What does it mean to say that the photodetector has a bandwidth?

Question

In computing an expression for the shot noise associated with a photocurrent, we are often told to consider a photodetector that averages a signal over time T. Consequently, the shot noise scales as $ ~ 1/\sqrt{T}. $ The bandwidth of the detector is defined as $$ \triangle \omega = \frac{2\pi}{T} $$ and the noise is often expressed in terms of the bandwidth instead of the integration time. I am trying to wrap my head around the notion of bandwidth:

1. Why is it useful to think about detectors in the context of their bandwidth?
2. What limits the bandwidth (and consequently the integration time) of detectors?

Answer 1

Any real device has a bandwidth. There are two ways to think about it: in time domain and in term of spectrum (frequency domain).

Frequency domain
In terms of spectrum it means that any real device has limited precision, i.e., it does not detect exactly frequency $\omega_0$, but rather frequencies in a small range $[\omega_0-\Delta\omega/2,\omega_0+\Delta\omega/2]$. In practice the spectrum of the device is usually not rectangular, but a smooth function (such as Lorentz shape or Gaussian curve), with $\Delta\omega$ characterizing its width. However, rectangular spectrum (i.e., a band that is uniform everywhere in $[\omega_0-\Delta\omega/2,\omega_0+\Delta\omega/2]$ and zero outside this interval) is often easier to deal with.

Time domain
If we now go to the time domain, then width of the spectrum corresponds to the frequency of the lowest harmonics of the signal. In order to measure frequency of a signal, it should be observed (often we say "averaged") over a time interval much longer than its period - otherwise we cannot claim that the signal is truly periodic. This means that the bandwidth $\Delta\omega$ is actually determined by the averaging time of the device, $T$. This averaging time is always finite - at least for the reason that our experiment lasts finite time, the scientists performing experiments are mortal, the PhD lasts only a few years, and even the lifetime of the Universe is finite - whatever applies to a particular situation. E.g., @ThePhoton has given a correct description of how finite bandwidth arises in photodetectors, but the constraints outlined here are valid for any kind of detectors.

Answer 2

What limits the bandwidth (and consequently the integration time) of detectors?

Generally, a PN (or PiN) junction has a capacitance associated with the charge separated in the space-charge region, which combines with the effective output resistance of the device to form a basic RC low-pass filter.

There is also a transit time associated with sweeping carriers out of the junction after they're generated by photoabsorption events.

Depending on the device design either of these effects might be the dominant bandwith limiting one. We often use P-i-N junctions rather than simple PN junctions for photodetectors because by increasing the length of the junction we reduce the junction capacitance. But at the same time the transit time is increased, so there's only so much benefit that can be achieved this way.

Why is it useful to think about detectors in the context of their bandwidth?

For example, if we want to use a photodetector in a high speed communication system, or to determine the time of arrival of a signal very precisely, then the bandwidth of the detector will be a critical limitation on our ability to do that.