Geometrical Optics Approach Research Articles

Stabilizing the carrier–envelope phase of an amplified Ti:sapphire laser pulse to a noise level of sub-100 mrad

To generate an intense femtosecond pulse with an energy of several mJ or more, the pulse must be sufficiently stretched and compressed using grating-based optics in a chirped pulse amplification system. However, the stability of the carrier–envelope phase (CEP) is readily degraded by movement of the diffraction gratings due to vibration caused by pump lasers and other equipment. By suppressing the vibration and acoustic waves, we reduced the CEP noise to much less than 100 mrad (rms). With feedback control using a wedge pair with a single shot f - 2 f interferometer (1 ms integration time), we were able to stabilize the CEP, achieving a CEP noise of 58 mrad (rms) at a pulse energy of 0.5 mJ and 99 mrad (rms) at a pulse energy of 7 mJ and a repetition rate of 1 kHz. We also analyzed the effects of vibration on the pulse stretcher and compressor using a geometrical optics approach and found that roof mirrors were the most influential.

Quasinormal modes in two-photon autocorrelation and the geometric-optics approximation

In this work, we study the black hole light echoes in terms of the two-photon autocorrelation and explore their connection with the quasinormal modes. It is shown that the above time-domain phenomenon can be analyzed by utilizing the well-known frequency-domain relations between the quasinormal modes and characteristic parameters of null geodesics. We found that the time-domain correlator, obtained by the inverse Fourier transform, naturally acquires the echo feature, which can be attributed to a collective effect of the asymptotic poles through a weighted summation of the squared modulus of the relevant Green’s functions. Specifically, the contour integral leads to a summation taking over both the overtone index and angular momentum. Moreover, the dominant contributions to the light echoes are from those in the eikonal limit, consistent with the existing findings using the geometric-optics arguments. For the Schwarzschild black holes, we demonstrate the results numerically by considering a transient spherical light source. Also, for the Kerr spacetimes, we point out a potential difference between the resulting light echoes using the geometric-optics approach and those obtained by the black hole perturbation theory. Possible astrophysical implications of the present study are addressed.

Numerical simulation of the performance of a spaceborne Offner imaging hyperspectrometer in the wave optics approximation

We propose a method for calculating the point scattering function (PSF) of an Offner imaging hyperspectrometer with a diffraction grating in the approximation of scalar diffraction theory. The method consistently takes into account limitations and diffraction of a light beam by elements of the hyperspectrometer system in accordance with the physics of image formation. The PSF of the Offner imaging hyperspectrometer is numerically simulated at various beam parameters and wavelengths. The simulation results are verified using analytical relationships, a geometrical optics approach, as well as a comparison with the related works of other researchers.

Caustics of Non-Paraxial Perfect Optical Vortices Generated by Toroidal Vortex Lenses

In this paper, we consider the comparative formation of perfect optical vortices in the non-paraxial mode using various optical elements: non-paraxial and parabolic toroidal vortex lenses, as well as a vortex axicon in combination with a parabolic lens. The theoretical analysis of the action of these optical elements, as well as the calculation of caustic surfaces, is carried out using a hybrid geometrical-optical and wave approach. Numerical analysis performed on the basis of the expansion in conical waves qualitatively confirms the results obtained and makes it possible to reveal more details associated with diffraction effects. Equations of 3D-caustic surfaces are obtained and the conditions of the ring radius dependence on the order of the vortex phase singularity are analyzed. In the non-paraxial mode, when small light rings (several tens of wavelengths) are formed, a linear dependence of the ring radius on the vortex order is shown. The revealed features should be taken into account when using the considered optical elements forming the POV in various applications.

Ka-Band Radar Cross-Section of Breaking Wind Waves

The effective normalized radar cross section (NRCS) of breaking waves, σwb, is empirically derived based on joint synchronized Ka-band radar and video records of the sea surface from a research tower. The σwb is a key parameter that, along with the breaker footprint fraction, Q, defines the contribution of non-polarized backscattering, NP =σwbQ, to the total sea surface NRCS. Combined with the right representation of the regular Bragg and specular backscattering components, the NP component is fundamental to model and interpret sea surface radar measurements. As the first step, the difference between NRCS values for breaking and non-breaking conditions is scaled with the optically-observed Q and compared with the geometric optics model of breaker backscattering. Optically-derived Q might not be optimal to represent the effect of breaking waves on the radar measurements. Alternatively, we rely on the breaking crest length that is firmly detected by the video technique and the empirically estimated breaker decay (inverse wavelength) scale in the direction of breaking wave propagation. A simplified model of breaker NRCS is then proposed using the geometric optics approach. This semi-analytical model parameterizes the along-wave breaker decay from reported breaker roughness spectra, obtained in laboratory experiments with mechanically-generated breakers. These proposed empirical breaker NRCS estimates agree satisfactorily with observations.

Observation of gravitational waves by light polarization

We provide analysis to determine the effects of gravitational waves on electromagnetic waves, using perturbation theory in general relativity. Our analysis is performed in a completely covariant manner without invoking any coordinates. For a given observer, using the geometrical-optics approach, we work out the perturbations of the phase, amplitude, frequency and polarization properties–axes of ellipse and ellipticity of light, due to gravitational waves. With regard to the observation of gravitational waves, we discuss the measurement of Stokes parameters, through which the antenna patterns are presented to show the detectability of the gravitational wave signals.

A Modified Geometrical Optical Model of Row Crops Considering Multiple Scattering Frame

The canopy reflectance model is the physical basis of remote sensing inversion. In canopy reflectance modeling, the geometric optical (GO) approach is the most commonly used. However, it ignores the description of a multiple-scattering contribution, which causes an underestimation of the reflectance. Although researchers have tried to add a multiple-scattering contribution to the GO approach for forest modeling, different from forests, row crops have unique geometric characteristics. Therefore, the modeling approach originally applied to forests cannot be directly applied to row crops. In this study, we introduced the adding method and mathematical solution of integral radiative transfer equation into row modeling, and on the basis of improving the overlapping relationship of the gap probabilities involved in the single-scattering contribution, we derived multiple-scattering equations suitable for the GO approach. Based on these modifications, we established a row model that can accurately describe the single-scattering and multiple-scattering contributions in row crops. We validated the row model using computer simulations and in situ measurements and found that it can be used to simulate crop canopy reflectance at different growth stages. Moreover, the row model can be successfully used to simulate the distribution of reflectances (RMSEs < 0.0404). During computer validation, the row model also maintained high accuracy (RMSEs < 0.0062). Our results demonstrate that considering multiple scattering in GO-approach-based modeling can successfully address the underestimation of reflectance in the row crops.

Shock-wave thickness influence to the light diffraction on a plane shock wave

This study is devoted to the numerical analysis of the result of light distribution after passing it through a shock wave, depending on the degree of gas rarefaction. The obtained numerical results allowed reproducing the experimental shadowgraph images obtained in our study. The range of shock wave thickness (from 0 mm to 20 mm) allowed considering the qualitative change in the light distribution on the screen during switching from the regime where the wave nature of light has the greatest influence on the distribution of light to the regime of the geometric optics approach. As a result, the criteria for the applicability of the shadowgraphy technique for the experimental description of the shock wave internal structure were obtained.

Tilt (in)variant lateral scan in oblique plane microscopy: a geometrical optics approach

Oblique plane microscopy (OPM) is a single objective light-sheet microscopy that performs three dimensional (3D) imaging by axial scan of the generated light-sheet. Recently, multiple techniques for lateral scanning of the generated light-sheet in OPM have emerged. However, their suitability for geometrically distortion free 3D imaging, which essentially requires a constant tilt light-sheet scan, has not been evaluated. In this work, we use a geometrical optics approach and derive analytical relationship for the amount of tilt variance in planar mirror based scanned oblique plane illumination (SOPi) arrangement. We experimentally validate the derived relationship and use it to arrive at an optimized scanner geometry and to understand its associated limitations. We discuss the effects of scanning on optical aberrations and 3D field of view in optimized, tilt invariant, lateral scanning OPM systems. We also provide experimental strategies enabling precise scanner alignment for tilt invariance, as well as an open source platform for rapid design of new oblique light-sheet microscopes.

Canopy modeling of aquatic vegetation: A geometric optical approach (AVGO)

Aquatic vegetation is an important component in the ecosystems of wetlands and coastal waters, for example it can take up harmful substances and pollutants from the water. Constructing a remote sensing model of aquatic vegetation can deepen our understanding of the spectral characteristics and directional reflection characteristics of the aquatic vegetation canopy under various environmental conditions. However, the existing aquatic vegetation models are limited to continuously distributed canopies. This paper proposes an aquatic-vegetation- geometric-optical (AVGO) model at canopy scale for discrete aquatic vegetation. After dividing the reflectance of water background into specular components and diffuse components, the reflectances of all reflecting surfaces are classified into different reflectance components based on the differences in their materials and on whether the reflectors are illuminated by direct sunlight. Then the scene bidirectional reflectance factor (BRF) is regarded as an area-weighted sum of different reflectance components. Detailed verifications were conducted to ensure that the simulation model operates as intended. The performance of the AVGO model was qualitatively validated with the WCRM model and showed the physical plausibility of the model prediction. Quantitative validations against an artificially designed experiment were consistent with the bidirectional reflectance and the nadir-viewed spectral reflectance. This model can be used for canopy modeling of discrete emergent or submerged aquatic vegetation. Potential applications include retrieval of biophysical or biochemical parameters and aquatic vegetation abundance based on hyperspectral data.

A geometrical optics approach to encircled-flux compliant source modeling for multi-mode fiber illumination and connection attenuation

Abstract We construct ray-based encircled-flux compliant sources for multi-mode optical fiber launches using classic geometrical optics. In particular, we make an LP-mode conforming launch that excites only guided rays by distributing the β ¯ ray invariant according to a (scaled) counterpart of the modal power distribution of LP-modes. Propagation of the ray distribution is spatially stable and simulations of attenuation due to fiber connections and 1 mm far-field behavior using this source agree well with measurements. We caution against models that configure ray positions and ray directions independently and show that they are prone to produce unstable ray distributions. Lastly, we show that modal noise due to laterally misaligned fibers can be replicated using geometrical optics by configuring our LP-mode conforming sources to match the radial power distribution of 3000 vectorial full-wave spatio-temporal coherent speckle patterns.

Analytical calculation of diffraction order intensities for a hyperspectrometer

We employ a geometric optics approach to calculate the eikonal of a light field at the output of a diffraction grating in the immediate microrelief vicinity. Analytic relations to calculate the diffraction order intensities as functions of the grating step height, line spacing, wavelength of incident light, and incidence angle are deduced. By way of illustration, intensities of the diffraction orders for a blazed reflection grating with a 555-nm operating wavelength utilized in hyperspectrometers are calculated.

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

We establish spreading properties of the Lotka-Volterra competition-diffusion system. When the initial data vanish on a right half-line, we derive the exact spreading speeds and prove the convergence to homogeneous equilibrium states between successive invasion fronts. Our method is inspired by the geometric optics approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. Our main result settles an open question raised by Shigesada et al. in 1997, and shows that one of the species spreads to the right with a nonlocally pulled front.

Geometrical-optics approach to measure the optical density of bacterial cultures using a LED-based photometer.

We develop a suitable geometrical-optics approach and demonstrate that it is possible to measure the optical density (OD) of bacterial cultures using a light emitting diode (LED)-based photometer. We measure both attenuation and spot-size variation, and we compensate for diffraction and stray-light impairment related to the incoherent source and large detection area. The approach is validated for different concentrations of two bacterial species, Escherichia coli and Staphylococcus aureus, that present different shapes and clustering organization.

Modeling of laser ponderomotive self-focusing in plasma within the paraxial complex geometrical optics approach

Laser ponderomotive self-focusing in an underdense homogeneous plasma is studied within the paraxial complex geometrical optics (PCGO) approach implemented in a hydrodynamic code in two-dimensional planar geometry. The self-focusing of a PCGO Gaussian beam is compared to simulations performed with a paraxial electromagnetic code. Good agreement has been found for beam powers less than three times the critical power and for plasma densities 5%–10% of the critical density. Besides Gaussian beams, PCGO allows to reproduce spatially modulated beams by superposition of Gaussian beams, mimicking a speckle pattern. Although the statistics of speckle patterns generated with PCGO reproduces well the speckle statistics of optically smoothed beams, a PCGO speckle is larger than optical speckles, carrying thus higher power such that they overestimate self-focusing effects. To overcome this issue, an algorithm is proposed within PCGO framework: it consists of superposing several Gaussian beams forming a speckle such that self-focusing effects are eventually well controlled. It is found that the superposition of three Gaussian beams with appropriate initial conditions leads to a reduction of the PCGO speckle intensity enhancement.

Morphological properties of 2D symmetric Airy beams extracted from the stationary wave approximation

We explore the morphological properties of symmetric Airy beams in the paraxial and nonparaxial regimes. We consider a 2D electromagnetic realization with a single transverse component of the electric field, and in the nonparaxial regime, the longitudinal component along the optic axis. The general structure of these beams is analyzed with the combination of several approaches: geometrical optics through the use of caustics, the asymptotic wave properties of the light field using the stationary wave approximation and numerical integration. The geometrical optics approach involves locating the critical points that are later used in the stationary phase approximation. In the paraxial regime the highest order of the roots is 3, while in the nonparaxial regime, the order can be of up to 6. The technique yields conditions to identify interesting features on the beam, like the number of waves interfering constructively/destructively at the critical positions. The results are confirmed by the numerical simulations. In this way it is possible to distinguish and classify phase singularities like optical vortices and dislocations. The developed algorithm could be used to study any structured light field.

Geometrical-optics approach to increase the accuracy in LED-based photometers for point-of-care testing.

A geometrical-optics approach is proposed to increase the accuracy in photometric measurements, using a point-of-care testing (POCT) LED-based sensor. Due to stray-light effects, the measurement accuracy depends on the dimension of the CMOS area, where the radiation is detected. We propose two image processing approaches and evaluate the influence of the sensor area. In addition, we demonstrate that with the same measurement, both absorption coefficient and refractive index can be determined, measuring the beam attenuation and the spot-size enlargement due to ray refraction.

Local origin of mode-B secondary instability in the flow past a circular cylinder

We perform a three-dimensional, short-wavelength stability analysis on the numerically simulated two-dimensional flow past a circular cylinder for Reynolds numbers in the range $50\le Re\le300$; here, $Re = U_{\infty}D/\nu$ with $U_\infty$, $D$ and $\nu$ being the free-stream velocity, the diameter of the cylinder and the kinematic viscosity of the fluid, respectively. For a given $Re$, inviscid local stability equations from the geometric optics approach are solved on three distinct closed fluid particle trajectories (denoted as orbits 1, 2 & 3) for purely transverse perturbations. The inviscid instability on orbits 1 & 2, which are symmetric counterparts of one another, is shown to undergo bifurcations at $Re\approx50$ and $Re\approx250$. Upon incorporating finite-wavenumber, finite-Reynolds number effects to compute corrected local instability growth rates, the inviscid instability on orbits 1 & 2 is shown to be suppressed for $Re\lesssim262$. Orbits 1 & 2 are thus shown to exhibit a synchronous instability for $Re\gtrsim262$, which is remarkably close to the critical Reynolds number for the mode-B secondary instability. Further evidence for the connection between the local instability on orbits 1 & 2, and the mode-B secondary instability, is provided via a comparison of the growth rate variation with span-wise wavenumber between the local and global stability approaches. In summary, our results strongly suggest that the three-dimensional short-wavelength instability on orbits 1 & 2 is a possible mechanism for the emergence of the mode B secondary instability.

On the ionospheric propagation of VLF waves generated by currents in the lower ionosphere

The consistent description of ELF/VLF wave propagation in the ionosphere is essential for understanding the properties of the wave phenomena observed both at the ground and satellites. Due to strong inhomogeneity of the lower ionosphere coupled with collisional damping of a wave, the geometrical optics approach is not valid, while the full-wave approach suffers from the problem of numerical instability. In this paper, we implement a rather general and powerful method avoiding the numerical instability to study propagation of the wave generated in the lower ionosphere. Our calculations demonstrate the quasi-periodic behaviour of the field in the upper ionosphere, which is associated with the Earth-ionosphere waveguide. The field distribution varies significantly with frequency, from being nearly sign-definite to oscillating with comparable positive and negative values, producing the ring structure in wave-vector space. The field distribution at the ground is studied and shown to have resonant structure defined by two parameters: number of the resonance, which depends on wave frequency, and spatial scale of the Earth-ionosphere waveguide.

Optical tweezers for trapping in a microfluidic environment.

Optical tweezers use the force from a light beam to implement a precise gripping tool. Based purely on an optical principle, it works without any bodily contact with the object. In this paper we describe an optical tweezers that targets an application within the framework of nuclear magnetic resonance (NMR) spectroscopy of small objects, which are embedded inside a microfluidic channel that will be integrated in a micro-NMR detector. In the project's final stages, the whole system will be installed within the wide bore of a superconducting magnet. The aim is to precisely maintain the position of the object to be measured, without the use of susceptibility disturbing materials or geometries. In this contribution we focus on the design and construction of the tweezers. For the optical force simulation of the system we used a geometrical optics approach, which we combined with a ray fan description of the output beam of an optical system. By embedding both techniques within an iterative design process, we were able to design efficient optical tweezers that met the numerous constraints. Based on details of the constraints and requirements given by the application, different system concepts were derived and studied. Next, a highly adapted and efficient optical trapping system was designed and manufactured. After the components were characterized using vertical scanning interferometry, the system was assembled to achieve a monolithic optical component. The proper function of the optical tweezers was successfully tested by optical trapping of fused silica particles.