Role of the tunneling ray in near-critical-angle scattering by a dielectric sphere

The scattering far zone for light transmitted through a sphere following p - 1 internal reflections by a family of near-grazing incident rays is subdivided into a lit region and a shadow region. The sharpness of the ray theory transition between the lit and the shadow regions is smoothed in wave theory by radiation shed by electromagnetic surface waves. It is shown that when higher-order terms in the physical optics approximation to the phase of the partial-wave scattering amplitudes are included, the transition between the lit and the shadow regions becomes a two-ray-to-zero-ray transition, called a superweak caustic in analogy to the more familiar scattering caustics and weak scattering caustics. One of the merged rays is a tunneling ray.     

Comparison of electromagnetic theory and various approximations for computing the absorption efficiency and single-scattering albedo of hexagonal columns

The applicability of various approximations for computing the absorption efficiency and single-scattering albedo of a randomly oriented hexagonal column is tested versus electromagnetic theory. To calculate the absorption efficiency and single-scattering albedo of the hexagonal column from electromagnetic theory we used a generalization to the separation-of-variables method, which enables continuous calculation of optical properties up to size parameters of 86. We found that the asymptotic absorption efficiency is independent of particle shape, and that, as the size parameter increases, the hexagonal column tends to its asymptotic absorption value more quickly than Mie theory. The asymptotic absorption limit of the hexagonal column is calculated accurately (to within 1%) and rapidly by use of the complex-angular-momentum approximation, indicating that this approximation could be used to calculate the absorption limit of nonspherical particles. The equal-volume sphere best approximates the hexagonal column single-scattering albedo at a strongly absorbing wavelength (e.g., 11.9 mum for an ice particle). However, in the resonance region (e.g., 80 mum for an ice particle) Mie theory fails to approximate the single-scattering albedo of the hexagonal column, but as the size parameter exceeds 10 the error in the sphere approximation reduces to within 2%. At 80-mum wavelength there is a characteristic ripple structure superimposed on the hexagonal column absorption efficiency solutions between size parameters from approximately 1 to 4. The ripple structure is indicative of surface-wave interference and is similar to the sphere but less pronounced on the hexagonal column. We investigated the applicability of ray tracing for calculating the single-scattering albedo at absorbing wavelengths relevant to remote sensing of ice particles in the atmosphere and found it to be within 4% for size parameters between 3 and 42 at 3.7-mum wavelength. At mid-infrared wavelengths (e.g., 8.5 and 11.9 mum) ray tracing is within 5% of electromagnetic theory for size parameters exceeding 10. We also tested the Bryant and Latimer absorption approximation to anomalous diffraction theory by using the separation-of-variables method.     

Field quantization,photons and non-Hermitean modes

Field quantization in unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes. We define non-Hermitean modes and their adjoints in both the cavity and external regions and make use of the important bi-orthogonality relationships that exist within each mode set. We employ a standard canonical quantization procedure involving the introduction of generalized coordinates and momenta for the electromagnetic (EM) field. Three-dimensional systems are treated, making use of the paraxial and monochromaticity approximations for the cavity non-Hermitean modes. We show that the quantum EM field is equivalent to a set of quantum harmonic oscillators (QHOs), associated with either the cavity or the external region non-Hermitean modes, and thus confirming the validity of the photon model in unstable optical systems. Unlike in the conventional (Hermitean mode) case, the annihilation and creation operators we define for each QHO are not Hermitean adjoints. It is shown that the quantum Hamiltonian for the EM field is the sum of non-commuting cavity and external region contributions, each of which can be expressed as a sum of independent QHO Hamiltonians for each non-Hermitean mode, except that the external field Hamiltonian also includes a coupling term responsible for external non-Hermitean mode photon exchange processes. The non-commutativity of certain cavity and external region annihilation and creation operators is associated with cavity energy gain and loss processes, and may be described in terms of surface integrals involving cavity and external region non-Hermitean mode functions on the cavity-external region boundary. Using the essential states approach and the rotating wave approximation, our results are applied to the spontaneous decay of a two-level atom inside an unstable cavity. We find that atomic transitions leading to cavity non-Hermitean mode photon absorption are associated with a different coupling constant to that for transitions leading to photon emission, a feature consequent on the use of non-Hermitean mode functions. We show that under certain conditions the spontaneous decay rate is enhanced by the Petermann factor.     

Scalar and electromagnetic diffraction point-spread functions

We describe an accurate technique for computing the diffraction point-spread function for optical systems. The approach is based on the combined method of ray tracing and diffraction, which implies that the computation is accomplished in a two-step procedure. First, ray tracing is employed to compute the wave-front error in a reference plane on the image side of the system and to determine the shape of the vignetted pupil. Next the Rayleigh-Sommerfeld diffraction theory, combined with the Kirchhoff approximation and the Stamnes-Spjelkavik-Pedersen method for numerical integration, is applied to compute the field in the region of the image. The method does not rely on small-angle approximations and works well for a pupil of general shape. Both scalar and electromagnetic computations are discussed and numerical results are presented.     

Recursive exact ray trace equations through the foci of the tilted off-axis confocal prolate spheroids

Abstract

Simple, exact ray trace equations are derived for a general ray propagating through the foci of tilted, coupled, confocal spheroids. In this off-axis optical system, there does not necessarily exist a (real) ray for which the paraxial approximation applies.     

Nanoantenna-enhanced gas sensing in a single tailored nanofocus

Metallic nanostructures possess plasmonic resonances that spatially confine light on the nanometre scale. In the ultimate limit of a single nanostructure, the electromagnetic field can be strongly concentrated in a volume of only a few hundred nm(3) or less. This optical nanofocus is ideal for plasmonic sensing. Any object that is brought into this single spot will influence the optical nanostructure resonance with its dielectric properties. Here, we demonstrate antenna-enhanced hydrogen sensing at the single-particle level. We place a single palladium nanoparticle near the tip region of a gold nanoantenna and detect the changing optical properties of the system on hydrogen exposure by dark-field microscopy. Our method avoids any inhomogeneous broadening and statistical effects that would occur in sensors based on nanoparticle ensembles. Our concept paves the road towards the observation of single catalytic processes in nanoreactors and biosensing on the single-molecule level.     

Three-dimensional polarization ray-tracing calculus II: retardance

The concept of retardance is critically analyzed for ray paths through optical systems described by a three-by-three polarization ray-tracing matrix. Algorithms are presented to separate the effects of retardance from geometric transformations. The geometric transformation described by a "parallel transport matrix" characterizes nonpolarizing propagation through an optical system, and also provides a proper relationship between sets of local coordinates along the ray path. The proper retardance is calculated by removing this geometric transformation from the three-by-three polarization ray-tracing matrix. Two rays with different ray paths through an optical system can have the same polarization ray-tracing matrix but different retardances. The retardance and diattenuation of an aluminum-coated three fold-mirror system are analyzed as an example.     

Multidimensional calculation of ray path in a twisted nematic liquid crystal cell

In general, light propagating an inhomogeneous liquid crystal (LC) cell can be modeled as ‘bundle rays’ because the LC cell consists of many birefringence layers. In order to calculate the optical path of the propagating light in the inhomogeneous LC cell, we multidimensionally calculated the wavevector, k, and the Poynting vector, S, of an ordinary and an extraordinary ray at LC grid interfaces, which are isotropic to a uniaxial medium and a uniaxial-to-uniaxial medium, by using the phase matching method. Furthermore, we also investigated the transmission coefficients and transmittance of the ordinary and the extraordinary rays as a function of difference of the optical axes of the facing birefringence medium at the interface to obtain the significant rays in the LC cell. Finally, we could calculate the exact path of the significant rays in the inhomogeneous LC cell, and compared the ray path in an electrically controlled birefringence (ECB) mode and a twisted nematic (TN) LC mode.     

Asymptotic behavior of the spatial frequency response of an optical system with defocus and spherical aberration

Asymptotic expressions are derived for the two-dimensional incoherent optical transfer function (OTF) of an optical system with defocus and spherical aberration. The two-dimensional stationary phase method is used to evaluate the aberrated OTF at large and moderately large defocus and spherical aberration. For small aberrations, the OTF is approximated by a power series in the aberration coefficients. An accurate approximation (in elementary functions) to the OTF is obtained for a defocused optical system with a circular pupil. We experimentally demonstrate the validity of the OTF approximations in sharp-focus image restoration from two defocused images. A digital focusing method is presented.     

Observation of sub-kilohertz resonance in Rf-optical double resonance experiment in rare earth ions in solids

Abstract

We have observed kilohertz and sub-kilohertz resonance structures in RF-optical double resonance experiments of rare-earth-doped solids, when the frequency of the RF field is scanned across the hyperfine transitions while monitoring the resonant optical absorption of a CW laser. The effect is observed only when the laser spectral width is broad compared to the hyperfine structure. The observed line widths are apparently free of the inhomogeneous widths of hyperfine levels and the line shape has peculiar double peak structure. The effect is modelled with a resonance involving three atomic levels interacting with three electromagnetic fields, two optical and one RF, in a triangular or “delta’ configuration. While the ordinary optical-RF two-field resonance is limited by spin inhomogeneous width, the simultaneous excitation of three coupled transitions leads to narrow and highly nonlinear resonance structures that are not averaged by the inhomogeneous distribution of hyperfine transition.     

General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media

Uniaxial optical anisotropy in the geometrical-optics approach is a classical problem, and most of the theory has been known for at least fifty years. Although the subject appears frequently in the literature, wave propagation through inhomogeneous anisotropic media is rarely addressed. The rapid advances in liquid-crystal lenses call for a good overview of the theory on wave propagation via anisotropic media. Therefore, we present a novel polarized ray-tracing method, which can be applied to anisotropic optical systems that contain inhomogeneous liquid crystals. We describe the propagation of rays in the bulk material of inhomogeneous anisotropic media in three dimensions. In addition, we discuss ray refraction, ray reflection, and energy transfer at, in general, curved anisotropic interfaces with arbitrary orientation and/or arbitrary anisotropic properties. The method presented is a clear outline of how to assess the optical properties of uniaxially anisotropic media.     

Transport Equations for the Wigner Distribution Function in an Inhomogeneous and Dispersive Medium

The Wigner distribution function of an optical signal, which can be considered as the momentary temporal and local spatial spectrum of the signal, is defined. Equations are derived which describe the transport of the Wigner distribution function in a medium, e.g. a plasma, that is weakly inhomogeneous in space and time, and exhibits weak dispersion for the temporal as well as the spatial frequency variable. The transport equations are compared with the eikonal equation in geometrical optics giving, as a geometrical-optical formulation of the transport equations, that along the path of a geometrical-optical light ray the Wigner distribution function has a constant value.     

Achromatic optical fourier transformer with planar-integrated free-space optics

We address the problem of achromatization of an optical system for the realization of planar-integrated, free-space optics. In particular we demonstrate an integrated optical Fourier transformation module that was achromatized for the visible spectrum by means of a diffractive lens doublet. The optical system design is studied by using the parabolic approximation of the scalar diffraction theory, including terms related to astigmatism. Based on the method of ABCD ray matrices, the optical specifications of the lens doublet are derived and the chromatic correction effect is quantified. For experimental confirmation the diffraction patterns of various grating structures are evaluated.     

Comparison of geometric optics approximation and integral method for reflection and transmission from microgeometrical dielectric surfaces

The effects of incidence angle, geometrical shape, and optical properties of dielectric rough surfaces on reflectivity and transmissivity are discussed. Radiative properties for various surface geometries are calculated. Since the integral method is computationally expensive, a geometric optics approximation is developed. The regions of validity of the approximation compared with the integral method are quantified. Curves are presented that show these radiative properties versus the correlation length at incidence angle for a fixed rms deviation of the surface. The surface geometry, incidence angle, multiple scattering, shadowing effects, and dielectric permittivity contributions to the domains of validity of the approximation method are discussed for both TE and TM polarizations.     

Scattering by simple and nonsimple shapes by the combined method of ray tracing and diffraction: application to circular cylinders

The combined method of ray tracing and diffraction (CMRD) is an efficient and accurate technique for computing the scattered field in focal regions of optical systems. Here we extend the CMRD concept so it can be used to compute fields scattered by objects of simple as well as nonsimple shapes. To that end we replace the scattering object by an equivalent, planar phase object; use ray tracing to determine its location, aperture area, amplitude distribution, and phase distribution; and use standard Kirchhoff diffraction theory to compute the field scattered by the equivalent phase object. To illustrate the practical use of the CMRD we apply it to a two-dimensional problem in which a plane or cylindrical wave is normally incident upon a circular cylinder. For this application we determine the range of validity of the CMRD by comparing its results for the scattered field with those obtained by use of an exact eigenfunction expansion.     

Dynamics of ultrashort light pulses in a semiconductor superlattice in the presence of a magnetic field

A system of equations that describes the propagation of ultrashort light pulses (optical solitons) in a semiconductor superlattice in the presence of a magnetic field is obtained using coupled Maxwell equations for the electromagnetic field and the Boltzmann equation written in the relaxation time approximation for the one-electron distribution function. It is shown that an initial linearly polarized light pulse induces a field with the orthogonal polarization in the sample. The dynamics of joint propagation of the initial and induced pulses in the sample is studied.     

Evaluation of the validity of the scalar approximation in optical wave propagation using a systems approach and an accurate digital electromagnetic model

The cause and amount of error arising from the use of the scalar approximation in monochromatic optical wave propagation are discussed using a signals and systems formulation. Based on Gauss’s Law, the longitudinal component of an electric field is computed from the transverse components by passing the latter through a two input single output linear shift-invariant system. The system is analytically characterized both in the space and frequency domains. For propagating waves, the large response for the frequencies near the limiting wave number indicates the small angle requirement for the validity of the scalar approximation. Also, a discrete simulator is developed to compute the longitudinal component from the transverse components for monochromatic propagating electric fields. The simulator output helps to evaluate the validity of the scalar approximation when the system output cannot be analytically calculated.     

Image Perturbations and Optical Flow in Time-dependent Refraction

Abstract

The ray formulation for light propagation is applied to the problem of image perturbations (‘shimmering’) resulting from perturbations in the refractive index profile of the medium between an object and an observer. Expressions are derived relating the image diagram perturbation and the instantaneous image motion (optical flow) to the change in the refractive index profile, in the paraxial ray approximation. In special circumstances the image perturbation and optical flow can be estimated directly. These results are applied to two simple examples.     

A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media

A finite-element method is presented that is particularly suited for the computer modeling of three-dimensional electromagnetic fields in inhomogeneous media. It employs a new type of linear vectorial expansion functions. Across an interface where the constitutive coefficients are discontinuous, they have the following properties: (1) the continuity of the tangential components of the electric and the magnetic field strengths is exactly preserved, (2) the normal component of the electric and the magnetic field strengths are allowed to jump and (3) the electric and the magnetic fluxes are continuous within the pertaining degree of approximation. The system of equations from which the expansion coefficients are obtained is generated by applying a Galerkin-type weighted-residual method. Numerical experiments are described that illustrate the efficiency of our elements, and the computational costs of the method.