We present a reflectivity-based method for measuring the in-plane refractive index of transparent van der Waals (vdW) materials. The approach enables the characterization of as small as $3 \times 3$~{\textmu}m$^2$ exfoliated flakes on a non-transmissive substrate without assuming any specific spectral shape of the refractive index. Exfoliated flakes are most commonly obtained through mechanical exfoliation, which generally produces vdW flakes with tens-of-micron lateral dimensions. As a result, conventional ellipsometry - which depends on large, uniform areas and specific spectral models - becomes challenging to apply. Our method determines the refractive index directly from the spectral position of reflectivity minima, provided the flake thickness and the substrate complex refractive index are known. We demonstrate the technique on hafnium disulfide (HfS$_2$), a vdW semiconductor with high refractive index and low absorption, retrieving its in-plane refractive index across the visible range. The results both validate previous ellipsometry measurements and establish this method as an accessible and spectral-model-free alternative for refractive index characterization of vdW materia
The potential to precisely control both the linear and nonlinear index of refraction through optical manipulation of the atomic states has recently pushed warm alkali vapors to the forefront of research in the field of quantum sensors, quantum memories, and quantum fluids of light. Rubidium (Rb) vapor in centimeter-scale glass cells or millimeter-scale MEMS cells has proven to be a very promising platform for these applications, yet only a handful of research works have been dedicated to the investigation of the (non)linear refractive index of Rb vapor. We present results of theoretical calculations of the (non)linear refractive index of warm Rb vapor, based on the optical Bloch equations for 6-level Rb atoms interacting with a probe laser. They are compared to the experimental results obtained using an interferometric technique, showing excellent quantitative agreement. A Kerr nonlinear refractive index $n_2$ of up to $10^{-4}$ cm$^2$/W is obtained. Python scripts for all theoretical calculations presented in this work are provided, including the refractive index calculation, that can readily be used in practical implementations for simulating the (non)linear refractive index of R
Metalenses have emerged as a powerful platform for compact wavefront engineering; however, their performance stability under refractive index fluctuations induced by environmental perturbations, such as temperature shifts, remains a critical concern. Here, we demonstrate the intrinsic refractive index robustness of dielectric metalenses and elucidate its physical origin. By parametrically sweeping the refractive index, we observe that the metalens maintains a stable focal profile with negligible deviations in best-focus position and spot size over a broad range of variations. We identify that this robustness arises from the structural invariance of the zone boundaries: despite index-induced local phase deformations, the spatial locations of the 2π phase-reset boundaries remain stationary, thereby maintaining the effective wavefront gradient. Furthermore, we reveal that this robustness enables a "quasi-scale-invariant" focusing behavior, where the focusing performance follows a predictable linear trend under uniform geometric scaling even in the presence of material dispersion. Our findings suggest that metalenses can maintain stable focusing behavior against refractive index variat
The use of nanostructures to enhance the emission of single-photon sources has withdrawn some attention in last decade due to the development of quantum technologies. In particular, the use of metallic and high refractive index dielectric materials has been proposed. However, the utility of moderate refractive index dielectric nanostructures to achieve more efficient single-photon sources remains unexplored. Here, a systematic comparison of various metallic, high refractive index and moderate refractive index dielectric nanostructures has been performed to optimize the excitation and emission of a CdSe/ZnS single quantum dot at visible spectral region. Several geometries have been evaluated in terms of electric field enhancement and Purcell factor, considering the combination of metallic, high refractive index and moderate refractive index dielectric materials conforming homogeneous and hybrid nanocylinder dimers. Our results demonstrate that moderate refractive index dielectric nanoparticles can enhance the photoluminescence signal of quantum emitters due to their broader electric and magnetic dipolar resonances compared to high refractive index dielectric nanoparticles. However,
White light spectral interferometry is applied to measure the refractive index in absorbing liquids in the spectral range from 400 to 1000 nm. We analyze the influence of absorption on the visibility of interferometric fringes and, accordingly, on the measurement of the refractive index. Further, we show that the refractive index in the absorption band can be retrieved by a two step process. The procedure requires the use of two samples of different thickness, the thicker one to retrieve the refractive index in the transparent region and the thinnest to obtain the data in the absorption region. First, the refractive index values are retrieved with good accuracy in the transparent region of the material for 1 mm thick samples. Second, these refractive index values serve also to precisely calculate the thickness of a thinner sample (about 150 micrometers) since the accuracy of the method depends strongly on the thickness of the sample. Finally, the refractive index is recovered for the entire spectral range. Copyright 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement. See corresponding link in the pdf file.
Gallium antimonide (GaSb) is a key material for near- and mid-infrared photonics, enabling high-performance laser architectures and detectors. Design and simulation of such devices depend on accurate optical material data, especially the complex refractive index $n^*_{\text{GaSb}} = n_{\text{GaSb}} +ik_{\text{GaSb}}$, consisting of the real part $n_{\text{GaSb}}$ (refractive index) and the imaginary part $k_{\text{GaSb}}$ (extinction coefficient). However, GaSb refractive index values are based either on theoretical models, typically informed by legacy experimental data, or on experimental measurements without quantified uncertainties. This limits their reliability for state-of-the-art devices. Here, we present measurement results of $n^*_{\text{GaSb}}$ in the near- to mid-infrared range from \SIrange{1}{3.1}{\micro \metre} with a relative uncertainty < um{7.8e-5} for $n_{\text{GaSb}}$, and < um{2.0e-3} for $k_{\text{GaSb}}$. As a side result of our method, we also report $n_{\text{AlAsSb}}$ for aluminium arsenide antimonide ($\mathrm{AlAs_{0.08}Sb_{0.92}}$) with a relative uncertainty < um{3.9e-4}. Our results are based on two complementary measurements on a GaSb/AlAsSb-b
Camera calibration is an essential process in photogrammetry, serving as a crucial link between the 2D image coordinate system and the 3D world coordinate system. However, when observations are conducted through refractive interfaces, the refraction effects at these interfaces render traditional calibration methods ineffective, significantly compromising measurement accuracy. To address this challenge, we propose a novel camera calibration method based on the analytical refractive imaging (ARI) equation. The ARI method facilitates accurate estimation of camera parameters from distorted images and enables in-situ joint calibration of both the camera and the refractive interface. The experimental results indicate that the proposed method reduces the error to only 10% of that produced by conventional ray-tracing (RT) method. Moreover, while maintaining comparable computational accuracy and efficiency, it effectively mitigates the local convergence issues that may arise in the polynomial fitting (PF) approach. Finally, reconstruction experiments further confirm the accuracy of the proposed method. Experimental results demonstrate that the proposed method outperforms existing refractive
This paper studies the near field refraction problem with loss of energy in negative refractive index material. Based on the relative refractive index $κ$, the analysis is categorized into two cases, namely $κ< -1$ and $-1 < κ< 0$. For each case, we give the definition of the refractor and discuss some crucial properties of it. The properties of Fresnel coefficients are also discussed. Based on these properties, the existence of the weak solution when the target measure is either discrete or a finite Radon measure are proved. Besides, the critical case $κ= -1$ is also discussed briefly at the end of this paper.
This paper presents a general refractive camera model and online co-estimation of odometry and the refractive index of unknown media. This enables operation in diverse and varying refractive fluids, given only the camera calibration in air. The refractive index is estimated online as a state variable of a monocular visual-inertial odometry framework in an iterative formulation using the proposed camera model. The method was verified on data collected using an underwater robot traversing inside a pool. The evaluations demonstrate convergence to the ideal refractive index for water despite significant perturbations in the initialization. Simultaneously, the approach enables on-par visual-inertial odometry performance in refractive media without prior knowledge of the refractive index or requirement of medium-specific camera calibration.
This paper studies the far field refraction problem in negative refractive index material with loss of energy, which is a remaining problem in E. Stachura, Nonlinear Anal. 2017;157:76-103. The analysis is divided into two cases according to the relative refractive index $κ$, that is, $κ<-1$ and $-1<κ<0$. For each case, we use the Minkowski method to establish the existence of the weak solution when the target measure is either discrete or a finite Radon measure. Eventually, the inequality involving a Monge-Ampère type operator satisfied by the solution of the problem is derived, which is useful to understand this complex optical phenomenon.
This work presents a camera model for refractive media such as water and its application in underwater visual-inertial odometry. The model is self-calibrating in real-time and is free of known correspondences or calibration targets. It is separable as a distortion model (dependent on refractive index $n$ and radial pixel coordinate) and a virtual pinhole model (as a function of $n$). We derive the self-calibration formulation leveraging epipolar constraints to estimate the refractive index and subsequently correct for distortion. Through experimental studies using an underwater robot integrating cameras and inertial sensing, the model is validated regarding the accurate estimation of the refractive index and its benefits for robust odometry estimation in an extended envelope of conditions. Lastly, we show the transition between media and the estimation of the varying refractive index online, thus allowing computer vision tasks across refractive media.
Due to interactions with dark matter, neutrinos can acquire refractive masses which explain the data from oscillation experiments. We study the effects of relic neutrinos with refractive masses on structure formation in the Universe. In the model with a light fermionic mediator, above the resonance energy, $E_R$, associated with the mass of the mediator, refractive masses have all the properties identical to the usual vacuum masses. Below the resonance, refractive masses decrease with neutrino energy, however, they cannot be used in the same way as usual masses. We study the dispersion relations and group velocities of such neutrinos and their dependence on redshift. We show that in the epoch of structure formation, relic neutrinos were ultrarelativistic and essentially massless particles for $E_R = (10 - 10^5)\,$eV. This allows to reconcile the values of masses extracted from oscillation experiments with the stringent bounds on sum of neutrino masses from cosmological surveys.
Refractive Index Tomography is the inverse problem of reconstructing the continuously-varying 3D refractive index in a scene using 2D projected image measurements. Although a purely refractive field is not directly visible, it bends light rays as they travel through space, thus providing a signal for reconstruction. The effects of such fields appear in many scientific computer vision settings, ranging from refraction due to transparent cells in microscopy to the lensing of distant galaxies caused by dark matter in astrophysics. Reconstructing these fields is particularly difficult due to the complex nonlinear effects of the refractive field on observed images. Furthermore, while standard 3D reconstruction and tomography settings typically have access to observations of the scene from many viewpoints, many refractive index tomography problem settings only have access to images observed from a single viewpoint. We introduce a method that leverages prior knowledge of light sources scattered throughout the refractive medium to help disambiguate the single-view refractive index tomography problem. We differentiably trace curved rays through a neural field representation of the refractiv
In this paper, we present a complete refractive Structure-from-Motion (RSfM) framework for underwater 3D reconstruction using refractive camera setups (for both, flat- and dome-port underwater housings). Despite notable achievements in refractive multi-view geometry over the past decade, a robust, complete and publicly available solution for such tasks is not available at present, and often practical applications have to resort to approximating refraction effects by the intrinsic (distortion) parameters of a pinhole camera model. To fill this gap, we have integrated refraction considerations throughout the entire SfM process within the state-of-the-art, open-source SfM framework COLMAP. Numerical simulations and reconstruction results on synthetically generated but photo-realistic images with ground truth validate that enabling refraction does not compromise accuracy or robustness as compared to in-air reconstructions. Finally, we demonstrate the capability of our approach for large-scale refractive scenarios using a dataset consisting of nearly 6000 images. The implementation is released as open-source at: https://cau-git.rz.uni-kiel.de/inf-ag-koeser/colmap_underwater.
Snell's law dictates the phenomenon of light refraction at the interface between two media. Here, we demonstrate arbitrary programming of light refraction through an engineered material where the direction of the output wave can be set independently for different directions of the input wave, covering arbitrarily selected permutations of light refraction between the input and output apertures. Formed by a set of cascaded transmissive layers with optimized phase profiles, this refractive function generator (RFG) spans only a few tens of wavelengths in the axial direction. In addition to monochrome RFG designs, we also report wavelength-multiplexed refractive functions, where a distinct refractive function is implemented at each wavelength through the same engineered material volume, i.e., the permutation of light refraction is switched from one desired function to another function by changing the illumination wavelength. As experimental proofs of concept, we demonstrate permutation and negative refractive functions at the terahertz part of the spectrum using 3D-printed materials. Arbitrary programming of refractive functions enables new design capabilities for optical materials, dev
Recently, differentiable volume rendering in neural radiance fields (NeRF) has gained a lot of popularity, and its variants have attained many impressive results. However, existing methods usually assume the scene is a homogeneous volume so that a ray is cast along the straight path. In this work, the scene is instead a heterogeneous volume with a piecewise-constant refractive index, where the path will be curved if it intersects the different refractive indices. For novel view synthesis of refractive objects, our NeRF-based framework aims to optimize the radiance fields of bounded volume and boundary from multi-view posed images with refractive object silhouettes. To tackle this challenging problem, the refractive index of a scene is reconstructed from silhouettes. Given the refractive index, we extend the stratified and hierarchical sampling techniques in NeRF to allow drawing samples along a curved path tracked by the Eikonal equation. The results indicate that our framework outperforms the state-of-the-art method both quantitatively and qualitatively, demonstrating better performance on the perceptual similarity metric and an apparent improvement in the rendering quality on sev
We have analytically determined the refractive index for the mechanical refraction of a relativistic particle for its all possible speeds. We have critically analysed the importance of Descartes' metaphysical theory and extended it in this regard. We have considered the conservation of the tangential component of the relativistic momentum and the relativistic energy of the particle in the process of the mechanical refraction within the optical-mechanical analogy. Our result for the mechanical refractive index exactly matches with the forms of both the Fermat's result on Snell's law of optical refraction at the ultra-relativistic limit and the Descartes' metaphysical result on the pseudo-Snell law of optical refraction at the non-relativistic limit.
In this tutorial we summarize the physics and mathematics behind refractive electromagnetic wave bending and delay. Refractive bending and delay through the Earth's atmosphere at both radio/millimetric and optical/IR wavelengths are discussed, but with most emphasis on the former, and with Atacama Large Millimeter Array (ALMA) applications in mind. As modern astronomical measurements often require sub-arcsecond position accuracy, care is required when selecting refractive bending and delay algorithms. For the spherically-uniform model atmospheres generally used for all refractive bending and delay algorithms, positional accuracies $\lesssim 1^{\prime\prime}$ are achievable when observing at zenith angles $\lesssim 75^\circ$. A number of computationally economical approximate methods for atmospheric refractive bending and delay calculation are presented, appropriate for astronomical observations under these conditions. For observations under more realistic atmospheric conditions, for zenith angles $\gtrsim 75^\circ$, or when higher positional accuracy is required, more rigorous refractive bending and delay algorithms must be employed. For accurate calculation of the refractive bendi
There are two main physical properties needed to fabricate 1D photonic structures and form perfect photonic bandgaps: the quality of the thickness periodicity and the refractive index of their components. Porous silicon (PS) is a nano-structured material widely used to prepare 1D photonic crystals due to the ease of tuning its porosity and its refractive index by changing the fabrication conditions. Since the morphology of PS changes with porosity, the determination of PS's refractive index is no easy task. To find the optical properties of PS we can use different effective medium approximations (EMA). In this work we propose a method to evaluate the performance of the refractive index of PS layers to build photonic Bragg reflectors. Through a quality factor we measure the agreement between theory and experiment and therein propose a simple procedure to determine the usability of the refractive indices. We test the obtained refractive indices in more complicated structures, such as a broadband Vis-NIR mirror, and by means of a Merit function we find a good agreement between theory and experiment. With this study we have proposed quantitative parameters to evaluate the refractive in
We consider in detail a possibility that the observed neutrino oscillations are due to refraction on ultralight scalar boson dark matter. We introduce the refractive mass squared, $\tilde{m}^2$, and study its properties: dependence on neutrino energy, state of the background, etc. If the background is in a state of cold gas of particles, $\tilde{m}^2$ shows a resonance dependence on energy. Above the resonance ($E \gg E_R $), we find that $\tilde{m}^2$ has the same properties as usual vacuum mass squared. Below the resonance, $\tilde{m}^2$ decreases with energy, which (if realised) allows us to avoid the cosmological bound on the sum of neutrino masses. Also, $\tilde{m}^2$ may depend on time. We consider the validity of the results: effects of multiple interactions with scalars, and modification of the dispersion relation. We show that for values of parameters of the system required to reproduce the observed neutrino masses, perturbativity is broken at low energies, which border above the resonance. If the background is in the state of coherent classical field, the refractive mass does not depend on energy explicitly but may show time dependence. It coincides with the refractive ma