Can Sheared Surfaces Emit Light?
JB Pendry
The Blackett Lab
Imperial College
London SW7 2BZ
UK
Abstract
We apply a new formalism to calculating emission of light from sheared dielectric
surfaces. When the surfaces are smooth, emission of light requires extreme velocities,
but introducing surface roughness with short-period components reduces the extreme
demands on velocity so that for atomic scale roughness shear velocities as small as
10+4ms-1 will result in light emission. Some order of magnitude estimates are given.
to appear in:
Journal of Modern Optics 45 2389-408 (1998).
PACS: 42.50.-p
quantum optics
78.45.+h
stimulated emission
78.60.Mq
sonoluminescence, triboluminescence
C:/word/light/qlight1.doc at 05/11/98
page 1
Can Sheared Surfaces Emit Light?
1. Introduction
Emission of light from accelerated mirrors or dielectrics, famously raised by
Schwinger in the context of sonoluminescence [1] has been the subject of many papers
[2]. Less commonly studied is light emission from two sheared dielectrics or two
sheared half-silvered mirrors [3]. The conclusion of these studies is that the amount of
radiation produced is extremely small, almost too small to be observed except in
extreme circumstances [4, 5]. In this paper we return to the question with a new
formalism and ask if surface roughness can influence the amount of light emitted.
In an earlier paper [6] I proposed a new formula for the work necessary to keep two
sheared surfaces in relative motion. Although introduced in the context of frictional
forces the theory is equally applicable to light emission. A review of previous work can
be found in [7], and related approaches have been pursued by Polevoi [8], Annett and
Echenique [9, 10] and Persson [11]. The new theory gives a central role to the
reflection coefficients of the surfaces in question and identifies the key process as
emission of pairs of correlated photons with equal and opposite momentum, one into
each medium. Dodonov and Klimov [12] have also commented on the queezed nature
of light emitted by moving surfaces. The two frequencies are not necessarily equal but
obey the sum rule,
ω 1 + ω 2 = k xv
(1)
where hk x is the momentum of one of the photons in the direction of the velocity, v. It
is immediately evident that the maximum frequency of emission is limited by the
velocity and the shortest wavelength of light available. Because of this restriction it is
impossible to emit visible light by shearing smooth surfaces, unless the shear velocity is
comparable to the velocity of light.
The new ingredient in this paper is surface roughness whose effect is to couple very
short wavelength modes to outgoing light. This enables high frequencies to be
generated by the short wavelength component of two photons grating against one
another. To be effective roughness on a scale of the order of 10-9m or less must be
present, then visible light can be generated when the shear velocity is comparable to
the speed of sound in solids. If we assume this to be the maximum speed with which
two surfaces can sensibly be sheared, light emission becomes a practical proposition.
We show that the amplitude of roughness need only be of the same order as the
roughness period: indeed greater amplitudes serve no purpose. So for very short
periods roughness could be due to the atomic constitution of the surface.
In the following section we introduce the formalism developed in a previous paper and
in the first instance apply it to emission of light from sheared smooth surfaces. We
choose to examine the case of two transparent materials defined by their dielectric
functions, ε . The limitations of this mechanism, especially the extreme shear velocities
required, are emphasised. The amount of energy emitted as light is extremely small
compared to the likely amounts of energy dissipated by more conventional friction
forces. In the section following surface roughness is introduced into the calculation
which alleviates the demands of extreme velocities. We consider the specific example
of a very thin short period grating drawn over the surface of an absorber and conclude
that measurable amounts of light may be emitted for shear velocities of the same order
as the velocity of sound in materials. In the appendix we show how to relate reflection
coefficients to the amplitude of surface modulations in a dielectric.
C:/word/light/qlight1.doc at 05/11/98
page 2
Can Sheared Surfaces Emit Light?
2. Light Emission from Smooth Surfaces - an Unlikely Story
In a previous paper we considered the possibility that shearing two surfaces results in
work being dissipated, treating lossy dielectrics with an imaginary component to their
dielectric permittivity that dissipated energy through internal processes. Such materials
are necessarily opaque and therefore light emission is not an issue. Here we
demonstrate that lossless dielectrics can produce light when sheared against one
another as shown in figure 1, provided that some rather stringent conditions are met
which we shall now derive.
Figure 1. Perfectly smooth dielectric surfaces shear against one another with
relative velocity v and emit light as correlated pairs of photons. The spherical
shape of the dielectrics allows all emitted light to escape. We assume that the
two halves of the system have the same dielectric permittivity.
If the surfaces are perfectly flat total momentum parallel to the surface must be
conserved therefore photons are emitted in symmetric pairs with equal and opposite
moment, k, parallel to the surface. The frequencies of the two photons need not be the
same, but they must obey the sum rule given above in equation (1), where the shear
velocity is assumed parallel to the x-axis. The origin of this rule is that the two modes,
into which light is emitted, grate against one another as the surfaces are sheared
generating a small contribution to the Hamiltonian of frequency k x v . The sum rule
then follows from first order perturbation theory.
The sum rule generates a conflict: in order to emit optical photons high frequencies
must be generated requiring that k x v be large. On the other hand if k x is too large
light cannot escape from the surface into the dielectric. In order to do so it must have a
real component to the z-wave vector inside dielectric 1. If we choose,
ω2 = ω
(2)
then the condition becomes,
Kz2 = εω 2 c0− 2 − k x2 − k 2y > 0
(3)
We also require the other photon to escape from the surface into medium 1, the
moving medium, and a similar condition must be fulfilled before this can happen,
C:/word/light/qlight1.doc at 05/11/98
page 3
Can Sheared Surfaces Emit Light?
K1'2z = εω 1'2 c0− 2 − k x'2 − k y'2 > 0
(4)
A complication arises because the condition has to be applied in the rest frame of the
moving dielectric. The primes denote that these quantities have been transformed into
the rest frame of medium 1:
k1' x = γ
k1x − βγ
c0− 1ω 1,
k1' y = k1 y ,
K1' z = K1z ,
b
ω 1' = γω 1 − vk1x
g
(5)
where,
−
γ= 1 − v 2 c0− 2
e
β = vc0− 1 ,
j
1
2
(6)
b
g
Note that the ‘clock’ in the moving frame registers a frequency of ω 1' = γω 1 − vk1x
rather than ω 1 − vk1x due to the relativistic dilation of time. After some manipulation
we can rewrite the two conditions,
b
g
1 − v 2 c0− 2 j
−2 2e
εc ω
2
LM
MN
b g OP e
PQ
2
c0− 2 ε − 1 vω
2 −2
− 1 − v 2 c02 k y2 ,
< εv c0 − 1 k x −
2 −2
2 −2
εv c0 − 1
εv c0 − 1
0
j
εc0− 2ω 2 > k x2 + k y2 ,
(7)
(8)
The first of these conditions represents an hyperbola centred on,
b g
c − 2 ε − 1 vω
k x = k0 = 0 2 − 2
εv c0 − 1
(9)
and with asymptotes,
b
k y = k x − k0
g
εv 2 c0− 2 − 1
(10)
1 − v 2 c02
Figure 2 gives a graphical representation of these conditions. Note that the left hand
part of the hyperbole is ruled out by the condition,
k xv > ω
(11)
It is only possible to satisfy these conditions simultaneously if a critical velocity is
exceeded,
v > vc =
2 ε
c0
1+ ε
(12)
If it is planned to excite visible radiation, this is a very restrictive condition because of
the relatively small values of ε available. Assuming that,
ε = 3.0
(13)
the critical velocity becomes,
vc = 0.866c0
(14)
which is an unrealistically large shear velocity especially as the surfaces must approach
to within a wavelength or so to have any effect. At much longer wavelengths larger
C:/word/light/qlight1.doc at 05/11/98
page 4
Can Sheared Surfaces Emit Light?
values of ε can be obtained, and the distance of approach can be relaxed because of
the longer length scale, but even in the case,
ε = 1000.0
(15)
we have to cope with extreme velocities much greater than the velocity of sound in
even the hardest materials,
vc = 0.063c0 = 19
. ×107 ms-1
(16)
It can been seen from figure 2 that,
c0− 2ω 2 < k x2 + k y2
(17)
which implies that radiation between the two surfaces has an imaginary wave vector
perpendicular to the surfaces.
Figure 2. Geometric conditions to be satisfied before light can be emitted from
a sheared interface. The figure shows the two components of wave vector
parallel to the surface. The allowed region for wave vectors, shaded in the
figure, lies between the two circles and to the right of the right hand hyperbola
defined by equation (7).
In an earlier paper [6] we derived a formula for the work done in shearing two
surfaces,
vFx = 4hv
b
gz
L2
2π
+∞
3 0
k x dk x
z
+∞
−∞
b
g
exp − 2 kd dk y
k v
X
LMIm R1ss bk , ω 1 = k xv − ω gIm R2ss bk , ω g OP dω
×Y
YZ 0 MN + Im R1pp b k, ω 1 = k xv − ω gIm R2 pp b k, ω gPQ
x
(18)
The formula is correct to first order in the reflection coefficients of the surfaces.
Although derived with frictional forces in mind this formula is general and can be used
to calculate the energy emitted as light from lossless dielectrics. We note that
C:/word/light/qlight1.doc at 05/11/98
page 5
Can Sheared Surfaces Emit Light?
dissipation of energy depends on both the reflection coefficients having an imaginary
component. It is not necessary for energy dissipation to take place in the dielectric
itself provided that work done can be carried away from the surface as light as we
discussed above. Therefore the question of whether light of frequency ω can be
emitted boils down to the conditions,
b g
Im R1b k , ω 1 = k x v − ω g > 0
Im R2 k , ω > 0
(19)
From the well known formulae for s-, and p-polarised reflectivities [13],
−2 2
2
2
K1z − K2 z i + k x + k y − c0 ω −
Rss k , ω =
=
K1z + K2 z i + k 2 + k 2 − c − 2ω 2 +
x
y
0
b g
b g
R pp k , ω = −
K2 z − ε2 K1z
=−
K2 z + ε2 K1z
c0− 2 εω 2
−
k x2
−
c0− 2 εω 2 − k x2 − k y2
,
c0− 2 εω 2 − k x2 − k y2
k 2y
− iε2 +
k x2
+
k y2
−
c0− 2ω 2
(20)
c0− 2 εω 2 − k x2 − k y2 + iε2 + k x2 + k y2 − c0− 2ω 2
where K1z and K2 z are z-components of the wave vector in vacuum and in dielectric
respectively. Clearly the imaginary parts of both these reflectivities are non zero if
inequalities (3), (4) and (17) are obeyed.
Reflection coefficients of the moving surface are more complicated but can be
calculated by transforming the incident wavefield to a frame in which the surface is
stationary, using the conventional formula to calculate the reflected wave, and finally
transforming back to the original frame of reference. The only complication is that in
general the s- or p-polarised nature of the radiation is not preserved under a Lorentz
transformation. This is only the case if,
ky = 0
(21)
However inspection of figure 2 will show that k y is small most of the time, especially
if the velocity is near to the critical value, and we shall assume that the polarisation is
approximately preserved. Then we may write,
b
g e
j
R1 pp b k , ω 1 = k x v − ω g ≈R1' pp e k ' , ω 1' = γk x v − ω j
R1ss k , ω 1 = k x v − ω ≈R1' ss k ' , ω 1' = γk x v − ω
(22)
where we can now calculate,
C:/word/light/qlight1.doc at 05/11/98
page 6
Can Sheared Surfaces Emit Light?
b
g
e
j
R1ss k , ω 1 = k x v − ω ≈R1' ss k ' , ω 1' = γk x v − ω =
=
b
K1' z − K2' z
K1' z + K2' z
i + k x'2 + k y'2 − c0− 2ω 1'2 −
c0− 2 εω 1'2 − k x'2 − k y'2
i + k x'2 + k y'2 − c0− 2 ω 1'2 +
c0− 2 εω 1'2 − k x'2 − k y'2
g
e
j
R1 pp k , ω 1 = k x v − ω ≈R1' pp k ' , ω 1' = γk x v − ω = −
=−
,
K2' z − ε2 K1' z
K2' z + ε2 K1' z
c0− 2 εω 1'2 − k x'2 − k y'2 − iε2 + k x'2 + k y'2 − c0− 2ω 1'2
c0− 2 εω 1'2 − k x'2 − k y'2 + iε2 + k x'2 + k y'2 − c0− 2ω 1'2
(23)
We are now in a position to calculate the rate of emission of light. The work done by
exciting s-polarised radiation is given by,
zb
L2
k expb − 2 kd g
g3 x
k v
×z0 Im Rss b k , ω gIm Rss b k , ω 1 = k x v − ω g
vFxs = 4hv
2π
x
×εω 2 c0− 2 cos θ dω dΩ ×εω 2c0− 2 cos θ dω dΩ
L2
e
j
b 2πg
k v
×z w Im Rss e εω c0− 1 sin θ, ω j
0
×Im Rss e εω c0− 1 sin θ, ω 1 = εω c0− 1 sin θ cos φv − ω j
= 4 hv
3
εω c0− 1 sin θ cos φexp − 2 εω c0− 1 sin θd
(24)
x
×εω 2 c0− 2 cos θ dω dΩ
where we have defined a set of polar co-ordinates: θ is the polar angle made by the
wave vector inside the dielectric with the surface normal, φ the azimuthal angle
defined relative to the k x axis, and dΩ is an element of solid angle.
We wish to extract the fraction of work that creates s-polarised photons of frequency
ω,
ω
=
k xv
ω
εω c0− 1 sin θ cos φv
(25)
b g
travelling in a direction given by the polar co-ordinates, θ, φ :
C:/word/light/qlight1.doc at 05/11/98
page 7
Can Sheared Surfaces Emit Light?
∂3
∂ω∂2Ω
b g
Ps Ω dω dΩ
L2
e
j
b 2πg
×Im Rss e εω c0− 1 sin θ, ω j
×Im Rss e εω c0− 1 sin θ, ω 1 = εω c0− 1 sin θ cos φv − ω jdω dΩ
= 4h
3
εω 3c0− 2 cos θ exp − 2 εω c0− 1 sin θd
(26)
with a similar expression for p-polarised photons,
∂3
∂ω∂2Ω
b g
Pp Ω dω dΩ
L2
= 4h
b 2πg
×Im R pp e
×Im R pp e
3
e
εω 3c0− 2 cos θ exp − 2 εω c0− 1 sin θd
εω c0− 1 sin θ, ω
j
(27)
j
j
εω c0− 1 sin θ, ω 1 = εω c0− 1 sin θ cos φv − ω dω dΩ
and the units are: watts per steradian per radian of frequency. The expressions are valid
only inside the allowed angles of emission.
Now let us estimate how much energy is radiated. We shall assume,
ε = 3.0
2 ε
c0 = 0.866c0
1+ ε
v = 0.9c0
vc =
(28)
d = 2 ×10 − 7 m
Although the choice of ε is reasonable, the choice of velocity and surface separation is
an unreasonable combination. Given these values we estimate the range of values taken
by each of the three variables,
∆k x =
b ε + 1gvc0− 1 − 2
εvc0− 1 − 1
ε −1 −1 −1
c0 2c0 d
= 6.08 ×105 m-1
e
j
L
O
∆k y = 2 Mb1 + εgv 2c0− 2 − 2 − 2e1 − c0− 2 v 2 j P
N
Q
1
2
1
2
e
v − 1 2c0− 1d
−1
j
(29)
= 3.37 ×106 m-1
e
∆ω ≈ 2c0− 1d
−1
j
= 7.5 ×10+ 14 rad s-1
The imaginary parts of the reflection coefficients can be shown to be of the order of
unity. We can now estimate the amount of energy radiated,
C:/word/light/qlight1.doc at 05/11/98
page 8
Can Sheared Surfaces Emit Light?
z b g
b gz
k v
×z0 Im Rss b k , ω gIm Rss b k , ω ' = k x v − ω gdω
vFxs = 4 hv
L2
+∞
+∞
k x dk x
exp − 2 kd dk y
3 0
−∞
2π
x
≈4hvc0− 1ω
1
b2πg3
(30)
εdk x dk y dω
≈2.90 ×106 wm -2
Bearing in mind the phenomenal amount of energy stored in translational motion at
relativistic velocities, this is a very small amount of light.
3. Emission of Light from Rough Surfaces
If our objective is to observe light emission from surfaces we must find some way
around the problem of smooth surfaces. This arises essentially because only light with a
small component of wave vector parallel to the surface can escape as a free photon.
However the small parallel wave vector limits the frequency through (11). Small k x
implies low frequencies unless v is impossibly large. The way around this problem is to
use a rough surface. Roughness has the capacity to change k x by diffraction and hence
light initially created with large k x , therefore high ω , may subsequently scatter from
the roughness into a state of lower k x , free to escape as visible radiation. In figure 3
we show as an unshaded circle the regions of k-space where escape from the surface
into free space is allowed, i.e. where,
k <
ω
c0
(31)
In the presence of periodic disorder other regions of k-space, related by a reciprocal
lattice vector, couple to the escape process and hence states inside the shaded circles
can now escape.
Figure 3. Reciprocal space picture of three Brillouin zones for our periodic
surface. A periodic surface will scatter light changing the wave vector parallel
to the surface, k, in units of a reciprocal lattice vectors, g. Only light with
sufficiently small k can escape from the surface i.e. inside the open circle in
the middle of the figure. However in addition some higher momentum states
can be scattered into an escaping state by the surface corrugations: those states
inside the shaded circles.
C:/word/light/qlight1.doc at 05/11/98
page 9
Can Sheared Surfaces Emit Light?
Calculating the emission of light
Figure 4. A rough surface, represented here by a periodic grating, shears past a
second surface. Short wavelength electromagnetic excitations are created
which normally will not leave the surface, but in this instance deposit enough
momentum in the grating to escape from the surface as observable light.
We calculate the light emission by using our formula to calculate the rate of working
and then identify the component of that work used to create light,
vFx = 4hv
XY
Z0
×
b
gz
L2
2π
k xv
+∞
3 0
k x dk x
bg
bg
z
+∞
−∞
b
g
exp − 2 kd dk y
ε ω − 1
Im 1
Im R2 k , ω ' = k x v − ω dω
ε1 ω + 1
b
(32)
g
We have assumed that one of the surfaces is a simple flat homogeneous material with
dielectric function ε1 ω with absorption at the relevant frequencies so that,
bg
ε b ω g− 1
R1b k,ω g ≈ 1
ε1b ω g+ 1
(33)
The other surface we shall assume to be the simplest possible diffraction grating
consisting of a layer of material sinusoidally modulated in density with a very short
wavelength, and having a thickness much less than the wavelength of the modulation.
Its reflection coefficient is now a matrix but it is simple to show that to first order in
multiple scattering between the surfaces we require only the diagonal elements in our
formula which are given as R2 k, ω ' = k x v − ω . In the appendix we show how to
calculate R2 .
b
g
Our formula shows that energy dissipation occurs only if both reflection coefficients
have an imaginary component at the relevant wave vector and frequency. For R1 k,ω
this can be arranged by choice of material, but by itself results in no light emission. On
the other hand the second surface can only dissipate energy by radiation of light. This
can only happen if k lies inside one of the two shaded circles in figure 3. There are two
contributing processes because light can be emitted either in the p- or s- polarised
state,
b g
C:/word/light/qlight1.doc at 05/11/98
page 10
Can Sheared Surfaces Emit Light?
b
g
b
g
b
R2 k , ω ' = k x v − ω = R ppp k , ω ' = k x v − ω + R psp k , ω ' = k x v − ω
g
(34)
Where R ppp and R psp are defined in the appendix. As we discussed before only ppolarised evanescent waves make a contribution.
For simplicity we shall assume that the velocity is parallel to g and hence almost
parallel to all the relevant k vectors,
b
vFx = 4hv exp − 2 gd
gb2Lπg3 z0+ ∞ k x dk x z−+ ∞∞ dk y
2
(35)
gv
ε b ω g− 1
X
×Y Im 1
Z 0 ε1bω g+ 1 Im R2 b k, ω ' = gv − ω gdω
substituting in the case of emission of p-polarised light:
g 2π 3 z0+ ∞ k xdk x z−+ ∞∞ dk y
b g
gv
2
X
2
Sg2
ε1b ω g− 1
O
L
'
2 '4 O L 1
×Y Im
YZ 0 ε1b ω g+ 1 4gk '2 Kg1z MNe k ⋅g Kg1z j + g k PQ MN ε2 − 1PQ dω
b
vFxp = 4 hv exp − 2 gd
L2
(36)
where Sg is the amplitude of the gth Fourier component as defined in the appendix.
We now choose a simple form for the first surface reflectivity, assuming that ε1 has a
plasma type of dispersion,
bg
ε1 ω = 1 −
ω 2p
(37)
ω2
hence,
bg
bg
ω 2p
ω 2p
ε ω − 1
Im 1
= − Im
= − Im
ε1 ω + 1
2ω − ω p + iγ 2ω + ω p + iγ
2ω ω + iγ− ω 2p
πω p
δω − 1 ω p − δω + 1 ω p
=+
2
2
2
b g
MNL FH
IK FH
d
IKOP
Q
id
i
(38)
Remembering that,
FH
IK
Kg1z k , gv − 1 ω p = +
2
FH
c0− 2 gv − 1 ω p
2
IK2 − k '2
(39)
and,
k x ≈g
(40)
Substituting into the formula for the rate of working:
C:/word/light/qlight1.doc at 05/11/98
page 11
Can Sheared Surfaces Emit Light?
b
g FH
vFxp = 4 hv exp − 2 gd θ gv − 1 ω p
×
×
L2
z
c − 1 gv − 1 ω p
g 0
3 0
2π
b g
FH
2
2
πω p
IK ' '
k dk
IK
z
2π
0
dφk '
Sg2
(41)
FH
IK2 − k '2
2
2
L
O
L
O
− 2F
2 '2
2
'2 O
'2 L 1
I
1
× g k M cos b φk ' gM c0 gv −
MN H 2 ω p K − k PPQ + k PPQ MN ε2 − 1PQ
MN
2
4 gk '2 c0− 2 gv − 1 ω p
2
simplifying,
g FH
b
IK ω p g2 LM 1 − 1OP 2 Sg2
16 2 N ε2 Q
2
c0− 2 F gv − 1 ω p I + k '2
H 2 K
2
c0− 2 FHgv − 1 ω p I
K − k '2
vFxp = 4 hv exp − 2 gd θ gv − 1 ω p
2
z
FH
L2 c0− 1 gv −
×
2π 0
1 ω
2 p
IK ' '
k dk
(42)
2
We wish to extract the fraction of this work that creates light in the vacuum, given by
the ratio of the two frequencies:
FHgv −
fraction of work as light =
1
2
gv
ωp
IK ω '
=
(43)
gv
where,
ω ' = gv − 1 ω p
2
(44)
Hence
∂2
∂2Ω
ω'
vFxp
gv
b g
Pp Ω dΩ =
LM OP
g
N Q
cos2 b φk ' ge c0− 2 ω '2 − k '2 j + k '2
2
ω '2 c − 2 cos θ dΩ
2
ωp 2 1
ω'
=
4 hv exp − 2 gd
g
− 1 Sg2
gv
ε2
16 2
b
×
L2
b 2πg2
(45)
0
c0− 2ω '2 − k '2
LM
N
OP
Q b g
2
2
ωp 2 1
ω'
2 L
=
− 1 Sg
8hv exp − 2 gd
g
2
gv
ε2
16 2
2π
b
b ge
g
j
cos2 φk ' c0− 2 ω '2 − c0− 2ω '2 sin 2 θ + c0− 2ω '2 sin 2 θ
×
ω '2 c0− 2 cos θ dΩ
c0− 2ω '2 − c0− 2ω '2 sin 2 θ
C:/word/light/qlight1.doc at 05/11/98
page 12
Can Sheared Surfaces Emit Light?
or,
∂2
∂2Ω
b g
hg F gv −
H
Pp Ω dΩ
=
1 ω
2 p
8 2 π2 c03
IK4 ω p
2
O
L
1
expb − 2 gd gM − 1P Sg2 e cos2 b φk ' gcos2 θ + sin 2 θj L2 dΩ
N ε2 Q
(46)
in watts per steradian.
Similarly for s-polarised emission,
g 2π 3 z0+ ∞ k xdk x z−+ ∞∞ dk y
b g
X gv ε1b ω g− 1 b gv − ω g2 Sg2 1 ' ' 2 L 1 O 2
×Y Im
YZ 0 ε1b ω g+ 1 4c02 Kg1z g k '2 k x g y − k y g x MN ε2 − 1PQ dω
L2
b
vFxs = 4 hv exp − 2 gd
(47)
or,
×
4 gc02
F
I
b gb 2Lπ 3 πω2p g z0c Hgv − ω K k 'dk ' z02π dφk '
g
FHgv − 1 ω p IK2 Sg2
2
2L 1
O
1 '
2
k g sinb φk ' g M − 1P
'2
2
k
N ε2 Q
'2
c0− 2 FHgv − 1 ω p I
k
−
K
−1
0
2
vFxs = 4 hv exp − 2 gd
1
2
p
(48)
2
which simplifies to,
b
vFxs = 4hv exp − 2 gd
z
FH
L2 c0− 1 gv −
2π 0
ωp
g16
1 ω
2 p
2
g
IK ' '
k dk
2
LM 1 − 1OP 2 Sg2
N ε2 Q
2
c0− 2 F gv − 1 ω p I
H 2 K
2
c0− 2 FHgv − 1 ω p I
− k '2
K
2
(49)
Working out the fraction radiated as light:
C:/word/light/qlight1.doc at 05/11/98
page 13
Can Sheared Surfaces Emit Light?
∂2
∂Ω
2
b g
Ps Ω dΩ =
ω'
vFxs
gv
LM
N
OP
Q
2
ωp 2 1
ω'
4 hv exp − 2 gd
=
g
− 1 Sg2
ε2
gv
16 2
b
g
FH
c0− 2 gv − 1 ω p
2
L
b 2πg
2
IK2
ω '2 c0− 2 2 sin 2 φcos θ dΩ
FH
IK − k '2
2
ω p 2L 1
O
ω'
4 hv expb − 2 gd g
=
g M − 1P Sg2
gv
16 2 N ε2 Q
2
c0− 2 gv − 1 ω p
2
c0− 2ω '2
L2
b 2πg
2
2
c0− 2ω '2
−
c0− 2ω '2 sin 2 θ
ω '2 c0− 2 2 sin 2 φcos θ dΩ
LM
N
OP
Q b g
2
ωp 2 1
1
L2
=
− 1 Sg2
4 hv exp − 2 gd
g
ω '4c0− 3 2 sin 2 φdΩ
2
gv
ε2
16 2
2π
b
g
(50)
or,
4
F
I
1
2
ω
ωp
hg Hgv −
L
O
∂2
1
2 pK
Ps b Ω gdΩ =
expb − 2 gd gM − 1P Sg2 L2 sin 2 φ dΩ
2
2 3
∂Ω
8 2π c
N ε2 Q
(51)
0
Some numbers
If we assume that,
hω p = 1eV = 16
. ×10− 19 J ,
g=
2π
10
− 10
m-1,
gv ≈ω p ,
hω p
16
16
. ×10− 19
. ×10− 19
v≈
=
=
= 2.5 ×104 ms− 1
− 34
10
− 34
10
hg
10
10
×2 π ×10
×2 π ×10
k ' ≈c0− 1gv =
16
. ×10− 19
10
− 34
8
×3 ×10
= 5.3 ×106 m − 1,
(52)
Sg ≈g − 1
then we can estimate the order of magnitude of the power,
3
L2 '3
. ×10− 19 L2
4 16
vFxp = 4 hv
× k = 4 ×2.5 ×10 ×
× × 5.3 ×106
(53)
2π
16 2 2 π
16 2
ωp
e
j
= 1675
.
×104 L2 watts
vFxs = 4 hv
ωp
16 2
×
L2 '3
k = vFxp
2π
(54)
= 1675
.
×104 L2 watts
Note that this is by no means a large quantity - ordinary friction at these speeds would
be generating of the order of 109 wm − 2 which makes the luminescent component only
C:/word/light/qlight1.doc at 05/11/98
page 14
Can Sheared Surfaces Emit Light?
one in 10-5 of the total at the very most, and most probably much less. Note also that
the velocity assumed is modest, of the same order as a typical longitudinal sound
velocity, 104 ms− 1 .
4. Conclusions
It is clear that emission of light from smooth sheared surfaces is only a theoretical
possibility: the enormous shear velocities required preclude any practical realisation.
Rough surfaces are a completely different proposition. As we have seen roughness
brings down the shear velocity to the same order as the velocity of sound in solids
although at the price of requiring that the surfaces be in close contact. The latter
condition will inevitably result in further sources of frictional dissipation so that the
light energy will be a very small fraction of the total.
How likely is it that surfaces of the required degree of roughness can be found? In our
calculation in the previous section we assumed roughness with a periodicity of 10-10m,
roughly an atomic diameter. Because of the exponential decay of the fields, the
roughness is sampled only on the same length scale as the periodicity. As a result our
assumptions are consistent with a close packed layer of atoms supplying the roughness.
Obviously every surface is provided with this facility! Remarkably it does seem as
though the atomic structure of a material can affect emission of light.
Next we must address the question of how surfaces might be sheared at speeds
approaching the velocity of sound. We imagine that this may happen when a solid is
fractured. Although details of the fracture process vary with the material and with
forces applied, the appearance of cracks is associated with a sudden release of energy
and crack propagation at the velocity of sound. A shear component to this propagation
would couple to our proposed mechanism for light emission. The phenomenon of
triboluminescence is well established, and Walton [14] states that something like half
of all known materials will show some degree of light emission when subjected to a
grinding action. However in many cases the light is clearly due to charge transfer
between the surfaces followed by electrical discharge, and several other mechanisms
have also been invoked. The amount of light involved is always small, visible only in a
darkened room, and therefore not inconsistent with our estimate that very much less
than 10-5 of the total work is emitted as light.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Schwinger J 1992 Proc. Natl. Acad. Sci. USA 89 4091.
Barton G and Eberlein C 1993 Annals of Physics 227 222.
Barton G 1996 Ann. Phys. (NY) 245 361.
Eberlein C 1996 Phys. Rev. Lett. 76 3842.
Weninger KR, Barber BP, Putterman SJ 1997 Phys. Rev. Lett. 78 1799.
Pendry JB 1997 J. Phys.: Condens. Matter 9 10301.
Persson BNJ and Tosatti E (Eds) 1995 Physics of Sliding Friction: NATO ASI
Series, series E, vol 311 (Kluwer: Dordrecht).
[8] Polevoi VG 1990 Sov. Phys. JETP 71 1119.
[9] Annett JF and Echenique PM 1986 Phys. Rev. B 34 6853.
[10] Annett JF and Echenique PM 1986 Phys. Rev. B 36 8986.
[11] Persson BNJ Phys. Rev. B 1998 to appear.
[12] Dodonov DD and Klimov AB 1996 Phys. Rev. A 53 2664.
C:/word/light/qlight1.doc at 05/11/98
page 15
Can Sheared Surfaces Emit Light?
[13] Nordling C, and Österman J 1980 Physics Handbook (Studentlitteratur, Lund).
[14] Walton AJ 1977 Advances in Physics 26 887.
C:/word/light/qlight1.doc at 05/11/98
page 16
Can Sheared Surfaces Emit Light?
Appendix
Reflection of light from a rough surface
Figure 5. A smooth interface located at z=0 dividing regions 1 and 2.
Suppose that we have a p-polarised wave in medium 1 incident on medium 2 as shown
in figure 5. The wavefield will be,
b
g
+
× e$ p e K 01
jexpb + iK1z z g+ R ppe$ p eK 01− jexpb − iK01z zg
−
+
= L expb ik . r/ / − iω t g Tpp e$ p e K 02
j expe + iK02z zj,
−
Ekω p = L
3
3
where,
2
exp ik . r/ / − iω t
2
LM
N
±
= Lk x , k y , ±
K 02
MN
z < 0,
(A1)
z> 0
OP
Q
ω 2 ε2 c0− 2 − k x2 − k y2 + iδO
PQ
±
K 01
= k x , k y , ± ω 2 ε1c0− 2 − k x2 − k y2 + iδ
(A2)
The wave field obeys the following equation,
LMω 2 − c02 ∇ ×∇ ×OPE = 0
MN εbrg
PQ
(A3)
and the Green’s function,
LMω 2 − c02 ∇ ×∇ ×OP G+ b r, r' , ω g = + δb r − r 'g
MN εbrg
PQ
(A4)
Here we make the assumption that the dielectric function is local. This will not always
be true especially on the shortest length scales, but is known to be a good
approximation for metals where non locality only intrudes on length scales less than a
screening length, usually an Ångstrom or so. It is also a good approximation for
transparent materials: because of the large band gap, electronic valence orbitals are
well localised within the unit cell implying that the dielectric function is also local to
this same degree.
In the presence of a flat surface if both z and z’are in medium 1 (i.e. <0), then the pp
Green’s function becomes,
C:/word/qfric/qlight2.doc at 05/11/98
page 17
Can Sheared Surfaces Emit Light?
+
G surf
LM + e$ p e K 01+ je$ Tp e K 01+ jexpc + iK01z b z − z'gh OP
b ω , k , z , z 'g = 2 c 2 K M
P,
T
−
+
01z M + e$ p e K 01 je$ p e K 01 j expc − iK01z b z + z 'ghR pp P
N
Q
−i
LM + e$ p e K 01− je$ Tp e K 01− jexpc − iK01z b z − z'gh OP
+
G surf
,
b ω , k , z , z 'g = 2 c 2 K M
P
T
−
+
01z M + e$ p e K 01 je$ p e K 01 j expc − iK01z b z + z 'ghR pp P
N
Q
−i
0 > z > z'
(A5)
0 > z' > z
(A6)
where we assume that medium 1 occupies the negative half space so that,
ε −
R pp = 2
ε2 +
ε1
ε1
(A7)
Otherwise,
b
g
G +surf ω , k , z , z ' =
e j
−
− i e$ Tp K 02
2c 2 K02 z
e j c
b
gh e j c
b
gh
−
+
× e$ p K 02
exp − iK02 z z − z ' + e$ p K 02
exp + iK02 z z + z ' R 'pp ,
0 < z < z'
(A8)
where,
ε −
R 'pp = 1
ε1 +
ε2
ε2
b
(A9)
g
Note that the sign of z − z ' affects the value of e$ p .
Next we suppose that the surface is not flat but instead has a profile,
b g
b
g
b
z s = 2 Sg cos g ⋅r = Sg exp + ig ⋅r + exp − ig ⋅r
g
(A10)
as shown below in figure 6.
Figure 6. A small degree of roughness is introduced at the interface between
regions 1 and 2.
C:/word/qfric/qlight2.doc at 05/11/98
page 18
Can Sheared Surfaces Emit Light?
This introduces a perturbation into Maxwell’s equations,
LMω 2 − c02 c1 + ∆br gh∇ ×∇ ×OP E + ∆E = 0
MN εbr g
PQ
(A11)
Hence we have to first order perturbation,
bg
2
b gεcb0r g∇ ×∇ ×E = + G+surf ∆br gω 2E
+
∆ 1 E ≈+ G surf
∆r
(A12)
The variation of ∆ is shown below in figure 7.
Figure 7. The roughness can be described as a perturbation of the smooth
interface.
The effect can be that waves which previously had too much momentum to escape
from the surface can now do so by depositing momentum into the surface perturbation.
This opens a new channel for dissipation: the escape of radiation from the surface.
However to calculate the force we need to calculate how this new channel acts on the
original wave and this will require a second order perturbation calculation:
b 2g
bg
bg
∆ E = ω 4 G +surf ∆ r G +surf ∆ r E
(A13)
In particular we are interested in the second order contribution to the specular
reflectivity of the surface. There are two components to this calculation: the product of
vectors and the algebraic integrations. First the vectors. We remind ourselves,
b
g
+
× e$ p e K 01
jexpb + iK1z zg+ Rppe$ p e K 01− j expb − iK01z zg z < 0
LM + e$ p e K g+ 1je$ Tp e K g+ 1j expd + iKg1z b z − z'gi OP
−i
+
G surf b ω , k − g, z, z'g = 2
,
2c Kg1z MM + e$ p e K g− 1 je$ Tp e K g+ 1j expd − iKg1z b z + z 'giR pp PP
N
Q
−
Ekω p = L
3
2
exp ik . r/ / − iω t
(A14)
0 > z > z'
(A15)
C:/word/qfric/qlight2.doc at 05/11/98
page 19
Can Sheared Surfaces Emit Light?
+
G surf
LM + e$ p e K 01− je$ Tp e K 01− jexpc − iK01z b z − z'gh OP
b ω , k , z , z 'g = 2 c 2 K M
P,
T
−
+
01z M + e$ p e K 01 je$ p e K 01 j expc − iK01z b z + z 'ghR pp P
N
Q
−i
0 > z' > z
(A16)
so that the relevant vectors involved give:
e j e j e j
e j e j
+
×e$ Tp e K g+ 1 j e$ p e K 01
j + Rppe$ p eK 01− j
−
−
+
e$ p K 01
e$ Tp K 01
+ e$ Tp K 01
R pp e$ p K g+ 1 + e$ p K g− 1 R pp
(A17)
If we neglect the refractive index difference to medium 2 and suppose that only a
grating is present, this simplifies to,
e j e j e j e j e j
− $T
−
+
e$ p K 01
e p K 01
⋅e$ p K g+ 1 e$ Tp K g+ 1 ⋅e$ p K 01
(A18)
Alternatively for emission of s-polarised light,
e j e j e j e j e j
− $T
−
+
e$ p K 01
e p K 01
⋅e$ s K g+ 1 e$ Ts K g+ 1 ⋅e$ p K 01
(A19)
Substituting,
d
i d
±
K 01
≈ k x , k y , ±K01z ≈±ik ≈ g x , g y , ±ig
d
e j
i
i
d
c
c
+
e$ p K 01
≈ 0 k x , k y , ±ik ≈ 0 g x , g y , ±ig
ω ε1
ω ε1
(A20)
i
and,
d
i
K g±1 = k x − g x , k y − g y , ±Kg1z ,
c0
2
e$ p K g+ 1 =
k x − g x Kg1z , k y − g y Kg1z , − k − g ,
k − g ω ε1
eb
g
d
i
1
e$ s e K g+ 1 j =
e − d k y − g y i, b k x − gx g, 0j
k− g
e j
c0− 2 ε1ω 2 − k − g
Kg1z = +
j
(A21)
2
We obtain,
e j e j
+
e$ Tp K +g1 ⋅e$ p K 01
=
eb
g
c0
k x − g x Kg1z ,
k − g ω ε1
d
c
⋅ 0 g x , g y , ig
ω ε1
=
c02
2
k − g ω ε1
d k y − g y iKg1z ,
− k− g
2
j
i
b k x − g x gg x Kg1z + d k y − g y ig y Kg1z − ig k − g 2
(A22)
C:/word/qfric/qlight2.doc at 05/11/98
page 20
Can Sheared Surfaces Emit Light?
e j e j
e$ Tp K −01 ⋅e$ p K g+ 1 =
⋅
=
d
c0
g x , g y , − ig
ω ε1
eb
i
g
c0
k x − g x Kg1z ,
k − g ω ε1
c02
2
k − g ω ε1
d k y − g y iKg1z ,
− k− g
2
j
b k x − g x gg x Kg1z + d k y − g y ig y Kg1z + ig k − g 2
(A23)
Hence,
e j e j e j e j
c04
22
k
g
g
K
k
g
g
K
ig
k
g
=
−
+
−
+
−
b
g
d
i
x
x
x
g
1
z
y
y
y
g
1
z
2
k − g ω 4 ε2
+
e$ Tp K −01 ⋅e$ p K g+ 1 e$ Tp K g+ 1 ⋅e$ p K 01
1
(A24)
Similarly,
e j e j
ed
c
⋅ 0 d gx ,
ω ε1
c0
=
ω k − g ε1
e j e j
(A25)
c0
k x g y − k y gx
ω k − g ε1
=
−
e$ Tp K 01
⋅e$ s K g+ 1 =
i b k x − gx g, 0j
g y , ±ig i
− g x d k y − g y i+ g y b k x − g x g
1
− k y − gy ,
k− g
+
=
e$ Ts K +g1 ⋅e$ p K 01
d
c0
g x , g y , − ig
ω ε1
i
i b k x − gx g, 0j
ed
⋅
1
− ky − gy ,
k− g
=
c0
k x g y − k y gx
ω k − g ε1
(A26)
Hence,
e j e j e j e j
+
e$ Tp K −01 ⋅e$ s K g+ 1 e$ Ts K g+ 1 ⋅e$ p K 01
=
c02
2
ω 2 k − g ε1
k x g y − k y gx
2
(A27)
If the amplitude of the perturbation is not too great we can approximate,
b g LMN εε21 − 1OPQ δb zgSg expb + ig ⋅r g+ expb − ig ⋅r g
∆r =
(A28)
We have assumed that the perturbation is entirely in region 1.
C:/word/qfric/qlight2.doc at 05/11/98
page 21
Can Sheared Surfaces Emit Light?
b g b g e j b g expbik. r/ / − iω t g
LM + e$ Tp e K 01− j⋅e$ p eK g+ 1je$Tp e K g+ 1j⋅e$ p e K 01+ jOP
=M
MN + e$ Tp e K 01− j⋅e$ s eK g+ 1je$Ts eK g+ 1j⋅e$ p e K 01+ j PPQ
2
L
−i
−i
ε1 O 2 −
4
×ω
M − 1P Sg L expbik. r/ / − iω t g
2c 2 Kg1z 2c 2 K01z N ε2 Q
LM + e$ Tp e K 01− j⋅e$ p eK g+ 1je$Tp e K +g1j⋅e$ p e K 01+ jOP
=M
MN + e$ Tp e K 01− j⋅e$ s eK g+ 1je$Ts eK g+ 1j⋅e$ p e K 01+ j PPQ
LM ε1 − 1OP 2 Sg2 L− expbik. r/ / − iω t g
− ε2
×ω 4 4 1
4 c K K N ε2 Q
−
+
ω 4 G +surf ∆ r G +surf ∆ r e$ p K 01
exp + iK1z z L
3
3
3
2
2
(A29)
2
0 g1z 01z
Therefore the reflection coefficient from this contribution is,
R pp = R ppp + R psp
(A30)
where,
e j
R ppp k ' , ω
2
L
ε1 O
−
1
e j e j e j e j 4c4 K K MN ε2 PQ
0 g1z 01z
2
L
ε1 O
c04
− ε12
'
'2 2 4 2
= '2 4 2 k ⋅g Kg1z + igk ω Sg 4
M − 1P
k ω ε1
4c0 Kg1z K01z N ε2 Q
2
2
− Sg2
O
L
'
2 '4 O L ε1
= '2
Mek ⋅g Kg1z j + g k PQ MN ε2 − 1PQ
4 k Kg1z K01z N
2
2
+ i Sg2 L '
O
2 '4 O L ε1
≈ '2
Mek ⋅g Kg1z j + g k PQ MN ε2 − 1PQ
4 k Kg1z g N
R psp e k ' , ω j
2
L
ε1 O
− ε12
−
+
+
+
4
2
T
T
= e$ p e K 01 j ⋅e$ s e K g1 je$ s e K g1 j ⋅e$ p e K 01 j ×ω Sg
M − 1P
4c04 Kg1z K01z N ε2 Q
2
2 4 2
L
c02
− ε12
ε1 O
'
'
= 2 ' 2 k x g y − k y g x ω Sg 4
M − 1P
4c0 Kg1z K01z N ε2 Q
ω k ε1
2
2 Lε
+ i ε1ω 2 Sg2 1 '
O
'
1
≈ 2
k x g y − k y g x M − 1P
4c0 Kg1z g k '2
N ε2 Q
−
+
ω 4 Sg2
= e$ Tp K 01
⋅e$ p K g+ 1 e$ Tp K g+ 1 ⋅e$ p K 01
− ε12
(A31)
(A32)
In (A33) and (A34) we have defined a new variable,
C:/word/qfric/qlight2.doc at 05/11/98
page 22
Can Sheared Surfaces Emit Light?
k' = k − g
(A33)
We should also remind ourselves that,
b g
Kg1z b k , ω g = +
K01z k , ω ≈ig
c0− 2 ε1ω 2 − k − g
2
C:/word/qfric/qlight2.doc at 05/11/98
(A34)
page 23