Ring resonator with gap / varactor. How to tune second mode?

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Terminator3

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I can't understand how to use varactor for tuning ring resonator.
First, there is a ring with all resonances: n=1,2,3,...
Then, we add a gap, and odd modes dissapear, so we have n=2,4,8,...
And the last, we add varactor, and what happens? We need a second gap to maintain biasing.
As i do not want second gap, we can put varactors in (anti)series, so ring is GND, and varactor connecting point is bias.
After adding varactors, what would be tuned, n=2,4,8? Beacuse as i read from one article it says that there would be some half modes (1.5, 2.5, etc.), and only they would be tuned, while 2,4,8 are not affected.

Main questions, can n=1 resonant frequency be tuned? I think not, because gap killed resonance at 1,3,5,etc. odd peaks
Is it possible to tune n=2 resonant by adding varactor? Or only some "half modes" can be tuned?
 

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