Optical Fibers

One of the great things about graduate studies is the many hats we get to wear.  I've mostly written about my experiences teaching and researching education this year, but I've also had the opportunity to be a student.  I think it is a very helpful thing to be on both sides of the desk at the same time.  

This semester, I enrolled in a quantitative research methods over in the Education Faculty and an electrodynamics course here in physics.  The courses couldn't have been more different from each other, and I'd say I quite enjoyed both courses for different reasons.  Last year when I was in teacher's college, I blogged a bit about how much I missed a really challenging physics problem (this post: http://birefringencemms.blogspot.ca/2011/01/reminiscing.html).  Education classes can sometimes feel a bit like chocolate - they're sweet, fun, and can be very well taught.  But it seems that there's not a great deal of education classes which you could really classify as 'hard' or 'challenging'.  For me, I find that physics classes are more like steak - the content can be a bit dense for me to understand, but you finish the course feeling very satisfied - like you really learned something challenging and interesting.  

My electrodynamics course this semester was exactly what I needed.  Problem sets were typically only 3 or 4 questions, but working out the solution would take sometimes 60 pages.  When you jump through that many algebra manipulations and integrals and you finally arrive at an answer, it can be a real thrill.  There is something just fabulous about stretching your brain for hours, and then suddenly on the horizon, seeing a promising solution like an oasis in the dessert (and hoping it's not a mirage!)  Perhaps this explains why physics can be so addictive? 

The electrodynamics professor taught his course in a fairly traditional manner, but for our final assessment, he kindly agreed to let us do a project instead of an exam! I was so pleased.  The only sad part of this assignment was handing in the essay, completing my presentation, and realizing that I had no one to share all my pretty physics with.  But I have a solution to this sadness!  I am sharing the physics-love with you my dear friends :)  

Therefore, the following is the non-physicsy edition of my essay on optical fibers.  If you enjoy the read, and would like to read the physicsy-edition, just leave a comment for me, and I would love to explain the "why?" behind everything here or even send you my real report if you are interested. 

Optical Fibers: 

You talk to a friend across the ocean over skype with almost no delay in the conversation. Militaries send confidential data around the globe in fractions of a second. You watch a youtube video of a really cute puppy.  You turn on your sweet 90's fiber optic lamp for 'mood lighting'.   How are all these technologies possible?  Optical fibers of course! With the exception of the 90's mood lighting and some other light delivery applications, optical fibers are mostly used to connect our world by allowing almost instantaneous hard-to-intercept sharing of huge amounts of data. Under the ocean, we have many submarine cables which can send data around the world encrypted in pulses of light. Here's a map of where these cables were 5 years ago:

How does light actually travel along these optical fibers? Well, we can get a pretty good sense of what's going on using a ray optics model. Whenever a ray of light hits an interface between two different clear mediums, the light ray changes direction (or refracts). If you hit this interface at the right angle (called the critical angle), the ray will change direction so much that it will end up just travelling right along the fiber. The light is trapped! So our message will stay stuck in the fiber, and will therefore arrive wherever we want to send it. This is called total internal reflection, and here is a sketch of it:

We can use this "ray model" to describe light propagation in big fibers ('big' refering to fibers around 0.0001m in diameter and bigger). An example of a really big optical fiber is a stream of water. Water is not a very practical material for a real optical fiber (especially in the ocean...), but I think it is a rather pretty demo. You can guide a ray of light inside a stream of water using total internal reflection. Here's a picture of this demo in my kitchen:

Kinda looks like the diagram above, eh? A little? To do this at home, you just need a bottle with a hole in the side near the bottom, a laser pointer, some water, and an anneke to ramble on about how beautiful the total internal reflection is.

Unfortunately, if we send data down big fibers we run into all sorts of problems. Pulses of light spread out in time and smear on top of each other so much that by the time they've traveled a long distance (eg. under the ocean), they are completely unintelligible. So instead, we like to make really tiny fibers to get rid of this problem (by tiny, I mean a fiber with a diameter of around 0.00001m and smaller). The trouble is that with a fiber this small, the diameter of the fiber is not much bigger than the wavelength of light, and our ray approximation doesn't work anymore.

What do we do when an approximation lets us down? Do we throw up our hands? No! It's time for us to travel...

Yes Maxwell, not the future. But equally awesome. This is in fact one of the things I find particularly beautiful about physics: its complex simplicity. An incredibly complex system can be described by the smallest fundamental concepts. The fundamentals which describe light propagation are Maxwell's equations. Physics folks like to represent Maxwell's equations like this:

I know this might look a little weird if you're not familiar with the "nabla" symbol (it's actually nothing fancy - just a slope in three spatial directions). But definitely do take a minute to revel in the amazing complex simplicity: those four little unassuming equations just explained every x-ray machine, MRI, computer, light bulb, stove, lightning bolt, sunny day... and the list could go on for literally days of typing! Maxwell's equations describe pretty much anything and everything about electromagnetism. I think this is pretty awesome.

Now you might not feel quite so enthusiastic about these four equations, and this is okay. So for the non-equation-lovers out there, here are Maxwell's equations in words:

i) You can make electricity just by having a changing magnetic field (eg. moving a magnet around).  If you've take a first year physics course, you can connect this to what you've learned: this Maxwell equation leads to a fun little law you might remember by Mr. Faraday.

ii) You can create little magnetic field loops if you have either a current or a changing electric field or both. Take the 'steady-state' case where the electric field isn't changing - what does this give you? Ampere's Law!

iii) Electric charges can be single, or they can find the love of their life and become a dipole (the physics word for happily married couple). If you know where charges are, then you can figure out the electric field they create. This is useful for predicting how the relationship will progress - who will feel forces of attraction? Who won't? 

iv) So far, no one has ever seen a north pole all alone without a south pole companion. Even if a magnet is really big (eg. the earth) and the poles seem far far away from each other, they're always connected by magnetic field lines. It's actually a pretty adorable romance. 

Okay, so now you know all about Maxwell's equations and how pretty they are. But why did we go back to Max in the first place? Remember we wanted to find out how light propagates in a tiny fiber (the kind that they actually use to send data under the ocean).  So next, we do a bunch of mathy cartwheels which combine Maxwell's equations, then we shake things up a bit, and something called the "wave equations" will appear.  Once you've got wave equations, you should be a very happy camper because now all you have to do is solve the wave equations and you can describe exactly how electric and magnetic waves (that's just a fancy way of saying 'light') travel down the fiber.

Unfortunately, when we do our mathy cartwheels, we end up with what's called "coupled wave equations" (quiver in fear). Physically, this means that the electric field wave is influencing the magnetic field wave and vice versa. Practically, this means we have two really beastly equations to solve. We're saved by some approximations though! Most tiny fibers are "weakly guiding" - this means that 'critical angle' we talked about is really small, so the light is pretty much going straight down the fiber. This lets us 'decouple' our wave equations, so we can solve them! When we do this, we get pretty pictures of the electric and magnetic fields. There's a lot of different possibilities (called 'modes') for what you can get - here are two:

This picture is a cross section of the fiber (the dark purple part is the main core where the light mostly travels). And the electric field is represented by the red lines (longer line = stronger electric field). Imagine this as a cross section of a wave which is bopping up and down, while moving directly towards you. Here's another way to draw the second picture: 

In the picture above, I'm representing a strong magnetic field as a bigger height on the 3D graph. The electric field gets small as you go out to the edges, and follows this pretty cosine as you go around the circle. 

That is a snapshot in time of one of the electric field waves, which is travelling down an optical fiber right now to send this blog post to you! There's also a magnetic field wave - I didn't mention him because he's not too hard to find once you know the electric field. He's always just 90 degrees to his buddy, the electric field.  

And that's it - a brief intro to sending data in light pulses inside optical fibers! Now you can dream of total internal reflection and pretty wave patterns the next time you send an email or make an overseas phone call :)  I hope you enjoyed the read!  If you'd like to see the actual math behind light travelling in optical fibers, just leave me a note in the comments and I'd be happy to share!