1. X-Ray Propagation Simulation Student: Jing Yee Chee Advisors: Kenneth D. Finkelstein David Sagan Georg H. Hoffstaetter 6/17/20151

2. Motivation The ultimate goal of BMAD – Track both charged particles and photons from source (electron gun) through the beamline to the sample – Build up a subroutine library that users can easily use to write programs 6/17/20152

3. Tracking particles from electron gun to beamlines! Need to extend into photon optics and weakly relativistic particles Mirror element already implemented by previous REU student and David This project aims to add 1 more element to BMAD’s library 6/17/20153

4. Crystal Element Perfect crystals are an important element in x- ray optics Frequently used as monochromators Diffracts (reflects) according to Bragg’s Law However, this is only an approximate law 6/17/20154

5. Dynamical Diffraction A more accurate account of diffraction was developed in the 20 th century – Some pioneers include Sir Charles Galton Darwin, Max von Laue, Paul Peter Ewald Summarized in Batterman and Cole, a very well cited 1964 paper Basically it involves solving Maxwell’s equations in a region of periodic electron density 6/17/20155

6. Some results from the theory 6/17/20156 For symmetric Bragg reflection, Diagram greatly exaggerates deviations from Bragg’s Law Bragg’s Law:

7. 6/17/20157 Some results from the theory For symmetric Bragg reflection,

8. 6/17/20158 Some results from the theory For symmetric Bragg reflection, Equation still holds, but for wave vectors inside crystal. Dispersion surface gives allowed wave vectors Reflectivity can be determined by determining which point on dispersion surface is active

9. Some results from the theory (cont’d) However, things are complicated by the fact that the crystal is rarely cut perfectly – i.e. n s is not in the direction of H The case where H, k in, n s, are coplanar is known as asymmetric Bragg reflection – Very well studied, eg. by Matsushita and Kaminaga 6/17/20159

10. How do we describe single photons? Similar description to charged particle optics – Or equivalently, if defined relative to some reference particle with energy λ – Add in ϑ for polarization and I for intensity of radiation. 6/17/201510

11. 2 stage algorithm Firstly, find the center of reflection – Output the required graze angle of the crystal to achieve the center of reflection – Output orientations of the outgoing reference basis Track individual photons as they diffract – Coordinates to be transformed into the outgoing basis 6/17/201511

12. Current Progress 1 st stage of the algorithm has been written and debugged – Results for the simpler cases compared to output from xop 2 nd stage is on its way… 6/17/201512

13. Future Work Curved optical elements – Eg. capillaries Time-evolution of short pulses of radiation Tracking of phase fronts 6/17/201513

14. Questions? 6/17/201514