Using rays better. I. Theory for smoothly varying media

Caustic Interpretation of the Abruptly Autofocusing Vortex beams

We propose an effective scheme to interpret the abruptly autofocusing vortex beam. In our scheme, a set of analytical formulae are deduced to well predict not only the global caustic, before and after the focal plane, but also the focusing properties of the abruptly autofocusing vortex beam, including the axial position as well as the diameter of focal ring. Our analytical results are in excellent agreement with both numerical simulation and experimental results. Besides, we apply our analytical technique to the fine manipulation of the focusing properties with a scaling factor. This set of methods would be beneficial to a broad range of applications such as particle trapping and micromachinings.

Ray-based method for simulating cascaded diffraction in high-numerical-aperture systems

The electric field at the output of an optical system is in general affected by both aberrations and diffraction. Many simulation techniques treat the two phenomena separately, using a geometrical propagator to calculate the effects of aberrations and a wave-optical propagator to simulate the effects of diffraction. We present a ray-based simulation method that accounts for the effects of both aberrations and diffraction within a single framework. The method is based on the Huygens-Fresnel principle, is entirely performed using Monte Carlo ray tracing, and, in contrast to our previously published work, is able to calculate the full electromagnetic field. The method can simulate the effects of multiple diffraction in systems with a high numerical aperture.

Simulating multiple diffraction in imaging systems using a path integration method

We present a method for simulating multiple diffraction in imaging systems based on the Huygens-Fresnel principle. The method accounts for the effects of both aberrations and diffraction and is entirely performed using Monte Carlo ray tracing. We compare the results of this method to those of reference simulations for field propagation through optical systems and for the calculation of point spread functions. The method can accurately model a wide variety of optical systems beyond the exit pupil approximation.

Ray-based diffraction calculations using stable aggregates of flexible elements

Diffraction effects are incorporated into a ray-based method for wave propagation, referred to as stable aggregates of flexible elements (SAFE). SAFE is based on the assignment of a Gaussian field contribution to each ray, where these contributions are not independent beam solutions of the wave equation. The effects of diffraction by planar opaque obstacles (within the Kirchhoff approximation) are accounted for by introducing rays emanating from the obstacle's edges. The two leading asymptotic terms to the complex amplitudes for these contributions are derived. It is shown that this scheme leads to field estimates that remain valid and accurate at caustics and shadow boundaries, as illustrated by two examples, corresponding to a focused wave in free space and a field propagating in a layered inhomogeneous medium. For simplicity, two-dimensional propagation is considered.

Framework for computing the spatial coherence effects of polycapillary x-ray optics

Despite the extensive use of polycapillary x-ray optics for focusing and collimating applications, there remains a significant need for characterization of the coherence properties of the output wavefield. In this work, we present the first quantitative computational method for calculation of the spatial coherence effects of polycapillary x-ray optical devices. This method employs the coherent mode decomposition of an extended x-ray source, geometric optical propagation of individual wavefield modes through a polycapillary device, output wavefield calculation by ray data resampling onto a uniform grid, and the calculation of spatial coherence properties by way of the spectral degree of coherence.

On the interference of two Gaussian beams and their ABCD matrix representation

Gaussian beam propagation is well described by the q-parameter and the ABCD matrices. A variety of ABCD matrices are available that represent commonly occurring scenarios/components in optics. One important phenomenon that has not been studied in detail is the interference of two optical beams with different q-parameters undergoing interference. In this paper, we describe the effect of interference of two Gaussian beams. We derive an ABCD matrix for the addition of two beams that takes into account both the amplitude and phase difference between two beams. This ABCD matrix will help greatly in determining the propagation of beams inside complex interferometers and finding the solutions for the coupled cavity Eigenmodes.

Modeling the interferometric radius measurement using Gaussian beam propagation

We model the interferometric radius measurement using Gaussian beam propagation to identify biases in the measurement due to using a simple geometric ray-trace model instead of the more complex Gaussian model. The radius measurement is based on using an interferometer to identify the test part's position when it is at two null locations, and the distance between the positions is an estimate of the part's radius. The null condition is observed when there is no difference in curvature between the reflected reference and the test wavefronts, and a Gaussian model will provide a first-order estimate of curvature changes due to wave propagation and therefore changes to the radius measurement. We show that the geometric ray assumption leads to radius biases (errors) that are a strong function of the test part radius and increase as the radius of the part decreases. We tested for a bias for both microscaled (<1 mm) and macroscaled parts. The bias is of the order of parts in 10(5) for micro-optics with radii a small fraction of a millimeter and much smaller for macroscaled optics. The amount of bias depends on the interferometer configuration (numerical aperture, etc.), the nominal radius of the test part, and the distances in the interferometer.

Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method

We use Hamiltonian optics to design and analyze beam propagation in two-dimensional (2D) periodic structures with slowly varying nonuniformities. We extend a conventional Hamiltonian method, adding equations for calculating the width of a beam propagating in such structures, and quantify the range of validity of the extended Hamiltonian equations. For calculating the beam width, the equations are more than 10(3) times faster than finite difference time domain (FDTD) simulations. We perform FDTD simulations of beam propagation in large 2D periodic structures with slowly varying nonuniformities to validate our method. Beam path and beam width calculated using the extended Hamiltonian method show good agreement with FDTD simulations. By contrasting the method with ray tracing of the bundle of rays, we highlight and explain the limitations of the extended Hamiltonian method. Finally, we use a frequency demultiplexing device optimization example to demonstrate the potential applications of the method.

Stable aggregates of flexible elements give a stronger link between rays and waves

A recently proposed ray-based method for wave propagation is used to provide a meaningful criterion for the validity of rays in wave theory. This method assigns a Gaussian contribution to each ray in order to estimate the field. Such contributions are inherently flexible. By means of a simple example, it is shown that superior field estimates can result when the contributions are no longer forced to evolve like parabasal beamlets.

Using rays better. IV. Theory for refraction and reflection

A new ray-based method is extended to include the modeling of optical interfaces. The essential idea is that the wave field and its derivatives are always expressed as a superposition of ray contributions of flexible width. Interfaces can be analyzed in this way by introducing a family of surfaces that smoothly connects them. Even though the ray-to-wave link may appear to be obscured at caustics, the standard Fresnel coefficients (for plane waves at flat interfaces between homogeneous media) are shown to be universally applicable on a ray-by-ray basis. Thus, in the interaction at the interface, the surface's curvature and any gradients in the refractive indices influence only the higher asymptotic corrections. Further, this method finally gives access to such corrections.

Using rays better. III. Error estimates and illustrative applications in smooth media

A new method for computing ray-based approximations to optical wave fields is demonstrated through simple examples involving wave propagation in free space and in a gradient-index waveguide. The analytic solutions that exist for these cases make it easy to compare the new estimates with exact results. A particularly simple RMS error estimate is developed here, and corrections to the basic field estimate are also discussed and tested. A key step for any ray-based method is the choice of a family of rays to be associated with the initial wave field. We show that, for maximal accuracy, not only must the initial field be considered in choosing the rays, but so too must the medium that is to carry the wave.

Using rays better. II. Ray families to match prescribed wave fields

A key step in any ray-based method for propagating waves is the choice of a family of rays to be associated with the initial wave field. We develop some basic prescriptions for constructing initial ray families to match two particular types of waves. Various Gaussian and Bessel beams are separately given special treatment because of their general interest. These ideas are directly useful for a newly developed method for ray-based wave modeling. The new method expresses the wave as a superposition of ray contributions that is independent of the width of the field element associated with each ray. This insensitivity is investigated here even when the elemental width varies from ray to ray. The results increase the applicability of the new wave-modeling scheme.