We developed a type of toroidal multi-pass cell with multi-layer patterns based on the off-axis model. The effective path length of the original toroidal multi-pass cell is extended several roundtrips in comparison with the single-layer pattern, since the inner surface of the toroidal multi-pass cell is more efficiently utilized. The light pattern has been achieved by using the simple ring surface, which is easy to fabricate. The exact analytical equations for the design of the toroidal multi-pass cell were derived based on analytical vector calculations. A series of numerical ray tracing simulations is presented, and the maximum theoretical optical path length that can be reached is 30 m with a setup of 5 cm column radius. Furthermore, two practical spot patterns are demonstrated with a path length of 8.3 m for a two-layer pattern and 10 m for a three-layer pattern, with respective effective volumes of 63 mL and 94 mL. Furthermore, the fringe effect is substantially reduced to less than 0.5% by the usage of our designed mask.

We propose to utilize the ${}^1\mathrm{\Sigma }{-}^1\mathrm{\Sigma }$ electronic transition system for direct laser cooling of heteronuclear diatomic molecules. AgH, as well as its deuterium isotopolog AgD, is used as an example to illustrate the cooling schemes. Potential-energy curves and relevant molecular parameters of both AgH and AgD, including the spin-orbit constants and the electronic transition dipole moments, are determined in internally contracted multiconfiguration-reference configuration interaction calculations. The highly diagonal Franck-Condon matrices of the $A{}^1{\mathrm{\Sigma }}^{+}-X{}^1{\mathrm{\Sigma }}^{+}$ transitions predicted by the calculations suggest the existence of quasi-closed-cycle transitions, which renders these molecules suitable for direct laser cooling. By solving rate equations numerically, we demonstrated that both AgH and AgD molecules can be cooled from 25 K to 2 mK temperature in approximately 20 ms. Our investigation elucidates and supports the hypothesis that molecules in the simplest ${}^1\mathrm{\Sigma }{-}^1\mathrm{\Sigma }$ system can serve as favorable candidates for direct laser cooling.