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I am trying to modulate an optical signal such that I create a pair of sidebands with arbitrary phase and amplitude with respect to each other. For example I would be like to be able to modulate my carreir signal, centered at 100, to have one of the two resultant spectra after my modulation.

specrtum 1

spectrum 2

(Blue is the real component of the spectrum and yellow is imaginary).

So that the 110Hz component has a different phase and amplitude form the 90Hz component.

If I take the inverse fourier transform of these two spectra I get real signals which look like this. signal 1signal 2

Both of these two signals aren't unphysical, however I don't know how to analyse them in what way I'd need to modulate my carrier to create them. Is it possible to recreate these time signals with Amplitude and phase modulation alone?

Thanks! :)

Edit: In response to Dan Boschen. This is what my spectrum looked like when I did phase modulation. I am using a large modulation "depth" of 0.5. When I reduce it the harmonic components do become negligible, so I guess in that case I do get just two sidebands.

Mathematically what I'm doing. Where fc is the carrier frequency and fm is the modulating frequency.

s(t) = cos(2\pi f_ct + 0.5\times cos(2\pi f_mt))

My python code to generate that signal:

timeStep =0.0001
t = misc.stepRange(0,1,timeStep)

#carrier freq
carrierFreq = 100

#modulating freq
modulatingFreq=10


#carrier= 10*np.cos(t*2*np.pi*carrierFreq)

modulatingSig= np.cos(t*2*np.pi*modulatingFreq)

modIndex = 0.5
resultantSig1 = np.cos(t*2*np.pi*carrierFreq+modIndex*modulatingSig)

Phase modulation

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  • $\begingroup$ Some additional information might help: (a) why are you trying to do this, (b) what are the properties of the data signal you're trying to modulate, (c) what operations can you perform on that signal. $\endgroup$ – MBaz Oct 28 '19 at 0:30
  • $\begingroup$ 1) It's an optical laser signal which will be used for an optical communication protocol. 2) I want to be able to be able to select the amplitude and phase of the sidebands to create a digital multibit character with each side band individually. 3) As it's an optical signal and the operations I can do are AM and PM modulation. $\endgroup$ – generic purple turtle Oct 28 '19 at 7:04
  • $\begingroup$ Yes but to control this accurately you will need phase coherency between your AM and PM modulations. AM for a sinusoidal waveform of a single frequency has two complex conjugate (same magnitude opposite phase) sidebands, PM does as well, but with the two sidebands in quadrature phase related to the AM. It is easy to see this on a vector diagram, and how only with both AM and PM present can you achieve two sidebands that are not the same magnitude. $\endgroup$ – Dan Boschen Oct 28 '19 at 17:04
  • $\begingroup$ @genericpurpleturle Are you asking how to easily translate from the magnitude and phase of your two sidebands to the baseband modulation signal (I and Q) that would create those? If so, I can elaborate as the technique to do that is quite simple and straightforward (no need for IFFT etc). $\endgroup$ – Dan Boschen Oct 30 '19 at 3:47
  • $\begingroup$ Yes that is what I'm asking! However, from my understanding of using I and Q quadratures in electronic signals, you need to split your carrier into two and delay one by pi/2, modulate both parts separately and then recombine in a summation. However when it comes to optics, I believe that the only summation you can do is with a beam splitter which has two output ports. Although that may still be okay, if you take just one output and just model it as a summation which also halves the amplitude? $\endgroup$ – generic purple turtle Oct 31 '19 at 16:38
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If I understand correctly, you want to use each side band as an independent information-bearing signal. In other words, you want to be able to generate $$a_k e^{2\pi f_i t}$$ with the information in carried by the amplitude $a_k$. There are several ways to do this, but I don't know which are feasible using optical processing. All of the following methods are feasible if you start with discrete signals and then convert them to optical.

  • Single-sideband modulation. The idea is that $$s(t)\cos(2\pi f_i t) \pm j\hat{s}(t)\sin(2\pi f_i t),$$ where $\hat{s}(t)$ is the Hilbert transform of $s(t)$, is a single side band signal or, in other words, one of the two sidebands you want to create. By choosing $+$ or $-$ you select the lower or the upper sideband.

  • Use the IFFT to go from any given spectrum to the time domain. This is the way OFDM works.

  • Use quadrature modulation (QAM). With this technique, you don't get to control each sideband independently; rather, you transmit two independent bitstreams by mixing two different versions of the sidebands. I have heard that QAM is possible to implement optically.

I hope this answer points you in some fruitful direction.

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  • $\begingroup$ Both on the amplitude a_k and on the phase of that corresponding peak (which you haven't included) as well. It is helpful to know that QAM is doable in optics, as I assumed it wasn't as there no direct analog of summation in optics. There are beam splitters, which can add two signals and have to output ports. So you could probably do it afterall, and compensate for the fact that you have a halving of amplitude during your summing operation. Single-sideband I also came across, but OFDM is something I'll look at. Thanks. $\endgroup$ – generic purple turtle Oct 31 '19 at 17:12
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Theoretically, yes. In the worst case, make your prototype wave in digital-land using whatever method you want. Then if your signal is $x(t)$ just calculate phase and magnitude: $\phi(t) = \arg \left (x(t) \right)$ and $m(t) = \left | x(t) \right |$.

Your biggest difficulty is there could be signals (e.g. if you have just two tones beating against each other with similar magnitudes) that would cause phase instantaneous phase changes of $180^\circ$ -- but these would coincide with $m(t) = 0$.

If your phase change is agile enough -- no problem! If your carrier is strong enough that the sum of the sideband signal never has magnitude greater than the carrier -- no problem!

I can anticipate all sorts of horrible practical problems involving keeping your phase and amplitude modulators synchronized. I suspect that the physical results will always have strange little deviations from ideal -- you'll have to assess just how bad that would be, and how to accommodate the issues.

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  • $\begingroup$ If I've understood you, with the case you've described, that's how to create the arbitrary wave from scratch right (without carrier)? I can't do that optically unfortunately. I will have a carrier frequency, which is fixed, and then I will be optically modulating that carrier frequency. $\endgroup$ – generic purple turtle Oct 31 '19 at 17:33
  • $\begingroup$ Well, yes, but if I understand you that's pretty much what you want -- what limitation am I leaving out of my answer that you see in your hardware? $\endgroup$ – TimWescott Oct 31 '19 at 19:42
  • $\begingroup$ As in, if I create an arbitrary signal in digital land. The signal could be such that I won't be able to recreate it in the real world, by simply using AM and PM on my carrier signal. Maybe I'm unclear what x(t) is. Is it the final encoded signal, with sidebands and the carrier frequency? If so applying that to a carrier, is surely not correct? If it's the signal which creates the desired encoded signal of the carrier + sideband, my issue is I don't know how to analytically find x(t), to create the desired encoded signal which has the correct frequency spectrum. $\endgroup$ – generic purple turtle Nov 4 '19 at 10:42
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For small angles (see further explanation at end for further details on small angle approximation) the sidebands for phase modulation are closely related to the sidebands for amplitude modulation as revealed in the IQ phasor diagrams below.

AM and PM

Both diagrams show large carrier AM and PM modulation being modulated by a single sinusoidal tone, resulting in two sidebands in each case. The carrier is represented by a fixed phasor along the real axis, and each sideband is represented by the two rotating phasors which would rotate at the angular rate given by the modulation (the phasor rotating counter-clockwise represents the upper sideband, and the phase rotating clockwise represents the lower sideband, and the relative magnitudes of the phasors to the fixed carrier is the relative magnitude for each of those sidebands).

The net result is the addition of all the phasors shown.

In the case of the AM diagram shown, the two rotating phasors will always have imaginary components equal and opposite, which cancel, resulting in a real vector sinusoidally varying in magnitude that adds to the carrier (thus only the amplitude is modulated).

In the case of the PM diagram shown, the two rotating phasors will always have real components equal and opposite, which cancel, resulting in an imaginary vector sinusoidally varying in magnitude that adds to the carrier. Using the small angle approximation, the ratio of the magnitude of this vector to the fixed carrier vector is the angle in radians.

From this we see how we can control the amplitude of each of these sidebands from coherent AM and PM modulations:

AM and PM

Here we have the summation of AM and PM modulation components, in this case with the upper sideband (counter-clockwise rotation) of the AM in phase with the upper sideband of the PM, and the lower sideband of the AM in anti-phase with the lower sideband of the PM, the upper sideband of the combined waveform is the sum of the AM and PM amplitude components while the lower sideband is the difference.

Further notes on small angle approximation:

The AM modulation as shown will always have two sidebands for the case of a single tone sinusoidal modulation, as given by:

$$1 + k(e^{j\omega t} + e^{-j \omega t})$$

Where $ke^{j\omega t}$ represents a phasor of magnitude k and angle dependent on time ($\omega t$). Here the carrier is represented as magnitude 1 and each sideband has a magnitude k.

Using Euler's identity this is equivalent to:

$$1 + 2k\cos(\omega t)$$

Which is completely real, and thus only the amplitude of the signal is changing while the phase remains equal to 0.

The simplification of just having two sidebands for the PM case is only an approximation that holds well for small angles. The PM equation for this case is

$$1 + k(e^{j\omega t} - e^{-j \omega t})$$

Which reduces to

$$1 + j 2k\sin(\omega t)$$

This is a much more complicated formula which described in terms of magnitude and phase components is

$$\sqrt{1-4k^2\sin^2(\omega t)}e^{j2k\sin(\omega t)}$$

For small angles $\phi$, the $\sin(\phi) \approx \theta$ and thus the magnitude as given by the above formula in these cases is approximately 1 and the the phase modulation $\phi(t)$ is equal to $2k\sin\omega t$. For larger angles, the phase modulation is the same but an incidental AM will be introduced if we are limited to having only two sidebands. (Thus in pure PM which must stay on the unit circle meaning no AM, we will see additional sidebands appear as the angles increase which serve the purpose of keeping the net sum of all the phasors involved on the unit circle. The magnitude of each of these sidebands, which occur at multiples of the modulation rate, are given by Bessel Functions of the First Kind).

This is made clearer by observing the next figure where we see the desired PM in addition to incidental AM that will exist if we are limited to just two sidebands. The desired PM is shown as a single phasor which for a sinusoidal modulation resulting in only two sidebands would move vertically up and down in sinusoidal fashion. As it thus moves up and down, the phase will modulate as desired, however due to the restriction of remaining vertical (as constrained by the two sidebands as depicted in the first figure for PM), an incidental AM modulation will also result. This AM will be non-linear and contain many spectral components, initially the second harmonic will be dominant. Because the pure PM (which would result in the phasor staying on unit circle as the phase cycled sinusoidally) also contains many spectral components, the result of the mixed AM annd PM in this case is such that all the higher harmonics cancel, resulting in only two sidebands (and a mixed AM/PM modulation). For small angles (sidebands < -20dB) this effect is negligible, and a two sideband estimate of pure PM can used in most cases (refer to Bessel functions to determine the strength of each sideband; here is a good reference for further reading on that: https://www.zhinst.com/blogs/michele/files/downloads/2012/12/AMFM.pdf?file=downloads/2012/12/AMFM.pdf

incidental AM

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