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Surface plasmons propagating parallel to the grooves of a large amplitude grating
240
Surface Science I 14 ( 1982) 240-250 North-Holland Publishing Company
SURFACE PLASMONS PROPAGATING PARALLEL TO THE GROOVES OF A LARGE AMPLITUDE GRATING *
and A.A. MARADUDIN
Received
**
30 June 1981
The dispersion relation is obtained for a surface plasmon propagating parallel to the groo~ea of a large amplitude grating. Two variants of the Rayleigh method arc used: the ftrat requircb the evaluation of the zeros of a determinant; the second yields the dispersion relatton in terms of the solution of a matrix eigenvalue problem. The dispersion relations are solved numerically for a sinusoidal and for a symmetric sawtooth grating profile. yielding in each case an inftnity of dtscretc branches.
1. Introduction Various theoretical studies have recently been undertaken on the propagation of electromagnetic waves across a large amplitude surface grating [l-3]. Interest in the interaction of electromagnetic radiation with large amplitude surface structure has been particularly acute due to its apparent role in the giant enhancement of the Raman signal from molecules adsorbed on a metal surface [4]. Since the randomly rough surface is very difficult to treat, except in the small roughness limit, the simple deterministic surface profile provided by a one-dimensional grating is taken to serve as a model for a rough surface. In refs. [2] and [3], surface polaritons and plasmons [S], respectively, which propagate normal to the grooves of a grating ruled on a free electron metal, described by dielectric constant C(W), are shown to have dispersion relations with an infinite number of branches. The existence of these branches is * UC1 Technical Report No. 81-39. ** Permanent address: Department of Physics, USA
0039-6028/82/0000-0000/$02.75
University
of California,
0 1982 North-Holland
Irvine. California
927 17.
N. E. Gluss, A.A. Murududin / Surfuce phsmons
241
understood as follows [2]: The surface polariton propagating with wave vector k across the grating of period a, undergoes Bragg reflections that mix in components k + (2m/a)n (n = * 1, * 2,. . . ) to produce a Bloch wave, the dispersion relation of which is described by folding the flat-surface dispersion curve into the first Brillouin zone (cut-off at the light-line), where the reflections at the zone boundary cause gaps to open there. These gaps promote some branches above the flat-surface plasmon frequency, ~r/fi, so that the dispersion relation consists of an infinity of discrete branches which, in the nonretarded limit (k Z+w/c), are nearly symmetrically disposed above and below the line w = up/&? and converge into that line. Since the existence of a closely spaced set of surface plasmon frequencies may play a role in surface enhanced Raman scattering [6], it is interesting now to pose the question of whether such frequency bands also arise when a plasmon propagates with wave vector parallel to the grooves of a grating. In this case there is no Bragg scattering and no Brillouin zone - hence, no zone boundary gaps. Nevertheless, a set of discrete frequency bands can be expected from consideration of a different problem: electromagnetic excitations localized at the apex of a dielectric wedge surrounded by vacuum. It was shown several years ago [7] that such modes exist in the electrostatic approximation. They are wavelike in the direction parallel to the edge of the wedge and decay in amplitude with distance from the interface, both into the vacuum and into the dielectric, and thus may be called edge plasmons. In ref. [7] a sharp dielectric wedge was considered, whose boundary was formed by the intersection of two semi-infinite planes. The resulting modes were found to have frequencies that are independent of the wave vector q, characterizing propagation along the edge, but that are functions of a continuously varying quantum number. Moreover, the electric fields associated with these modes are singular. In subsequent work, Davis [8] studied a wedge whose boundary is a hyperbolic cylinder, while Eguiluz and Maradudin [9] considered one whose boundary is a parabolic cylinder. In both cases, it was found that rounding the edge removes the singularity in the electric fields and changes the eigenvalue spectrum from being continuous and independent of q to being discrete and a function of q. Since a grating can be thought of as being produced by arranging a large number of dielectric wedges with their edges parallel to each other and in one plane, each groove should possess a discrete spectrum of plasmon frequency branches, affected by the presence of neighboring grooves. On this basis, we expect to find an infinite number of branches in the surface plasmon dispersion relation for propagation parallel to the grooves, as well as for propagation normal to the grooves. In this paper we obtain the dispersion relation for a surface plasmon propagating along the grooves of a large amplitude grating by two variants of Rayleigh’s method [lo].
2. Theory
We consider a surface plasmon propagating parallel to the grooves of a grating, whose profile is given by x3 = [(xi), where {(xl) is a periodic surface profile function with perioda. The region defined by x3 > {(x,) is vacuum, while the region x3 < {(x,) is a dielectric medium characterized by the dielectric function e(w). We seek the solution of Laplace’s equation V2zcp(xlw)
=o
(2.1)
that (1) satisfies x2: CPHJ)
the infinitesimal
of the problem
along (2.2)
q2 f(.y+)=O;
(2.3)
eq. (2.1) becomes
-$+-$i
invariance
Ia),
= exptiqxMx,x,
whereby
translational
I
3
I
as Ix3 ( ---t00, and that (3) satisfies
that (2) vanishes
the boundary
conditions: (2Aa)
cp~~/~~1,~=~(x,)_=(P~~/~~/~~=~~x,~~’
(2.4b) In eq. (2.4b)
(2.5) is the derivative surface. The solutions
“(x,x,
the unit
vector
directed
of eq. (2.3) that give vanishing
normally
outward
from
the
F(X) at Ix3 ) - co are:
: A m exp[i%x, VI= -‘x
- a,(q)
~31 I
x3
‘Snmx~
(2&a)
10) =I: $ B, exp[i%x, r?,=--30
+ a,,(q)
x.J],
x3
<
(2.6b)
f”(X,X,l~>
f
along
=
Lniw
where G,, = (2v/a)
m,
(2.7)
a,(q)
+ q2)1’2.
(2.8)
= (G;
We now assume that the solutions
(2.6). which are valid outside the selvedge
region, can be continued into the surface itself - the Rayleigh hypothesis; and we substitute them into the boundary conditions (2.4), thus obtaining a pair of homogeneous equations for the expansion coefficients: $
{ -exp[iG,x,
+exp[iG,,x,
+a,(q)
5 {exp[iG,,x, )n= -33 +~(a>
-a,,(q) ((x,)1
-a,(q)
exp[iG,x,
s(x,)]
A,,
%)
@r)&,(q)
+ a,(q)
(2.9a)
==O, +iGJ’]
l(xr)][a,(q)
A,
- iGJ’]
%j
=O,
(2.9b)
where the prime denotes differentiation with respect to x,. From eqs. (2.9) we shall proceed in two different ways to obtain the dispersion relation for the surface plasmons. In the first method we introduce into eqs. (2.9) the following expansions: exp[ --a S(xl>]
S’(x,)
=
(2.10a)
5 $,(a> exp(iG,xt)+ il=-30
exp[ --a! [(x,)]
=
-?$(a) g n= --m
sxp(iG,,x,),
(2.10b)
where
%(4 =;i:::,
dx, exp[ -a
{(x,)
(2.11)
- iG,,x,] ;
and we equate to zero the pth Fourier coefficient in each resulting Fourier series, to obtain the infinite homogeneous linear equations for the {A,} and {&J:
mzz_ (-$?-,,b%m -4, + &?k-~,(q))
R?,}= 0,
p=O,
‘1,
-1-2,...,
(2.12a)
p=o,
I-1,
*2,
(2.12b)
.. . .
The solvability condition for eqs. (2.12), namely that the determinant of the coefficients of the unknowns vanish, provides the dispersion relation. Equations exactly analogous to eqs. (2.12) have been obtained from the Rayleigh hypothesis to provide the dispersion relations of p-polarized surface polaritons [ I,1 I], plasmons [I 11, and Rayleigh elastic waves [ 121, all propa-
gating pe~e~~cu~~~y to a grating’s grooves. In each of these three cases it has been shown analyticahy [I 1,121 that precisely the same secular determinant equation derives from the formally exact extinction theorem form of Green’s theorem - independent of Rayleigh’s hypothesis. This supports the idea (first stated in ref. f I)) that Rayleigh’s hypothesis is generally correct - at least for ~orn~ut~ng dispersion relations - and that any uon~onv~~gent behavior in its numerical application is only due to the choice of the particular basis set in the expansion. In fact, refs. [2], [3] and [ 121show that convergent numerical results are obtained, for large roughness, by use of the extinction theorem equations which are equivalent to the full set of Rayleigh equations. Since the full set of Rayleigh equations (2.12) for the present problem are sbown in section3 to lead to highly convergent numericaf results, even fur Iarge roughness, it is possible to assume by analogy that they are correct. Nevertheless, a second method to obtain the dispersion relation from eqs. (2.9) is pursued, because it offers greater ease jn the numerical calculations, although it leads to convergent. results only for relatively smaJl roughness. We first rn~t~p~y eq. (2.9a) by
--e(a)-’
exp[a,(q)
i(xl)
- iG,,xl],
add the resulting equations, and then integrate the sum on xt over the interval (-rt/2, a/2)* This finafly feads to a set of “reduced” Rayleigh equations (so-called by Toigo et ai. [I]) for the {A,) alone:
~exp~~~~(q~-~~~(q)~ %(x,)f ewt--iGj-,,x,l&, =o.
@IS)
The term containing {‘(x1 ) is nonzero only for M f n, on the assumption made here that &a/2 _) = [(-(a/2 + ), in which case it can be integrated by parts to yield the matrix equation: f2.14) where %Aq)=o,
n=m,
where !#Jcx) was defined by eq. (2.11).
(2.1Sa)
N. E. Gluss, A.A. Murcrdudin / Surfuce plusn~ons
245
If we denote the eigenvalues of M(q) by {X,(q)}, the dispersion relation surface plasmons propagating parallel to the grating grooves becomes:
for
(2.16) giving a separate branch for each distinct eigenvalue, labeled s. Equations analogous to eqs. (2.14)-(2.16) were found previously [3], also from the reduced Rayleigh equations, for the case of surface plasmons propagating normal to the grooves of a grating (although in that case, unlike the present, the matrix and hence dispersion curves are periodic in 2a/a).
3. Numerical results The numerical solution of the dispersion relation given by the two methods in section2, is now described for the case in which the dielectric medium is a free electron metal: e(w) = 1 - ,;/GJ2,
(3.1)
where the bulk plasma frequency wP will be taken as AU, = 15.3 eV for Al. Two different surface profile functions are considered: the first is the sinusoidal profile, (3.2) and the second is the symmetric Z(X,) =A+
(4W
LXX,) = h - (4h/a)
sawtooth,
x,,
-a/2Gx,
Xl,
O&x,
GO,
(3.3a) (3.3b)
The integral $,J (Y), defined by eq. (2.11) and needed for both the full Rayleigh equations (2.12) and for the reduced Rayleigh eigenvalue equations (2.1.Q is given analytically as follows. For the sinusoidal profile: Sl( &LX)= (7 1)” I,(@),
(3.4)
where 1,,(x) is the modified
.$(a) =
4ha n2n2 + 4h2a”
%b> = r2n2
-4hcu + 4h2a2
Bessel function,
sinh ha,
cash ha,
and for the sawtooth:
n even,
(3.5a)
n odd.
(3Sb)
3.1. Method of the reduced Rayleigh
equations
With the dielectric function of eq. (3.1) the dispersion relation from eq. (2.16) for the reduced Rayleigh method can be written: %(4)
= (a,/@
that follows
(3.6)
(1 -k(9))“*.
The eigenvalues X,(q) of the infinite matrix /V(q) of eqs. (2.15) are found, for a fixed q, by diagonalizig a matrix of finite dimension N (corresponding to n, m = -N/2,..., 0, . N/2 - 1 in eqs. (2.1 S)), and searching for convergence as N is increased. For the sinusoidal profile, the eigenvalues are found in pairs =h,( q), which decrease rapidly in magnitude (1A,(q) I> 1X2(q) I> . . ), and i-h,(q),..., thus correspond to an infinity of branches of the dispersion relation which are nearly symmetrically disposed above and below the dispersionless flat-surface curve at w = wP/ fi and which merge into that flat-surface dispersion curve. The difference between the frequency of the flat-surface plasmon and the frequency of the surface plasmon propagating parallel to a one-dimensional grating can be expressed as: (3.7) In fig. 1 we have plotted AU,, ( wP/ fi)) ’ for the three branches of highest frequency above the flat-surface frequency, for the case where SO/a = 0.064. Convergence with increasing N is most rapid for the largest eigenvalue pair, ‘h,(q), and becomes slower for each successively smaller X,(q). Moreover, the convergence becomes slower as currugation strength lo /a is increased. Convergence for the sinusoidal profile is found for lo/a up to 0.12, at which point h, is divergent and h, obtains an unphysical imaginary part. For the sawtooth profile, the results are all qualitatively the same as for the sinusoidal profile; however, divergent results appear almost immediately for this nonanalytic profile function. For h/a = 0.016, the results for the first three branches of the dispersion relation are converging, but reach only two-figure accuracy with N up to 52. For h/a = 0.04, divergence begins at N = 28, after convergence to no better than one-figure. 3.2. Method of the full Rayleigh
equations
The zeros, ws( q), of the infinite-dimensional determinant of the coefficients in eqs. (2.12) are found by solving the N-dimensional determinant equation (p,m= -N/4+ l,..., 0, . . . N/4 in eqs. (2.12)) and searching for convergence as N is increased. For the sinusoidal profile, convergence using the full Rayleigh equations is found for lo/a up to 0.6 with matrices of dimension up to N = 52. Within this range, the rate of convergence becomes slower with increasing {,,/a and also
247
Sinusoidal
0.05
Grating
0.04
;&
0.03
% 3” 0.02 -cl 0.0 i
0.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
k/(2m’o) Fig. I. Dispersion curves for surface plasmons propagating along the grooves of a sinusoidal profile with &,/0=0.064. PIotted as the normalized shift from the flat-surface frequency ,=,,/A. for s+ = l +-3+, the three largest shifted branches that are above Calculated by either the reduced or full Rayteigh methods.
w,/fi
k/(2n/a) Fig. 2. Dispersion curves for surface plasmons propagating along the grooves of a sinusoidal profile with y,/u=O.25. Plotted as the normalized shift from the flat-surface frequency w=cJ,/&, for s+=l+-5+, the five largest shifted branches that are above w,/v’!?. Calculated by the full Rayleigh method.
with increasing q (e.g. in the first branch for q(2a/a)mm' = 0.5,with {o/a = 0.064, there is convergence to nine-figure accuracy with N = 20, but with lo/a = 0.6 convergence to five-figure accuracy requires N = 48). At lo/a = 0.7 the results have converged to four-figure accuracy, when at N = 44 divergence begins. The dispersion curves for lo/a = 0.064 are identical to those found with the reduced Rayleigh equations (fig. l), but only three pairs of branches are resolved. For To/a = 0.25, five branches of the dispersion relation are resolved and plotted in fig. 2, as the normalized frequency difference with respect to wP/fi, for the branches above the flat-surface curve. When the sawtooth profile is employed with the full Rayleigh equations, convergence is found for h/a = 0.04 to four-figure accuracy (in the highest frequency branch at q(2s/a)- ' = 0.5) with N = 44. For h/u = 0.06 divergence of the results is seen (in the highest frequency branch at q(2v/a)- ' = 0.5). beginning at N = 28 after an initial convergence to three-figure accuracy; while at [o/u = 0.1 divergence begins almost immediately.
4. Conclusions With respect to the two methods of solution, we may conclude that the computationally simpler method is the one based on the reduced form of the Rayleigh equations, which can be formulated as an N-dimensional matrix eigenvalue problem, immediately yielding N branches of the surface plasmon dispersion relation. Using the full Rayleigh equations, we must search individually for each zero of a determinant, and can resolve only a few of the branches. The reduced Rayleigh eigenvalue approach, however, yields convergent results only for small roughness (lo/a 5 0.1) when applied to an analytic profile function, and is severely restricted for a non-analytic profile. The full Rayleigh method, on the other hand, yields results convergent to high accuracy for large amplitudes (&/a d 0.6) for the analytic case, and also yields convergent results for the nonanalytic, sawtooth profile, though only for small roughness (lo /a 5 0.06). This breakdown of the reduced Rayleigh eigenvalue approach at much smaller corrugation strengths than those with the full Rayleigh method is the same behavior found in ref. [3], for the case of plasmons propagating across the grating (note that the formally exact extinction theorem method used in ref. [3] is equivalent to the full Rayleigh method). The numerical results confirm the expectation based on the findings with edge plasmons, as cited in the introduction, that there exists an infinite number of discrete branches of the dispersion relation for surface plasmons propagating parallel to a one-dimensional grating. Comparison of the present results with those of ref. [9] reveals that for both the rough-surface plasmons and edge plasmons, the branches approach o = or /fi as q + m, but that at q - 0 the
249
two cases are qualitativeIy dissimilar: the surface plasmon frequencies preach w = wr/$?, or close to it, while the edge plasmon frequencies w = o for the modes above up/ u’z and go to w = 0 for the modes
reapgo to below
“p/ P2. It is understood that for q + cg, the penetration depth of the fields into the medium goes to zero, so that the curvature of the interface in either case is not seen; and all modes are degenerate at the w,/a of a flat surface. With decreasing q, as the penetration depth increases, the fields localized on the wedge decay with distance into the medium along a direction initially normal to the interface such that the field on one side of the wedge eventually interferes with the field on the other side. Thus, as with surface modes on thin films, interference of the fields from opposite faces causes the edge mode frequencies to be pushed up to or or down to zero. But for the modes on the surface grating of a semi-infinite medium, the field decays into the medium along a direction normal to the average, that is, flat surface; and so there is no interference. As these fields penetrate vertically into the bulk, they simply see less of the grating and so behave like flat-surface plasmons as ~1-f 0: their frequencies return to wP We therefore can understand physically how the infinite set of discrete branches in the dispersion relation arises for plasmons propagating along the grooves of a grating, and can calculate their frequencies with high accuracy for large corrugation strengths.
,/a.
Acknowledgements This research was partially supported by the US Air Force Office of Scientific Research, through Contract No. F49620-78-C-0019. The work of one of the authors (A.A.M.) was carried out under the auspices of an Alexander von Humboldt Senior US Scientist Award, whose support is gratefully acknowledged.
References Ill F. Toigo, A. Marvin, V. Celli and N.R. Hiil, Phys. Rev. B15 (1977) 5618. 121B. Laks, D.L. Mills and A.A. Maradudin, Phys. Rev. B23 (198 1) 4965. Phys. Rev. B24 (1981) 595. [31 N.E. Glass and A.A. Maradudin, Ed. W.F. Murphy [41 E. Burstein and C.Y. Chen, in: Proc. 7th Conf. on Raman Spectroscopy, (North-Holland, New York, 1980) p. 346. wave in a solid. resulting [51In this paper we use the term plasmon to denote the electromagnetic from the linear coupling of a photon to an electric or magnetic dipole active excitation in the solid, in the limit that retardation effects can be neglected. Chem. 84 (1977) 1. L61 D.L. Jeanmaire and R.P. Van Duyne. J. Electroanal. and A.A. Maradudin, Phys. Rev. Bh (I 972) 38 IO. 171 L. Dobnynski PI L.C. Davis, Phys. Rev. Bl4 (1976) 5523.
[9] A. Eguiluz and A.A. Maradudin, Phys. Rev. 014 (1976) 5526. [IO] Lord Rayleigh, Phil. Mag. 14 (1907) 70; Theory of Sound. Vol. 2. 2nd cd. (Dobcr. Neu York. 1945) p. x9. [I I] A.A. Maradudin, in: Surface Polaritons. Eda. V.M. Agranovich and D.L. Milk (North-Holland. Amsterdam, 1982). [ 121 N.E. Glass, R. Loudon and A.A. Maradudin. Phys. Rev. B, to bc published.
Surface Science I 14 ( 1982) 240-250 North-Holland Publishing Company
SURFACE PLASMONS PROPAGATING PARALLEL TO THE GROOVES OF A LARGE AMPLITUDE GRATING *
and A.A. MARADUDIN
Received
**
30 June 1981
The dispersion relation is obtained for a surface plasmon propagating parallel to the groo~ea of a large amplitude grating. Two variants of the Rayleigh method arc used: the ftrat requircb the evaluation of the zeros of a determinant; the second yields the dispersion relatton in terms of the solution of a matrix eigenvalue problem. The dispersion relations are solved numerically for a sinusoidal and for a symmetric sawtooth grating profile. yielding in each case an inftnity of dtscretc branches.
1. Introduction Various theoretical studies have recently been undertaken on the propagation of electromagnetic waves across a large amplitude surface grating [l-3]. Interest in the interaction of electromagnetic radiation with large amplitude surface structure has been particularly acute due to its apparent role in the giant enhancement of the Raman signal from molecules adsorbed on a metal surface [4]. Since the randomly rough surface is very difficult to treat, except in the small roughness limit, the simple deterministic surface profile provided by a one-dimensional grating is taken to serve as a model for a rough surface. In refs. [2] and [3], surface polaritons and plasmons [S], respectively, which propagate normal to the grooves of a grating ruled on a free electron metal, described by dielectric constant C(W), are shown to have dispersion relations with an infinite number of branches. The existence of these branches is * UC1 Technical Report No. 81-39. ** Permanent address: Department of Physics, USA
0039-6028/82/0000-0000/$02.75
University
of California,
0 1982 North-Holland
Irvine. California
927 17.
N. E. Gluss, A.A. Murududin / Surfuce phsmons
241
understood as follows [2]: The surface polariton propagating with wave vector k across the grating of period a, undergoes Bragg reflections that mix in components k + (2m/a)n (n = * 1, * 2,. . . ) to produce a Bloch wave, the dispersion relation of which is described by folding the flat-surface dispersion curve into the first Brillouin zone (cut-off at the light-line), where the reflections at the zone boundary cause gaps to open there. These gaps promote some branches above the flat-surface plasmon frequency, ~r/fi, so that the dispersion relation consists of an infinity of discrete branches which, in the nonretarded limit (k Z+w/c), are nearly symmetrically disposed above and below the line w = up/&? and converge into that line. Since the existence of a closely spaced set of surface plasmon frequencies may play a role in surface enhanced Raman scattering [6], it is interesting now to pose the question of whether such frequency bands also arise when a plasmon propagates with wave vector parallel to the grooves of a grating. In this case there is no Bragg scattering and no Brillouin zone - hence, no zone boundary gaps. Nevertheless, a set of discrete frequency bands can be expected from consideration of a different problem: electromagnetic excitations localized at the apex of a dielectric wedge surrounded by vacuum. It was shown several years ago [7] that such modes exist in the electrostatic approximation. They are wavelike in the direction parallel to the edge of the wedge and decay in amplitude with distance from the interface, both into the vacuum and into the dielectric, and thus may be called edge plasmons. In ref. [7] a sharp dielectric wedge was considered, whose boundary was formed by the intersection of two semi-infinite planes. The resulting modes were found to have frequencies that are independent of the wave vector q, characterizing propagation along the edge, but that are functions of a continuously varying quantum number. Moreover, the electric fields associated with these modes are singular. In subsequent work, Davis [8] studied a wedge whose boundary is a hyperbolic cylinder, while Eguiluz and Maradudin [9] considered one whose boundary is a parabolic cylinder. In both cases, it was found that rounding the edge removes the singularity in the electric fields and changes the eigenvalue spectrum from being continuous and independent of q to being discrete and a function of q. Since a grating can be thought of as being produced by arranging a large number of dielectric wedges with their edges parallel to each other and in one plane, each groove should possess a discrete spectrum of plasmon frequency branches, affected by the presence of neighboring grooves. On this basis, we expect to find an infinite number of branches in the surface plasmon dispersion relation for propagation parallel to the grooves, as well as for propagation normal to the grooves. In this paper we obtain the dispersion relation for a surface plasmon propagating along the grooves of a large amplitude grating by two variants of Rayleigh’s method [lo].
2. Theory
We consider a surface plasmon propagating parallel to the grooves of a grating, whose profile is given by x3 = [(xi), where {(xl) is a periodic surface profile function with perioda. The region defined by x3 > {(x,) is vacuum, while the region x3 < {(x,) is a dielectric medium characterized by the dielectric function e(w). We seek the solution of Laplace’s equation V2zcp(xlw)
=o
(2.1)
that (1) satisfies x2: CPHJ)
the infinitesimal
of the problem
along (2.2)
q2 f(.y+)=O;
(2.3)
eq. (2.1) becomes
-$+-$i
invariance
Ia),
= exptiqxMx,x,
whereby
translational
I
3
I
as Ix3 ( ---t00, and that (3) satisfies
that (2) vanishes
the boundary
conditions: (2Aa)
cp~~/~~1,~=~(x,)_=(P~~/~~/~~=~~x,~~’
(2.4b) In eq. (2.4b)
(2.5) is the derivative surface. The solutions
“(x,x,
the unit
vector
directed
of eq. (2.3) that give vanishing
normally
outward
from
the
F(X) at Ix3 ) - co are:
: A m exp[i%x, VI= -‘x
- a,(q)
~31 I
x3
‘Snmx~
(2&a)
10) =I: $ B, exp[i%x, r?,=--30
+ a,,(q)
x.J],
x3
<
(2.6b)
f”(X,X,l~>
f
along
=
Lniw
where G,, = (2v/a)
m,
(2.7)
a,(q)
+ q2)1’2.
(2.8)
= (G;
We now assume that the solutions
(2.6). which are valid outside the selvedge
region, can be continued into the surface itself - the Rayleigh hypothesis; and we substitute them into the boundary conditions (2.4), thus obtaining a pair of homogeneous equations for the expansion coefficients: $
{ -exp[iG,x,
+exp[iG,,x,
+a,(q)
5 {exp[iG,,x, )n= -33 +~(a>
-a,,(q) ((x,)1
-a,(q)
exp[iG,x,
s(x,)]
A,,
%)
@r)&,(q)
+ a,(q)
(2.9a)
==O, +iGJ’]
l(xr)][a,(q)
A,
- iGJ’]
%j
=O,
(2.9b)
where the prime denotes differentiation with respect to x,. From eqs. (2.9) we shall proceed in two different ways to obtain the dispersion relation for the surface plasmons. In the first method we introduce into eqs. (2.9) the following expansions: exp[ --a S(xl>]
S’(x,)
=
(2.10a)
5 $,(a> exp(iG,xt)+ il=-30
exp[ --a! [(x,)]
=
-?$(a) g n= --m
sxp(iG,,x,),
(2.10b)
where
%(4 =;i:::,
dx, exp[ -a
{(x,)
(2.11)
- iG,,x,] ;
and we equate to zero the pth Fourier coefficient in each resulting Fourier series, to obtain the infinite homogeneous linear equations for the {A,} and {&J:
mzz_ (-$?-,,b%m -4, + &?k-~,(q))
R?,}= 0,
p=O,
‘1,
-1-2,...,
(2.12a)
p=o,
I-1,
*2,
(2.12b)
.. . .
The solvability condition for eqs. (2.12), namely that the determinant of the coefficients of the unknowns vanish, provides the dispersion relation. Equations exactly analogous to eqs. (2.12) have been obtained from the Rayleigh hypothesis to provide the dispersion relations of p-polarized surface polaritons [ I,1 I], plasmons [I 11, and Rayleigh elastic waves [ 121, all propa-
gating pe~e~~cu~~~y to a grating’s grooves. In each of these three cases it has been shown analyticahy [I 1,121 that precisely the same secular determinant equation derives from the formally exact extinction theorem form of Green’s theorem - independent of Rayleigh’s hypothesis. This supports the idea (first stated in ref. f I)) that Rayleigh’s hypothesis is generally correct - at least for ~orn~ut~ng dispersion relations - and that any uon~onv~~gent behavior in its numerical application is only due to the choice of the particular basis set in the expansion. In fact, refs. [2], [3] and [ 121show that convergent numerical results are obtained, for large roughness, by use of the extinction theorem equations which are equivalent to the full set of Rayleigh equations. Since the full set of Rayleigh equations (2.12) for the present problem are sbown in section3 to lead to highly convergent numericaf results, even fur Iarge roughness, it is possible to assume by analogy that they are correct. Nevertheless, a second method to obtain the dispersion relation from eqs. (2.9) is pursued, because it offers greater ease jn the numerical calculations, although it leads to convergent. results only for relatively smaJl roughness. We first rn~t~p~y eq. (2.9a) by
--e(a)-’
exp[a,(q)
i(xl)
- iG,,xl],
add the resulting equations, and then integrate the sum on xt over the interval (-rt/2, a/2)* This finafly feads to a set of “reduced” Rayleigh equations (so-called by Toigo et ai. [I]) for the {A,) alone:
~exp~~~~(q~-~~~(q)~ %(x,)f ewt--iGj-,,x,l&, =o.
@IS)
The term containing {‘(x1 ) is nonzero only for M f n, on the assumption made here that &a/2 _) = [(-(a/2 + ), in which case it can be integrated by parts to yield the matrix equation: f2.14) where %Aq)=o,
n=m,
where !#Jcx) was defined by eq. (2.11).
(2.1Sa)
N. E. Gluss, A.A. Murcrdudin / Surfuce plusn~ons
245
If we denote the eigenvalues of M(q) by {X,(q)}, the dispersion relation surface plasmons propagating parallel to the grating grooves becomes:
for
(2.16) giving a separate branch for each distinct eigenvalue, labeled s. Equations analogous to eqs. (2.14)-(2.16) were found previously [3], also from the reduced Rayleigh equations, for the case of surface plasmons propagating normal to the grooves of a grating (although in that case, unlike the present, the matrix and hence dispersion curves are periodic in 2a/a).
3. Numerical results The numerical solution of the dispersion relation given by the two methods in section2, is now described for the case in which the dielectric medium is a free electron metal: e(w) = 1 - ,;/GJ2,
(3.1)
where the bulk plasma frequency wP will be taken as AU, = 15.3 eV for Al. Two different surface profile functions are considered: the first is the sinusoidal profile, (3.2) and the second is the symmetric Z(X,) =A+
(4W
LXX,) = h - (4h/a)
sawtooth,
x,,
-a/2Gx,
Xl,
O&x,
GO,
(3.3a) (3.3b)
The integral $,J (Y), defined by eq. (2.11) and needed for both the full Rayleigh equations (2.12) and for the reduced Rayleigh eigenvalue equations (2.1.Q is given analytically as follows. For the sinusoidal profile: Sl( &LX)= (7 1)” I,(@),
(3.4)
where 1,,(x) is the modified
.$(a) =
4ha n2n2 + 4h2a”
%b> = r2n2
-4hcu + 4h2a2
Bessel function,
sinh ha,
cash ha,
and for the sawtooth:
n even,
(3.5a)
n odd.
(3Sb)
3.1. Method of the reduced Rayleigh
equations
With the dielectric function of eq. (3.1) the dispersion relation from eq. (2.16) for the reduced Rayleigh method can be written: %(4)
= (a,/@
that follows
(3.6)
(1 -k(9))“*.
The eigenvalues X,(q) of the infinite matrix /V(q) of eqs. (2.15) are found, for a fixed q, by diagonalizig a matrix of finite dimension N (corresponding to n, m = -N/2,..., 0, . N/2 - 1 in eqs. (2.1 S)), and searching for convergence as N is increased. For the sinusoidal profile, the eigenvalues are found in pairs =h,( q), which decrease rapidly in magnitude (1A,(q) I> 1X2(q) I> . . ), and i-h,(q),..., thus correspond to an infinity of branches of the dispersion relation which are nearly symmetrically disposed above and below the dispersionless flat-surface curve at w = wP/ fi and which merge into that flat-surface dispersion curve. The difference between the frequency of the flat-surface plasmon and the frequency of the surface plasmon propagating parallel to a one-dimensional grating can be expressed as: (3.7) In fig. 1 we have plotted AU,, ( wP/ fi)) ’ for the three branches of highest frequency above the flat-surface frequency, for the case where SO/a = 0.064. Convergence with increasing N is most rapid for the largest eigenvalue pair, ‘h,(q), and becomes slower for each successively smaller X,(q). Moreover, the convergence becomes slower as currugation strength lo /a is increased. Convergence for the sinusoidal profile is found for lo/a up to 0.12, at which point h, is divergent and h, obtains an unphysical imaginary part. For the sawtooth profile, the results are all qualitatively the same as for the sinusoidal profile; however, divergent results appear almost immediately for this nonanalytic profile function. For h/a = 0.016, the results for the first three branches of the dispersion relation are converging, but reach only two-figure accuracy with N up to 52. For h/a = 0.04, divergence begins at N = 28, after convergence to no better than one-figure. 3.2. Method of the full Rayleigh
equations
The zeros, ws( q), of the infinite-dimensional determinant of the coefficients in eqs. (2.12) are found by solving the N-dimensional determinant equation (p,m= -N/4+ l,..., 0, . . . N/4 in eqs. (2.12)) and searching for convergence as N is increased. For the sinusoidal profile, convergence using the full Rayleigh equations is found for lo/a up to 0.6 with matrices of dimension up to N = 52. Within this range, the rate of convergence becomes slower with increasing {,,/a and also
247
Sinusoidal
0.05
Grating
0.04
;&
0.03
% 3” 0.02 -cl 0.0 i
0.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
k/(2m’o) Fig. I. Dispersion curves for surface plasmons propagating along the grooves of a sinusoidal profile with &,/0=0.064. PIotted as the normalized shift from the flat-surface frequency ,=,,/A. for s+ = l +-3+, the three largest shifted branches that are above Calculated by either the reduced or full Rayteigh methods.
w,/fi
k/(2n/a) Fig. 2. Dispersion curves for surface plasmons propagating along the grooves of a sinusoidal profile with y,/u=O.25. Plotted as the normalized shift from the flat-surface frequency w=cJ,/&, for s+=l+-5+, the five largest shifted branches that are above w,/v’!?. Calculated by the full Rayleigh method.
with increasing q (e.g. in the first branch for q(2a/a)mm' = 0.5,with {o/a = 0.064, there is convergence to nine-figure accuracy with N = 20, but with lo/a = 0.6 convergence to five-figure accuracy requires N = 48). At lo/a = 0.7 the results have converged to four-figure accuracy, when at N = 44 divergence begins. The dispersion curves for lo/a = 0.064 are identical to those found with the reduced Rayleigh equations (fig. l), but only three pairs of branches are resolved. For To/a = 0.25, five branches of the dispersion relation are resolved and plotted in fig. 2, as the normalized frequency difference with respect to wP/fi, for the branches above the flat-surface curve. When the sawtooth profile is employed with the full Rayleigh equations, convergence is found for h/a = 0.04 to four-figure accuracy (in the highest frequency branch at q(2s/a)- ' = 0.5) with N = 44. For h/u = 0.06 divergence of the results is seen (in the highest frequency branch at q(2v/a)- ' = 0.5). beginning at N = 28 after an initial convergence to three-figure accuracy; while at [o/u = 0.1 divergence begins almost immediately.
4. Conclusions With respect to the two methods of solution, we may conclude that the computationally simpler method is the one based on the reduced form of the Rayleigh equations, which can be formulated as an N-dimensional matrix eigenvalue problem, immediately yielding N branches of the surface plasmon dispersion relation. Using the full Rayleigh equations, we must search individually for each zero of a determinant, and can resolve only a few of the branches. The reduced Rayleigh eigenvalue approach, however, yields convergent results only for small roughness (lo/a 5 0.1) when applied to an analytic profile function, and is severely restricted for a non-analytic profile. The full Rayleigh method, on the other hand, yields results convergent to high accuracy for large amplitudes (&/a d 0.6) for the analytic case, and also yields convergent results for the nonanalytic, sawtooth profile, though only for small roughness (lo /a 5 0.06). This breakdown of the reduced Rayleigh eigenvalue approach at much smaller corrugation strengths than those with the full Rayleigh method is the same behavior found in ref. [3], for the case of plasmons propagating across the grating (note that the formally exact extinction theorem method used in ref. [3] is equivalent to the full Rayleigh method). The numerical results confirm the expectation based on the findings with edge plasmons, as cited in the introduction, that there exists an infinite number of discrete branches of the dispersion relation for surface plasmons propagating parallel to a one-dimensional grating. Comparison of the present results with those of ref. [9] reveals that for both the rough-surface plasmons and edge plasmons, the branches approach o = or /fi as q + m, but that at q - 0 the
249
two cases are qualitativeIy dissimilar: the surface plasmon frequencies preach w = wr/$?, or close to it, while the edge plasmon frequencies w = o for the modes above up/ u’z and go to w = 0 for the modes
reapgo to below
“p/ P2. It is understood that for q + cg, the penetration depth of the fields into the medium goes to zero, so that the curvature of the interface in either case is not seen; and all modes are degenerate at the w,/a of a flat surface. With decreasing q, as the penetration depth increases, the fields localized on the wedge decay with distance into the medium along a direction initially normal to the interface such that the field on one side of the wedge eventually interferes with the field on the other side. Thus, as with surface modes on thin films, interference of the fields from opposite faces causes the edge mode frequencies to be pushed up to or or down to zero. But for the modes on the surface grating of a semi-infinite medium, the field decays into the medium along a direction normal to the average, that is, flat surface; and so there is no interference. As these fields penetrate vertically into the bulk, they simply see less of the grating and so behave like flat-surface plasmons as ~1-f 0: their frequencies return to wP We therefore can understand physically how the infinite set of discrete branches in the dispersion relation arises for plasmons propagating along the grooves of a grating, and can calculate their frequencies with high accuracy for large corrugation strengths.
,/a.
Acknowledgements This research was partially supported by the US Air Force Office of Scientific Research, through Contract No. F49620-78-C-0019. The work of one of the authors (A.A.M.) was carried out under the auspices of an Alexander von Humboldt Senior US Scientist Award, whose support is gratefully acknowledged.
References Ill F. Toigo, A. Marvin, V. Celli and N.R. Hiil, Phys. Rev. B15 (1977) 5618. 121B. Laks, D.L. Mills and A.A. Maradudin, Phys. Rev. B23 (198 1) 4965. Phys. Rev. B24 (1981) 595. [31 N.E. Glass and A.A. Maradudin, Ed. W.F. Murphy [41 E. Burstein and C.Y. Chen, in: Proc. 7th Conf. on Raman Spectroscopy, (North-Holland, New York, 1980) p. 346. wave in a solid. resulting [51In this paper we use the term plasmon to denote the electromagnetic from the linear coupling of a photon to an electric or magnetic dipole active excitation in the solid, in the limit that retardation effects can be neglected. Chem. 84 (1977) 1. L61 D.L. Jeanmaire and R.P. Van Duyne. J. Electroanal. and A.A. Maradudin, Phys. Rev. Bh (I 972) 38 IO. 171 L. Dobnynski PI L.C. Davis, Phys. Rev. Bl4 (1976) 5523.
[9] A. Eguiluz and A.A. Maradudin, Phys. Rev. 014 (1976) 5526. [IO] Lord Rayleigh, Phil. Mag. 14 (1907) 70; Theory of Sound. Vol. 2. 2nd cd. (Dobcr. Neu York. 1945) p. x9. [I I] A.A. Maradudin, in: Surface Polaritons. Eda. V.M. Agranovich and D.L. Milk (North-Holland. Amsterdam, 1982). [ 121 N.E. Glass, R. Loudon and A.A. Maradudin. Phys. Rev. B, to bc published.