# Confusion between a signal sent at a given frequency and the Fourier components?

I have been reading about phase and group velocity (maybe the context will be useful here) and was attempting a question when I became rather stumped. The question was

Square pulses are sent along optical fibres with an infrared laser of frequency f= 2x10^14 Hz, and each pulse has a duration of 10^-10 seconds. The time interval between successive pulses is 10^-9 seconds. The refractive index of the fibres in the frequency range is given by $n(f) =n_0 +\tau (f-f_0)$

a) sketch the power spectrum of the signal and estimate the range of frequencies present in the signal.

Now this has really confused me! How can there be more than one frequency in the signal if we are using a laser of a particular frequency? But on the other hand, a square wave signal has to have multiple frequencies present in its Fourier spectrum (sinc-like, I think)! The only thing I can think of is that the apparatus producing the blinking infrared light actually produces a range of frequencies centred on the central frequency stated. However, I can still imagine having a steady single frequency infrared source, and moving an opaque sheet in front of it periodically to produce the series of pulses. Now I can't figure out how that could somehow introduce additional frequencies to a single frequency signal!!! The sheet doesn't produce its own light after all!

• Have you drawn a picture of your signal? a train of square pulses modulating your light waves? What is its Fourier transform? – Cosmas Zachos Jan 10 '18 at 16:37
• The laser produces an almost-single frequency, but whatever is modulating the signal changes the spectrum. – garyp Jan 10 '18 at 16:44
• Oh so the signal representation is $f(t)=F(t)cos(\omega t)$ where $F(t)$ is the square wave pulses and $\omega$ is the laser frequency! I was only considering the square wave signal before! Would you also be able to comment on the case of covering the light source with an opaque sheet periodically? How does this modulation produce additional LIGHT frequencies if it doesn't emit light? – Meep Jan 10 '18 at 16:58
• As the previous comments say, its all about Fourier transform. Imagine you have a segment of harmonic wave. The only fact that it is finite makes its fourier transform is not a Dirac Delta but a sum of so many frequencies. IT doesn't mean they appear out of the blue. It means that the EFFECT of having a cut harmonic wave is the SAME as if you had all those frequencies. Physically, if both descriptions are equivalent, you should work with the easier/more useful one. – FGSUZ Jan 10 '18 at 17:08
• A bit of of an obtuse interpretation of your opaque sheet modulation is that the light is to interpret it as if the medium through which the light is travelling (air and sometimes opaque sheet) as a nonlinear medium. It is non-linear by virtue of the fact that the transfer function (dielectric function, susceptibility, whatever you prefer) is not constant in time but is time-varying. A time-varying susceptibility gives you a non-linear medium which is able to take light incident at one frequency and mix it to other frequencies. – jgerber Jan 10 '18 at 17:20

The fact that a modulated signal cannot be purely single frequency comes from the modulation property of the Fourier transform:

$$\mathcal{F}\left\{x(t)y(t)\right\}=\mathcal{F}\left\{x(t)\right\}\star\mathcal{F}\left\{y(t)\right\}$$

where $\star$ indicates the convolution operator.

However, I can still imagine having a steady single frequency infrared source, and moving an opaque sheet in front of it periodically to produce the series of pulses. Now I can't figure out how that could somehow introduce additional frequencies to a single frequency signal!!! The sheet doesn't produce its own light after all!

If your concern is that the light source emits photons only at a single frequency, and you don't think it's possible for the modulation process to change the frequency of these photons, remember the time-energy version of the Heisenberg uncertainty principle:

$$\Delta E \Delta t \ge \frac{h}{2\pi}$$

By gating the times when photons can pass through an aperture, your opaque-paper modulator has reduced the uncertainty in the time at which the photons pass through the aperture. Therefore increases the uncertainty in the energy of the photons, which is directly related to the their frequency by $E=h\nu$.

• This particular convolution can be done explicitly. Why not do it and plot the results? – DanielSank Jan 10 '18 at 17:25
• @DanielSank, because I'm just trying to explain the conceptual issue, not do OP's homework for them. – The Photon Jan 10 '18 at 17:35
• Thank you for your reply. I think I get the convolution effect now, but I am not entirely convinced by the quantum explanation for the opaque sheet case. Surely a classical physicist would have some way of explaining the phenomenon? It just seems like we are in the classical, and not quantum, realm. A commenter on my original post also had a different explanation... Thank you! – Meep Jan 10 '18 at 19:18
• @21joanna12, The classical explanation is that blocking and unblocking a light beam is a form of modulation and so the modulation property of the F.T. tells you that this process will change the spectrum. – The Photon Jan 10 '18 at 19:21

First off, I agree with The Photon's answer and other comments about that the modulated spectrum is the convolution of the original spectrum (monotone at the lasers optical frequency - this is the carrier tone) and the signal, that is the pulse train at much lower frequency.

21joanna12 then raises what I think be an interesting question which is how can we get NEW frequencies of light simply by modulating a laser beam? I think this question is motivated by the knowledge that it is typically difficult to shift the frequencies of light. We must send the light through some non-linear medium which typically involves high intensities, careful alignment, etc., yet this seems to be a way to change the frequency of light without needing a non-linear medium.

Here is an attempt to reconcile these two notions of frequency shifting of light. A non-linear medium is characterized by having a non-linear electric susceptibility. If the susceptibility is linear then you know that passing through the medium will not shift the frequency of light. That is why you need a non-linear susceptibility for non-linear optics.

There are other ways to think about the non-linearity however. One way to think about the non-linearity is that it means a pulse of light entering the medium alters the medium in a way that light coming in shortly after the first pulse will experience the medium differently. This shows that there is some time dependence of the response of the medium.

In other words, I am trying to draw an equivalence between a time varying response function of an optical medium and a non-linear susceptibility for the medium.

I'll summarize in the next section explaining how the frequency shifting properties of both a non-linear medium and an opaque modulated sheet can be viewed in both of these ways.

First, viewing both as a time varying response. -In other answers/comments you have learned how modulating the signal gives rise to new frequencies because of the Fourier components of the new signal. This is how the time varying response of the opaque sheet gives you new frequencies. -I have argued above that the non-linearity of a non-linear medium means the medium actually has a time-varying response. If you followed everything carefully you could see how the time-varying response of the medium is processing the incident light field and misshaping it from being a perfect sinusoid at one frequency to being a sinusoid at two more different optical frequencies. Clearly the material must be able to response very quickly to alter the light field on such short time scales, but electrons in materials can respond very quickly.

Second, viewing both as an optical non-linearity -In the case of a non-linear medium there are many references explaining how a non-linear susceptibility function "mixes" two signal and gives you new signals at sum and difference frequencies. This is the usual explanation for optical non-linearities. -How do we view the opaque sheet as a non-linear medium? Well, above I was trying to draw some kind of equivalence between a non-linear medium and a medium with a time-varying susceptibility. Consider the volume of air in the region where the opaque sheet is being modulated. Much of the time this volume of air has the response function of air, basically it doesn't change anything about the light field. But some of the time it has the response function of the opaque material. That is to say, the response function of this volume of space is modulated in time. We could certainly cast this is a time-varying susceptibility and it may be possible to derive some sort of non-linear frequency susceptibility for this sort of system.

In any case, the reason I've gone into such detail here is to try to illustrate that (perhaps with some extreme stretching of the formalism/imagination) it is actually the same thing going on when either a modulated opaque sheet or a non-linear medium changes the frequencies of a beam of light. That is, both processes can be analyzed in either the time or frequency domain. This may be a bit confusing but maybe it can shed light on both why a modulated sheet CAN add new frequencies to a light beam and also WHY a non-linear medium adds new frequencies to a beam of light.