Significance and analysis of x-ray pupil diffraction

We study analytically and numerically in which way the width of ring aperture containing a phase jump affects the size and intensity of the focal spot generated with a radially polarized beam. It is shown that by means of destructive interference of beams coming from the different-phase rings it becomes possible to overcome the scalar diffraction limit corresponding to the first zero of the zero-order Bessel function. The minimal focal spot size (FWHM=0.33λ) is found to be attained when the annular aperture width amounts to 20% of the full-aperture radius. In this case, the side-lobe intensity is not larger than 30% of the central peak. A wider annular aperture with the phase jump introduced is also shown to form a focal spot not exceeding the diffraction limit for a narrow annular aperture, simultaneously providing a nearly six times higher intensity. In this case, the side lobes amount to 35% of the central peak.

We show that for any rotationally symmetric apodizer on the full aperture, there is a family of apodizers on the annular aperture with the same functional Strehl ratio vs defocus (W(20)) and vs spherical aberration (W(40)). However, in the latter case, the coefficients W(20) and W(40) are reduced by the factors (1 - epsilon(2)) and (1 - epsilon(2))(2) respectively, where is the central obscuration ratio. We indicate that the best focal plane is shifted from W(20) = - W(40) to W(20) = -(1 + epsilon(2)) W(40). These general results allow us to design and to compare novel apodizers on annular apertures which reduce the influence of W(20) and W(40). The Strehl ratios of a novel family of apodizers are discussed to illustrate our general results.

A new optical system with a resolution that is superior to the resolution of the usual optical systems with diffraction limit is presented. We introduce a newly generated narrow light beam that propagates for a long range while almost maintaining its beam width and show that the beam width is narrower than that of the diffraction limit of normal optics. Thus a super high resolution is achieved for a long range, e.g., a range of a few kilometers, by the use of a 10-cm-diameter telescope. The high resolution for long-range imaging can be obtained by a Galilean telescope with a negative eyepiece that has a spherical aberration. We demonstrate theoretically high-resolution imaging by using simple objects and assuming a telescope 10 cm in diameter and a visible wavelength. A comparison of simulation results by the conventional optical system and by the special optical system clearly shows the superiority of the new system.

In order to obtain the intensity distribution of a 351 nm focal spot and smoothing by spectral dispersion (SSD) focal plane profile of a SGII-upgraded facility, a type of off-axis imaging system with three spherical mirrors, suitable for a finite distance source point to be imaged near the diffraction limit has been designed. The quality factor of the image system is 1.6 times of the diffraction limit tested by a 1053 nm point source. Because of the absence of a 351 nm point source, we can use a Collins diffraction imaging integral with respect to λ=351 nm, corresponding to a quality factor that is 3.8 times the diffraction limit at 351 nm. The calibration results show that at least the range of ±10 mrad of view field angle and ±50 mm along the axial direction around the optimum object distance can be satisfied with near diffraction limited image that is consistent with the design value. Using this image system, the No. 2 beam of the SGII-upgraded facility has been tested. The test result of the focal spot of final optics assembly (FOA) at 351 nm indicates that about 80% of energy is encompassed in 14.1 times the diffraction limit, while the output energy of the No. 2 beam is 908 J at 1053 nm. According to convolution theorem, the true value of a 351 nm focal spot of FOA is about 12 times the diffraction limit because of the influence of the quality factor. Further experimental studies indicate that the RMS value along the smoothing direction is less than 15.98% in the SSD spot test experiment. Computer simulations show that the quality factor of the image system used in the experiment has almost no effect on the SSD focal spot test. The image system can remarkably distort the SSD focal spot distribution under the circumstance of the quality factor 15 times worse than the diffraction limit. The distorted image shows a steep slope in the contour of the SSD focal spot along the smoothing direction that otherwise has a relatively flat top region around the focal spot center.

Optical systems designed to utilize extreme ultraviolet (EUV) and x-ray photons from synchrotron radiation (SR) light sources with grazing incidence optics generally have entrance apertures that are long in the horizontal plane and narrow in the vertical plane. Apertures that are 1 mrad high by several milliradians wide at a distance of 10 meters from the source are typical. A point source illuminated by a red He-Ne laser beam imaged through a system with this aperture and focal length results in a 1.2 mm high image that is severely broadened by diffraction. Component alignment with visible light is difficult when the diffraction limit is so severe. The use of visible light for system alignment is, however, a necessity, because alignment under actual operating conditions and at operating wavelengths in ultra high vacuum chambers is totally impractical. Except for rare instances, components are not accessible for alignment adjustments. How, then, can we make use of the information available in the severely diffraction-limited visible image to assess the performance of our system at x-ray wavelengths?

Smallest focal hole

Experiments of focusing coherent laser beams being diffracted on annular aperture are carried out in order to verify issues of a two-dimensional Fast Fourier Transform simulation of coherent pure (dark) annular flat beams done and published before. The beam pattern at focal plane (aka the far-field pattern) is a prominent central peak and faint concentrical rings around it. In the course of simulation we calculate the fraction of the central peak power to the whole power of beam that gives a notion of power spread within the focal spot. This fraction is a function of beam annularity i.e. ‘inside diameter /outside diameter’ ratio. The experimentally-measured dependence of the central peak power to the whole power of beam versus the annularity of pure annular laser beams was the major target of the report. The purpose was to verify experimentally the issues of our earlier simulation. We found that from the five experimental points four of them are within 16% error as to the simulated dependence which is tolerable in such a measurement.

The sub-diffraction-limit spatially structured light patterns have attracted more and more attention for their important applications in many frontier scientific fields. The present paper aims at developing sub-diffraction-limit spatially structured beam patterns which might have great potential to improve the light performance in fields such as super resolution imagery, optical tweezer, micro/nano lithography, etc. Here, a variety of spatially structured beam patterns are obtained by the phase modulation of polarized beams and studied in detail experimentally and numerically. Firstly, a new kind of phase plate, which combines the merits of circular and vortex 2 phase plates, is proposed based on the wave front design; it is composed of two spiral-shaped phase plates with their phases changing from 0 to 2 and - to , respectively. Later, the phase plate is applied to the circularly polarized Gaussian beam modulation in a high NA system. By combining a self-made circular with a commercial vortex 2 phase plate, the designed new phase plate is implemented in the experiment. The morphology of the spatially structured light pattern, which is generated on the focal plane, is observed by a CCD camera in the experiment. The beam pattern presents a donut shape on the focal plane, while the dimension of the donut-shaped pattern becomes smaller as the imaging plane axially deviates from the focal plane. It is found that the beam patterns captured in experiment highly consist with the numerical simulation results carried out by the vectorial diffraction integral theory. It can be deduced that the spatially structured beam is capillary-shaped. Meanwhile, at the two ends of the capillary-shaped beam, the inner diameter is smaller than the diffraction limitation. Furthermore, the structured beam pattern presents a spatial voxel distribution with center and axis symmetry. Finally, the characteristics of the spatially structured beam patterns, which are generated by modulating circular, linear, radial and azimuthal polarized beams with the new designed phase plate, are analyzed and discussed in detail. It is found that for circular, linear, radial and azimuthal polarization, the full widths at half maximum (FWHMs) of the minimum dark spots in the horizontal direction are 0.31, 0.32, 0.24 and 0.36, respectively. On the optical axis, the FWHMs of the dark spots created by linearly, radially and azimuthally polarized light, are 0.8, 0.78 and 0.76 , respectively, and no axial intensity is found with circularly polarized beam incidence.

We show that the contribution of the electric field components into the focal region can be controlled using binary phase structures. We discuss differently polarized incident waves, for each case suggesting easily implemented binary phase distributions that ensure a maximum contribution of a definite electric field component on the optical axis. A decrease in the size of the central focal spot produced by a high numerical aperture (NA) focusing system comes as the result of the spatial redistribution of the contribution of different electric field components into the focal region. Using a polarization conversion matrix of a high NA lens and the numerical simulation of the focusing system in Debye's approximation, we demonstrate benefits of using asymmetric to polar angle ϕ binary phase distributions (such as arg[cos ϕ] or arg[sin 2ϕ]) for generating a subwavelength focal spot in separate electric field components. Additional binary structure variations with respect to the azimuthal angle also make possible controlling the longitudinal distribution of light. In particular, the contribution of the transverse components in the focal plane can be reduced by the use of a simple axicon-like structure that serves to enhance the NA of the lens central part, redirecting the energy from focal plane. As compared with the superimposition of a narrow annular aperture, this approach is more energy efficient, and as compared with the Toraldo filters, it is easier to control when applied to three-dimensional focal shaping.

Digital holography and coherent imaging are explored in the terahertz frequency region at 0.480 Terahertz with highly coherent, frequency-tunable continuous wave sources. The long coherence length of microwave sources suggests that interferometric imaging techniques such as Fresnel off-axis holography can achieve sub-micrometer depth resolution of surfaces and through materials and structures that are transmissive in the terahertz spectral region. Research in this area will provide an important non-destructive imaging tool for the rapidly expanding field of additive manufacturing and composite fabrication. Unfortunately, imaging optics as they are used in confocal imaging and optics in general adversely affect the performance of terahertz imagers as their aperture size if comparable to the wavelength, which artificially limits image resolution according to the diffraction limit and can cause undesirable coherence effects in the image. We report on imaging methods that minimize the use of optics but use signal processing techniques that form images digitally from recorded holograms. This approach is directly applicable to focal plane arrays in contrast to confocal imaging modalities. Furthermore, the performance of image reconstruction at multiple aspect angles compiled into videos as user friendly inspection tool is investigated.

We study the effect of polarization and aperture geometry on the focal spot size of a high numerical aperture (NA) aplanatic lens. We show that for a clear aperture geometry, illuminating the lens by linear or circular polarization is preferable over radial polarization for spot size reduction applications. For annular aperture and objective lenses of 0.85 NA and above we give the sizes of the inner annulus which constitute the transition points to a state where the radial polarization illumination gives smaller spot size. We analyze the evolution, the profile and the effect of transverse and longitudinal field components in the focal plane, and show that they play an opposite role on the spot size in the cases of circular and radial polarization illumination. We show that in the limit of a very thin annulus the radial polarization approaches the prediction of the scalar theory at high NA, whereas the linear and circular polarizations deviate from it. We verify that the longitudinal component generated by radially polarized illumination produces the narrowest spot size for wide range of geometries. Finally, we discuss the effects of tight focusing on a dielectric interface and provide some ideas for circumventing the effects of the interface and even utilize them for spot size reduction.

for a high throughput (approximately 30-40 wafers/h), high resolution (< 0.25 [mu]m) lithography system that is competitive with more complex x-ray and short wavelength optical systems. 31 refs., 5 figs.

Planar lenses are attractive photonic devices due to its minimized size and easy to integrate. However, planar lenses designed in traditional ways are restricted by the diffraction limit. They have difficulties in further reducing the focal spot size beyond the diffraction limit. Super-oscillation provides a possible way to solve the problem. However, lenses based on super-oscillation have always been affected by huge sidelobes, which resulted in limited field of view and difficulties in real applications. To address the problem, in the paper, a far-field sub-diffraction lens based on binary amplitude-phase mask was demonstrated under illumination of linearly polarized plane wave at wavelength 632.8 nm. The lens realized a long focal length of 148λ (94 µm), and the full width at half maximum of the focal line was 0.406λ, which was super-oscillatory. More important is that such a flat lens has small sidelobes and wide field of view. Within the measured range of [-132λ, + 120λ], the maximum sidelobe observed on the focal plane was less than 22% of the central peak. Such binary amplitude-phase planar lens can also be extended to long focal length far-field sub-diffraction focusing lens for other spectrum ranges.

Low temperature detectors have been a subject of intense interest to the scientific community over the last decade. These detectors work at very low temperatures, often well below 1 Kelvin, to minimize the noise in the measurement of photons. This leads to very powerful detectors applicable to a broad wavelength range. Since these detectors are so sensitive even single pixels and small arrays (up to several hundred pixels) enable deeper explorations of the cosmos than ever before. Instruments based on these technologies have been used at submillimeter, optical, and X-ray wavelengths. The scientific prospects for these detectors increase as they grow in pixel count. For some applications, especially for Cosmic Microwave Background (CMB) polarization work, a large focal plane will not only increase efficiency but will also enable new and vital science. Current superconducting technologies, such as Transition Edge Sensors (TESs), can currently deliver extremely high sensitivity in the submillimeter and read-noise free imaging spectroscopy at Optical/UV and X-ray wavelengths, but the largest arrays contain less that 100 pixels. In order to make real progress these arrays must contain many thousands of pixels. This is a formidable technical challenge. This thesis will explore a promising emerging technology called Microwave Kinetic Inductance Detectors (MKIDs). MKIDs make use of the change in the surface impedance of a superconductor as incoming photons break up Cooper pairs. This is accomplished by making the strip of superconductor part of a microwave resonant circuit, and monitoring the phase of a signal transmitted through (or past) the resonator. The primary advantage of this technology is that by using resonant circuits with high quality factors, passive frequency domain multiplexing will allow up to thousands of resonators to be read out through a single coaxial cable and a single HEMT amplifier. This eliminates the cryogenic electronics (SQUIDS) and wiring problems associated with current superconducting devices. Inexpensive and powerful room-temperature readout electronics can leverage the microwave integrated circuits developed for wireless communications. When this project started four years ago MKIDs were just a concept. In this thesis I will recount the progress we have made in taking this concept and turning it into a detector technology, building on the work we published in Nature in 2003. I will demonstrate that we have overcome the major technical obstacles and shown conclusively that MKIDs work, and should provide much larger arrays than are possible with other technologies. Due to our work at Caltech and JPL, MKIDs are considered one of the leading technologies to reach the ambitious goals set for future ground and space missions. The noise performance is already good enough for sky-limited, ground-based, submillimeter astronomy. Despite MKIDs acceptable noise performance for ground-based astronomy, there are many applications which require lower detector noise. The simple MKIDs we measure in this thesis still exhibit a noise higher than theory would predict which could limit their usefulness for some applications. In Chapters 7-9 we perform several experiments which identify the source of the excess noise as the substrate. In the course of these experiments to characterize and identify the noise, we successfully demonstrate two distinct approaches which dramatically reduce the noise excess.

In contrast to the conventional adaptive optics (AO) system, the wavefront sensorless (WFSless) AO system doesn’t need a WFS to measure the wavefront aberrations. It is simpler than the conventional AO in system architecture and can be applied to the complex conditions. The model-based WFSless system has a great potential in real-time correction applications because of its fast convergence. The control algorithm of the model-based WFSless system is based on an important theory result that is the linear relation between the Mean-Square Gradient (MSG) magnitude of the wavefront aberration and the second moment of the masked intensity distribution in the focal plane (also called as Masked Detector Signal-MDS). The linear dependence between MSG and MDS for the point source imaging with a CCD sensor will be discussed from theory and simulation in this paper. The theory relationship between MSG and MDS is given based on our previous work. To verify the linear relation for the point source, we set up an imaging model under atmospheric turbulence. Additionally, the value of MDS will be deviate from that of theory because of the noise of detector and further the deviation will affect the correction effect. The theory results under noise will be obtained through theoretical derivation and then the linear relation between MDS and MDS under noise will be discussed through the imaging model. Results show the linear relation between MDS and MDS under noise is also maintained well, which provides a theoretical support to applications of the model-based WFSless system.