# [10GMMF] Please delete Refractive index profiles and DMD

```Dear All,

I would like you to delete and not use my document of last week entitled "Refractive index profiles and DMD scaling".

Reason:

Whilst the derived scaling relationships between DMD and refractive index profile is correct (within perturbation theory) per the contribution presented at the May interim "More information on statistical modelling of MMF optical links" (cam_1_0504.pdf).  Expressions 6, 7, 8, 10, 11 for the modal delay times ignore the k dependence of the modal wave functions and should not be used to calculate modal delay times. I made this approximation to simplify the explanation however I have become aware that the implication is that equation 6 is a general result of perturbation theory which it is not.

Instead of using my withdrawn document (which I assume you have now deleted) I suggest using the following reference:

R. Olshansky, "Pulse Broadening caused by deviation from the optimal index profile" Applied Optics, Vol.15, No.3, pp782-788,March 1976.

Equation 46 of that paper, gives the correct form of the average delay time for index perturbations relative to the optimum power law refractive index profile.  The second term in the equation is the relative delay as a function of: the mode power distribution, the relative index perturbation and a weighting function.  From this equation it is clear that if the scanning spot DMD is calculated using an unscaled relative perturbation and the DMD is then scaled by a factor, S, then the equivalent scaled index perturbation is just the unscaled perturbation scaled by the same factor S.

Therefore, the scaling equations presented at the May interim are correct (within perturbation theory). Specifically, for the 81 fibre model:

To find the scaled refractive index profiles simply scale the relative refractive index perturbation by the DMD scaling factor.  For the power law perturbation segments this leads to the relationship between the power law exponents gs, g and go as stated in May.

Regards,
David

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