Original Wavefront Research Articles

Describing ocular aberrations with wavefront vergence maps

A common optometric problem is to specify the eye's ocular aberrations in terms of Zernike coefficients and to reduce that specification to a prescription for the optimum sphero‐cylindrical correcting lens. The typical approach is first to reconstruct wavefront phase errors from measurements of wavefront slopes obtained by a wavefront aberrometer. This paper applies a new method to this clinical problem that does not require wavefront reconstruction. Instead, we base our analysis of axial wavefront vergence as inferred directly from wavefront slopes. The result is a wavefront vergence map that is similar to the axial power maps in corneal topography and hence has a potential to be favoured by clinicians. We use our new set of orthogonal Zernike slope polynomials to systematically analyse details of the vergence map analogous to Zernike analysis of wavefront maps. The result is a vector of slope coefficients that describe fundamental aberration components. Three different methods for reducing slope coefficients to a spherocylindrical prescription in power vector forms are compared and contrasted. When the original wavefront contains only second order aberrations, the vergence map is a function of meridian only and the power vectors from all three methods are identical. The differences in the methods begin to appear as we include higher order aberrations, in which case the wavefront vergence map is more complicated. Finally, we discuss the advantages and limitations of vergence map representation of ocular aberrations.

Detection of the tagged or untagged photons in acousto-optic imaging of thick highly scattering media by photorefractive adaptive holography

We propose an original adaptive wavefront holographic setup based on the photorefractive effect (PR), to make real-time measurements of acousto-optic signals in thick scattering media, with a high flux collection at high rates for breast tumor detection. We describe here our present state of the art and understanding on the problem of breast imaging with PR detection of the acousto-optic signal.

A New Wave-Front Reconstruction Method for Adaptive Optics Systems Using Wavelets

A new technique for wave-front reconstruction from gradient measurements on a square data set is presented. This technique is based on obtaining the Haar wavelet image decomposition of the original wave-front by appropriate filtering and down-sampling of the gradient measurement data. The use of the wavelet decomposition leads to an algorithm with complexity of O(N), where N is the number of data points in the reconstruction, and at the same time allows denoising of the data using the wavelet coefficients. The proposed technique is illustrated with two examples and its reconstruction accuracy, computational speed and denoising performance are discussed. Results indicate that the proposed technique is a computationally efficient and accurate technique for wave-front and/or image reconstruction from gradient measurements.

Instantaneous Phase-Shifting Lateral-Shearing Interferometer

Instantaneous Phase-Shifting Lateral-Shearing Interferometer Jae Bong Song, Yun Woo Lee and Ho Soon Yang Space Optics Research Center, Korea Research Institute of Standards and Science, Daejeon 305-600 (Received 15 January 2008) A simple instantaneous phase-shifting lateral-shearing interferometer is described. This interferometer uses only one wedge plate and two identical transmission gratings. The two identical gratings generate equally spaced identical multiple test beams, which are necessary for instantaneous phase-shifting interferometry. The multiple test beams generated by the two gratings are incident on the wedge plate along the wedge direction. The optical path di erence between the original wavefront and the sheared wavefront depends on the incident position of the test beam on the wedge plate because the wedge thickness varies along the wedge direction. This optical path di erence gives a phase shift corresponding to each interferogram. A numerical expression is derived and an experimental result is shown. PACS numbers: 07.60.Ly, 42.87.Bg

The adjoint method in seismology: I. Theory

The adjoint method is a powerful tool in geophysics that permits the computation of the exact first derivative of a physical observable or an associated objective function with respect to its parameters. Typical observables are displacement fields and flow patterns at the surface. Possible geophysical parameters are density, viscosity or elastic coefficients. When the observable can be modelled by solving a differential equation, the computation of the derivative only consists in solving the forward problem and its adjoint problem. Therefore, the adjoint method is far more efficient than any finite difference approximation. Here we present a mathematical formalism that generalises the derivation of the adjoint problem. In order to connect to work by Tarantola [Tarantola, A., 1984. Inversion of seismic reflection data in the acoustic approximation. Geophysics 49 (8), 1259–1266] we first give a derivation of the adjoint equations for the scalar wave equation in two dimensions. As objective function we choose the L 2 distance between the modelled wave field and real data. In this case the adjoint problem coincides with the original forward problem, the only difference being that the adjoint field satisfies terminal rather than initial conditions. A numerical example in two dimensions demonstrates that the adjoint field focusses near the location of a parameter perturbation at the same time when the original wavefront reaches that location. Based on this simple example, we introduce a generalised formalism for the adjoint method. It is independent of the existence of Green’s functions and their spatio-temporal reciprocity relations. Moreover, the formalism applies to non-linear equations such as the Navier–Stokes equations. This may become important in mantle flow reconstructions. The source term of the adjoint equations depends only on the specific objective function. Choosing the objective function to coincide with the observable itself allows us the computation of Jacobians, i.e., the derivative of the observable with respect to the model parameters. To demonstrate the consistency of our formalism with earlier analyses, we consider the anisotropic elastic wave equation with attenuation, which is of major interest in seismology.

Iterative zonal wave-front estimation algorithm for optical testing with general-shaped pupils

An iterative zonal wave-front estimation algorithm for slope or gradient-type data in optical testing acquired with regular or irregular pupil shapes is presented. In the mathematical model proposed, the optical surface, or wave-front shape estimation, which may have any pupil shape or size, shares a predefined wave-front estimation matrix that we establish. Owing to the finite pupil of the instrument, the challenge of wave front shape estimation in optical testing lies in large part in how to properly handle boundary conditions. The solution we propose is an efficient iterative process based on Gerchberg-type iterations. The proposed method is validated with data collected from a 15 x 15-grid Shack-Hartmann sensor built at the Nanjing Astronomical Instruments Research Center in China. Results show that the rms deviation error of the estimated wave front from the original wave front is less than lambda/130-lambda/150 after approximately 12 iterations and less than lambda/100 (both for lambda = 632.8 nm) after as few as four iterations. Also, a theoretical analysis of algorithm complexity and error propagation is presented.

Wavefront Generation Using a Dense Array and Multidimensional ΔΣ Modulation

We propose a new digital to analogue (DA) conversion technique for acoustical transducer arrays. This technique utilizes the multidimensional band-limited property of array signals. Using the technique, a large dynamic-range wavefront is converted to multiple 1-bit waveforms which are over-sampled temporally and spatially. The 1-bit waveforms are fed into a dense array of transducers. The acoustical wavefront transmitted from the array is approximately equal to the original wavefront, because the spatial high frequency components are eliminated by band-limitation due to wave propagation. This paper describes an over-sampling technique using two dimensional (2-D) ΔΣ modulation for 1-D and 2-D arrays. Results of simulations are presented for evaluation and demonstration of the applicability of the proposed method.

Analysis of lateral shearing interferograms by use of Zernike polynomials

A modal phase-reconstruction method for wave-front analysis in lateral shearing interferometry is presented. Pseudo-Zernike polynomial functions describe the differential wave fronts and are related to a Zernike polynomial description of the original wave front. We show that this reconstruction is robust for shear ratios in the range 0.15-0.50. The error propagation properties of this differential Zernike polynomial matrix-inversion method are discussed on the basis of both analysis and simulation. It is concluded that the method allows wave-front analysis with an absolute inaccuracy of 2 mλ rms for diffraction-limited wave fronts and with 1% relative inaccuracy for more strongly aberrated wave fronts.

Mechanism of spontaneous termination of functional reentry in isolated canine right atrium. Evidence for the presence of an excitable but nonexcited core.

According to the spiral wave hypothesis of reentry, the core of functional reentry remains excitable but not excited. We sought to determine whether the core remains excitable and whether excitation of the core by an outside wave front leads to termination of the reentry in the atrium. In nine isolated canine right endocardial atrial tissues (3.8 by 3.2 cm wide), reentry was induced by a premature point stimulus (S2). The isochronal activation maps and dynamics of the activation patterns were visualized with the use of 509 bipolar electrodes (1.6-mm resolution). The S2 applied after 8 regular beats induced reentry with a mean cycle length of 162 +/- 20 ms (15 episodes). Reentry had a large excitable gap (93 +/- 26 ms) as determined by early capture with twice the level of threshold stimuli. The central area (core) around which the wave fronts rotated had a mean surface area of 12 +/- 3 mm2. The electrograms located in the core of the reentry registered no or very low amplitude potentials. In 13 of 15 episodes, reentry terminated when an outside new wave front merged with the original wave front and excited the core. Core excitation caused disruption of the original wave front, and the newly formed wave front(s) vanished at the tissue border within 77 +/- 18 ms. In 2 episodes, reentry terminated abruptly when an outside new wave front propagating in a direction opposite to the reentrant wave front collided with the leading edge of the reentrant wave front. Functional reentry in the atrium is compatible with a spiral wave of excitation with an excitable but nonexcited core and a large excitable gap. Reentry may be terminated either by direct excitation of the core that displaces the wave front to the tissue border or by collision with an outside new wave front.

The study of sound field control using reproduction of plane-wave sound

A system for teleconferences using a medium-size screen with 3-D sound reproduction having proper incoming direction of the talker’s speech by using the wavefront reproduction technique is discussed. In the ordinary configuration, the multiple loudspeakers are spanned in front of the screen. Therefore, it is expected to reproduce desired wavefronts by a small number of loudspeakers. The number of loudspeakers may depend on the sizes of the screen and service area for the listeners. On the other hand, the sizes of the filter coefficients of the equalizer are also very important to simplify the system configuration, in particular, the capability of computation tasks of the digital signal processor. In this paper, the simulations and the experiments on the wavefront reproduction from the viewpoint of number and location of the loudspeakers and filter for the equalizer were demonstrated. The experimental results, in which the achievements on retrieving the original wavefront are evaluated, will be very useful in constructing the actual system.

Circus movement atrial flutter in canine sterile pericarditis model. Activation patterns during entrainment and termination of single-loop reentry in vivo.

Recently, we used a custom designed "jacket" electrode with 127 bipolar electrodes in a flexible nylon matrix to map the total atrial epicardial surface in the in situ canine heart. Atrial flutter in dogs with sterile pericarditis was shown to be due to a single wave front circulating around a combined functional/anatomic obstacle, with the arc of functional conduction block contiguous with one or more of the atrial vessels. In the present study, this model was used to analyze the activation pattern during pacing-induced entrainment and termination of single reentrant loops in a syncytium without anatomically predetermined pathways. Sustained atrial flutter was induced in five dogs with 3-5-day-old sterile pericarditis. Atrial pacing at a cycle length 5-30 msec shorter than the spontaneous cycle length entrained the arrhythmia and could result in a "classical" activation pattern, characterized by an antidromic stimulated wave that collided with the reentrant orthodromic wave front of the previous beat at a constant site. However, two variations of this classical activation pattern were also observed: 1) Pacing at short cycle lengths could lead to localized conduction block in antidromic direction, forcing a change in the pathway of the antidromic wave front. This could prevent the expected shift of the site of collision in antidromic direction. 2) The stimulated orthodromic wave front could also use a pathway different from that of the original reentrant impulse, so that a different circuit was active during the pacing period. Termination of atrial flutter by rapid atrial stimulation was associated with progressive slowing and finally blocking of the paced orthodromic wave front and a progressive shift of the site of collision in antidromic direction. The occurrence of conduction block was determined by the cycle length of stimulation and the number of stimulated beats. A longer train at the critical cycle length or the critical number of beats at a shorter cycle length could reinduce the same reentrant circuit or a different reentrant circuit, respectively, during stimulated cycles following the beat that terminated reentry. The epicardial activation sequence during entrainment of reentrant arrhythmias does not necessarily follow a standard activation pattern. Instead, the stimulated orthodromic as well as the antidromic wave front might use a pathway different from that of the original reentrant wave front. The mechanisms of termination, failure of termination, and reinitiation of single-loop reentry are similar to those in the "figure-eight" reentrant circuit.

Method of Characterizing Hologram-Recording Materials*

A new method of characterizing hologram-recording materials is presented. This method is applicable to all types of holograms. An ideal hologram-recording material permits the reconstruction of a wave front linearly proportional to the original subject wave front. Based on this concept of linear reconstruction, a material can be characterized by the properties of holograms recorded in the material with two interfering plane waves. If η is the hologram diffraction efficiency, E0 the average exposure, and V the visibility of the interference fringes, the ideal material is shown to be characterized by the relation √η = SE0V, where S is a constant. For the ideal material, curves of √η vs E0 with constant V as parameter and curves of √η vs V with constant E0 as parameter are families of straight lines. Since η, E0, and V are measurable quantities, similar curves can also be plotted for any real material. However, for a real material these curves are straight lines only within limited ranges of η, E0, and V. Within these ranges, the material may be considered ideal. Besides indicating the linearity, a set of √η−E0 or √η−V curves for a particular material also indicates explicitly the maximum hologram diffraction efficiency, the optimum average exposure value, and the holographic sensitivity of the material. √η−V characteristics of some common hologram-recording materials are shown, and the properties of the holograms recorded in these materials are compared.

Diagnostic Experiments on Converging Detonations

Photographic observations of the formation of a cylindrical detonation wave from the “patching up” of a number of planar detonation wavelets are reported. It is shown that “Mach type” reflections are responsible for the smoothing out of the curvature of the original polygonal wave front. There is evidence that stable cylindrical convergence is achieved on the basis that large-scale vorticity in the flow field behind the reflected shock wave after collapse is absent, and an extremely high temperature lasting for a few microseconds is detected. Spectroscopic studies show that a temperature of 1.89 × 105 °K is obtained when the imploding wave reflects from the center of symmetry.