# Time multiplexed detector - pulse dispersion - Fourier limits

I'm currently in my second semester in physics and a couple of weeks ago I saw a job offer in the optics department of my university. Although I don't know much about optics, since we didn't deal with it yet I still applied for the job since they said that they didn't require any background knowledge. And that was the case for the first couple of weeks. All I did was work in the lab - doing measurements which could be done by anyone once they got the routine. Anyway, but now I'm supposed to look into the theory behind the measurements, which truth be told I have zero knowledge about. A key question they wanted me to think about was: How does a light (pulse) behave in a fiber?

Alright, before I go on I want to explain the measurements I did.

Basically I'm detecting the number of photons in a light pulse. So there is a light pulse which is shot periodically and it then goes through a beam splitter, and then through another and so on. The point in using beam splitters is to minimize the amount of detectors used (due to monetary and spatial reasons). Basically we're talking about time multiplexed detectors. Here's a figure illustrating the concept pretty good:

My job is to use loopy simulations and then doing the measurements while varying the burst rate, repetition rate, number of pulses, the mean value of photons per pulse and so on.

I will try and explain the terms above as best as I can (truth be told I'm still not quite familiar with those terms):

As far as I understood from the briefing for the job there are a number of pulse trains (not really the scientific term but bear with me) per measurement and the terms I mentioned above define that either directly or indirectly.

The repetition rate defines the distance between pulse trains whereas the burst rate defines the distance between one pulse to the next one in a pulse train. Again, here is a figure to illustrate it:

I forgot to mention that the x-axis in the figure is time.

Anyway, as said before I can change basically anything with the function generator. Usually I don't change the mean value of photons per pulse, but that's because so far I didn't have to.

I'm bad with words so I think that talking about my first measurement would tell a better than any words I could put together:

The first measurement was pretty straightforward. I decided to go with a repetition rate of 100 kHz, meaning that the distance between pulse trains is $$10^{-5}$$s. At that point I decided to keep that value as well the mean value of photons per pulse constant. All I did then was changing the burst rate and recording the results for each increase. I started with 1 MHz, meaning $$10^{-6}$$s distance between pulses to each other in a pulse train and increased it to 20 MHz. From 1-10 MHz I increased it 1 MHz at a time and then I went for 0.4 MHz at a time, because I saw some fluctuations in the countrate registered by the detectors (not important right now, but wanted to mention it either way).

Btw. before I forget: The reason why I started with 1 MHz is because I was told that going under that value for this particular repetition rate would cause latching (don't ask me about that term, no idea what it is. I just took it as it is.).

Well, that was a long prologue. Anyway, I was told to think about how the light travels through the fibers. And for that they gave me two hints (not really hints, but key words to look up): Fourier limit and pulse dispersion. I tried borrowing some books about nonlinear fiber optics from the library and reading about it, but truth be told I couldn't understand it fully (mainly because I don't have the background knowledge on that). At some point I came across the terms chromatic dispersion and polarization mode dispersion. I asked around and I'm supposed to neglect anything about polarization mode dispersion since it doesn't have an influence on the set-up I'm measuring with.

And this is the point where I'm lost. I mean I get the concept that due to different wavelengths of light the refractive index changes accordingly and the velocity is then something like $$c(\lambda)=\frac{c_0}{n(\lambda)}$$. I was even told to look up some dispersion values of single mode fibers (a value I found was for a SMF 28 fiber which was like 18 $$\frac{ps}{nm\cdot km}$$) and some calculations but I don't know how - I don't know any formulas for that.

Could someone more knowledgeable about it help me here? Especially about Fourier limit - I couldn't find anything about it in the books I borrowed.

I posted this before as well but I had troubles with the edits, so I deleted the first post.

There's a lot here... let me break it down a little bit into some intuitive chunks. I hope that tackling the math will be less scary after that.

It's not quite clear from your description whether you are using single mode fiber, or multimode. Let's first look at multi mode.

When light travels through an optical fiber, it can choose many different paths. The shortest path would be "straight down the middle", but since the fiber is usually not straight, that won't work very well. As soon as it starts bouncing off the walls, it is no longer taking the shortest possible path. And as the path length increases, so does the time taken.

So if you have a pulse of 100 photons all starting at the same time, and traveling through a fiber, they will each take a path with a slightly different path length - and after a while they will be "out of step" in the same way that a pack of runners will be closely bunched at the start of the race, but gradually spread out as the race gets longer. This is called "dispersion".

Now if you have a pulse train, the spacing between the individual pulses can be thought of as groups of runners in a race. If you are organizing a big race (like the New York Marathon), you don't start all runners at the same time: instead you start the fastest group first, wait a few minutes, then start the next group, etc. Because, in principle, you sorted the runners by speed, you expect that this makes the race more enjoyable (less crowded) for everyone. Instead, if you mixed the runners up, then that crazy fast Kenyan in group five will come barreling through the much of middle aged guys with something to prove to themselves in group 4 in no time at all. Soon, the field, instead of spreading out, will become a huge mess. If you were trying to detect "bunches of runners" (photons), you wouldn't be able to do this once they were a few miles into the race.

Back to optics. The spacing between the bunches is going to be more or less constant, but the width of each bunch will spread. When it spreads by a sufficient amount, it will become impossible to tell the bunches apart. You can use Fourier analysis to model the pulse shape, and this is one way to calculate the dispersion.

Now if you have single mode fiber, such a fiber is constructed so light can really only travel at one angle, and therefore, in principle, might not show dispersion. However, as a light pulse becomes shorter, it becomes less monochromatic. You can prove this with Fourier analysis: the Fourier Transform of a Gaussian pulse (product of sine wave and Gaussian) is the convolution of the FT of the sine wave (a delta function) and the Gaussian (another Gaussian). So as the pulse becomes shorter (narrow Gaussian), its Fourier Transform (frequency components in the pulse) becomes wider.

And the wider the frequency content of the pulse, the more you become susceptible to chromatic dispersion: light of different wavelengths travels at slightly different speeds. You found a value of 18 $$\frac{\rm{ps}}{\rm{nm\cdot km}}$$ which tells us that a change in wavelength of 1 nm will change the transit time through 1 km of fiber by 18 ps.

If chromatic dispersion is the thing you need to worry about, then you have to think about the width of your pulse and the effect this has on the spread in wavelengths. The factor that ties it together is the speed of light: a pulse of 1 ps has a width of $$3\cdot 10^8 \cdot 10^{-12}\ \rm{m} = 0.3\; \rm{mm}$$. Compare this to the wavelength of light, maybe 600 nm, and you have about 500 wavelengths in your pulse. This means (I am doing this in my head) that the wavelength spread will be about 1/500th of the center wavelength or a little over 1 nm, and so your pulse will spread by about 18 ps in one km. At 40 MHz repetition rate your pulses are about 25 ns apart - you would need a lot of fiber before they overlap.

So if you find the pulses overlapping sooner, you would have to conclude you don't have mono mode fiber.

This is all a bit hand-wavy intuitive. Let me know through the comments if things make a bit more sense for you.

Incidentally - in my experience APDs are not particularly fast detectors. I would want to use an SiPM (silicon photomultiplier) or some such if I wanted to really push an experiment like this.

• Thanks for your answer. I understood a fair bit more about the concept thanks to you (although I'm not quite sure how you got to 0.3mm - what's 1 ps in your example? The width of the pulse or the distance defined by the burst rate?). As far as I know we're dealing with single mode fibers in the lab (SMF 28, PM and such). I don't want to sound rude because you already did so much, but could you walk me through the math? I have never read about Fourier Analysis nor am I familiar with the math for this. Sep 26, 2015 at 8:32