Majorized iPADMM for Nonseparable Convex Minimization Models with Quadratic Coupling Terms

In this paper, we consider nonseparable convex minimization models with quadratic coupling terms arised in many practical applications. We use a majorized indefinite proximal alternating direction method of multipliers (iPADMM) to solve this model. The indefiniteness of proximal matrices allows the function we actually solved to be no longer the majorization of the original function in each subproblem. While the convergence still can be guaranteed and larger stepsize is permitted which can speed up convergence. For this model, we analyze the global convergence of majorized iPADMM with two different techniques and the sublinear convergence rate in the nonergodic sense. Numerical experiments illustrate the advantages of the indefinite proximal matrices over the positive definite or the semi-definite proximal matrices.

An Improved Imaging Algorithm for Multi-Receiver SAS System with Wide-Bandwidth Signal

When the multi-receiver synthetic aperture sonar (SAS) works with a wide-bandwidth signal, the performance of the range-Doppler (R-D) algorithm is seriously affected by two approximation errors, i.e., point target reference spectrum (PTRS) error and residual quadratic coupling error. The former is generated by approximating the PTRS with the second-order term in terms of the instantaneous frequency. The latter is caused by neglecting the cross-track variance of secondary range compression (SRC). In order to improve the imaging performance in the case of wide-bandwidth signals, an improved R-D algorithm is proposed in this paper. With our method, the multi-receiver SAS data is first preprocessed based on the phase center approximation (PCA) method, and the monostatic equivalent data are obtained. Then several sub-blocks are generated in the cross-track dimension. Within each sub-block, the PTRS error and residual quadratic coupling error based on the center range of each sub-block are compensated. After this operation, all sub-blocks are coerced into a new signal, which is free of both approximation errors. Consequently, this new data is used as the input of the traditional R-D algorithm. The processing results of simulated data and real data show that the traditional R-D algorithm is just suitable for an SAS system with a narrow-bandwidth signal. The imaging performance would be seriously distorted when it is applied to an SAS system with a wide-bandwidth signal. Based on the presented method, the SAS data in both cases can be well processed. The imaging performance of the presented method is nearly identical to that of the back-projection (BP) algorithm.

Optomechanically induced transparency and possible application in a quadratic-coupling optomechanical system with an induced electric field

Tunable optomechanically induced transparency (OMIT) with an induced electric field (IEF) in a quadratic-coupling optomechanical system is theoretically investigated. The system transmission rate under different controlling parameters has been discussed. It is revealed that both phase and group delay of the probe field can be adjusted by the IEF and pump field. Such a system may be used in tunable optical buffer, IEF detector, modulator or other optical devices.

Optomechanically induced transparency, amplification, and Fano resonance in a multimode optomechanical system with quadratic coupling

We explore the optical response of a multimode optomechanical system with quadratic coupling to a weak probe field, where the cavity is driven by a strong control field and the two movable membranes are, respectively, excited by weak coherent mechanical driving fields. We study the two cases that the two movable membranes are degenerate and nondegenerate. For the degenerate case, it is shown that only one transparency window occurs and the transition between optomechanically induced transparency and Fano resonance can be realized by tuning the cavity-control field detuning. For the nondegenerate case, two transparency windows are observed and the absorption spectrum can switch between a single Fano resonance and double Fano resonances. Furthermore, we show that the output probe field can be greatly amplified or completely suppressed due to the complex interference effect by tuning the amplitude and phase of the mechanical driving fields. Our results can be extended to the optomechanical system with multiple membranes, which enables us to control the light propagation more flexibly.

Strong quadratic acousto-optic coupling in 1D multilayer phoxonic crystal cavity

Four methods are applied to calculate the acousto-optic (AO) coupling in one-dimensional (1D) phoxonic crystal (PXC) cavity: transfer matrix method (TMM), finite element method (FEM), perturbation theory, and Born approximation. Two types of mechanisms, the photoelastic effect (PE) and the moving interface effect (MI), are investigated. Whether the AO coupling belongs to linear or quadratic, the results obtained by the perturbation theory are in good agreement with the numerical results. We show that the combination method of FEM and perturbation theory has some advantages over Born approximation. The dependence of linear and quadratic couplings on the symmetry of acoustic and optical modes has been discussed in detail. The linear coupling will vanish if the defect acoustic mode is even symmetry, but the quadratic effect may be enhanced. Based on second-order perturbation theory, the contribution of each optical eigenfrequency to quadratic coupling is clarified. Finally, the quadratic coupling is greatly enhanced by tuning the thickness of the defect layer, which is an order of magnitude larger than that of normal defect thickness. The enhancement mechanism of quadratic coupling is illustrated. The symmetry of the acoustic defect mode is transformed from odd to even, and two optical defect modes are modulated to be quasi-degenerated modes. This study opens up a possibility to achieve tunable phoxonic crystals on the basis of nonlinear AO effects.

Manipulating the spectral collapse in two-photon Rabi model

We have investigated the eigenenergy spectrum of the two-photon Rabi model with a full quadratic coupling, particularly the special feature “spectral collapse”. The critical coupling strength is reduced by half from that of the two-photon Rabi model, implying that the spectral collapse can now occur at a more attainable value of the critical coupling. At the critical coupling some discrete eigenenergy levels still survive below the continuous energy spectrum, i.e. an incomplete spectral collapse, and the set of discrete eigenenergies has a one-to-one mapping with that of a particle of variable effective mass in a finite potential well. Since the energy difference between the two atomic levels specifies the depth of the potential well, the number of bound states available (or the extent of the “spectral collapse”) can be straightforwardly monitored. Obviously, this bears a great resemblance to the spectral collapse of the two-photon Rabi model, at least qualitatively. Moreover, since the full quadratic coupling includes an extra term proportional to the photon number operator only, our analysis indicates that one may manipulate the critical coupling of the two-photon Rabi model by incorporating an adjustable proportionality constant to this extra term.