Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region

Optics Communications North-Holland

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86 ( 199 1) 245-254

Full length article

Diffraction and scattering by lamellar gratings in the X-UV region A. Sammar,

amplitude

multilayer

J.-M. AndrC

Laboratoire de Chimie Physique, UniversitP Pierre et Marie Curie, CNRS URA 176, I I rue Pierre et Marie Curie, 75231 Paris Cedex 05, France

and B. Pardo Institut d’optique ThPorique et AppliquPe. UniversitP Paris Sud. Bcitiment 503. 91405 Orsay Cedex, France Received

4 April 199 1; revised manuscript

received

6 June 199 1

A dynamical theory is given to calculate the diffraction effkiencies of lamellar amplitude multilayer gratings (LAMGs). The model is applied to the LAMGs recently realized for the X-UV region. The unique diffraction properties of these devices are discussed. Roughness-induced scattering effects resulting from the pattern in relief of the grating are predicted and evaluated.

1. Introduction The combination of the microfabrication technology developed for highly integrated electronic chips with the recent progress in this film deposition makes it possible to realize a new class of optical devices synthetized for the X-UV region. These optics are referred to as combined microstructures when diffracting properties are mainly concerned [ 1 ] or as Bragg-Fresnel optics when focusing properties are involved [ 21. Until now, in the field of combined microstructures, lamellar gratings coated with multilayer coating [ 1,3-5 ] or etched into multilayer structure [ 6-81 have received a large interest. Nevertheless poor attention has been paid to modelize these optics in the framework of a dynamical theory though pioneering works in the field of multilayer gratings of various types were carried out using numerical methods [ 91. The diffraction by a lamellar homogeneous grating was investigated in the framework of the Electromagnetic theory by several authors [ lo-121 and their analysis was extended to treat the problem of diffraction by gratings of arbitrary shape [ 13,141. The purpose of the paper is to specialize this model to the particular case of lamellar amplitude multilayer gratings (LAMGs) and to analyze with the help of this approach the behavior of the LAMGs in the X-UV region.

2. Dynamical theory The principle of the calculation is based on the extension of the matricial formalism previously used in describing the optics of stratified homogeneous media [ 15,161 to optics of a pile of slabs, where the complex dielectric constants vary laterally in a periodic manner. Fig. 1 gives the geometry and the relevant notations. The multilayer structure consists of an alternating arrangement of a material denoted a and a material denoted b. Each slabj is homogeneous in depth but all of them have a complex dielectric constant t, varying along the x-axis with the spatial frequency g= 2x/D, according to 0030-4018/9

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B.V. All rights reserved.

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j+l

1

2 1

Fig.

t,(x)=t,(x+pD) and within

,

I. Schematic of a LAMG and geometry.

(1)

pan integer,

a period

c,(x) =&H(X)

H(TD-x)

+ct,,H(x-TD)

H(D-x)

)

(2)

where H(x) is the Heaviside function, t, and E,, are the dielectric constants in material a and in the space between two zones of material a, respectively. A relation similar to eq. (2) applies to the layers made up with material b. Let us note that the medium where the incident wave propagates (vacuum) and the substrate can be regarded as periodic media with any period and particularly with the period D. Consequently the whole system has the spatial frequency g and it will be possible to apply the Floquet theorem to the field equations everywhere in the system. A plane and monochromatic wave is incident at an angle 0, with respect to the x-axis on a stack 246

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of slabs unbounded in the x and y directions. The incident wavevector k,, is assumed to be normal to the yaxis. The (x, z) plane is the plane of incidence. From Maxwell’s equations written in the Gauss system, one can derive the following equations for the electric field E and magnetic field H in each slab j, assuming that the materials are non magnetic VIE, - V(P;E,)

= -t,(x)

k; E, ,

k;H,,

V’H,-P,O(V@H,)=-e,(x)

(3)

where P,=Vt,/e,. To treat the problem in its generality, it is sufficient to consider two polarization cases: (i) the electric field is normal to the plane of incidence (TE mode), (ii) the magnetic field is normal to the plane of incidence (TM mode). In these two cases the problem is simplified because there is only one component of the electric field or magnetic field in the Cartesian reference frame considered, according to whether the mode is TE or TM respectively. Both modes can be treated with the same formalism so that we restrict the presentation to the TE case using the following notations .&.= U(x, z) ,

H.,. = V(x, z) ,

Hz= W(x,z)

with H,=E,=E,=O.

(4)

Let us note that the treatment of the TM case involves the introduction of the dielectric constant instead of the unity in some parts of the wave equation [ 121. It is well-known that in the spectral range of interest, the refractive index generally is slightly less than unity; consequently in the region of small glancing angles (less than 25 degrees) where our discussion will be focussed, the effects of polarization are small. At larger glancing angle (around 45 degrees) effects similar to Brewster’s extinction can take place even in the X-UV regime and the two polarization cases must be treated; the phenomenom was used to realize X-UV polarizers by means of multilayer mirrors [ 171. Using the notations given by (4), the problem is then restricted to the scalar equation V’U,(x, Z) +t,(_u) k: U,(x, z) =O . We expand the solutions exp( *ik,,;.z)

(5)

of this equation

on the basis of the functions

P,S(x)

(6)

namely 1 Ty exp( +ik,,,,z) n

Bzn(x)

+

C

n

R; exp( -ik,j:.z)

BJ~”(x)

.

(7)

In the following, we refer to T” and R” as the excitations. The wavenumber kl,_ is the normal component of the nth wavevector in the jth slab and it has to be determined. By inserting eq. (6) in eq. (5), we obtain (d’/dx’)

B,(X)+hj(X)

B,(X)=0

9

with hj(X)

EC,(X)

kz-k$j.

(8)

Since h,(x) is a periodic function with period D, eq. (8) is a Hill-type According to the Floquet theorem, the functions B, can be written B,:,(X) =exp(iKx)

equation.

b/;,(X) ,

(9)

where K is merely equal (modulo g) to the parallel component 6,:n(x) are periodic and can be expanded in a Fourier series bJ,,(x) = C b,:N,pexp(iPgx) P

kilo of the incident

wavenumber.

.

The problem now is to determine the functions we consider a layer j made up with the material

The functions

(10) b,;,(x)

and more precisely the coefficients a (and vacuum), and we define

b,i,.o. To do that

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h,.,,(,y)

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=xl.,,(~~-)

ifO
=Xab:n(~~)

ifrD
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(11)

We define &,(.x) in a similar way for a layer with material b. Using eq. (9) and eq. (8 ) it comes that XaLn(x) fulfills the following (d’/dx’)X,,,,(x)+ZiK(d/dx)X,:,(x)+(E,k:,-K’-k:,;,) and XaV,n(x) fulfills a similar (d2/dx’)

1991

differential

equation

X,:,(x)=0,

(12)

equation

x,,:,(x)+2iK(dldx)X,,;,(x)+(t,,k~-K’-k:,;,)X,,,,(x)=O,

(13)

k 1a.n is the perpendicular component of the wavevector in the layer j made up with the material a. Similar equations operate in a layer with material b so that in the following we consider only equations for layers with material a. The solutions of eq. ( 12) and eq. ( 13) are respectively X,:,,(x) =A&, exp(kLx)

+A;,

(14)

,

exp(k;,x)

and JLn(x)

=A,+,:, cxp(kLx)

+A;;,

exp(k&,,x)

(15)

,

where k& = -iK? (-e,kg+k:,:,)“*. We have the same relations with the subscript av instead of a for k& and E,. We apply the continuity relations for the field and its first derivative at the point x=TD, AC, exp(k&J’D)

+A;,

exp(k&JD)

=A,+,:, exp(k&J’D)

+A&,

exp(k&JD)

then (16)

,

and Ai,k&exp(k&I’D)+A;,k;,exp(k;,I’D)=A+.

av.nk+. av,nexp(k&J’D)+A-.

Moreover

since the field and its first derivative

are periodic,

AL, +A;,,

=A,+,:, exp(kL,D)

av,nk-. a”,” exp(k,,,I’D).

(17)

we get (18)

+A&,, exp(k&,,D)

and AC,k,+, +A&k;,

=A&,k&,exp(kZ.,:,D)

Finally we obtain

an homogeneous

exp(kLJD) k&, exp(kZJD) 1

exp(kz,JD) k:, exp(k;JD) 1

k;,

G,

+A&,k;,,exp(k;,,D)

system of four equations -exp(kLJD) -k&, exp(k,f,JD) -exp(k&,D) -kh, ew(kL,D)

A,, 0 I[II

(19)

. which can be written

in the matricial

-exp(k&JD)

A&

-k&, exp(k;,JD) -exp(k,,,D) --kin

eWk,,D)

AA;,

A,,,

form

0

=

0

’

0

(20)

A necessary condition for the existence of a non trivial solution, it yields the dispersion relation which reads [-(~,+t,,)ki+2k:.;,]

sin[TD(e,ki-k:.;.)‘/*]

+2{cos[TD(e,k2-k:,,,)“2] x [ (t,,k~-k:,;,)“2(E,k~-kk:,;,)“2] 248

of the matrix vanishes;

sin[(l-I’)D(e,,k$-k:,;,)‘/2]

cos[ (1 -T)D(t,,k;-k:,;,)“2] =O .

is that the determinant

-cos(KD)} (21)

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This transcendental equation whose unknowns are kL,;,, can be solved by using a numerical method. The homogeneous system (20) is then solved by using standard algebra. We apply the Fourier transform to the analytic expression of b,,,(x) that we have found to obtain the Fourier coefficients b,;n,p We now face the problem of determining the excitations of the Bloch waves from the boundary conditions. These conditions stipulate that the tangential components of the fields are continuous. This leads to alJ/az =- ik,V. Let us consider an interface at the depth z,+ ,,, separating two neighboring slabs referred to as j and j+ 1. The boundary conditions give

Expanding

wwzi.+,,,=

)

v,(z,+,,,)=li,+,(z,+,,,)

(T:b;“.,,+R;bJTn.,)=

;

( TJ’ q,L

C n

-R;q,k)=

(22)

.

( 10) of b,:,(x), we obtain

U, and using eqs. (22) and the expansion

;

after some algebra

(T,=.,b,+,,;,,~+R;+,b,,...) . . ,

C (TJ”c,q:+,:n,p-R:+rq;+,:n.p) n

with q,n.p = blfm.gkll:n. This set of equations (23), F

[w+,mlz,+,,,

(24) can be regarded

(23) (24)

>

as the product

of vectors

V,= (Tjq R,) by a super matrix

[b,,l [ -G.pl > ’

= [b,Ll I ( [4/LJl

(25)

that is, F,++I?;+,*q+, It follows that a vector V,, , can be deduced from the former V, by means of a matrix generalizing matrix involved in the problem of homogeneous media

the Fresnel

P,;‘, .F,.

<+I,,=

To obtain the value of the field’U,( z,+ r,, ) at the top of the slab j from its value U,(z,,,_ , ) at the bottom has to operate with the propagator Gj G

/

=

[&.Pl

[Ol [~/%I > ’

LOI

with gJ?n.p = L+w(

one

* kij:nd,> ,

where S,,, denotes the Kronecker symbol and d, is the thickness of the jth slab. Finally the vector V,, in vacuum is related to the vector Vsub in the substrate through the matricial

relation

N+I V”,

=A?-

K”b

1

where A?=

n G, ‘Fj>j- 1 , ,=I

and N is the actual number of slabs. This relation is in fact a linear equation that connects the transmitted diffracted waves in the substrate, the reflected diffracted waves in vacuum and the incident plane wave. This equation must be solved assuming that there is no reflected wave in the substrate. The resolution allows us to calculate the amplitude rP, fP of the pth order reflected or transmitted waves by

r,= 1 R”b,,, n

to= C T”b;,. ,I 249

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Finally the grating is characterized by the diffraction efficiencies of the reflected orders defined as r,r; calculated at the vacuum level and the efficiencies of the transmitted orders defined as t,,tz calculated at the substrate level.

3. Numerical results and discussion Fig. 2 show the diffraction efficiency of the - 1, 0 and + 1 orders of the grating versus the glancing angle computed at 150 eV for a LAMG with the following characteristics: molybdenum/carbon multilayer with period d= 22.5 nm (thickness of MO layer= 4.05 nm, thickness of C layer= 18.45 nm), number of bilayers= 10, period of the grating D= 240 nm and ratio r of the multilayer bar to the grating period= 7/8. The glancing angles range from 6 to 25 degrees. A large number of peaks and structures appear and our purpose now is to identify the nature of these structures. The structures denoted by the symbol B( i, j) correspond to the diffraction of the incident wave at the order i of the multilayer and at the order j of the grating. The structures B( 1, - 1 ), B( 2, - 1 ) and B( 1, + 1) appear mainly as a doublet while B( 1,O) and B( 2,0) appears as a single peak. In a genera1 way no simple explanation, particularly in the framework of a kinematical theory, can be given to account for the manifold structures. The shape depends on various parameters as the thickness ratio of the materials within a bilayer, the nature of the materials (dielectric constants). The number of peaks can also be changed by modifying the number of materials in the multilayer unit cell: thus we found that for a threefold unit cell the number of peaks in the diffraction pattern increases. Indeed only the dynamical theory can precisely describe diffraction by a multilayer grating. Nevertheless a very important point can be understood by means of a kinematical approach: diffraction does not occur at the same position in terms of glancing angle for a given grating diffraction order, i.e. the mean glancing angle of B( i, j) depends on j. In passing it must be emphasized that consequently a LAMG does not work as a standard bulk grating as it has been evidenced by several works [ 18,191. To understand this fact, a LAMG must be regarded as a double periodic structure: there are an in-depth periodicity due to the multilayer mirror (period d) and a lateral periodicity due to the grating pattern (period D); consequently the diffraction of an electromagnetic wave fulfills the two scalar Laue conditions written in direct space M=sin(B)+sin(B;;T)

P=c0s(8)-c0s(~),

(26)

with m order of multilayer

diffraction

,

P=p & ,

p order of grating diffraction

where fi is the average refractive index in the grating medium and 0;, related to the angles BO,8 in vacuum by the Snell-Descartes law; 13is the The first condition eq. (26) given by Warburton [ 181 in a somewhat as the generalized Bragg condition for the multilayer structure and the well-known grating rule for the lateral grating. These two relations can for which diffraction by the whole system occurs sin(0&)=f{M*

[M2+2L(1-P’/L’)]“‘},

,

e the angles within the grating medium outgoing angle in the plane of incidence. different formulation can be considered second condition is nothing else but the be combined to yield the glancing angle

whereL=f(M’+P’),

and the positive sign for p positive and the negative sign for p negative. This formula can be viewed as the modified Bragg law for the LAMGs. It is easy to check numerically that the position of the peaks B( i, j) given by this relation is in close agreement with the position predicted by the dynamical model. Since the grating of interest was actually realized, an experimental characterisation was carried out at 150 eV [20] which has given positions of the diffraction peaks in fair agreement with both the kinematical approach and the dynamical model. It must be noted that experiments reveals only one peaks for 250

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-1

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0.00075~. 0.00050.00025-.

6

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6

glancing

angle

(deg)

25

0 order 0 order

(b)

04

6 6

glancing

angle

glancing

angle +l

+l

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25

(deg)

order

order 0. 008..

0.006..

0 . 0 l0.008.. 0. 00 6.. 0.004..

6 6

glancing

angle

(deg)

25

Fig. 2. Diffraction efficiencies versus glancing angle at 150 eV for a LAMG whose characteristics are given in text, computed by means of the dynamical theory; (a) - 1grating order, (b) 0 grating order, (c) + 1 grating order.

glancing

angle

(deg)

25

Fig. 3. Diffraction efficiencies versus glancing angle at 150 eV for a lamellar grating (without multilayer structure) with the same grating characteristics as the LAMG of fig. 2 computed by means of the dynamical theory; the grating medium is homogeneous and its dielectric constant has the value of the average dielectric constant of the multilayer structure of the LAMG; (a) - I grating order, (b) 0 grating order, (c) + 1 grating order.

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B( 1, - 1) and B( 1, + 1) because to so-called grating scan mode used by the authors does not follow exactly the grating rule. It would be necessary to check our theoretical predictions that accurate experiments be carried out using a process which obeys both conditions (26). The structures denoted by Yu) are related to the pattern in relief of the grating and remains when the indepth stratified structure disappears. This fact is illustrated by fig. 3 which give the variation of diffraction efficiences in the same conditions as fig. 2 but with the multilayer structure replaced by an homogeneous medium whose dielectric constant has the mean value of the corresponding stratified structure. The physical origin of these structures is a roughness-induced scattering (RIS); the roughness is formed by the relief pattern of the grating. The phenomenon presents a great similarity with the so-called Yoneda effect observed with surfaces offering a random roughness [21,22]. In our case the roughness is periodic. Though a detailed study of the phenomenon is beyond the scope of a paper devoted to multilayered grating, it is worth to compare our results with the state-of-art in the field of the RIS. Several authors have modelized the RIS for a semi-infinite homogeneous medium in the framework of a vector perturbation theory. Thus Maradudin and Mills [ 23 ] give for the cross-section of the scattering at the outgoing angle 13in the plane of incidence Z,,,,=(kz/n”)

sin(&)

sin2(8)

wQ(t,

I!?,, 19))

where II’ is the so-called power spectral density function (PSDF) of the surface roughness which can be regarded as the Fourier transform of the autocovariance function of the surface roughness, Q is a polarization dependent function of the angles and of the dielectric constant E of the medium. For a TE incident wave the term Q for scattering in the plane of incidence without change of polarization is ]e-112 Q

TE+rE=

(27)

]sin(8)+J~]2]sin(8,)+J~]2’

By expliciting of the grating

W in the case of interest

(lamellar

profile),

we find for the intensity

scattered

at the pth order

where h is the depth of the grooves. 0, is the outgoing angle given by the grating rule for a diffraction at the pth order and for a glancing angle 0,. In fig. 4 are plotted the diffraction efficiencies in the same conditions as fig. 3 computed by means of eq. (28). It is interesting to note that the shape of the curves are very similar but the absolute value of the efficiencies given by eq. (28) are nonsensical. This is due to the fact that the conditions required for the perturbative theory be valid, are not obeyed. Comparison between our exact model and the perturbative approach shows that the latter becomes valid provided that the condition k,h < 10 is approximately satisfied. Thus fig. 5 gives in comparison the diffraction efficiency of the + 1 grating order for the same homogeneous grating as in figs. 3 but with a groove depth equal to 2,=2x/ 10ko (0.82 nm), computed from the dynamical theory and the perturbative theory eq. (28). It is worth noting that the intensity of the structure Y( + 1) can be increased and its width decreased by a judicious choice of the ratio rand of the depth of the grooves h (see fig. 6 ).

4. Conclusion A rigorous dynamical theory is presented which accounts for the various scattering and diffraction effects occurring when a plane electromagnetic wave interacts with a LAMG. When applied to the X-UV region, the theory enables us to evaluate the diffraction efficiencies of the LAMGs recently developed for this spectral domain, and consequently to determine the optimal parameters for practical purposes. Among the various applications it can be mentioned filtering, beamsplitting, monochromatization. The RIS effects could also find 252

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6

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+l Order

25

0 order 0.0003~. 0.00025,. 0. 0002.. 0.00015~~

glancing angle

(deg)

(deg)

25

25 +l order

+l order

2-5 (c) 6

angle

glancing

6

6

I

glancing angle

(deg)

25

Fig. 4. Diffraction efftciencies versus glancing angle at 150 eV for the same grating (monogeneous medium) as in fig. 3, computed from a perturbative vector theory of RIS (eq. (46)). (a) -I grating order, (b) 0 grating order, (c) + I grating order.

6

glancing

angle

(deg)

25

Fig. 5. Diffraction efftciency versus glancing angle at 150 eV of the + I grating order for the same homogeneous grating as in fig. 4 but with a groove depth equal to A/IO (0.82 nm); (a) computed from the perturbative theory, (b) computed from the dynamical theory.

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order

0.1 1 0 . 0 8.. 0 . 0 6,. 0. 04.. 0. 02..

6

glancing

practical applications applications.

angle

(deg)

but a more sophisticated

Fig. 6. Diffraction efficiency versus glancing angle at I50 eV for the same laminar grating as in fig. 3 but with a ratio f equal to 1/ 12 (instead of 7/S) computed from the dynamical theory.

25

theoretical

study is required

to establish the conditions

of such

References [ I ] T.W. Barbee, Rev. Sci. Instr. 60 (1989) 1588. [2] A.I. Erko, J. X-ray Science and Techn. 2 ( 1990) 297. [3] T.W. Barbee, Proc. SPIE 91 I (1988). [4] Y. Utsumi, J. Takahashi and T. Urisu, Rev. Sci. Instr. 60 (1989) 2024. [S] R.G. Cruddace, T.W. Barbee, J.C. Rife and W.R. Hunder, Phys. Scripta 41 ( 1990) 396. [6] H. Berrouane, J.-M. Andre, C. Khan Malek, S. Fouchet, F.R. Ladan, R. Rivoira and R. Barchewitz, Proc. SPIE 1160 ( 1989). [ 71 H. Berrouane, C. Khan Malek, J.-M. Andre, D. Lesterlin, F.R. Landan, R. Rivoira, Y. Lepetre and R. Barchewitz, Proc. SPIE 1343 (1990). [ 81 H. Berrouane, J.-M. Andre, R. Barchewitz, C. Khan Malek and R. Rivoira, Optics Comm. 76 ( 1990) I Il. [ 91 B. Vidal. P. Vincent, P. Dhez and M. Neviere, Proc. SPIE 563 ( 1985). [lO]K.Knop,J.Opt.Soc.Am.68(1978) 1206. [ 1 I ] I.C. Botten, M.S. Craig, R.C. McPhedran, J.L. Adams and J.R. Andrewartha, Optica Acta 28 5 ( 1981) 413, 1102; I.C. Botten, M.S. Craig and R.C. McPhedran, Optica Acta 28 ( 198 I ) I 103. [ 121 Ping Sheng, R.S. Stepleman and P.N. Sanda, Phys. Rev. B 26 (1982) 2907. [ 131 J.Y. Suratteau, M. Cadilhac and R. Petit, J. Optics (Paris) 14 (1983) 273. [ 141 Ki-Tung Lee and T.F. George, Phys. Rev. B 31 (1985) 5106. [ 151 F. Abel&, Ann. Phys. Fr. 12 (1950) 596. [ 161 B. Pardo, T. Megademini and J.-M. Andre, Rev. Phys. Appliquee 23 ( 1988) 1579. [ 171 A. Khandar, these de troisieme cycle, Universite Paris Sud ( 1987). [ 181 W.K. Warburton. Nucl. Instr. Meth. A 29 1 ( 1990) 278. [ 191 H. Berrouane, J.-M. Andre, R. Barchewitz, T. Moreno, A. Sammar, Khan Malek, B. Pardo and R. Rivoira, Experimental and theoretical performances of an etched multilayer grating in the I keV region, submitted to Nucl. Instr. and Methods (February 1991). [ 201 P. Troussel, D. Schirmann, J.-M. Dalmasso, H. Berrouane, R. Barchewitz and C. Khan Malek, Tests of 240 nm period multilayer mirror gratings in the X-UV low energy range on the Super-AC0 storage ring at the LURE, submitted to Rev. Sci. Intrum. ( 1991 ). [21]Y.Yoneda,Phys.Rev. 131 (1963)2010. [ 221 N. Alehyane, M. Arbaoui, R. Barchewitz, J.-M. Andre, F.E. Christensen, A. Hornstrup, J. Palmari, M. Rasigni, R. Rivoira and G. Rasigni, Appl. Optics 28 ( 1989) 1763. [23] A.A. Maradudin and D.L. Mills, Phys. Rev. B 11 ( 1975) 1392.

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