Laser Photonics Time-harmonic waves

Time-harmonic waves

We saw that the solutions to the scalar wave equation were of the form

\displaystyle E_x = f(z \pm v_p t)

where f was any twice-differentiable function.  I’m going to start using sinusoidal  or harmonic functions for f – functions like cosine, sine, and complex exponentials.  Let’s start with cosine.  We should certainly be allowed to replace f with a cosine, as it is twice differentiable.  If we do so, we have

\displaystyle E_x = A \cos {(k(z - v_p t))}

I decided to go with the minus sign for the forward-propagating wave (wave traveling in the +z direction), and I slipped in a few new constants: A out front, which is just the amplitude of the cosine wave; and a k inside the cosine, which I’ll discuss below.  Neither of these constants changes the fact that this harmonic wave is a solution to the wave equation, because it’s still twice-differentiable, and it still has the necessary (z - v_p t) grouping inside.  I’m going to rewrite the harmonic wave a bit without really changing it.  First I’ll bring the k inside the parentheses:

\displaystyle E_x = A \cos {(kz - k v_p t)}

The quantity in front of t in a harmonic function is just the angular frequency \omega, so we can rewrite this as

\displaystyle E_x = A \cos {(kz - \omega t)}

where we must have

\displaystyle \omega = k v_p

You probably have some intuitive feel for the angular frequency \omega = 2 \pi \nu, where nu is the ordinary frequency, often given in Hertz (Hz).  For example, as \omega gets bigger, the harmonic function is oscillating faster in time.  The period of oscillation – the time it takes for the oscillation to undergo one complete cycle – is just

\displaystyle T = \frac{1}{\nu} = \frac{2 \pi}{\omega}

Slightly newer is k, which is called the wavenumber.  Looking at our harmonic wave, notice that k sits in front of z, just as \omega sits in front of t.  In fact, the wavenumber k plays the same role as \omega does, except that it applies to the wave’s oscillation in space, rather than time.  So, as k gets bigger, the harmonic wave is oscillating faster in space.  We can think of the wavenumber k as being an angular spatial frequency.

The wavelength \lambda is the spatial distance between peaks of the wave; i.e. it’s the distance over which the oscillation undergoes one complete cycle.  It’s analogous to the period, but applied to space, rather than time.  Similar to how the period T =  2 \pi / \omega, the wavelength

\displaystyle \lambda = 2 \pi / k.

We can do a little manipulation here to come up with a very famous equation.  We already know that

\displaystyle \omega = k v_p

and we know \omega = 2 \pi \nu, and k = 2 \pi / \lambda.  In the next section we will see that the phase velocity v_p = c / n.  Putting all of this together and rearranging things,

\displaystyle \lambda \nu = \frac{c}{n}

In words, wavelength times frequency equals the speed of light divided by the refractive index.  This holds for all electromagnetic waves, including light.

Below are some animations to illustrate the connections between \lambda, v_p, and \omega.

Animation showing a wave with amplitude A = 1, \lambda = 1, and v_p = 1:


Animation showing a wave with amplitude A = 2, \lambda = 1, and v_p = 1:


Animation showing a wave with amplitude A = 1, \lambda = 0.5, and v_p = 1:


(Notice how the frequency \omega has increased, because the faster spatial oscillations translate into faster time oscillations when the wave passes at a fixed speed.)

Animation showing a wave with amplitude A = 1, \lambda = 1, and v_p = 2:


(Notice how the frequency \omega has increased, even though the spatial frequency k have not, because the spatial oscillations are passing by faster.)