Reflection and Refraction at an Interface,
Total Internal Reflection, Brewster’s Angle
z
n = 1 – δ + iβ
Vacuum
n=1
φ
k
φ′
φ′′
Refracted
wave
k′
x
k′′
Incident
wave
Reflected
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
Vφ =
Professor David Attwood
Univ. California, Berkeley
ω
ω′
c
=
, thus k′ = |k′| = c (1 – δ + iβ)
k′
n
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)
(3.34a)
(3.34b)
k′ sinφ′
n = 1 – δ + iβ
Vacuum
n=1
φ′
k
φ
k sinφ
(3.36)
(3.35a)
Professor David Attwood
Univ. California, Berkeley
ω
ω′
nω
k = c and k′ = c/n = c
sinφ = n sinφ′
(3.35b)
The angle of incidence equals
the angle of reflection
k′′
φ′′
x
k′′ sinφ′′
Conclusions:
Since k = k′′ (both in vacuum)
∴
k′
(3.38)
Snell’s Law, which describes
refractive turning, for complex n.
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
sin φc
1=
1–δ
φ′
2
θ
φ
θ + φ = 90
(3.40)
Glancing incidence (θ < θc) and
total external reflection
(3.41)
The critical angle for total
external reflection.
Professor David Attwood
Univ. California, Berkeley
θc
θ < θc
l ray
ica
Crit
Totally
reflected
wave
Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007
Exponential
decay of the
fields into the
medium
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 f10is approximated by Z. Note that f10is 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
0
D
0
0.5
1
E
1.5
θ/θc
2
2.5
• finite β/δ rounds the sharp angular
dependence
• cutoff angle and absorption edges
can enhance the sharpness
• note the effects of oxide layers
and surface contamination
Professor David Attwood
Univ. California, Berkeley
(b)
3
(c)
(d)
Reflectivity (%)
0.5
Reflectivity (%)
Reflectivity
A: β/δ = 0
B: β/δ = 10–2
C: β/δ = 10–1
D: β/δ = 1
E: β/δ = 3
Reflectivity (%)
A
B
C
1
(a)
Reflectivity (%)
. . . for real materials
Glancing incidence reflection
as a function of β/δ
100
Carbon (C)
30 mr
80
60
40
80 mr
20
0
100
80
30 mr
60
40
20
0
100
Aluminum (Al)
80 mr
Aluminum Oxide
(Al2O3)
80
60
40
80 mr
(4.6)
20
0
100
30 mr
(1.7)
Gold (Au)
80
60
30 mr
40
20
0
80 mr
100
1,000
10,000
Photon energy (eV)
(Henke, Gullikson, Davis)
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
reflectivity
(“low-pass”)
Absorption
edge
Filter
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
(3.43)
φ′
H′ cosφ′
tangential magnetic fields continuous
(3.44)
H′
n = 1 – δ + iβ
n=1
H
φ
H cosφ E
E′
φ′
x
φ φ′′
H′′
E′′
φ′′
H′′ cosφ′′
(3.45)
Snell’s Law:
Professor David Attwood
Univ. California, Berkeley
Three equations in three unknowns
(E0′ , E0′′, φ′) (for given E0 and φ)
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:
Professor David Attwood
Univ. California, Berkeley
Nickel @ 300 eV (4.13 nm)
From table C.1, p. 433
f10= 17.8
f20= 7.70
δ = 0.0124
β = 0.00538
R⊥ = 4.58 × 10–5
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
A
B
C
1
Reflectivity
where
For n = 1 – δ + iβ
A: β/δ = 0
B: β/δ = 10–2
C: β/δ = 10–1
D: β/δ = 1
E: β/δ = 3
0.5
D
0
0
0.5
1
E
1.5
θ/θc
2
2.5
3
E. Nähring, “Die Totalreflexion der
Röntgenstrahlen”, Physik. Zeitstr.
XXXI, 799 (Sept. 1930).
Professor David Attwood
Professor
David
AST
210/EECS
213Attwood
Univ.
California,
Berkeley
Univ.
California,
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)
φ′
(3.55)
The reflectivity for parallel (p) polarization is
H′
E′ cosφ′
φ′
n = 1 – δ + iβ
n=1
x
φ
(3.56)
E′
φ′′
E
φ
E cosφ
E′′
H
H′′
φ′′
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′
E′
0
(3.56)
(3.58)
0
φB
E
Squaring both sides, collecting like terms
involving φB, and factoring, one has
Reflectivity
the condition for a minimum in the reflectivity,
for parallel polarized radiation, occurs at an angle
given by
(3.59)
For complex n, Brewster’s minimum occurs at
10–2
S
10–4
P
10–6
0
Professor David Attwood
Univ. California, Berkeley
k′′
1
or
or
90
0
There is a minimum in the reflectivity
where the numerator satisfies
E′
0′ =
n = 1 – δ + iβ
n=1
k
sin2Θ
radiation
pattern
(3.60)
W
4.48 nm
45
90
Incidence angle, φ
Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007
(Courtesy of J. Underwood)
Ch03_BrewstersAngle.ai
Focusing with Curved, Glancing Incidence Optics
The Kirkpatrick-Baez mirror system
(Courtesy of J. Underwood)
•
•
•
•
•
Professor David Attwood
Univ. California, Berkeley
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
Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007
Ch03_FocusCurv.ai
Determining f10 and f20
• f20 easily measured by absorption
• f10 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