All of you have looked at rainbows, but very few of you
have ever seen one. Looking at something is very different
from seeing it. And today I will make you
see the rainbow in a way that goes way
beyond the beauty that we can all experience, a way that you will always remember. And I would like to start asking you fifteen
perhaps simple questions about the rainbow. The first question then is would any one of you
remember if you see a bow whether the red color is outside or whether the-- the red color is inside? And then I wonder about the radius of the bow. If this is a bow in the sky,
something like this, here is the horizon,
it's clearly a perfect circle and so the perfect circle
has somewhere a center. And so that means
there must be a radius R. You can measure that radius in terms
of how many degrees and so what is roughly that radius. You've never measured it but is it ten degrees,
is it twenty, thirty, fifty, sixty? The length of the bow. Is there a difference, do you sometimes see
a very long bow, sometimes a very short one? What is the width of the bow? You see colors here. How wide is that strip of colors
in degrees? Perhaps some of you have noticed-- that there is a difference in light intensity
between inside the bow and outside the bow. Maybe you've never seen it
and if there is a difference, where is it brighter,
inside the bow or outside the bow? What time of the day would you see bows? Would you see rainbows in the north, east,
south or west? Is there perhaps a second bow in the sky? And if there is a second one, where should you look for the second bow? And if there is a second one
what is the color sequence of the second bow? Is the red on the outside or is the red
on the inside? And then you can ask the same question,
what would be the radius of the second bow? And what would be the width
of the second bow? All these first twelve questions in principle you should have been able to
answer if you really have seen a rainbow. The last three is more difficult. The question is are the bows polarized? In what direction are they polarized? And are they weakly polarized
or are they strongly polarized? Who knows the answer to twelve questions,
to the first twelve questions? Who knows the answer to more than ten? Who knows the answer to nine? Eight? Seven? Six? Five? Four, do I see a hand at four?
Good for you. Five, four, three? Three, good, that's already good. Two? One? Who knows the answer to zero? Most of you, right? I haven't seen a lot of hands though. All right. So I've made my point. You've looked at rainbows
but you've really never seen them. And I'm going to make you see them today. What you see here on the blackboard
is one drop of water. I put the sun for simplicity at the horizon. Later I will put it a little bit higher
in the sky. Light from the sun hits this raindrop. I've only drawn one narrow beam which hits
the raindrop right there. And you see here the angle of incidence, which
with Snell's law we call theta one. I call it I here because it's nicer for me,
more descriptive, it means incidence angle. Right at that point A some of the light
will be reflected and some of the light will go
into the water. We call that refraction. And Snell's law will tell me
this angle r. Whatever goes in there reaches point B
where there is a transition back to air and so some of that light
will come out here and some of that light
will be reflected inside. And then when it reaches point C again there
is a transition from water to air. Some of that light will be reflected
inside the water. And some of it will come out. And as far as the geometry is concerned,
if this angle is R, then this angle is also R, this is also R and this is also R. And the angle here is i. That follows from Snell's law
and I'll leave you with that. Notice that the light came in like this
but it comes back like this. So the direction has changed
over the angle delta. And the angle delta is very easy to calculate
in terms of i and R. Delta is a hundred eighty degrees
plus two i minus four R. I want you to check that at home. The four r's come in here. One, two, three, four and the two i's come in here and there. If now I think of all possible narrow
beams of light that can strike this raindrop, one that would strike it here
would have an i of zero degrees. And then here would be ten degrees
and twenty degrees and thirty and forty. And the largest value for i is when the light
strikes here, would be ninety degrees. And so I can calculate for all these values of i,
which obviously all of them occur, sunlight strikes this raindrop
and all these angles for i are present. So I can calculate now for all these angles
of i what the value is for r and then I can calculate what delta is. R follows from Snell's law and delta follows
from this geometric relationship. And what you will find now
very much to your surprise, that there is a minimum value for delta
which is about hundred and thirty-eight degrees. That means this angle phi here
has a maximum value which is very roughly
about forty-two degrees. And I will show you some numbers. You can download this, by the way,
this is on the Web, under lecture supplements. Here all I have done I've taken i
to be from zero to ninety degrees, all these angles are possible,
with Snell's law, using an index of refraction
of one point three three six, that you see at the bottom,
I calculate r and then in the last column
using that relationship I calculate delta. And indeed you see that delta starts at
a hundred eighty degrees when i is zero. And then goes to a minimum of roughly a hundred
thirty-eight, after which it increases again. And this now is crucial, is key to
an understanding of the rainbow. Imagine now that I have
one drop of water here. And sunlight comes in at all angles of i,
not just at one, but all angles of i. Whatever you see here
has of course axial symmetry. It is a spherical drop. And the light comes in like this. So light can go this way
but it can also go this way. And it can also go this way and this way,
so there's complete axial symmetry, so this whole drawing you can rotate
about this line here. And everything holds then in axial symmetry. So therefore if phi maximum, if this angle phi
maximum is forty-two degrees, then the light that will go back
in the direction of the sun, the light that goes through the journey
A, B, C and then comes out of the raindrop,
that's all I'm talking about, now, I'm not talking about this light
that sneaks out here, it is this journey, A, refraction at A, reflection at B
and then coming out at C. That light comes out
in the form of a cone. And the half-- top angle of the cone
must be roughly forty-two degrees. And so I will go-- I'm going to draw that
cone for you. Like so. And like so. And you have to think of this
as a cone. It's completely symmetric, axial symmetric,
about this line. And this angle here then is roughly forty-two
degrees. No light can go here. Because that would mean that phi
would be larger than forty-two degrees and that's not allowed. Now comes something very important. The index of refraction of red light for water
is about one point three three one. And that translates into an angle for phi
max which I can calculate now-- it translates into an angle of phi max which is about forty-two point four degrees. But blue light has a slightly different index
of refraction, therefore has a slightly different angle
for phi max and the blue light has an index of refraction
of something like one point three four three. Notice I have blue light
and I don't use violet light. Violet is much harder to see
with our eyes. So I always refer to it as blue light. And that has a value for phi max which is
approximately forty point seven degrees. A different index of refraction means of course
that if you know i that r is slightly different, using Snell's law, so you get slightly different
values for r and so you get slightly different values
for delta, so you get a slightly different value
for delta minimum, you get a slightly different value
for phi max. What does this mean now? That means if you look at this cone of light
that goes back into the direction of the sun, that the outer edge,
the outer surface of that cone, which has the largest possible angle,
this angle is now forty-two point four degrees, must be red light, because blue light cannot come
out in that direction. Because the maximum angle for blue is forty
point seven, is not forty-two point four. So it's only red light that can come out when
the angle-- the half angle of the cone
is forty-two point four degrees. And the blue light is not going to make it
until this angle here-- I'll put it in here--
is forty point seven degrees. If you look inside the cone with an half top angle
of forty point seven degrees all colors can come back. All this is saying is that for red light phi
maximum is forty-two point four. It can also therefore be forty point seven. It can be thirty. It can be twenty. It can be ten. It can be zero. Light that comes in here at i equals zero
reaches point P, comes straight back. That's allowed. Phi would be zero then. If all the colors can make it back inside
this cone that I have given here a blue color, that would mean that your brains will tell
you that it is white light. Because you see red light, you see blue light,
you see green light, you see yellow light and so your brains will tell you
that that is white light. If I had a screen here with a small opening
to let the sunlight in and I asked you what would you see
on this screen having only one water drop, then you would see the intersection
of this cone of light with this screen. And it would look as follows. The outer edge,
the outer circle, which is the intersection
of the cone with your screen, would be red. And then there would be an inner portion
whereby all colors can come back. So that would be white light. And then as you go further in from the red,
until you reach that white portion, then the last color that will be added
is the blue. And you can already of course sense that all
the action about the rainbow occurs here. And here there is no light. It's going to be dark
because there is no way that phi can be larger than forty-two
point four degrees. And if light would appear on the screen
outside the red it means that phi would be larger than forty-two point four degrees and that's not allowed. So it's dark here. It's white light here. It's red light here. And as you go further in you will finally
see the other colors. And this really is now the key to the geometry
of the-- of the rainbow. I'm going to put you here,
so here you're standing and let the sun again
be near the horizon. It's always nice when you make a picture. The easy reference that you have. You're standing here. Sunlight is coming in in this direction. And there is rain here. If it's also raining here you won't see a
rainbow. Because the sun will not be able to hit these
raindrops. So it's essential that it's raining
in the direction away from the sun but that you can still see the sun. So here are these raindrops. All right. You're looking in this direction in the sky. You're looking up in the sky like so. And I pick here one raindrop
and only one but what I'm telling you holds for all the raindrops
in that direction. What will that raindrop do? That raindrop will produce
a cone of light which goes back
in the direction of the sun whereby the edge of the cone
is red light and this angle is forty-two degrees,
forty-two point four, whatever. What do you see? Nothing. Because there is no light that can come from
this raindrop in your direction. There is no light because that would mean
that phi is larger than forty-two degrees and that's not allowed. So you look high up in the sky, you will not see
any light coming back from that raindrop. And think of the whole thing
as axial symmetric, right? Not only there, but there and there
you will not see any light. Now I am looking say at some raindrops
which are here. I pick one. It holds for any one in that direction. And so now I draw a line to this point. This is what I'm looking. I look in this direction. I take this raindrop. I could have picked this raindrop. I could have picked this one. Would have made no difference. What is that raindrop doing? Well, that raindrop is throwing back at the sun
light in the form of a cone and the cone has this angle forty-two degrees,
forty-two degrees. What do you see now? Look. You're looking straight into that cone. You're nowhere near the edge. So you see white light. Because green light comes back at you,
red light comes back at you, everything comes back at you. So you will say, "Ha, I see white light." Not only there, low on the horizon,
but the whole thing is axial symmetric, also there and there. So you haven't seen a rainbow yet. But now suppose I ask you to look up
in the sky at a very specific angle and we'll make the angle
forty-two point four degrees. So you're looking at the sky in the direction--
hm... somewhere here, I'm not quite sure
that I have the angle just right. So I'm looking in the sky here. Pick one raindrop. But you could pick any one,
any other other one. There's nothing special
about that one. So the sunlight comes in like so. What is that raindrop doing? Well, it's throwing a cone of light
back to the sun. And it just so happens that you are only looking
at the outer surface of the cone where only red light can go because this is the
famous angle of forty-two point four degrees. And so you see red light. And since the whole problem
is axial symmetric, you would see red light if you look forty-two
point four degrees in this direction but also forty-two degrees in this direction, away from this direction to the sun. And so now you see how the bow is being formed
by zillions and zillions of small water drops which each of them in their own way contribute
to your rainbow. And we'll now put the sun a little higher
in the sky. So I'll put you here now. And let's say the sun is now like so. So, you look at your shadow
here on the ground. The sun is there. There's your shadow. Here's your shadow. Here's your head--
your shadow. And this is the reference line
that we're talking about. That is this line. That was this line. And so where would you see a rainbow? You have to look now forty-two degrees
away from that line. Forty-two degrees away from that line. That would be your red then and a little lower
would be your-- your blue. And so if I sketch in there the bow,
it would be sort of like this and then the blue would be on the inside
and here you would have white light and this would then be the red. It's always relative to this reference line. This would be roughly--
I call it forty-two degrees, but according to my calculation there,
it would be be forty-two point four. So whenever you know where the sun is,
you look at the shadow of your own head and you have to go forty-two point four degrees
away from that direction from your eye to the shadow. And so what you're seeing now is that
if the sun is low in the horizon then the bow will be high above the horizon
and when the sun is rising then the bow goes down and goes down
and goes down and by the time that the sun
is forty-two degrees above the horizon, you're not going to see a rainbow anymore
unless the water is right where you are. If the water is at a distance,
you won't see a rainbow. So the higher the sun in the sky, the less likely it is
that you will ever see a rainbow. Before I show you some--
some slides and before we're going to answer
some of the questions, uh, I want to mention to you that what I went
through here is refraction at A, reflection at B-- reflection at B
and coming out at C. You can go through the same geometry
and allow for one more reflection at C and then let the light come out,
so it would come out at point D and if you did that, you can convince yourself
that there is indeed a second rainbow and we call that the secondary. We call this the primary. And the secondary has a radius-- this one of course has a radius
of forty-two point four degrees for the red and the blue would have a smaller radius. The secondary has a radius for the red
of fifty point four degrees. So in the red, it's fifty point four degrees
and in the blue, it's larger. So the blue is outside fifty-two point five
degrees. Is that what it is? Fifty-three point five. The secondary bow is fainter
and it is also wider because you see this separation in terms of angle is larger than the separation there. That's only one point seven degrees
and this is, uh, more than three degrees. And so there is a secondary bow and the secondary bow is only about roughly
ten degrees above the primary. So if you see a primary at forty-two degree
angles away from your shadow there and there and there,
go another ten degrees and you should see a second bow. And you should see
the colors reversed. Red is there on the outside
and blue is on the inside. Red is on the outside
and blue is on the inside. Red is on the outside and blue is on the inside,
but here it is reversed. The secondary bow, the red is on the inside
and the blue is on the outside. So let's answer some questions now. We'll put the questions back up again
and you will see that you can now without any difficulty
already answer twelve questions. The first twelve. Red is outside. That is non-negotiable, right? That follows immediately from what
we'd talked about, but that's not an issue anymore. The radius is about forty-two degrees
and the length of the bow, well that depends on if the sun is high in the sky then the length will be very small because this whole s-- arc will then go down
and there will only be a little bit left. It could also be that it's only raining here
and it's not raining there. So depending upon where there is rain
and how high the sun is in the sky, that will determine the length of the bow. The width of the bow, you would think,
perhaps naively, that if you subtract the forty-two from the forty--
forty point seven from the forty two point four that that would give you
the width of the bow which would then be one point seven
degrees in angle. However, you would overlook then
that the sun is not a point, but that the sun in the sky has a dimension
of half a degree and each point of the sun, of course,
makes its own little rainbow, so you really have to add
roughly half a degree. So, the width of the bow is more like,
something like two degrees rather than the one point seven, maybe two--
two point two degrees. It's a little wider than what you would think. It has to do with the finite size of the sun. A comparison of the light inside and outside,
clearly inside the bow, there must be a lot of white light
which you don't see outside the bow-- big difference. You will see a slide shortly. The time of the day-- well the sun has to be
a little bit low, so you don't expect it at midday
and you want rain as well, so late afternoon,
early morning would be ideal and you can figure out uh
in what direction to look. If it's early morning, the sun rises in the east,
you would see the bows in the west and late in the afternoon
when the sun is in the west, you would see the bows
in the east. Yes there is a second bow. It is about ten degrees higher in the sky
than the primary. And the colors are reversed. The blue is outside and the red is inside. And the radius is about fifty-two degrees. And the width of the bow is again
the difference between these two numbers which you have to add about half a degree. So you would think on the basis of this
that it is more like three degrees and you have to add roughly the half a degree,
so it's about three-- three-and-a-half degrees. So let's now look at some slides. And the first slide that you will see is a--
is a drawing that I made. Which uh is meant to repeat some
of what I told you. So you see a person standing there
and here you see the direction to the sun. Your shadow would be this long. This would be your head of the shadow,
here would be your feet. And if there are water drops here
and if there's no interference between the sunlight
and the water drops then in this direction forty-two degrees
you would see this water drop red. And then the water drops which were here
you would see white light from those water drops and maybe from this one you would see
red as well as blue. And if you have water near your feet
there's no reason why you can't see the rainbow there either
but of course you would have to lead some garden hose to produce water there. The next slide then. This is a drawing made by the maestro himself,
Newton, who was the first to fully understand
the workings of the rainbow. You see here the primary bow
and for those of you who are sitting close you can see the direction of the sunlight
going in at A, reflects at B coming out at C. Which he calls E,
but that's a detail. And here you see then the secondary
whereby the light comes in, one reflection,
two reflections, and now it comes back at you
and that gives you the secondary. And so this is red and this is blue
whereas here this is blue and this is red. The next slide. So if the sun is high in the sky,
I've done this many times when I was uh watering my garden,
you get this for free, you might as well do this,
it's great fun, you're standing there,
gives you an immense feeling of power and it's completely clear,
there's not a -- not a cloud in the sky, there's no rain anywhere,
but you spray water around you and you see a beautiful rainbow
encircling your feet. And always what matters is this forty-two
degree angle relative to the direction
from the sun to you and this is where the shadow
of my head would be. Very easy to do and I would recommend
that you try that when you get a chance. The next slide is a painting
from the eighth century, Turkey, I see a hand here-- probably makes reference to the Bible, but I think it reads in the Bible
I do set my bow in the clouds, it's very nice and very dandy,
the fact that the colors are wrong, the sequence of the colors, that's a detail,
it's a nice picture, but red has to be on the outside
and blue has to be on the inside. Small detail. Next slide. Ah yeah. A-- a few years ago,
actually it's more than a few, when I was first lecturing 803
I knew I was going to talk about the rainbow, ideal for 803. And so I wanted to make some rainbows myself
in my backyard in Winchester. And so there is water
coming out of a water hose and the sun was behind me
and I took this picture and you see indeed all the ingredients
that we just discussed. Notice the red is on the outside
and the blue is on the inside. You see all this white light. That is that light that comes back
from the water drop but if you go here where the angle of phi
would be larger than forty-two degrees, which is not allowed,
you don't see any light coming back at you, so you look straight through
and see the forest without any white light. So you really see already here sky so to speak
is bright here and darker there. I needed help from my daughter. And the next slide will show you that. The poor darling was suffering badly. It was January, it was freezing cold,
she was crying. She was really crying and I felt guilty
but I said look, you know I really need these slides
for my 803 class, it'll only take you an hour or so
and she still remembers it, I had email exchange with her yesterday
and she said dad I was crying, it was an awful thing
what you did to me. But look, but look. You know, you got to do something
and when you're daughter of a scientist occasionally you have to suffer a little bit. And so she's holding here the--
the water hose and you see the same thing
red outside blue inside, and clearly the white light, you see this is that white light
that I discussed with you earlier. The next slide will show you then
a real rainb-- oh no, this is the um this is the artificial
one I made uh over my driveway, if you want to see the secondary,
you need a real dark background, because the secondary
is quite faint. And so that's why I did it
over my driveway and you see here the primary, red on the outside,
blue on the inside, and here you see the colors
reversed. You can also see perhaps
that it's a little wider. But it's always much s-- it's much fainter,
so it's-- it's hard to tell. And then the next slide is one that's a wonderful
slide made by Doug Johnson in New Mexico, Socorro, it is where the radio telescopes are,
the very large array, now look, red on the outside,
blue on the inside. The sky you got to admit is a hell of a lot
brighter here than it is there. And you may never have noticed that, you've--
you've looked at it but you've never seen it and then here
you see the secondary red on the inside blue on the outside. There is a phenomenon that we have never discussed uh in 802 yet, it's coming up I think next week. And that phenomenon
we call diffraction. Snell's law cannot deal with diffraction. That occurs when water drops are very very
small, say, smaller than a tenth of a millimeter. What you then get over and above the bow
you get areas in the bow whereby you get as we call that
destructive interference, the waves begin to kill each other
and you get dark bands and you can actually see that here. With a little bit of imagination
you see here a dark band. And when that is the case the water
is always extremely small in size and we give this a name,
we call this supernumerary bows. It's not so uncommon. If the water drops become exceedingly small,
let's say, smaller even than fifty microns, then this diffraction phenomenon becomes so
important that in addition to the dark bands all the colors are beginning to wash out
over each other and that creates then a white rainbow. Plus the dark bands. And a student of mine who was in my 803
lectures sent me years later the next slide when he was--
his name was Carl Wales, this picture he took, it's three hundred forty miles
from the North Pole, he was at Fletcher Island at the time, this picture was taken at two AM at night
in July when the sun is above the horizon. And so this must be the result
of very fine water drops which somehow are there
in the atmosphere. Doesn't look like it's raining but there must
have been small water drops. Fifty microns or less. And here you see the white call-it-rainbow and you also see beautifully the supernumerary bows, you see the-- the dark bands in there
and the next slide is a close-up of that. So you see here the white rainbow
plus the dark bands which is the phenomenon,
the result of uh of diffraction. Now before I will discuss the polarization
of the bows, because I still owe you answers
to the last three questions, now that I'm at it, I want to show you
some phenomenon which are quite common
which you may never have seen even though they're quite common
and they're rather spectacular. The first slide that comes now shows you what
we call the twenty-two degree halo. It's very common around the sun,
it's very common around the moon. The red is inside. It has nothing to do with water. It is the result of ice crystals
way up in the atmosphere. You can see it both in the summer
as well as in the winter because it's very cold way up there
also in the summer. This is very common. The reason why you and I
don't see it that often, because who wants to look
in the direction of the sun. This angle is only twenty-two degrees,
it's not so far. But I would advise you to at least
keep an eye on the moon. Because the moon also has this
twenty-two degree halo. And of course it's much easier
to look in the direction of the moon. That doesn't uh-- any--
no problems for your eyes. This is very common that I see this at least
three or four times per month and I always look for it of course. The next um slide shows you both the twenty-two
degree halo as well as the forty-six degree halo which is way less common,
it's very rare to see the forty-six degree-- I've seen it only a few times. It's very rare. In addition to the twenty-two degree halo
and the forty-six degree halo you sometimes see bright spots here. You see them here and you see them here. They're really not circles. They are arcs. And they have names, they call them sun dogs,
mock suns, they have various names. I've seen this often in-- in Boston,
the forty-six degree halo is rare. All of this as the result by the way of um
ice crystals. Ice crystals way up in the earth atmosphere. And there is a phenomenon that many of you
may have seen from an airplane. Uh if your airplane flies over clouds and you look at the shadow
of your airplane onto the cloud you may have noticed colorful rings
around the shadow of your airplane. And the next slide is such an example. I took this picture several years ago. You are always right at the center
of the circle. So you can see that I was sitting
just behind the wings. This is the result of diffraction. Got nothing to do with Snell's law. It is the result though of very very fine
water drops but not in the sense of refraction and reflection the way that we discussed it
with the rainbow. This is the result of diffraction. It's extremely difficult,
the explanation of the glory. And even ten years ago,
I read an enormous article about it, I think it was in Scientific American. Which made an utter attempt to be very
quantitative about the explanation, which is not easy. The radius of this bow,
if you want to call it a bow, depends on the size of the water drops. If the water drops are very small
the radius is substantially larger. So as you fly over clouds you may fly over
different clouds with different size of water drops and on a very short time scale
will you see the radius of this glory change. It can be very rapidly. It could be on a time scale
of seconds to minutes. It's very dramatic. It's very clear. And these rings can extend,
there's just not one ring, but you can have several rings. And so this is the result
of what we call diffraction. Now I want to mention to you another phenomenon
which is um also the result of diffraction. And that you can see when there is fog. If you have fog-- fog consists of extremely
small water drops, just like the ones in clouds. I mean fog is a cloud, let's face it. Then if you for instance had the headlights
of a car you could see your own shadow onto the fog
away from the direction of the car. And that means you could see then around your
head the same kind of stuff that you saw here. A glory. And so you see this around your head,
so it is sort of the-- the radiation that you would
expect from a saint. Quite a few years ago I was invited
to visit the Soviet Union. And they took me to the Caucasus,
to show me the six-meter telescope which was at the time the largest telescope
on earth. And I went there for a few days. And I noticed much to my surprise
that every night at five o'clock there would be fog coming out of the valley overtaking the telescope so you could make
no observations at night but that was a detail and what I realized is
that the fog was coming up, the sun was there, that was the west
and the valley was at the east and so I said to myself my goodness if only at the right moment of time
I am there then the sun is my light beam and my shadow
will fall onto this wall of fog coming towards me, this is my chance
to make a wonderful picture. And I'll show you that I succeeded. I will first show you a picture
of this bizarre observatory. This is the six-meter telescope
in the Caucasus. And this is indeed you could every night around
five-thirty really like a wall the fog will just coming towards you,
overtake you. You can see the timing of my picture
was extremely crucial. If you wait a little bit too long
then the fog overtakes you and then the sun would obviously
not be effective anymore. I wouldn't see my shadow on the fog. So I only had a small time window,
maybe thirty seconds, maybe one minute. But I succeeded and you will see w--
Saint Walter coming up. There is Saint Walter. You can see here my arm
and you can see here my camera. The camera must be exactly
at the center of the glory. And that's where it is. I have the camera right here and here you
see the rest of my body, isn't that terrific? This can be used as proof
sooner or later. OK. Back to our last three questions. The last three questions deal
with the polarization of the bow. Are the bows polarized? And the answer is, "Yes,
they are enormously polarized." Why are they polarized? You can answer that
question. But I will answer it for you. The light that contributes
to the rainbow is the light whereby
the minimum angle for delta is about a hundred thirty-eight degrees and if you go back
to that first transparency you will see that that's the case when the angle of incidence
is about sixty degrees. You can download that again. And that means that the angle of refraction
is about forty degrees. Forty degrees is very close
to the Brewster angle. The Brewster angle if you go from water to air,
you-- medium where you are is water and you bounce it off this medium,
so you go from water to air, the tangent of the Brewster angle
is n two divided by n one. N two is where you're going. That's air. So this is one. N one is where you are. That's the water. Let's just call that the index of refraction
one point three three. If you calculate now
what the Brewster angle is you will find that the Brewster angle theta
Brewster is about thirty-seven degrees. That is only three degrees different
from this angle which is the crucial one
which contributes to the rainbow. So you're only three degrees away
from the Brewster angle. So what happens as this light comes in
I will make a a separate drawing So as this light comes in, here,
practically unpolarized you should remember the meaning of the parallel and the perpendicular component the way I discussed that with you earlier. But, right here, you are so close
to the Brewster angle that almost everything that reflects
is now polarized in this direction perpendicular to the blackboard and so what comes out here is also polarized
perpendicular to the blackboard. And that would mean then,
that if you really think through what that means, that direction, of the polarization. It means that the bow is polarized
here in this direction, here in this direction, here in this direction and so on. And so if you take your linear polarizors,
you can hold them like this, And you can actually see
that the bow is polarized. And if you put your linear polarizors at ninety degrees angles, then you can kill the rainbow. Very highly polarized, nearly hundred percent,
because of the-- the fact that the Brewster angle
is so very close to the forty degrees. I like to go to Plum island
which is one hour drive north of here. Near Ipswich, it's a beautifull beach
and you have the ocean in the east The sun, late afternoon, in the west. And then the waves come in
and the waves splash up water. And I always look for the rainbow. I look at my shadow. The sun is there. Look at my shadow. Go fourty-two degrees away from the shadow,
wherever there is splashing water. [wssshht] For a split second,
you see the rainbow. Red outside, blue inside,
just for a split second. Because then,
there is no water anymore. You have to wait
for the next wave to come in. Water splashes up... [wsshht] Rainbow, gone. And so a few years ago I took my friend
Bill Priedhorsky to the beach. And I saw the rainbow,
I always look for the rainbow right, it's clear. And I pointed it out to Bill and didn't make
a big deal out of it because he is a physicist
so he knows that stuff. And so I said to Bill, "Here, look, you know,
look forty-two degrees away from your shadow." "You see, [wsshht] [psshht]" He looked at it. "I see nothing." Looked at it again... [wsshht] Beautifull rainbow. Bill, nothing. I got extraordinarily annoyed with Bill. I said, "Man, look at it, the rainbow is as crystal
clear as you can have it, what's wrong with you?" He says, "I don't think there is anything wrong with
me, maybe there is something wrong with you." Yeah, just imagine. And then I looked at Bill and I remember
what he looked like. He looked like this. And then I said to myself, "Oooooh, I understand
why he doesn't see the rainbow" He is wearing polarized sunglasses. And the polarization of the sunglasses
is in this direction. So every time that I saw this beautifull rainbow here... [psshht] He had his glasses in this direction,
he killed the rainbow. So I said Bill would you please
take your glasses off. He takes his glasses off and he said, "There
is nothing wrong with you, I see the rainbow.". So whenever you want to see the rainbow
at the beach make sure you don't wear your polarized sunglasses. Now I will demonstrate to you the rainbow and what I will do
is use one drop of water. This is exactly what we discussed earlier. One drop of water, light goes on to the drop,
comes back. Here is that one drop of water. And here is the light,
I'm going to blind you. You see this water drop? You've seen it. I'm going to put this, to make sure that your eyes
don't get blinded. And this light will come in your direction. And in this water drop,
all these things are going to happen. All these angles of i occur and so light will come
in this direction. And I will project it on the screen. And what will you see? You will see white in the inside,
you will see red on the outside. And that light that comes out near the edge
is very strongly polarized. I will show that to you. You can't see that with your polarimeters
because once it hits the screen and it comes back at you it loses
the direction of polarization. So you can not use your personal polarizors. So I will lower the screen. I will also make it dark and give you thirty seconds for your eyes to adjust to the light because you can imagine
I have only one drop of water. So don't expect too much, you can't do much
with one drop of water. And what you're looking at is really not the rainbow itself of course. The angle of forty-two degrees is only as measured from this one water drop. Not from where you are sitting if you are sitting
very far in the audience the bow looks smaller
than when you are close. Because you only see the intersection
of that cone with my screen. But it has all the ingredients
of the rainbow. And with a little bit of imagination you may be thinking that you see the rainbow. That would be fine with me. So, we'll make it dark. So that your eyes will adjust. We'll give it a few seconds,
for your eyes to adjust. In the meantime I can find my polarizor,
it's here. Look at it, Look at it! There it is! Isn't it fantastic! You see the beautifull arc here. All the way, red on the outside,
exactly as I predicted. Blue on the inside, I admit the blue is not as blue as you would like to see. But I can see the blue quite well
and look at all this white light. And you know now where all this white light
comes from. And so the bow is polarized
tangentially to the arc. So the polarization is in this direction here,
but the polarization is in this direction here. And I will show you that,
I have a polarimeter here. I turn the polarimeter so that the light
goes through. Now the light goes through,
now I'm going to turn it. And you will see it gets completely dark. Not only is-- are the colors gone
but the white light is also gone because the white light is also polarized. I'll do it here again,
the angle is now a little differrent. You see the light goes through,
you can see that very clearly. I'm going to rotate--
and you see-- it gets dark--
it gets dark-- the colors are gone
and the white light is gone. And now, for the first time in your life,
you understand why this is the case. Now the next time that you are going
to see a rainbow, you will look at that rainbow
in a very, very different way. For one thing you will check that the red is outside. You will make sure
that it's bright inside the bow. That it's dark outside. You will look for the secondary. And you will convince yourself that the color sequence of the secondary is different from the primary. This is a contagious disease
which you can not resist anymore. Your life will never be the same. And I'll be very proud of you,
if from now on, you will always carry
a linear polarizor with you. [laughter] And I gave you three,
so you better. So that every time
that you see a rainbow, you can be absolutely sure
that the bows are indeed polarized. And because of your knowledge
you will be able to see way more than just the bo-- beauty of the bows
that everyone else can see. Your knowledge will add
something very, very special and maybe you will be thinking of me. Thank you,
see you friday. [applause]
