O Star!

star-jesus-birth-154378-galleryStars are giant balls of hot gas, and they’re really far away. The closest one (after the sun of course) is more than 4 light-years away: that means it takes 4 years for its light to get to us.

What does a ball look like from that distance? It should look like a circle, like the moon does. But since stars are so far away, the circle is so tiny it appears as a dot on the sky.

But then why do we draw stars like this:

starOr as depicted in the nativity scene above, why do we think we see the long spikes radiating from bright stars?

The answer comes from the structure of the eye itself, and in particular the lens.

The lenses of our eyes have a network of thin, usually invisible lines in them called “suture lines”.

When we look at a bright point of light, like a star, the light gets scattered by these lines and forms an image with spiky radiating lines. These radiating lines are called “diffraction spikes”.

So it’s more than artistic license — this really is the image formed in our eyes by a bright dot of light.

I hope to write some posts explaining both diffraction and the anatomy of the human eye in the new year. Until then happy holidays!

 

 

Rainbow

The rainbow is a beautiful example of an interplay between ray optics and the wave nature of light — both are crucial in producing the familiar bright arc of colours.

640px-Double-alaskan-rainbowYou may know that rainbows are usually seen when rain is falling during bright sunshine. In this post I explained that when white light passes into a medium like glass, the various wavelengths (i.e. colours) bend by different amounts, causing the light to fan-out into the rainbow colours. So when a ray of sunlight enters a drop of rain this fanning-out happens as well.

It would be natural to think that the light just goes through the rain drops like this:

throughWell — this does happen, but it is not the rainbow we see! Why not? It just isn’t bright enough to compete with the brightness of the sky itself. It is washed out.

You may have noticed that to see a rainbow, one’s back is turned toward the sun. In the figure above you would need to be looking towards the sun, another clue that this is not the explanation for the rainbow.

In fact what happens is that the light rays bounce off the back of the raindrop, and emerge from the front. Let’s look at one ray:

bounceSo why doesn’t the rainbow still just get washed out, like the rays that emerge through the back of the raindrop? It’s because, as we’ll see below, the rays get focused and this makes the rainbow very bright — the reason you can see the rainbow is because it is a caustic!

This focusing effect has nothing to do with the different colours having different wavelengths. It comes from considering rays of any colour. If a ray of light enters the drop in the center, it just bounces off the back and comes out along the same line it went in on. But a ray that comes in higher up ends up coming out angled downward.

The sun is very far away and this means we should think of the rays of light coming from the sun as being parallel. To help us understand the rainbow we’ll think about horizontal incoming rays of sunlight.

Here is an animation showing horizontal rays in black entering the raindrop. Everything depends on how high up an incoming ray strikes the front of the drop. The reflection off the backside of the drop is shown in red, and finally the emerging ray in blue. (This just to aid the eye and isn’t meant to indicate the colour of the light.)

mov

The crucial thing to notice is that the emerging rays point further and further downwards as the incoming ray scales the raindrop — but then they suddenly turn around and start going the other way!

The place where this turn around happens is the angle we see the rainbow at, approximately 42 degrees away from the line joining the sun to your eye. The turn around is a caustic — where the light folds back on itself and produces a focusing effect.

drawingHere is a diagram with the rays in the animation above shown all together:

imgNotice how the emerging rays never get beyond a certain line — this is the caustic where we see the rainbow:

img copySo why does the light fold back? You can see it happening inside the raindrop: look at the black lines above. The horizontal rays coming in higher up naturally strike the back of the drop higher up — but note that they are getting more and more bent by the refraction as they enter the drop. Eventually this bending overtakes the situation and the place where the rays strike the back of the raindrop actually starts to go down.

Now, if we followed rays of different colours, they would get refracted by slightly different amounts, and so the rainbow angle of 42 degrees is slightly different for each colour. Purple is bent the most and so appears on the inside of the rainbow, while red is bent the least and so appears on the outside of the bow.

 

Perspective

One of the things that took a long time for humanity to master was the art of perspective — how to draw a picture on a flat piece of paper which really looks like the 3D scene you view with your eye.

Look at this relief from Mesopotamia:

IMG_0432We see that they understood the obvious fact that closer things obscure the further things they stand in front of. Here everything in the scene is about the same distance away so they didn’t need to think about how things appear smaller when they’re farther away.

A couple thousand years later we get this from the 15th century: persp15thcwhere we see the clear attempt at getting the rule “the farther something is away the smaller it appears” right. But they haven’t quite understood how to do it yet and you can see that it looks wonky.

So how does perspective really work?

The answer can be summed-up simply as follows: we see angles. When an object is close by like this:fig2

the rays of light from the object make a large angle when they enter the eye. Move the object further away like so:

fig1 and now the angle corresponding to the same object is smaller. We see distant objects as small because they make small angles at our eyes.

We can think of what we see as a visual screen where distances are measured in degrees (or whatever unit you like to measure angles in, see my post about angles). In the pictures above we’re looking at someone’s eye looking at a sail boat. Let’s now draw what that person sees: on the left when the object is close, and on the right when it is farther away

fig3fig4

Now let’s think of a picket fence with a straight row of equally spaced posts. Here is a bird’s eye view of a person (just their head, in blue) receiving a light ray from each post:

fenceWe see that the angles between the rays from neighbouring posts are getting smaller and smaller as the posts get farther away. This means that the posts will appear to bunch-up as they recede into the distance. Combined with the effect we saw with the sail boat, which makes the distant posts appear shorter, we get a familiar picture like this:

fence_perspEach post has a height given by the angle between the rays which travel from the top and bottom of that post to the eye, like we saw with the sail boat. The horizontal position of each post is given by the angles from the bird’s eye view figure above.

So one just needs to position things on the paper according to the angles they make with the eye. The rules of perspective are nothing but the fact that the eye sees angles, not distances!

Multiple reflections in bubbles

I was struck when I first saw this fantastic phenomenon, photographed beautifully by Richard Heeks

bubble3

When the sun — or any bright object for that matter — shines on a spherical soap bubble, one sees first and foremost two bright spots on a diameter of the bubble (a line which passes through the bubble’s center). But look more closely, or on a bright sunny day there is no need to squint, it is obvious that there is a whole bunch of bright spots on the diameter and they tend to bunch up at the ends.

This is a wonderful example of multiple reflections. We’ve all had the experience of looking at ourselves in a mirror, when there is another mirror behind us. We see our reflection repeated many, many times, stretching out into the distance, and getting darker and usually differently colored the further out we look.

Let’s start with the two bright spots — one is a single reflection off the front face of the bubble, the other a reflection off the back face. Below I’ve shown two rays of light coming from the sun, reflecting off the front and back of the bubble, and landing together on someone’s eye.

mainspots

And here are the corresponding spots

bubble3ann

Every time a light ray reflects it looses some of its power. This is because with a translucent thing like a soap bubble, most of the light just shines right through. Only a small percentage reflects back. This means that the reflection off the front face is the brightest one. You can see this in Richard Heeks’ photograph, one of the two brightest spots is less bright — this is the reflection off the back face, labelled by a “2” above.

What about all those other little spots? Well spots “3” and “4” come from these rays, which bounce off the back of the bubble twice, they are called multiple reflections:

nextspots

Since they bounce twice they are less bright. As you can imagine the remaining spots come from rays which bounce three times, four times, etc. The spots bunched on one side all come from rays like the red ray, where the ray crosses itself once, but bounces many times off the inside of the bubble. The spots on the other side come from rays like the black ray “3” above, which do not cross themselves.

Colours and Prisms

In the last post I introduced the idea that light is made of waves of electric and magnetic fields. The one and only difference between orange and green (or any colour) light is in the “wavelength”; this is the distance between crests on the wave. Your eye responds to different wavelengths differently and this results ultimately in the sensation of colour. I’ll detail the physiology of colour perception in a later post; this one will concentrate on seeing the beautiful rainbow of colours from a prism. Here’s a shot I took of sunlight shining through a water glass on my dining room table:

prism

White light from the sun is a mixture of light of different wavelengths, and that means a mixture of colours. When all the colours in the light are in roughly equal amounts, the light looks white.

In the post about refraction we saw that light bends when it goes from air into something else, like glass or water. It turns out that different wavelengths (and so different colours) bend different amounts. Red light, which has the longest wavelength, bends the least while blue/purple light, which has the shortest wavelength, bends the most. So when a ray of white light hits a block of glass, something like this happens

Untitled 1

The colours separate out, and we get the rainbow effect. The order of the colours is the order of their wavelengths!

Waves vs. Rays

So far the things I’ve written about in this blog have been about light behaving as “rays” — straight lines that bounce off of things or change direction when they pass through them. There is a bunch of other things that light does which has to do with the fact that light is really a wave. In this post I’m going to describe what kind of wave light is and what is “waving”. I’ll need to start with some perhaps unfamiliar and seemingly unconnected physics. So get ready for a bit of a longish detour!

Electricity

Light is an electromagnetic wave — but what does electromagnetic mean? The word encompasses two things everybody has experienced. The first is electricity, the second magnetism.

Electricity is familiar — we are all know about plugging things into the wall socket. Lightning is another example (here’s a fantastic image from lightning’s Wikipedia article)

Lightning_hits_treeWhat causes the lightning bolt? Have you ever pulled off a sweater and had your hair go all frizzy and stick to it? Or touched something and got a little shock or spark? What about rubbing a balloon on your head and then sticking it to a wall? Or seen a little spark when plugging something in?

Even though most of the time you never notice it, all stuff is “charged” in one of two ways: either positively (+) or negatively (-). Positively charged things feel a repelling force between them and so do negatively charged things. Oppositely charged things feel an attractive force between them. This force is called the “electric” force. Most stuff has as much positive as negative charge in it — the overall charge is zero, i.e. the positive and negative charges cancel each other out. So a bowl and spoon don’t attract or repel each other because they each have as much + as – charge and so are effectively uncharged or “neutral”.

Any and every material — air, the earth, the keyboard I’m typing on, food, really just about anything in your day to day experience is made of atoms, and atoms have a positively charged core called a nucleus and a region of negatively charged things called electrons which are attracted to the nucleus and sort of stick to it, or around it.

atomLightning happens when something causes the electrons on the surface of the earth to be stripped-off by the clouds — leaving the ground positively charged and the clouds negatively charged. The attractive force between the separated charges gets so strong that eventually something dramatic happens: a conduit is forced open where the charges can return to their proper locations. This conduit is the lightning bolt. The charges wreak havoc on the atoms in the air as they travel, causing them to get hot and glow.

OK — so we have learned that an electric force acts on charges. That’s one half of the physics of an electromagnetic wave. Now for the other half.

Magnetism

What’s magnetism all about? Probably the most familiar magnetic things are the fridge magnet and perhaps the compass needle. What was important for the electric force was charges, positive and negative ones. The magnetic force acts instead between the poles of other magnets (in fact it also acts on moving charges, but let me leave that aside for now as I think it will confuse things). The poles of magnets also come in two varieties north and south:

magneticNorth poles repel each other and south poles repel each other. North and south poles attract each other. The fridge magnet works by “inducing” the opposite pole in the metal door on the fridge — but that’s a little advanced for our purposes so I won’t explain that here.

The compass needle is a magnet and the planet earth is also a giant magnet! There is a magnetic force from the poles of the earth on the poles (the two ends) of the compass needle, so that it aligns its south pole with the earth’s north pole and points north.

Electromagnetic Waves

Electromagnetic waves are electric and magnetic forces which travel through space without any charges or magnets in them. Of course we need to have some charges and/or magnets somewhere to get the waves going — think of ripples on a pond when a stone is thrown in. We need the stone to get the wave going, but after that the wave just keeps going out from that spot on its own.

rippleA water wave makes things floating on the surface move up and down. An electromagnetic wave makes charged things move up and down and it makes magnets  turn back and forth, in an attempt to align their poles with the passing magnetic force or “magnetic field”.

One example of electromagnetic waves is the signal (wifi or otherwise) on your phone or device. These waves make the electrons in the atoms of the antenna shake — and that shaking is processed by the computer in your device and turned into data.

Another example is your microwave oven — here the shaking is converted into heat.

The purpose of this post is to underscore that another example is light itself! Light is an electromagnetic wave!

Radio waves, microwaves, wifi and mobile phone signals, light, X-rays, and gamma rays are all the same thing: waves of electric and magnetic forces.

What makes these waves different? Look again at the water ripples:

wavelength

The distance between successive ripples is called the “wavelength”. It is simply the wavelength which makes the difference between gamma rays, X-rays, light, and radio waves. Green light is around 550 nanometers in wavelength. A nanometer is one millionth of a millimeter — so the wavelength of light is really, really short. The wifi signal on my 5 GHz home router is around 6 cm — much, much longer, but in every other way exactly the same as light.

Rays

So how is the wave connected to the ray? The wave travels at right angles to the crests of the wave or “wavefronts”. Below I have drawn a ray with the wave crests superposed — this is like looking down at a water wave:

raySometimes the wave nature of light isn’t important — this is when ray optics or “the ray approximation” can explain what is seen. Other times the wave nature is essential to understanding what is seen. Keep reading the blog to see some examples of when wave optics comes into play.

Caustics

Caustics are bright shapes produced by light which is, in some sense, focused either through reflection or refraction. You may have noticed this shape in the bottom of your cup:

caustic

Let’s look at some incoming light rays which are striking the circular surface of the cup; below I’ve drawn these in white:

cccIN

We know that the rays will be reflected according to the rule that the incoming and outgoing angles are equal, see my previous post here.

Let’s just concentrate on the rays on the left side of the circle. The reflected rays look like this:

cccPART

Notice that in the lighter grey region on the left the various rays cross each other. Notice also that there is a dark region below where no light reaches. In between these lighter grey and black regions the rays are piling-up on each other, producing a bright arc — this is the caustic.

Below I’ve added the rays on the right side and increased the number of rays so that they blur together and are no longer visible individually:

ccc

This picture is literally just a bunch of reflected rays which came in vertically. It’s amazing how well it reproduces the real thing:

caustic

But that’s what ray optics is all about!

Below are some beautiful caustics I shot on my wall:

caustics

… produced by this dish:source