Surface plasmon dispersion relation and local field enhancement distribution for a deep sinusoidal grating

230

Surface Science 172 (1986) 230-256 North-Holland, Amsterdam

SURFACE PLASMON DISPERSION RELATION AND LOCAL FIELD ENHANCEMENT DISTRIBUTION FOR A DEEP SINUSOIDAL GRATING Dan AGASSI Naval Surface Weapons Center, White Oak, Silver Spring,, Maryland 20903, USA

and T h o m a s F. G E O R G E Departments of Chemistry and Physics, State University of New York at Buffalo, Buffalo, New York 14260, USA

Received 9 August 1985; accepted for publication 4 December 1985

Two features of light scattering from a deep lossless metallic sinusoidal grating are considered in the limit g / d ~ oo, where g and d are the height and periodicity of the grating, respectively. It is found that the surface plasmon dispersion relation is comprised of two flat bands with a frequency gap At0/~p = ( 8 - 1 ) / 2 , where ~p is the volume plasmon frequency. For the local field enhancement distribution at the grating's surface, the results show the existence of two qualitatively distinct domains, i.e., h :~ d and A .~ d, where ~ denotes the radiation wavelength. In both domains, however, the local field enhancement is larger at the bottom of the troughs than at the top of the peaks. The dressed Rayleigh expansion is used throughout for the analysis.

1. Introduction Light scattering f r o m a d e e p grating, i.e., a g r a t i n g for which fl = 2 ~ r g / d >> 1 w h e r e g a n d d are the m a x i m u m ( a n d m i n i m u m ) extension a n d p e r i o d i c i t y of the grating, respectively (see fig. 1), is a relatively n e w l y s t u d i e d p h y s i c a l p h e n o m e n o n . M o d e r a t e l y d e e p gratings can n o w b e fabricated, e.g., b y p h o t o d i s s o c i a t i o n o f o r g a n o m e t a l l i c molecules on a s u b s t r a t e [1,2]. T h e e x t r e m e l y d e e p g r a t i n g case (fl ~ o0), which is at the focus of this work, should be c o n s i d e r e d as an idealized limit. This limit is c o m p l e m e n t a r y to the shallow g r a t i n g limit (fl << 1), which is realized in m a n y p h y s i c a l systems a n d has been extensively e x p l o r e d over the last century. T h e e x t r e m e l y d e e p g r a t i n g regime is q u a l i t a t i v e l y different from the shallow g r a t i n g regime [3]. It can be c h a r a c t e r i z e d as a strong c o u p l i n g situation in the sense that the m a n y possible Bragg reflections (fig. 1) interfere 0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

1), Agassi, T.F. George / Surface plasmon dispersion

g

~.

231

d

T

p

-~- . . . . . . . . .

Y.~>;(<

b~ o

"4///>;~? ' / / / / A ///,", ........

:A

R ; eGiON 1 ~

-1

0

I

x.,; ~:<~ ~; J / / , J.~'j,j;:.S,,/. . . . . ,. ~././,,,////)'//.// / ~ ~ < .'~/yV///..x/C, ~ / ~ I~

2

3

4

X Fig. 1. Schematic display of the sinusoidal [z = g cos(2~rx/d)] grating and the notation used in this work. The incident light Effp and the Bragg reflected waves are indicated.

strongly with each other, giving rise to a scattered field of a new qualitative nature. By contrast, the shallow grating regime c a n b e analyzed in terms of a few "weak" (in comparison with the specular reflected light) Bragg reflections. Unfortunately, attempts to solve the Maxwell equations for deep gratings using the Rayleigh expansion, which works beautifully when fl << 1, have encountered numerical instabilities [4]. These difficulties have motivated the introduction of several alternative schemes [1,5,6], all of which, however, share difficulties in the fl >>> 1 regime. In another paper [3] we have suggested yet another scheme - dubbed as the dressed Rayleigh expansion - which overcomes these difficulties. It has been explicitly demonstrated [3] that this scheme converges for a sinusoidal grating (SG) of arbitrary ft. Here we apply the dressed Rayleigh expansion to study two aspects of light scattering from a very deep metallic SG, namely, the surface plasmon (SP) dispersion relation and the distribution of the local field enhancement along the grating's surface. Our results are valid in the fl --* ~ limit and under several approximations which are spelled out as we go along. For the SP dispersion relation we find two flat bands (see fig. 2). The frequency band gap Ato is given by Ato/~0p = ( v ~ - - 1)/2 = 0.207, where ~0p is the volume-plasma frequency of the metal. We interpret the two bands to reflect two classes of localized surface modes those for which the field maxima are in the peaks and in the troughs of the grating respectively. A similar situation occurs in standard band structure studies. The fact that we obtain dispersion bands rather than the usual dispersion curves for fl << 1 is in keeping with studies of moderately deep gratings, fl < 1 [7,8]. In that limit the infinitely many dispersion curves cluster

D. Agassi, T.F George / Surface plasmon dispersion

232 1

V3/2 ,~-/2

/

1/2

k(O)/Kp

Ktl

KG/2

Fig. 2. Display of the surface-plasmon flat bands, eq. (4.4). The light cone is indicated left of the diagonal line, and kp = top/c is the volume plasma wave vector.

densely around the £d//02p ~---1/¢2- line. As fl increases SP dispersion branches pop out in pairs from the ~0/% = 1/¢~- line. As /3 keeps increasing, one branch of the pair is pushed upward and the other branch is pushed downward to be followed by the emergence of a new pair of branches. Since the available frequency interval is confined both from below (~o/~0p = 0) and from above (~0/% = 1), the accumulation of the branches into bands as/3 --, oe is expected. However, the particular band gap and widths we derive [eqs. (4.4) and (4.5)] may be, in part, an artifact of our approximations. SP dispersion bands have also been derived for an infinitely large stack of thin metallic films [91. We also evaluate a simple figure-of-merit representing the local field enhancement distribution. The local field enhancement R is defined as R = IE//Einc 12, where E is the total electric field and Ei, c is the incident electric field [6,10]. The value and distribution of R are of prime importance in determining optical and chemical processes near the grating [11]. To remain within reach of a simple analytical result, we evaluate only R ( T / B ) = I R ( T ) / R ( B ) I 2, where the point T (top) and B (bottom) are defined in fig. 1, and the incident light is normal to the grating. The result [eq. (5.3)] indicates the existence of two distinct regimes, i.e., X >> d and X << d, where X is the wavelength of the incident light. The R ( T / B ) ratio in the former is smaller than in the latter, and in both regimes R ( T / B ) = 1//3. These findings are understood in terms of the following physical picture: The incident field induces the electrons in the metal into an oscillatory motion, and the electrons respond. Since the grating is deep, the motion of the electrons at the grating's

D. Agassi, T.F. George / Surface plasmon dispersion

233

peak is inhibited by the grating's boundaries, much less than at the troughs' bottoms. Hence R ( T / B ) << 1. The distinction between the ~, >> d and h << d regimes can also be understood in the same vein: The response of the electrons at the bottoms is slightly inhibited by the grating's boundaries and hence it depends only mildly on ~. The response of the electrons at the peak's top, however, is strongly dependent on X: For h >> d, the electrons respond very poorly due to the confinement of the grating's boundaries, while, as ~, ever decreases, the top electrons can respond ever more efficiently. Consequently, the ratio R ( T / B ) is larger for ~, << d than for ~ >> d. The paper is organized as follows: We introduce in section 2 the dressed Rayleigh expansion [3] in the context of the SG. In section 3 we present the central approximations which enable a simple solution for the ligth scattering problem in the asymptotic limit fl ~ o¢. The results of section 3 are then applied in section 4 to analyze the SP dispersion relations and in section 5 to evaluate the local field enhancement distribution for a particularly simple configuration. Section 6 is a discussion and summary.

2. Dressed Rayleigh expansion and sinusoidal grating We introduce in this section the dressed Rayleigh expansion [3] and the pertinent formulas of the SG. As has been demonstrated elsewhere, the dressed Rayleigh expansion converges for an SG of arbitrary fl and therefore provides a convenient framework to analyze the fl + oo limit. All expressions in this section are exact. The notations throughout are defined in fig. 1 The dressed Rayleigh expansion for the electromagnetic fields is [3]

eo( ,

e,(x, z)=

oo £ { ~t0(/)P0,- (l)

Vl(1) Pl,--(/)

e i[ktx-W°(lXz+g)]

e i{ktx-Wl
(2.1 .a)

l=-oo

and { 70(t ) eiIk,x - Wo(l)(g+z)] + Cto(l) eilk,x+ wo(/Xg+z))}, ~q( l) e i[k'x-iw'(I)tz-*)] ,

(2.1.b)

I= --o¢~ where 70( l ) = 8,,o G-Tp eiW°(')*,

(2.2)

234

D. Agassi, T.F. George / Surface plasmon dispersion

and

k(i)=~ik,

k=wo/C ,

kG=2cr/d,

1 k tz-T-W/(I)~:], ^ ~i,~:(l)--k---~[

~=.~Xe,

fori=0,1.

(2.3)

In (2.1)-(2.3) the subscripts of the fields refer to the domains (0 or 1), the time dependence of the fields is assumed to be exp ( - i~00t), k His the x-momentum component label of the field, .~, $ and ~ are unit operators in the obvious three directions, E@ is the arbitrary amplitude of the p-polarized incident light normal to the grating's grooves, and we always choose Im[W~(l)] or Re[145~(/)] __>0. The height g (positive) is half the maximum z-extension of the grating (fig. 1), and the central parameter fl is again

fl = gk o.

(2.4)

The dressed Rayleigh expansion coefficients satisfy

M,,,.tao(l)=#(m), 1= -oo

£

N..,, y l ( l ) = v ( m ),

(2.5.a)

I= - ~

where the matrix ~£¢, for instance, can be dubbed as the "Fresnel matrix". It is given by M,,,,, = ,.'~itWl(m)+Wo{t)lgMa,,,.t, tx(m) = eiW'tm)g#B(m),

gin, ! = eitW°(m)+ Wl(l)]gN2d,

v( rn ) = e iw°(m)g vB( m ),

(2.5.b)

and

W°(l)W'(m) + ktk"i'-t

Wo(t)- wl(m)

N,,a t =

W,(l)Wo(rn ) +

k,k,,, i " - '

w,(t)- Wo(m)

J,,,_t{ g[Wo(l ) - W,(m)] ), J m - I ( g [ -- 14/1(1) 4" W 0 ( m ) ] ) ,

#"(rn) = - W°(O)Wl(rn) + k ° k " i" J,,(-g[W0(0) + Wl(rn)] }Eop, W0(0 ) + W,(m) va(m ) = 2 % q W0(rn)8=.0 Eop, El -- E0

(2.6)

where Jr denotes the Bessel function of order v. It has been shown elsewhere [3] that the set of equations (2.5) can be safely truncated at a maximum I l I-value L of the order/3, since the matrix elements of ~ ' decay exponentially with 1 or m and become insignificant beyond that point. A more detailed examination reveals that the important matrix elements

D. Agassi, T.F. George / Surfaceplasmon dispersion

235

of Jr' are distributed in two lobes: one lobe is for m "small" and Ill "large" and the other for l "small" and Iml "large". The diagonal terms diminish exponentially with 1 = m faster than either lobe's length. Once the set of equations (2.5) is solved, all properties of the scattered fields are known. The difficulty when fl ~ oo, however, is that the relevant matrix .At (or .A/') is very large and highly non-diagonal. We introduce in the next section an asymptotic approximate solution of (2.5) and then explore its implications with regard to the SP dispersion relation and light scattering.

3. Approximate solution for fl >>> 1

To discuss the fight scattered by the grating, it is necessary to invert the Fresnel matrix ~ [eq. (2.5)], which has an effective dimensionality on the order of fl when fl >> 1. The construction of a simple approximation to .At'-1 is the content of this section. The sequence of approximations that lead to the final results [eqs. (3.8)-(3.10)] is valid for fl >>> 1, i.e., very deep gratings. At some points we resort to additional approximations motivated by the gained mathematical simplicity. We believe that the plausibility of the results derived in sections 4 and 5 based on the approximate ¢¢¢-1 vindicate the appoximations. To gain perspective on the fl >> 1 regime considered here, we give the analysis pertaining to shallow gratings in appendix A. An attractive feature of .,¢¢, or J t 'B, is its partial separability in the " m " and "'1 "' indices [eq. (2.6)]. As will become obvious shortly, separable matrices are convenient mathematically. We are therefore motivated to devise a separable approximation to the other factors in (2.6), in particular to the Bessel function factor. This turns out to be possible provided [m 1, [l [ and [ m - 1 [ are very large compared to unity, which is the case for most of the elements of the highly nondiagonal matrix ~g. Using (2.6), the ensuing quasi-separable approximation is (appendix B) Mm(S,l,

"~ Co

1 i'-' [Wo(l ) - W l ( m ) ] 3/2

[ W o ( l ) W , ( m ) + k,k~]

X e iglW°(I)+ W'(m)l (e i• + e-i~),

(3.1.a)

where Co =

kv~2~ ~/(fl)

1 (1 + •2) 1/4'

[ W0(l) - W1(m)]

[-~

, ( f l ) = ~ + f12 + ln[ fl/(1 + +1 / 1 - ~ ) ] .

(3.1.b)

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D. Agassi, T.F. George / Surface plasmon dispersion

It can be easily checked by comparison with (2.6) that (3.1) provides a surprisingly good approximation even for the diagonal element MI. r Note that M ts~ has a separable form except for the denominator [W0(l ) Wl(m)] 3/2. This is t o be expected since a fully separable form for M (s~ does not possess an inverse. To maintain the simplicity of a separable approximation on the one hand and yet to secure the existence of a simple inverse on the other hand, we introduce the following additional approximation: Consider first the off-diagonal matrix elements of M (s~. These elements are grouped into two lobes (section 2), i.e., when [m I is "small" and 1l[ is "large", and vice versa. When [m [ is "small" and 111 is "large", for example, the fastest [l I-changing factor in M (s) is the exponential. Consequently, the above mentioned non-separable denominator, which changes with [11 much slower is, approximately, a normalizing factor. It can be therefore approximated by a suitably chosen constant, denoted by C r For the sake of simplicity, we further approximate [1410(l) - W l ( m ) ] 3/2 ~ C 1 for both the [m [ << I11 and l l[ << Ira[ elements. The diagonal elements of M (s~ comprise the complementary (minority) group of elements to the off-diagonal elements just discussed. In this case, the above argument concerning the fastest 111= [m [-changing factor in M ts~ is still valid. However, the corresponding constant approximating the non-separable denominator is obviously quite different, i.e., [ W o ( l ) W1(l)] 3/2 _ C2" The foregoing discussion suggests therefore the approximation M,,,., = M(S~ -- D, 8,,,.t + V,,.,,

(3.2)

V,,,.,= C i ' ' - t [ W o ( l ) W , ( m ) + ktkm] e iglw°(')+ W'('')l (ei~ d- e - i ~ ) , C = Co/C , ,

(3.3)

and C 1 has been defined in the discussion above. The diagonal elements D t in eq. (3.2) depend, of course, on the choice of C2. However, regardless of the particular choice, due to the convergence factor exp {ig[Wo(l ) + Wl(m)]}, the number of significant diagonal dements is of the order of unity. Therefore, there are only very few significant diagonal terms in comparison with the large number of off-diagonal terms, yet if they are neglected, eq. (3.2) ceases to have an inverse. We have therefore a situation where a few diagonal terms are not important in terms of physical results, yet due to the particular mathematical structure (separability) it is necessary to keep them to assure the existence of an inverse. Under such conditions, it is probably immaterial as to how the diagonal terms are chosen, provided the calculated observable does not depend on the specifics of the choice. We therefore simplify eq. (3.2) further by the approximation D t = D, where D is a very small number of the order e -~ or less. Again, to be consistent with the arguments above, we can use the ensuing approximation only to calculate observables that do not depend on D, such as those analyzed in the next two sections. We would like to emphasize that the Dt = D approximation is not

D. Agassi, T.F. George / Surface plasmon dispersion

237

essential, i.e., it is possible to invert eq. (3.2) as it stands. The substantial simplification gained by adopting this approximation, however, warrants an examination. Therefore, eq. (3.2) is replaced by

AIm,, = D 8m,t + I'm,t,

(3.4)

where D is a very small number of the order exp ( - fl). The inversion of eq. (3.4) can be obtained now in a straightforward manner. Note from eq. (3.3) that V is comprised of four separable terms, 4

V,,,t = E aj(m)by(l),

(3.5)

j=l

where the factors aj and bj are extracted from eqs. (3.1.b) and (3.3) [see appendix C, eq. (C.2)]. The form of eq. (3.5) has the attractive property that V to the n th power is given by 4

(V"),,,,=

E

aj(m)[A"-l]i,jbj(l),

(3.6)

i,j=l

where the auxiliary interaction matrix ~¢ is of dimension 4 with the elements

Ai,j=

~ l ~

bi(l)aj(l).

(3.7)

--oo

The auxifiary matrix ~ (appendix C) plays a central role in all subsequent manipulations. Consequently, expanding the inverse of eq. (3.4) in a geometric series and using eq. (3.6) yields

(

)m,t = "-~Sm,t DE

ai(m ) 1 + 1

--'DI[ 8,,,,,-

A

i,jbj(l)

"=

i,j=~lai(m)[A-1]i'jbj(l) ]

(3.8)

The second line in eq. (3.8) follows since ( I ~¢ I ) / D >> 1, where ( [ ~ I) is a typical average value of the elements of ~ [see (C.18) and (C.20)]. The approximate inverse (3.8) is a key result of this work. The unknown parameter D appears as a multiplicative factor, implying that (3.8) can be used only to evaluate ratios so that D drops out from the final result. Note that (3.8) expresses the reduction of an inversion problem of a fl × fl matrix to the inversion of a 4 × 4 matrix. Based on the analytical expressions for the matrix elements of ~¢ (appendix C), the inverse ~ 1 can be further simphfied. The key remark is that when /3 ~ oo, the matrix elements denoted by " a " in table 1 are substantially

D. Agassi, T.F. George / Surface plasrnon dispersion

238

Table 1 Outlay of the Ai,y elements according to the classification defined in eq. (C.7). The discussion in appendix C implies that for fl ---, oo, the class-a terms can be neglected in comparison to the other entries 1

2

3

4

fl fl

-/ a Y

8

ot

1

fl

2

8

3 4

fl

y a V

8

a

8

smaller than the rest of ~¢ [see (C.11) and following discussion] and therefore can be neglected. Consequently, det(a~¢) = det( ~¢~ ) det( -~¢8), det(~)

= A1,2A3,4

-

-

A1,4A3,2,

(3.9)

det(a~'8) = A2,1A4, 3 - A2,3A4,1, where the symbol "det" indicates a determinant, and A3,4

°°° /

0 -A3, 2

0 1

aet( ,)

1

+ det(~8-------~

0

0

-A1, 4

0

0

0

0

0

A1,2

0/

A4, 3

0

0

0

0

--A4,1

0

A2,1

0

0

0

+ mixed y and 8 terms.

-A2, 3

(3.10)

The simplicity of (3.9) and (3.10) in conjunction with (3.8) is most gratifying. It vindicates an expectation that in the limit/3 >>> 1, as in the/3 << 1 limit, the physics should simplify. All that is needed to obtain J t '-1 are eight matrix elements, combined in the simple form of (3.10)!

4. Surface plasmon dispersion relation Surface plasmon excitations are collective, surface-attached oscillations of the metal's electron gas. For the case of a lossless shallow metallic grating [7,8], the dispersion relation is comprised of many branches which appear in pairs above, below and on the line k/kp = ¢~-, where kp = % / c and % is the volume plasmon frequency. As /3 is ever increased, an ever larger number of

D. Agassi, T.F. George /Surfaceplasmon dispersion

239

branch pairs seperates off the k/kp = ~/2 line. In this section we derive the dispersion relation for a very deep grating, i.e. when fl ---, oo. The pairing of branches can be understood using arguments employed in interpreting the opening of a band gap for electrons moving in a periodic ionic potential [12]. In the latter case there are two interfering combinations of the unperturbed electron wave and the umklapp-reflected wave. Since one combination is centered at the ions' sites and the other between ions, the associated energy of these two combinations are split and a band gap opens. In the present context the umklapp processes correspond to the Bragg reflections and the unperturbed electron waves to the flat-surface SP mode (see appendix A). The remaining issue therefore is how do these pairs of branches accumulate as fl---, oo, given that the available frequency domain is bounded from below by ~0/tOp = 0 and from above by t o / % = 1. By definition, the SP excitations decay exponentially in both z-directions, fig. 1, i.e., they correspond to Effp = 0 in (2.6). Consequently, to have a non-trivial solution of the Fresnel equations (2.5), it follows that det(..Cf) = 0,

(4.1)

which determines a function of k(0) versus kll provided that --kG/2 < kll < k~/2, i.e., kll is confined to the first Brillouin zone [7]. Condition (4.1) is tantamount to finding the points where ¢¢¢-1 does not exist (poles). Hence, in the context of the approximate inverse (3.8), eq. (4.1) transcribes into det(.~¢) = 0. The determinant of ~¢, however, is given in the leading asymptotic order by (3.9). Consequently, since det ( ~ 8 ) :# 0 outside the light cone [see eq. (C.22)], the SP dispersion relation is simply det(~¢,) = Aa,2A3,, -

Aa,4As,2 =

0.

(4.2)

Inserting into (4.2) the explicit expressions for the matrix elements (C.20) yields [3k2(0) + k2(1)] [kZ(O) + 3k2(1)] 3[ k2(O) + k2(1)] 2

{ sin[L(a+ y)] sin[l(a-y)] -sin[L(a-y)]sin[½(a+ y)] } 2, =

sin[L(a+V)]

~--~] ~~--~]

sin[½(,,,+~,)] (4.3.a)

where

k(i) = f~i k

y = 2~rk/k o,

from (2.2) and ~"

a = ~--~c( - [ k 2 ( 0 ) + k2(1)] )a/2

(4.3.b)

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D. Agassi, T.F. George / Surface plasmon dispersion

In (4.3.a) L is a very large number of the order fl [see the discussion after eq. (C.12)], the precise value of which is immaterial (see below). The analysis of (4.3) is quite simple. Consider first the k(0) =< kp/[1 + c0] 1/2 regime, where a and obviously "t of (4.3.b) are real and hence the RHS of (4.3.a) is real and positive. It can be easily shown that for (4.3.a) to have a solution it is necessary that k < kp/[3c 0 + 1] 1/2. When k(0) >__kp/[1 + c0] 1/2 then a is imaginary and hence the RHS of (4.3.a) is real and negative. Simple considerations show again that to make the LHS of (4.3.a) negative it is necessary that k < kp[3/(3 + c0)] 1/2. Therefore, for a solution to (4.3) to exist, it is necessary that (assuming c o = 1 for simplicity)

0 <_k <_½kp, ½Gkp <=k <=½vl3kp.

(4.4)

Condition (4.4), however, is also sufficient. Since L = fl >>> 1, the RHS exhibits oscillations which are ever increasing in amplitude and frequency as fl is increased. Therefore, any k-value within the admissible ranges (4.4) generates a solution of (4.3), i.e., the dispersion relation constitutes two bands (fig. 2). The frequency gap At0 between the bands is according to (4.4) A~o/a~p

=

½(v~- - 1).

(4.5)

The results (4.4) and (4.5) describe the accumulation of dispersion branch pairs mentioned in the beginning of this section. The interpretation of the two bands is identical to that of the two members of a dispersion branch pair: The two bands are comprised, respectively, of modes in which the field is mostly localized either in the troughs or in the peaks of the grating. We cannot be sure at this point to what degree the particular band gap (4.5) and band widths are an artifact of our approximations. At the very least, (4.4) and (4.5) indicate domains of high density of dispersion-relation curves. Note that the dispersion relation (4.3) or (4.4) does not depend on g, k G or kll. The independence of g or k c can be understood as follows: If fl ~ oo is interpreted as g ~ 0o and d finite, then obviously the result of any observable cannot depend on g in the asymptotic region. By the same token, the limit fl ---, oo can be construed as keeping g finite but letting d --* 0, which leads again to the conclusion that the result cannot depend, to the leading order, on ko. The independence of the band structure of kll may be an artifact of our approximations. The behavior of the matrix elements of ¢¢/ near the light cone, for instance, is controlled by the Wj(I) factors [eq. (2.2)], with the square-root feature playing an essential role (apendix A). On the other hand, in all manipulations employed here, we expand the square root Wj(I) as I,Vj(1) -- i [ I k , I - l k 2 ( j ) / ] k, I], which destroys the retardation effects near the light cone.

(4.6)

D. Agassi,T.F.George/Surfaceplasmondispersion

241

5. faecal field enhancement distribution for normal incident light

As a second application of our central result (3.8), we turn now to the calculation of the local field enhancement distribution at the grating's surface. The local field enhancement R(x, z) is defined as

R(x, z)=

leo(X,

2

It measures the collective response of the electrons in the metal to an incident electromagnetic field, and is of obvious importance in determining optical and scattering processes near the grating [11]. To simplify the analysis, we pick the simplest configuration. We assume normal incidence of the p-polarized light and consider the ratio of the local field enhancements at the top of the peak to that at the bottom of the trough (points T and B in the notation of fig. 1). This ratio, denoted by R(T/B), is a convenient figure-of-merit describing the local field enhancement distribution. Since R ( T / B ) is a ratio of two fields, the unknown constant D in (3.8) does not enter the final expressions. Combining (2.1), (2.5), (3.8) and (3.10), it is straightforward to obtain for the electric field in region 0 (fig. 1)

Co(X, z )

=

o(O) e

-t-l([,=~_ooPO.+(l) ei[ktx+W°(l'z]~(l)]

"[

]

-- E £ p0.+(/) e i[ktx+W°(l)z] aj(l) (.~¢-')j., j,n= l 1=--oo

X[m=~_oobj(m)p'(m)]} =Ein¢(x, g) + Ef(x, g) + Es(x , z). (5.1) The decomposition of Eo(x, z) in the last line of (5.1) reflects the structure of (3.10): The first term in (3.10), involving the matrix elements A],2, A1,4, etc. (or class-v elements in the terminology of appendix C and table 1) gives rise to field components that vary very rapidly with small changes in either k, kll or k G, etc. This feature is manifested in the explicit expressions for these matrix dements [eq. (C.20)] and has been exploited already in conjunction with the SP dispersion relation (section 4). The corresponding component in the total field is denoted by El(x, z), where the subscript " f " indicates "fast". The second term in (3.10) involving the elements A2j , A2,3, etc. (class-8 in the terminology of appendix C and table 1) changes slowly with small changes in k, k,, etc. and gives rise to the "slow" component in the total field, denoted by Es(x, z) in (5.1). The incident field, denoted by Eii~c(x, z), is the first term

242

1). Agassi, T.F. George / Surface plasmon dispersion

on the LHS of (5.1). The decomposition (5.1) of the total field into a fast oscillating component El(x, z) and a slow envelope Es(x, z) is expected due to the many interfering components (Bragg reflections) of which Eo(x, z) is comprised. In terms of the decomposition (5.1), we identify the ratio of the local field enhancements at points T and B in fig. 1 with the field envelope, or average: Einc(T ) + £ s ( T ) 2 Es(T) 2 R ( T / B ) = Ei,c(B ) + E s ( B ) = E~(B) '

(5.2)

where the arguments " T " and " B " in (5.2) indicate the coordinates of points T and B. Straightforward yet somewhat lengthy algebra (appendix D) gives R ( T / B ) --- { r/nfl' for r<< 1, r7//4fl, for r >> 1,

(5.3)

and (c o = 1 for simplicity)

r = k(O)/k G = d/X.

(5.4)

The complete expressions are given in appendix D, whereas (5.3) represents an order-of-magnitude estimate. The result (5.3) is interpreted in the following way: The fact that there are two regimes, c r u d d y defined by X << d and X >> d, can be understood in terms of the electrons' relative response to an incident field at the top of the peaks and the bottom of the troughs. When h >> d (r << 1), the electrons at the peaks' tops respond poorly by comparison to the electrons at the bottom of the troughs, simply because the former are confined to the metallic fingers of the grating. On the other hand, when ~ << d (r >> 1), the response of the electrons at the peaks' top is impeded to a much lesser degree by the boundaries of the grating's fingers, while the electrons at the troughs' bottoms respond in about the same manner as in the ~ >> d regime. This consideration implies that R ( T / B ) is smaller for X >> d than for X << d, which is born out by (5.3). The fact that in both regimes R ( T / B ) = fl-1 << 1 is in keeping with the above picture: Since the grating is deep, for normal incidence it is always easier for the electrons to respond when they are located at the bottoms of the troughs than when they are at the tops of the peaks. Note that these considerations do not depend on g, since they pertain to relative magnitude. It is interesting to notice that the same trends are present in numerical studies o f shallow gratings [10].

6. Discussion and summary

The main message of this work is that for the very deep sinusoidal grating limit (fl >>> 1), the physics simplifies considerably: By introducing a sequence

D. Agassi, T.F George / Surface plasmon dispersion

243

of reasonable approximations, the problem of inverting the Fresnel matrix (2.6) of order fl has been reduced to the calculations of eight matrix elements (3.10). The simplicity of the outcome is in keeping with a related study of the surface-plasmon dispersion for an infinite stack of metallic thin films [9]. With regard to the surface-plasmon dispersion relation, we find two bands of branches, separated by a frequency band gap given by At~/O~p--- 0.2. These two bands represent two classes of localized SP excitations: Those for which the fields are maximized in the troughs and those where the fields are maximized inside the peaks. The latter class of modes correspond to the upper band since they represent an enhanced oscillation of the electrons. This interpretation is in keeping with the interpretation of dispersion branch pairs already observed for shallow gratings. It is also consistent with the analogy of the two-adjacent-band structure of electrons moving in a periodic potential [12]. With regard to the local field enhancement distribution, we find that the field at the bottom of the throughs is always larger than at the tops of the peaks. However, the relative magnitude of the "top/bottom" enhancement exhibits different behavior in the X >> d and X << d regimes. The existence of these regimes reflects primarily the response of the electrons at the peaks' tops. The electrons there are restricted in their motion by the grating's boundary, and therefore respond poorly when ~ >> d and much better when X << d. The simple outcome of this study will hopefully stimulate further studies in the same vein. To clarify to what degree the specifics of our results do not depend on an artifact of the approximations, it would be desirable to attempt some refinements, such as straight inversion of the quasi-separable approximation (3.1) or (3.2). Another possibility is to carry out large-scale exact calculations of the SG using the dressed Rayleigh expansion as described in section 2 [3]. It is interesting in particular to substantiate the quantitative values we obtain for the width and gap of the SP bands [eqs. (4.4) and (4.5)]. These results hopefully will motivate interest in corresponding experimental work. Acknowledgments We are grateful to A.C. Beri for his assistance in preparing the figures and to J. Rouse, R. Ferron and L.W. Edwards for their typing the manuscript. This work has been supported by the Office of Naval Research and the Air Force Office of Scientific Research (AFSC), United States Air Force, under Contract F49620-86-C-009. Appendix A: Plasmon dispersion relation for a shallow grating surface To highlight the particular features of the fl << 1 regime and to appreciate the role of the various factors in (2.6), we analyze here the shallow grating SP

244

D. Agassi, T.F. George /Surfaceplasmon dispersion

dispersion relation. The result [5,7,8] obtained here in a direct and transparent manner. Consider first the case of a flat surface, i.e., g = 0 in the notation of fig. 1. Since [14] J~(g = 0) = 6~,0, eq. (2.6) yields m = l and the matrix ~ is diagonal. Consequently, for the determinant of ~ to vanish, eq. (4.1), one or more of the diagonal elements must vanish. Hence, the dispersion relation reads

Wo(I)WI(I ) + kt2 = 0,

(A.1)

or, by inserting the definitions (2.3), (A.1) yields

k, =

+ ,1).

(A.2)

For a flat surface d--* ~ , i.e. k c = 0, so that k I --* kll and (A.2) is the well-known dispersion relation [7,8]. Thus the numerator in (2.6) involving Wo(l)Wl(m ) + ktk,, entails the retardation effects, namely, that the SP dispersion cannot cross the light cone. Consider now the case of a very shallow grating, i.e., g is finite yet fl = gk G << 1. Here the argument of the Bessel function factor is small, yet finite. Therefore, since [14] lim J , ( z ) = z ~0

[ (z/n)n' [ 1,

for n ~ 0, for n = 0,

(m.3)

to first order in /3, the only non-diagonal terms that contribute are when I m - 11 < 1. Thus, .~' is tri-diagonal, with a dominating diagonal and the two first-off-diagonal of order ft. The/3 = 0 and fl << 1 examples discussed so far clearly indicate that the Bessel function factor in (2.6) determines the amount of mixing of the m and l Bragg reflections. The non-vanishing denominator of (2.6) affects only the normalization and the fall-off in m, 1 rate of M,,a. To determine the conditions for (4.1) to hold when/3 << 1 is no simple task in general. However, for special (k, kll ) combinations the problem can be easily solved. Let us thus consider, in particular, the boundary of the first Brillouin zone kll = ½k G and the crossing of the l = 0 and l = - 1 dispersion curves given by (A.2). If there were not off-diagonal terms in J ( , the crossing of the curves at kll = ½k G would define a (k, kll ) point for which (4.1) is satisfied. For finite g, however, the two relevant off-diagonal elements M0,_ 1 and M a,0 change the eigenvalues of .At'. The condition for the perturbed eigenvalues of ~ to vanish [so that (4.1) is satisfied] is therefore that the corresponding determinant vanishes: M~l,0

M0-1 "

= 0.

(A.4)

Eq. (A.4) is an equation for k which has two solutions, whose difference is the band gap. This equation can be easily solved provided that at the Brillouin

D. Agassi, T.F. George/Surface plasmondispersion

245

zone boundary (kll = ½kG) we are allowed to neglect retardations, i.e., k o >> k(0), [ k(1) 1. In this case, W0(0) -- 1411(- 1) --- W0(- 1) -- ½ik G, and simple algebra gives the frequency gap Ao: as [7] A~o

%

-

~ g

(A.5)

~ d"

Appendix B: Separable approximation for the Bessel function factor in (2.6)

We derive a separable approximation to the Bessel function factor in the matrix ~¢:

Bm.,=Jm_t[g(Wo(l) - W,(m))] = Jm_t[ifl( I l l - [ml)],

(B.1)

which is valid for fl >> 1 and I l I, I m I >> 1. The second line in (B.1) follows from (4.6). We analyze (B.1) by considering separately the pertinent asymptotic domains and then introduce a simplifying separable expression. When 1 << I l - m I <<
J~(z) = (2/~rz) 1/2 cos[z - (½n + ¼¢r)], n=m-l,

(B.2.a)

z=i/3(lll-lml)=g[Wo(l)-

Wl(m)].

(B.2.b)

When/3 -- I I - m I, the argument of the Bessel function (B.1) and its order are large and proportional each other. The asymptotic expression in this regime is

[]4]

J~(pz)

] =

-

1

-

[1

-

Z2] 1/4

eg(z)=l/l_zZ+ln(

e,~,(,, + ~(~,-3/2), z

)

for [argz I < r r ,

1,---}oe.

(B.3)

1 + l~-~z 2 ' To apply (B.3) to (B.1), we consider separately the three possible subdomains: (a) l,m>O, (b) l,m>1, Ira[ >>1 and m, l of opposite signs. Using [14] J , ( z ) = ( - 1 ) " J,(z)and J , ( - z ) = ( - 1 ) " J,(z), simple analysis gives B~,t

= e:r_~i,rl/_ml

1

~2~'1l-ml

~(/3) = ~1 +/32 + ln(

/3 1 + +1~ - ~

1 e it_,,in(# ) [1+/32] 1/4 )

(B.4.a)

(B.4.b) '

where B + refers to domain (a) and B - to domain (b). When ! and m have

246

D. A g a s s i , T.F. George / S u r f a c e p l a s m o n

dispersion

different signs [domain (c)], the argument of the Bessel function (B.1) is much smaller than its order, and hence the corresponding asymptotic expression is [14] J,(z)= l ( e z ) -~v ", v = m - l , z=ifl(lll-lm[). (B.5) Consequently, z / v -- fl I(l + m ) / ( l - m) [ << 1 and hence the m, l combinations. Given (B.2)-(B.5), the approximation ikG Jm-t{g[W°(l)-

Wl(m)]} =

2rr

Bin, 1 "~

1

(1..{_82)1/4[ W o ( l )

-

1 Wl(m)]l/2

X (e iq' -4- e-i°), (/) = *l(fl)[ W o ( l ) _ W,(m)] - [ l ( m kG unifies all relevant domains.

0 for most of

(B.6.a)

l ) + ¼1 ~r

(B.6.b)

Appendix C: Reduced auxiliary matrix .~

We calculate some and estimate the remaining matrix elements of the auxiliary matrix d defined by A,.j=

~

bi(l)aj(l ),

i , j = l ..... 4,

(C.1)

I ~ -- oo

where, from (3.1) and (3.3), aa(m ) = Wl(m ) e x p { i [ W l ( m ) ( g - 71(fl)/kG) -- ~r/4] }, (C.2.a) bl( l ) = CWo(1 ) exp( i[ Wo( l)( g + ~ ( f l ) / k G ) ] ), (C.2.b) a2(m ) = Wl(m ) e x p ( i [ W l ( r n ) ( g + T l ( f l ) / k c ) + ~rm + ~r/4] }, (C.2.c) b2(/) = CWo(l ) exp{i[Wo(l)(g - , ( / 3 ) / k G ) -- ~rl] }, (C.2.d) a3(m ) = k,, exp( i[ W~( rn )( g - ~( fl )/kG ) -- rr/4] }, (C.2.e) b3(l ) = Ck t exp{i[ Wo(1)(g + ~/(fl)/kG)] }, (C.2.f) a4(m ) = k,, e x p ( i [ W l ( m ) ( g + , ( B ) / k G ) + ~rm + tr/4] }, (C.2.g) b4(l ) = Ck l exp(i[Wo(l)(g - 7/(fl)/kG) -- qrl] }, (C.2.h) and C is defined in (3.3). Note that in (C.1) all the/-dependence enters via the combintion k I = kll + lk c [see (C.2)]. Hence, the summation in (C.1) can be evaluated by using the Poisson formula [15]: G ( y + la) = ~1 l z --00

g n ~

--o0

e -2~'i'y/'a,

g(t) =

du G(u)e"", --o0

(c.3)

D. Agassi, T.F. George /Surface plasmon dispersion

247

with the identifications y ~ kip A --. k o and u ~ kl. Furthermore, since for most terms in (C.1) I11 >> 1, it is justified to evaluate Ai, j using the asymptotic expression of Wj(I), eq. (4.6). Keeping the leading term in (4.6) is not sufficient since all references to k ( j ) enter in the second term. It is therefore the first small connection to Ai,j, beyond the leading term, that carries all the physics. By inserting (4.6) into (C.2), it is easily seen that all Fourier transforms g ( t ) mandated by (C.3) take the form

f~

g,J~') = _ d k l b i ( k l ) a j ( e l ) e ik'' = foo d k t F , ( k t ) e itfn(kt)+kttl, O0

for: i , j = l

. . . . 4,

--00

n = l . . . . 4,

and

~ / = a , fl, 7 , 3 ,

(C.4)

where F,(k,) = Wo(I)W~(I ) ~ C{-k

2 + ½[k2(0) + k2(1)] },

(C.5.a)

Fz( k t ) = C[ - i k t I k' I + ½ik2(0) sgn(k,)],

(C.5.b)

F3 (kt) -- C[ - i k , I k, I + ½ik2(1) sgn(kt)],

(c.5.c) (C.5.d)

F, (k,) = C k 2 , and

L = - ~ + Wo(O[g- ~ ( ~ ) / ~ o ] + Wa(O[g+~(~)/ko] 2ig [ k, [ - i g k 2 (1)/I k , I,

A= 1 ~

+ Wo(l)[g+*i(fl)/kG]

(C.6.a) + W,(l)[g--Tq(fl)/kG]

+ 2ig I k, I - i g k 2 ( O ) / I k, I,

(C.6.b)

fv = l~r + W o ( l ) [ g + r l ( f l ) / k G ] + W,(I)[ g + *l(fl)] + ~rl .~ ~¢r - ¢rkll/k c + 7rkt/k 6 + 4 i g l k ' I - i g [k2(O) + k 2 ( 1 ) ] / I kt l,

A = -~

+ w0(t)[g- ~(~)/ko] +

~rkll - ¼~r + k----o

~rkt

k----o + ~

i

Wl(t)[g- ~ ( ~ )

Iko] - ~t

i[k2(0) + k2(1)] I k' I

4flkG

(C.6.c)

1

(C.6.d)

I k, I "

The sixteen Fourier transforms g i j ( t ) [eq. (C.4)] are obtained from all combinations of the F m and the fn" The explicit correspondence in (table 1) is f~ ft~ fv f8

with with with with

A2.2, A2A, A4,2, A4,4, A1 a, hi, 3, A3,1, h3, 3, A1,2, A1,4, A3,2, h3, 4, A2 a, A2,3, A4,1, -44,3.

(C.7.a) (C.7.b) (C.7.c) (C.7.d)

D. Agassi, T.F. George / Surfaceplasmon dispersion

248

The procedure of calculating Ai, j is now clear. We first evaluate the transforms (C.4) for all t-values mandated by the Poisson formula (C.3), and then sum the contributions. As we now show, these two operations can be carried out in closed form. An important bonus of having analytical expressions is the possibility to identify the dominating matrix elements of ~ . Combining (C.5) and (C.6) with (4.6) impfies that all integrals (C.4) are combinations of [16]

fo°°dxx ~-1 exp[½i#(x-72/x)]

= 2 ) ,~ exp(½i~rv) K_~(7#)

f0mdx x p-' exp[½i#(x + 72/x)] =i~ry ~ e x p ( - ½i~rv) H ~ ( y # ) for Im(/~) > 0,

Im(y2/x) > 0,

any J,,

(Case I), (Case IX), (C.8)

where K~ and H (1) denote the modified Bessel and Hankel functions of the first kind of order p, respectively. As will become clear shortly, we shall need only the asymptotic or near the origin behavior of the Bessel function, given by [14]

K_~(z)

= [~r//2z] 1/2 e - z ,

H~_l~( z ) = [2/rrz ],/2 eitz +,,./2_./41,

for lzl ~ oo,

(C.9.a)

and = -

=

for [z [ ~ O.

±r(=)

e'-

-=,

(C.9.b)

Before proceeding further, we mention that the conditions attached to (C.8), i.e., I m ( # ) > 0, etc. are always fulfilled. Also because the krintegration in (C.3) is over the whole kraxJs while in (C.8) it runs over only half the axis, we subdivide the krintegration into - o¢ < k t < 0 and 0 < k t __
ik2(1) ]1/2 ~ _~ t ]

(Case II),

(C.10.a)

-I-

II

~.

,~" '--'-'

°

v

+

H-

~

+1

II

"~l+

II

~

~

~,

÷

'

¥

~

+

~

,

\

^

"~

~

~

~

I

+

o

,~, -"

I@

~

÷

II

II

II

~

o

V

I+ NI

+

I

H-

I÷ ~ ,

-t-

II

o

A

II

÷

o

1- I

I , -.0

b~

i- I

~

I '-"

II

II

~.

~,~ i ~ - ~

~

CJ2.

II

II

II

•

250

D. Agassi, T.F. George / Surface plasmon dispersion

"t~ = { -

i[k2(O) + k2(1)]

/'/2

4flk~ (i/flk G +_t T rr/k G) j i[k2(O) +2k2(1) ]

{ =

(Case II),

for k < k p / ~ - ,

(Case I),

for k > kpv~,

}1/2

+ 4#ko (i/Bko + t • ~/kc)

(C.lO.i)

{(i

),?/~:=2

~

+t-T-~

i

2

-4flkG(k (0)

+ k2(1))]1/2,1~ / f o r k < kpf~-, ] ] 1/2

=2 for all

~-~c +t-T~--~

+ k2(1))])

, for k > kp/g/2,

t,

(C.10.j)

where kp = tOp//C and where the square roots in (C.10) and throughout [eq. (2.3)] are chosen so that the real or imaginary parts are positive. We have also used in (C.10), and in the rest of the appendix, c o = 1, and the notation "Cases I or II" are defined in (C.8). Expressions (C.10) yield a quick order-of-magnitude estimate for gio(t). Consider first the g >> t regime. Since gk(O) ~ gk(1) --- gkp = fl >> 1, the elements belonging to classes a, fl and ,/call for the asymptotic limit (C.9.a), while those of class 8 call for the limit (C.9.b). Therefore, the leading t-independent factors in (C.10) yield the following dominating exponents in &j(t), classified according to (C.7): class a ~ exp(2i[2 g2k2(1)] 1/2},

(C.11.a)

class fl = exp{ - 2 [ - 2gEk2 (0)] 1/2},

(C.11.b)

class'g~exp{Ei[4g2(k2(O)+k2(1))] 1/2 } ~exp(-E[-4g2(k2(O)+k2(1))]1/2}, class 8 --. 1,

for g >> t.

fork<=kp/v~, fork>kp/x/~,

(C.11.c) (C.11.d)

The symbol "1" is the class-8 entry indicates that no exponential factors appear, only powers. Estimates (C.11) imply that as f l ~ oo, the class-a elements are always the smallest [k2(1) < 0, always]. The class--y elements are larger than the class-a elements by a factor exp[2i[4gEkE(1)]l/2k2(O)/k2(1)] >> 1 for k < kp/V~, and by yet a larger factor when k > kp//1/~-. It is therefore justified to discard (set to zero) the class-a elements for the g >> t regime.

D. Agassi, T.F. George /Surfaceplasmon dispersion

251

Consider now the complementary g << t regime. Considerations similar to those above yield class a -- exp{ 2i[itgk2(1)] 1/2}, class fl --- exp( -

(C.12.a)

2[itgk2(O)]'/2),

(C.12.b)

class ~-~exp{2i[itg(k20) + k2(O))]l/2}, ~.exp{-E[itg(k2(O)+k2(1))]l/2}, class 8 ----1,

for k <=kp/~/2, fork>__kp/~/2,

(C.12.c)

for g << t.

(C.12.d)

Since the real and imaginary parts of the square root are always chosen to be positive, (C.12) clearly indicates that the dements belonging to classes a, fl and y are exponentially small, while the class-8 elements are of the order 1 (i.e., fall off as an inverse power). Therefore, in conjunction with the Poisson sum (C.3), there is a sharp cutoff in the n-summation for the g i j ( t ) pertaining to classes a, fl and ~,, whereas for the class-8 element it is necessary to sum all the way to infinity. The cutoff L (in the n-summation) ocurs [from (C.10)] when 2~rL/k G = t = 2g or L ~ fl/cr>>> 1. It is sharp in the sense that over a range An/L << 1 the n-dependent contributions diminish as e x p ( - Vcfl). The asymptotic (fl >> 1) vanishing of the class-a elements implies that in the evaluation of ~¢-1 only the eight elements of clases ~/ and 8 contribute [17]. We therefore turn now to the calculation of these entries for a fixed t and then consider the summation mandated by the Poisson formula (C.3). Straightforward algebra gives for the class-), elements

ga,2(t)= pr~(g, kll)(3[k2(O)+ k2(1)]}cos [v~ (t)], g3,4(t) =

P~(g,kl{){l[k2(O) + k2(i)]

(C.13.a)

} cos[v~(t)],

(C.13.b)

gl.4 (t) = Pv~ (g, kll){ ¼[3k2(0) + k2(1)] }i sin[ v ~ ( t ) ] ,

(C.13.c)

g3.2(t)=Pv~(g,

(C.13.d)

kll){l[k2(0) +

3k2(1)]}isin[v~(t)],

where the labels > and < refer to the domains where respectively, and

kp/v~,

Pv>(g'k)=--~[

4

× exp[i~'(~ -

k >=kp/V~

or k __<

{ _ 492 [ ; 2 ( - ~ ~ k 2 (1)]) 1/2

kll/k c)]

exp{ - 2 [ - 4 g 2 ( k 2 ( O ) + k2(1))] 1/2}, (C.14.a)

D. AgassL T.F. George/ Surfaceplasmon dispersion

252

~,>(t)=2{_4g2[k2(O)+k2(1)])a/2

i

t+C

)

(C.14.b)

( exp, Pv< ( g' k ) = --~o [ -

4

{ 4 g 2-(k-~)) + k 2 (1) l } l /2

X exp[i~r (~ - kll/k o)] exp{ 2i[4g/(k2(0) + k2(1))] 1/2}, (C.14.c)

u< ( t ) = Ei( 4g2[ k2(O) + k2(1)] ,]a/2 -~g i(t+rr/ko).

(C.14.d)

Note that y~(t) do not depend on g! Similarly, the class-8 elements g2,1(t) =P~(kll)(Ee 3'~i/2 [ I f ( g , t ) + I f ( g ,

t)]

+½[k2(O) +k2(1)] ei'~/2[I~-(g, t)+I~-(g, t)]},

g2,3(t)=P~(kll)(2e3"~i/2 ( - i ) [ I 3 ( g , t ) - I ~ ( g ,

(C.15.a)

t)]

+ 1ik2(O)ei=/2[If(g, t ) - I f ( g ,

t)] },

(C.15.b)

+½ik2(1) ei"/2[ I i ( g, t) - I~-(g, t)]},

(C.15.c)

g4.1(t)=Pn(kll)(-Zie3"i/2

[If(g, t)-If(g,

g4.3(t) = Ps(kll)2e 3"i/2 [ I f ( g , t ) + I f ( g ,

t)]

t)],

(C.15.d)

where

P~( k ) = C exp[i(-¢r/4 + ~rkll/k G) ],

(C.16.a)

I+ ( g, t) = [i/flk o +_t -T-~r/kG ]"

(C.16.b)

Expressions (C.15) and (C.16) are valid for the whole k < kp domain. The last step is to perform the n-summation implied by (C.3). As pointed out before, for the class-y dements (C.14), there is a sharp cutoff at t = 27rn/k o ~ 2g, whereas for the class-8 elements (C.15) all values of t = 2~rn/k c must be summed. The summations pertaining to (C.13) and (C.15) are simply caried out by using the identities [18] L

Ic(a , ~,) =

•

e -iv"cos[a(n+ 1/2)]

n~--L

1

=~e

i v / 2f , s i nI, L ,fa + V ) ~ 1+

[ s-~n [a-~a + ~,) l

s. .i .n. [. .L ( a - y)] ]

sin[½(a-,/)l

]'

(C.17.a)

D. Agassi, T.F. George / Surface plasmon dispersion

253

L

Is(a , y) =

Y'. e -ivn sin[a(n + 1/2)] n ~ --L

• [ sin[L(a +'r)] s i n [ L ( a - ~,)] ] = ½i e 'v/2 [ sin[½(----~+V)--] - s i n [ ½ ( a - V)] ~' i i ( a ' y) =

~ e_iv n 1 _ ~" e +iav ,,= -oo a +n sin (act) ' 1 d2

- - I i ( a , y). I3(a, ~) - 2 da:

(C.17.b)

(C.17.c) (C.17.d)

Using (C.17) in (C.3) gives for the class-y matrix elements

A , , 2 = P 2 ( g , kll ) {~[k2(O) +k2(1)]} I ¢ ( a ~ , y ) ,

(C.18.a)

A 3 . a = P ? ( g , kll ) (~[k2(0) +k2(1)]} I¢(a~, y),

(C.18.b)

A 1 , 4 = P v ~ ( g , kll ) {¼13k2(0)+ k:(1)] }i Is(a ~ , y),

(C.18.c)

A3,2=PvX(g, kll ) (¼[k2(0)+ 3k2(1)] }i Is(aX, y),

(C.18.d)

where a < = i[k2(O) + k2(1)] 1/2¢l'/kG,

for

k < kplv/2,

(C.18.e)

.

for k > kp/~/2,

(C.18.f)

'"

>=

tr/K G,

and

= 2~rkll/kG.

(C.19)

For the class-6 matrix elements we obtain A2a = 0

[of order (1/fl)],

A2.3 = ½Ps(k,l) e-iY/2 [2( ~r2 + y2)(kc/2cr)2 _ k2(0)], A4,, = ½Ps(k,i) e-iv/2 [2( ~r: + y2 ) ( k c , / 2 . ) 2 _ k2(1)], A4, 3 0 [of order (1/fl)], (C.20) In evaluating (C.20) we used a = - 1 / 2 + d~(1/fl) = -0.5 in Ii(a, y) and I3(a, ~,) of eq. (C.17), and 3' is defined in (C.19). Note that A4.1 never vanishes and A2.3 vanishes along a curve within the light cone k(O) = k,. =

Appendix D: Local field enhancement

Our starting point is the projection of the slow component in the field, Es(X, Z) [eq. (5.1)], along a vector b, which is either ~ or L From (5.1), (3.10)

D.Agassi,T.F.George/ Surfaceplasmondispersion

254

and (C.20) one observes

~ . E s ( x , 2) = -~(,=~_oogll-t(

l)- Z [ ~ I g'al(l)][~l b4(l,l't(l)] A23 ~Xta3(l) , tt

x, = ~.~o,+(t/exp(;[~,~

Ebz(l)l~(l)

,

Jt/

+ Wo(t)z] ).

(D.1)

For fl >> 1 and normal incidence (kll = 0), the entries to (D.1) are [eqs. (3.3), (4.16) and (C.2)] #(1)=

ik(0)

1

1

exp[-111/2fl],

I11>>1,

(D.2.a)

B1/2 2¢Tg a 1( l ) = ik G exp( - iqr/4) I 11 exp[ - 11 I/2fl ],

(D.2.b)

a3(l ) = k G e x p [ - i ~ r / 4 ] l e x p [ - II 12fl],

(D.2.c)

b2(l) =

Ciko I l l ( - 1)' exp[ - [1 I / 2 f l ] ,

(D.2.d)

b,(/) =

CkGI ( -- 1)' exp[ -- II I / 2 f l ] ,

(D.2.e)

and from (2.3) one deduces [14] ~t(0) = - i e x p [ - i ( g k ( 0 )

+ 7r/4)]

k(0)k(1)

1

(D.3)

[1,(o) + I,(1)] ~/~ 2 ¢ ~ We now specialize (D.1) to points T and B in the notation of fig. 1. Using the definitions (2.3), one obtains Xt(T;'~)

Wo( l) k(0)

exp[2iW°(l)g] -- - O ( [ r ] -

Ill)exp[2ik(O)g], (D.4.a)

Xt(T;~ ) =

x,(s;~) Xz(B;e) =

k, exp[2iWo(l)g ] -- / O ( [ r ] -

Il l ) e x p [ 2 i k ( 0 ) g ] ,

W°(l)exp[ik,d/2] = - 7i I / 1 ( - 1 ) ' , k(0) kt

.

l

t

- ~ e x p [ , k t d / 2 ] = r ( - 1) ,

(D.4.b) (D.4.c) (D.4.d)

where, for instance, X/(T;.~) indicates that in (D.1) one substitutes b = ~ and the coordinates of point T, r = k ( 0 ) / k o = ,(/Sod/X,

(D.5)

D. Agassi, T.F. George / Surface plasmon dispersion

255

[r] is the integer closest to r from below, and ~ is the wavelength of the incident light. With the help of (D.2), (D.3) and (D.4), the field projections (D.1) can now be calculated in a straightforward manner. Note first that since b4(l ) = - b 4 ( - 1) and # ( 1 ) = # ( - l), the term is (5.1) involving h4,1 does not contribute. The other terms give .~-Es(T)

F1 ( ~

=

Xt(T '

k)l~(l)},

(D.6.a)

/~o¢

~-Es(T) = 1 ( - ~ 2 . 3

~)a3(l,][~b2(l)t~(,)]),

[~Xt(T,

1{ 5

(D.6.b)

,o6o,

k. E~(B) = ~

l=c,o

e. E~(B) =

E X,(B, ~)~( l)] [ E b2( ~),(1)]

(D.6.d)

or explicitly ].~.Es(T ) 1 2 _

2

0(I

1 k2(0)

D 2 2~rflr

-

-

-

3/2

-

1 k2(0) D 2 2~rfl I2,x'

I£.Es(T) 12 . . . . .1

4k2(0)

for r << 1,

(D.7.a)

for r >> 1,

(D.7.b)

I2I 2 2 T.z

(D.7.c)

D2 r2[~_r2] 2 2~B'

i~. Es(B ) f2 =

i

k2(0) -i

~i2x,

(O.7.d)

D 2 2~rfl r

le'Es(B) l 2 = 1 4k2(0) 1 1212 D 2 2¢rfl r 2 ( ½ _ r 2 ) 2 2 B,z,

(D.7.e)

where

12=

V/[I[ ( - 1)' e x p [ - ]l

]/fl]

1 = -V/~,

(D.8.a)

[r] IT, x =

1 - - - ~ e x p [ - [l [/2fl] = V~,

[1[>1 ~ -

1 << r < 2fl,

(D.8.b)

D. Agassi, T.F George / Surface plasmon dispersion

256

[#] IT,z=

E 12 expI-Ill/2fl]

=r3,

(D.8.c)

t= - [ r l IB,x =

~ ~T(-1)Zexp[-1ll/2fl]=l/~, /~-oo

IB,z = ~ 12(-1)'exp[-1ll/2fl]=fl.

(D.8.d) (D.8.e)

I~ -- o¢

Using (D.8), we obtain (5.3). References [1] [2] [3] [4] [5]

[6] [7] [8] [9] [10]

[11]

[12] [13] [14] [15] [16] [17] [18]

S.R.J. Brueck and D.J. Ehrlich, Phys. Rev. Letters 48 (1982) 1678. C.J. Chen, H.H. Gilgen and R.M. Osgood, Opt. Letters 10 (1985) 173. D. Agassi and T.F. George, Phys. Rev. B, in press. R. Petit, Ed., Electromagnetic Theory of Diffraction Gratings (Springer, New York, 1980). See, for example, F. Toigo, A. Marvin, C. Celli and N.R. Hill, Phys. Rev. B15 (1977) 5618; N. Garcia, V. Celli, N.R, Hill and N. Cabrera, Phys. Rev. B18 (1978) 5184; N. Garcia, G. Diaz, J.J. Saenz and C. Ocal, Surface Sci. 143 (1984) 338. P. Sheng, R.S. Stepleman and P.N. Sanda, Phys. Rev. B26 (1982) 2907; K.T. Lee and T.F. George, Phys. Rev. B31 (1985) 5106. B. Laks, D.L. Mills and A.A. Maradudin, Phys. Rev. B23 (1981) 4965; N.E. Glass and A.A. Maradudin, Phys. Rev. B24 (1981) 595. V.M. Agranovich and D.L. Mills, Eds., Surface Polaritons (North-Holland, New York, 1982). E.N. Economou, Phys. Rev. 182 (1968) 539. See, for example, M. Weber and D.L. Mills, Phys. Rev. B27 (1983) 2698; E.V. Albano, S. Daiser, G. Ertl, R. Miranda, K. Wandelt and N. Garcia, Phys. Rev. Letters 51 (1983) 2314. See, for example, M. Hutchinson, K.T. Lee, W.C. Murphy, A.C. Beri and T.F. George, in: Laser-Controlled Chemical Processing of Surfaces, Eds. A.W. Johnson, D.J. Ehrlich and H.R. Schlossberg (North-Holland, New York, 1984). N.D. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Rinehardt and Winston, New York, 1976) ch. 9. See, for example, P.K. Aravind and H. Metiu, Surface Sci. 124 (1983) 506. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964) ch. 9. F. Oberhettinger, Fourier Expansions (Academic Press, New York, 1973) p. 5. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1965) p. 340, relations 10 and 11. See, for example, M.L. Mehta, Elements of Matrix Theory (Hindustan Publishing, Delhi, 1977) p. 21. Ref. [16], p. 30, relations 3 and 4; p. 40, relations 5 and 6.