It's About Time, Part I

(trying my hand at putting latex into posts. Since the preview function doesn’t work, this may not go well)

I’ve already commented that, within the timing community, optical frequency combs are basically everywhere, and that this is driving a lot of investigation into optical transitions for clocks/frequency standards.

Why are these such a big deal? Let’s start with what we’re measuring. Time, as Albert had put it, is what is measured by a clock. Put another way, in terms of what a clock is doing: time is the phase of an oscillation. To build a clock, you need an oscillator and something to count the oscillations, or fractions thereof, i.e. the phase, in chunks of 2\(pi \) or possibly smaller. If your oscillator is running at some frequency \(omega\), you have

\(phi = omega t \), or put another way, \(omega = frac{dphi}{dt} \) and you integrate the frequency, or count the oscillations, in order to get the phase (time)

That is, the phase (the time you measure) is just the counting of all those oscillations over some period of time. If the frequency is 1 cycle/second, then you are ticking once a second, which is pretty freaking obvious and you’re wondering where you can sign up to get paid a fair wage for something that’s so utterly, blatantly true.

But … what if the frequency isn’t constant? How can you possibly tell what’s going on if your oscillator isn’t running at a constant frequency? How do you even know if your oscillator’s frequency is changing? That’s where the work starts, and one reason this line of work isn’t simply a tautology (the “man with one clock always knowing what time it is” aside). What we want is t, but we’re measuring \(phi \), and now we have to figure out how to get one from the other.

Let’s assume that the frequency of our oscillator is changing linearly, and we’ll call this the drift.

\(D = frac{domega}{dt} \) or, integrating, \(omega = D t \)

But we want the phase, so we have to integrate again

\(phi = omega t + frac{1}{2}D t^2 \)

These look exactly like kinematics equations, which should be no surprise, because we’re doing the same math as something undergoing constant acceleration. You can go to higher order, but really you’d want to just get a better oscillator — in practice you put up with drift if you’ve got a precise clock, or if there is some other reason for it. And none of this discussion accounts for other noise that your oscillator would naturally have (and white noise on an oscillator, integrated, becomes a random walk in phase) which will further complicate your analysis; this and other noise sources that depend on fractional powers of the frequency.

But more in line with this review, how can you tell what your oscillator is doing? You have to compare to another oscillator, and things get even more interesting. I’ll pick up here in the next installment.

0 thoughts on “It's About Time, Part I

  1. My solution (not perfect) to the equation problem is to make equations with latex and then take a screen shot of the output. I tried some of the WordPress plugins, but without latex on the server, the output was poor. Also, I found that a white background helps. But the above IS readable.