Ch03_ReflctnRefrctn_2007.aiProfessor David AttwoodUniv. California, Berkeley

Reflection and Refraction at an Interface, Total Internal Reflection, Brewster’s Angle

z Refractedwave

ReflectedwaveIncident

wave

x

k′

k′′k

φ′

φ′′φ

n = 1 – δ + iβVacuum

n = 1

incident wave:

refracted wave:

reflected wave:

(3.30a)

(3.30b)

(3.30c)

(1) All waves have the same frequency, ω, and |k| = |k′′| =(2) The refracted wave has phase velocity

ωc

ωc

ω′k′

cnVφ = = , thus k′ = |k′| = (1 – δ + iβ)

Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_BndryConditns.ai

Boundary Conditions at an Interface

• E and H components parallel to the interface must be continuous

(3.32a)

(3.32b)

(3.32c)

(3.32d)

• D and B components perpendicular to the interface must be continuous

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_SpatialContin.ai

Spatial Continuity Along the Interface

Continuity of parallel field components requires

(3.34a)

(3.33)

(3.34b)

(3.35a)

(3.35b)∴

(3.36)

(3.38)

Conclusions:Since k = k′′ (both in vacuum)

k = and k′ = =

sinφ = n sinφ′

The angle of incidence equalsthe angle of reflection

Snell’s Law, which describesrefractive turning, for complex n.

z

xk′

k′ sinφ′

k′′ sinφ′′k sinφ

k k′′

φ′

φ′′φ

n = 1 – δ + iβVacuum

n = 1

ωc

ω′c/n

nωc

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_TotalExtrnlRflc1.ai

Total External Reflectionof Soft X-Rays and EUV Radiation

Snell’s law for a refractive index of n 1 – δ, assuming that β → 0

Consider the limit when φ′ →

Glancing incidence (θ < θc) andtotal external reflection

π2

1 = sin φc1 – δ

The critical angle for totalexternal reflection.

(3.41)

(3.39)

(3.40)

φ′ > φ

θ + φ = 90

φ′

φθ

θ < θcθc

Critical rayTotallyreflectedwave

Exponential decay of the fields into the medium

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_TotalExtrnlRflc2.ai

Total External Reflection (continued)

The atomic density na, varies slowly among the naturalelements, thus to first order

(3.41)

(3.42a)

(3.42b)

where f1 is approximated by Z. Note that f1 is a complicatedfunction of wavelength (photon energy) for each element.

0 0

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_TotalExtrnlReflc3.ai

Total External Reflection with Finite b

Glancing incidence reflectionas a function of β/δ

. . . for real materials

• 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

1

0.5

00 0.5 1 2 31.5 2.5

θ/θc

ABC

D E

A: β/δ = 0B: β/δ = 10–2

C: β/δ = 10–1

D: β/δ = 1E: β/δ = 3

Ref

lect

ivity

100

(Henke, Gullikson, Davis)

1,000 10,000

100

0

100

0

10080604020

80604020

80604020

0

10080604020

0

Ref

lect

ivity

(%)

Ref

lect

ivity

(%)

Ref

lect

ivity

(%)

Ref

lect

ivity

(%)

Photon energy (eV)

Gold (Au)

Aluminum Oxide(Al2O3)

Aluminum (Al)

Carbon (C)30 mr

30 mr

30 mr

80 mr

80 mr

80 mr(4.6) (1.7)

80 mr

30 mr

(a)

(b)

(c)

(d)

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_NotchFilter.ai

The Notch Filter

• Combines a glancing incidence mirror and a filter• Modest resolution, E/∆E ~ 3-5• Commonly used

Mirrorreflectivity(“low-pass”)

Absorptionedge Filter

transmission(“high-pass”)

Photon energy

1.0

Filter/reflectorwith responseE/∆E 4

Professor David AttwoodUniv. 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

E0 perpendicular to the plane of incidence (s-polarization)

tangential electric fields continuous(3.43)

(3.44)

(3.45)

tangential magnetic fields continuous

Snell’s Law:Three equations in three unknowns(E0, E0, φ′) (for given E0 and φ)′ ′′

H′′ cosφ′′H cosφ

H′ cosφ′

z

x

H′

E′

E′′E

H′′H

φ′

φ′

φ′′φ

φ φ′′

n = 1 – δ + iβn = 1

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_ReflecInterf2.ai

Reflection at an Interface (continued)

E0 perpendicular to the plane of incidence (s-polarization)

The reflectivity R is then

With n = 1 for both incident and reflected waves,

Which with Eq. (3.46) becomes, for the case of perpendicular (s) polarization

(3.47)

(3.46)

(3.48)

(3.49)

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_NormIncidReflc.ai

Normal Incidence Reflection at an Interface

Normal incidence (φ = 0)

For n = 1 – δ + iβ

Example: Nickel @ 300 eV (4.13 nm)From table C.1, p. 433f1 = 17.8 f2 = 7.70δ = 0.0124 β = 0.00538

R⊥ = 4.58 × 10–50 0

Which for δ << 1 and β << 1 gives the reflectivity for x-ray and EUVradiation at normal incidence (φ = 0) as

(3.49)

(3.50)

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_GlancIncidReflc.ai

Professor David AttwoodAST 210/EECS 213Univ. California, Berkeley

Glancing Incidence Reflection (s-polarization)

For

For n = 1 – δ + iβ

where

(3.49)

A: β/δ = 0B: β/δ = 10–2

C: β/δ = 10–1

D: β/δ = 1E: β/δ = 3

E. Nähring, “Die Totalreflexion derRöntgenstrahlen”, Physik. Zeitstr.XXXI, 799 (Sept. 1930).

1

0.5

00 0.5 1 2 31.5 2.5

θ/θc

ABC

D E

Ref

lect

ivity

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_ReflecInterf3.ai

Reflection at an Interface

E0 perpendicular to the plane of incidence (p-polarization)

(3.54)

(3.55)

(3.56)

The reflectivity for parallel (p) polarization is

which is similar in form but slightly differentfrom that for s-polarization. For φ = 0 (normalincidence) the results are identical.

E′′ cosφ′′E cosφ

E′ cosφ′H′

z

x

E′

E′′

H′′H

E

φ′

φ′

φ′′φ

φ φ′′

n = 1 – δ + iβn = 1

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_BrewstersAngle.ai

Brewster’s Angle for X-Rays and EUV

For p-polarization

(3.56)

(3.58)

(3.59)

(3.60)

There is a minimum in the reflectivitywhere the numerator satisfies

Squaring both sides, collecting like termsinvolving φB, and factoring, one has

or

the condition for a minimum in the reflectivity,for parallel polarized radiation, occurs at an anglegiven by

For complex n, Brewster’s minimum occurs at

or

k′

E′

n = 1 – δ + iβ

sin2Θradiationpattern

n = 1

k′′

k

0

E 0

φB

E′′ = 0

0

90

S

P

W4.48 nm

0

Ref

lect

ivity

45 90

1

10–2

10–4

10–6

Incidence angle, φ

(Courtesy of J. Underwood)

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_FocusCurv.ai

Focusing with Curved, Glancing Incidence Optics

The Kirkpatrick-Baez mirror system

• 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

(Courtesy of J. Underwood)

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007

Ch03_Determining.ai

Determining f1 and f20 0

• f2 easily measured by absorption• f1 difficult in SXR/EUV region• Common to use Kramers-Kronig relations

(3.85a)

(3.85b)

• 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)

as in the Henke & Gullikson tables (pp. 428-436)

0

0

Professor David AttwoodUniv. California, Berkeley Reflection and Refraction at an Interface,Total Internal Reflection, Brewster’s Angle, EE290F, 1 Feb 2007