Solid State Communications, Vol.27, pp. 1265—1267.
©Pergamon Press Ltd. 1978. Printed in Great Britain~
0038—1098/78/0922—1265 $02.OO/O
*
NONLINEAR MIXING OF OPPOSITELY TRAVELLING SURFACE PLASMONS
M. Fukui,t J.E.
Sipe, V.C.Y. So and G.I. Stegeman
Department of Physics, University
of Toronto, Toronto, Canada
MSS lA7
Received 17 July 1978 by A.A. Maradudin
An analysis of the nonlinear mixing of oppositely travelling surface
plasmons on a semi—infinite metal is discussed within the framework of
Maxwell’s eciuations. Coupling to free—space radiation fields, and
transverse bulk waves in the metal is predicted under certain conditions
at the sum frequency of the two surface plasinons.
1.
Introduction
Recently a number of nonlinear surface
polariton and plasmon interactions have been
analysed.”~ Attention has been concentrated
1”~ produce
In this second
note we
primarily on geometries which
extend
analysis
to the mixing of oppositely
harmonicthe
surface
p1asmons.
propagating surface plasmons employing the non—
linear mechanisms discussed by Rudnik and
Stern.7 Steady state fields are found which
satisfy the driven wave equation, the non—
linear surface currents and the nonlinear
boundary conditions at the sum frequency of the
interacting plasnons. Under various conditions
these fields correspond to a photon field
emitted into the air, and bulk transverse waves
generated into the metal. The power radiated
into the air is estimated and shown to be
measurable under present conditions,
2.
Nonlinear Current Sources
and ~BoundaryConditions
The various nonlinear mechanisms
appropriate to surface plasmon interactions have
7 Their
results
are
been discussed
in aanalytical
comprehensive
paper
by for
Rudnik
second
harmonic generation and are not directly
and Stern.
applicable to the present case. However it is
relatively simple to extend their work to show
that the appropriate nonlinear currents for the
sum frequency case (ww
14u2) are given by
and
for the integral over z of the surface current
due to the rapid variation of the normal compon—
ent of the electric
field at the surface; and
2(w
—
jew
1~2)
—
—
(c) J
V {E(w1)~E(w2)}
(3)
B
32 wm* W1 2 2
for the bulk current due to the nonlinear res—
ponse of the electron gas to an electromagnetic
field. In these equations Wp is the usual bulk
plasma frequency, m* the effective electron
mass~f the direction normal to the surface,
and E(w
1) and E(w2) are the electric fields in
the metal associated with incident surface plas—
mons at the frequencies w1 and w2. The para—
meters “a” and “b” are phenomenological constants
of the order of unity which depend on the details
of the charge current densities at the surface.
These currents act as sources for electro—
magnetic fields and can be expressed in an intermediate form in terms of driven fields and dis—
continuities in the usual electromagnetic bound
ary conditions. Since the shielding of the surface currents is negligible?they are approximated
by a current sheet placed just outside the metal.
For the bulk current terms the driven wave equa—
the
the follow—
tion surface.
was solvedThese
and analyses
the fieldslead
wereto evaluated
at
ing ‘jiscontinuities
=
y
~‘
—
—
1-fw2)
EZ (w1)E(w2)f
22
32wm w1 w2
*
H
ya
=
4w
c
x
(4)
S
4irkxjz+ 4wi
x
w
S
we(w) ~B
(1)
The subscripts “a” and “m” refer to the air and
metal sides of the interface respectively.
Furthermore, x corresponds to the surface plas—
mon propagation direction, k~ k~1k~2is the
difference in the incident surface plasmon wave—
vector components parallel to the surface and
the surface currents in equations 4 and 5 con—
tam contributions from both (a) and (b).
3. Nonlinearly Generated Fields
(2)
analysis is completed by adding solutions of the
for the normal surface current due to the
breaking of inversion symmetry at the surface;
(b)
—
yn
~Ex =E ~i —E xa
iaew 2(w
(a)
H
ibew 2
2 2 {wjE(w1)V’E(~2)
p
w1
8irm*
The nonlinear electromagnetic excitation
+ w2E(w2)V’E(wj))
*
1’On
research
leave from
the National
University
of Tokushima,
Research
supported
by the
Research
Council Tokushima,
of Canada. Japan.
1265
1266
NONLINEAR MIXING OF SURFACE PLASMONS
source—free Maxwell equations (in the metal and
air) as required to ensure the continuity of H~
and
and E~. The various nodes generated can best 6e
discussed with reference to Fig. 1. Curve I
t(w)
J
III
No.12
wk k k E(w)
I4(a
2w1-4-n1w2)
x_XI__x~
_____________
b + 4~5(k ~1—w2k a2)
x2
~
kw(n1a~—k k )
x1 x7
+1+a~—
46(w~k a2~)1k ~1)
xl
X2
________________________________
(10)
.
The corresponding power flow per unit area is
given in the usual way by
2IF I2IEx(t~i)(2~x(1~
2_
S(~)=~—R (E5H*) = c
2)I
81T e
a a
8irw
(11)
O)p
—
=
Vol.27,
____________
I
P
______________________________________
Kx
Fig. 1
where k — ~
—
k~af.Assuming a beam width “d”
and an overlap distance “L” along the propagation
direction for the incident surface plasnons, the
cross—sectional area of the radiation field is
Ldk
5a/k. Therefore the total radiated power Pa(u)
can be expressed in terms of the incident surface
plasmon powers P(w1) and P(w2) as
2
z
dwu
~a() — 32wc
1u2k
k
Xl
x~
35 3a
3ct 3
61 2 1
2
x
Photon and plasmon dispersion curves for a
semi—infinite metal.
Curve I, surface plasmons;
II, light line for wckx; III, bulk transverse
waves
with k0
IV,longitudinal bulk plasnons
with k5=O.
incident
surface
plasmons,
and relation
the sun frequency
corresponds
to the
dispersion
for the
solution fields l~eto the left of this line and
have a maximum possible frequency of l/~ Wp.
Coupling can occur to free—space radiation
modes, and to transverse bulk waves in the
metal; these lie in the regions to the left of
curves II and III respectively. The radiation
fields and transverse bulk waves are trans—
verse excitations which are driven by the AllY
and AE~boundary conditions.
2LP(wi)P(u
x
2)
2
~a()
we
2dtn*2
a
k 2k
x
(w
1—w2)
Z
A ~ P(w1)P(w2)
H
ya
The
F E (wl)E(w2)eit_1(~~u1(zZ
ox
=
(6)
where
26(w
eu
2k~1a2_wlk~2al)
F
2m * ca1a2w1 2 w2 2
0
(7)
,
(13)
k 2
=
w2~(w)/c2, a
—
+ a
=
2
,
X
5,
A(w)
2X
=
~
6 + ic(w)k
=
k
—
2
=
k~
—
c(w
2/c2
,
1)w1
~(w
2
2)w2
Ic2
coefficient
A
was
(14)
.
evaluated
for
silver and copper using
m* = In0 (free
electron mass) and Wp = 9.2 eV for
silver and m~= 1.4 m0 and Wp = 9.3 eV for
copper. Fixing A1(:2wc/w1) at the CO2 laser
wavelength
in A
A2 is shown ofin 10.6
Fig. pm,
2. the
As variation
A2 decreases,
the with
dir—
ection of the radiation field becomes progressively more parallel to the surface and the
solutions approach the light—line in Fig. 21. over
Assuming incident laser powers of 10 MW/cm
a spot diameter of ~0.2 mm with 10% input prism
coupling efficiency, for silver we find
Pa(A.
1PIn)
2 x 10—8 w.
A(w)
with
112
L P(w1)P(w2)
Radiation Fields
A radiation field produced when
has an amplitude given by
62
(12)
2/c2 and B 2 =42 — u)22/c2.
The = feasibility
of detecting
this radiation
with ~2
41—w1
2
field depends strongly on the frequencies of the
two input plasnons. To achieve propagation
lengths of at least millimetres it is necessary6
that Wp2 >> ü) 2, w22 and we Impose this condition
on subsequent 1calculations. In this limit the
total power radiated into the air is
=
(i)
IFI
(9)
detectability
of existing
instrumentation.
This power level
is well within
the range of
Vol.27,
No.12
NONLINEAR MIXING OF SURFACE PLASMONS
(ii)
X1=1O.6lLm
i~23
10
I
I
1
Cu~
I
I
_____
for this nonlinear interaction.
In summary, the nonlinear interaction
between oppositely propagating surface plasnons
can lead under the appropriate wavevector and
frequency matching conditions to the generation
of a radiation field into the vacuum (air) and
transverse bulk waves into the metal. The radi—
tion
fieldto should
be
current
coupling
instrumentation.
longitudinal
No observable
coupling
bulk plasmons
tousing
longitudinal
is zero
Ag
_____
Cu
U
I
-20
10
10-25
(
Bulk Transverse Waves
Bulk transverse waves which travel down
into the metal are generated when
2 = w2~(w)/c2 — k~2> 0. These fields are of
k!
the sane form as equations 6—10 with k
2~replaced
by —k2~.
Longitudinal bulk plasnions can be described
by scalar potential fields and can also, in the
general case, be driven by discontinuities in
electromagnetic boundary conditions. In the
present analysis, AD2 and llBz (=0) were also
evaluated. However, it was found that continuity
of all of the usual boundary conditions, i.e. E~,
H~, D2 and B2 was obtained by using radiation
fields in the air and metal only. Therefore the
I
.LAg
-18
1267
bulk plasmons is predicted. A more detailed
treatment of this problem including the differ—
II
ence frequency case will be published elsewhere.
9 have
predicted an enhancement
in
Note added
prior also
to publication:
Chen and
Burnstein
the generation of an electromagnetic field radi—
ated into the air for oppositely directed light
beams incident at the surface plasmon coupling
angles In an ATR geometry.
It~.1
II
ici22
~
0
5
10
15
20
Fig. 2
Nonlinear coefficient
A
versus A~with A
11O.6 pm.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
D.L. MILLS, Solid State Comm. 24, 669 (1977)
L. BONSALL & A.A. MA~RADUDIN, J. Appi. Phys. 49, 253 (1978).
N. FUKUI & G.I. STEGENAN, Solid State Comm., 26, 239 (1978).
N. FUKUI, V.C.Y. SO & G.I. STEGENAN, Phys. Rev. 815, in press.
H.J. SIMON, D.E. MITCHELL and .J.G. WATSON, Phys. Rev. Lett. 33, 1531 (1974).
F. DEMARTINI & Y.R. SHEN, Phys. Rev. Lett. 36, 216 (1976); F. DEMARTINI, C. CUILIANA,
P. MATALONI, E. PLANANGE & Y.R. SHEN, Phys. Rev. Lett. 37, 440 (1976).
J. RUDNIK & E.A. STERN, Phys. Rev. BA, 4274 (1971).
—
J. SCHOENWALD & E. BURSTEIN, Proc. of the Taormina Conference on Polaritons edited by
E. Burstein (Pergamon, New York, 1974) p. 89.
Y.J. CHEN & E. BURSTEIN, Nuovo Cimento, 39B, 807 (1977).