We solve a popular effective Hamiltonian of competing d-density-wave and d-wave superconductivity orders, self-consistently at the mean-field level for a wide range of doping and temperatures. The theory predicts a temperature dependence of the d-density wave order parameter seemingly inconsistent with the neutron-scattering and muon scattering resonance experiments of the cuprates. We further calculate thermodynamic quantities, such as chemical potential, entropy, and specific heat. Their distinct features can be used to test the existence of the d-density-wave order in cuprates.

Triangular planar ring resonant cavities are widely used in cavity ring-down spectroscopes (CRDS) because of the unique optical path advantage. Hermite-Gaussian theory is the classical method used to describe the beam in the resonant cavity. Since this theory can express the transverse electromagnetic mode (TEM) in the meridian plane and sagittal surface separately in an astigmatic triangular cavity, it has a unique advantage in the analysis of triangular cavities. However, among the many analyses of triangular ring-down cavities, existing theories of Hermite-Gaussian modes have not been able to reasonably explain the many phenomena observed in alignment and ring-down experiments. In this paper, the mechanisms of the effects of first-order Hermite-Gaussian mode on mounting alignment in CRDS instruments are investigated in depth.The frequency separation characteristics of the lower-order mode in the triangular cavity are verified in the mounting alignment stage. The separation characteristics of first-order mode can guide the misalignment of the two orthogonal planes separately. The resonant frequency separation property of the Hermite-Gaussian mode in a triangular ring-down cavity can also be applied to more complex odd-numbered cavity mirror resonant systems.

The development of space surveillance capabilities has received increased emphasis with time. Suitable evaluation parameters are investigated to analyze the effects of multiple factors on short-arc angles-only initial orbit determination. This investigation is crucial for designing surveillance system parameters. In this paper, the orbit determination equations where the target state is linear were derived by assuming that Lagrangian coefficients are constant and taking the minimum residual sum as the optimization objective. From this paper’s analysis, it was demonstrated that because the position uncertainty distribution primarily lies in the line-of-sight vector, the upper limit of relative positioning error (RPE) can be used as the evaluation parameter of the determination accuracy in scenarios where a high-orbit target is observed by a low-orbit sensor observes. An empirical relationship was also found, indicating that the upper limit of RPE is proportional to the observation error, −2.5 power of the observation arc length, and the 0.5 power of the observation interval.

We propose a method for obtaining the centroid locating accuracy (CLA) of a monocentric imager with a fiber optic plate (FOP) as a relay image transmission element in order to reduce the loss of CLA due to the addition of FOP. We constructed a two-stage image transmission coupling model of spherical focal surface (FOP) image sensor. By analyzing the influences of FOP parameters, including the fill factor and the fiber diameter, and FOP in-plane displacements, including rotation and translation on CLA, the loss of the lowest CLA that the monocentric imager can withstand caused by the addition of the FOP was reduced by 20%.

Triangular cavities are increasingly used in cavity ring-down spectrometers (CRDS) to compensate for the existence of optical feedback and other disadvantages of straight cavities. However, at the same time, the difficulty of the cavity mounting alignment process is increased. Consequently, the fast and stable establishment of resonance between the optical source and the triangular ring-down cavity has been widely explored for CRDS-based applications. The Hermite-Gaussian theory has unique advantages for the description of the Gaussian resonant beams characteristics in triangular cavities, while the theory is of great importance for the alignment in high finesse triangular ring-down cavities. Along these lines, in this work, the eigenfrequency separation characteristics of the Hermite-Gaussian modes in the triangular cavity, in the meridional and sagittal planes, as well as the excitation and transmission characteristics of the first-order mode in the high finesse cavity were verified. A static alignment method was established for the high-finesse triangular ring-down cavity, and the transmission suppression of the Hermite-Gaussian modes by the high-finesse cavity was overcome in the experiment. Finally, by leveraging the advantageous features of the fundamental and the first-order mode that are complementary to each other, the tilt and axis shift errors of each specified plane with directional high accuracy can be aligned. The static alignment scheme of the triangular cavity based on the Hermite-Gaussian theory that was established here paves the way for the mounting alignment of the CRDS instrument before the ring-down experiment, whereas the system accuracy of CRDS is further improved.