A \(T=20\) ps transform-limited pulse propagates through \(10 \mathrm{~km}\) of a dispersive medium for which \(\beta_{2}=12 \mathrm{ps}^{2} / \mathrm{km}\). The pulse then propagates through a second \(10 \mathrm{~km}\) medium for which \(\beta_{2}=-12 \mathrm{ps}^{2} / \mathrm{km}\). Describe the pulse at the output of the second medium and give a physical explanation for what happened.

Imagine a wave of light as a group of runners, each runner representing a different color of the spectrum. In a race without any obstacles, they would all finish at the same time. In the world of optics, this race is a 'transform-limited pulse,' where all the colors of light (wavelengths) that make up the pulse reach the finish line (the end of the medium) together. However, when we introduce second-order dispersion, it is as if we add hurdles of varying heights that each runner (color) must clear.

Some runners will clear the hurdles easily, while others will struggle, causing them to finish at different times. In optics, the 'hurdles' are the medium's characteristics that cause different colors to travel at slightly different speeds, leading to 'pulse broadening.' The pulse that was once sharply defined now becomes a washed-out smear of colors across time, much like our runners straggling across the finish line at different times.

The coefficient \( \beta_{2} \) determines the extent of this effect—positive \( \beta_{2} \) values cause a longer race for the high-frequency runners, while negative values speed them up. In terms of light, this corresponds to the delay or advancement of the different spectral components of the pulse as it propagates through the medium. The exercise showcases this effect dramatically, with the pulse initially spreading out through a positive \( \beta_{2} \) medium, and then subsequently being compressed back to its original sharp form by an equal and opposite \( \beta_{2} \) medium.

A 'transform-limited pulse' is the optical equivalent of a perfectly coordinated group all arriving at once. This is the ideal scenario for optical communications, where clarity and precision are paramount.

The term 'transform-limited' refers to a pulse of light that is as short in duration as the uncertainty principle allows for its bandwidth. In other words, it is the 'shortest possible pulse' for a given range of colors, with the spectral components perfectly aligned in time such that the pulse does not spread out as it travels (in the absence of dispersion). This is crucial, as any spreading in time (temporal dispersion) can lead to information loss or corruption in optical signals.

In our exercise example, the initial pulse is transform-limited, meaning it starts as a tight, undispersed pack of spectral components. However, even the most neatly aligned group will face challenges when hitting those hurdles mentioned in the previous section. The ultimate goal in optical communication is to maintain the transform-limited status throughout propagation, which, as we will see in the discussion on 'pulse propagation,' can be a delicate balancing act.

Much like sending a paper boat down a stream, pulse propagation involves sending a pulse of light down an optical fiber or through a similarly dispersive medium. The shape of the boat, the speed of the stream, and any disturbances in the water will determine how the boat looks when it reaches its destination.

In the case of our pulse of light, dispersion within the medium it travels through determines how the pulse changes over distance. If the medium is uniform and has zero dispersion, our pulse would travel without changing shape, just like a boat on a still pond. But in reality, optical fibers and other media have dispersion that can either stretch or compress the pulse, like streams with varying currents and eddies.

The concept of pulse propagation takes center stage in our exercise. Our transform-limited pulse starts its journey pristine. As it enters the first medium with positive \( \beta_{2} \) (the broadening stream), it loses its sharpness and broadens. Then, it enters the second medium with negative \( \beta_{2} \) (the compressing stream), which undoes the broadening, restoring the pulse to its initial state. This dynamic journey highlights the complexities of propagating pulses in dispersive media and the ingenious ways to counteract dispersion to maintain signal integrity.