Radial Mode Indices Research Articles

On the dispersion management of fluorite whispering-gallery mode resonators for Kerr optical frequency comb generation in the telecom and mid-infrared range.

Optical whispering gallery mode (WGM) resonators have been very attracting platforms for versatile Kerr frequency comb generations. We report a systematic study on the material dispersion of various optical materials that are capable of supporting quality factors above 109. Using an analytical approximation of WGM resonant frequencies in disk resonators, we investigate the effect of the geometry and transverse mode order on the total group-velocity dispersion (GVD). We demonstrate that the major radii and the radial mode indices play an important role in tailoring the GVD of WGM resonators. In particular, our study shows that in WGM disk-resonators, the polar families of modes have very similar GVD, while the radial families of modes feature dispersion values that can differ by up to several orders of magnitude. The effect of these giant dispersion shifts are experimentally evidenced in Kerr comb generation with magnesium fluoride. From a more general perspective, this critical feature enables to push the zero-dispersion wavelength of fluorite crystals towards the mid-infrared (mid-IR) range, thereby allowing for efficient Kerr comb generation in that spectral range. We show that barium fluoride is the most interesting crystal in this regard, due to its zero dispersion wavelength (ZDW) at 1.93 μm and an optimal dispersion profile in the mid-IR regime. We expect our results to facilitate the design of different platforms for Kerr frequency comb generations in both telecommunication and mid-IR spectral ranges.

Remote state preparation of three-dimensional optical vortices.

We propose a feasible scheme to remotely prepare three-dimensional (3D) optical vortex lines. Our scheme relies on the complete description of high-dimensional orbital angular momentum (OAM) entanglement in terms of the Laguerre-Gaussian modes. It is theoretically demonstrated that by simply changing the pump beam waist, we can remotely prepare the target photons in the vortex states of 3D interesting morphology, appearing as twisted vortex strands, separated vortex loops, and vortex link or knot. Furthermore, we employ the biphoton Klyshko picture to illustrate the conservation law of the OAM index ℓ and the spreading effect of the radial mode index p, where the Schmidt numbers are calculated to show the high-dimensional capacity of the quantum channels involved in the present remote state preparation.

Autocorrelation properties of fully coherent beam with and without orbital angular momentum

The spatial correlation singularity of a partially coherent vortex beam was demonstrated recently [Phys. Rev. Lett. 92, 143905 (2004)], and it was shown that the cross-correlation singularity disappears when the spatial coherence is high. In this paper, we demonstrate that the spatial autocorrelation function of a fully coherent beam in the far-field is equivalent to the Fourier transform of its intensity in the source plane. Our theoretical and experimental results show that, depending on both the radial and azimuthal mode indices (p, λ) of the incident light beam, the distribution of the far-field autocorrelation function displays a series of concentric, alternate bright and dark rings. This phenomenon may be used to determine the topological charge (the azimuthal index) of light beam with a nonzero radial index.

Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity

The existence of a spatial correlation singularity or a ring dislocation in the spatial coherence function when a vortex is present has been demonstrated recently. Here, we investigate how the spatial correlation singularity is affected by both the radial and azimuthal mode indices (p, ℓ) in a partially coherent light field. Theoretically, we find that the spatial correlation singularity may exist even in a non-vortex beam (ℓ = 0) due to the radial index. Numerical simulations show the number of ring dislocations in the far-field cross correlation function is equal to 2p + |ℓ| for the low coherence cases. This is confirmed by our experimental results. This phenomenon may occur in any partially coherent vortex wave.

Controllable vector bottle-shaped fields generated by focused spatial-variant linearly polarized vector beams

We demonstrate that the optical bottle-shaped fields can be controllably generated by the focused spatial-variant linearly polarized vector beams. Based on the vectorial Rayleigh–Sommerfeld formulas under the paraxial approximation, we present theoretically the analytical expression for the focused field of the vector beam and predict the evolution of the sate of polarization (SoP) in the focal region. Experimentally, we observe the vector bottle-shaped field that is in agreement with the numerical simulations. In particular, we validate that both the SoP and the size of the optical bottle field are manipulated easily by varying the azimuthal topological charge and the radial mode index.

Modal Characterization using Principal Component Analysis: application to Bessel, higher-order Gaussian beams and their superposition

The modal characterization of various families of beams is a topic of current interest. We recently reported a new method for the simultaneous determination of both the azimuthal and radial mode indices for light fields possessing orbital angular momentum. The method is based upon probing the far-field diffraction pattern from a random aperture and using the recorded data as a ‘training set'. We then transform the observed data into uncorrelated variables using the principal component analysis (PCA) algorithm. Here, we show the generic nature of this approach for the simultaneous determination of the modal parameters of Hermite-Gaussian and Bessel beams. This reinforces the widespread applicability of this method for applications including information processing, spectroscopy and manipulation. Additionally, preliminary results demonstrate reliable decomposition of superpositions of Laguerre-Gaussians, yielding the intensities and relative phases of each constituent mode. Thus, this approach represents a powerful method for characterizing the optical multi-dimensional Hilbert space.

Measurements of transverse lateral and longitudinal angular shifts of high-azimuthal-mode Laguerre–Gaussian beams reflected at a dielectric interface near critical incidence

Laguerre–Gaussian (LG) beams with a radial mode index p = 0 and azimuthal mode indices l from −5 to 5 are reflected at a glass–air interface near critical incidence, and the transverse lateral and longitudinal angular shifts of the intensity center of gravity are measured relative to the LGp=0,l=0 beam. The LG beams are generated with a programmable spatial light modulator that displays computer-generated holograms encoding phase and amplitude information. The measured relative shifts are greatly enhanced near the critical angle and agree with the theoretical values numerically calculated using the wavenumber-space representation.

Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum

A wide array of diffractive structures such as arrays of pinholes, triangular apertures, slits, and holograms have all recently been used to measure the azimuthal index of individual Laguerre-Gaussian beams. Here, we demonstrate a powerful approach to simultaneously measure both the radial and azimuthal indices of pure Laguerre-Gaussian light fields using the method of principal component analysis. We find that the shape of the diffracting element used to measure the mode indices is in fact of little importance and the crucial step is training any diffracting optical system and transforming the observed pattern into uncorrelated variables. The method is generic and may be extended to other families of light fields such as Bessel or Hermite-Gaussian beams.

Improvement of optical trapping effect by using the focused high-order Laguerre–Gaussian beams

We investigate the optical trapping effect of high-order Laguerre–Gaussian (LG) beams acting on a dielectric sphere in Rayleigh regime. For LG beams with the azimuthal mode index l=0, it is found that under the same input power, the transverse trapping effect can be enhanced several times with increasing the radial mode index p, compared with that of the Gaussian beam; while its axial trapping effect is exactly the same as that of Gaussian beam, although the central trapping region reduces as p increases. For LG beams with l≥1, we find that the maximal transverse gradient forces increase with the increasing of p and the axial radiation forces reduces slightly, therefore an optimal choose on p and l is necessary for obtaining an optimal optical guiding. Our result is useful for analyzing the trapping efficiency of LG beams applied in micromanipulation technologies.

Radial mode dependence of optical beam shifts

It is known that orbital angular momentum (OAM) couples the Goos-Hänchen and Imbert-Fedorov shifts. Here, we present the first study of these shifts when the OAM-endowed LG(ℓ,p) beams have higher-order radial mode index (p>0). We show theoretically and experimentally that the angular shifts are enhanced by p while the positional shifts are not.

Focusing of high order cylindrical vector beams

We present the results of calculations of focusing high order cylindrical vector beams in the limit of high numerical aperture. We derive a form of the vectorial diffraction integrals for arbitrary radial and azimuthal mode indices and evaluate these numerically for a number of different modes. We identify combinations of mode indices and lens filling factors that produce focal volume shapes that may be of interest for a number of applications such as optical trapping, two-photon lithography or optical super-resolution. Finally we evaluate the effect of spherical aberration on the focusing.

An improved multimodal method for sound propagation in nonuniform lined ducts

An efficient method is proposed for modeling time harmonic acoustic propagation in a nonuniform lined duct without flow. The lining impedance is axially segmented uniform, but varies circumferentially. The sound pressure is expanded in term of rigid duct modes and an additional function that carries the information about the impedance boundary. The rigid duct modes and the additional function are known a priori so that calculations of the true liner modes, which are difficult, are avoided. By matching the pressure and axial velocity at the interface between different uniform segments, scattering matrices are obtained for each individual segment; these are then combined to construct a global scattering matrix for multiple segments. The present method is an improvement of the multimodal propagation method, developed in a previous paper [Bi et al., J. Sound Vib. 289, 1091-1111 (2006)]. The radial rate of convergence is improved from O(n(-2)), where n is the radial mode indices, to O(n(-4)). It is numerically shown that using the present method, acoustic propagation in the nonuniform lined intake of an aeroengine can be calculated by a personal computer for dimensionless frequency K up to 80, approaching the third blade passing frequency of turbofan noise.

Generation of self-imaged optical bottle beams

We propose a novel technique to generate self-imaged optical bottle beams by interfering two Bessel beams with varying radial propagation constant or radial frequency. The optical bottle beams have on-axis intensity minima along the propagation direction surrounded with regions of higher intensity. We introduce the phenomena of self-imaging in an optical bottle beam to possess numerous bottles embedded in a single beam. This removes the constraint on optical bottle beams to possess p bottles or null intensity points, where p is radial mode index of a Laguerre–Gaussian beam used to generate it, as reported by McGlaoin [Opt. Commun. 225 (2003) 215]. This technique also provides additional freedom on the bottle size and its location, which can be pre-determined. Both simulation and experimental results for the generation of such beams are reported in the paper.

Three-dimensional arrays of optical bottle beams

We demonstrate the production of an optical beam with a number of embedded optical bottles. We generate this beam by passing a Laguerre–Gaussian beam, with a radial mode index of p>1, through an axicon. Such beams are found to possess the “non-diffracting” properties of a Bessel beam while having unusual intensity distributions along the propagation distance. A beam with a single radial node has an on-axis intensity with an intensity minima within the propagation distance, surrounded by a high intensity region. This creating an optical bottle beam with cylindrical symmetry. The production of non-diffracting multiple-cavity beams – a three-dimensional array of optical bottles – is also shown.

The initiation, termination, and evolution of nonparaxial optical vortices: High-order singular beams

Exact solutions to the scalar Helmholtz wave equation describing nonparaxial high-order mode beams are considered. To match these solutions to those of the parabolic wave equation, soft approximate boundary conditions are formulated. According to these conditions, in the paraxial limit kz 0≫1, paraxial and nonparaxial wave beams transform into each other either far off the focal caustic (at z≫z 0) or near it. The behavior of the mode beams with different azimuthal and radial mode indices is described.

Mode chart of electrically thin parallel-plate circular resonators

Modes in electrically thin circular parallel-plate resonators are discussed in detail. It is shown that the widely used mode indices (found from the field solutions) do not indicate the radial numbers of the field maximums in a straightforward manner. The diameter of the disc, not the radius, accommodates integer numbers of the field maximums (antinodes). These diametrical numbers are expressed in terms of radial mode indices. Emphasis is placed on the first axially symmetric mode TM020, which was incorrectly regarded as TM010 in the past. Some applications of the mode chart, including interpretation of experimental observations, are also discussed. Comparison is made with microstrip and annular resonators.

Initial Algebraic Growth of Small Angular Dependent Disturbances in Pipe Poiseuille Flow

The evolution of small, angular dependent velocity disturbances in laminar pipe flow is studied. In particular, streamwise independent perturbations are considered. To fully describe the flow field, two equations are required, one for the radial and the other for the streamwise velocity perturbation. Whereas the former is homogeneous, the latter has the radial velocity component as a forcing term. First, the normal modes of the system are determined and analytical solutions for eigenfunctions, damping rates, and phase velocities are calculated. As the azimuthal wave number (n) increases, the damping rate increases and the phase velocities decrease. Particularly interesting are results showing that the phase velocities associated with the streamwise eigenfunctions are independent of the radial mode index when n = 1, and when n = 5 the same is obtained for phase velocities associated with the eigenfunctions of the radial component. Then, the initial value problem is treated and the time development of the disturbances is determined. The radial and the azimuthal velocity components always decay but, owing to the forcing, the streamwise component shows an initial algebraic growth, followed by a decay. The kinetic energy density is used to characterize the induced streamwise disturbance. Its dependence on the Reynolds number, the radial mode, and the azimuthal wave number is investigated. With a normalized initial disturbance, n = 1 gives the largest amplification, followed by n = 2 etc. However, for small times, higher values of n are associated with the largest energy density. As n increases, the distribution of the streamwise velocity perturbation becomes more concentrated to the region near the pipe wall.

Asymptotic expansion for spherical mode phase with application to near-field measurements

The exact-circuit representation of spherical modes allows the derivation of a convenient asymptotic relationship for the phase difference between a plane wave and a spherical wave as a function of the radial position and spherical mode index. One application of this asymptotic form is to allow the effects of range uncertainties on patterns measured in a near-field antenna range to be related simply to the more familiar finite range effects in a direct far-field measurement.

Role of scattering in the design of optimal duct treatments

The effects of scattering at the beginning of a finite length, cylindrical duct wall treatment was studied. Single frequency attenuation and optimal impedance of a locally reacting lining was considered. The effect of backward scattering was found to be small, but that of forward scattering substantial. Optimal wall impedance is presented as a function of the ratio of treatment length to duct radius and the mode cutoff ratio, which has been introduced as a design criterion (E. J. Rice, NASA TMX‐73411). Although the optimal impedance can be reasonably well described by the cutoff ratio, the attending attenuation cannot, but must be presented as a function of both azimuthal and radial mode indices.