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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
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 2
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
rT (output aperture)
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)
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 4
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)
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 5
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))
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 9
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
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 10
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
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 11
(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
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 12
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 ?
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 13
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
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 14
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)
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 15
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)
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 16
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)
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 17
)()(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)
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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)
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 19
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
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 20
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
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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
rT (output aperture)
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
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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)
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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)
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 24
)(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)
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( ) ( ) ( ) "/'/ 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)
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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 ?
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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)
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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))
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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 ?
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 30
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
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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
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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)
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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))
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 35
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
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 36
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
37TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project
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
38TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project
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)
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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)
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(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)
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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)
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 42
- 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
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 43
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 .
TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 44
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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