Can anyone give me a reference that I may find a closed-form formula for the output electric field (not power) in terms of the input small-signal electrical waveform (not input power) of an electroabsorption modulator?
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1\$\begingroup\$ "the community of fier-optic communications seems not to be mathematically oriented" I know a lot of people who would that statement personal. It's definitely wrong – all progress in fiberoptical communications in the last twenty year is extremely math-heavy, from modelling of nonlinear effects, finding mutual information-optimizing constellations for that, to very large iterative channel codes and their theory, over to quantum communications and the algebraic codes there: I don't think you'll find many disciplines within electrical engineering with more math. \$\endgroup\$– Marcus MüllerCommented May 18, 2023 at 20:05
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\$\begingroup\$ So, anyways. What kind of EA Modulator are you referring? Generally, field strength is root-proportional to power, and the input field is defined by whatever you put into the modulator; the output field will heavily depend on geometry. If you want the physical modelling of EA modulators, you'd go back to the late 1950's original publication of Keldysh & Franz, after whom that effect got its name. But honestly, the whole math behind that is "simply" based on photons being particles when they interact with appropriately prepared materia (i.e., by exciting an electron); so the loss effect is \$\endgroup\$– Marcus MüllerCommented May 18, 2023 at 20:15
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\$\begingroup\$ an effect on intensity (photon count); only as you go back to the propagating wave, you see the wave, now with reduced intensity, and hence square-root reduced field strength. So, could you elaborate on what you need? "A formula" is "take the intensity formula of your specific EA modulator, and calculate the field strength that gives you that intensity", with phase being dictated by geometry. \$\endgroup\$– Marcus MüllerCommented May 18, 2023 at 20:17
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\$\begingroup\$ @MarcusMüller Thanks for your comments. I deleted the first paragraph. However, I am not looking for the magnitude (intensity) of the output power. I am lloking for the complex-valued electric field E(t) at the output of EML in terms of the complex-valued electrical valued input v(t). The functional that maps v(t) tp E(t) may itself depends on physical parameters of the modulator. \$\endgroup\$– CLAUDECommented May 18, 2023 at 20:21
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\$\begingroup\$ I addressed that. the amplitude of the E-field is dictated by the intensity transfer function, the phase by the geometry. \$\endgroup\$– Marcus MüllerCommented May 18, 2023 at 20:24
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The amplitude of the E-field is dictated by the intensity transfer function, the phase by the geometry of your specific EA modulator. Hence, no general formula exist, other than just taking the square root of the absorbtion coefficient as amplitude factor, and setting an arbitrary phase as given by how the modulator is constructor, i.e. its phase length.
For a proper derivation of the effect underlying EA modulation, and thus a mathematical discussion of the E-field,
C. Hamaguchi, Basic Semiconductor Physics. in Graduate Texts in Physics. Cham: Springer International Publishing, 2023. doi: 10.1007/978-3-031-25511-3.
Section 5.1.2 discusses the Franz-Keldysh effect, which is how your modulator works, and describe the E-field relations (well, it derives the imaginary part of the dielectric constants, which is just as good, through trivial application of the wave equation). Notice how this is solid-state physics: you cannot make statements on fields without looking at the particle side of your photon, so modelling as intensity effect is inherent – you derive the expressions from the energy operator; intensity is not an afterthought, it's the "cause" (if you can speak of that in this context), because it's individual photons interacting with electrons that cause the absorption.
answered May 18, 2023 at 20:36
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\$\begingroup\$ Note that I'm not a physicist. These texts are heavy for me as well; but you keep insisting that deriving the E-field strength from the intensity transfer function and tacking a phase to it isn't good enough for you, so, well, semiconductor physics is the answer to that. \$\endgroup\$ Commented May 18, 2023 at 20:38
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\$\begingroup\$ Thanks Marcus for your expansive reply. Based on your explanation and the fact that the phase of EML modulators are highly dependant to their geometry I guess that these modulators are only used for IMDD and intensity detection, not for IQ modulation using some pulse shaping, e.g., roor raised cosine etc. Am I right? \$\endgroup\$– CLAUDECommented May 19, 2023 at 16:45
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\$\begingroup\$ Not sure that is the case! I also don't think your implication makes much sense: phase will always depend, for any kind of device, on geometry. \$\endgroup\$ Commented May 20, 2023 at 12:00