根來表示

This is not an equal sign this essentially says the in the complex index of refraction Essentially is related to alpha and beta, but is not equal to alpha and beta So let me just clarify that since that's a mistake in the slide Now if the material is very lossy or if it's something like a metal then then most of the incident radiation is going to be Reflected, but if it's a dielectric material then the transmitted wave which essentially has a k vector K sub T here is coming in a different angle So theta sub T is not necessarily equal to theta sub I or theta sub R the transmitted wave goes in a different direction than the incident wave and the reflected wave and We can calculate the angle of that transmitted wave using something called Snell's law, which is given right here that the the Index of refraction in I of the wave coming in times the sine of the incident angle incident angle is equal to the index of refraction for the material the waves getting transmitted into times the angle of the Transmitted wave so if we know in sub I in sub T and the incident angle theta we can do a simple algebraic manipulation to calculate the angle of transmission And so those really are the basics of waves and incident on surfaces if you understand these basics the rest of it Although there are some long equations follows pretty straightforwardly now the first complication we run across is that the Reflection and transmission that we see depend on the direction of the electric field And so the first direction of electric field we're going to talk about is called s or perpendicular Polarization and the way we we define this is we think as we've talked about before of a plane of incidence and this plane of incidence is the plane that contains both the incident and the In the plane of incidence the normal to the surface lies in the plane of incidence and if the electric field is sticking up out Of the plane of incidence so there's essentially a 90-degree angle Between the electric field and the plane of incidence then we call it s or perpendicular polarization where perpendicular means It's perpendicular to the plane of incidence Now if the electric field is perpendicular to the plane of incidence We can calculate the magnitude of the reflected field and the transmitted field given the magnitude of the incident field so essentially this incident field is going to come in like this some of its going to bounce off and there will also be an electric field pointing in the same direction that we're going to call e Sub r or the reflected field and the relationship between the incident field e sub i Right here and the reflected field e sub r right here is The reflected field and notice we have this little perpendicular sign that says it's perpendicular to the plane of incidence is equal to the coefficient r perpendicular times the the incident field and Essentially without deriving it the reflection coefficient for perpendicularly or s polarized radiation is given by this it depends both on the incident index of refraction the index of refraction of the material that the waves reflecting from are getting transmitted into and the cosine of both the incident angle and The angle the waves being transmitted at and remember Snell's law says in sub i sine of theta sub i is Equal to the index of refraction of the material that the waves getting transmitted into Times the sine of that angle and so we calculate theta sub t from this equation right here So overall this gives the reflected field Once we know all of these things similarly, there's going to be an electric field pointing in the same direction That's transmitted into the material We can call it e sub t here and the relationship between the transmitted field and the incident field is given by a very similar expression, but we use the term t perpendicular for the We find it depends on the incident and transmitted indices of refraction the incident and transmitted angles And if we plug it into this equation, we'll know what fraction of the electric field gets transmitted in other words the ratio Between the incident field the magnitude of the incident field on the transmitted field and because they're vectors the vectors are going to point in the same direction Now I'm using a lot of Lines here, but we have to remember that these are actually waves coming in as shown in the bottom figure down here So they oscillate up and down and they're plane waves So we assume that everywhere on these planes that are represented by these little green squares The electric field is a vector field the vectors all point in the same direction But that direction can vary with time and space as we saw in previous talks And again, the angles are just given by the K vectors K incident K transmitted K reflected we have exactly the same situation or very close to the same situation if the Notice that instead of the electric field pointing up and down like before Perpendicular to the plane of incidence now the electric field lies in the plane of incidence and we call this P or parallel polarization Because the electric field is in the plane of incidence.

這不是一個等號，本質上說的是復折射率，本質上與α和β有關，但不等於α和β，所以讓我澄清一下，因為這是幻燈片中的一個錯誤、我們可以利用斯涅耳定律來計算透射波的角度、是以，如果我們知道子 I 子 T 和入射角度 theta，我們就可以通過簡單的代數運算來計算透射角。所以我們要討論的第一個電場方向叫做垂直極化，我們定義垂直極化的方法就是我們之前討論過的入射平面，入射平面是包含入射和透射的平面，表面的法線位於入射平面內。如果電場垂直於入射面，我們就可以根據入射場的大小計算出反射場和透射場的大小。入射電場會像這樣射

It's parallel to the plane of incidence Again exactly the same thing happens.

它與入射面平行，又會發生完全相同的情況。

We have a reflected electric field also lying in the plane of incidence e sub R We have a transmitted electric field We're going to call e sub T and the relationship between the incident field and the reflected field is given by those sets of equations The relationship between the incident field and the transmitted field is given by those sets of equations again The fields point in pretty much the same direction The electric fields in both cases are parallel to the plane of incidence Except the magnitude of the field the amount that gets through varies depending on the reflection coefficient And the transmission coefficient again very very similar the equations are slightly different but you do the calculation once you know the the Index of refraction on either side of the material the incident angle and from Snell's law you calculate the transmitted angle And again, let's stress is shown in the bottom that this is a wave coming in It's not just a line although we can represent it as a line and that the wave essentially maps an electric field to every point in space Given by planes that are perpendicular to the propagation direction of the wave So what you'll see in a lot of books or figures that look like this for s polarization We have the electric field sticking up out of the screen right at you for perpendicular Or excuse me for parallel or p polarization.

我們有一個反射電場，也位於入射面上 e sub R 我們有一個透射電場，我們稱之為 e sub T 入射電場和反射電場之間的關係由這兩組方程給出 入射電場和透射電場之間的關係也由這兩組方程給出這兩種情況下的電場都平行於入射面 除了場的大小外，透過的量取決於反射係數和透射係數 同樣非常相似，公式略有不同，但只要知道材料兩側的折射率和入射角，就可以進行計算，並根據斯涅耳定律計算出透射角、讓我們強調下圖中顯示的，這是一個波，雖然我們可以將其表示為一條線，但它並不只是一條線，而且從本質上講，波將電場映射到空間中的

We have the electric field in the plane of the screen We have essentially a distance between peaks We're going to call the wavelength when we go into a material the wavelength changes because the phase changes when we go into the material and Essentially through Snell's law we can relate the incident angle and the transmitted angle We know the incident angle is equal to the reflected angle and we can calculate the magnitudes of e sub i well If we're given e sub i and we can calculate the magnitudes of e sub t and e sub r through the reflection and transmission Coefficients for either the perpendicular polarization case or the parallel polarization case quick summary And these are the types of figures that you're going to see in most of your textbooks So let's consider the gray plane here in this figure being the plane of incidence What happens if we have an electric field that's not in the plane of incidence in other words this green line that we're going to Doesn't stick in either the s-plane perpendicular to the plane of incidence or the p-plane parallel to the plane of incidence This is pretty straightforward.

我們有螢幕平面上的電場，我們有峰值之間的距離，我們稱之為波長，當我們進入材料中時，波長會發生變化，因為我們進入材料中時，相位會發生變化，基本上，通過斯涅爾定律，我們可以將入射角和透射角聯繫起來，我們知道入射角等於反射角，我們可以計算出 e sub i 的大小。我們可以通過垂直極化或平行極化情況下的反射係數和透射係數來計算 e sub t 和 e sub r 的大小。平面或平行於入射平面的 p 平面上 這很簡單。

This is just like we worked with the parallelization essentially We're going to break it into two components We're going to say there is one component that is s polarized or perpendicular to the plane of incidence We're going to add a second component.

我們要把它抽成兩個部分，一個是偏振的，另一個是垂直於入射面的。

That's parallel polarized parallel to the plane of incidence or the electric field points and Simply by doing the summation of these two components.

這就是平行於入射面或電場點的平行極化，簡單來說就是對這兩個分量進行求和。

We can apply the Fresnel equations for the reflection or transmission for each component separately and use superposition to sum up and get the overall electric field Quick message here if you are given a problem where you have an electric field that's not polarized Perpendicular or parallel to the plane of incidence use superposition to break it up Essentially if you know this angle then the parallel component is going to be the cosine of this angle.

我們可以分別應用菲涅爾方程來計算每個分量的反射或透射，然後使用疊加法求和，得到整體電場。 如果您遇到的問題是電場不極化，垂直或平行於入射面，請使用疊加法來分解電場。

Let's call it phi the Perpendicular component is going to be proportional to the sine of phi through simple sine of phi through simple geometry and Essentially we just do our calculation twice one for parallel one for perpendicular Sum it up at the end to get the overall field superposition works here as well So I've taken you on kind of a whirlwind tour of Fresnel's equations Let's actually stop for a minute use some real numbers put in some Representative values and try to figure out what we're doing here What I'm going to assume is that I've got essentially a block of material The index of refraction on this side is 1 so we assume the incident wave is going through free space the index of refraction e sub in sub t is 2 in the material and I just chose those numbers I could have chosen any set of numbers But these are the ones I chose so our incident wave is going to come in here at some angle theta sub I to the normal and We want to know how strong or what the magnitude of the reflected and transmitted fields are In essence, what I've done is I went into the computer program I like to use for plotting called MATLAB and I've essentially plotted the equations for the reflection coefficient and the transmission coefficient as a function of incidence angle theta I and so theta I is given in degrees right here and Let's first look at the electric field that's reflected So if the electric field is P polarized, so our electric field vector points in that direction right there Then the green line essentially represents the strength of the electric field that's reflected So you can see if you come in at normal incidence in a material Whose indices of refraction are given by one on the incident wave and two on the transmitted side about 0.35 of the electric field or 35 percent is going to be reflected if the incidence angle is zero or the beams coming in straight On in that direction, or if you've come out to about a 30 degree angle that hasn't dropped very much But by the time you get to about 50 degrees Only about maybe 0.16 of the incident field 16 percent is reflected And as you drop down to 90 degrees, you can see that pretty much all the light So I've taken you kind of on a whirlwind tour of Fresnel's equation So let's stop for a minute and essentially do the calculations let's actually calculate the reflection and transmission coefficients and do it for a Set of materials where essentially the incident wave is coming in with an index of refraction 1 So we're in free space over on this side and the index of refraction of the transmitted wave for the index of refraction the material the waves going into is 2 and Essentially if we do that we can use Fresnel's equations given the incident electric field to calculate Relatively how strong the reflected and transmitted electric fields are so let's first look at the reflected electric field If we do this calculation and essentially plug in theta sub I and I've represented the angle of incidence here in degrees So this is a 30 degree angle of incidence a 70 degree angle of incidence a zero degree angle of incidence down here is Essentially going to be when the wave is coming in at normal incidence because the angle between the incident k vector and the surface normal is Zero, there's no difference between them for normal incidence.

垂直分量將與 phi 的正弦成正比，通過簡單的幾何原理，我們只需計算兩次，一次是平行分量，一次是垂直分量。這邊的折射率是 1 所以我們假設入射波穿過自由空間 材料的折射率 e sub in sub t 是 2 我選擇了這些數字 我可以選擇任何一組數字 但這是我選擇的數字我所做的就是進入我常用的計算機程序 MATLAB，繪製出反射係數和透射係數與入射角θ I 的函數關係式、那麼我們的電場矢量就指向這個方向，然後綠線基本上代表了反射電場的強度。如果入射角度為零，或者光束直射過來，或者入射角度為 30 度，那麼

You can see that for the electric field.

你可以看到電場的情況。

That's polarized S or perpendicular to the plane of the electric field in other words We're talking about electric fields that are pointing up in this direction in this case What you're going to see is that for normal incidence?

換句話說，我們所說的電場是指向上方的電場。

You're going to get a reflection coefficient of about minus point three five now What does that minus sign mean it simply means there's a hundred and eighty degree phase shift in the electric field so s or?

這個負號是什麼意思？ 它簡單地表示電場有 180 度的相移。

perpendicular polarization Sees a phase shift when it hits the interface However as the angle gets larger and larger as theta incident becomes further and further away from the normal the reflection coefficient Gets larger and larger or the magnitude does it actually gets more and more negative?

然而，隨著角度越來越大，入射角離法線越來越遠，反射係數就會越來越大，或者說反射係數的大小實際上會越來越負？

Until down here at 90 degrees the reflection coefficient is close to 1 which essentially is saying That all the light is reflected What happens if we take a look at the p polarized case or the when the electric field lies?

直到下面 90 度的地方，反射係數接近 1，也就是說所有的光都被反射了。

parallel to the plane of incident In this case we go ahead and erase these electric fields and put our electric fields in that direction You can see that we get about a thirty five percent reflection at normal incidence that Drops off until at some angle we get zero reflection here And then eventually the reflection coefficient becomes negative and we start to get that hundred eighty degree phase shift again and so essentially what this curve is telling us is that it's telling us what the phase shift and The strength of the reflected electric field is as a function of the incident angle Similarly if we want to know the the strength of the transmitted field We can plot the transmission coefficient as a function of the angle of incidence Which I've done here for perpendicular s and parallel or p polarized and essentially if we take a look at that Maybe 65% of the wave gets through This makes a lot of sense since it was only 35% over on the other side But as the wave comes in at steeper and steeper angles eventually we drop to zero transmission and all the wave gets reflected It makes a lot of sense.

在這種情況下，我們繼續擦除這些電場，並將電場置於該方向。可以看到，在正常入射時，我們會得到大約百分之三十五的反射率，然後逐漸下降，直到某個角度時，我們會得到零反射率，最後反射係數變為負值，我們又開始得到一百八十度的相移。同樣，如果我們想知道透射電場的強度，我們可以繪製透射係數與入射角度的函數關係圖，我在這裡繪製的是垂直 s 極化和平行或 p 極化的透射係數圖。

So this is simply what happens if you plot those reflection and transmission coefficients All you do is you simply look up the value of the strength of the reflected and transmitted field And we know if the value is positive.

是以，這就是繪製反射和透射係數圖的簡單過程。

There's no phase shift if the value is negative There's a hundred and eighty degree phase shift and that's pretty much about it.

如果數值為負，則沒有相移。

We've got a few special cases.

我們有一些特殊情況。

We need to talk about One of these cases is something called total internal reflection It turns out that if you have a wave coming from a material that has a high index of refraction So in this case in sub I is greater than in sub T As long as your angles of incidence are small or the the direction of propagation is Is pretty much close to the normal then things come out as you would expect you get a reflected field and transmitted field But as theta I increases and gets bigger as you're going from the material with higher index to the material with lower index essentially the direction case of T of This vector is going to move that way as the incident angle increases and at some point it's going to lie along the surface This means that all the radiation that comes in this direction is going to get reflected You're going to get a hundred percent reflection of the radiation and there's going to be no propagating radiation that goes out and this is called total internal reflection for the obvious reason that the total amount of the radiation gets Reflected from the surface going from a material of higher index into lower index The place this is most commonly used as an optical fibers because this is what keeps the light inside fibers to go very very long distances and Essentially if you want to calculate what the angle is the incident angle is where total internal reflection Starts to occur you simply use Snell's law you basically set theta T is equal to 90 degrees or greater and Essentially you can find that critical angle Let's call it theta C there is given the by the equation the sine of the critical angle is the Insub T the transmitted divided by the incident indices of refraction the second case we've also mentioned very briefly, but that's called Brewster's angle, and that's essentially the point where the Reflection coefficient of the parallel or P polarized electric field component is equal to zero right here in this case You get no reflection whatsoever And so for theta I equal to Brewster angles and Brewster angles given by that equation right there where theta B is Brewster angle so for theta sub I equal Theta sub B.