In the conventional visible light microscope the magnified image of the object is produced by combination of the refractive lenses. That allows to achieve resolution 7 close to the Abbe diffraction limit∆, established by [25]
∆ = 0.61λ
nsinθ. (2.10)
Herenis the refractive index of the medium andθis half the angle subtended by the objective lens. The larger the aperture of the lens, and the smaller the wavelength, the finer the resolution of an imaging system. Considering green light around 500nm wavelength and a numerical aperturensinθof1, the Abbe limit can be estimated as roughly300nm.
To increase the resolution, the radiation of shorter wavelengths, such us ultraviolet and X-rays can be used. X-ray microscopes offer up to 12nm of resolution [52, 53] but are suf-fer from lack of contrast. In addition, available focusing elements of X-ray optics, such as Kirkpatrick-Baez (KB) mirrors [54], compound refractive lenses (CRL) [55] and Fresnel zone plates (FZP) [56] do not allow to achieve high magnification and efficiency comparable with the those in optics of the visible light. However, there is an alternative approach that can produce high-resolution image of a sample without using any optics on the way to detector. Methods of Coherent X-Ray Diffractive Imaging (CXDI) [57, 58] exploit diffraction data from coherently illuminated specimen and reconstruct the sample image relying on the relationship between the wave field in the object plane and detector plane. More specifically, the complex amplitude of the wave field on the exit surface of the object is reconstructed.
There are two distinguished concepts of experimental realization of these methods8. First one is based on the forward scattering where the diffraction pattern is recorded by a two-dimensional (2D) detector positioned in the transmission geometry (see Figure 2.2). This ap-proach is typically used for imaging of non-crystalline objects. Distinct to that, the second concept exploits the Bragg geometry where a crystalline sample is oriented to satisfy the Bragg
7The smallest distance at which two points can still be uniquely resolved.
8In this thesis we do not discuss the Grazing Incidence Small Angle X-ray Scattering methods.
condition and the detector records the scattered intensity in the vicinity of the selected Bragg peak. This section will focus on the CXDI in the transmission geometry, while the Bragg CXDI will be discussed next section.
Figure 2.2: The concept of a coherent X-ray scattering experiment in case of a non-crystallographic object. A coherent X-ray beam illuminates the sample producing the modulations in the wave field on the exit surface. The diffraction pattern is measured under far-field conditions by a two-dimensional detector, protected from the direct beam with the beamstop.
Figure 2.2 illustrates the general principles of a coherent X-ray diffraction experiment with a non-crystallographic sample. A coherent beam is incident on the sample and results in two-dimensional diffraction pattern on the detector, positioned at distancez downstream in the far-field. In order to block the direct beam, which is usually much brighter than the scattered signal and can damage the detector, a beamstop may be used. By back-propagation to the object plane one retrieves the two dimensional complex amplitude of the exit surface of the sample.
Generally speaking, the propagation through the free space, which establishes the relationship between the wave fields in the object plane and detector plane, is done by integration of the point source function over the irradiating surface [59]. However, conventional CXDI schemes assume the detector to be located in the far-field which allows to reduce this integral to a simply Fourier transformation. On the other hand, if the sample is thin and weakly diffracting, the projection approximation is valid. According to that, the exit surface wave (ESW) is a product of the incident illumination function and the projection of the refraction coefficient, given by the equation (1.30). Therefore, the reconstruction of the ESW provides full information about projected electron density, if the illumination function is known a priorior can be determined by the reconstruction, as in ptychography [60].
The concept of the projection approximation can also be conveniently described in terms of basics of kinematical theory of X-ray diffraction. According to this theory, a two-dimensional
diffraction pattern measured in the far-field corresponds to the cross-section in reciprocal space which contains information about electron density distribution of the scattering object. Gener-ally speaking, that cross-section in reciprocal space is described by the Ewald sphere. However, in many practical cases the curvature of the part subtended by the detector is small. When the full detector side lengthD is much smaller than the sample-to-dectector distancez, the small angle approximation
sin D 2z
≈ D
2z (2.11)
is valid and the subtended part of the Ewald sphere can be considered as being flat. Conse-quently the scattering vector can be approximated by
Q= 2π λ [x
z,y
z,0], (2.12)
wherexandyare coordinates in the detector plane. From the basic properties of Fourier trans-formation a two-dimensional amplitude distributionA(qx, qy, qz = 0)within this flat reciprocal space cross-section corresponds to the projection of the object in real space on thexy-plane. By collecting diffraction patterns at different orientations of the sample the whole reciprocal space can be measured and thus the CXDI method is extended to three dimensions so that
A(q)∝ Z
ρ(r)eiq·rdr. (2.13)
Hereρ(r)denotes electron density distribution in real space.
Since the diffraction image recorded by a detector represents discrete intensity distribution with a certain pixel size, the recovery of the scattering object requires sufficient sampling cri-terion. In essence, it is based on the Nyquist-Shannon-Kotelnikov sampling theorem [61,62].
It states that a band-limited analog signal that has been sampled can be perfectly reconstructed from an infinite sequence of samples if the sampling rate exceeds 2samples per period of the highest frequency in the original signal. In imaging that means that to recover a feature of size
∆l the corresponding fringes in reciprocal space has to be sampled with at least two pixels per interference fringe. In other words the resolution in reciprocal space must not fall below
∆q=π/∆l. In addition, from the sampling theorem it follows that the achievable resolution in real space is theoretically limited by the highest measured spatial frequencies and therefore can be calculated from the detector size as
∆x= λz
D. (2.14)
In practical cases, however, the effective resolution depends on the biggest scattering angle where a significant signal can be measured, which in its turn depend on the scattering ability of the specimen and the detector efficiency. Unfortunately, in many cases the lower signal can not be simply compensated by the longer exposure, since the atomic structures can only withstand a limited dose before they are destroyed. This effect is called radiation damage and occurs
already at low and medium doses in case of biological materials consisting of large molecules with weak chemical bonds and predominantly light elements [63].