Modern Developments in X-Ray and Neutron Optics Episode 13 doc

Figure 28.10 shows the experimental a and calculated b curves of the diffracted X-ray radiation intensity, I, as a function of the detector scanning angle, ΔΘd, obtained at the X-ray inci

orders However, this method is valid only for a lamellar grating It is notapplicable, for example, for a sawtooth profile grating In the present chapter,the differential technique, developed in [21], is used for multilayer merid-ional Bragg–Fresnel grating calculations These calculations are carried outlayer-by-layer, making the method suitable for any kind of profile Lamel-lar gratings have been studied by several authors both theoretically andexperimentally [22–25].

The nature of diffraction on a three-dimensional grating/multilayer ture strongly depends on the optical properties of the materials and the

struc-characteristic size of a grating period with the ‘lattice parameter’ a1 and

a multilayer period with the ‘lattice parameter’ a2 As a consequence, two

different approximations can be used for different limiting conditions

Multilayer Etched Meridional Grating

In the first approximation, the ‘double dispersion’ phenomena of multilayergratings can be described as a combination of Bragg diffraction on reflectinglayers and surface diffraction on a planar grating In this simplest case, Braggdiffraction limits the output energy and the angular spectra of the reflectedbeam, and the planar grating produces an additional angular dispersion Themeasurements of meridional gratings with a ‘large’ period, which exceeds cri-teria described later, show a simple combination of grating and multilayer, as

if they were used separately, one after other, similar to a surface grating Thedetector scan spectrum at the fixed Bragg angle, ΘB, shows several diffractionpeaks from the surface grating inside the broad Bragg peak of the multilayermirror The property of the ‘short’ period, etched volume grating is not thesimple ‘overlapping’ of the two independent structures Instead, one must refer

to the theory of crystal diffraction The characteristics of a volume gratingcan be demonstrated by the dependence of the absolute efficiency of the +1order on the depth of the grating profile with a lamellar grating period beingtaken as the variable parameter These curves are shown in Fig 28.5 For

Fig 28.5 Maximum efficiency of the +1 diffraction order vs the number of etched

periods The W/Si multilayer mirror is performed with 100 bilayers with a period

The value of the ‘resonance phase’ depends on the mean value of the

refrac-tive index of a multilayer structure, δ, and corresponds to the extinction depth

in a multilayer Properties of such gratings are the same as for conventionalreflection phase gratings, except for the Bragg selectivity For short periodgratings, i.e with lateral periods of less than 10μm, the behavior is different.The aforementioned W/Si multilayer with the spacing of 3 nm is an example.According to the differential model the ‘phase peak’ of efficiency is shifted intothe depth of the multilayer, and the +1 order intensity continuously increaseswith the increasing depth of the grating profile (Fig 28.5) This phenomenoncannot be explained without involving volume diffraction effects

Let us describe a multilayer grating as a two-dimensional crystal with two

different translation vectors, a1 in the direction along a surface (X) and a2

in the depth of a multilayer (crystal) (Z) Such a macro-crystal has two main

crystallographic directions along the z and x axes The multilayer crystal

structure is shown schematically in Fig 28.6 As in a natural crystal withtwo different lattice parameters, one can define crystallographic directions

Fig 28.6 Schematic representation of a ‘multilayer crystal’

corresponding to different translation vectors of the lattice with an absolute

value of ha1 and ka2

Using crystallographic indexes one can define diffraction parameters d h,k

for a short period grating (a2 ∼ = a1) as

d h,k=  a1a2

For a long period grating (a2 << a1) an effective diffraction occurs onlybetween waves diffracted from the top and the bottom of the grooves A phaseshift between these waves depends only on the different optical paths in themultilayer groove and vacuum Multilayer mirrors act like a monochromaticreflector with the phase reflecting grating on the top The properties are the

same as for a sagittal grating with a period of d h,k ≈ a1 Looking at theefficiency dependence vs the depth of the etched profile (Fig 28.5), one cansee an increase in the absolute reflectivity up to 0.3, which corresponds to thephase maximum (π phase shift) between diffracted waves Diffraction ordersare located inside of a multilayer Bragg peak and cannot be observed withoutzero order diffraction

One can define these two limiting cases even more precisely taking intoaccount the extinction depth of a multilayer or crystal structure (28.11) Asalready mentioned, for a sagittal grating the depth of profile is optimal if it

is equal to the value of extinction depth, t z

ext Extending the definitions, one

can introduce an extinction depth value for the grating along the X direction,

which could be defined as

t xext≈ t zext

The volume properties of a meridional etched grating become essential if the

period of a lateral grating is less than t x For example, a W/Si multilayer

with a period of 3 nm and γ = 0.3 has an extinction depth on the order of

50 nm at 0.154 nm wavelength The corresponding ‘volume effect’ parameter

In this mode, the diffracted field was scanned with the detector slit at theangle Θd for each incident angle, Θi, in order to record the intensity of all

‘n’ orders.

The diffracted efficiency distribution in three dimensions vs incident Θi

as well as the diffracted 2Θi angle was measured and plotted As an example,Fig 28.7 represents the results of the grating measurements in the detectorscan mode for the 230 nm profile depth and 4μm grating period The angle

of incidence was in the range of 1.5 ◦ –1.65 ◦ For each incident angle, Θ

i, thediffracted field was scanned with the detector slit in the same range Usingsuch a method, diffraction orders−2, −1, 0, +1, +2 can easily be resolved

(see Fig 28.7)

As can be seen from this plot, the maximum intensity for the minus firstdiffraction order corresponds to the minimum of the zero order A similarresult was described by Neviere [26] for another type of multilayer grating,one coated on large period blaze echelette grating In that paper, the structure

Fig 28.7 Θi− Θd plot in the detector scan mode for the meridional multilayergrating with a profile depth of 230 nm

SAW (2,000–4,000 m s ) is much lower than the speed of the X-rays, theacoustic deformation can be considered as quasi-static and characterized byits wavelength and amplitude However, the use of an ultrasonic super-lattice

in the X-ray wavelength range has some limitations First, it is necessary toapply acoustic waves with a very short wavelength (Λ∼ 1–10 μm) in order to

produce a large angular dispersion between diffraction satellites [27–32] Thisrequirement is related to the large Bragg angles, ΘB, for the real piezoelectriccrystals such as quartz, LiNbO3, LiTaO3, La3Ga5SiO14, La3Ga5.5Ta0.5O14,which lie between 3◦ and 40◦ Therefore, it is attractive to use a multilayer

X-ray mirror under the Bragg angle on the order of 1◦ [33–36] or in a total

external reflection mode (αi ∼ 0.1 ◦ –0.3 ◦) [37, 38], where the SAW with a

wavelength of Λ = 10–40μm produces considerable angular dispersion withX-rays Total external reflection is interesting for two other reasons First, ahigh reflectivity, typically 90%, is possible Second is the high efficiency ofscattering by the surface acoustic waves, the amplitude of which is nearlycomparable to the depth of the penetration of the evanescent X-ray wave ofthe order of 10 nm

It is also possible to control both the wavelength and the amplitude of adynamic SAW grating by changing the amplitude of the input high-frequencyelectric signal and the excitation frequency These possibilities can be used

to optimize the space–time modulation based on X-ray diffraction by surfaceacoustic waves [39, 40]

28.3.1 The SAW Device

Figure 28.8a, b show the SAW device based on a piezoelectric crystal To excite

a Rayleigh SAW, an interdigital transducer (IDT) is deposited on the crystalsurface by photolithography or e-beam lithography An IDT transforms thehigh-frequency signal into acoustic oscillations of the crystal lattice, whichpropagate along the crystal surface The SAW amplitude on the crystal sur-face can be changed linearly from zero to several angstroms by varying theamplitude of the high-frequency electrical voltage supplied by a high-frequency

(a) (b)

Fig 28.8 (a) SAW device (b) SAW propagation in the YZ-cut of a LiNbO3crystal

Λ = 30μm

generator to the IDT Figure 28.8b presents the scanning electron microscopy

image of the SAW propagation in the YZ-cut of a LiNbO3 crystal with the

velocity of V = 3,488 m s −1 The SAW with wavelength Λ = 30μm was

excited at the resonance excitation frequency f = 116.3 MHz It is seen that

SAW behaves like a strongly periodic sinusoidal diffraction grating

SAW propagation causes a sinusoidal deformation of the crystal latticeand crystal surface in the first approximation A Rayleigh SAW is actuallyelliptically polarized, but in the case of a symmetric reflection geometry, in-plane displacements of the crystal lattice do not influence diffraction Thedeformation involved in the diffraction process can be written as

where K = 2π/Λ is the SAW wave vector and h0 is the SAW amplitude onthe crystal surface, which can be controlled by varying the input signal onthe IDT

28.3.2 Total External Reflection Mirror Modulated by SAW

The diffraction of light by ultrasound has been investigated theoretically[41, 42] and experimentally [43–45] Theoretical curves (see Figs 28.10–28.12)show excellent agreement with experimental results A detailed description ofthe diffraction theory on a surface grating can be found in [38]

Figure 28.9 depicts a double-crystal X-ray diffractometer used to study

X-ray diffraction on the surface of the YZ-cut of a LiNbO3 crystal ulated by surface acoustic waves under total external reflection An X-raytube with a rotating copper anode (Cu Kα radiation, λ = 0.154 nm, run-

mod-ning at 40 kV and 60 mA) was used as the source of X-ray radiation A planeX-ray wave behind a double Si(111) crystal-monochromator was collimated

by a 10μm slit For diffraction studies under total external reflection, thecrystal surface was treated by chemical dynamic polishing so that the rough-ness does not exceed 1 nm This treatment is very important because theroughness decreases the value of the critical angle An IDT with an 8μm fin-ger width that corresponds to a Λ = 32μm SAW was deposited on the surface

of the sample so that the SAW propagates along the Z axis with a velocity

V = 3.488 km s −1 The resonance frequency of the IDT was f0 = 109 MHz.

For the experiment described here the collimated plane X-ray wave falls onthe crystal surface modulated by the surface acoustic wave at the incidentangle Θi = 0.22 ◦, slightly below the experimentally measured critical angle of

the YZ-cut of a LiNbO3 crystal, αc = 0.30 ◦ The X-ray plane wave diffracts on

the ultrasonic superlattice so that the angular position of diffraction satellitescan be determined from the grating equation:

k cos Θ m = k cos Θi + mK, (28.16)

where k = 2π/λ, K = 2π/Λ and m is the diffraction order.

According to (28.16), the X-ray radiation is expected to diffract on thecrystal surface modulated by the SAW so that the angular divergence should

be 0.090 ◦ and 0.063 ◦ for m = 0 and m = +1 ( −1), respectively The diffracted

X-ray radiation is recorded by a scintillation detector behind a 10μm slit Inall results, the diffracted X-ray intensity was normalized to the intensity ofthe incident beam

Figure 28.10 shows the experimental (a) and calculated (b) curves of the

diffracted X-ray radiation intensity, I, as a function of the detector scanning

angle, ΔΘd, obtained at the X-ray incident angle ΔΘi = 0.22 ◦ The

reso-nance excitation frequency of the SAW was f0= 109 MHz and values of the

amplitude of the input sinusoidal signal on the IDT ranged from U = 2–17 V The sinusoidal amplitude of the SAW, h, is a linear function of the amplitude

of the input signal on the IDT In the calculated curves, h is assumed to be

between 0.2 and 1.7 nm In Fig 28.10, diffraction satellites are observed at theangles ΔΘ1 = 0.090 ◦ and ΔΘ−1 = 0.063 ◦ from the intense reflected beam.

These values are in a good agreement with those calculated from expression(28.15) for the −1 and +1 diffraction orders The maximum intensity of the

m = −1 diffraction order makes up 10.5% of the intensity of the incident X-ray beam for an amplitude of the input signal on the IDT U = 17 V The great difference in the diffraction order intensities (E −1 > E1) and angular

(a) (b)

Fig 28.10 Experimental (a) and calculated (b) diffracted X-ray intensity I as a

function of the detector scanning angle ΔΘd, obtained at the X-ray incident angle

Θi= 0.22 ◦ , resonance excitation frequency of the SAW, f0= 109 MHz and at different

amplitudes of the SAW: U = 2–17 V

Fig 28.11 Experimental (a) and calculated (b) diffracted X-ray radiation intensity,

I, as a function of the detector scanning angle, ΔΘd, obtained at the resonance

excitation frequency of the SAW f0 = 109 MHz, amplitude of the input signal on

the IDT U = 17 V and at different values of the incident angle Θi= 0.15 ◦ –0.37 ◦

divergences between the diffraction orders (ΔΘ−1 < ΔΘ1) is a consequence of

the small X-ray incident angle ΔΘi= 0.22 ◦ It is observed that the linewidth

is larger for the m = +1 peak, which is closer to the surface This is an effect

of the divergence of the incident beam and can be understood by calculating

dΘm /dΘi from (28.16)

Figure 28.11 shows the experimental (a) and calculated (b) diffracted

X-ray radiation intensity, I, as a function of the detector scanning angle, ΔΘd, obtained at the resonance excitation frequency of the SAW, f0 = 109 MHz,

with an amplitude of the input signal on the IDT, U = 17 V, and at

differ-ent values of the X-ray inciddiffer-ent angle, ΔΘi = 0.15 ◦ –0.37 ◦ In the calculated

dependence, the sinusoidal amplitude, h, is assumed to be 1.7 nm Figure 28.12

represents the experimental dependence and theoretical curves (full lines) of

the diffracted X-ray intensity, I, as a function of the incident angle, ΔΘi,

obtained at the resonance excitation frequency of the SAW, f0= 109 Hz, and

at an amplitude of the input signal on the IDT, U = 17 V These dependencies (Figs 28.11 and 28.12) demonstrate that the m = +1 ( −1) diffraction order

tude of the input signal on the IDT U = 17 V The full line shows the calculated

values

Fig 28.13 Formation of the reflecting pseudo-lattices

has a maximum intensity at an incident angle of the X-ray beam of Θi= 0.22 ◦

and 0.28 ◦, respectively.

28.3.3 Multilayer Mirror Modulated by SAW

The main restriction of the total external reflection technique is the generallylow efficiency of the diffraction satellites (around 20%) [38] By using an X-raymirror this efficiency can be increased

The next considerations help to predict which incident angle is likely tofavor a given diffraction order Because of the presence of the acoustic wave,

the incident angle on the surface varies between ω − ϕ and ω + ϕ (Fig 28.13).

Therefore, a strong Bragg reflection occurs if the incident angle, Θ, fulfills theinequality

ΘB− ϕ < ω < ΘB+ ϕ · · · (28.17)This situation is indeed possible, since at an incident angle corresponding

to (28.17), some parts of the acoustic wave form a new family of reflectingpseudo-planes, for which the Bragg condition is fulfilled (Fig 28.13)

The multilayer interference X-ray mirror modulated by SAW thus acts

as a diffraction grating, reflecting the maximum intensity in the direction

determined by the angle β (Fig 28.13):

The incident angle, ω, giving a maximum intensity in the mth diffraction

order, can be determined from (28.15)

Solving (28.18) and (28.15), we obtain

cos ω − cos(2ΘB− ω) = mλ/Λ (28.20)

or, for small incident angles,

which is in a good agreement with experimental results Note that (28.20)

is similar to the equation that gives the position of the peaks of maximumintensity on the rocking curve

The propagation of an X-ray wave in a multilayer interference X-ray mirrormodulated by surface acoustic waves can be investigated using the dynamicdiffraction theory in distorted crystals presented in [46–48] In this case, themultilayer acts as an artificial crystal The deformation field in the crystal

(as in the theory of elasticity) is described by the vector  u, representing the

displacement of the atoms from the equilibrium position in the perfect crystal.This displacement must satisfy some limitations, the same as in the case ofelastic wave propagation in a crystal The next expression can be used todescribe the polarizability of the distorted crystal [49–51]

Fig 28.14 Coordinate system for the calculation of the wave field in the multilayer

Fig 28.15 Coordinate systems for the determination of the boundary conditions

are the Fourier coefficients of the polarizability;  s0 and  s h are the unit

vec-tors along the refracted and diffracted waves (Fig 28.14), α = k2

h− k2&

k2;

and V is the volume of the unit cell It is necessary to take into account

some boundary conditions: the continuity of the wave on the surface gives(Fig 28.15):

where indices i,d and 0,h correspond to the incident and the diffracted waves

in the vacuum and in the crystal, respectively and  re is the radius-vector ofthe input surface In (28.25) it is also assumed that the incident angle is large

enough to neglect the reflection The following two equations are obtainedfrom (28.25): ⎧

inter-The calculations of the real diffraction pattern are carried out using themethod described in [38] Thus, we suppose that all the surface elements

on which the plane wave,  E d, falls act as secondary sources of sphericalwaves:

"

cos



 n ∧ , kd

Using (28.22) and taking into account that we investigate the diffracted

wave in the far field region (1/r ≈ const)

E hexp{−i [(k hx − k cos α) x + (k hz − k sin α) z]}dS, (28.28)

where β is the diffraction angle, k = 2π/λ, z = h sin Kx the amplitude of the modulated surface, h the SAW amplitude, K = 2π/Λ, S the modulated

surface The exponential term describes the phase shift due to the refraction

on the crystal–vacuum interface

The following equation is used to calculate the diffracted X-ray intensity

I = E(p)E ∗ (p) · · · (28.29)

In the calculated intensities, the sinusoidal amplitude, h, of the SAW is

assumed to be 1.4 nm Theoretical curves (see Figs 28.18 and 28.19) showgood agreement with experiments

The multilayer mirror was deposited on the YZ-cut of a LiNbO3 crystaltreated initially by chemical dynamic polishing to decrease the roughness to0.5 nm The W/C multilayer was produced by magnetron sputtering It ismade of 60 bilayers of 5.3 nm each The Bragg angle of this multilayer is

ΘB= 0.83 ◦.

An IDT for the SAW excitation with a 4μm finger width corresponding

to Λ = 16μm SAW wavelength was also deposited on the free surface of theLiNbO3 crystal The SAW propagates along the Z axis with a velocity V = 3.488 km s −1 The resonance excitation frequency of the IDT is f0= 218 MHz.

Fig 28.16 Diffracted intensity as a function of the detector scanning angle ΔΘdobtained at the Bragg incident angle for different amplitudes of the input signal onthe IDT

The diffracted X-ray intensity is recorded by a scintillation detector with

a 10μm input slit

Figure 28.16 shows the experimental curves of the diffracted X-ray

radi-ation intensity, I, as a function of the detector scanning angle, ΔΘ, where

ΔΘB = 0.823 ◦ and for different values of the amplitude of the input

sinu-soidal signal on the IDT in the range of 1–17 V The diffraction satellites areobserved at the angular deviations ΔΘ+1= 0.039 ◦ and ΔΘ−1 = 0.038 ◦ from

the intense reflected beam These values can be precisely predicted from thegrating equation

The intensities of the m = +1, −1 satellites increase and the zero order

satellite decrease with the SAW amplitude The maximum intensities of thesediffraction orders make up 7% of the intensity of the diffracted beam with-out SAW excitation The difference in the angular deviations between thediffraction orders ΔΘ+1> ΔΘ −1 is in agreement with (28.10).

Figure 28.17 presents rocking curves obtained with the detector placed inthe Bragg position ΔΘB = 0.823 ◦, and for different amplitudes of the input

signal on the IDT The satellite intensities increase rapidly with the SAW

amplitude while the zero order peak decreases In the case of U = 17 V, the intensities of the +1, −1 satellites become higher than the zero order intensity.

At this voltage, the intensity in the +1 and −1 satellites reaches 68% of the

Bragg peak without SAW excitation

Figure 28.18 shows the (a) experimental measurements and (b) tions based on the model developed below for the diffracted X-ray intensity,

calcula-E, as a function of the detector scanning angle, ΔΘd, for different incident

Fig 28.17 Rocking curves with the detector in the Bragg position for different

amplitudes of the input signal on the IDT

Fig 28.18 Experimental (a) and calculated (b) diffracted intensity as a function

of the detector scanning angle ΔΘ obtained for U = 17 V (h = 1.4 nm) and for various incident angles Θ between 0.792 ◦ and 0.872 ◦

angles The m = ±1 diffraction orders reach a maximum intensity ∼58% for

an incident angle of Θi = 0.808 ◦ and 0.848 ◦, respectively.

The same phenomena take place for the±2 order satellites: the maximum

intensity ∼22% was obtained for an incident angles Θi = 0.792 ◦ and 0.872 ◦.

This means that, to obtain the maximum energy diffracted towards a precisesatellite, it is necessary to lightly shift the incident angle away from the exactBragg angle of the multilayer

In Fig 28.19 are shown the experimental and calculated maximal ties of the±1 and 0 order peaks for various incident angles, Θi The maximumintensity in the diffraction orders corresponds to the minimum intensity of thezero order, as was demonstrated previously for the volume gratings, etched in

intensi-a multilintensi-ayer mirror (Fig 28.7)

Fig 28.19 Maximum diffracted intensity of the +1, 0, −1 peaks as a function of

incident angle Θi, f0= 218 MHz, U = 17 V The solid line shows calculated values

28.3.4 Crystals Modulated by SAW

In contrast to acoustically modulated multilayer mirrors, a crystal, modulated

by SAW, acts much more effectively because, for the crystal, the value ofthe SAW amplitude can exceed the interplanar spacing The same diffractionefficiency can be obtained with much lower acoustic amplitude than requiredfor an X-ray mirror In this section the X-ray diffraction by langasite (LGS)crystal, (La3Ga5SiO14), excited by SAW is presented

Figure 28.20 shows the calculated amplitude of the crystal lattice placements in LGS caused by SAW propagation vs crystal depth Thecalculation [55] is based on the elastic and piezoelectric properties of theLGS [56] It is seen that the SAW penetration depth inside the crystal is

dis-approximately one SAW wavelength (see component u1normal to the crystal

surface) The longitudinal component, u2, is parallel to the direction of the

SAW propagation The presence of the transverse displacement component,

u3, suggests that the propagation direction of the acoustic energy flow doesnot coincide with the SAW wave vector direction

X-ray diffraction on acoustically modulated atomic planes gives rise todiffraction satellites on both sides of the Bragg peak In case of symmet-ric Bragg reflections, the angular position of diffraction satellites can bedetermined from the grating (28.15)

The angle between adjacent satellites measured on a rocking curve can bededuced from (28.15)

δΘ mRC = mλ/2Λ sin ΘB = md/Λ, (28.30)

where d is the interplanar spacing.

Fig 28.20 Calculated SAW amplitudes vs crystal depth: normal (u1), longitudinal

(u2) and transverse components (u3)

LGS is a piezoelectric crystal of space group symmetry 32 The crystal

lattice is similar to that of quartz with the parameters a = 0.817 nm and

c = 0.5095 nm [57].

An X-cut, (110) atomic planes parallel to the crystal surface, of LGS wasused for this experiment To excite a Rayleigh SAW, IDT was deposited onthe crystal surface by photolithography At the resonance excitation frequency,

f0 = 192.5 MHz, the SAW wavelength was λ = 12μm and the propagation

velocity was V = 2,310 ms −1.

Rocking curves were measured at various SAW amplitudes The X-rayenergy was 11 keV The interplanar spacing for the (110) reflection in LGS is

d = 0.4087 nm In the kinematic approximation the X-ray penetration depth

inside the crystal depends on the absorption in LGS as a function of energygiven by

μ −1

where μl is the linear absorption coefficient and ΘB is the Bragg incident

angle This dependence is shown in Fig 28.21 The K-edge of Ga at 10.47 keV

causes a drastic change in the absorption coefficient At the energy of 11 keV,

the X-ray penetration depth reaches only μ −1

z = 0.48μm, which is much less

than the SAW penetration depth inside the crystal (μ −1

z /μ −1

SAW< 1).

Figure 28.22 shows selected rocking curves for LGS (reflection (110))excited by a Λ = 12μm SAW at 11 keV measured at various input volt-

ages (U ) supplied to the IDT The Bragg incident angle is ΘB = 7.92 ◦ The

FWHM of the Bragg peak without SAW excitation is 3.2 arcsec (Fig 28.22a).Figure 28.22 shows that the number of diffraction satellites observed onthe rocking curve increases with the amplitude of the input signal supplied tothe IDT, i.e with the SAW amplitude

(a) (b)

Fig 28.22 Rocking curves measured for different amplitudes of the input signal

supplied to the IDT: (a) U = 0 V, (b) U = 8.5 V, (c) U = 14 V, (d) U = 18 V

Fig 28.23 Intensities of the diffraction satellites (m = 0, 1, 2, 3) vs amplitude of

the input signal supplied to the IDT Black circles, squares, triangles and diamonds:

experimental data Solid lines: calculated data E = 11 keV; Λ = 12μm; (110)reflection

The angular divergence between two neighboring diffraction satellites,

δΘ mRC, is 6.8 arcsec, which agrees quite well with the value 6.9 arcseccalculated from (28.30)

Intensities of selected diffraction satellites (m = 0, 1, 2, 3) as a function of

the input voltage on the IDT are shown in Fig 28.23 The intensity of the

diffraction satellites, except for satellite m = 0, develops as soon as the

acous-tic amplitude reaches a threshold value, which increases with the diffractionorder After rapidly reaching a maximum, the satellite intensity decreasessmoothly and oscillates

It can be seen (Fig 28.22b) that the intensity of the m = 0 diffraction satellite is equal to zero for U = 8.5 V For this specific SAW amplitude, the

phase shift of the X-ray radiation diffracting into the zero satellite from theSAW minima and maxima regions (where the atomic planes are still parallel

to the surface) is equal to π This phenomenon can be observed only if the

acoustic wave field probed by X-rays is very homogenous in amplitude This

is, therefore, only possible if the X-ray absorption is strong enough to avoidany interaction with deep regions of the crystal where the acoustic ampli-tude is strongly damped In the case of LiNbO3, the absorption is never highenough to achieve the complete extinction of a satellite except in the case of

an asymmetric reflection for which the incident angle can be very small [29]

For U = 14 V, the extinction of the m = +1( −1) diffraction satellites

is observed and can also be explained by the π-phase shift between crystal

regions diffracting towards this satellite (Fig 28.22c) The maximum value of

tensors of the langasite crystal and on the Rayleigh wave characteristics arecorrect These results are much better than those for the case of SAW propaga-tion in a LiNbO3crystal where kinematic simulations were useless especially

at low acoustic amplitudes and for 0 and 1 order satellites [29] This ference can be explained by the fact that for the X-ray penetration depth,being so small in a langasite crystal at 11 keV, the X-rays interact only withstrongly distorted regions of the crystal If this was not the case, dynami-cal theory should be necessary to take into account the contribution to thediffracted intensity coming from deep non-distorted (i.e perfect) regions ofthe crystal [31]

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