lecture 3 interference, diffraction, coherence, refraction ... · lecture 3 interference,...

$: Lecture 3 Interference, diffraction, coherence, refraction ... · Lecture 3 Interference, diffraction, coherence, refraction, ... related to interference, diffraction, coherence,$
Lecture 3Interference, diffraction, coherence, refraction,

O tli

f , ff , , f ,reflection and absorption do affect x-ray lasers

Outline• Basic wave phenomena involved in the operation and application of x-ray lasers

(sources)F W ti t i t f f F W ti t diff ti• From Wave equation to interference of x-rays. From Wave equation to diffractionof x-rays. Weakest, diffraction-less and strong diffractions of x-rays. Longitudinal(temporal) and transverse (spatial) coherences of x-rays by statistic optics.Longitudinal coherence-length lc is determined by the line spectral bandwidth Δλ.Longitudinal coherence length lc is determined by the line spectral bandwidth Δλ.From the Fresnel integral to degree of transversal coherence of x-rays (forsimplicity, 1-D case). Van Cittert-Zernike theorem for 2-D x-ray source. The useof Van Cittert-Zernike theorem for an incoherent circular x-ray source. The use ofV Citt t Z ik th f i d ti di fVan Cittert-Zernike theorem for passive and active mediums of x-ray sources.

• Manipulation of x-rays (waves) by refraction, reflection and absorption. Fromrefraction index n(ω) to refraction, reflection and absorption of x-rays.From Maxwell equations to Wave equation and then to refractive index n( ) The• From Maxwell equations to Wave equation and then to refractive index n(ω). Thevelocity vi of an oscillating electron of an atom driven by an incident x-ray. Thecurrent density JT in Wave Equation yields refractive index n(w). Refractive indexn(w) in x-ray spectral region. Different forms of n(w) in x-ray spectral region.( ) y p g ( ) y p g

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 1

Outline ctd.

• Atomic scattering factors f01 and f02 for Carbon and Aluminum (examples). Refractiveindex n(ω) from the IR to X-ray region. Snell’s law for the total (100%) external reflectionof x-rays at an interface. Total external reflection of x-rays and the glancing x-ray optics.From [Wave equation + refractive index n(ω)] to Fresnel reflection coefficient R FromFrom [Wave equation + refractive index n(ω)] to Fresnel reflection coefficient R. Fromrefraction index n(ω) to Fresnel reflection and refraction of x-rays at an interface.Reflection coefficient Rs for the S-polarization Fresnel reflection coefficient Rs at normalincidence. Fresnel Rs at the glancing incidence Q ~ Qc. The values Rs at the glancingincidence Q~Qc. . Reflection coefficient Rp for P-polarization.

• Modification of x-rays (waves) by reflection at glancing incidence. The guiding (focusing)by the Kirkpatrick-Baez glancing incidence, curved-mirror x-ray optics. Guiding andfocusing by the capillary x ray optics “Focusing” x ray femtosecond pulses by a taperedfocusing by the capillary x-ray optics. Focusing x-ray femtosecond pulses by a taperedcapillary on a nanometer scale. The free-space virtual source of a system of the “fs-nmfocusing” of x-rays. The model shows feasibility of the “fs-nm focusing” of x-rays. Guidingand focusing by Kumakhov’s “lens”. Focusing of incoherent and coherent x-rays by ag y g y y”Kumakhof’s lens”.

• The model results. Guiding and focusing of x-rays by a gradient refraction index plasma-based guide. Summary of guiding and focusing of x-rays by a gradient refraction indexl b d id G idi d f i f b l b d id t i tplasma-based guide. Guiding and focusing of x-rays by a plasma-based guide: transient

modes m (m=0,1,2 …). The use of Van Cittert-Zernike theorem for the x-rays focused bya plasma-based guide.

• Problems as home assignments• Problems as home assignments• References


Basic wave phenomena involved in the operation d li i f l ( )and application of x-ray lasers (sources)

In the following lectures 4-12, we will consider how interference, diffraction, g , , ,coherence, refraction, reflection and absorption of x-rays affect operation of x-ray sources (lasers), manipulation of x-rays and their application

Manipulation of x-rays

Application of x-rays

x-ray sourceei

EM wave of x-raysrj

x-ray source

L ri

rT (output aperture)

Fig. 1 Interference, diffraction, coherence, refraction, reflection and absorption of x-rays are involved in the operation of x-ray sources, manipulation of x-rays and their applications

T ( p p )

Most equations related to interference, diffraction, coherence, refraction, reflection and absorption of x-rays are described by equations of conventional optics in the visible spectral region, but some of them require additional consideration !!!

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 3

g q

From Wave equation to interference of x-raysrj From the questions (What is x-ray

medium of x-ray radiation

L ri

eiEM wave or x-ray?

j From the questions (What is x-ray radiation? Is it an EM wave or a ray?) to the interference of x-rays.

Fig. 2 A source of incoherent or coherent EM waves of x-rays

L


Interference ?

Wave equation yields the principle of superposition (Fourier- decomposition) for EM waves.That is to say EM waves are given by E(r t) = Σ E (r t)

Maxwell’s equations Wave equation of x-rays?

(1)That is to say EM waves are given by E(r, t) = Σi Ei (r, t) Ei (r, t) ~ ei aTi (t – rj /c + ϕj) /rj ~ ei aTi (kiri - ωit + ϕj) /rj,

whereω = kc = (2π/λX-ray)c , and the field E(r, t) ~ Σi ei aTi (kiri - ωit + ϕj) /rj

(1)(2)

(3)y T j j

is induced by incoherent (ϕi = ϕj, aTi = aTj , ri = rj ) or coherent (ϕi = ϕj, aTi = aTj , ri = rjor ri = rj) accelerated and/or de-accelerated (|aTi | = 0) electrons of the x-ray medium.

Intensity (I) of the Fourier-like field E(r, t) ~ Σi ei aTi (kiri - ωit + ϕj) /rjis given by I ~ |E|2 = ΣiE2

i + ΣiΣj Ei Ej

E E > 0 constructive interference

(4)(5)


Interference termEi Ej > 0 – constructive interferenceEi Ej < 0 – destructive interference

From Wave equation to diffraction of x-raysMaxwell’s equations Wave equation Diffraction

?Maxwell s equations Wave equation Diffraction

of x-raysWave equations:

tt T∂⎟⎞

⎜⎛

∇∂ ),()(22

2 rJE 1 (6)

X-ray waveΕinc(r’)

rΕΤ(r)

r’ttc

tT

T ∂=⎟⎟

⎠⎜⎜⎝

∇−∂

),(),(222 rE

0

-ε

0)(222

=⎟⎟⎞

⎜⎜⎛

∇−∂ tc rE

(6)

(7)

ΕΤ(r,t) = ΕΤ(r)ei(ωt) , ω = kc = (2π/λX-ray)c

r

Sh d

Fig. 3

0),(2 =⎟⎟⎠

⎜⎜⎝

∇∂

tct T rE

Monochromatic x-rays:

(7)

(8)

ΕΤ(r) + k2ΕΤ(r) = 0

Helmholtz differential equation: Εinc(r’)r’

Shadow

X-ray wave(9)

Integral form of Helmholtz differential equation:r

ΕΤ(r)Fig. 4

'')'()('

ddeik

∫∫ ⎥⎤

⎢⎡ −

EErr

(10)

Fresnel-Kirchhoff diffraction integral = secondary spherical waves

Diffraction-freeparallel beam

Diffracted x-ray in the shadow region (diffraction)

'''4

)'()(.

.) dydxapert

inc∫∫ ⎥⎥⎦⎢

⎢⎣ −

=rr

rErE T (T π(10)


Fresnel Kirchhoff diffraction integral secondary spherical waves, Huygens-Fresnel principle [1]

Weakest, diffraction-less and strong diffractions of x-raysof x rays

ΕΤ(r,t) = ΕΤ(r) e-iωt , where ω = <ω>, k = <k>, λ = <λ>Quasi-monochromatic waves:

We now get an answer to the questions (What is x-ray radiation? Is it an EM wave or a ray?).

Τ( , ) Τ( ) , , ,

'''4

)'()(),(.

'

.) dydxeeetapert

i

inctiti ∫∫ ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

−==

−−−

rrrErErE

rrk

T (TT πωω

λ

Δλ

<λ> (11)

r’

abWeakest diffraction (smallest b) if

ΕΤ(inc.)(r,t) = ΕΤ(r,t)exp (-ωt+ϕ(r,t)) = const. Θ

p ⎦⎣

rΕΤ(r)

r’

-a-b

b ~ Lfap(λ/a)--------------------------------------------------------“Diffraction-less” = (near-field) = ( ) ( t i ti ) if

Lfap

Θ

Fig 5

X-ray wave

(12)(wave = x-ray) = (geometric optics) if

b ~ a--------------------------------------------------------St t diff ti if

Fig. 5

Weakest dif. [ΕΤ(r,t) = const., ϕ(r,t) = const.] Diffraction-less [near field, Lfap << b ~ a2 /λ]

(13)

Strongest diffraction if

Ειnc(r,t) = ΕΤ(r,t)exp (ωt+ϕ(r,t)) is incoherent,

respectively ∞b

pStrongest dif. [incoherent field, ΕΤ(r,t) const. and/or ϕ(r,t) const )

ΔxΔp > h - Heisenberg uncertainty

≠≠

(14)

6

respectively

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project

∞b ΔxΔk > 1

( )

Longitudinal (temporal) and transverse (spatial) coherencies of x-rays by statistic opticscoherencies of x-rays by statistic optics

a P1transverse

-a

ΘP2

(spatial) coherencewidth D

X-ray source

Lflongitudinal (temporal) coherenceMutual coherence factor For incoherent source:

Fig. 6?

Lfap

length lcΓ12 (τ) = <E1(P1,t+τ)E*2(P2,t)>

Degree of spatial coherence (coherence factor mutual intensity)

Longitudinal (temporal) full-coherence length

Simple explanation:(coherence factor, mutual intensity)J12 (P1,P2) = Γ12(0) = <E1(P1,t)E*2(P2,t)>

Normalized degree of spatial

lc = λ2/Δλ

Transverse (spatial)f ll h idth

S p e e p a at oThe phase difference between any two waves (rays) in the l it di l

(15)

coherence (coherence factor)μ12=<E1(P1,t)E*2(P2,t)> / <|E1(P1,t)|2>1/2

<|E2(P2,t)|2>1/2

full-coherence width

Θ = λ/4πaD = Lfap Θ

longitudinal or transverse direction is less than ~ 0.5 period(16)

(17)

7TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project

( )

Longitudinal coherence-length lc is determined b th li t l b d idth Δλby the line spectral bandwidth Δλ

lSpectral bandwidth

λ

Δλ

lc

λ

Δλ

Fig. 7 The line spectral bandwidth and the longitudinal coherence length by the phase shift

Δλλ

with wavelength λ

N

The phase difference between the two waves is equal to 0.5 period.

(18)λcoh = Nλ

with wavelength λ+Δλ

(18)

λcoh = (N - 0.5)(λ+Δλ)

N = λ/2Δλ

(19)

(20)

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 8lc = λ2/Δλ (21)

From Fresnel’s integral to degree of transversal coherence of x-rays (for simplicity 1-D case)coherence of x-rays (for simplicity, 1-D case)

xix”iMutual intensity (see, p.6): Jij (Pi Pj) = Γ12(0) =

Θi

xjx”jFig. 8 Propagation of the wave of x-rays induced by an incoherent 1-Dimensional

f

Jij (Pi,Pj) Γ12(0) <E1(P1,t)E*2(P2,t)>

Θj

Calc. by Fresnel’s integral (11)x-ray source source of x-rays

For 1D incoherent source: Jij (P”i,P”j) = Α Ι(P”i) δ12(|P”i - P”j|)Jij (x”i,x”j) = Α Ι(x”i) δ12(|x”i - x”j|)

Calc. by Fresnel s integral (11)

(22)(23)

source

( ) ( )[ ] ( ) ( )∫ ∫ − ′′′′

′′Θ′′

′′Θ′′

′′−′′−′′′′=h

h

h

hji

j

j

i

i

jijiji xdxdrr

rrcinxxJxxJλ

χλ

χω1000 exp,),(ij

ij ( i, j) ( i) 12(| i j|) ( )

(24)− −h h ji

( ) ( )( ) ⎥

⎤⎢⎡ ⎟

⎠⎞⎜

⎝⎛ ′′Δ λπ /2sin 2x

normalized degree of spatial coherence

The value in the box is

( ) ( )( ) ⎥

⎥⎥

⎦⎢⎢⎢

⎣⎟⎠⎞⎜

⎝⎛ ′′Δ

⎠⎝′′′′=′′′′λπ /2

2),(2x

xIxIxxJ jijiij(coherence factor)

μij= μij (x”i,x”j)(25(a))

~

(25(b))


Van Cittert-Zernike theorem for 2-D x-ray sourceη

ξyP1(x1,y1)

Observation plane

ξ x

Θ1

1( 1,y1)

Fig. 9 For derivation and application of the theorem to 2-Dimensional x-ray source)X-ray source

pΘ2P(ξ,η)

P2(x2,y2)

Van Cittert-Zernike theorem: μ12 is the Fourier transform of the Intensity distribution of incoherent

( ) ( )[ ] ( )∫ ∫∫ ∫+∞

∞−

+∞

∞−

− Δ+Δ= ηξηξηξηξηξμ ψ ddIddzyzxkiIeyxyx i ,//exp,),;,( 221112 (26)

μ12emitters (ψ is a phase parameter that is irrelevant for |μ12| )

For circular (radius R) surface of incoherent (uncorrelated) emitters

1

Θ∞

∫ dkJI )()( 0 ρρρρ

)(),( 220 RcircII ηξηξ += (27)

|μ12|

ΘΘ

=Θ

= −∞

∫

∫Rk

RkJe

dI

dkJIi )(2

)(

)()(10

0

12ψ

ρρρ

ρρρρμ (28)

0 0 10<k>RΘ


0Fig. 10 Degree of spatial coherence of the circular x-ray source |μ12 (<k>RΘ )|

The use of Van Cittert-Zernike theorem for an incoherent circular x-ray sourceincoherent circular x-ray source

2R dOutput aperture of an x-ray source

Fig. 11 Schema for the use of theorem in case of a 2-Dimensional (circular, radius R) source of incoherent uncorrelated emittersZ

Θ

of an x-ray source

source of incoherent, uncorrelated emitters

An incoherent x-ray source produces partially coherent or fully transversally coherent x-rays (waves) in the regionrays (waves) in the region

For a circular (radius R) surface of incoherent (uncorrelated) emitters

ΘRkJ )(2 1 (29)Θ

Θ=

RkJ )(1

12μ

Thus the use of Van Cittert-Zernike theorem yields

(29)

(30)(31)

Transverse coherence

- Incoherent radiation: 2R >> <λ>Z / dPartially coherent radiation: 2R ~ <λ>Z / d


(31)(32)

- Partially coherent radiation: 2R ~ <λ>Z / d- Coherent radiation: 2R << <λ>Z / d

The use of Van Cittert-Zernike theorem for i d ti di fpassive and active mediums of x-ray sources

Incoherent passive X ray1 2 From Van Cittert-Zernike theorem:R dmedium

Θ

X-ray (wave)

From Van Cittert Zernike theorem: - Incoherent radiation: 2R >>λ Lfap / d- Partially coherent radiation: 2R~λLfap /d- Coherent radiation: 2R << λ Lfap / d

R

-R

d

X-ray source

Fig. 12 Incoherent, partially coherent and coherent X-rays from incoherent source

L

(output aperture) rT

Lfap.

R

Active medium

Θ From Van Cittert Zernike theorem:

RX-ray ( )

X-ray source

L

(output aperture) rT

From Van Cittert-Zernike theorem:- Incoherent radiation: 2R >>λ L / R- Partially coherent radiation: 2R~λL / R- Coherent radiation: 2R << λ L / R

-R(wave)

y

Fig. 13 Incoherent, partially coherent and coherent x-rays from the passive and active mediums of x-ray

( p p ) T


coherent x-rays from the passive and active mediums of x-ray sources

Manipulation of x-rays (waves) by refraction, reflection and absorption

Manipulation of x-raysby refraction

Application of x-rays

D d

Θ

Output aperture of an x-ray source

by refraction, reflection and absorption

ZΘ

Fig. 13 X-ray source

M i l ti fManipulation of x-raysby refraction, reflection and L i ti tt t filt d

Optical models and formulas for the x-ray spectral region?

reflection and absorption

Lenses, mirrors, gratings, attenuators, filters and waveguides for incoherent and/or coherent x-rays ?


From refraction index n(ω) to refraction, reflection and absorption of x raysreflection and absorption of x-rays

refracted wavereflected

z

wavewave

xEM wave of x-raysrj

ϕ

ϕ’ ϕ”

x

incident (wave)

n (λ) n (λ)

x-ray source

L ri

r (output aperture)

eiEM wave of x rays ϕ

Fig. 14 Refraction and reflection of an incident wave of x-rays

boundaryn1 (λ) n2(λ)rT (output aperture)

I(λ) = I0(λ) e- ρμL ; μ = μ(n2)incident wave b b d

n1 (λ)

g

I0(λ)absorbed(attenuated) wave

n2 (λ)

L


Fig. 15 Absorption (attenuation) of an incident wave of x-rays

From Maxwell equations to Wave equation d th t f ti i d ( )and then to refractive index n(ω)

Maxwell equations t

ttct

TT ∂

∂=⎟⎟

⎠

⎞⎜⎜⎝

⎛∇−

∂∂ ),(),(22

2

2rJ

rE

0

1-

εTransverse Wave Equation (33)

In the case of the refraction, reflection and absorption of x-rays, the positions of electrons in atoms and the positions of electrons are irrelevant under the forward scattering of x-rays by the atom mediums. Then, for an atom with Z electrons, we get

∑−=Z

iitT tgent ),(),( rvrJ (34)∑i

iiatT tgent ),(),( rvrJwhere

∑=Z

igZ

( )

(35)

vi(r,t) ?

iThen we should find the velocity vi(r,t). (36)


The velocity vi of an oscillating electron of t d i b i id tan atom driven by an incident x-ray

vi(r,t) ?

We use the Lorentz classical model of an oscillating bound electron of an atom. The EM force of the incident wave of x-rays changes the electron coordinate (r)

2 2

i( , )

and velocity (v(r,t) = dr/dt) and acceleration (a(r,t) = d2r/dt2) :

B )vE- ×+=++ (22

2

emdtdm

dtdm i r

rr ωγ (37)dtdt

In the case of time-harmonic incident wave with the field E given by ~0

tie ω−= 0EE

we get a solutionee

it

tiω−

+= 0E

22

1)(r

(38)

(39)g mii γωωω +− 22

which yieldsddet Evi = 22

1)((40)


ydtim i

i γωωω +− 22)(

Current density JT in Wave equation yields refractive index n(ω)refractive index n(ω)

Thus Eqs. (34) and (40) yieldZ dE2

∑ +−−=

Z

i

T

i

iatT dt

di

gmnet ErJ

γωωω 22

2

),( (41)

The use of Eq (40) in Eq (33) yields the equationThe use of Eq. (40) in Eq. (33) yields the equation

∑=⎟⎟⎞

⎜⎜⎛

∇−∂ Z

Tiat dgnetc22

222

)( ErE (42)∑ +−⎟⎟⎠

⎜⎜⎝

∇∂ i i

T dtimtc

t 2222 ),(rEγωωωε0

2222 ⎤⎡ ∂⎟

⎞⎜⎛

∑ gne Z

(42)

and the equation 0),(1 22222 =⎥

⎦

⎤⎢⎣

⎡∇−

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

− ∑ tcti

gmne

Ti i

iat rEγωωωε0

(43)

which can be presented in the conventional form of the Wave equation

0),()(

22

2

2

2

=⎥⎦

⎤⎢⎣

⎡∇−

∂∂ tc

T rE(44)


)()(22 ⎥

⎦⎢⎣∂ nt Tω

Refractive index n(ω) in x-ray spectral region

The comparison Eqs. (43) and (44) yields the refraction index 2/12 ⎞⎛ Z gne

221)( ⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−= ∑Z

i i

iat

ig

mnen

γωωωεω

0(45)

In the case of x-rays, ω >> e2nat/ε0m. That yields the approximation

Z gne21 ∑ +−−=

i i

iat

ig

mnen

γωωωεω 222

11)(0

(46)

If ω > ωi, then n(ω) < 1 and the phase velocity v = c/n > c. (47)

If ω < ω2i, then n(ω) > 1 and the phase velocity v=c/n < c. (48)


Different forms of n(ω) in x-ray spectral regionRefractive index n(ω) is usually represented by using atomic real (f01) and imaginary (f02 ) scattering factors or the respective real (δ) and imaginary (β) parameters. For the representation of Z gne21 ∑ +−

−=i i

iat

ig

mnen

γωωωεω 222

11)(0

we note that for forward wave scattering the scattering factor is given by

(49)

we note that for forward wave scattering, the scattering factor is given by

∑ +−=−=

Z

i i

i

igifff

γωωωωωωω 22

20

20

10 )()()( (50)

where

that yields [ ])()(2

1)( 02

01

2

ωωπλω iffrnn eat −−=

2e

(51)

where

is the classical electron radius. To this end, the definition

24 mcere0πε

=

)(2

01

2. ωπ

λδ frn eat= (52)

yields the conventional form used in the x-ray spectralregion and the research fields related to x-ray lasers

2π

)(2

02

2. ωπ

λβ frn eat= (53)


βδω in +−=1)( (54)

Atomic scattering factors f01 and f0

2 for Carbon and Aluminum (examples)Carbon and Aluminum (examples)

Carbon Aluminum

Fig. 16 Scattering factors of C and Al (from B.L. Henke, E.M. Gullikson, J.C. Davis, Atomic Dat. And Nucl. Dat.

Carbon Aluminum


gTables 54, 181 (1993)

Refractive index n(ω) from the IR to X-ray regionA simple analysis yields the schematic dependence n=n( ) from IR to X ray region

ts]

A simple analysis yields the schematic dependence n=n(ω) from IR to X-ray region

b. u

nit

ω) [

arb

1

n(ω 1

IR Visible UV X-ray0

ωV ωUV ωX-ray

IR Visible UV X ray

ωIR

Fig. 17 Schematic behavior of the refractive index vs frequency from the IR to X-ray spectral region

V UV X ray


Snell’s law for the total (100%) external reflection of x-rays at an interfacereflection of x-rays at an interface

Most equations in the x-ray optics are described by equations of conventional optics in the visible spectral region, but total reflection of x-rays at an interface requires additional

id ti !consideration !

refractedz

Snell’s law:2

1

sin"sin

nn

=ϕ

ϕ(55)

refracted wave

ϕ”=π/2

In the visible region: n1 (λ) >1, n2 (λ) = 1In the x-ray region: n1 (λ) =1, n2 (λ) = 1-δ+β ∼1−δ<1

(56)(57)

x

incident x-ray source

L

eiEM wave of x-rays

rj ϕ

Θ

boundary

(wave)n1 (λ) =1 n2(λ) =1-δ

L ri


Fig. 18 Total external reflection of the waves of x-rays

The visible region: (from (56)) Total internal reflection Θ = π/2−ϕ

Total reflection at ϕc”=π/2


The x-ray region: (from (57)) Total external reflection

Total external reflection of x-rays and glancing x-ray opticsglancing x-ray optics

δπϕ

=Θ 1

1)2/sin(

"sin c

Θ /2

Total reflection at ϕc”=π/2=

11 Θc << π/2

δπ −Θ− 1)2/sin( cΘ = π/2−ϕ δ−Θ 1cos c

=1

11

δ−+Θ− 1...)211( 2

c δ2=Θc

2

(58)

Using in Eq. (58) yields )(2

01

2. ωπ

λδ frn eat= Zfrn eatc λω

πλ ~)(0

1

2.=Θ (59)

zΘ<Θctotal reflection

Glancing incidence x-ray opticsK-P Kum.

x

ΘcΘ

reflection

Fig. 20 Glancing Incidence x-ray optics (Kirkpatrick-Baez and Kumakhov optics)

Fig. 19 Glancing Incidence (Θ<Θc)and total reflection of x-rays

(Kirkpatrick-Baez and Kumakhov optics)


From “Wave equation + refractive index n(ω)”t F l fl ti ffi i t Rto Fresnel reflection coefficient R

refracted reflected

zFrom Snell”s law: δ2,1 =Θ<Θ= cifR

wavereflected wave

EM wave of x raysrj

ϕ’ ϕ”)?RR Θ=

x

incident (wave)

x-ray source

L ri

eiEM wave of x-rays ϕ

Θ

Fig. 21 Refraction and reflection of an incident wave of x-raysboundary

( )n1 (λ) n2(λ)rT (output aperture)

Boundary conditions )( tie ω−−== krEEE

Wave Equation: + reflection coefficient R

(60)Dirichlet: f(1)(z=0) = f(2)(z=0) Neumann: df(1)(z=0)/dz = df(2)(z=0)d/z

)"(

)'(0.

0.

"

'ti

tireflect

inc

e

e

ω

ω

−−

−−==

==

rk

rk

EEE

EEEEEE

"

'

"

'(60)

(62)

(61)(63)

(64)


)(0. " ti

refract e ω== rkEEE " (62)

From refraction index n(ω) to Fresnel reflectiond f i f i fand refraction of x-rays at an interface

Wave equation Linear processes

)(0.

)(0.

' tireflect

tiinc

e

eω

ω

−−

−−

==

==kr

kr

EEEEEE'

"' ωωω hhh ==

ckk /' ω==

(65)

(66)

(68)

(69)

)(0.

0.

" tirefract

reflect

e ω−−== rkEEE ""( )βδωω icnc

k +−== 1/

"2

(67) (70)

Continuity of E and H components which are parallel to the interface

( ) ( ) ( )"sin"'sin'sin ϕϕϕ kkk ==

'ϕϕ = Incidence angle = Reflection angle

(75)

(76)( ) ( ) ( ) "/'/ 00 EzzEEzz ×=+× 0

( ) ( ) ( ) "/'/ 00 HzzHHzz ×=+× 0

ϕϕ

2

1sin

"sinn

=ϕ

ϕg g

Snell’s law:(71)

(72)

( )

(77)

Continuity of the D and B components which are parallel to the interface

( ) ( ) ( ) "/'/ 00 DzzDDzz ×=+× 0(73)


( ) ( ) ( ) "/'/ 00 BzzDBzz ×=+× 0 (74)

Reflection coefficient Rs for S-polarization

Continuity of tangential components of E and H and Eq. (40) from Lecture 2 yields

S-polarization: E is perpendicular to the plane of incidence

ϕϕ

ϕϕ22

2

222

0

0

sincos

sincos'−+

−−=

n

nEEFor reflected x-rays (78)

ϕϕ 2

ϕ22

0

icos2"

=EEFor refracted x-rays (79)

ϕϕ 2220 sincos −+ nE

( ) ( ) 20 '''Re2/1'' EI ×HES

Then Fresnel reflection coefficient Rs is given by

( ) ( )( ) ( ) 2

0

0

00

00

00 Re2/1Re2/1

E

EII

Rs =××

===HEHE

SS

2

(80)

222

222

2

sincos

sincos

ϕϕ

ϕϕ

+

−−=

n

nRs

(81)


02 sincos ϕϕ −+ n

Fresnel reflection coefficient Rs at normal incidenceincidence

S-polarization and normal incidence: ϕ = 02

222 1sincos n ϕϕ

(81)

2

2

22

022

2

2

1

1

sincos

sincos

n

n

n

nRs

+

−=

−+

−−=

ϕϕ

ϕϕ(82)

βδ in +−=12The refraction index is given by Eq. (54) as(83)

( )( ) 2211 βδ +∗

(84)

( )( )( )( ) ( ) 22

22

22

21111

βδβδ+−

+=

++−−

= ∗nnnnRsThe use of (83) in Eq. (82) yields

(85)For δ << 1 and β << 1, we finally get

4

22 βδ +=sR

An example: Al at λ = 4 nm.pf01=12 and f02=4.7 (from B.L. Henke, E.M. Gullikson, J.C. Davis, Atomic Dat. And Nucl. Dat. Tables 54, 181 (1993)).That yields δ 10-2 and β 5x10-3

1105 5 <<×= −sR Mirrors for x-rays ?


That yields δ ~10 2 and β~ 5x10 3

Fresnel Rs at the glancing incidence Θ ∼ ΘcS-polarization and glancing incidence: Θ <Θc (86)

zΘ<Θctotal

S polarization and glancing incidence: Θ Θc ( )2

222 sincos ϕϕ −−

=n

R (87)x

Θ

total reflection

2

022

2 sincos ϕϕ −+=

nRs

( )122/ ΘΘ δ

(87)

(88) ΘcΘ

Fig. 22 Total reflection of x-rays at Θ<Θc

( )122/ <<=Θ<−=Θ δϕπ c

Θ≈Θ=sincosϕ2222 1sin1cos1sin ΘΘϕϕ R

(88)

(89)

(90) S ll’ R1

2222 1sin1cos1sin Θ−≈Θ−=−= ϕϕ( ) ( ) ( ) ( ) 2222

2 1211 βδβδβδ −−+−=+−= iin

Rs(90)

(91)

Snell’s RsΘ/Θc10-2

10-1

Fresnel’s Rs

( )( ) 2

22 2 βiR

c +Θ−Θ−Θ=Θ (92)

1010-0

Fig. 23 Rs calculated by (92) for glancing incidence Θ ∼ Θ

0 1 2 3

( )( ) 2

22 2 βiR

c

s

+Θ−Θ+Θ=Θ

Θ /Θc

(92)


incidence Θ ∼ Θc

Values Rs at the glancing incidence Θ∼Θc

Fig. 24 The dependences of Rs for x-rays at the glancing (Θ ∼ Θc ) incidence for Carbon (f B L H k E M G llik J C D i(from B.L. Henke, E.M. Gullikson, J.C. Davis, Atomic Dat. And Nucl. Dat. Tables 54, 181 (1993))

Fig. 25 The dependences of Rs for x-rays at g s the glancing (Θ ∼ Θc ) incidence for Aluminum(from B.L. Henke, E.M. Gullikson, J.C. Davis, Atomic Dat. And Nucl. Dat. Tables 54, 181 (1993))


Reflection coefficient Rp for P-polarization

Continuity of tangential components of E and H and Eq. (40) from Lecture 2 yields

P-polarization: E is parallel to the plane of incidence

ϕϕ

ϕϕ22

222

222

22

0

0

sincos

sincos'−+

−−=

nn

nnEE

For reflected x-rays (93)ϕϕ 22

ϕ222

0

icos2"

=EEFor refracted x-rays (94)

ϕϕ 222

220 sincos −+ nnE

( ) ( ) 20 '''Re2/1'' EI ×HES

Then Fresnel reflection coefficient Rp is given by

( ) ( )( ) ( ) 2

0

0

00

00

00 Re2/1Re2/1

E

EII

Rp =××

===HEHE

SS

2

(95)

Notice for normal Mi

222

222

222

22

sincos

sincos

ϕϕ

ϕϕ

−+

−−=

nn

nnRp (97)

(96)

Notice, for normal Incidence

14

22

<<+

==βδ

sp RR

Mirrors for x-rays ?


022 sincos ϕϕ + nn 4

Modification of x-rays (waves) by reflectionat glancing incidence

M difi ti fModification of x-rays: glancing (Θ~π/2) reflectionof x-raysby a solid-state

Application of x-rays by a solid state

materialof x rays

D d

Θ

Output aperture of x-ray source

ZΘ

Fig. 26 Modification of x-rays (waves), which are produced by an incoherent or coherent x-ray source, via the reflection of x-rays from a solid-state material at glancing incidence


Glancing incidence, curved-mirror x-ray optics?

The guiding (focusing) by the Kirkpatrick-Baez glancing incidence curved-mirror x-ray opticsglancing incidence, curved-mirror x-ray optics

The Kirkpatrick-Baez glancing-incident, curved-mirror x-ray optics

R ~ 100%GG

x-rays R ~ 100%

G

Fig. 27 Guiding and focusing of x-rays by the Kirkpatrick-Baez glancing-incident, curved-mirror x-ray optics


Mono-guide (capillary):

Guiding and focusing by the capillary X-opticsR ~ 100%

focusing “lens”?Mono guide (capillary): R 100%

(a)Kumakhov’soptics

(b) (e)

p

collimating “lens”?( ) ( )

(c) Kumakhov’soptics

(d)(f)

Poly-guides(capillaries):

R 100% (d)

33TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project Fig. 28 Capillary optics [4]

R ~ 100%

“Focusing” x-ray femtosecond pulses by a tapered capillary on a nanometer scaletapered capillary on a nanometer scale

Femto-nano “focusing” of an x-ray (λ=10nm) pulse

Input x-ray fs-pulse tapered capillary Output x-ray fs-pulse

Lcap=10cm.λ=10nm

t

Δτ ∼ 20fstapered capillary

t

Δτ ∼ 20fs

2b=20 nm

2a=20 μmSiO2

μ

D d

Θ

Output aperture of x-ray source

Importance of the wave phenomena at

Fig. 29 “Focusing” x-ray femtosecond pulses by a tapered capillary on a nanometer scale. The phenomenon was shown by S V Kukhlevsky et al ˝Interference and diffraction in capillary x-ray optics˝

ZΘ

Lcap. > ab /λ (98)


phenomenon was shown by S.V. Kukhlevsky et al., Interference and diffraction in capillary x-ray optics , X-Ray Spectr. 32: 223-228 (2003)

Free-space virtual source of a system of th “f f i ” fthe “fs-nm focusing” of x-rays

X

Fig. 30 A free-space virtual source of a system of the fs-nm focusing” of x-rays (from S.V. Kukhlevsky and G. N it Ph L t A 291 459 (2001))


Nyitray, Phys. Let. A, 291, 459 (2001))

The model shows feasibility of the “fs-nm focusing” of x-raysfocusing of x-rays

(a) (b)

The feasibility of “focusing” of x-rays

Fig 31 The model results demonstrate feasibility of the “fs-nm focusing”

(c)y g y

on the 20 fs x 20 nm scale is useful result for X-ray FELs


Fig. 31 The model results demonstrate feasibility of the fs nm focusing of x-rays with λ=10 nm. The phenomenon was shown by S.V. Kukhlevsky et al. in Phys. Let. A, 291, 459 (2001)

Kumakhof’s polycapillary “lens”

Guiding and focusing by Kumakhov’s “lens”

X-ray source

Image

P1’

p y p yComputations

P1

P2’~10μm

P2

Fig. 32 Computation of the focusing of incoherent and coherent x-rays by a polycapillary lens (from S.V. Kukhlevsky, “Focusing of incoherent and coherent X-ray radiation by a polycapillary lens”, NURT-2006, y g y y y yHavana, Cuba, April 3-7, 2006)

Photo from M A Kumakhov(a)

Coherent x-ray sources: Synchrotrons of 3rd and 4th generation; FEL; Plasma-based lasers; HHG;

Photo from M.A. Kumakhov

Fig. 33 (a) An original Kumakhof’s polycapillary ‘lens’. (b) A modern Kumakhov’s optics (from http://www.xos.com)

(a) (b)

y y g ; ; ; ;Point-like laboratory and astrophysical plasma sources


Focusing of incoherent and coherent x-rays by a Kumakhov’s “lens”: The model resultsa Kumakhov s lens : The model results

Non-coherent input x-rays: | μ12(P1, P2) | = 0 Eph = 0.1 keV,

Νph 20 000 000

Coherent input x-rays: | μ12(P1, P2) | = 1 Eph = 0.1 keV,

Νph = 20 000 000Νph = 20 000 000Ζ = 0.5 cm (focus) DFWHM~15μm

Νph = 20 000 000DFWHM~7μm , Ζ = 0.5 cm (focus)

μm

Fig. 34 The model results for the focusing of incoherent and partially coherent x-rays by a Kumakhof’s “lens”.The improvement was shown by S.V. Kukhlevsky et al. in S.V. Kukhlevsky, “Focusing of incoherent and

μmμm


p y y y, gcoherent X-ray radiation by a polycapillary lens”, NURT-2006, Havana, Cuba,April 3-7, 2006: In the cooperation (L. Vincze, S.V. Kukhlevsky, K. Janssens),SPIE Vol. 5536, 81-85 (2004))]

Guiding and focusing of x-rays by a gradient refraction index plasma based guiderefraction index plasma-based guide

D dOutput aperture of x-ray source

ZΘ

(99)

(100)

Fig 35 Guiding and focusing” of an x-ray beam by a gradient-index plasma

(100)(101)(102)


Fig. 35 Guiding and focusing of an x ray beam by a gradient index plasma guide (from S.V. Kukhlevsky, L. Kozma, Contributions to Plasma Physics 38: 583-597 (1998)

Summary of guiding and focusing of x-rays by a gradient refraction index plasma based guidegradient refraction index plasma-based guide

Within MHD approximation, WE yields (103)

The ray-tracing approximation of Eq. (103) yields

The paraxial-envelope approximation of Eq. (103) yields

(104)

(105)p p pp q ( ) y

From Eq. (104), we get the ray trajectories in paraxial approximation (small angles) with:

( )

(106)

(107)From Eq. (106), we see that the beam intensity distribution is periodically reproduced with period z : (108)

(Nc=π/reλ2):

is periodically reproduced with period zp: (108)

The plasma guide of length L supports transient mode m (m = 0,1,2, …) if

(109)

The plasma does guide x-rays if the value L/zp> 0.1, which yields condition (110)


(Transient modes in a plasma waveguide were introduced by S.V. Kukhlevskyet al. in Contr. Plasma Phys. 38: 583-597 (1998))

Guiding and focusing of x-rays by a plasma-based guide: transient modes m (m=0,1,2 …)based guide: transient modes m (m 0,1,2 …)

(a) (b)

(e)

(c)(c) (d) (f)


Fig. 31 Guiding and focusing x-rays by a plasma guide. Transient modes m. (from S.V. Kukhlevsky et al., Contr. Plasma Phys. 38: 583-597 (1998))

The use of Van Cittert-Zernike theorem for f d b l b d idx-rays focused by a plasma-based guide

2RdOutput aperture of an x-ray source

based on plasma guide Fig. 32 Schema for the use of theorem in case of an x-ray source based on plasma guide (2-Dimensional. circular, radius R) source composed from incoherent

2R

ZΘ

p g

(uncorrelated) emitters

Plasma guide produces partially coherent or fully transversally coherent x-rays (waves) inthe regionthe region.

For a circular (radius R) surface of incoherent (uncorrelated) emitters of a plasma-guide, we have

ΘΘ

=Rk

RkJ )(2 112μ (111)

ΘRk

Thus the use of the Van Cittert-Zernike theorem for an x-ray source based on plasma-guide yields

Transverse coherence

- Incoherent radiation: 2R >> <λ>Z / dPartially coherent radiation: 2R ~ <λ>Z / d

(112)(113)


- Partially coherent radiation: 2R ~ <λ>Z / d- Coherent radiation: 2R << <λ>Z / d

(113)(114)

Problems as home assignments (A)1. Show that the principle of superposition (Fourier-like decomposition) for EM waves does

not contradict wave equation.2. How does the principle of superposition for EM waves yield interference phenomenon?3. How does Fresnel-Kirchhoff diffraction integral relate to wave equation, Helmholtz

equation and the secondary spherical waves of the Huygens-Fresnel principle?4. Explain the weakest, “diffraction-less” and the strongest forms of diffractions.5. What are the relations between the mutual coherence factor, the degree of spatial

coherence (coherence factor, mutual intensity) and the normalized degree of spatialcoherence (coherence factor)?

6. Calculate the longitudinal coherence-length for the spontaneous transition determined bythe Einstein coefficient Aul = 1/Δτul = 1010 s-1.

7. For a point source having the intensity distribution I(ρ) = I0δ(ρ)/2πρ demonstrate that theradiation is fully coherent with |μ12|=1.

8. For an incoherent source having the Gauss intensity distribution I(ρ) = I0exp(-ρ2 / 2a2

demonstrate that the value |μ12|=0.88.9. Show the difference between applications of the van Cittert-Zernike theorem for passive

and active mediums.10.Use the van Cittert-Cernike theorem and the three forms of diffraction for the explanation

of the angle divergence of the incoherent x-ray source of Fig. 5 (Lecture 2) radiating thesoft X-rays with the wavelength λ << rT if the aperture dimension rT= 0.1 mm and Lap =0.5m

11.For the incoherent source of Fig. 5 (Lecture 2) radiating the soft x-rays with thewavelength λ = 50nm find the aperture dimension rT, when the incoherent source will betransversally coherent (let Lap = 0.5m). Explain the result by using the van Cittert-Cerniketh d th th f f diff ti


theorem and the three forms of diffraction.

Problems as home assignments (B)12. For the source of coherently accelerated electrons of Fig. 7 (Lecture 2) radiating the softx-rays with the wavelength λ = 50nm find the aperture dimension rT that provides thetransversally coherent x-rays (let L = 0 5m) Use the van Cittert-Cernike theorem and the onetransversally coherent x-rays (let L = 0.5m). Use the van Cittert-Cernike theorem and the oneof the three forms of diffraction.13. How are the x-rays (waves) modified by refraction, reflection and absorption?14. Show relations between the refraction index n and refraction, reflection and absorption.15 How do Maxwell’s equations and wave equation explain the refractive index (n)?15. How do Maxwell s equations and wave equation explain the refractive index (n)?16. What is the connection between the Lorentz model and the refraction index?17. Show the role of the atomic scattering factors f0 , f01 and f02.18. Explain the variation of the refractive index from IR to the x-ray spectral region.19 Explain the difference between total internal and external reflections19. Explain the difference between total internal and external reflections.20. Compare reflections of x-rays for S- and P-polarizations.21. What is the difference between the Kirkpatrick-Baez and Kumakhov glancing x-ray optic?22. Discuss the “focusing” x-ray femtosecond pulses by a tapered capillary on a nanometerscalescale.23. Explain the guiding and focusing of x-rays by a gradient refraction index plasma-basedguide .


References1. J. W. Goodman, Statistical physics, Wiley, New York (1985)2. David Attwood, Soft-X-rays and Extreme Ultraviolet Radiation, Cambridge University

Press, 2000.3 E Náhring “Die totalreflexion der Röntgenstrahlen” Physik Zeistr XXXI 799 (Sept3. E. Náhring, Die totalreflexion der Röntgenstrahlen , Physik. Zeistr. XXXI, 799 (Sept.

1930).4. S.V. Kukhlevsky, “Focusing of incoherent and coherent X-ray radiation by a

polycapillary lens”, NURT-2006, Havana, Cuba, April 3-7, 2006.

For additional information see: 1. R. C. Elton, X-ray lasers, Academic Press, 1990.2. David Attwood, Soft-X-rays and Extreme Ultraviolet Radiation, Cambridge University

Press 2000; David Attwood Soft-X-rays and Extreme Ultraviolet Radiation (www coePress, 2000; David Attwood, Soft X rays and Extreme Ultraviolet Radiation (www.coe. berkeley.edu).

3. J.J. Rocca, Review article. Table-top soft x-ray lasers, Rev. Sci. Instr. 70, 3799 (1999)4. H. Daido, Review of soft x-ray laser researches and developments, Rep. Prog. Phys.

65 1513 (2002)65, 1513 (2002) 5. A.V. Vinogradov, J.J. Rocca, Repetitively pulsed X-ray laser operating on the 3p-3s

transition of the Ne-like argon in a capillary discharge, Kvant. Electron., 33, 7 (2003)6. S. Suckewer, P. Jaegle, X-Ray laser: past, present, and future, Laser Phys. Lett. 6,

411 (2009)411 (2009).


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