Back in our plane wave solution to the wave equation, I was trying to convince you all that using plane waves (i.e. monochromatic waves) is worth your time. While plane waves are definitely worth your time, like I said then, plane waves are not a physical picture: you cannot have waves that begin and end at infinity. It turns out there are a lot of problems that would occur if you did try to make such a wave, a detector with infinite detection time and an infinite supply of energy being just a couple of issues you may run into. Real waves have a finite bandwidth. As we will learn in greater detail later on (head to the “Coherence Theory” section if you are curious now), a finite bandwidth means a finite coherence length which in turn means a finite length over which the wave propagates. Thus, a real solution! It is useful then to develop some intuition of the properties of nonmonochromatic waves and pulses, as I have attempted to do in the following paragraphs:

*Superposition* – If two plane waves $E_1(r_1,t)$ and $E_2(r_2,t)$ occurring at the same frequency are separately solutions to the wave equation, then so is their superposition. Superposition covers the idea of *spatial* overlap; this is covered in greater detail in the “Interference” section of this portion of the website.

*Beats* – Consider the example of two waves occurring at the same points in space:

\begin{equation} E(r,t) = E_1(r)e^{i\omega_1t} + E_2(r)e^{i\omega_2t}\end{equation}

Taking the real part and equating the amplitudes ($E_1 = E_2$):

\begin{equation} E(r,t) = 2E_1(r)cos(\frac{\omega_1+\omega_2}{2}t)cos(\frac{\omega_1-\omega_2}{2}t) \end{equation}

where the first term is the average frequency and the second term is the beat frequency. The intensity is then proportional to:

\begin{equation} I = |E^2| = 4E_1^2cos^2(\frac{\omega_1+\omega_2}{2}t)cos^2(\frac{\omega_1-\omega_2}{2}t) \end{equation}

If the frequencies are close i.e. $\omega_1 \approx \omega_2$, then $\frac{\omega_1+\omega_2}{2} >> \frac{\omega_1 – \omega_2}{2}$. What this means for measurements then is that the fluctuations of the average frequency will be so fast that they are constant in the time-averaging view of the detector. You can see this easily if you plot each “cos” term with realistic frequency values in the argument. If you do decide to plot the terms, you will notice then that the frequency period of the plot corresponds in fact to the *difference* frequency; in the optics field, we call this beating! This is actually exactly analogous to the case when one says “Drop me a beat.” :-).

*Power Spectrum* – Now that we have considered the case where the frequencies of consideration are very close in value, what happens when they are further apart? Well, let’s think about what happened in the last case. With a high value in the argument for the average frequency, the “cos” function became constant. When the difference between the two frequencies also grows, as it would in the case where $\omega_1$ and $\omega_2$ are very far apart, then the argument for this $cos$ term also grows very large and becomes constant in time. Thus, with two distinct frequencies, the power spectrum becomes constant in time as t goes to $\pm \infty$, which we know from our extensive knowledge of Fourier transforms (ha) means that the frequency spectrum would show two delta functions at each $\omega_1$ and $\omega_2$.

*Pulses* – Now that we have extensively considered the cases at two frequencies, let’s consider the case where we add a continuum of pulses around a central frequency. As we add the amplitudes of each frequency, which are of course shifted with respect to one another in time, at each point in time, we see that something that resembles a pulse is formed! Mathematically, we can write this as:

\begin{align} E(r,t) & = \sum_j E_j(r)e^{i\omega_jt} \\ & = \int^{\bar{\omega}+\Delta\omega/2}_{\bar{\omega}-\Delta\omega/2}E(\omega)e^{i\omega{t}}d\omega \\ & = \int^{\infty}_{-\infty}E(\omega)e^{i\omega{t}}d\omega\end{align}

In the first step above, the summation was replaced with an integral over the central frequency $\bar{\omega}$ $\pm$ half of the the bandwidth, which we have assumed goes to infinity. Then the next step is the assumption that E($\omega$) is only substantial in the range of $\bar{\omega}$. With this last step, we have inadvertently derived the Fourier relation between the frequency and time domain.

Most times in optics when describing laser pulses, you will see pulses from lasers represented by an envelope multiplied by a carrier wave, i.e.

\begin{equation} E(z,t) = A(z,t)e^{i(\bar{\omega}t-\bar{\beta}z)} \end{equation}

where “$A(z,t)$” describes the shape of the envelope that makes this the electric field of a pulse, and the exponential term is the carrier wave contained within the envelope. It is important to note that without the envelope, the carrier wave alone describes the electric field for a continuous wave. The continuous wave may have a Gaussian profile in the transverse direction.

Typically, in most of the literature on optical pulses, you will find that the envelope function of the pulse is given by a Gaussian,

\begin{equation} E(t) = E_0e^{-t^2/\tau_p^2}e^{i(\bar{\omega}t-\bar{\beta}{z})} \end{equation}

where the term “$e^{-t^2/\tau_p^2}$” is the slowly varying (in comparison to the carrier frequency) envelope function.