Airy Disk Research Articles

On the focussing of light, as limited by the uncertainty principle

Equations indicating the minimum size of the focal spot in an optical system appear in various forms. Many of these are approximations to special cases, such as the size of the Airy ring pattern produced by a circular lens as a function of its aperture size, wavenumber, k 0, and focal length, f. Various forms of focal point engineering enable arbitrarily narrow central maxima. However, a simple application of the uncertainty principle suggests an underlying limit to the spread of the intensity distribution in the focal plane that cannot be bettered. For cylindrically symmetric focussing elements and intensity distributions, the root mean square radius, r rms, in the lens plane is related to that in the focal plane, R rms, by r rms R rms ≥ f/k 0.

Resolution of ghost imaging for nondegenerate spontaneous parametric down-conversion

We examine the resolution of a ghost imaging system for the case in which the entangled photons are nondegenerate in frequency. The signal photon illuminates the object and is detected by a bucket detector. The idler photon is detected by a point detector. We consider the two cases in which the imaging lens is in the same arm as the object and when it is in the other arm. In the first case, the Airy disk is shown to depend on the wavelength of the idler photon; however, because the magnification depends on the two wavelengths, the resolution of a pair of point scatters on the object depends on the wavelength of both photons. In cases in which the position of the object can be controlled, the minimum resolvable distance can be optimized. In the case in which the object is in the far field, the resolution depends only on the signal wavelength. In the second case, the Airy disk depends on the wavelength of both photons but the resolution depends on the signal photon. While the ghost imaging system has flexibility for certain applications, the resolution of the nondegenerate ghost imaging system does not offer improvement over the classical illumination of the object.

Interference and diffraction effects in folding mirror schlieren diffraction interferometer

Experimental investigations on interference and diffraction phenomenon occurring in a folding mirror schlieren diffraction interferometer have been presented. It is shown that the well-known Lloyd mirror interferometer can easily be converted into a folding mirror schlieren diffraction interferometer. Various intermediate stages occur as the basic two-beam interference pattern in Lloyd mirror interferometer is transformed into a schlieren diffraction pattern. Study of these intermediate stages gives a good insight in understanding the transition behaviour between the fundamental phenomenon of interference and diffraction taking place as the mirror-edge approaches the Airy disk.

爱里斑的结构

The description of a plane wave diffracted by a circular aperture is directly started from the Maxwell's equations. Based on the vector angular spectrum representation of Maxwell's equations, the diffracted plane wave is decomposed into the TE and TM terms. The analytical TE and TM terms in the far field are presented by stationary phase. As the TE and TM terms are orthogonal to each other in the far field, their sum constitutes the so-called Airy disc pattern. Therefore, this research reveals the composition of Airy disc, which is beneficial to deepen and enhance the recognition of the classical diffraction problem.

Beam-Profiling and Wavefront-Sensing of THz Pulses at the Focus of a Substrate-Lens

We report combined wavefront detection and beam profiling of single-cycle terahertz (THz) pulses. In our system, the electric field is recorded highly resolved in two spatial and one temporal dimension before and after propagation through an optical component. Using this approach, we examine the imaging properties of a hyperhemispherical silicon lens as it is commonly used in THz dipole antennas. We observe an asymmetric spatiotemporal field dynamic in the focus, which can be attributed to distortion of the incident wavefront in combination with the image properties of the lens. Diffraction on the lens aperture influences the spectral beam profile at the focus. The frequency dependence of the Airy pattern indicates a rapidly degrading Strehl ratio with increasing frequency.

True polycoria or pseudo‐polycoria?

Editor, True polycoria is extremely rare (Duke-Elder 1964; Loewenfeld 1993). An intact sphincter muscle is required in each of two or more pupils in a given iris. Patients with intensely miotic pupils, with or without polycoria, are at risk of amblyopia. To our knowledge this is the only reported and photographed case of polycoria and miosis, without profound amblyopia. A 55-year-old man with presbyopia was referred by his local optician. His visual acuity (VA) was 6/6 in the right eye and 6/9 in the left. He had had the observed iris abnormality all his life, and had no previous ocular inflammation, surgery or trauma. His right pupil was normal and measured 3 mm in diameter, whereas both left pupils presented within a 2-mm central zone (Fig. 1) and measured 1.3 mm and 0.5 mm in diameter, respectively. The subject had dimmer vision in his left eye, without monocular diplopia. The photographs in Fig. 1 show that the left eye sphincter segments dilated to an equal degree around both pupils. Our patient declined all further imaging of the iris sphincter and pigment epithelium. The patient's left eye in (A) light and (B) dim light. (C, D) The pupillary margins are no longer puckered up following dilatation with guttae tropicamide 1% and phenylephrine 2.5%. Iris holes occur with considerable frequency, but in the great majority of cases they represent colobomata, iris dehiscence or iris diastasis. They are simple defects in the iris, have no pupillary sphincter and are not true pupils (Duke-Elder 1964). Pseudopolycoria is distinguished by the passive constriction of the accessory pupil when the true pupil is dilated. The mechanism of true polycoria remains uncertain. The most likely cause in our case (Fig. 2A) was the snaring off or segregation of a portion of the pupil margin (Mann 1957). Alternatively, the partial closure of a coloboma (Fig. 2B) may be caused by a bridge containing ectoderm as well as mesoderm (Mann 1957). Thirdly, residual adherence of the lens to the cornea after the separation of the two tissues may lead to the formation of a bridge across this area when the anterior chamber forms, resulting in two pupillary apertures lined with muscular fibres (Jaffe & Knie 1952). Fourthly, the anomaly may date from an early stage when the cells of the neural ectoderm are pluripotential and differentiate into muscle fibres in abnormal situations. Possible formation of two pupils by (A) snaring off a portion of the pupillary margin, or (B) secondary closure of a coloboma. Duke-Elder (1964) noted three such cases published between 1899 and 1920, and stated that a sphincter surrounding a supernumerary pupil ‘has never been demonstrated to occur’. Additional pupils associated with iridoschisis tend to be elliptical rather than circular. Drawings of polycoria often show the extra pupils to be at some distance from the principal pupil (Loewenfeld 1993). In such instances it is clear that separate iris sphincters have developed independently. In our case, the two sphincters were adjacent on retro-illumination and iris trans-illumination. We can only hypothesize that the iris anomaly did not result from in utero exposure to a toxic agent, as there was no pertinent maternal obstetric history and no other systemic medical or ocular conditions, such as anterior capsular cataract or deep corneal opacities, were found. Visual acuity decreases linearly with pupil diameter below 1.5 mm (Miller & Judisch 1979); this may account for our patient's VA of 6/9, which did not increase with pinhole or refraction. Glare can be defined as a contrast-lowering effect of stray light on a visual scene (Fig. 3). Our patient's dimmer left eye vision or decreased retinal illumination can be explained by Airy disc effect or diffraction rings and interference fringes induced by the second pupil. Patients may not convey glare symptoms clearly and this complaint can often be ignored. The second pupil allows for a contrast-lowering effect by enabling stray light at the retinal surface to obscure the macular image ‘E’. We present and review this rare phenomenon in order to provide further insight into the embryology of the pupil and to encourage others to investigate any known or future similar cases.

Very high contrast integral field spectroscopy of AB Doradus C: 9-mag contrast at 0.2 arcsec without a coronagraph using spectral deconvolution†

We present an extension of the spectral deconvolution (SD) method to achieve very high contrast at small inner working radii. We apply the method to the specific case of ground-based adaptive optics fed integral field spectroscopy (without a coronagraph). Utilizing the wavelength dependence of the Airy and speckle patterns, we make an accurate estimate of the point spread function that can be scaled and subtracted from the data cube. The residual noise in the resulting spectra is very close to the photon noise from the starlight halo. We utilize the technique to extract a very high signal-to-noise ratio H- and K-band spectrum of AB Doradus (AB Dor) C, the low-mass companion to AB Dor A. By effectively eliminating all contamination from AB Dor A, the extracted spectrum retains both continuum and spectral features. The achieved 1σ contrast is 9 mag at 0.2 arcsec, 11 mag at 0.5 arcsec, in 20 min exposure time, at an effective spectral bandwidth of 5.5 nm, proving that the method is applicable even in low-Strehl regimes. The SD method clearly demonstrates the efficacy of image slicer based integral field units in achieving very high contrast imaging spectroscopy at small angular separations, validating their use as high-contrast spectrographs/imagers for extreme adaptive optics systems.

Diffraction-induced spectral anomalies for information encoding and information hiding – Possibilities and limitations

It has been shown both theoretically and experimentally that drastic spectral changes take place around phase singularity and the shift in the spectral lines gives rise to the phenomenon called spectral switching. The shift of spectral lines towards the blue end or towards the red end of the spectrum might perhaps be used in free space communication applications. In more recent investigations, the possibility of information encoding with spectral anomalies have been discussed for two different optical systems. In this paper, we present schemes of information encoding and information hiding in the Airy pattern formed by diffraction of spatially coherent light after passing through a circular aperture. We propose a communication system that might be used to realize a new communication technique.

Alfvénic phenomena triggered by resonant absorption of an O-mode pulse

A simulation and modeling study is made of the nonlinear interaction of an electromagnetic pulse, in the O-mode polarization, with a magnetized plasma having a cross-field density gradient. For small amplitudes, the pulse propagates up to the cutoff layer where an Airy pattern develops. Beyond a certain power level, the ponderomotive force produced by the standing electromagnetic fields carves density cavities. The excess density piled up on the side of the cavities causes secondary, field-aligned plasma resonances to arise. Strong electron acceleration occurs due to the short scale of the secondary resonant fields. The fast electrons exiting the new resonant layers induce a return current system in the background plasma. This generates a packet of shear Alfvén waves of small transverse scale and increasing frequency. The results provide insight into microscopic processes associated with a recent laboratory investigation in which large-amplitude Alfvén waves have been generated upon application of high-power microwaves [B. Van Compernolle et al., Phys. Plasmas 13, 092112 (2006)].

Overcoming the Rayleigh Criterion Limit with Optical Vortices

We experimentally and numerically tested the separability of two independent equally luminous monochromatic and white light sources at the diffraction limit, using optical vortices (OV). The diffraction pattern of one of the two sources crosses a fork hologram on its center generating the Laguerre-Gaussian (LG) transform of an Airy disk. The second source, crossing the fork hologram in positions different from the optical center, generates nonsymmetric LG patterns. We formulated a criterion, based on the asymmetric intensity distribution of the superposed LG patterns so created, to resolve the two sources at angular distances much below the Rayleigh criterion. Analogous experiments in white light allow angular resolutions which are still one order of magnitude below the Rayleigh criterion. The use of OVs might offer new applications for stellar separation in future space experiments.

Are High‐Redshift Quasars Blurry?

It has been suggested that the fuzzy nature of spacetime at the Planck scale may cause lightwaves to lose phase coherence, and if severe enough this could blur images of distant point-like sources sufficiently that they do not form an Airy pattern at the focal plane of a telescope. Blurring this dramatic has already been observationally ruled out by images from Hubble Space Telescope (HST), but I show that the underlying phenomenon could still be stronger than previously considered. It is harder to detect, which may explain why it has gone unseen. A systematic search is made in archival HST images of among the highest known redshift quasars. Planck-scale induced blurring may be evident, but this could be confused with partially resolved sources.

Maximum fiber coupling efficiency and optimum beam size in the presence of random angular jitter for free-space laser systems and their applications

Free-space laser communication systems use optical-fiber-based technology such as optical amplifiers, receivers, and high-speed modulators. In these systems using single-mode fibers, the fiber coupling efficiency is one of the most significant issues to be solved. Optimum relationships between a focused optical beam and mode field size of the optical fiber in the presence of random angular jitter are discussed in relation to fiber-coupled optical systems. Maximum fiber coupling efficiency is analytically derived with the optimum Airy disk radius normalized by the mode field radius as a function of random angular jitter. The fade level of fiber-coupled signals at desired fade probability is investigated. It is shown that the average bit error ratio significantly degrades with the random angular jitter normalized by the mode field radius larger than about 0.3 when the Airy disk size is optimally selected.

Information encoding by spectral anomalies of spatially coherent light diffracted by an annular aperture

Extensive experimental study is carried out to investigate spectral behavior of spatially polychromatic coherent light diffracted by an annular aperture. It is found that the spectrum at the observation plane exhibits anomalous behavior after passing through the annular aperture. Experimental observations show that at a particular diffraction angle, the diffracted spectrum splits into two halves, while at other diffraction angles the spectrum either shifts towards lower wavelengths (blue-shift) or shifts towards higher wavelengths (red-shift). These spectral changes take place in the vicinity of the dark ring of the Airy pattern. Moreover, the behavior of the spectrum in the Fraunhofer region around a small circular loop centered at a critical direction is also studied and it is found that these spectral changes are qualitatively different from the spectral changes of the spectrum of the focused waves on a circle centered at the singular point. It is discussed in detail that such spectral anomalies of spatially coherent light may be exploited for information encoding and information hiding for free-space communications. The possibility to entangle bits in the spectral anomalies is also proposed. Such technique might be very useful to improve the throughput of information transmission in free space.

Method to mosaic gratings that relies on analysis of far-field intensity patterns in two wavelengths

We propose an experimental method to coherently mosaic two planar diffraction gratings. The method uses a Twyman–Green interferometer to guarantee the planar parallelism of the two sub-aperture gratings, and obtains the in-plane rotational error and the two translational errors from analysis of the far-field diffraction intensity patterns in two alignment wavelengths. We adjust the relative attitude and position of the two sub-aperture gratings to produce Airy disk diffraction patterns in both wavelengths. In our experiment, the repeatability of in-plane rotation adjustment was 2.35μrad and that of longitudinal adjustment was 0.11μm. The accuracy of lateral adjustment was about 2.9% of the grating period.

Exoplanet Imaging with a Phase‐induced Amplitude Apodization Coronagraph. III. Diffraction Effects and Coronagraph Design

Properly apodized pupils can deliver point-spread functions (PSFs) free of Airy rings and are suitable for high dynamical range imaging of extrasolar terrestrial planets (ETPs). To reach this goal, classical pupil apodization (CPA) unfortunately requires most of the light gathered by the telescope to be absorbed, resulting in poor throughput and low angular resolution. Phase-induced amplitude apodization (PIAA) of the telescope pupil combines the advantages of classical pupil apodization (particularly low sensitivity to low-order aberrations) with full throughput, no loss of angular resolution and little chromaticity, which makes it, theoretically, an extremely attractive coronagraph for direct imaging of ETPs. The two most challenging aspects of this technique are (1) the difficulty of polishing the required optics shapes and (2) diffraction propagation effects, which, because of their chromaticity, can decrease the spectral bandwidth of the coronagraph. We show that a properly designed hybrid system combining classical apodization with the PIAA technique can solve both problems simultaneously. For such a system, the optics shapes can be well within today's optics manufacturing capabilities, and the 10-10 PSF contrast at ≈1.5λ/D required for efficient imaging of ETPs can be maintained over the whole visible spectrum. This updated design of the PIAA coronagraph maintains the high performance of the earlier design, since only a small part of the light is lost in the classical apodizer(s).

MEMS-based adaptive optics scanning laser ophthalmoscopy

We have developed a compact, robust adaptive optics (AO) scanning laser ophthalmoscope using a microelectromechanical (MEMS) deformable mirror (DM). Facilitated with a Shack-Hartmann wavefront sensor, the MEMS-DM-based AO operates a closed-loop modal wave aberration correction for the human eye and reduces wave aberrations in most eyes to below 0.1 microm rms. Lateral resolution is enhanced, and images reveal a clear cone mosaic near the foveal center. The significant increase in throughput allows for a confocal pinhole whose diameter is less than the Airy disc of the collection lens, thereby fully exploiting the axial resolution capabilities of the system.

SiC-based optical interferometry at high pressures and temperatures for pressure and chemical sensing

Crystalline silicon carbide is a chemically inert wide band gap semiconductor with good mechanical strength and oxidation-resistant properties at elevated temperatures, which make it a good sensor material for harsh environments such as combustion chambers and turbine systems. For such cases, optical sensors are generally superior to electrical sensors in many aspects such as responsivity, detectivity, and sensitivity. This paper presents a wireless technique for pressure and chemical sensing based on the pressure-and temperature-dependent refractive indices of silicon carbide. A helium-neon laser with a wavelength of 632.8nm was used as a probe laser to obtain the complementary Airy pattern of the laser power reflected off a silicon carbide wafer segment at high temperatures (up to 300°C) and pressures (up to 400psi). The interference patterns revealed unique characteristics for nitrogen and argon test gases. This pattern is different at the same pressure and temperature for the two gases, indicating the chemical sensing selectivity capability of silicon carbide. Also the pattern changes with pressures for the same gas, indicating the pressure sensing capability. The refractive index of silicon carbide has been obtained for different pressures and temperatures using the interference pattern. A three-layer model has been employed to determine the refractive indices of the gases using the reflected power data.

Evolution of the hyperbolic umbilic diffraction pattern from Airy rings

If the diffraction pattern of the hyperbolic umbilic diffraction catastrophe is produced by an optical system of increasing aperture, it passes continuously from the two-dimensional system of Airy rings in the focal plane, made by a very small aperture, to the full three-dimensional pattern corresponding to infinite aperture. The paper studies this transition by examining the truncated diffraction integral and following the evolution of the wave dislocation lines (phase singularities) on which the pattern is based. The seed of the evolution from a two-dimensional to a three-dimensional pattern turns out to be already present asymptotically even for the smallest aperture: namely, a column of small dislocation rings very close to the axis that stream in procession towards the focal plane, and become dislocations lying in the Airy fringe surfaces that run parallel to the main fold caustic, only to disappear ultimately by retreat to infinity. The evolution into the final dislocation pattern takes place by a sequence of primitive local topological events, such as reconnection (hyperbolic interchange) and ring creation.

A213 PTVにおいて同一画像から複数の種類の粒子を認職する方法に関する研究

A particle-mask-correlation (PMC) method is developed to identify the multi-sized particles for Particle Tracking Velocimetry (PTV). The main principle of the technique is to utilize the difference of the correlation value obtained from correlation calculation between the particle mask and the particle image with multi-sized particles. The accuracy and the validity of the PMC method have been studied in the present research for different mask widths, M, and standard deviation δ of the mask. In the particle image, each particle is large enough so that the airy disk can be neglect and the particle diameters are easy to be distinguished. Investigation of the PMC method has been started with the artificial particle images, and then testified using actual PTV images. It has been shown that the two diameters dp, and d'can be well distinguished if d'/dp>2.5; correlation value is most decreased when the distance between two particles 1 is identical to the particle diameter dp.

IRAN: Interferometric Remapped Array Nulling

IRAN is a method of beam-combination in the hypertele-scope imaging technique recently introduced by Labeyrie in optical interferometry. We propose to observe the interferometric image in the pupil plane, performing multi-axial pupil plane interferometry. Imaging is performed in a combined pupil-plane where the point-source intensity distribution (PSID) tends towards a pseudo Airy disc for a sufficiently large number of telescopes. The image is concentrated into the limited support of the output pupil of the individual telescopes, in which the object-image convolution relation is conserved. Specific deconvolution algorithms have been developped for IRAN hypertelescope imagery, based upon Lucy-like iterative techniques. We show that the classical (image plane) and IRAN (pupil plane) hypertelescope imaging techniques are equivalent if one uses optical fibers for beam transportation.