I have been studying the paper you mentioned previously ‘compact focusing grating couplers for silicon-on-insulator integrated circuits’ and tried to draw it in FDTD.

The equations from this paper do not agree with the equations from the first post of @gusdnr00211.
The paper yield a, b, center values of

When drawing these equations I did not get neither a smooth grating, nor the required fill factor.
Can you guide me what I am doing wrong?

The elliptical grating equation from the reference can be converted into a parameterized form:

It should be noted that the formula above assumes that the grating supports only a single order (m=1). The pitch of the grating is automatically determined based on the fiber angle, target wavelength, effective index of the waveguide mode and the refractive index of the environment.

Here’s the simulation file (6025.fsp) (394.6 KB) containing the structure group for the elliptical surface grating.

Hi skim
I have a question about the focusing grating coupler. How do you get the effective index neff of the grating coupler? I hope to get your feedback. Thanks in advance.

Hi @jbwei
I hope @skim will answer and confirm/correct the following information.
neff of the grating is calculated from the following law:
neff=ff*neff1+(1-ff)*neff2
where ff is called the fill factor defined as (the tooth width/grating period). neff1 and neff2 is the effective indeces of two slabs of thickness h1 and h2 (the tooth height and slot height respectively).

Hi omnia.nawwar
Thanks for your reply.As you said ,the effective index neff is equal to ff*neff1+(1-ff)*neff2.So If I calculate the neff1 and neff2, then I can get the neff value.the picture below is what I draw.Is my understanding right?

furthermore If the GC is apodized,which means that the pitch and ff is different in differen period.Then how can I draw the focused GC structure?
Hope to get your feedback.Thanks a lot.

dear gsun,skim and omnia.nawwar
Thanks for your reply.firstly I want to tell the reason why I want to calculate the neff of different FF.
The reason is that now I have designed an apodized GC, which means that the FF in each pitch is different.Now I want to draw the focus GC structure with the different FF and pitch. So do I should calculate the neff with different pitch.
furthemore I have a question about the script of the focus GC.In the script there is a sentence I dont understand.

I think the formula provided in the reference should not be taken too seriously, especially when you have varying fill-factors and pitches in your grating. The various parameter values you choose based on this formula can probably make a good initial guess towards an optimized design. But there is no guarantee that the simple formula will give you the desired performance of the grating. In that sense, the choice of neff value for the formula and hence the resulting pitch of the grating can be a bit arbitrary, as @gsun has mentioned.
It might be probably fine to start with an neff value of the unetched region. Or you can use an average for the etched and unetched region. For the appodized FF and pitch, you might want to try the average of the whole region. Whatever you choose for your initial guess for the neff, you will eventually need to sweep the pitch, fill factor, fill factor variation rate, etch depth, etc. in your simulation to be able to achieve an optimized grating.

For the question about extra terms in the definition of the ‘Vout’, please have a look at the following post in Korean category. The figure there will probably provide you with enough information.

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