(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

Three-dimensional crystals of air spheres in titania (TiO 2 ) with radii between 120 and 1000 nanometers were made by filling the voids in artificial opals by precipitation from a liquid-phase chemical reaction and subsequently removing the original opal material by calcination. These macroporous materials are a new class of photonic band gap crystals for the optical spectrum. Scanning electron microscopy, Raman spectroscopy, and optical microscopy confirm the quality of the samples, and optical reflectivity demonstrates that the crystals are strongly photonic and near that needed to exhibit band gap behavior.

https://science.sciencemag.org/content/sci/281/5378/802.full.pdf

Preparation of photonic crystals made of air spheres in titania

Coherent diffractive imaging (CDI) and scanning transmission x-ray microscopy (STXM) are two popular microscopy techniques that have evolved quite independently. CDI promises to reach resolutions below 10 nanometers, but the reconstruction procedures put stringent requirements on data quality and sample preparation. In contrast, STXM features straightforward data analysis, but its resolution is limited by the spot size on the specimen. We demonstrate a ptychographic imaging method that bridges the gap between CDI and STXM by measuring complete diffraction patterns at each point of a STXM scan. The high penetration power of x-rays in combination with the high spatial resolution will allow investigation of a wide range of complex mesoscopic life and material science specimens, such as embedded semiconductor devices or cellular networks.

https://science.sciencemag.org/content/sci/321/5887/379.full.pdf

High-Resolution Scanning X-ray Diffraction Microscopy

心臓病診療プラクティス 16. 心不全を癒す

Phase-contrast x-ray imaging (PCI) is an innovative method that is sensitive to the refraction of the x-rays in matter. PCI is particularly adapted to visualize weakly absorbing details like those often encountered in biology and medicine. In past years, PCI has become one of the most used imaging methods in laboratory and preclinical studies: its unique characteristics allow high contrast 3D visualization of thick and complex samples even at high spatial resolution. Applications have covered a wide range of pathologies and organs, and are more and more often performed in vivo. Several techniques are now available to exploit and visualize the phase-contrast: propagation- and analyzer-based, crystal and grating interferometry and non-interferometric methods like the coded aperture. In this review, covering the last five years, we will give an overview of the main theoretical and experimental developments and of the important steps performed towards the clinical implementation of PCI.

http://iopscience.iop.org/article/10.1088/0031-9155/58/1/R1/pdf

X-ray phase-contrast imaging: from pre-clinical applications towards clinics

THE development of techniques for focusing X-rays has occupied physicists for more than a century. Refractive lenses, which are used extensively in visible-light optics, are generally considered inappropriate for focusing X-rays, because refraction effects are extremely small and absorption is strong. This has lead to the development of alternative approaches1,2 based on bent crystals and X-ray mirrors, Fresnel and Bragg–Fresnel zone plates, and capillary optics (Kumakhov lenses). Here we describe a simple procedure for fabricating refractive lenses that are effective for focusing of X-rays in the energy range 5–40 keV. The problems associated with absorption are minimized by fabricating the lenses from low-atomic-weight materials. Refraction of X-rays by one such lens is still extremely small, but a compound lens (consisting of tens or hundreds of individual lenses arranged in a linear array) can readily focus X-rays in one or two dimensions. We have fabricated a compound lens by drilling 30 closely spaced holes (each having a radius of 0.3 mm) in an aluminium block, and we demonstrate its effectiveness by focusing a 14-keV X-ray beam to a spot size of 8 μm.

A compound refractive lens for focusing high-energy X-rays

The manufacture and properties of compound refractive lenses (CRLs) for hard X-rays with parabolic profile are described. These novel lenses can be used up to ∼60 keV. A typical focal length is 1 m. They have a geometrical aperture of 1 mm and are best adapted to undulator beams at synchrotron radiation sources. The transmission ranges from a few % in aluminium CRLs up to about 30% expected in beryllium CRLs. The gain (ratio of the intensity in the focal spot relative to the intensity behind a pinhole of equal size) is larger than 100 for aluminium and larger than 1000 for beryllium CRLs. Due to their parabolic profile they are free of spherical aberration and are genuine imaging devices. The theory for imaging an X-ray source and an object illuminated by it has been developed, including the effects of attenuation (photoabsorption and Compton scattering) and of the roughness at the lens surface. Excellent agreement between theory and experiment has been found. With aluminium CRLs a lateral resolution in imaging of 0.3 µm has been achieved and a resolution below 0.1 µm can be expected for beryllium CRLs. The main fields of application of the refractive X-ray lenses are (i) microanalysis with a beam in the micrometre range for diffraction, fluorescence, absorption, scattering; (ii) imaging in absorption and phase contrast of opaque objects which cannot tolerate sample preparation; (iii) coherent X-ray scattering.

/pdf/imaging-by-parabolic-refractive-lenses-in-the-hard-x-ray-335k2lxryz.pdf

Imaging by parabolic refractive lenses in the hard X-ray range

Based on nanofocusing refractive x-ray lenses a hard x-ray scanning microscope is currently being developed and is being implemented at beamline ID13 of the European Synchrotron Radiation Facility (Grenoble, France). It can be operated in transmission, fluorescence, and diffraction mode. Tomographic scanning allows one to determine the inner structure of a specimen. In this device, a monochromatic (E=21keV) hard x-ray nanobeam with a lateral extension of 47×55nm2 was generated. Further reduction of the beam size to below 20 nm is targeted.

Hard x-ray nanoprobe based on refractive x-ray lenses

We describe refractive x-ray lenses with a parabolic profile that are genuine imaging devices, similar to glass lenses for visible light. They open considerable possibilities in x-ray microscopy, tomography, microanalysis, and coherent scattering. Based on these lenses a microscope for hard x rays is described, that can operate in the range from 2 to 50 keV, allowing for magnifications up to 50. At present, it is possible to image an area of about 300 μm in diameter with a resolving power of 0.3 μm that can be increased to 0.1 μm. This microscope is especially suited for opaque samples, up to 1 cm in thickness, which do not tolerate sample preparation, like many biological and soil specimens.

A microscope for hard x rays based on parabolic compound refractive lenses

We address the question of what is the smallest spot size that hard x rays can be focused to using refractive optics. A thick refractive x-ray lens is considered, whose aperture is gradually (adiabatically) adapted to the size of the beam as it converges to the focus. These adiabatically focusing lenses are shown to have a relatively large numerical aperture, focusing hard x rays down to a lateral size of $2\text{ }\mathrm{n}\mathrm{m}$ (FWHM), well below the theoretical limit for focusing with waveguides [C. Bergemann et al., Phys. Rev. Lett. 91, 204801 (2003)].

http://xrm.phys.northwestern.edu/research/pdf_papers/2005/schroer_prl_2005.pdf

Bruno Lengeler

Papers

A compound refractive lens for focusing high-energy X-rays

Imaging by parabolic refractive lenses in the hard X-ray range

Hard x-ray nanoprobe based on refractive x-ray lenses

A microscope for hard x rays based on parabolic compound refractive lenses

Focusing hard x rays to nanometer dimensions by adiabatically focusing lenses.