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Introduction and Fundamental Theory

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High-Resolution Extreme Ultraviolet Microscopy

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

This chapter covers the underlying theory and introduces concepts for imaging using extreme ultraviolet radiation. The concept of high harmonic generation is briefly introduced. This is followed by general aspects, geometric considerations, and numerical reconstruction procedures for diffraction-based imaging and digital in-line holography. The chapter is concluded by a summary and an overview of the state-of-the-art techniques in the field of lensless imaging.

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Notes

  1. 1.

    Throughout this thesis vectors and vectorial quantities will be denoted by a boldface typeset, e.g. \(\mathbf {r}\). The modulus of \(\mathbf {r}\) is denoted simply as \(r\).

  2. 2.

    In literature this field is also often referred to as the exit surface wave.

  3. 3.

    One can compare Eq. 2.22 with the well-known momentum transfer equation from elastic scattering \(\triangle \mathbf {p}=\mathbf {p}-\mathbf {p}_0\) by multiplying both sides in Eq. 2.22 with \(\hbar =h/2\pi \) where \(h\) is Planck’s constant. If one now uses the de Broglie relation \(\mathbf {p}_0=\hbar \mathbf {k}_\mathrm{in }\) and \(\mathbf {p}=\hbar \mathbf {k}_\mathrm{out }\) for the incident and scattered particle, respectively, it becomes clear that \(\hbar \mathbf {q}=\hbar \mathbf {k}_{\mathrm {out}}-\hbar \mathbf {k}_{\mathrm {in}}\) is the momentum transfer vector [26].

  4. 4.

    For the sake of being consistent with most of the literature on CDI. Hence, \(|\mathbf {k}|=1/\lambda \) is used.

  5. 5.

    This equation comes from an isosceles triangle, where \(c=2a\sin (\gamma /2)\). Comparing it to the well-known Bragg’s law one finds that Eq. 2.23 is equivalent to a volume grating being tilted about half the diffraction angle \(\theta \).

  6. 6.

    Provided that diffraction data is measured to sufficiently high momentum transfers, such that in principle a higher resolution could be obtained considering perfect coherence properties. See the following pages for the fundamental resolution limit.

  7. 7.

    It is assumed that the zero deflection point, i.e. \(|q|=0\), resides in the center of the detector. The factor \(2\) comes from the fact that the pattern is sampled from \(-[(N_x-1)/2]\triangle q_x\) to \([N_x/2]\triangle q_x\) on the detector and that the FFT conserves the total amount of pixels, i.e. the object plane is also sampled by \(N_x\) times \(N_y\) pixels.

  8. 8.

    This refers to the midpoints from the diagonal edges of the CCD. Sometimes in literature the highest momentum transfer in the edges of the detector is mentioned. It is thus a factor of \(\sqrt{2}\) larger, but this yields no additional resolution because the highest momentum transfer with respect to the \(x\)- and \(y\)-axis is still the same.

  9. 9.

    Due to shot noise and electronic noise of the readout electronics.

  10. 10.

    Alternatively, one can threshold the PRTF at e.g. \(0.5\) or \(1/e\) to determine the highest \(|q|\).

  11. 11.

    An Airy pattern or Airy disk is the diffraction pattern caused from a round aperture featuring a bright central maximum, i.e. zeroth diffraction order, surrounded by dark and bright rings, i.e. higher order diffraction terms.

  12. 12.

    Other iterative methods reported in literature make use of the Gerchberg-Saxton algorithm [73] or use a support constraint to retrieve the hologram [74, 75]. The important difference is that here the full illumination field, which is well characterized by the pinhole being illuminated with a XUV beam having good coherence properties, is used at every step, in contrast to the algorithms reported in literature, which mostly switch between the object plane and the detector plane and disregard the illumination field.

  13. 13.

    More precisely it would be \(L_\mathrm{det }+L_\mathrm{ph }\), however, for all experiments presented in this thesis \(L_\mathrm{det }\gg L_\mathrm{ph }\), hence simply \(L_\mathrm{det }\) is used.

  14. 14.

    For typical XUV wavelengths and pinhole sizes in the order of a micron the Fraunhofer condition (Eq. 2.12) is already fulfilled a few tens of microns behind the pinhole. Hence, the Fraunhofer condition is not limiting in XUV DIH.

  15. 15.

    This is analogous to the Rayleigh criterion.

  16. 16.

    This is because the high flux of such sources allows extremely short exposures down to single shot measurements which circumvents the stability problem.

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  1. Institute of Optics and Quantum Electronics, Friedrich Schiller University Jena, Jena, Germany

    Michael Werner Zürch

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Zürch, M.W. (2015). Introduction and Fundamental Theory. In: High-Resolution Extreme Ultraviolet Microscopy. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-12388-2_2

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