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A surface wave dispersion relation for non-local media
~
Solid State Communications, Vol. 28, pp. 927-929. Pergamon Press Ltd. 1978. Printed in Great Britain.
A SURFACE WAVE DISPERSION RELATION FOR NON-LOCAL MEDIA Deva N. Pattanayak and Joseph L. Birman Department
of Physics,
City College,
CUS~, New York, N.Y. 10031
(Received 20 October 1978 by A.A. Maradudin)
Conditions are obtained for the existence of surface waves at the interface of vacuum and a semi-infinite non-local medium whose inverse d~electric function is assumed to be symmetric in spatial coordinate in the direction perpendicular to the interface. It is shown that these conditions reduce to those obtained by Maradudin in the appropriate limits: for the isotropic case; and for no retardation.
Introduction Recently Maradudin I obtained a dispersion relation neglecting retardation for surface plasmons for a semi-infinite dielectric medium (e~bedded in vacuum) whose dielectric function E(k I,~]x3, X 3 ) is assumed to be isotropic and symmetric in x~ and x~. Here kll is a two- dimensional wave-vector-parallel to the interface (defined by the equation x 3 = 0)and ~ is the frequency of the electromagnetic field. One major advantage of his analysis is that it is applicable to a wide class of systems and is not based on any particular assumption about the boundary condition on the elementary excitations of the medium. This is so because the analysis is based on a response function which is only assumed to be symmetric in its spatial arguments. His derivation of the dispersion relation is however indirect and rests upon a method which employs series expansions of various field quantities in terms of a complete and orthonormal set of eigenfunctions. In this paper we present a derivation of a dispersion relation for surface waves at the interface between a non-local medium and vacuum. We assume the nonlocal medium to be uniaxial, to occupy the half space x^ > 0, and to be characterized by an inverse d~electric function -i ~ij
(~I
l,~Ix3,x~)which
has components
applying the Maxwell continuity conditions at the interface we then obtain a dispersion relation for the surface waves. We start by characterizing the material medium by the constitutive relation 3 ÷ _l + + ~+) (x',~0) d 3÷, E~(.L~ "(x,m) = I eij(x,x';C0)D x ,
(i)
v
where -i -i -i -i -i E.. 13 = £.. ii 6 ij and 811 = e22 # e33 " -> ->
(2)
-~ ->
and E(x,60) and n(x,~) are th~ ~ourier time transform of the electri$ ~ield E(x,t) and the displacement field D(x,t) respectively. The integration in Eq. (i) goes from 0 < x_ < oo and
-In v a
-> -~
!
and therefore £ij(x'x';~)
depends on Xl-Xl, x 2-
x" only. On taking a 2-dimensional Fourier z transfrom of both sides of Eq.(1) with respect to x I and x 2 we obtain
;o
~f+)(~, i ,I ,~Ix 3) =
(~ll,~Olx:p
0
^-l cij (~I
' Dj -~(+) l,~olx3,x31 (3)
dx~ ,
~ij = where we have introduced
6 i j and E l l = E22 # e33' and which is syrmnet-
the notation that ->
ric in x and x'. Our analysis includes retardation an~ is th~s applicable to a wider class of situations including that of surface polarltons. We thus simplify the derivation and extend the applicability.
~(~I I'°~I~3)
Representation of the Fields in Terms of Green's Function Inside the Medium As is well known 2 surface waves are regular solutions of Maxwell equations which are outgoing at infinity and are localized at the interface. Such solutions satisfy Maxwell continuity conditions at the interface. In order to obtain conditions under which surface waves exist for our problem, we proceed first to obtain a suitable representation for the electromagnetic field inside the material medium which is additionally localized at the interface. We then obtain a representation of the fields outside the medium (in vacuum) also localized at the boundary. On
Vf~,~) =
e-
I "0
F(x,~)
(2.1T) 2
(4)
dx 1 dx 2
F(kll,~Ix3)eikll
~II
dudv,
(S)
= (u,v,O),
(6)
0 = (x I, x 2, 0),
(7)
and F stands for any function. We assume -1 ÷ £1J(kl['~Ix3' x~) to be symmetric in x 3 and x~. From Maxwell equations we obtain the wave tion ÷ Vx(vx~ (+) (r,el)
927
-
±C+) k 20 o--
÷ (r,~)
-
0
equa-
(8)
A SURFACE WAVE DISPERSION RELATION FOR NON-LOCAL MEDIA
928
On using Eq.(8) and Eq.(1) we find that the cartesian component of the displacement field in the x 3 direction obeys the integrodifferentlal equation
^-1 (~1
a a~3
x~) :(+) (~1
^
x3(k Il,ml% x~) = 0.
dx;
B~+) (~ii,~Ix3)=- i3 (~]l,~l°,x3)
ax~
d%
0 2 k0
~(+) ('~l D3
= O.
I,~°lx3 )
^-] g33 (kl I,~l
I
' x3'x3)
X3 (kl I,~l'x3'x3 " )
dx~
0 A
a
I:
ax 3
^-i
ell
+ (kl l'mlx3'x3 )
- ~02 x3 ([l]'~lXB'X~)
~3(~ll,mlx3,x~)_ is
+
!
ax~
(15)
On using t~e constitutive relation Eq. (1) and the fact that D is dlvergenceless We may rewrite Eq. (15) as follows ^
B~+)(all,~l× a)
= ~
->
-~-
x3(kll,~lo,x3) kll
I!
ax3(kll'tO]x3'x3) ~x"
= 8(x3- x~) .
dx~
(lo)
assumed to satisfy outgoing
boundary condition at infinity and it can be shown from Eq.(10) and the fact that
~-)(tll,~lx3)
I~0^lql(~II,~10,xp
(9)
In order ~o ~olve Eq.(9) we introduce the Green function X (kI[,~Ix3,x~)a which satisfies the 3 equation
k
(14)
with a similar condition for D.(~,l,0JIx~). On using the boundary condition E~. d~) an~ the boundary conditions at infinity we obtain the following equation
aB~+) (~1 i,o~lx~)
^-i + ,{~lx 3, xl) qi (kll
Vol. 28, No. ii
+
= %°(-)'~IIm't [~3) for x% o
(II)
Eq. (16) expresses the cartesian component of the displacement field in th~ x 3 direction in terms of the Green +function X- and the boundary 5 value of kll" Eli. We shall use this result later to obtain surface wave dispersion relation. Fields in the Vacuum Region For x_3 < O, i.e. in the vacuum region the components of the vector fields satisfy the He lmholt z equation
that (V 2 + k )
^-1 ql (~1 I ,~lo,x~)
a~3 (~11'~1%'5) ax~
d,i: o,
for x 3 > 0. (12) Eq.(12) acts as a^bo~ndary condition for the Green function X3(kli,mlx3). From Eqs. (9), (i0) and (12) one can show after some mathematical manipulations that
~(x,(,o) = 0 ,
(17)
+
where ~(x,~) stands for any component of the vector field. We write down the following angular spectrum representatlon 4 of the fields in the vacuum region which are outgoing at infinity and are localized at the surface x 3 = 0.
~(~,~o) =
~(~ii,~olx 3)
I'P dudv,
(18)
k2 >.2
II ~0
°- [¢) where
~(~ll,~lx3) = ¢(~ii,~Io) eWllX3 ,
~^-l (~.II ,~I~3, x~) ql "
(19)
and
0
2 w,l = + (kf,-k~) I/2 for k~l > k 0. x3
Using the representation Eq.(~8)+we obtain the following representaiton for D3(kj1,~Ix 3) in the vacuum region
Ix3 (Kil'~lx~'x3) I~ ^-l %1 (~ II ,~l~,xp 0 aB~+)(~ll,~l% ) ~]x~~ ~x~
dx
(k]] 'wlx3) =
(13)
In obtaining Eq.{13) we have used the symmetric property of @i}(~[l,~Ix3, x;. Since the fields are localized at the interface x3=O we impose the boundary condition that
(20)
(k]l'wIx3)
=
wJl
]I
• 17) (ll
(21)
In obtaining equation (21) we used the transverse nature of the field in the vacuum region. Dispersion Relation for Surface Waves As we stated earlier the conditions under whlch localized fields at the interface exist
A SURFACE WAVE DISPERSION RELATION FOR NON-LOCAL MEDIA
Vol. 28, No. ii
are the dispersion relations for the surface waves. Previously we have obtained representations for the x^ component of the dielectric displacement fle~id both inside and outside the medium in terms of the boundary value of the tangential component of the electric field. At the interface x3=0 we have the Maxwell continuity ~ondltlons that the tangential component of ~ and H are continuous and the normal component of the displacement field ~ is continuous. We therefore obtain from Eq. (16) and Eq. (21) that ^
-+
[I + wiix3(kli,~I00)]
- 0
(22)
For non~r~[vi~l solutions of Eq. (22) we require that ~ & - ) (kjl,~ol0) # 0 and thus the expression in the-square bracket vanishes i.e. ^
-~
[I + w llX3(kll,~m0,0)] = 0.
(23)
Equation (23) expresses the relationship between the wave vector kll and the frequency ~0 and is the dispersion rel~tlon for the surface wave
including dispersion.
2> 2
that kll
k O.
in vlew of
929
We also have the condition
We n o t e t h a t t f
^(-) ÷
E3
(ktl,~mo)
= 0
Eq.(21) ^--~(-)(~II,~I0)= O''andthe
field outside is zero. Therefore for the half space geometry the surface waves can only exist if the z component of the electric field associated wlth them in the vacuum region is nonzero. 5 Conclusion We have derived in a direct manner the dlsperslon relation for surface waves at the interface of vacuum and a semi-inflnlte non-local medium possessing uniaxlal symmetry and characterized by an lnverse dielectric function which is symmetric in spatial arguments perpendicular to the interface. The surface wave dispersion relation is given In terms of the boundary value of a Green function. In the llmlt c ÷ ~ l.e. when retardation is ignored and for the case when the dielectric function is Isotroplc our dispersion relation goes over to that due to Maradudln. I
REFERENCES I. 2.
3. 4.
5.
MARADUDIN, A.A., Technical Report #78-16, Department of Physics, University of California, Irvlne, and to appear in Surface Science. For a discussion of surface plasmons see for example the review article by ECONOMOU, E. and NGAI, K.L. in Aspects of the Study of Surfaces~ Advances in Chemical Physics, Vol XXVII, Editors I. Prlgoglne and S.A. Rice, (John Wiley & Sons, New York 1974), p. 265. The superscripts + and - in the field variables are used to distinguish fields in the medl ,,m (x3>0) and in the vacuum (x3<0) respectively. For a discussion of the angular spectrum representation, see, for example, CLEMMOW, P.C., The l~la-m Wave Spectrum Representations of Electromagnetic Fields, (Pergamon Press, Oxford, 1966); GOODMAN, J.W.,Introductlon to Fourier Optics, (McGraw-Hill, New York 1968) Sec. 3-7; S}{EWELL, J.R. and WOLF, E., J.O.S.A. 58, 1596 (1968). For a similar observation see AGARWAL, G.S., Phys. Rev. B8, 4768 (1973); after Eq.(2.11).
Solid State Communications, Vol. 28, pp. 927-929. Pergamon Press Ltd. 1978. Printed in Great Britain.
A SURFACE WAVE DISPERSION RELATION FOR NON-LOCAL MEDIA Deva N. Pattanayak and Joseph L. Birman Department
of Physics,
City College,
CUS~, New York, N.Y. 10031
(Received 20 October 1978 by A.A. Maradudin)
Conditions are obtained for the existence of surface waves at the interface of vacuum and a semi-infinite non-local medium whose inverse d~electric function is assumed to be symmetric in spatial coordinate in the direction perpendicular to the interface. It is shown that these conditions reduce to those obtained by Maradudin in the appropriate limits: for the isotropic case; and for no retardation.
Introduction Recently Maradudin I obtained a dispersion relation neglecting retardation for surface plasmons for a semi-infinite dielectric medium (e~bedded in vacuum) whose dielectric function E(k I,~]x3, X 3 ) is assumed to be isotropic and symmetric in x~ and x~. Here kll is a two- dimensional wave-vector-parallel to the interface (defined by the equation x 3 = 0)and ~ is the frequency of the electromagnetic field. One major advantage of his analysis is that it is applicable to a wide class of systems and is not based on any particular assumption about the boundary condition on the elementary excitations of the medium. This is so because the analysis is based on a response function which is only assumed to be symmetric in its spatial arguments. His derivation of the dispersion relation is however indirect and rests upon a method which employs series expansions of various field quantities in terms of a complete and orthonormal set of eigenfunctions. In this paper we present a derivation of a dispersion relation for surface waves at the interface between a non-local medium and vacuum. We assume the nonlocal medium to be uniaxial, to occupy the half space x^ > 0, and to be characterized by an inverse d~electric function -i ~ij
(~I
l,~Ix3,x~)which
has components
applying the Maxwell continuity conditions at the interface we then obtain a dispersion relation for the surface waves. We start by characterizing the material medium by the constitutive relation 3 ÷ _l + + ~+) (x',~0) d 3÷, E~(.L~ "(x,m) = I eij(x,x';C0)D x ,
(i)
v
where -i -i -i -i -i E.. 13 = £.. ii 6 ij and 811 = e22 # e33 " -> ->
(2)
-~ ->
and E(x,60) and n(x,~) are th~ ~ourier time transform of the electri$ ~ield E(x,t) and the displacement field D(x,t) respectively. The integration in Eq. (i) goes from 0 < x_ < oo and
-In v a
-> -~
!
and therefore £ij(x'x';~)
depends on Xl-Xl, x 2-
x" only. On taking a 2-dimensional Fourier z transfrom of both sides of Eq.(1) with respect to x I and x 2 we obtain
;o
~f+)(~, i ,I ,~Ix 3) =
(~ll,~Olx:p
0
^-l cij (~I
' Dj -~(+) l,~olx3,x31 (3)
dx~ ,
~ij = where we have introduced
6 i j and E l l = E22 # e33' and which is syrmnet-
the notation that ->
ric in x and x'. Our analysis includes retardation an~ is th~s applicable to a wider class of situations including that of surface polarltons. We thus simplify the derivation and extend the applicability.
~(~I I'°~I~3)
Representation of the Fields in Terms of Green's Function Inside the Medium As is well known 2 surface waves are regular solutions of Maxwell equations which are outgoing at infinity and are localized at the interface. Such solutions satisfy Maxwell continuity conditions at the interface. In order to obtain conditions under which surface waves exist for our problem, we proceed first to obtain a suitable representation for the electromagnetic field inside the material medium which is additionally localized at the interface. We then obtain a representation of the fields outside the medium (in vacuum) also localized at the boundary. On
Vf~,~) =
e-
I "0
F(x,~)
(2.1T) 2
(4)
dx 1 dx 2
F(kll,~Ix3)eikll
~II
dudv,
(S)
= (u,v,O),
(6)
0 = (x I, x 2, 0),
(7)
and F stands for any function. We assume -1 ÷ £1J(kl['~Ix3' x~) to be symmetric in x 3 and x~. From Maxwell equations we obtain the wave tion ÷ Vx(vx~ (+) (r,el)
927
-
±C+) k 20 o--
÷ (r,~)
-
0
equa-
(8)
A SURFACE WAVE DISPERSION RELATION FOR NON-LOCAL MEDIA
928
On using Eq.(8) and Eq.(1) we find that the cartesian component of the displacement field in the x 3 direction obeys the integrodifferentlal equation
^-1 (~1
a a~3
x~) :(+) (~1
^
x3(k Il,ml% x~) = 0.
dx;
B~+) (~ii,~Ix3)=- i3 (~]l,~l°,x3)
ax~
d%
0 2 k0
~(+) ('~l D3
= O.
I,~°lx3 )
^-] g33 (kl I,~l
I
' x3'x3)
X3 (kl I,~l'x3'x3 " )
dx~
0 A
a
I:
ax 3
^-i
ell
+ (kl l'mlx3'x3 )
- ~02 x3 ([l]'~lXB'X~)
~3(~ll,mlx3,x~)_ is
+
!
ax~
(15)
On using t~e constitutive relation Eq. (1) and the fact that D is dlvergenceless We may rewrite Eq. (15) as follows ^
B~+)(all,~l× a)
= ~
->
-~-
x3(kll,~lo,x3) kll
I!
ax3(kll'tO]x3'x3) ~x"
= 8(x3- x~) .
dx~
(lo)
assumed to satisfy outgoing
boundary condition at infinity and it can be shown from Eq.(10) and the fact that
~-)(tll,~lx3)
I~0^lql(~II,~10,xp
(9)
In order ~o ~olve Eq.(9) we introduce the Green function X (kI[,~Ix3,x~)a which satisfies the 3 equation
k
(14)
with a similar condition for D.(~,l,0JIx~). On using the boundary condition E~. d~) an~ the boundary conditions at infinity we obtain the following equation
aB~+) (~1 i,o~lx~)
^-i + ,{~lx 3, xl) qi (kll
Vol. 28, No. ii
+
= %°(-)'~IIm't [~3) for x% o
(II)
Eq. (16) expresses the cartesian component of the displacement field in th~ x 3 direction in terms of the Green +function X- and the boundary 5 value of kll" Eli. We shall use this result later to obtain surface wave dispersion relation. Fields in the Vacuum Region For x_3 < O, i.e. in the vacuum region the components of the vector fields satisfy the He lmholt z equation
that (V 2 + k )
^-1 ql (~1 I ,~lo,x~)
a~3 (~11'~1%'5) ax~
d,i: o,
for x 3 > 0. (12) Eq.(12) acts as a^bo~ndary condition for the Green function X3(kli,mlx3). From Eqs. (9), (i0) and (12) one can show after some mathematical manipulations that
~(x,(,o) = 0 ,
(17)
+
where ~(x,~) stands for any component of the vector field. We write down the following angular spectrum representatlon 4 of the fields in the vacuum region which are outgoing at infinity and are localized at the surface x 3 = 0.
~(~,~o) =
~(~ii,~olx 3)
I'P dudv,
(18)
k2 >.2
II ~0
°- [¢) where
~(~ll,~lx3) = ¢(~ii,~Io) eWllX3 ,
~^-l (~.II ,~I~3, x~) ql "
(19)
and
0
2 w,l = + (kf,-k~) I/2 for k~l > k 0. x3
Using the representation Eq.(~8)+we obtain the following representaiton for D3(kj1,~Ix 3) in the vacuum region
Ix3 (Kil'~lx~'x3) I~ ^-l %1 (~ II ,~l~,xp 0 aB~+)(~ll,~l% ) ~]x~~ ~x~
dx
(k]] 'wlx3) =
(13)
In obtaining Eq.{13) we have used the symmetric property of @i}(~[l,~Ix3, x;. Since the fields are localized at the interface x3=O we impose the boundary condition that
(20)
(k]l'wIx3)
=
wJl
]I
• 17) (ll
(21)
In obtaining equation (21) we used the transverse nature of the field in the vacuum region. Dispersion Relation for Surface Waves As we stated earlier the conditions under whlch localized fields at the interface exist
A SURFACE WAVE DISPERSION RELATION FOR NON-LOCAL MEDIA
Vol. 28, No. ii
are the dispersion relations for the surface waves. Previously we have obtained representations for the x^ component of the dielectric displacement fle~id both inside and outside the medium in terms of the boundary value of the tangential component of the electric field. At the interface x3=0 we have the Maxwell continuity ~ondltlons that the tangential component of ~ and H are continuous and the normal component of the displacement field ~ is continuous. We therefore obtain from Eq. (16) and Eq. (21) that ^
-+
[I + wiix3(kli,~I00)]
- 0
(22)
For non~r~[vi~l solutions of Eq. (22) we require that ~ & - ) (kjl,~ol0) # 0 and thus the expression in the-square bracket vanishes i.e. ^
-~
[I + w llX3(kll,~m0,0)] = 0.
(23)
Equation (23) expresses the relationship between the wave vector kll and the frequency ~0 and is the dispersion rel~tlon for the surface wave
including dispersion.
2> 2
that kll
k O.
in vlew of
929
We also have the condition
We n o t e t h a t t f
^(-) ÷
E3
(ktl,~mo)
= 0
Eq.(21) ^--~(-)(~II,~I0)= O''andthe
field outside is zero. Therefore for the half space geometry the surface waves can only exist if the z component of the electric field associated wlth them in the vacuum region is nonzero. 5 Conclusion We have derived in a direct manner the dlsperslon relation for surface waves at the interface of vacuum and a semi-inflnlte non-local medium possessing uniaxlal symmetry and characterized by an lnverse dielectric function which is symmetric in spatial arguments perpendicular to the interface. The surface wave dispersion relation is given In terms of the boundary value of a Green function. In the llmlt c ÷ ~ l.e. when retardation is ignored and for the case when the dielectric function is Isotroplc our dispersion relation goes over to that due to Maradudln. I
REFERENCES I. 2.
3. 4.
5.
MARADUDIN, A.A., Technical Report #78-16, Department of Physics, University of California, Irvlne, and to appear in Surface Science. For a discussion of surface plasmons see for example the review article by ECONOMOU, E. and NGAI, K.L. in Aspects of the Study of Surfaces~ Advances in Chemical Physics, Vol XXVII, Editors I. Prlgoglne and S.A. Rice, (John Wiley & Sons, New York 1974), p. 265. The superscripts + and - in the field variables are used to distinguish fields in the medl ,,m (x3>0) and in the vacuum (x3<0) respectively. For a discussion of the angular spectrum representation, see, for example, CLEMMOW, P.C., The l~la-m Wave Spectrum Representations of Electromagnetic Fields, (Pergamon Press, Oxford, 1966); GOODMAN, J.W.,Introductlon to Fourier Optics, (McGraw-Hill, New York 1968) Sec. 3-7; S}{EWELL, J.R. and WOLF, E., J.O.S.A. 58, 1596 (1968). For a similar observation see AGARWAL, G.S., Phys. Rev. B8, 4768 (1973); after Eq.(2.11).