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Imagine two scenarios: two cladding materials with the same real refractive index, but one has absorption while the other does not. Now, these materials act as the top cladding of a waveguide (for simplicity, let's assume there is no absorption in the core material).

Is the real part of the propagation constant of the waveguide, beta, the same for both scenarios? Or does the presence of absorption change the real value of the propagation constant of the waveguide?

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I'm not sure if this is what you're asking, so just maybe: the overall absorption of the signal is the combination of loss in the core and loss of the fraction of the signal in the cladding. I don't think that these absorption losses affect the phase propagation of the "real" wave directly.
You could make the argument that, since the index of refraction of the cladding differs from that of the core, and thus the cladding-portion of the signal is dispersing from the core-portion, if there is or isn't absorption in the cladding, the magnitude & spectral distribution of the dispersion of the summed signal (cladding and core portions) will depend on the amount of loss in the cladding.

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