wave propagation and refractive index at euv and … · ch03_normincidreflc.ai professor david...

WAVE PROPAGATION AND REFRACTIVE INDEXAT EUV AND SOFT X-RAY WAVELENGTHS

Chapter 3

n = 1 – δ + iβn = 1 φ

k

k′

k′′

Ch03_F00VG.ai

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

Ch03_WavEq_RefrcIndx1.ai

The Wave Equation and Refractive Index


The transverse wave equation is

(3.1)

(3.2)

where na is the average density of atoms, and

For the special case of forward scattering the positions of the electronswithin the atom (∆k ? ∆rs) are irrelevant, as are the positions of theatoms themselves, n(r, t). The contributing current density is then

Ch03_WavEq_RefrcIndx2.ai

The Wave Equation and Refractive Index (Continued)


The oscillating electron velocities are driven by the incident field E

such that the contributing current density is

Substituting this into the transverse wave equation (3.1), one has

Combining terms with similar operators

(3.2)

(3.4)

(3.5)

Ch03_RefracIndex1.ai

Refractive Index in the Soft X-Rayand EUV Spectral Region


Written in the standard form of the wave equation as

The frequency dependent refractive index n(ω) is identified as

For EUV/SXR radiation ω22 is very large compared to the

quantity e2na/e0m, so that to a high degree of accuracy theindex of refraction can be written as

which displays both positive and negative dispersion, dependingon whether ω is less or greater than ωs. Note that this will allowthe refractive index to be more or less than unity, and thus thephase velocity to be less or greater than c.

(3.6)

(3.7)

(3.8)

Ch03_RefracIndex2.ai


Refractive Index in the Soft X-Rayand EUV Spectral Region (continued)

Noting that

and that for forward scattering

where this has complex components

The refractive index can then be written as

which we write in the simplified form

(3.8)

(3.9)

(3.12)

Ch03_RefrcIndxIR.XR.ai


Refractive Index from theIR to X-Ray Spectral Region

•λ2 behavior•δ & β << 1•δ-crossover

(3.12) (3.13a)

(3.13b)

1

0Infrared Visible Ultraviolet

Ultraviolet

Ref

ract

ive

inde

x, n

X-rayωIR ωUV ωK,L,M

Ch03_PhasVelo_Refrc1.ai


Phase Velocity and Refractive Index

The wave equation can be written as

(3.10)

(3.11)

The two bracketed operators represent left and right-running waves

where the phase velocity, the speed with which crests offixed phase move, is not equal to c as in vacuum, but rather is

z z

Right-running waveLeft-running wave

EE

Vφ = – Vφ = cn

cn

Ch03_PhasVelo_Refrc2.ai


Phase Velocity and Refractive Index (continued)

Recall the wave equation

(3.10)

Examining one of these factors, for a space-time dependence

Solving for ω/k we have the phase velocity

ET = E0 exp[–i(ωt – kz)]

Vφ =

Ch03_PhaseVarAbsrb.ai


Phase Variation and Absorptionof Propagating Waves

For a plane wave (3.14)

(3.15)

(3.16)

(3.17)

in a material of refractive index n, the complex dispersion relation is

Solving for k

or

where the first exponential factor represents the phase advance had thewave been propagating in vacuum, the second factor (containing 2πδr/λ)represents the modified phase shift due to the medium, and the factorcontaining 2πβr/λ represents decay of the wave amplitude.

Substituting this into (3.14), in the propagation direction definedby k ? r = kr

Ch03_IntenstyAbsrp.ai


Intensity and Absorption in a Materialof Complex Refractive Index

For complex refractive index n

The average intensity, in units of power per unit area, is

or

orthe wave decays with an exponential decay length

Recalling that

(3.18)

(3.19)

(3.20)

(3.17)

(3.21)

(3.22)

Ch03_AbsorpLngths.ai


Absorption Lengths

(3.23)

Recalling that β = nareλ2f2(ω)/2π°

In Chapter 1 we considered experimentally observed absorptionin thin foils, writing

where ρ is the mass density, µ is the absorption coefficient, r is thefoil thickness, and thus labs = 1/ρµ. Comparing absorption lengths,the macroscopic and atomic descriptions are related by

where ρ = mana = Amuna , mu is the atomic mass unit, and A isthe number of atomic mass units

(3.22)

(3.24)

(3.26)

Ch03_PhaseShift.ai


Phase Shift Relative to Vacuum Propagation

(3.23)

(3.29)

For a wave propagating in a medium of refractive index n = 1 – δ + iβ

the phase shift ∆φ relative to vacuum, due to propagation througha thickness ∆r is

•Flat mirrors at short wavelengths•Transmissive, flat beamsplitters•Bonse and Hart interferometer•Diffractive optics for SXR/EUV

Reference wave

Object wave

Object∆r

M

M BS

BS

Filmor CCD

Ch03_ReflctnRefrctn.ai


Reflection and Refraction at an Interface

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′c′

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

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

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

x

k′

k′ sinφ′

k′′ sinφ′′k sinφ

k k′′

φ′

φ′′φ

n = 1 – δ + iβVacuum

n = 1

ωc

ω′c/n

nωc

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 ray

Totallyreflectedwave

Exponential decay of the fields into the medium

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.

° °

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 = 0

B: β/δ = 10–2

C: β/δ = 10–1

D: β/δ = 1

E: β/δ = 3

Ref

lect

ivity

100

(Henke, Gullikson, Davis)

1,000 10,000

100

0

100

0

100

80

60

40

20

80

60

40

20

80

60

40

20

0

100

80

60

40

200

Ref

lect

ivity

(%

)R

efle

ctiv

ity (

%)

Ref

lect

ivity

(%

)R

efle

ctiv

ity (

%)

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)

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

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



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)

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 × 10–5° °

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)

Ch03_GlancIncidReflc.ai


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

1

0.5

00 0.5 1 2 31.5 2.5

θ/θc

A

BC

D E

Ref

lect

ivity



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

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)

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)

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)

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wave propagation and refractive index at euv and … · ch03_normincidreflc.ai professor david...

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