Help with understanding of RF theory-Kinetic inductance parametric amp

[dp20051]

TL;DR Summary

I really need help understanding the principles behind the microwave engineering/electrical engineering aspect of structure and principles behind why it works. I have performed multiple simulations using Sonnet but it is very apparent that my understanding of the fundamentals are wrong and I have tried to figure out where my understanding is going wrong but I can't seem to wrap my head around what is missing. This is ultimately in relation to a Kinetic Inductance Parametric amplifier (KIPA)

So this might be long question that requires some literature review but I will try condense it as much as possible such that hopefully I can get some help without the reader having to review the related paper.

So I will start off by saying that I am involved in a honours thesis in which I need to come up with a design that replaces a certain component. In this case that component is a stepped impedance filter(SIF)/(Bragg mirror). These seem to be used interchangeably as I see the effectively produce the same outcome.

This Bragg mirror is one component used in an overall system called a kinetic inductance parametric amplifier (KIPA). Now my task is to design a filter that has a smaller footprint than that of the SIF/brag mirror yet still produce the same overall functionality of the KIPA.

I will include a picture here for reference:

So I am only interested in the purple box area. This includes the bragg mirror/SIF galvanically connected to a 1/4 wavelength resonator.

Lack in my understanding:

My undertanding of the basic principles of how the microwave enegineering works is that the bragg mirror/SIF effectively acts as a mechanism to "trap" the signal we want to amplify in the 1/4 wavelength resonator for a longer duration allowing further amplification or I guess a higher quality factor Q.

This 1/4 wavelength resonator is shorted to ground on the other side.

The signal that we want to resonate in the 1/4 wavelength resonator is 7.2GHz and the pump signal for PA is double that 14.4GHz

All of this is designed in a CPW structure. (This is getting lengthy so I may include the paper i'm working off if anyone is kind enough to skim the parts that I may have missed.

Now I have designed a CPW filter that has a notch at 7.2GHz through sonnet to replace that of the bragg mirror. My understanding is that if it is to produce a similar effect as the brag mirror then we need the |S11| = 0dB at 7.2GHz or close to meaning that it will reflect the desired signal and trap it in the resonator for longer as does the bragg mirror.

Design with |S12| :

|S11| graph: I thought this would be enough information to deduce that this design should at least emulate the affect of the SIF/Bragg mirror and thus be a viable substitute.

I have been told by my advisor that the |S11| parameter is not useful in making such a conclusion and that I have not shown that this filter connected with the 1/4 resonator will even perform resonance.Would someone kind enough please if they have the time explain to me why this is not the right conclusion and that I need to simulate this filter with the 1/4 resonator attached. I've also been told if it does not resonate then try open circuited 1/4 or try 1/2 wavelength etc until resonance occurs.

For the 1/4 wavelength resonator that is shorted to ground on one end my knowledge is that it is inductive in nature. Either way this segment should act as a resonator at the desired frequency from using Vp?

I was advised I would need to check the phase to see if resonance is occurring yet the whole point I thought of the bragg mirror was just to increase the quality factor. Hence why my design with a |S11| = 0dB at 7.2GHz and a |S12| = -40dB at 7.2GHz would suffice. This would allow the pump frequency in and the keep the 7.2 GHz signal trapped in the 1/4 resonator for longer. I guess could someone please explain to me what parameters I need to see what is going on with this 1/4 resonator and its interaction with my filter to be able to conclude it allows resonance to occur?

So to summaries I thought the 1/4 resonator was the component in which the amplification/resonance occurs, and the filter is just there to increase quality factor Q.

Maybe my understanding on shorter/open 1/2&1/4 wavelength resonators are wrong. Please explain it to me as simple as possible.

I also understand that I could effectively use the S parameters of the 1/4 resonator and compute the ABCD matrix then go from there? I'm not sure exactly where to though, the Reff ?I also understand there are way more details in how this whole thing works with kinetic inductance and superconductivity etc. Yet I need to understand this before I move forward.

This is a detailed question, and I am truly grateful if anyone takes the time to help me understand this. Thanks guys.

Here is the research paper for reference:

https://arxiv.org/pdf/2108.10471.pdf

 

[f95toli]

I'm not sure where to start here.
Firstly, forget about the parametric amplifier part here; it is not actually important.
The EM problem (the bit you can do in Sonnet) here is that you are trying to design a high-Q microwave resonator which is galvanically (=DC) coupled to the outside world (this DC coupling is needed because of the way the amplifier works, but as mentioned the reason is not really relevant)

Now, there are several ways to do this. Bragg mirrors is one way, but there are also others.
Two terms worth googling is "Stepped impedance notch filter" and "photonic bandgap filter"

A "regular" lamda/2 or lambda/4 resonator will just be capacitively or inductively coupled; but that won't work here since you need DC coupling. Instead, what you are trying to do is to create a huge impedance mismatch at the frequency of interest This mismatch will then reflect the MW signal while still allowing of galvanic coupling.

I am not actually not quite sure what your question is, you seem to be on the right track.

But my suggestion for a "design flow" would be to start with a simple circuit model where you model the resonator as an LCR circuit (try LTSpice) . Once you have some idea of the parameters, you can then move on to using a ABCD matrix.
Finally, you should be able to model the whole resonator in Sonnet; assuming you are using one of the paid versions. You can model the superconducting resonator as being a very good conductor (say a couple of order of magnitude better than gold). If you have the full version of Sonnet it can actually handle kinetic inductance, but it is very slow and not needed for this design.