Chapter 3 WAVE PROPAGATION AND REFRACTIVE INDEX AT EUV AND SOFT X-RAY WAVELENGTHS k′ n = 1 – δ + iβ n = 1 φ k k′′ Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_F00VG.ai The Wave Equation and Refractive Index The transverse wave equation is (3.1) For the special case of forward scattering the positions of the electrons within the atom (∆k ? ∆rs) are irrelevant, as are the positions of the atoms themselves, n(r, t). The contributing current density is then (3.2) where na is the average density of atoms, and Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_WavEq_RefrcIndx1.ai The Wave Equation and Refractive Index (Continued) The oscillating electron velocities are driven by the incident field E (3.2) such that the contributing current density is (3.4) Substituting this into the transverse wave equation (3.1), one has Combining terms with similar operators (3.5) Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_WavEq_RefrcIndx2.ai Refractive Index in the Soft X-Ray and EUV Spectral Region Written in the standard form of the wave equation as (3.6) The frequency dependent refractive index n(ω) is identified as (3.7) 2 For EUV/SXR radiation ω 2 is very large compared to the quantity e2na/e0m, so that to a high degree of accuracy the index of refraction can be written as (3.8) which displays both positive and negative dispersion, depending on whether ω is less or greater than ωs. Note that this will allow the refractive index to be more or less than unity, and thus the phase velocity to be less or greater than c. Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_RefracIndex1.ai Refractive Index in the Soft X-Ray and EUV Spectral Region (continued) (3.8) Noting that and that for forward scattering where this has complex components The refractive index can then be written as (3.9) which we write in the simplified form (3.12) Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_RefracIndex2.ai Refractive Index from the IR to X-Ray Spectral Region (3.12) (3.13a) (3.13b) n , x e d n i e v i t 1 c a r f e Infrared Visible Ultraviolet X-ray R 0 ωIR ωUV ωK,L,M Ultraviolet • λ2 behavior • δ & β << 1 • δ-crossover Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_RefrcIndxIR.XR.ai Phase Velocity and Refractive Index The wave equation can be written as (3.10) The two bracketed operators represent left and right-running waves c c V = – V = φ n φ n E E z z Left-running wave Right-running wave where the phase velocity, the speed with which crests of fixed phase move, is not equal to c as in vacuum, but rather is (3.11) Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_PhasVelo_Refrc1.ai Phase Velocity and Refractive Index (continued) Recall the wave equation (3.10) Examining one of these factors, for a space-time dependence ET = E0 exp[–i(ωt – kz)] Solving for ω/k we have the phase velocity Vφ = Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_PhasVelo_Refrc2.ai Phase Variation and Absorption of Propagating Waves For a plane wave (3.14) in a material of refractive index n, the complex dispersion relation is (3.15) Solving for k (3.16) Substituting this into (3.14), in the propagation direction defined by k ? r = kr or (3.17) where the first exponential factor represents the phase advance had the wave been propagating in vacuum, the second factor (containing 2πδr/λ) represents the modified phase shift due to the medium, and the factor containing 2πβr/λ represents decay of the wave amplitude. Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_PhaseVarAbsrb.ai Intensity and Absorption in a Material of Complex Refractive Index For complex refractive index n (3.18) The average intensity, in units of power per unit area, is (3.19) or (3.20) Recalling that (3.17) or (3.21) the wave decays with an exponential decay length (3.22) Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_IntenstyAbsrp.ai Absorption Lengths (3.22) 2 Recalling that β = nareλ f2°(ω)/2π (3.23) In Chapter 1 we considered experimentally observed absorption in thin foils, writing (3.24) where ρ is the mass density, µ is the absorption coefficient, r is the foil thickness, and thus labs = 1/ρµ. Comparing absorption lengths, the macroscopic and atomic descriptions are related by (3.26) where ρ = mana = Amuna , mu is the atomic mass unit, and A is the number of atomic mass units Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_AbsorpLngths.ai Phase Shift Relative to Vacuum Propagation For a wave propagating in a medium of refractive index n = 1 – δ + iβ (3.23) the phase shift ∆φ relative to vacuum, due to propagation through a thickness ∆r is (3.29) Reference wave M • Flat mirrors at short wavelengths BS • Transmissive, flat beamsplitters • Bonse and Hart interferometer Film ∆r or CCD • Diffractive optics for SXR/EUV Object M Object wave BS Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_PhaseShift.ai Reflection and Refraction at an Interface 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 c′ n c Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_ReflctnRefrctn.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 AST 210/EECS 213 Univ. California, Berkeley 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 AST 210/EECS 213 Univ. California, Berkeley 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 ray ical wave external reflection. Crit Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley 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) where f1° is approximated by Z. Note that f1° is a complicated function of wavelength (photon energy) for each element. Professor David Attwood AST 210/EECS 213 Univ. California, Berkeley Ch03_TotalExtrnlRflc2.ai Total External Reflection with Finite b Glancing incidence reflection . for real materials as a function of β/δ 100 (a) ) % Carbon (C) ( 80 30 mr y t A i 60 v 1 i A: β/δ = 0 t B c 40 e –2 l B: β/δ = 10 f 20 C e 80 mr R y –1 t 0 i C: β/δ = 10 v i 100 ) t D: = 1 (b) c 0.5 β/δ % e Aluminum (Al) ( l 80 f E: β/δ = 3 y 30 mr t e i 60 v R i t c 40 E e 80 mr D l f e 20 0 R 0 0.5 1 1.5 2 2.5 3 0 100 θ/θc (c) ) % Aluminum Oxide ( 80 y t (Al2O3) i 60 v • finite β/δ rounds the sharp angular i t c 40 80 mr 30 mr e dependence l f e 20 (4.6°) (1.7°) • cutoff angle and absorption edges R 0 can enhance the sharpness (d) ) 100 % ( 80 Gold (Au) y • note the effects of oxide layers t i 60 v i and surface contamination t 30 mr c 40 e l f 80 mr e 20 R 0 100 1,000 10,000 Photon energy (eV) Professor David Attwood AST 210/EECS 213 (Henke, Gullikson, Davis) Univ. California, Berkeley 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 AST 210/EECS 213 Univ. California, Berkeley 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 AST 210/EECS 213 Univ.