Gas-surface scattering effects in the surface electromagnetic wave spectra

Surface S o e n c e 2 6 9 / 2 7 0 (1992) 169-174 North-Holland

SLlrfaCe s c i e n c e

Gas-surface scattering effects in the surface electromagnetic wave spectra V.G. B o r d o ln~Utute of General P;'ysws, Russtan Academy of Sciences, 117942 Moscow, ?ussza Recewed 15 A u g u s t 1991, accepted for p u b h c a t m n 23 S e p t e m b e r 1991

A ~.heoret~cal mvesttgat~on of the g a s - s u r f a c e scattering mamtestat~on m surface electromagnet,c wave ( S E W ) spectra ~s presented T h e S E W propagating along a g a s - s o h d interface is assumed to be in r e s o n a n c e w~th an atomic or molecular trans~tmn m the gas T h e time of passage of a n a t o m or molecule t h r o u g h t h e S E W held ts supposed to be comparable with that of relaxation m the gas volume. Two cases are c o n s i d e r e d 0 ) when the gas p a m c l e s fly through the S E W Ioeahzatlon area without colhslons with each o t h e r a n d (fi) when they pass this area m a dfffus:ve m a n n e r . In case (0 the spectral profiles of the S E W a b s o r p t m n coefhcmnt and the intensity o f the scattered h g h t are evaluated In case (u) the intensive S E W a b s o r p t m n saturahon is investigated. In both cases the S E W spectra are shown to b e markedly d e p e n d e n t on the parameters of the g a s - s o h d mteractmn

I. Introduction Surface electromagnetic waves (SEWs) are waves which propagate along the interfaces of media. T h e S E W field d e c r e a s e s rapidly with increasing distance from the interface making ~t highly sensitive to interface propertms and processes near the boundary. [1]. :n the c~se whore one of the media is a gas, o n e can expect g a s surface scattering effects in the S E W spectra. The stze of the SEW locahzation area is of the order of a wavelength. So the ~as molecule (or atom) zan pass the S E W propagation zone in a very short time. If th~s t~me is shorter than or comparable with that of the molecule dipole moment relaxation m the gas volume, the polarization of the gas ts determined to a great extent by the b o v n d a r y con&tlons at the surface. The ga~ polarization value near the surface, m t m n , is determined by the gas-molecule-surface scattermg process. Thus, the p a r a m e t e r s of gas-surface scattering can be displayed m the absorption and scattering S E W spectra. An analogous situatmn takes place in the case of selective reflection of

hght from the interface between a dieIecmc a~d an atomic vapor [2-6] The present ~ o r k deals wtth the theory oi SEW absorption and scattering at a gas-solid interface The S E W field ~s assumed to be m resonance ~lth the molccvlar or dtomlc trdn',lllol~ in the gas. G e n e r a l expressions for the SEW absorption coefficient and the spectral density of the scattered hght are gwen in section 2. In section 3 we evaluate the absorption and scattermg spectral lmes for the collisionless case. In section 4 we study the absorption saturatmn m the dfffuswe case C,mc[udmg remarks are gwcn in section 5.

2. SEW absorption and scattering in gas Consider the mtertace ot a ,,ohd occupying a half-space z < 0 and a gas occupying a half-space z > 0. Let the sohd be linear and lsotropm and have permittiv~ty ,o) =

I)1)39-6028/92/${)5 00 c, 1992 - Elsevl~.r Scmncc Publishers B V A'A right, reserved

+

V G Bordo / Gas-surface scattering effects m SEW spectra

170

Assume that in the frequency range considered the relation ['~(to)l >> ~i'(to) holds and the optical density of the gas is much less than unity. The condition of SEW existence at th,~ interface is as follows: ,~(to) ~.< - 1 . In the case where the solid is a metal, this condition holds for all frequencies less than t o v / v ~ , where top is the plasma frequency.

Z 1. Absorption coeffictent The SEW absorption coefficient, m view of the above assumptions, can be approximately presented as

q=

c

el(to ) + 1

where ao is the absorption coefficient at the solid-vacuum interface, ,~ is the gas contribution to SEW absorption. In carrying out the calculauon of at one can consider E~(to) as a real value. In turn, the coefficient a~ can be given by

f_ fl( z) dz where Q = g ' o ~ / o t is the power required for the wave field to periodically change the gas polarization ~ , S = (c/4~r) g' × , ~ is the Poynting vectt,,, the bar means time averaging over the field period. The direction of SEW propagation is taken along the x axis. The Intensities of the electric and magnetic fields, ~ and ~ , revolved in expression (1), correspond to the SEW propagating over the solid-vacuum interface and can bc given m the " - - [11 C

~,(r. t) = ~s × k - - E , exn~i~ r - ~oJt) -, c c . tO

"

T h e depths of field penetration into the contacting media are defined by the expressions K~

[

q2

=

to -

--

e,

,

ReK,>0.

C

The gas polarization in the steady-state limit can be given in the form

~ ( r , t) = ½P(z) e x p ( i k 2 e - itot) + c.c.. Making the necessary substitutions into eq. (1), we have [7] E1

a I = 87rqK2~ el+l

ff ~ 0~0 + ~: ~.

i

the hEW wave vector along the surface, q, is correlated with the frequency 0~ by the dispersion relation

1

Ez~

×J0 e x p ( - 2 K 2 z ) Im ~ ( z )

dz,

(2)

where E 2 =E2(c/to)s × k z. It should be noted that analogously to the case of total internal reflection at the iransparent dielectric-gas interface [8], the integral in expression (2) determines the gas contribution to the reflection coefficient m the Kretschmann geometry. The gas polarization can be expressed via the microscopic values. Suppose that the SEW fiequency is near that of the transition between the ground [I) and excited [2) states of the gas atom or molecule. Assume also that the interaction of the SEW field with the other states can be neglected so that the gas particle can be approximated by a two-level system. Then we may write

~ ( e, t) =N(p21(r , t)d12 ) + c.c., where N

is the n u m b e r density of the gas t~ t h o r n n l o t ' n l o cl~nr~ity m ; i t r i Y , and d is the molecule dipole moment operator. The angula* brackets ~mply avcragmg over the veloclties v and orientations of the gas particles. mnl~rl,l~,~

:¢;(r, t) = ~se,,.. ' ~ . x p ( l k r - iwt) + c.c., v~here i = 1 for the solid, t = 2 for the gas, k a= (q, 0, --IK~), k z = ( q , O , iK2), e 2 = 1, and s is thc unit vector along the 3' ax~s The componcnt of

2.2. Scattenng spectrum The gas molecule driven by the SEW field has an o~cillating dipole moment and, hence, can

V G Bordo / Ga~-surface scattermg effe~t~ tn SEW ,tmctra

emit radiatxon. In the wave zone the scattered field is given by 60 2

~(R,

cZR3[d(t)×R]×R.

t) =

(Striclty speaking, the field reflected by the surface also contributes :) 8"~ [9], but we will assume that the reflection can be neglected.) Let us introduce the raising and lowe;ing operators a + - 12)(11,

a--11)(21.

Then the spectral density of the scattered light is determined by the Fouri. r transform of the scattered field correlation function: [10]

l ( v , R, t ) = ldp(R)l

_ , g ( t ' t + r ) e-'"" dr'

where

and 3' : are the longitudinal and transverse relaxation rates in the gas volume We will confine the discussion to the case of a weak SE'~, held. where the transitmn saturation ts not essential. The equations of motion can be solved by means of a perturbation theory with ~ as a small parameter. The b o u n d a r y conditions for the solutions of the equations must be determined separately for particles moving to the surface (with c_ < 0) and .away from the surface (with c: > 0). '| nc corresponding solutions will be marked by - and + superscripts. Let us consider first order perturbation theory. Requiring for v_ < 0 vanishing of the off-diagonal matrix elements at infinite distance from the surface we get m

0) 2 c2R3(dl2XR

if2

°'21(v) = 2 F ( v )

g ( t , t') -- ( a + ( t ) a - ( t ' ) ) , dp(R) =

171

) XR.

2.3. Equatton of molton in the SEW]reid Now we n e e d a density matrix to evaluate the absorption and scattenng spectra. In the resonant approximation for the steady-state limit we can write P21( r, v, t ) = o"21( z, v) exp[i( k2r - tot - ~ )],

o,,(r,v, t) where the matrix o" obeys the optical Bloch equations with substitution ~/Ot--,v,O/Oz and the Rabi frequency O l z ) = g 2 e x p ( - K z z ) . We will investigate two qualitatively different cases: (i) the ,,,as particles fly through the S E W propagation zone without mutual collisions; (ii) the gas particles pass the S E W p r o p a g a u o n zone an a diffusive meaner.

where F(v) - 3' + i ( a + k : ' ) , and .1 = w2~ - o) is the detunlng. The boundary conditions for t . > 0 are specified at z = 0 and are determined by the character of the gas-molecule-surface scattering. The most simple and at the same t~me rather reahsttc assumptton concerning the scattering is the followmg The gas molecule approaching the surface has a defimte p r o b a h h t y r/ to bc ,,c:ttcrcd m a specular direction The quantum state and dipole moment of the molecule are unchanged, but the phase of its state can be changed [IlL On the other hand, the molecule can be adsorbed at the surface with probablhty 1 - rl ~stlcking probabdity). The d e s o r b e d molecules are in the ground state and ha,,o a Maxwelhan velocity distribution. Their polarization .s fully quenched. In accordance with these assumptions the ,,,~lu¢~on for ~r~ can be written in the form

~ r J l ( z ' v J = 2t" + ~r~l(0, v ) - - ~

i. I

exp - - - 2 t

t3)

3. Case (D: coL,sionless problem - spectral lines In this case we assume the relaxation to be purely r a d k ..ve, so that 3'11= 2T ± - 23', where 3't~

where o-+l(0, v) = cry(0, v ) exp(iAaS) for specular scattering and o-+t(0, v) = 0 for diffuse scattering. Here v, = (v,, c,, - t ' , ) , Ado = d)2 - 05l, d)t and

V G Bordo / Gas-surface scattenng effects m SEW spectra

172

the scattering phases of the molecule m the ground and excited states, respectively. In the second order of the perturbation theory we will confine ourselves to considering the excited state population. The solutions for cr~ can be given by

~2

are

o'2~(z, v ) = e x p [ - 2 T z t \

z

to'~(z±,

0.~

04

i

03

ot )

]

" f oxoI ('- )z,1

+ -V: "% +

1"

--K 2

O2

[ ~ Vz

× I m o'2T(z + ' , v) d z ' },

(4)

O

where z _ = o0, z + = 0, o-~(oo) = 0, o-~(0, v) = try,(0 v~) for specular scattering and a~2(0, v) = 0 for diffuse scattering. It should be n o t e d that the terms including the boundary, conditions in eqs. (3) and (4) are essential only if the molecule time-of-flight through the

~-,

I

"

-4

-2

0

2

4

6

8

10

(~ -*a)/¢ol) Fig. 2 The SEW scattering spectra hi arbltrarv units. The case of the forward scattering (kllq); K 2 = o / c , # = 0 , za= ~'*t) (1) -q=0, (2) 77=1 k and v are the wave vector and frequency of the scattered photon

06r

S E W localization area is less than or comparable with that of relaxation m the gas volume. 'The scattering phases 4', can be expressed via the adsorption po.~entials U, in the corresponding state. When the time of m o l e c u l e - s u r f a c e interaction is much greater than the reverse transition frequency to 21 -I the difference b e t w e e n phases can be found as [12]

'\,

j/)t

04

02

L

O~

i. 1

fz [ui(z)- u2(z)] =/3--', Uz w h e r e z I is the turning point for the m o t i o n in the potential U 1, v T is the thermal velocity, of a "3 " 2.

w-

I

I

-2

0

2

Fig 1 T h e S E W a b s o r p h o n c o e f f i c m n t a 1 in arbttrary u m t s for K2 = oa/c a n d for different v a l u e s o f r; a n d /3 (1) ~ = 0,

~ =o, (2~ n = l, fl=O, (3) n = l, fl= l

~ , ' ~ I ~,-.,, ! ~.

T h e evaluated spectrum a~(w) is asymmetrical with respect to the frequency a~21 if the specularly scattered molecules are present (7 ¢ 0 ) . The asymmetry of the s p e c t r u m is weak for both cases, ,8 << ! a n d / 3 >> t. Fig. 1 shows the S E W absorpo tion spectra for different values of -q and fl, and for T = 0 . l o ~ , where w o = ( o o / c ) v y .

V G Bordo / Gas-~urface scattering effi'c~s m S E W *pettra

Fig. 2 shows the scattering spectra in the case of forward scattering for different values of We can see that the spectrum of the scattered hght consists of two components" at the SEW frequency to and at the transition frequency toz~. The first component corresponds to SEW resonance scattering. The second one is caused by the transient behavior of the gas polarization near the surface.

173

i

4. Case (ii): diffusive problem - saturation effect \

"\"

Now we assume the inequalities

a)

.-').

<<

where r is the mean free time. Since the SEW

field is slightly nonuniform, and since the transitions between the levels of the molecule are slow, the evolution equations must take the form of diffusion equations for the ground and excited ~tates populations and they must include terms which incorporate the cha.nge in the state of tiae molecule. We may write [7] d2w

D--~z2 - yll(w-

1)

-a~,g22w

exp( -2K2z ) = 0,

(5 where w = [[cql(v)-o'z2(v)] dv, D is the seltdiffusion coefficient, and ao, is the shape of the absorption line of the molecule. Eq. (5) should be complemented by the boundary conditions. We assume here that the er, cited molecules approaching the surface have probabiliW e to lose excitation when being scattered by the surface. Then, we may write [13]

0

\oz/_.=0

3e 2(2-e)

1 i(1

15

I

2

25

3

35 I

Fig 3 T h e S E W a b s o r p t t o n c o e f f l o e n t d e p e n d e n c e u p o n the saturaUon p a r a m e t e r for different e (1) e = 1, (2) E = 10 1 (3) ~ = 10 -2, (4) e = 10- ~ a m Is the a b s o r p n o n c o e f f l o e n t of a w e a k S E W m gas

this condition eq. (5) has an approxtmate solutton [141: w(:)

---

to[g

exp( - K : : ) ]

X [1 +

F(g)

exp(--K2AZ)],

where g~_- - K-,

as the dimensionless parameter characterizing the transitton saturation, 1 +F(g)

(d.w t

05

,~.10( g ) +~z

=

w(O)),

= 1,

where l is the mean free path. We consider the case where the excited molecules pass the SEW propagation zone without excitauon losses, i.e. eD/TII >>K~-1. Under

/z-

3e

1

2(2 - e)

•2 I"

I,, ts the modified nth order Bessel function. Ftg. 3 presents the at(g) dependence for different 6

174

V G Bordo / Gas-surface scattermg effects m SEW spectra

at the following values of paramele1,~: ,% = 10 2 cm -I, l = 3.7 x 10 - 4 cm, D = 9.3 c m 2 / s , Yll = 2.7 X 10; s -I. The SEW absorption saturation ~s seen to exhibit an explicit dependence upon the probability E.

5. Conclusion We have presented here the theory of absorption and scattering of the SEW propagating along a gas-solid interface. The SEW was proposed to be in resonance with molecular or atomic transitions in gas. The essential point of our discussion was the strong spatial localization of the SEW field. We have considered two cases: (i) when gas particles fly through the SEW propagation area without collisions with each other, and (ii) when they pass this area in a diffusive manner. In case (i) we have assumed that the time-of-flight through the SEW field is comparable with that of the transverse relaxation in gas volume. In case (ii) we have supposed that the time of diffusion through the SEW field is less than that of the longitudinal relaxatmn. In the first case the spectral profiles of the SEW absorption coefficient and the intensity of the scattered light depend on the sticking probability and the difference bet,."ec.-, the ~"r,~tma mad ,~'w,t+d admnrptlon pment~als. In the second case the character of the

REW absorption ~amratior it ~eqsitlve to the probability of molecule deexcitation on collisions with the surface, but the shape of the absorption line is independent of the gas-surface interaction.

References [1] V.M Agranovtch and D L Mills, Eds., Surface Polarltons (North-Holland, Amsterdam, 1982) [2] L. Cojan, Ann. Phys. (Par~s) 9 (1954) 385. [3] J,P. Woerdman and M.F.H. Schuurmans, Opt. Commun. 16 (1975) 248, [4] A L J. Burgmans and J.P. Woerdman, J. Phys. (Paris) 37 (1976) 677 [5] M.F.H. Schuurmans, J. Phys. (Pans) 37 (1976) 469. [6] P Bmssel and F. Kerherve, Opt. Commun. 37 (1981) 397. [7] V.G Bordo, Zh Eksp. Teor. Fiz. 95 (1989) 594 [Soy. Phys. JETP 68 (1989) 334]. [8] G. Ntenhms, F Schuller and M Ducloy, Phys. Rev. A 38 (1988) 5197. [9] X.Y Huang, J T Lm and T F George, J. Chem. Phys. 80 (1984) 893 [10] B R Mollow, Ph~cs Rev 188 (1969)1969 [11] F O Goodman and I-~ Y Wachman, Dynamics of GasSurface S,'attermg (Academic Press. New York, 1976). [12] A Messiah, Quantum Mechanics, Vol 2 (North-Holland, Amsterdam, 1961 ). [13] Yu M Gershenzon, V B Rozenshtem and S Ya Umanskn, m Khlmlya Plazmy, Vol 4, Ed B.M. Smirnov (Atomlzdat, Moscow, 1977)p 61 [m Russian]. [14] V G Bordo, Phys. Lett. A 146 (1990) 447