# Calculation of Phase Delay of a Curved Planar Dielectric Waveguide (i.e., bend)

I am working on a very phase sensitive planar dielectric waveguide device and I need to calculate the phase delay induced by a bend of some radius, $r$ and of $\phi$ radians for a give planar waveguide structure so that I can correct for it.

For a uniform, unperturbed straight dielectric waveguide of some dimentions height, $h$, and width, $w$, we use Maxwell's equations and the boundary conditions for the give planar waveguide profile to define the characteristic equations to determine the propagation constants, $\beta_{pm}$, for the modes, $m$ of the polarization of interest, $p$. Using the equation for the effective index, $n_{eff_{pm}}=\frac{\beta_{pm}}{k_0}$, we can obtain the index for the mode of interest. Then, using the relation for the optical path length, $OPL= nd$ we can easily calculate the phase delay induced by a given waveguide structured compared to another for a given physical, $d=d_1=d_2$, by using the optical path delay relation, $OPD= n_2d_2 - n_1d_1$.

I have looked in the literature, but I have yet to find an analytical relationship or even a way of aproximating the phase delay induced by such a bend as aforementioned. All that I can find is that the bend efectively acts as a higher index region and therefore compared to a straight region of equal physical length, $d=r\phi$, the bend will suffer a longer delay.

I have an idea of how to simulate for it using MIT's MEEP for the waveguide structures I am interested in, but the sheer number of simulations I will need to run will eat up weeks of time just for the few cases I want to check (because I strongly suspect there is not only a radius dependance, but also an angular dependance).

Does anyone know an approximation I could use for a single mode waveguide bend? Thank you!

• If you are that sensitive to phase changes, wouldn't you have to simulate the entire structure at once? – CuriousOne Jun 3 '16 at 6:42
• @CuriousOne, because it is massive for FDTD and I am worried about the validity of BPM in this case because of the high refractive index contrast and sharp bends. I'm trying to simulate a horseshoe type Arrayed Waveguide Grating (AWG). I can easily ignore the $90\deg$ bends at the top of the array of waveguides because they are all the same. But in the transition from the radial to parallel components, I can't ignore them because they are all different angles and I need to correct for the phase delay induced by them so that the path length difference between each arm is an equal difference. – ansebbian0 Jun 3 '16 at 7:09
• I see. My concern is that if the structure is sensitive in simulation, then it will also be sensitive to geometry changes in real life. Is there a recipe to remove the change in effective index by e.g. varying the width of the guide? We certainly play those kinds of games with strip-line structures on PCBs. If you can spare the simulation time, then it should be used to optimize the structure to have the least sensitivity before you build it and it turns out to be hard to reproduce. Aren't there any methods that are between BPM and full FDTD that can be used? – CuriousOne Jun 3 '16 at 7:15
• @CuriousOne, Thank you for your comment & suggestions. You are absolutely right, there are many tricks here to counteract the change in the effective index (including a delay in each line, and the width). My end goal is to use FDTD to simulate the whole structure (i.e., the AWG) as you mentioned. But, the time to simulate each whole AWG is about five days with my current server. So, as you can see, it would make for a maddeningly long process to try to optimize the whole structure at once. That said, I have to break it up into several sections, optimizing each, and then give it a final go. – ansebbian0 Jun 3 '16 at 8:02
• cont~. @CuriousOne. Concerning simulation methods, EME (Eigenmode Expansion Method) is another possibility which is better than BPM for accuracy since it is bidirectional and rigorous concerning Maxwell's Equations, and it is better than FDTD because it doesn't take so much time. Unfortunately, however, I don't have any software to do such a simulation for an AWG and I don't have the time/know-how to try to write such a program at this moment. FEM is always a good way to go for accuracy, but time intensively like FDTD. – ansebbian0 Jun 3 '16 at 8:19