Pupil Apodization Research Articles

Optimal pupil basis set for telescope-coronagraph design and perturbation analysis based on the method of moments.

This paper concerns optimization and analysis of telescope-coronagraph systems for direct imaging of exoplanets. In this paper, the coronagraph system, with arbitrary pupil geometry, is theoretically considered, and the governing equation for the pupil design is derived. The method of moments is applied to solve the generalized energy-concentration eigenvalue problem to obtain the optimal pupil apodization and complete sets of orthonormal basis functions for arbitrary pupil geometries. The method yields eigenvalues indicating the fraction of starlight energy encircled in the area of the focal-plane mask (FPM), where starlight can be occulted and/or nulled. In other words, a higher eigenvalue implies less leakage/spillover of light outside of the FPM region and into the planet-discovery zone. Thus, a higher eigenvalue supports better starlight suppression for a given type of coronagraph. This methodology is useful for semi-quantitatively ranking different modes of perturbation with respect to energy spillage in the focal plane independent of coronagraph design details. A model-order-reduction-based sensitivity analysis is conducted to investigate the coupling between different pupil modes induced by aberrations. A pupil mode recovery scheme is presented to offer a theoretically rigorous and computationally efficient approach to reconstruct the optimal pupil mode under an arbitrary phase perturbation. The reconstruction coefficients and recovery-effectiveness factors are derived theoretically and demonstrated numerically. Several numerical examples, including the LUVOIR A and B pupils, are provided to validate and demonstrate the applicability of the proposed methods. The reported methodology enables model-order reduction based on degree of focal-plane energy concentration and reconstruction of optimal pupil apodization vis-á-vis phase aberrations using a precomputed basis set. These features should enhance computational efficiency for coronagraph design and sensitivity analysis.

Nonparaxial diffraction of the Cartesian oval

We study the nonparaxial diffraction of the well-known Cartesian oval in finite and infinite conjugate configurations. We started by expressing the refraction of the Cartesian oval by analytical closed-form equations. These equations are convenient to obtain the pupil apodization function of the Cartesian oval, which is needed to compute the diffraction pattern using Richard–Wolf theory. A comparison of the diffraction patterns of the Cartesian oval with finite/infinite objects, the aplanatic lens, and the parabolic mirror for radially polarized illumination is presented. From this comparison, we conclude that a Cartesian oval for a far object is not a good candidate for tight focusing and super-resolution applications and the performance of a Cartesian oval in a finite conjugate configuration is similar to the performance of an aplanatic lens and in some scenarios, it can perform better than the aplanatic lens.

Pupil apodization in digital holographic microscopy for reduction of coherent imaging effects

A dark-field Digital Holographic Microscope with a single lens for imaging is a potential candidate for future overlay metrology on semiconductor wafers. Aberrations caused by this single lens are computationally corrected allowing high-resolution imaging over a large wavelength range. However, the spatially-coherent imaging conditions in our microscope introduce coherent imaging artifacts that can limit the metrology performance. We present computational apodization of the optical field in the exit pupil of the lens as a potentially effective solution to mitigate these coherent imaging effects. A comparison of experimental data and simulations is presented that demonstrates the importance of this apodization in metrology applications. Moreover, our data also shows that exploiting the full potential of DHM requires an imaging lens with low optical scattering levels.

A family of phase masks for broadband coronagraphy example of the wrapped vortex phase mask theory and laboratory demonstration

Context. Future instruments need efficient coronagraphs over large spectral ranges to enable broadband imaging or spectral characterization of exoplanets that are 108 times fainter than their star. Several solutions have been proposed. Pupil apodizers can attenuate the star intensity by a factor of 1010 but they only transmit a few percent of the light of the planet. Cascades of phase and/or amplitude masks can both attenuate the starlight and transmit most of the planet light, but the number of optics that require alignment makes this solution impractical for an instrument. Finally, vector phase masks can be used to detect faint sources close to bright stars but they require the use of high-quality circular polarizers and, as in the previous solution, this leads to a complex instrument with numerous optics that require alignment and stabilization. Aims. We propose simple coronagraphs that only need one scalar phase mask and one binary Lyot stop providing high transmission for the planet light (> 50%) and high attenuation of the starlight over a large spectral bandpass (∼30%) and a 360° field-of-view. Methods. From mathematical considerations, we find a family of 2D phase masks optimized for an unobscured pupil. One mask is an azimuthal wrapped vortex phase ramp. We probe its coronagraphic performance using numerical simulations and laboratory tests. Results. From numerical simulations, we predict the wrapped vortex can attenuate the peak of the star image by a factor of 104 over a 29% bandpass and 105 over a 18% bandpass with transmission of more than 50% of the planet flux at ∼4λ/D. We confirm these predictions in the laboratory in visible light between 550 and 870 nm. We also obtain laboratory dark hole images in which exoplanets with fluxes that are 3 × 10−8 times the host star flux could be detected at 3σ. Conclusions. Taking advantage of a new technology for etching continuous 2D functions, a new type of mask can be easily manufactured opening up new possibilities for broadband coronagraphy.

Laboratory demonstration of a broadband six-level phase mask coronagraph.

The six-level phase mask (SLPM) can be used in a focal plane as an efficient coronagraph [Opt. Express 22, 1884 (2014)]. It has several advantages: high-contrast imaging in broadband with small inner working angle; easy fabrication at low cost by photolithography and reactive ion etching processes; easy implementation with no need of pupil apodization. We present in this paper the first laboratory results demonstrating the high performance of a SLPM with an unobscured pupil. The on-axis attenuation reaches 2 × 10-5 at λ = 800 nm and is better than 10-4 over a 10% spectral bandwidth and better than 10-3 over a 20% bandwidth. Finally, the detection of a planet can be achieved down to 1 λ/D.

Improvement of imaging characteristics by pupil apodization in a laser scanning system for electro-photography

It is well known that imaging characteristics such as beam spot size and focal depth can be improved by applying apodization of amplitude and phase at the entrance pupil of an optical system. As an optical system for the electro-photography continuously requires highly resolved dot image and extended focal length to obtain more delicate expression with adequate production stability. Advantages from apodization technique can improve system performance and supply high degree of reliability of the optical system. However, issues on reduction in absolute optical power and on side-lobe of beam spot profile have been technical barrier for its useful implementation to imaging optics for commercial devices. Thorough this paper, theoretical feasibility of application of a pupil apodization to a laser scanning optical unit is mainly discussed in the aspects of optical power reduction and side-lobe of spot profile, as well as enhancement of optical resolution and focal depth. In addition, fundamental experimental results on beam spot profile through focal region are followed to uphold feasibility analysis.

Theoretical and experimental studies on optical trapping using radially polarized beams

Optical trapping using radially polarized beams is studied theoretically and experimentally. First the geometric ray model is introduced to calculate the trapping efficiencies, and simulation results using three kinds of pupil apodization functions are presented to disclose the influences of pupil apodization functions of incident beams on trapping efficiencies, which indicates the better trapping performances can be achieved by modulating the polarization and amplitude distributions of incident beams using designated pupil filters. Furthermore, the optical tweezers using radially polarized beams are built up based on an inverted microscope and a spatial light modulator (SLM), where better trapping efficiencies can be achieved. Yeast cells about 10μm in diameter are trapped and manipulated using the optical tweezers and the cells can be trapped stably and shifted along the tracks of the focusing spot, which can be programmed by a computer. In addition, the values of trap stiffness for different pupil apodization functions are measured at different laser powers based on the Boltzmann statistics method, which indicates the AL-types pupil apodization function is a better choice for general trapping and manipulation of cells.

APODIZED PUPIL LYOT CORONAGRAPHS FOR ARBITRARY APERTURES. V. HYBRID SHAPED PUPIL DESIGNS FOR IMAGING EARTH-LIKE PLANETS WITH FUTURE SPACE OBSERVATORIES

ABSTRACT We introduce a new class of solutions for Apodized Pupil Lyot Coronagraphs (APLC) with segmented aperture telescopes to remove broadband diffracted light from a star with a contrast level of 1010. These new coronagraphs provide a key advance to enabling direct imaging and spectroscopy of Earth twins with future large space missions. Building on shaped pupil (SP) apodization optimizations, our approach enables two-dimensional optimizations of the system to address any aperture features such as central obstruction, support structures, or segment gaps. We illustrate the technique with a design that could reach a 1010 contrast level at 34 mas for a 12 m segmented telescope over a 10% bandpass centered at a wavelength of λ 0 = ?> 500 nm. These designs can be optimized specifically for the presence of a resolved star and, in our example, for stellar angular size up to 1.1 mas. This would allow one to probe the vicinity of Sun-like stars located beyond 4.4 pc, therefore, fully retiring this concern. If the fraction of stars with Earth-like planets is &eegr; ⊕ = 0.1 ?> , with 18% throughput, assuming a perfect, stable wavefront and considering photon noise only, 12.5 exo-Earth candidates could be detected around nearby stars with this design and a 12 m space telescope during a five-year mission with two years dedicated to exo-Earth detection (one total year of exposure time and another year of overheads). Our new hybrid APLC/SP solutions represent the first numerical solution of a coronagraph based on existing mask technologies and compatible with segmented apertures, and that can provide contrast compatible with detecting and studying Earth-like planets around nearby stars. They represent an important step forward toward enabling these science goals with future large space missions.

Numerical modeling of the proposed WFIRST-AFTA coronagraphs and their predicted performances

The WFIRST-AFTA 2.37 m telescope will provide the opportunity to host a coronagraph for the imaging and spectroscopy of planets and disks in the next decade. The telescope, however, is not ideal, given its obscured aperture. Only recently have coronagraph designs been thoroughly investigated that can efficiently work with this configuration. Three coronagraph designs, the hybrid Lyot, shaped pupil, and phase-induced amplitude apodization complex mask coronagraph have been selected for further development by the Astrophysics Focused Telescope Asset project. Real-world testbed demonstrations of these have just begun, so for now, the most reliable means of evaluating their potential performance comes from numerical modeling incorporating diffraction propagation, realistic system models, and simulated wavefront sensing and control. Here, we present the methods of performance evaluation and results for the current coronagraph designs.

Effect of the focal shaping generated from different double-mode cylindrical vector beams.

We investigate three-dimensional focus shaping generated from double-mode cylindrical vector beams with the Gaussian and Bessel-Gaussian pupil apodization functions by choosing the suitable polarization states of beams. Further, we compare them with that generated from the Laguerre-Gaussian pupil apodization function in the same situation. We find that the focus shaping generated from the Gaussian beam has the smallest zero intensity spot size. However, the situation of the Bessel-Gaussian beam not only possesses stability, which makes it suitable when applied in optical trapping, but also shows the best uniformity, which indicates its excellent performance in super-resolution fluorescence microscopy.

Coherent Response of a Human Eye: an Apodization Model of Stiles Crawford Effect of the first Kind

In Stiles Crawford effect ofthe first kind a beam entering a pupilfrom the edge shows a diminution in the effective brightness compared to an identical beam passing axially. The same reduction in brightness is also observed in a pupil becoming gradually less transparent from the centre toward its periphery, known as pupil apodization. In this study we have treated Stiles Crawford effect as pupil apodization and evaluated the modulation in the retinal image using a periodic object of infinite cycles with a saw-tooth wave transmission profile, both under incoherent and coherent illumination. Over a wide range of spatial frequencies, though the incoherent entering beam shows a gradual decrease in the modification, no change is observed for coherent illumination. Thus, the modeling of Stiles Crawford effect as pupil apodization is insensitive to coherent illumination and the retinal distribution of light is dependent on the coherence of the incident beam apart from pupil entry point of the beam in traditional Stiles Crawford effect. Hence, this study highlights the significance of employing coherence in the entering beam as an optical parameter in governing retinal response and understanding the spatial distribution of photon absorption in the photoreceptors of a human eye. Keywords: Apodization, Coherent Illumination, Modulation, Saw-Tooth Wave, Stiles-Crawford Effect

Chromatic aberration and polychromatic image quality with diffractive multifocal intraocular lenses

To evaluate the impact of target distance on polychromatic image quality in a virtual model eye implanted with hybrid refractive-diffractive intraocular lenses (IOLs). School of Optometry, Indiana University, Bloomington, Indiana, USA. Experimental study. A pseudophakic model eye was constructed by incorporating a phase-delay map for a diffractive optical element into a reduced eye model incorporating ocular chromatic aberration, pupil apodization, and higher-order monochromatic aberrations. The diffractive element was a monofocal IOL with a +3.2 diopter (D) diffractive power or 2 types of bifocal IOLs (nonapodized or apodized) with a +2.92 D addition (add) power. Polychromatic point-spread functions and image quality for white and monochromatic light were quantified for a series of target vergences, wavelengths, and pupil diameters using modulation transfer functions and image-quality metrics. Ocular longitudinal chromatic aberration was largely corrected by the monofocal design and by both bifocal designs for near targets. In the bifocal design, add power and the ratio of distance:near image quality changed significantly with wavelength and pupil size. Also, image quality for distance was better with the apodized design. Achromatization by the diffractive IOL provided significant improvement in polychromatic retinal image quality. Along with apodization and higher-order aberrations, it can significantly affect the near-distance balance provided by a diffractive multifocal IOL. No author has a financial or proprietary interest in any material or method mentioned.

Visualization of retinal vascular structure and perfusion with a nonconfocal adaptive optics scanning light ophthalmoscope

Imaging of the retinal vascular structure and perfusion was explored by confocal illumination and nonconfocal detection in an adaptive optics scanning light ophthalmoscope (AOSLO), as an extension of the work by Chui et al. [Biomed. Opt. Express 3, 2537 (2012)]. Five different detection schemes were evaluated at multiple retinal locations: circular mask, annular mask, circular mask with filament, knife-edge, and split-detector. Given the superior image contrast in the reflectance and perfusion maps, the split-detection method was further tested using pupil apodization, polarized detection, and four different wavelengths. None of these variations provided noticeable contrast improvement. The noninvasive visualization of capillary flow and structure provided by AOSLO split-detection shows great promise for studying ocular and systemic conditions that affect the retinal vasculature.

The Dependence of Spatial Distribution of Excitement of Receptors on Their Stiles-Crawford Effect

The spatial distribution of excitement of receptors depends on the magnitude of amplitude of light integrated over the wavefront passing through the pupil as conditioned by the receptor's acceptance lobe. Because the peripheral zones of the pupil are less efficient in eliciting visual excitation than the pupil's centre, otherwise known as Stiles-Crawford effect of the first kind (SCE I), the spatial distribution of excitement of receptors will differ depending on their Stiles- Crawford effect. Assuming the funneling of light into the receptors through a single entering aperture the SCE I is integrated into the calculation of retinal diffraction images, in the manner of an apodization effect. And, in the absence of aberrations, the pupil apodization is chosen twice as narrow as suggested by the SCE I to compute modulation in the retinal light distributions. But using coherent illumination, a periodic target of rectangular profile and incorporating SCE I as pupil apodization we got a result where neither the intensity nor the modulation changes indicating absence of SCE I for a full pupil admitting a single coherent beam.

HIGH PERFORMANCE LYOT AND PIAA CORONAGRAPHY FOR ARBITRARILY SHAPED TELESCOPE APERTURES

Two high performance coronagraphic approaches compatible with segmented and obstructed telescope pupils are described. Both concepts use entrance pupil amplitude apodization and a combined phase and amplitude focal plane mask to achieve full coronagraphic extinction of an on-axis point source. While the first concept, named Apodized Pupil Complex Mask Lyot Coronagraph (APCMLC), relies on a transmission mask to perform the pupil apodization, the second concept, named Phase-Induced Amplitude Apodization complex mask coronagraph (PIAACMC), uses beam remapping for lossless apodization. Both concepts theoretically offer complete coronagraphic extinction (infinite contrast) of a point source in monochromatic light, with high throughput and sub-lambda/D inner working angle, regardless of aperture shape. The PIAACMC offers nearly 100% throughput and approaches the fundamental coronagraph performance limit imposed by first principles. The steps toward designing the coronagraphs for arbitrary apertures are described for monochromatic light. Designs for the APCMLC and the higher performance PIAACMC are shown for several monolith and segmented apertures, such as the apertures of the Subaru Telescope, Giant Magellan Telescope (GMT), Thirty Meter Telescope (TMT), the European Extremely Large Telescope (E-ELT) and the Large Binocular Telescope (LBT). Performance in broadband light is also quantified, suggesting that the monochromatic designs are suitable for use in up to 20% wide spectral bands for ground-based telescopes.

Defocus-corrected analysis of the foveal Stiles–Crawford effect of the first kind across the visible spectrum

The Stiles–Crawford effect of the first kind describes a gradually diminished visibility of light that enters the eye towards the pupil rim. Although of retinal origin, it is commonly described by a Gaussian pupil apodization whose width is determined by a directionality parameter that depends on retinal eccentricity, wavelength and spatial coherence of the light. As the measurements are done psychophysically they are prone to subjective variations and difficult to obtain across the visible spectrum. In this work, requirements for accurate refractive correction when determining the directionality parameter at any given wavelength are discussed and we show that a current-controlled tunable liquid-polymer lens provides a convenient means to accomplish this without requiring mechanical readjustments. This may be the most convenient way to combat defocus across the visible spectrum in the analysis of the Stiles–Crawford effect as demonstrated through experiments and with a detailed Zemax eye-and-system analysis. The results obtained are discussed in relation to myopia and a reduced directionality for highly myopic eyes.

In vivo dark-field imaging of the retinal pigment epithelium cell mosaic

Non-invasive reflectance imaging of the human RPE cell mosaic is demonstrated using a modified confocal adaptive optics scanning light ophthalmoscope (AOSLO). The confocal circular aperture in front of the imaging detector was replaced with a combination of a circular aperture 4 to 16 Airy disks in diameter and an opaque filament, 1 or 3 Airy disks thick. This arrangement reveals the RPE cell mosaic by dramatically attenuating the light backscattered by the photoreceptors. The RPE cell mosaic was visualized in all 7 recruited subjects at multiple retinal locations with varying degrees of contrast and cross-talk from the photoreceptors. Various experimental settings were explored for improving the visualization of the RPE cell boundaries including: pinhole diameter, filament thickness, illumination and imaging pupil apodization, unmatched imaging and illumination focus, wavelength and polarization. None of these offered an obvious path for enhancing image contrast. The demonstrated implementation of dark-field AOSLO imaging using 790 nm light requires low light exposures relative to light safety standards and it is more comfortable for the subject than the traditional autofluorescence RPE imaging with visible light. Both these factors make RPE dark-field imaging appealing for studying mechanisms of eye disease, as well as a clinical tool for screening and monitoring disease progression.

Apodization in high-contrast long-slit spectroscopy

The spectroscopy of faint planetary-mass companions to nearby stars is one of the main challenges that new-generation high-contrast spectro-imagers are going to face. In a previous work we presented a long slit coronagraph (LSC), for which the presence of a slit in the coronagraphic focal plane induces a complex distribution of energy in the Lyot pupil-plane that cannot be easily masked with a binary Lyot stop. To alleviate this concern, we propose to use a pupil apodization to suppress diffraction, creating an apodized long slit coronagraph (ALSC). After describing how the apodization is optimized, we demonstrate its advantages with respect to the CLC in the context of SPHERE/IRDIS long slit spectroscopy (LSS) mode at low-resolution with a 0.12" slit and 0.18" coronagraphic mask. We perform different sets of simulations with and without aberrations, and with and without a slit to demonstrate that the apodization is a more appropriate concept for LSS, at the expense of a significantly reduced throughput (37%) compared to the LSC. Then we perform detailed end-to-end simulations of the LSC and the ALSC that include realistic levels of aberrations to obtain datasets representing 1h of integration time on stars of spectral types A0 to M0 located at 10 pc. We insert spectra of planetary companions at different effective temperatures (Teff) and surface gravities (log g) into the data at angular separations of 0.3" to 1.5" and with contrast ratios from 6 to 18 mag. Using the SD method to subtract the speckles, we show that the ALSC brings a gain in sensitivity of up to 3 mag at 0.3" with respect to the LSC, which leads to a much better spectral extraction below 0.5". In terms of Teff, we demonstrate that at small angular separations the limit with the ALSC is always lower by at least 100K, inducing an increase of sensitivity of a factor up to 1.8 in objects' masses at young ages. [Abridged]

Impact of primary spherical aberration, spatial frequency and Stiles Crawford apodization on wavefront determined refractive error: a computational study

We tested the hypothesis that pupil apodization is the basis for central pupil bias of spherical refractions in eyes with spherical aberration. We employed Fourier computational optics in which we vary spherical aberration levels, pupil size, and pupil apodization (Stiles Crawford Effect) within the pupil function, from which point spread functions and optical transfer functions were computed. Through-focus analysis determined the refractive correction that optimized retinal image quality. For a large pupil (7 mm), as spherical aberration levels increase, refractions that optimize the visual Strehl ratio mirror refractions that maximize high spatial frequency modulation in the image and both focus a near paraxial region of the pupil. These refractions are not affected by Stiles Crawford Effect apodization. Refractions that optimize low spatial frequency modulation come close to minimizing wavefront RMS, and vary with level of spherical aberration and Stiles Crawford Effect. In the presence of significant levels of spherical aberration (e.g. C(4)(0) = 0.4 μm, 7 mm pupil), low spatial frequency refractions can induce -0.7 D myopic shift compared to high SF refraction, and refractions that maximize image contrast of a three cycle per degree square-wave grating can cause -0.75 D myopic drift relative to refractions that maximize image sharpness. Because of small depth of focus associated with high spatial frequency stimuli, the large change in dioptric power across the pupil caused by spherical aberration limits the effective aperture contributing to the image of high spatial frequencies. Thus, when imaging high spatial frequencies, spherical aberration effectively induces an annular aperture defining that portion of the pupil contributing to a well-focused image. As spherical focus is manipulated during the refraction procedure, the dimensions of the annular aperture change. Image quality is maximized when the inner radius of the induced annulus falls to zero, thus defining a circular near paraxial region of the pupil that determines refraction outcome.

Adaptive optics scanning ophthalmoscopy with annular pupils

Annular apodization of the illumination and/or imaging pupils of an adaptive optics scanning light ophthalmoscope (AOSLO) for improving transverse resolution was evaluated using three different normalized inner radii (0.26, 0.39 and 0.52). In vivo imaging of the human photoreceptor mosaic at 0.5 and 10° from fixation indicates that the use of an annular illumination pupil and a circular imaging pupil provides the most benefit of all configurations when using a one Airy disk diameter pinhole, in agreement with the paraxial confocal microscopy theory. Annular illumination pupils with 0.26 and 0.39 normalized inner radii performed best in terms of the narrowing of the autocorrelation central lobe (between 7 and 12%), and the increase in manual and automated photoreceptor counts (8 to 20% more cones and 11 to 29% more rods). It was observed that the use of annular pupils with large inner radii can result in multi-modal cone photoreceptor intensity profiles. The effect of the annular masks on the average photoreceptor intensity is consistent with the Stiles-Crawford effect (SCE). This indicates that combinations of images of the same photoreceptors with different apodization configurations and/or annular masks can be used to distinguish cones from rods, even when the former have complex multi-modal intensity profiles. In addition to narrowing the point spread function transversally, the use of annular apodizing masks also elongates it axially, a fact that can be used for extending the depth of focus of techniques such as adaptive optics optical coherence tomography (AOOCT). Finally, the positive results from this work suggest that annular pupil apodization could be used in refractive or catadioptric adaptive optics ophthalmoscopes to mitigate undesired back-reflections.