Learning Forum

[Mohamed Selim]

https://optics.ansys.com/hc/en-us/articles/360036108194-Optical-Fiber-FIBER-INTERCONNECT-Element
I have a question regarding the optical fiber model, it is written that with differential group delay (DGD) the difference of 1/Vgx & 1/Vgy is denoted as Delta_Beta and is added to Beta_2 in the NLS equation.   


Something does not make sense to me, the units of Beta2 and Delta_Beta are different, Beta2 is related to the second deriviatve and it shows the change in group velocity with frequency while Delta_Beta shows difference of group velocity. Should not Delta_Beta be added to Beta_1 not Beta_2 ? 
Beta_1 is omitted from this equation to remove the fiber delay. 

Thanks

[Guilin Sun]

 

Thank you for posting this question. I am sorry to reply you late as my colleage is out of office longer than expected.

The equations are from references listed at the bottom of the page.

I do not have access to the papers.

Beta2 is the 2nd derivative of beta, whereas delta_beta is the difference of inversed group velocity (with coefficients). The group velocity is the inverse of 1st derivative of beta. So the delta_beta2 is simular to beta2 in the sense of unit, as it is a difference. however it may omitted delta_w or assuming it is unit or other assumption, or simply it missed some quantity.

I would strongly suggest that you refer to the reference papers to make sure the theory is correct. I believe the solver its self uses correct formula (and it is discrete), as customers use it to design optical circuits with agreement to the experiments.

 

 

 

 

 

[Mohamed Selim]

 

Let me make my point more clear, first regarding Beta_2, its equation is: 


The unit of Beta(w) is (1/m) where m is (meter) and units of omega is (1/s) where s is seconds, so the units of Beta_2 from the previous equation is (s^2/m) 

The defintion of Delta_Beta as you said is inverse of difference of group velocities : 


Which makes its  unit (s/m)

So you can’t add Beta_2 (s^2/m)  with delta_beta (s/m). 

I have checked the first reference “Nonlinear Fiber Optics, 4^th ed.”  and could not find this equation, actually the equation I found supports my argument and here it is: 


In the formulation provided in Lumerical, the term Beta_1 (which accounts for the simple fiber delay) is omitted to reduced simulation time, while Delta_Beta is embedded inside Beta1_x & Beta1_y and it is not related to Beta_2. 

I have also checked the second reference and it does not address the polarization dispersion in its formulation, so kindly tell me where is this equation in the paper?  

Maybe the simulation uses different equation than the one described in the model link. 

I have other question posted somehow related to polarization mode dispersion model in :  /forum/forums/topic/lumerical-interconnect-fiber-pmd-effects/ 
kindly have a look and answer it. 

Thanks 

 

[Guilin Sun]

For the first part, as I said prviously, it might losed a term delta_w or assumed that delta_w is unit with 1/s. So yes they cannot be directly added together without such delta_w. It uses delta_beta_2 with a purpose to distinguish it from delta_beta.

But seems Lumerical does not use beta1 (6.1.13).

From the Taylor series:

beta=beta0+beta1*delta_w+0.5*beta2*(delta_w)^2+...

where beta1=d(beta)/dw=1/vg.

so (beta1x-beta1y)/delta_w has the same unit as deta2. This is why I said there might missed a delta_w.

for the other post I have asked my colleage to help you as I am not expert for INTERCONNECT.