Reflection and Refraction at an Interface, Total Internal Reflection, Brewster’s Angle z Refracted wave k′ φ′ n = 1 – δ + iβ Vacuum x n = 1 φ φ′′ k′′ k Reflected Incident wave wave incident wave: (3.30a) refracted wave: (3.30b) reflected wave: (3.30c) ω (1) All waves have the same frequency, ω, and |k| = |k′′| = c (2) The refracted wave has phase velocity ω′ c ω V = = , thus k′ = |k′| = (1 – δ + iβ) φ k′ n c Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_ReflctnRefrctn_2007.ai Boundary Conditions at an Interface • E and H components parallel to the interface must be continuous (3.32a) (3.32b) • D and B components perpendicular to the interface must be continuous (3.32c) (3.32d) Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_BndryConditns.ai Spatial Continuity Along the Interface Continuity of parallel field components requires z (3.33) k′ sinφ′ φ′ k′ (3.34a) n = 1 – δ + iβ Vacuum x n = 1 k φ φ′′ k′′ (3.34b) k sinφ k′′ sinφ′′ Conclusions: Since k = k′′ (both in vacuum) (3.36) ω ω′ nω (3.35a) k = c and k′ = c/n = c sinφ = n sinφ′ ∴ (3.35b) (3.38) The angle of incidence equals Snell’s Law, which describes the angle of reflection refractive turning, for complex n. Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_SpatialContin.ai Total External Reflection of Soft X-Rays and EUV Radiation Snell’s law for a refractive index of n 1 – δ, assuming that β → 0 (3.39) φ′ > φ Consider the limit when φ′ → π φ′ 2 sin φ θ 1 = c φ 1 – δ θ + φ = 90 (3.40) Glancing incidence (θ < θc) and total external reflection Exponential decay of the (3.41) fields into the medium θ < θc Totally The critical angle for total θc reflected wave external reflection. Critical ray Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_TotalExtrnlRflc1.ai Total External Reflection (continued) (3.41) (3.42a) The atomic density na, varies slowly among the natural elements, thus to first order (3.42b) 0 0 where f1 is approximated by Z. Note that f1 is a complicated function of wavelength (photon energy) for each element. Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_TotalExtrnlRflc2.ai Total External Reflection with Finite b Glancing incidence reflection . for real materials as a function of β/δ (a) 100 80 30 mr Carbon (C) A 60 1 β/δ B A: = 0 40 –2 C B: β/δ = 10 20 80 mr C: β/δ = 10–1 Reflectivity (%) 0 100 D: β/δ = 1 (b) 0.5 80 Aluminum (Al) E: β/δ = 3 30 mr 60 Reflectivity 40 D E 80 mr 20 0 Reflectivity (%) 0 0.5 1 1.5 2 2.5 3 0 100 θ/θc (c) 80 Aluminum Oxide (Al2O3) • finite β/δ rounds the sharp angular 60 40 dependence 80 mr 30 mr 20 (4.6) (1.7) • cutoff angle and absorption edges Reflectivity (%) 0 can enhance the sharpness (d) 100 80 Gold (Au) • note the effects of oxide layers 60 and surface contamination 40 30 mr 20 80 mr Reflectivity (%) 0 100 1,000 10,000 Photon energy (eV) (Henke, Gullikson, Davis) Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_TotalExtrnlReflc3.ai The Notch Filter • Combines a glancing incidence mirror and a filter • Modest resolution, E/∆E ~ 3-5 • Commonly used 1.0 Mirror Absorption reflectivity edge Filter (“low-pass”) transmission (“high-pass”) Filter/reflector with response E/∆E 4 Photon energy Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_NotchFilter.ai Reflection at an Interface E0 perpendicular to the plane of incidence (s-polarization) z tangential electric fields continuous H′ (3.43) φ′ tangential magnetic fields continuous H′ cosφ′ E′ (3.44) φ′ n = 1 – δ + iβ n = 1 x H φ φ′′ H′′ φ φ′′ E′′ H cosφ E H′′ cosφ′′ (3.45) Three equations in three unknowns Snell’s Law: (E0′, E0′′, φ′) (for given E0 and φ) Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_ReflecInterf1.ai Reflection at an Interface (continued) E0 perpendicular to the plane of incidence (s-polarization) (3.47) (3.46) The reflectivity R is then (3.48) With n = 1 for both incident and reflected waves, Which with Eq. (3.46) becomes, for the case of perpendicular (s) polarization (3.49) Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_ReflecInterf2.ai Normal Incidence Reflection at an Interface Normal incidence (φ = 0) (3.49) For n = 1 – δ + iβ Which for δ << 1 and β << 1 gives the reflectivity for x-ray and EUV radiation at normal incidence (φ = 0) as (3.50) Example: Nickel @ 300 eV (4.13 nm) From table C.1, p. 433 –5 0 0 R⊥ = 4.58 × 10 f1 = 17.8 f2 = 7.70 δ = 0.0124 β = 0.00538 Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_NormIncidReflc.ai Glancing Incidence Reflection (s-polarization) (3.49) For 1 A β/δ B A: = 0 B: β/δ = 10–2 where C C: β/δ = 10–1 D: β/δ = 1 0.5 E: β/δ = 3 Reflectivity For n = 1 – δ + iβ D E 0 0 0.5 1 1.5 2 2.5 3 θ/θc E. Nähring, “Die Totalreflexion der Röntgenstrahlen”, Physik. Zeitstr. XXXI, 799 (Sept. 1930). Professor David Attwood ASTProfessor 210/EECS David 213 Attwood Univ.Univ. California, California, Berkeley Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_GlancIncidReflc.ai Reflection at an Interface E0 perpendicular to the plane of incidence (p-polarization) z (3.54) E′ φ′ H′ (3.55) φ′ E′ cosφ′ n = 1 – δ + iβ The reflectivity for parallel (p) polarization is n = 1 x φ φ′′ (3.56) E E′′ φ φ′′ E cosφ H H′′ E′′ cosφ′′ which is similar in form but slightly different from that for s-polarization. For φ = 0 (normal incidence) the results are identical. Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_ReflecInterf3.ai Brewster’s Angle for X-Rays and EUV For p-polarization k′ 2 (3.56) ′ sin Θ E 0 radiation n = 1 – δ + iβ pattern 90 There is a minimum in the reflectivity n = 1 where the numerator satisfies k (3.58) φB = 0 ′′0 0 E E Squaring both sides, collecting like terms k′′ involving φB, and factoring, one has or 1 –2 the condition for a minimum in the reflectivity, 10 S for parallel polarized radiation, occurs at an angle –4 P given by (3.59) 10 W For complex n, Brewster’s minimum occurs at Reflectivity 10–6 4.48 nm 0 45 90 Incidence angle, φ or (3.60) (Courtesy of J. Underwood) Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_BrewstersAngle.ai Focusing with Curved, Glancing Incidence Optics The Kirkpatrick-Baez mirror system (Courtesy of J. Underwood) • Two crossed cylinders (or spheres) • Astigmatism cancels • Fusion diagnostics • Common use in synchrotron radiation beamlines • See hard x-ray microprobe, chapter 4, figure 4.14 Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_FocusCurv.ai 0 0 Determining f1 and f2 0 • f2 easily measured by absorption 0 • f1 difficult in SXR/EUV region • Common to use Kramers-Kronig relations (3.85a) (3.85b) as in the Henke & Gullikson tables (pp. 428-436) • Possible to use reflection from clean surfaces; Soufli & Gullikson • With diffractive beam splitter can use a phase-shifting interferometer; Chang et al. • Bi-mirror technique of Joyeux, Polack and Phalippou (Orsay, France) Professor David Attwood Univ. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007 Ch03_Determining.ai.