To enable MRI-based nongated 3D depiction of forced expiration with both high spatio-temporal resolution and fidelity of lung dynamics. An undersampled 3D Fermat looped, orthogonally encoded trajectory (FLORET) was optimized for nonsegmented dynamic acquisitions during forced expiration and a corresponding model-based reconstruction pipeline promoting spatial and temporal total variation was implemented. The approach was first optimized and validated using a lung phantom, specifically constructed for this study and was then translated to in vivo measurements. This allowed high temporal resolution of 200 ms and isotropic spatial resolution of 5 mm. Volumes derived by segmenting the MR images of the phantom demonstrated excellent agreement with the air exchange measured directly at the phantom within the bore, therefore confirming the validity of the approach. In both phantom and in vivo experiments, the dynamic deflation process could be successfully depicted, confirming the potential for assessing forced expiration with 3D UTE FLORET MRI. From the segmented volumes in vivo, quantitative parameters were derived, including the duration of the dynamic phase (2.8 ± 0.2 s), the time to 90% deflation (1.8 ± 0.2 s), and the volume change within the first second (2.8 ± 0.2 L). The phantom and in vivo studies demonstrate, that the 3D FLORET trajectory can be utilized to study lung deflation during forced expiration with high temporal resolution of 200 ms and isotropic spatial resolution of 5 mm using a TV-regularized primal-dual SENSE reconstruction. Acquisitions were performed for discrete distinct time frames, that is, no segmented k-space filling across several respiratory cycles was used.
The conformal optical systems are tremendously applied in precision-guided weapons. A corrector is traditionally used to compensate for the dynamic aberrations of an ellipsoidal dome. In this paper, what we believe to be a novel method is proposed to design conformal optical system based on Fermat's principle. The corrector needn't be required in the conformal optical system. The optical path distances (OPD) between the incident spherical wavefront and the rotating center of the imaging system are all set to be equal when the rays entering from different look angles, so the ellipsoidal dome acts as a spherical dome. The inner surface of the dome is calculated to ensure an identical OPD. At last, a conformal optical system is designed with a fineness ratio of 1, a diameter of 120 mm, the field of regard of ±60°, an F-number of 1.2 for the mid-wave infrared 3.7-4.8µm. The example demonstrates the feasibility of designing a conformal optical system using Fermat's principle.
In recent years, wireless sensor networks have been widely used, especially in three-dimensional environments such as underwater and mountain environments. However, in harsh environments, wireless sensor networks may be damaged and split into many isolated islands. Therefore, restoring network connectivity to transmit data effectively in a timely manner is particularly important. However, the problem of finding the minimum relay nodes is NP-hard, so heuristics methods are preferred. This paper presents a novel connectivity recovery strategy based on boundary nodes and spatial triangle Fermat points for three-dimensional wireless sensor networks. The isolated islands are represented as the boundary nodes, and the connectivity recovery problem is modeled as a graph connectivity problem. Three heuristics algorithms-the variant Kruskal algorithm, the variant Prim algorithm, and the spatial triangle Fermat point algorithm-are proposed to solve this problem. The variant Kruskal algorithm and the variant Prim algorithm connect the isolated islands by constructing the minimum spanning tree to link all the boundary nodes and placing relay nodes along the edges of this tree. We derive an accurate formula to determine the coordinates of spatial triangle Fermat points. Based on this formula, the spatial triangle Fermat point algorithm constructs a Steiner tree to restore network connectivity. Extensive simulation experiments demonstrate that our proposed algorithms perform better than the existing algorithm.
Wireless Sensor Networks (WSNs) have emerged as a critical research frontier in the Internet of Things (IoT) domain, with widespread applications in three-dimensional environments. However, due to harsh environments (such as high temperature, high pressure, etc.), natural disasters (such as earthquakes, etc.), large-scale attacks (such as bombing), WSNs in a certain area is split into many isolated islands, and the regional network fails. In emergency scenarios, restoring network connectivity in a timely manner is essential for ensuring reliable data transmission. However, finding the minimum relay node to recover the network is an NP hard problem. To address this challenge, this paper proposes a novel connectivity recovery strategy for 3D wireless sensor networks by leveraging boundary nodes and tetrahedral approximate Fermat points. The proposed approach is evaluated through three distinct algorithms: (1) a variant of Prim's algorithm (VPrim), which iteratively connects pairs of isolated segments until full network connectivity is restored, (2) a tetrahedral approximate Fermat point algorithm (TAFP), which simultaneously connects four isolated segments in each iteration until the network is fully recovered, and (3) Hybrid TAFP and VPrim Algorithm (HybridTV) to restore the entire network. Extensive simulation experiments demonstrate that our strategy significantly reduces the number of required relay nodes, enhances connectivity in the recovered network topology, and improves overall fault tolerance, offering a robust solution for network recovery in complex 3D environments.
We propose a non-uniform Fermat spiral (NFS) array for small-channel coherent beam combining. Owing to the non-uniform array density and the changing distances between two consecutive channels, the combined beam quality is naturally improved by the NFS array. All the factors affecting the combined beam quality were analyzed in detail, comparing with the hexagon array and the uniform Fermat spiral (UFS) array. Experiments were carried out to verify the combined beam quality of the small-channel optical system. It is demonstrated that the combined beam quality of the NFS array is better than that of the UFS array. The experimental results matched well with the numerical results.
With the exponential growth of artificial intelligence-driven data centre traffic, next-generation data centre optical interconnects must deliver high-speed data transmission while ensuring low latency and power consumption. Here, we present an ultra-simple low-latency self-homodyne coherent interconnect solution through anti-resonant hollow core fibre and leverages the Fermat number transform to implement the entire digital signal processing. The Fermat number transform eliminates the round-off errors prevalent in the fast Fourier transform through modulo operations and replaces computationally intensive multiplications with simple cyclic shift and addition operations. As a proof of concept, we demonstrate bidirectional transmission through a 5.1-km anti-resonant hollow core fibre, achieving a data rate of 448 Gb·s-1. Our proposed scheme reduces complexity of digital signal processing by 90%, whereas the integration of the anti-resonant hollow core fibre reduces the propagation latency by 28.4%. This work establishes a promising path to push the energy-efficiency boundary of coherent structure and enables large scale deployment of coherent optical interconnects.
The vortex beam has been widely applied for micromanipulation, biomedicine, and optical communications. Based on the previous work on phase measurement of a vortex beam, we present an uncalibrated multi-coherent modulation imaging technique for quantitative characterization of a Fermat-spiral-based semi-transparent photon sieves. The proposed methodology effectively circumvents stringent optical alignment constraints, thereby streamlining the experimental configuration and enhancing operational reproducibility. Experimental measurements of both reflected and transmitted vortex beam exhibit remarkable consistency with simulations, validating the robustness of the methodology for light-field manipulation.
When using ray tracing for optical system design, it is often the case that the designer would like to implement simplified versions of one or more compound lens groups. This could be the case during initial layout when idealized versions of such compound lenses are needed or, perhaps alternatively, to mimic a well-corrected commercially available lens for which the prescription details are unavailable. One option is to use a paraxial thin lens as a proxy for the actual lens group, but doing so will yield a layout that is not consistent with Fermat's principle or the Abbe sine condition. For example, a paraxial lens version of a compound microscope objective typically produces the wrong numerical aperture for a given entrance pupil diameter, and vice versa. A better option is to use a lens model that provides perfect imaging for a specified paraxial magnification and obeys Fermat's principle. A variant of the model can yield a perfect Fourier transform lens. In addition, it is desirable to implement an idealized thick lens in which the principal planes are separated by a user-specified distance. This paper presents such a model, referred to as the Cardinal Lens, with implementation in Zemax OpticStudio via a user-defined surface.
Reinterpretation of the Fermat principle governing the propagation of light in media within the Ramsey theory is suggested. Complete bi-colored graphs corresponding to light propagation in media are considered. The vertices of the graphs correspond to the points in real physical space in which the light sources or sensors are placed. Red links in the graphs correspond to the actual optical paths, emerging from the Fermat principle. A variety of optical events, such as refraction and reflection, may be involved in light propagation. Green links, in turn, denote the trial/virtual optical paths, which actually do not occur. The Ramsey theorem states that within the graph containing six points, inevitably, the actual or virtual optical cycle will be present. The implementation of the Ramsey theorem with regard to light propagation in metamaterials is discussed. The Fermat principle states that in metamaterials, a light ray, in going from point S to point P, must traverse an optical path length L that is stationary with respect to variations of this path. Thus, bi-colored graphs consisting of links corresponding to maxima or minima of the optical paths become possible. The graphs, comprising six vertices, will inevitably demonstrate optical cycles consisting of the mono-colored links corresponding to the maxima or minima of the optical path. The notion of the "inverse graph" is introduced and discussed. The total number of triangles in the "direct" (source) and "inverse" Ramsey optical graphs is the same. The applications of "Ramsey optics" are discussed, and an optical interpretation of the infinite Ramsey theorem is suggested.
This paper presents novel design techniques for the Fermat spiral, considering a maximum side lobe level (SLL) reduction. The array system based on a Fermat spiral configuration considers techniques based on uniform and non-uniform amplitude excitation. The cases of uniform amplitude excitation are the golden angle and the optimization of the angular separations. The cases of non-uniform amplitude excitations consider a raised cosine distribution and the optimization of the amplitude excitations through the Fermat spiral array. In this study, the method of genetic algorithms (GA) was used in the cases to find the values of the angular separations and the amplitude excitations of the Fermat spiral array. A performance evaluation was conducted for all these design cases, considering the Fermat spiral geometry. These design cases were validated using electromagnetic simulation to take mutual coupling into account and consider the effect of the antenna element pattern in each proposed design case.
We propose a Fermat spiral laser array as illumination source in ghost imaging. Due to the aperiodic structure, the Fermat spiral laser array generates illuminating light field without spatial periodicity on the normalized second-order intensity correlation function. A single-pixel detector is used to receive the signal light from object for image reconstruction. The effects of laser array parameters on the quality of ghost imaging are analyzed comprehensively. Through experimental demonstration, the Fermat spiral laser array successfully achieves ghost imaging with high quality by combining with the compressive sensing reconstruction algorithm. This method is expected to be applied in remote sensing by combining with phased and collimated fiber laser array equipped with the high emitting power and high-speed modulation frequency.
Structured beams carrying orbital angular momentum (OAM) provide powerful capabilities for applications in optical tweezers, super-resolution imaging, quantum optics, and ad-vanced microparticle manipulation. However, it is challenging for generate and control the OAM beams at the extreme ultraviolet (EUV) region due to the lack of suitable wave front shaping optics arise from being limited to the strong absorption of most materials. Here, we use a modified Fermat-spiral photon-sieve splitter to simultaneously generate two focused doughnut beams with opposite helical phase. Our technique enables us to produce splitting focused vortex beams with different rotation directions at EUV wavelengths. Additionally, we provide experimental evidence showcasing the capabilities of our method and further detect the helical phase by self-reference interferometry. This work not only opens a route for OAM-driven applications in EUV radiation, but also paves the way to studies of holographic technique by EUV splitter.
In order to reduce the power consumption of digital signal processing (DSP) in a coherent optical communication system, a low complexity equalization scheme in DSP flow of a 400 Gb/s DP-16QAM system has been proposed. This scheme is based on Fermat number transform (FNT), which sequentially performs static equalization (SE) and dynamic equalization (DE) in the transform domain. For different distances, the proposed scheme finds the optimal solution under the condition that transform length and data bit width are mutually restricted under different transmission distances while achieving low complexity and optimal performance. The experimental results show that the adopted transform-domain equalization (TrDE) scheme has much lower computational complexity than the traditional frequency-domain equalization (FDE) and time-domain equalization (TDE) nearly without any performance loss. In the 80, 160, and 240 km scenarios, the number of multiplier is reduced by more than 72%, and the advantage becomes more obvious as the transmission capacity increases.
A desirable lanthanide-based ratiometric fluorescent probe was designed and integrated into a self-designed Fermat spiral microfluidic chip (FS-MC) for the automated determination of a unique bacterial endospore biomarker, dipicolinic acid (DPA), with high selectivity and sensitivity. Here, a blue emission wavelength at 425 nm was generated in the Fermat spiral structure by mixing the europium (Eu3+) and luminol to form the Eu3+/Luminol sensing probe. DPA in the reservoir can be used to specifically bind to Eu3+ under the negative pressure and transfer energy from DPA to Eu3+ sequentially via an antenna effect, thus resulting in a significant increase in the red fluorescence emission peak at 615 nm. According to the fluorescence intensity ratio (F615/F425), a good linearity can be obtained with increasing the concentration of DPA from 0 to 200 μM with a limit of detection as low as 10.11 nM. Interestingly, the designed FS-MC can achieve rapid detection of DPA in only 1 min, reducing detection time and improving sensitivity. Furthermore, a self-designed device integrated with the FS-MC and a smartphone color picker APP was employed for the rapid automatic point-of-care testing (POCT) of DPA in the field, simplifying complex processes and reducing testing times, thus confirming the great promise of this ready-to-use measurement platform for in situ inspection.
Wearable electrochemical sensors have shown potential for personal health monitoring due to their ability to detect biofluids non-invasively at the molecular level. Smart fibers with high flexibility and comfort are currently ideal for fabricating electrochemical sensors, but little research has focused on fluid transport at the human-machine interface, which is of great significance for continuous and stable monitoring and skin comfort. Here, we report an electrochemical sensing fiber with a special core-sheath structure, whose outer layer is wound by nanofibers with a hierarchical Fermat helix structure which has excellent moisture conductivity, and the inner layer is based on CNT fibers covered by three-dimensional reduced graphene oxide folds which have good sensing properties after modification of active materials such as enzymes and selective membranes. This kind of fiber enables efficient sweat capture, and thus only 0.1 μL of sweat is required to activate the device, and it responds very quickly (1.5 s). The fibers were further integrated into a garment to build a wireless sweat detection system, enabling stable monitoring of six physiological markers in sweat (glucose, lactate, Na+, K+, Ca2+, and pH). This work provides a feasible proposal for future personalized medicine and the construction of "smart sensing garments".
Beginning with Fermat's principle, we provide a detailed derivation of the generalized laws of refraction and reflection for a geometry realizing a metasurface. We first solve the Euler-Lagrange equations for a light ray propagating across the metasurface. The ray-path equation is found analytically, and the results are supported by numerical calculations. We get generalized laws of refraction and reflection that have three main features: (i) They are relevant in gradient-index optics and in geometrical optics; (ii) A collection of rays emerges from the metasurface as a result of multiple reflections inside the metasurface; and (iii) The laws, although derived from Fermat's principle, differ from previously published results.
Chromatic dispersion (CD) compensation is a critical, yet power-intensive operation in coherent optical communication systems. Transform-based methods, such as the FFT and FNT, often rely on overlap-save (OS) or zero-padding overlap-add (ZP-OA) techniques for blockwise equalization in the receiver's DSP. However, these methods only achieve (N-NOverlap) effective points with N-point transforms, limiting the efficiency. In this Letter, we propose the non-zero-padding overlap-add (NZ-OA) method, which achieves N effective points using weighted Fourier transform (WFT) and weighted Fermat number transform (WFNT). The NZ-OA method reduces computational complexity by 41.7% in the 400 km DFT/WFT scenario, and by 15.7% in the 50 km FNT/WFNT scenario when the non-prime Fermat modulus q=232+1 is employed.
Reliable manufacturing and high optical performance of plano-convex aspheric lenses necessitate an improved optical design characterized by an optimal refractive index of the lens material and the prevention of the slope surface shape oscillations, which provide an increased angular field of view and low sensitivity to misalignments. To formulate the criteria for the optimal refractive index, we propose an analytical representation of the optimal aspheric surface based on the Fermat's principle in the form of parametric functions of coordinates, allowing the surface to be approximated by the elliptical conic section expanded with high-order aspheric deformation terms starting from the 6th power. There are determined the ranges of optimal refractive indices for materials of plano-convex aspheric lenses: n=1.605…1.665 for an aplanatic design with the Abbe sine condition fulfilled, ensuring low lens sensitivity of the lens to angular misalignment; n=1.735…1.835 for sharp imaging of elongated longitudinal objects in accordance with the Herschel's condition.
The bonding integrity of corrugated interfaces in clad plates fundamentally governs their macroscopic performance, yet high-precision nondestructive evaluation (NDE) of these geometrically complex bonds remains a critical industrial challenge. Addressing the inspection demands for corrugated clad plates, this study investigates Cu/Al laminates as a representative system. A multi-parameter coupling framework integrating Rayleigh perturbation theory and critical slope criterion is established to enable quantitative prediction of imaging quality for corrugated interfaces. For the first time, the dimensionless parameter cp is proposed as a scale-invariant criterion to characterize imaging failure. Through systematic finite element simulations, the nonlinear coupling effects of corrugation wavelength, amplitude, and interface depth on imaging fidelity are quantified. The mechanism of imaging degradation at corrugation waists is elucidated via the relationship between local interface slope and ultrasonic reflection angle. Using Fermat's principle, the minimum-time acoustic path in a bilayer medium is derived to analyze systematic imaging offsets caused by excitation-receiver separation. Experimental validation is performed on a laser-ultrasonic NDE platform using eight specimens with precisely controlled corrugation parameters fabricated by wire EDM. Results demonstrate excellent agreement with simulations, confirming the predictive accuracy of the theoretical model. The study reveals that when the maximum interface slope exceeds a critical threshold, reflected signals from waist regions fall outside the detection aperture, causing severe imaging quality deterioration. Decreasing wavelength, increasing amplitude, and increasing depth collectively exacerbate signal attenuation and geometric distortion. The proposed framework provides a theoretical foundation for quality assurance of corrugated clad plates and establishes a generalized analytical methodology for NDE of irregular multi-layered interfaces.
Traditional design of underwater two-dimensional arrays focuses on local beam-pattern metrics, with insufficient attention paid to the overall imaging performance in practical acoustic environments. To mitigate this limitation, this paper proposes a collaborative array design framework that integrates physical acoustic simulation with Bayesian optimization. Employing a Gaussian process as the surrogate model, the framework incorporates a dual-stage delay optimization strategy and a composite loss function to enable the automatic search for optimal array configurations. Simulation experiment results demonstrate that within a 90° × 90° field of view, the proposed DSDO method reduces the maximum delay mean square error by approximately 66.7% compared with the Fresnel approximation. The segmentation performance metrics (peak signal-to-noise ratio, structural similarity, intersection over union, and accuracy) of the images obtained by the optimized array show improvements of 6.49%, 1.62%, 6.21%, and 2.17%, respectively, compared to the Fermat spiral array. This indicates that the method enhances the clarity and structural fidelity of targets within the images, laying a foundation for subsequent downstream tasks, such as target detection and recognition.