I have been fascinated by a Sixty Symbols YouTube video on the topic of the maximum bandwidth that can be sent down an optical cable. In summary, we are informed of how a laser, which emits red light of one frequency, only does so when it is permanently on. Turning it on or off causes nature to create new frequencies which somehow facilitate the wave being there when it's on, and not when it isn't.
Since I'm a musician, I can place a loudspeaker in a medium such as air, and connect it to a sine wave generator in series with a switch. When the switch is open, no sound is generated, but close the switch and the loudspeaker outputs a sine wave. If I were to close the switch precisely when the phase was zero, surely there are no added frequencies other than the one in question which "turns the sound on", and likewise for "turning it off".
I think that this would not be the case for a sine wave at any other phase, as a 'pulse' or 'jolt' is needed to force the loudspeaker into the correct position when opening or closing the switch. I believe this causes the loudspeaker itself to generate odd harmonics for a brief period of time, as though it were being driven by the rising edge of a square wave. (That click/snap sound when attaching DC to a loudspeaker, also known as a Dirac delta function?)
I do not understand how this fits into using a laser of one frequency and sending 'pulses'. One comment on the video seemed to contradict it by asking what would happen if a filter was applied to the laser for its exact frequency, and whether that would cause the filter to always be letting light through like the laser was on, even if it was actually off.
It sounds like he's trying to describe sending the kind of pulse analogous to a square wave of sound which requires you to drive the transducer with multiple frequencies. It must be easier to detect a pulse if the amplitude ramps up quickly, but surely you're definitely not using a one frequency laser anymore to achieve that.
Q: Please can you shed some light onto how a laser being 'pulsed' actually works with respect to bandwidth used? And is the analogy with sound valid?
Answer
If I were to close the switch precisely when the phase was zero, surely there are no added frequencies other than the one in question which "turns the sound on", and likewise for "turning it off".
I made more or less this same statement to the professor in one of my undergraduate EM classes.
He gently reminded me that this zero crossing switched on/off sinusoid is equivalent to the product of a genuine sinusoid (over all time, never stopping etc.) and a boxcar function.
As soon as he made that statement, I knew how to go the rest of the way since I had already studied Fourier transforms and, in particular, that (1) the Fourier transform of the product of two time domain signals is the convolution of their respective frequency domain representations and (2) the Fourier transform of the boxcar function extends over all frequencies,
Further, and simply put, a time domain signal with compact support (non-zero over a finite time) is necessarily non-zero over all frequencies. See, for example, this answer at the Signal Processing Stack Exchange site:
In other words, a (nonzero) time-limited signal cannot be also band-limited. In other words, a function and its (continuous) Fourier transform cannot both have finite support.
No comments:
Post a Comment