Creating uneven sidebands with AM + PM modulation? 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.





(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. 

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.



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)
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.