Resistance values under transformations in regular triangular grids

In [Evans, Francis 2022; Hendel] the authors investigated resistance distance in triangular grid graphs and observed several types of asymptotic behavior. This paper extends this work by studying the initial, non-asymptotic, behavior found when equivalent circuit transformations are performed, thus reducing the rows in the triangular grid graph one row at a time. The main conjecture characterizes when edge resistance values are less than, equal to, or greater than one after reducing an arbitrary number of times a triangular grid all of whose edge resistances are identically one. A special case of the conjecture is shown. The main theorem identifies patterns of repeating edge labels arising in diagonals of a triangular grid reduced $s$ times provided the original grid had at least $4s$ rows of triangles. This paper also improves upon the notation and concepts introduced by the authors previously, and provides improved proof techniques.Comment: Intent to submit to Discrete Applied Mathematic

Continuously tunable modulation scheme for precision control of optical cavities with variable detuning

We present a scheme for locking optical cavities with arbitrary detuning by many linewidths from resonance using an electro-optic modulator that can provide arbitrary ratios of amplitude-to-phase modulation. We demonstrate our scheme on a Fabry–Perot cavity, and show that a well-behaved linear error signal can be obtained by demodulating the reflected light from a cavity that is detuned by several linewidths.National Science Foundation (U.S.) (PHY-0757058