Download Consistent formalism for the momentum of electromagnetic waves in

Consistent formalism for the momentum of electromagnetic waves in
lossless dispersive metamaterials and the conservation of momentum
Yingran He1, Jianqi Shen1 and Sailing He1,2*
1
Centre for Optical and Electromagnetic Research, JORCEP[KTH-ZJU Joint Research Center of Photonics], East Building #5,
2
Department of Electromagnetic Engineering, School of Electrical Engineering, Royal Institute of Technology (KTH), S-100 44
Zijingang campus, Zhejiang University (ZJU), Hangzhou 310058, China
Stockholm, Sweden
* Corresponding author: [email protected]
Abstract- A new formalism for electromagnetic and mechanical momenta in a metamaterial is
developed by means of the technique of wave-packet integrals. The medium has huge mass
density and can therefore be regarded as almost stationary upon incident electromagnetic waves. A
clear identification of momentum density and momentum flow, including their electromagnetic
and mechanical parts, is obtained by employing this formalism in a lossless dispersive
metamaterial (including the cases of impedance matching and mismatching with vacuum). It is
found that the ratio of the electromagnetic momentum density to the mechanical momentum
density depends on the impedance and group velocity of the electromagnetic wave inside the
metamaterial. One of the definite results is that both the electromagnetic momentum and the
mechanical momentum in the metamaterial are in the same direction as the energy flow, instead of
in the direction of the wave vector. The conservation of total momentum is verified. In addition,
the law of energy conservation in the process of normal incidence is also verified by using the
wave-packet integral of both the electromagnetic energy density and the electromagnetic power
density, of which the latter is caused by the interaction between the induced (polarized) currents
and the electromagnetic wave.
1. INTRODUCTION
The expressions/definitions for the electromagnetic momentum density and momentum stress
tensor in vacuum have been clearly identified and widely accepted [1]. However, the expressions
for the electromagnetic momentum density and momentum stress tensor in a medium have been
debated for over a century. The earliest expressions for momentum density in a dielectric medium
were proposed by Abraham [2] as g = ε 0 µ0 E × H and by Minkwoski [3] differently as
g = D × B . Proceeding theoretical and experimental works have been carried out [4-14], trying
to solve this controversy. Early experiment by Jones and Richards found that the radiation
pressure on a mirror immersed in a refracting liquid medium is proportional to the refractive index
of the liquid [4]. Theoretical analyses by Gordon [5] and Peierls [6] have concluded that: (a)
Abraham’s momentum gives the electromagnetic part momentum of waves in the medium, but
excludes the mechanical part momentum carried by the medium itself; (b) Minkwoski’s
momentum is actually a pseudo-momentum, which results from the invariance of physical laws
with respect to the displacement of medium coordinates; This can be illustrated more clearly using
1
both spatial (Eulerian) and material (Lagrangian) coordinates [7, 8]. (c) The Minkwoski’s
momentum is mathematically useful in determining the radiation pressure exerted on a mirror by
incident light, although the radiation pressure origins from both electromagnetic momentum and
mechanical momentum of waves. Recent studies focus on the calculation of momentum of waves
in different media, from ordinary dielectrics [9] to magnetic media [10, 13], and even to dispersive
media [11, 12, 14]. A recent review paper can also be found for an overview on this subject [15].
Metamaterials with simultaneously negative permittivity and negative permeability may
achieve unprecedented electromagnetic properties, such as negative refraction [16, 17] and
potentially perfect imaging [18]. However, far less attention has been paid to the investigation of
the momentum for electromagnetic waves in metamaterials [12, 19, 20]. Scalora et al. [12] showed
that the Lorentz force in a metamaterial was consistent with the rate of change of Abraham’s
momentum, but they did not obtain the ratio of the mechanical part momentum to the
electromagnetic part momentum for electromagnetic waves. Kemp et al. [19] claimed that the
momentum flow in a left-handed material was opposite to the power flow direction, but we
believe this conclusion still requires further argument/research. Veselago has noticed the
incompatibility between photon momentum flow and energy transfer for electromagnetic waves in
metamaterials, and assumed that the momentum inside a metamaterial is in the same direction as
the wave vector [20], which is not correct as we show in the present paper.
2. MODELING
It has been pointed out that a complete expression for the momentum in a medium needs to
take both the electromagnetic part momentum and the mechanical part momentum into account
[15]. The continuity equations for them are
∂ g em + ∇iT em = − f mech
∂t
∂ g mech + ∇iT mech = f mech
∂t
(1)
(2)
where g em , T em , g mech and T mech represent the momentum density and momentum flow
density tensor (momentum stress tensor) for the electromagnetic part and the mechanical part,
respectively.
The sum of the above two equations yields the expression for the conservation of the total
momentum
d g
dV
+
d
S
tot
∫ S iT tot = 0
dt ∫V
(3)
Here, g tot and T tot represent the total momentum density and momentum flow density tensor
inside the medium, respectively.
By directly applying Maxwell’s equations, we can obtain the following expressions for the
momentum density of electromagnetic field g em , the generalized Lorentz force density f mech ,
and the electromagnetic momentum flow density T em ,
g em = ε 0 µ0 E × H
2
(4)
f mech = −(∇i P) E − µ0 (∇i M ) H + µ0 (∂ P / ∂t ) × H − µ0ε 0 (∂ M / ∂t ) × E
(5)
1 T em = (ε 0 E ⋅ E + µ0 H ⋅ H ) I − (ε 0 E E + µ0 H H )
(6)
2
where D = ε 0 E + P and B = µ0 ( H + M ) have been substituted. The momentum density of
electromagnetic field is in Abraham’s form, and the four terms in Eq. (5) represent the force on the
electrically polarized charge by the electric field, the force on the magnetically polarized charge
by the magnetic field, the force on the electrically polarized current by the magnetic field and the
force on the magnetically polarized current by the electric field, respectively. The electromagnetic
momentum flow density tensor relates to E and H , rather than D and B .
It can be verified that f mechδ V equals the mechanical momentum increase rate inside the
macroscopic small volume δ V [14]. In other words, the generalized Lorentz force plays the role
of converting the electromagnetic momentum into the mechanical momentum, while the total
momentum is conserved. Due to the action of the generalized Lorentz force, the unit cell of the
medium will gain certain mechanical momentum and, therefore, be set into motion.
In general, motion of unit cells inside the medium will not only give rise to mechanical
momentum and mechanical momentum flow, but also have an impact on the constitutive relation
of the medium [8]. Therefore, the mechanical properties of a dielectric material have to be
specified, in order to describe how it responses to the action of the generalized Lorentz force. To
simplify the calculation, in this paper we aim at deriving an explicit expression for momentum in a
homogenous isotropic medium, which is constituted by unit cells with very large mass, i.e., huge
mass density M , such that the motion of dielectrics is sufficiently small and the medium remains
almost stationary and consequently the constitutive relation of the medium remains the same upon
electromagnetic wave incidence.
Although velocity v of the mass center of the unit cell is infinitesimal as a result of huge
mass density M , their product Mv , representing the accompanied mechanical momentum
density g mech , is a comparable quantity with respect to electromagnetic momentum density g em .
Note that the mechanical momentum flow T mech , of the order ( Mv )v , is still negligibly small
compared with electromagnetic momentum flow T em .
Based on the above analysis, we can obtain a clear physical interpretation of the momentum
transfer and conversion for propagating waves inside our medium model. When the leading edge
of a travelling wave passes though a macroscopic small volume, the volume will not only gain a
momentum carried by the electromagnetic wave itself, but also acquire a mechanical momentum
carried by the infinitesimal movement of the huge mass, namely Mv , due to the presence of the
generalized Lorentz force. The force will decrease gradually as the electromagnetic field grows
inside this volume and eventually vanish when the electromagnetic field becomes time-harmonic.
After that, both the electromagnetic momentum density and the mechanical momentum density of
this small volume reach some steady values, and a constant momentum flow is carried by the
3
electromagnetic field, which transmits the required momentum to establish a time-harmonic field
at the new leading edge of the propagating wave.
The additional mechanical momentum density Mv can be evaluated by integrating the
Lorentz force over the time for establishing a time-harmonic field. For a plane wave in a
dispersion-free non-magnetic dielectric with refractive index n, it has been shown that [6, 9]
g mech =
n2 − 1
g em
2
(7)
In this paper, we would like to focus on the momentum density and momentum flow in a
metamaterial, which is intrinsically dispersive. Inside a dispersive medium, the medium’s response
to the driving of an incident field is time-dependent in a causal way, i.e., its response at time τ0 is
related not only to the driving field at time τ0, but also to the driving field at times previous to τ0.
To treat this dispersive property, it is convenient to construct a wave packet, and investigate the
increase of the mechanical momentum along with the establishment of this wave packet inside the
medium. In the limit that the wave packet is long enough, a correct expression for momentum
density can be derived (see [11] for a dispersive dielectric case). It is interesting to notice that the
momentum flow density tensor is independent of dispersion.
3. MOMENTUM DENSITY, MOMENTUM FLOW AND MOMENTUM CONSERVATION
Since the momentum of electromagnetic waves in vacuum is unambiguous, the fundamental
physical law of momentum conservation can be used to check the validity of our modeling theory
for the case of normal incidence of plane waves from vacuum into a semi-infinite medium. In this
paper the metamaterial is assumed to be lossless for the sake of simplicity and clarity.
3.1 Momentum for the case of matched impedance
Now we study the case when a plane wave normally impinges from vacuum on a semi-infinite
metamaterial with negative relative permittivity εr and negative relative permeability µr.
A metamaterial is always dispersive. First we assume εr and µr have the same dispersion, i.e.,
εr(ω)=µr(ω), so that its impedance matches with the vacuum impedance for all frequencies to
ensure total transmission. In order to deal with a dispersive material, it is convenient to construct a
wave packet by using two spectrum components whose frequencies are close to each other, similar
to the treatment of energy density inside a dispersive dielectric medium [1].
In vacuum, the incident field components are
 Exi = E0 sin(k0a z − ωa t ) − E0 sin(k0b z − ωbt )
 i
a
b
 H y = ε 0 / µ0 E0 sin(k0 z − ωa t ) − ε 0 / µ0 E0 sin(k0 z − ωb t )
ν
where k0 is the vacuum wave vector corresponding to
(8)
ων with ν = a, b .
In the metamaterial, the transmitted field components are
 Ext = E0 sin(ka z − ωa t ) − E0 sin(kb z − ωb t )
 t
 H y = ε 0 / µ0 E0 sin(ka z − ωa t ) − ε 0 / µ0 E0 sin(kb z − ωb t )
where kν = −
(9)
ε νr µνr k0ν = ε νr kν0 , ε νr and µνr = ε νr are the wave vector, relative permittivity
and relative permeability at frequency ων with ν = a, b . In the limit of ωa → ωb → ω , the
4
wave vector, relative permittivity and relative permeability corresponding to the central frequency
ω will be denoted by k , ε r and µr , respectively.
Now we focus on the momentum conversion process of this wave packet during a time
interval [0, τ]. τ = 2π / (ωa − ωb ) corresponds to the period of this wave packet envelope.
At t=0, the incident wave packet which locates at z∈[-cτ, 0] has the following momentum
0
0
− cτ
− cτ
Pvac = ∫ ( gem )in dz = ∫ ε 0 µ0 Exi H yi dz
=
=
ε0E
2
0
∫ τ ( sin(k
0
−c
c
a
0
z − ωa t ) − sin( k0b z − ωbt ) )( sin(k0a z − ωa t ) − sin(k0b z − ωbt ) ) dz
(10)
 cos(2k0a z − 2ωa t ) cos(2k0b z − 2ωb t )

−
1 −

2
2

 dz
∫
−
τ
c
c
a
b
a
b
 + cos (k + k ) z − (ω + ω )t  − cos (k − k ) z − (ω − ω )t  
0
a
b 
0
a
b 
 0
 0

ε 0 E02
0
where c is the optical velocity in vacuum.
It is noted that the one-packet spatial integral of the highly-oscillating terms cos(2k0 z − 2ωa t ) ,
a
cos(2k0b z − 2ωbt ) and cos (k0a + k0b ) z − (ωa + ωb )t  would vanish when we take the limit of
ωa → ωb → ω , and that the wave-packet spatial integral of the slowly-oscillating term
cos (k0a − k0b ) z − (ωa − ωb )t  is also zero, i.e.,
ε 0 E02
cos ( (k0a − k0b ) z − (ωa − ωb )t ) dz
c
ε E2 1 
=− 0 0 a
sin ( (k0a − k0b )(cτ ) + (ωa − ωb )t ) − sin ( (ωa − ωb )t ) 
b 
c k0 − k0
−
∫
0
− cτ
=0
Therefore, the incoming momentum is easily found as
Pvac = ε 0 E02τ
(11)
At t=τ, this wave packet will completely penetrate into the metamaterial at the location z∈[0,
vgτ],
vg ≡
where
vg ≡
dω ωa − ωb
=
dk
k a − kb
is
the
group
velocity.
In
this
case,
dω
dω
c
=
=
is always positive since a normal dispersion, i.e.,
dk d(ωε r / c) d(ωε r ) / dω
dε r / dω > 0 , is assumed in our investigation. The total momentum inside the metamaterial
consists of two parts, the electromagnetic part and the mechanical part. Similar to the calculation
of Pvac , electromagnetic part momentum Pem , and spatially averaged momentum density g em
5
are
Pem = ∫
vgτ
0
vgτ
gem dz = ∫ ε 0 µ0 Ext H yt dz = ε 0 E02 vgτ / c
(12)
0
gem = Pem / vgτ = ε 0 E02 / c
(13)
The mechanical part of the momentum is produced by the action of Lorentz force, so that it
can be calculated by integrating the Lorentz force exerted on this wave packet, i.e.,
Pmech = ∫
τ
0
∫
vg t
0
f mech ( z , t )dzdt
mech
In this situation, the z component of Lorentz force, f z
(14)
, turns out to be
f zmech = µ0ε 0 [ −ωa (ε ra − 1) E0 cos(ka z − ωa t ) + ωb (ε rb − 1) E0 cos(kb z − ωb t )]H yt
+ µ0ε 0 [ −ωa ( µ − 1) ε 0 / µ0 E0 cos( ka z − ωa t ) + ωb ( µ − 1) ε 0 / µ0 E0 cos(kb z − ωbt )]E
a
r
b
r
(15)
t
x
After calculation (see Appendix), it is found that the mechanical momentum is
Pmech = ε 0 µ0 ε 0 / µ0 E02τ [(ε ra − 1)ωa − (ε rb − 1)ωb ] / (ka − kb )
(16)
In the limit of ωa → ωb → ω , we obtain the following mechanical momentum inside this wave
packet
Pmech = ε 0 E02τ (
d(ωε r )
d(ωε r )
− 1) /
= ε 0 E02τ (1 − vg / c)
dω
dω
(17)
and subsequently the spatially averaged mechanical momentum density is obtained
g mech = Pmech / vgτ = ε 0 E02 (1/ vg − 1/ c)
(18)
It follows from Eqs. (11), (12) and (17) that Pvac = Pem + Pmech , which means that the law of
momentum conservation holds during the period when the electromagnetic wave penetrates into
the metamaterial from vacuum (see Fig. 1)
Figure 1. Illustration of momentum conservation at the interface between vacuum and an impedance-matched
metamaterial. The incident wave packet (in blue) carries momentum Pvac, while the transmitted wave packet (in
red) carries momentum Pem+Pmech.
Since a total momentum Pem + Pmech is transmitted into the metamaterial from the vacuum
6
during a time interval [0, τ], the total momentum flow in metamaterial is
pztot = ( Pem + Pmech ) / τ = ε 0 E02
(19)
This result is in agreement with the theoretical prediction by the momentum flow density tensor
Tzztot = Tzzem = ε 0 E02 in the metamaterial, according to Eq. (6).
In a brief summary, for a +z propagating plane wave in a metamaterial, our results show that
 k = − ε r µ r k 0 = ε r k0

 S z = ε 0 / µ0 E02
 tot
2
 pz = ε 0 E0

2
 g em = S z / c
 g mech = (c / vg − 1) g em

(20)
It is worth noticing that, inside the metamaterial the wave vector is in –z direction, but the
momentum density and momentum flow (both the electromagnetic part and mechanical part) of
the plane wave is in +z direction (as the energy flow). Therefore, we conclude that, within the
classical electrodynamics framework, the total momentum of electromagnetic waves in a
metamaterial has the same direction as the energy flow, instead of as the wave vector [19, 20].
3.2 Momentum for the case of mismatched impedance
In this situation, we would like to perform a calculation on a more general case, i.e.,
εr(ω)≠µr(ω) to show that our formalism is applicable for more general situations.
The components for the incident field, reflected field and transmitted field in this case are as
follows,
 Exi = E0 sin(k0a z − ωa t ) − E0 sin(k0b z − ωbt )
 i
a
b
 H y = ε 0 / µ0 E0 sin(k0 z − ωa t ) − ε 0 / µ0 E0 sin(k0 z − ωb t )
(21)
 Exr = ra E0 sin(k0a z − ωa t ) − rb E0 sin(k0b z − ωb t )
 r
a
b
 H y = − ε 0 / µ0 ra E0 sin(k0 z − ωa t ) + ε 0 / µ0 rb E0 sin(k0 z − ωb t )
(22)
 Ext = ta E0 sin(ka z − ωa t ) − tb E0 sin(kb z − ωb t )
 t
a
a
b
b
 H y = ε 0ε r / µ0 µr ta E0 sin(ka z − ωa t ) − ε 0ε r / µ0 µr tb E0 sin(kb z − ωb t )
(23)
Here, rν =
1 − ε νr / µνr
ν
ν
1 + ε r / µr
, tν =
2
, and kν = −
1 + ε r / µr
ν
ν
ε νr µνr kν0 are the reflection
coefficients, transmission coefficients (for the electric field), and wave vector corresponding to
ων with ν = a, b , respectively. ε νr and µνr are the relative permittivity and permeability
corresponding to ων with ν = a, b , respectively. In the limit of ωa → ωb → ω , the reflection
coefficient, transmission coefficient and wave vector corresponding to the central frequency
will be denoted by r0 , t0 and k , respectively.
7
ω
Again, we focus on the momentum conversion process of this wave packet during a time
interval [0, τ] with τ = 2π / (ωa − ωb ) .
At t=0, the incident wave packet which locates at z∈[-cτ, 0] has the following momentum
0
0
− cτ
− cτ
Pvac = ∫ ( g em )in dz = ∫ ε 0 µ0 Exi H yi dz = ε 0 E02τ
(24)
At t=τ, part of this wave packet will penetrate into the metamaterial at the regime z∈[0, vgτ],
while another part of this wave packet will be reflected backward to the regime z∈[-cτ, 0]. It is
noted that the group velocity vg ≡
dω
dω
=
is also positive in this case, since a
dk d( − ω ε r µ r / c)
normal dispersion, i.e., dε r / dω > 0 , and dµr / dω > 0 is assumed. The reflected wave packet
which locates at z∈[-cτ, 0] has the following momentum
0
0
− cτ
− cτ
Pvac2 = ∫ ( g em )reflection dz = ∫ ε 0 µ0 Exr H yr dz = −ε 0 E02 r02τ
(25)
The total momentum inside metamaterial consists of two parts, i.e., the electromagnetic part and
the mechanical part. They are shown to have the following expressions (see Appendix)
Pem = ε 0 E02τ
gem =
ε r 2 d(− ε r µr ω )
t0 /
µr
dω
(26)
Pem
ε
= ε 0 E02 r t02 / c
vgτ
µr
Pmech = ε 0 E02τ t02
g mech =
(27)
 d  εr
  d(− ε r µr ω )
1  ε r d  µr
(ε r − 1)ω  +
( µr − 1)ω   /



2  µ r dω  ε r
dω
 dω  µ r
 
 d  εr

Pmech
1  ε d  µr
= ε 0 E02t02  r
(ε r − 1)ω  +
( µr − 1)ω   / c


vgτ
2  µ r dω  ε r
 dω  µ r
 
(28)
(29)
The above expressions (24)-(29) are the limiting results when ωa → ωb → ω . Through
some tedious derivation, we can still show that Pvac = Pvac2 + Pem + Pmech (see Appendix), which
is the expression for momentum conservation in this situation (see Fig. 2). The mechanical
momentum can be rewritten in a compact way,
Pmech = ε 0 E02τ (1 + r02 − t02
ε r vg
)
µr c
(30)
g mech = ε 0 E02 (1 + r02 − t02
ε r vg
) / vg
µr c
(31)
8
Figure
2.
Illustration
of
momentum
conservation
at
the
interface
between
vacuum
and
an
impedance-mismatched metamaterial. The incident wave packet (in blue) carries momentum Pvac, while the
reflected wave (in green) carries momentum Pvac2, and the transmitted wave packet (in red) carries momentum
Pem+Pmech.
It is shown that the mechanical momentum is in the same direction as energy flow, for
ε r vg
)
µr c
Pmech = ε 0 E02τ (1 + r02 − t02
>ε 0 E02τ (1 + r02 − t02
1+
= ε 0 E τ (t
2
0
2
0
εr
µr
2
εr
)
µr
−t
2
0
(32)
ε 0 E02τ t02
εr
ε
)=
(1 − r )2 ≥ 0
2
µr
µr
It is noted that the total momentum flow in metamaterial is,
pztot = ( Pem + Pmech ) / τ = (1 + r02 )ε 0 E02
(33)
which is again consistent with momentum flow density tensor
 ε 
1
1
Tzztot = Tzzem = [ε 0 ( Ext )2 + µ0 ( H yt ) 2 ] = ε 0 E02t02 1 + r 
2
2
 µr 
(34)
by the aid of (1 + r0 ) / t0 = (1 + ε r / µr ) / 2 .
2
2
Based on the above calculation, we again conclude that inside a metamaterial, both the
electromagnetic momentum and the mechanical momentum are in the same direction as the energy
flow, instead of as the wave vector. Comparing with the impedance matching case, we also
conclude that the ratio of the electromagnetic momentum density to the mechanical momentum
density depends on the impedance and group velocity of electromagnetic waves inside the
metamaterial.
4. ENERGY DENSITY, ENERGY OF WAVE PACKET AND ENERGY CONSERVATION
In the previous section, we have utilized the formalism of Yaghjian [14], where the field
variables are E and H , to show that the law of momentum conservation is preserved when the
9
electromagnetic wave is launched into the metamaterial. We can derive the Poynting-type theorem
using the formalism of Yaghjian [14], which is given by
 ∂P ∂  1 2 1 2 
∂M 
⋅ E + µ0
⋅H 
 ε 0 E + µ0 H  + ∇ ⋅ E × H = − 
∂t  2
∂t
2

 ∂t

(
)
(35)
This means that the power density that leads to the mechanical energy of the induced (polarized)
electric and magnetic currents is (∂P / ∂t ) ⋅ E + ( µ0 ∂M / ∂t ) ⋅ H .
As
ε 0 E 2 / 2 = µ0 H 2 / 2 for the electromagnetic wave in vacuum, the incident
2
electromagnetic energy density in vacuum is wvac = ε 0 E , which can be expressed explicitly by
wvac = ε 0  E0 sin( k0a z − ωa t ) − E0 sin(k0b z − ωbt ) 
2
= ε 0 E02 − ε 0 E02 cos ( (k0a − k0b ) z − (ωa − ωb )t )
(36)
 cos 2(k0a z − ωa t ) cos 2(k0b z − ωbt )

− ε 0 E02 
+
− cos ( (k0a + k0b ) z − (ωa + ωb )t ) 
2
2


The electromagnetic energy of a wave packet which locates at z∈[-cτ, 0] is defined by
Ε vac = ∫
0
− cτ
wvac ( z , t )dz
(37)
It should be noted that the one-packet spatial integral of the highly-oscillating terms [where
cos 2( kν0 z − ων t ) or cos ( (k0a + k0b ) z − (ωa + ωb )t ) are involved] would vanish when we take the
limit of
ωa → ωb → ω , and that the wave-packet spatial integral of the slowly-oscillating term
−ε 0 E02 cos ( ( k0a − k0b ) z − (ωa − ωb )t ) in wvac ( z , t ) is also zero, i.e.,
− ε 0 E02 ∫ cos ( (k0a − k0b ) z − (ωa − ωb )t ) dz
0
− cτ
= −ε 0 E02
1
sin ( (k0a − k0b )(cτ ) + (ωa − ωb )t ) − sin ( (ωa − ωb )t ) 

k0a − k0b 
(38)
=0
Therefore, it follows that the incident electromagnetic energy of a wave packet is
Ε vac = ε 0 E02 cτ .
In the following part, we shall consider the problem of energy conservation during the
electromagnetic interaction when the electromagnetic wave is launched normally into the
metamaterial. We will address two cases, namely, impedance matching and mismatching cases, to
show that the energy in these processes is also conserved.
4.1 Energy conservation in the case of matched impedance
When the impedance of the metamaterial is matched with that of vacuum, the explicit
expression for electromagnetic energy Ε em =
∫
10
vgτ
0
wem ( z , t )dz in the metamaterial can be
obtained in a similar manner as for obtaining electromagnetic energy Ε vac in vacuum. It can be
readily verified that the wave-packet spatial integral of the highly-oscillating terms involved in the
electromagnetic energy density wem ( z , t ) in the metamaterial vanishes under the condition
of
ωa → ωb → ω . Besides, the wave-packet integral of the slowly-oscillating term
−ε 0 E02 cos ( (ka − kb ) z − (ωa − ωb )t ) in wem ( z , t ) can be expressed by
− ε 0 E02 ∫
vgτ
0
cos ( (ka − kb ) z − (ωa − ωb )t ) dz
= −ε 0 E02
1
vτ
sin ( (ka − kb ) z − (ωa − ωb )t ) |0g
k a − kb
= −ε 0 E02
1
sin ( (ka − kb )vgτ − (ωa − ωb )t ) − sin ( −(ωa − ωb )t ) 

k a − kb 
(39)
=0
Therefore, the electromagnetic energy of a wave packet propagating in the metamaterial is
Εem = ε 0 E02 vgτ .
Let us now consider the interaction energy, which is the mechanical energy acquired by the
induced electric and magnetic currents. The energy transferred by the electric interaction, of which
(
)
the power density is ∂P / ∂t ⋅ E , in one wave packet is
 vg t ∂P 
We = ∫  ∫
⋅ Edz dt
0
0
∂t


τ
(40)
where the induced electric current density is given by
∂P
= −ωa (ε ra − 1)ε 0 E0 cos(ka z − ωa t ) + ωb (ε rb − 1)ε 0 E0 cos( kb z − ωb t )
∂t
The explicit expression for the electric power density is
(41)
∂P
⋅ E = ( −ωa (ε ra − 1)ε 0 E0 cos(ka z − ωa t ) + ωb (ε rb − 1)ε 0 E0 cos( kb z − ωb t ) )
∂t
× ( E0 sin(ka z − ωa t ) − E0 sin( kb z − ωb t ) )
 −ωa (ε ra − 1)sin 2(ka z − ωa t ) − ωb (ε rb − 1)sin 2(kb z − ωbt ) 
(42)
= ε0E 

2



sin ( (ka + kb ) z − (ωa + ωb )t ) − sin ( ( ka − kb ) z − (ωa − ωb )t ) 
+ ε 0 E02 ωa (ε ra − 1)

2


2
0

sin ( (kb + ka ) z − (ωb + ωa )t ) − sin ( ( kb − ka ) z − (ωb − ωa )t ) 
+ ε 0 E02 ωb (ε rb − 1)

2


It should be noticed that the wave-packet integral of the highly-oscillating terms in ∂P / ∂t ⋅ E
(
vanishes when the limit process of
)
ωa → ωb → ω is taken. Thus, the only interesting term in
11
( ∂P / ∂t ) ⋅ E
that deserves consideration is the slowly-oscillating term
ε 0 E02
ωa (ε ra − 1) − ωb (ε rb − 1)  sin ( (ka − kb ) z − (ωa − ωb )t )
2 
Thus, the acquired mechanical energy of the induced electric current can be rewritten as
A( z, t ) = −
We = ∫
τ
0
(∫
vg t
0
(43)
)
A( z, t )dz dt
ε 0 E02
τ
vg t
ωa (ε ra − 1) − ωb (ε rb − 1)  ∫  ∫ sin ( (ka − kb ) z − (ωa − ωb )t )dz dt


0
0


2
2
ε E
−1 τ 
vt
= − 0 0 ωa (ε ra − 1) − ωb (ε rb − 1) 
cos ( (ka − kb ) z − (ωa − ωb )t ) |0g dt
∫

0
2
k a − kb
=−
=
ε 0 E02
1
ωa (ε ra − 1) − ωb (ε rb − 1) 
2
k a − kb
(44)
 sin ( (ka − kb )vg − (ωa − ωb ) )τ  sin ( (ω − ω )τ ) 
 

−
a
b


(
k
k
)
v
(
)
−
−
ω
−
ω
ω
−
ω
a
b
g
a
b
a
b


As the group velocity of the electromagnetic wave is vg = (ωa − ωb ) / ( ka − kb ) , the
ωa → ωb → ω limit of the term sin ( ( ka − kb )vg − (ωa − ωb ) )τ   ( ka − kb )vg − (ωa − ωb ) 
is τ . Thus, the mechanical energy transferred by the electric interaction reads
We =
when the
We
ε 0 E02  d (ωε r ) dω 
2 
−
dk
τ =
dk 
ε 0 E02
2
( c − vg )τ =
ε 0 E02 
vg
1 −
c
2 

 cτ

(45)
ωa → ωb → ω limit is taken into account.
now
consider
(
the
mechanical
energy
Wm
transferred
by
the
magnetic
)
interaction µ0 ∂M / ∂t ⋅ H . For the present case of impedance matching, we have the
(
)
(
)
relation µ0 ∂M / ∂t ⋅ H = ∂P / ∂t ⋅ E , and this implies that Wm = We . Hence, the total
mechanical
energy
Wmech
( ∂P / ∂t ) ⋅ E + µ ( ∂M / ∂t ) ⋅ H
0
caused
by
the
electromagnetic
interaction
in one wave packet (propagating in the metamaterial) is given by
 v 
Wmech = We + Wm = ε 0 E02 1 − g  cτ
c

(46)
It can be found that there exists a relation of Ε vac = Ε em + Wmech , i.e., the incident
electromagnetic energy in vacuum equals the sum of the mechanical energy of the induced
electromagnetic currents and the electromagnetic energy of the wave propagating in the
metamaterial.
4.2 Energy conservation in the case of mismatched impedance
12
The incident electromagnetic energy of one wave packet in vacuum is Ε vac = ε 0 E0 cτ .
2
Obviously, the electromagnetic energy of one wave packet reflected back into vacuum is
Ε vac2 = ε 0 E02 r02 cτ . We will calculate the transmitted electromagnetic energy density
ε 0 E 2 / 2 + µ0 H 2 / 2 in the metamaterial. The explicit expression for the electric energy density is
given by
2
1
1
ε 0 Et2 = ε 0 ta E0 sin( ka z − ωa t ) − tb E0 sin(kb z − ωb t ) 
2
2
1
1
= ε 0 E02 ( ta2 + tb2 ) − ε 0 E02ta tb cos ( (ka − kb ) z − (ωa − ωb )t )
4
2


cos
2(
k
cos 2(kb z − ωb t ) 2
1
a z − ωa t ) 2
− ε 0 E02 
ta +
tb − ta tb cos ( (ka + kb ) z − (ωa + ωb )t )  .
2
2
2


(47)
In the same fashion as the previous analysis, the spatial integrals of both the highly- and
slowly-oscillating terms in one wave packet vanish. Thus, the electric energy,
Εe = ∫
vgτ
0
1
ε 0 Et2 dz , of the wave propagating inside the metamaterial is
2
1
Ε e = ε 0 E02 ( ta2 + tb2 ) vgτ
4
(48)
On the other hand, the explicit expression for the magnetic energy density is found to be
1
1
1
µ 0 H t2 = µ 0 2
2
2 η0
=
1

1
 t a E 0 sin( k a z − ω a t ) − t b E 0 sin( k b z − ω b t ) 
η
η
b
 a

2
 1
1
1  1
t t
ε 0 E 02  2 t a2 + 2 tb2  − ε 0 E 02 a b cos ( ( k a − k b ) z − (ω a − ω b ) t )
4
ηb  2
η aη b
 ηa
 cos 2( k a z − ω a t ) 1 2 cos 2( k b z − ω b t ) 1 2 t a tb

1
− ε 0 E 02 
t +
t −
cos ( ( k a + k b ) z − (ω a + ω b ) t ) 
2 a
2 b
2
2
2
η
η
η
η

a
b
a b

µ0
µνr
where η0 ≡
is the vacuum impedance and ην ≡
ε0
ε νr
corresponding to
(49)
is the relative impedance
ων , with ν = a, b . In the limit of ωa → ωb → ω , the relative impedance
corresponding to the central frequency
ω will be denoted by ηr . As the spatial integrals of both
the highly- and slowly-oscillating terms in one wave packet vanish, the magnetic energy of a wave
packet in the metamaterial is
Εm = ∫
vgτ
0
 1
1
1
1 
µ0 H t2 dz = ε 0 E02  2 ta2 + 2 tb2  vgτ
2
4
ηb 
 ηa
(50)
Therefore, the total electromagnetic energy (for one wave packet) in the metamaterial is expressed
by


1
1 
1  
Εem = Εe + Ε m = ε 0 E02 1 + 2  ta2 + 1 + 2  tb2  vgτ
4
 ηb  
 η a 
13
(51)
which can be rewritten as Ε em =

1
1 
ε 0 E02 1 + 2  t02vgτ if we take the limit of ωa → ωb → ω .
2
 ηr 
Now we shall consider the mechanical energy acquired by the induced electric current. The
power density of the electric interaction is
 −ω (ε a − 1)ta2 sin 2(ka z − ωa t ) − ωb (ε rb − 1)tb2 sin 2(kb z − ωbt ) 
∂P
⋅ E = ε 0 E02  a r

∂t
2


sin ( (ka + kb ) z − (ωa + ωb )t ) − sin ( (ka − kb ) z − (ωa − ωb )t ) 

+ ε 0 E02 ωa (ε ra − 1)ta tb

2


(52)

sin ( (kb + ka ) z − (ωb + ωa )t ) − sin ( (kb − ka ) z − (ωb − ωa )t ) 
+ ε 0 E02 ωb (ε rb − 1)ta tb

2


Note that we only need to consider the slowly-oscillating term
B( z, t ) = −
ε 0 E02
ta tb ωa (ε ra − 1) − ωb (ε rb − 1)  sin ( (ka − kb ) z − (ωa − ωb )t )
2
 vg t ∂P 
⋅ Edz dt , caused by the electric interaction can
Thus, the transferred energy, We = ∫  ∫
0
0
∂t


τ
be explicitly expressed by
=
=
τ
(∫
)
ε 0 E02
τ
vg t
ta tb ωa (ε ra − 1) − ωb (ε rb − 1)  ∫  ∫ sin ( (ka − kb ) z − (ωa − ωb )t )dz dt
 0
0
0
0 

2
2
τ
ε E
1
 cos ( (ka − kb )vg t − (ωa − ωb )t ) − cos ( −(ωa − ωb )t ) dt
= 0 0 ta tb ωa (ε ra − 1) − ωb (ε rb − 1) 
∫

0
2
ka − kb 
We = ∫
vg t
B( z, t )dz dt = −
ε 0 E02
1
ta tb ωa (ε ra − 1) − ωb (ε rb − 1) 
2
ka − kb
 sin ( (ka − kb )vg − (ωa − ωb ) )τ  sin ( (ω − ω )τ ) 
 

−
a
b


(ka − kb )vg − (ωa − ωb )
ωa − ωb


ε 0 E02
The
1
ta tb ωa (ε ra − 1) − ωb (ε rb − 1) 
τ.
2
ka − kb
ωa → ωb → ω limit of the above result reads
We =
ε 0 E02
 d(ωε r ) dω 
t02 
−
τ
2
dk 
 dk
(53)
which can be rewritten as
 
1
 d  −ω µrε r
2
η
ε E
r
We = 0 0 t02  

2
dk


Here, relations k = −ω




2
 − dω  τ = ε 0 E0 t 2  c + ω µ ε 1 dη r − dω  τ
0 
r r
dk 
2
ηr2 dk dk 
η r


(
)
µrε r / c and d −ω µ rε r / dk = c have been used.
The energy density of the magnetic interaction is given by
14
(54)
µ0
 1

∂M
1
1
⋅ H = ε 0 E02  − 2 ωa ( µra − 1)ta2 sin 2(ka z − ωa t ) − 2 ωb ( µrb − 1)tb2 sin 2(kb z − ωbt ) 
∂t
2
ηb
 ηa


t t sin ( (ka + kb ) z − (ωa + ωb )t ) − sin ( (ka − kb ) z − (ωa − ωb )t ) 
+ ε 0 E02 ωa ( µra − 1) a b

2
η aηb



t t sin ( (kb + ka ) z − (ωb + ωa )t ) − sin ( (kb − ka ) z − (ωb − ωa )t ) 
+ ε 0 E02 ωb ( µrb − 1) a b

2
η aηb


Similarly, the wave-packet integrals of all the highly-oscillating terms vanish, and only the
slowly-oscillating term
C ( z, t ) = −
ε 0 E02 ta tb
ωa ( µ ra − 1) − ωb ( µrb − 1)  sin ( (ka − kb ) z − (ωa − ωb )t )
2 ηaηb 
will be integrated for obtaining the magnetic interaction energy. The result is as follows
Wm = ∫
τ
0
=−
(∫
vg t
0
)
C ( z, t )dz dt
vt
τ
ε 0 E02 ta tb
ωa ( µra − 1) − ωb ( µrb − 1)  ∫  ∫ sin ( (ka − kb ) z − (ωa − ωb )t )dz dt
 0
0 

2 ηaηb 
g
 sin ( (ka − kb )vg − (ωa − ωb ) )τ  sin ( (ω − ω )τ ) 
 

a
b
−


k
−
k
v
−
−
−
(
)
(
)
ω
ω
ω
ω
g
a
b
a
b
a
b


=
ε 0 E02 ta tb
1
ωa ( µra − 1) − ωb ( µ rb − 1) 
ka − kb
2 η aηb 
=
ε 0 E02 ta tb
1
ωa ( µra − 1) − ωb ( µ rb − 1) 
τ
ka − kb
2 η aηb 
The
ωa → ωb → ω limit of the above formula gives Wm =
(55)
ε 0 E02 t02  d(ωµ r ) dω 
−
τ . It
2 ηr2  dk
dk 
can be rewritten as
Wm =
(
)


ε 0 E02 t02  d −ω µrε rηr dω 
ε 0 E02 2  c
1 dη r 1 d ω 
−
τ
=
t0  − ω µ r ε r 2
−
τ (56)
2
2 ηr 
dk
dk 
2
ηr dk η r2 dk 
ηr


Hence, the total mechanical energy transferred from the electromagnetic wave to the induced
electromagnetic currents is
 1 1
1  vg 
Wmech = We + Wm = ε 0 E02t02  − 1 + 2   cτ
ηr 2  ηr  c 
(57)
Now we have obtained the incident electromagnetic energy Ε vac = ε 0 E0 cτ in vacuum, the
2
reflected electromagnetic energy Ε vac2 = ε 0 E0 r0 cτ , and the propagating electromagnetic energy
2 2
1
Ε em = ε 0 E02 (1 + 1 / ηr2 ) t02 vgτ in the metamaterial. By aid of relation r02 + t02 / ηr = 1 , one can
2
find that the law of energy conservation, Ε vac = Ε vac2 + Ε em + Wmech , is preserved in this process.
15
5. DISCUSSION AND CONCLUSION
The momentum of an electromagnetic wave in a medium has been studied for over a hundred
years, and yet contradictive conclusions have been obtained by different approaches. Concerning
the more complicated case of negative index metamaterial, which is not only polarized and
magnetized, but also intrinsically dispersive, some attempts have been made to study the
momentum of electromagnetic waves inside the metamaterial. In [20], the incompatibility between
the photon momentum flow and the energy transfer for electromagnetic waves in a metamaterial
has been pointed out. The inconsistency is supposed to stem from particle-wave duality [20].
Another formalism, based on classical electromagnetic theory, has been developed in [19]. The
definitions of momentum density, momentum stress tensor and Lorentz force, are different from
those in our present paper. Especially, the Lorentz force vanishes inside a lossless medium
according to their formalism, and the force arising from the sudden change of electromagnetic
stress tensor at the interface between vacuum and the metamaterial lacks a physical explanation
[19].
In contrast, the present formalism provides a clear illustration of the electromagnetic
momentum and the mechanical momentum, by defining the Lorentz force at the very beginning.
Momentum conversion from the electromagnetic part into the mechanical part has been clearly
demonstrated. The calculated results are consistent with the law of momentum conservation. Our
additional calculation on energy flow shows that our formalism also obeys the law of energy
conservation.
In summary, we have developed a theoretical approach for momentum in dispersive media
with huge mass density. A clear identification of momentum density and momentum flow, for both
the electromagnetic part and the mechanical part, has been obtained by implementing this
formalism for a lossless dispersive metamaterial. Contrary to pervious studies on this subject [19,
20], our results show that the momentum density and momentum flow inside a metamaterial
definitely have the same direction as the energy flow, rather than as the wave vector. We expect
that our formalism may pave the way for easing the long-lasting dispute, or solving the
incompatibility proposed recently in [20].
APPENDIX
Here we show the calculation of mechanical momentum and the proof of momentum
conservation for the case of mismatched impedance. The impedance matching case can be
regarded as a special case of impedance mismatching.
(1) Calculation of mechanical momentum for the case of mismatched impedance
For the case of normal incidence, the generalized force is simplified to
f mech = µ0 (∂ P / ∂t ) × H − µ0ε 0 (∂ M / ∂t ) × E
Now we define
(58)
f e = µ0 (∂ P / ∂t ) × H , f h = − µ0ε 0 (∂ M / ∂t ) × E , then the mechanical
momentum acquired by the induced electric current is given by
16
τ
Pe = ∫
0
=
=
=
=
ε0E
2
0
c
ε 0 E02
c
ε 0 E02
c
ε 0 E02
c
∫
vg t
0
f e ( z , t )dzdt
∂

a
b
 ∂t (ε r − 1)ta sin(ka z − ωa t ) − (ε r − 1)tb sin(kb z − ωbt )  
τ vg t 

dzdt
b
∫0 ∫0   ε ra

ε
× t

sin(ka z − ωa t ) − tb rb sin(kb z − ωbt ) 
  a µra

µr




 (−ωa )(ε ra − 1)ta cos(ka z − ωat ) − ( −ωb )(ε rb − 1)tb cos(kb z − ωbt )  


τ vg t 


a
b
dzdt
∫0 ∫0 × t ε r sin(k z − ω t ) − t ε r sin(k z − ω t )
 a

a
a
b
b
b
µra
µrb

 



εa
a
( −ωa )(ε r − 1)ta cos( ka z − ωa t )ta ra sin( ka z − ωa t ) 
µr




b
 +ω (ε a − 1)t cos(k z − ω t )t ε r sin( k z − ω t ) 
a
a
a
b
b
b
 a r

µrb
τ vg t 

dzdt
∫0 ∫0
ε ra


b
 +ωb (ε r − 1)tb cos( kb z − ωbt )ta µ a sin(ka z − ωat ) 
r




b
ε
 −ωb (ε rb − 1)tb cos(kb z − ωbt )tb rb sin( kb z − ωbt ) 
µr




ε ra 1
a
2
sin(2
k
z
−
2
t
)
ω
( −ωa )(ε r − 1)ta

a
a
µra 2




b
 +ω (ε a − 1)t t ε r 1 sin [ ( k − k ) z − (ω − ω )t ] − sin [ (k + k ) z + (ω + ω )t ] 
{
}
a b
b
a
b
a
b
a
b
a
 a r
µrb 2
τ vg t 

dzdt
∫0 ∫0 
a
ε
1


b
r
 +ωb (ε r − 1)tbta µ a 2 {sin [ ( ka − kb ) z − (ωa − ωb )t ] − sin [ ( ka + kb ) z + (ωa + ωb )t ]} 
r




b
ε
 −ωb (ε rb − 1)tb2 rb 1 sin(2kb z − 2ωbt )

µr 2


After integration over the spatial coordinate z, we obtain


εa 1
a
2
cos(2ωa t ) − cos(2ka vg t − 2ωa t ) 
(−ωa )(ε r − 1)ta ra

µ r 4k a




 cos [ (ωa + ωb )t ] − cos ( ka + kb )vg t − (ωa + ωb )t   






b
k a + kb
1
ε


a
r
+ωa (ε r − 1)ta tb



µrb 2  cos [ (ωb − ωa )t ] − cos (kb − ka )vgt − (ωb − ωa )t   



+

kb − ka
ε 0 E02 τ 


Pe =

dt
∫
0
c
 cos [ (ωa + ωb )t ] − cos  (ka + kb )vg t − (ωa + ωb )t   




a
k a + kb

+ω (ε b − 1)t t ε r 1 


b a
a
 b r
µr 2  cos [ (ωa − ωb )t ] − cos (ka − kb )vg t − (ωa − ωb )t   

 

+


k
−
k
a
b




b


1
ε
b
2
r
cos(2ωb t ) − cos(2kb vg t − 2ωb t ) 
−ωb (ε r − 1)tb

b

µ r 4 kb


17
It should be noted that, in the limit of
ωa → ωb → ω , all the above terms will vanish after
integrating over the time interval [0,τ ] , except the term cos[( kb − k a )vg t − (ωb − ωa )t ] ,
whose value is exactly one, given that vg =
ωb − ωa
kb − ka
. Thus, the mechanical momentum caused
by the electric current interaction is
Pe =
ε 0 E02
c

ε rb
ε ra 
a
b
ωa (ε r − 1) b − ωb (ε r − 1) a  ta tb
2(ka − kb ) 
µr
µ r 
τ

ε0E
µra
µrb  ε raε rb
τ
a
b
=
t a tb
ωa (ε r − 1) a − ωb (ε r − 1) b 
c 2(ka − kb ) 
εr
ε r  µ ra µ rb
(59)
2
0
ωa → ωb → ω , we get
In the limit
d 
µr 
ω
(
ε
−
1)


r
εr 
ε 0 E02τ 2 ε r dω 
Pe =
t0
2
µr
d(−ω ε r µ r )
dω
(60)
In a similar way, we obtain
Ph = ∫
τ
0
∫
vg t
0
f h ( z , t )dzdt =
ε 0 E02τ
2
t02

d  εr
( µ r − 1)ω 

dω  µ r

d(− ε r µ r ω )
dω
(61)
Finally, we obtain the momentum expression (28).
(2) The proof of Pvac = Pvac2 + Pem + Pmech for the case of mismatched impedance
In Appendix (1), the expression for the momentum inside the metamaterial, Eq. (28), is
decomposed into two parts,
Pmech = Pe + Pm
where Pe =
and k = −
ε 0 E02 2 d ω ( µ r − 1) / η r 
ε 0 E02 t02 d ω (ε r − 1)ηr 
µ
,
P
=
t0
τ with ηr = r
τ
m
2
2c
dk
2c η r
dk
εr
ω
c
(62)
ε r µr . And the two parts can be further rewritten as,
18
Pe =
(
)


ε 0 E02 t02  d −ω µrε r
d(ωηr ) 
−
τ
2c η r2 
dk
dk 

(

)
=


ε 0 E02 t02  d −ω µrε r
dω
dη r 
−
η
−
ω
τ
r
2c ηr2 
dk
dk
dk 
=
ε0E t 
dη 
c − vgη r − ω r  τ

2c η 
dk 

Pm =
2
0

2
0
2
r
(
)
(
)
 d −ω µrε r
d (ω / ηr ) 
−
t02 
τ

2c
dk
dk 


ε 0 E02
 d −ω µrε r

dω 1 ω dη r 
t02 
−
+ 2
τ

2c
dk
dk ηr ηr dk 


2
ε E  vg ω dη r 
= 0 0 t02 c − + 2
τ
2c
 η r η r dk 
=
(63)
ε 0 E02
(64)
Consequently, we have the total mechanical momentum
Pmech =
ε 0 E02
2v 

1
t02  (1 + 2 )c − g  τ
2c
ηr
ηr 

(65)
As we have obtained the electromagnetic momentum of one wave packet in vacuum and the
metamaterial, i.e., Eqs. (24), (25) and (26), which are rewritten here as,
Pvac = ε 0 E02τ
(66)
Pvac2 = −ε 0 E02 r02τ
(67)
Pem = ε 0 E02τ t02
vg
ηr c
(68)
By the aid of (1 + r0 ) / t0 = (1 + 1/ η r ) / 2 , we can show that
2
2
2
Pvac = Pvac2 + Pem + Pmech
(69)
This means that the momentum is conserved in this process.
ACKNOWLEDGMENT
This work is supported by the National Natural Science Foundation of China (No. 60990320,
and No. 60990322), and Natural Science Foundation of Zhejiang Province under Grant
No.Y6100280. We are also grateful for valuable discussion with Dr. Yi Jin and Dr. Linfang Shen.
One of the authors (JQ Shen) is also grateful to the support of the Fundamental Research Funds
for the Central Universities of China.
19
REFERENCES
1.
Jackson, J.D., Classical electrodynamics. 1999, New York: Wiley.
2.
Minkowski. Nachr. Ges. Wiss. Gottn Math.-Phys. KI., Vol. 53, No. 1908.
3.
Abraham. M. Rend. Circ. Matem. Palermo, Vol. 28, No. 1, 1909.
4.
Jones, R.V. and J.C.S. Richards. "The Pressure of Radiation in a Refracting Medium,"
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences,
Vol. 221, No. 1147, 480-498, 1954.
5.
Gordon, J.P. "Radiation Forces and Momenta in Dielectric Media," Physical Review A, Vol. 8,
No. 1, 14-21, 1973.
6.
Peierls, R. "The Momentum of Light in a Refracting Medium," Proceedings of the Royal
Society of London. Series A, Mathematical and Physical Sciences, Vol. 347, No. 1651,
475-491, 1976.
7.
Nelson, D.F. "Momentum, pseudomomentum, and wave momentum: Toward resolving the
Minkowski-Abraham controversy," Physical Review A, Vol. 44, No. 6, 3985-3966, 1991.
8.
Loudon, R., L. Allen, and D.F. Nelson. "Propagation of electromagnetic energy and
momentum through an absorbing dielectric," Physical Review E, Vol. 55, No. 1, 1071-1085,
1997.
9.
Mansuripur, M. "Radiation pressure and the linear momentum of the electromagnetic field,"
Opt. Express, Vol. 12, No. 22, 5375-5401, 2004.
10.
Kemp, B., T. Grzegorczyk, and J. Kong. "Ab initio study of the radiation pressure on dielectric
and magnetic media," Opt. Express, Vol. 13, No. 23, 9280-9291, 2005.
11.
Mansuripur, M. "Radiation pressure and the linear momentum of light in dispersive dielectric
media," Opt. Express, Vol. 13, No. 6, 2245-2250, 2005.
12.
Scalora, M., et al. "Radiation pressure of light pulses and conservation of linear momentum in
dispersive media," Physical Review E, Vol. 73, No. 5, 056604, 2006.
13.
Mansuripur, M. "Radiation pressure and the linear momentum of the electromagnetic field in
magnetic media," Opt. Express, Vol. 15, No. 21, 13502-13518, 2007.
14.
Yaghjian, A.D. "Internal Energy, Q-Energy, Poynting's Theorem, and the Stress Dyadic in
Dispersive Material," Antennas and Propagation, IEEE Transactions on, Vol. 55, No. 6,
1495-1505, 2007.
15.
Pfeifer, R.N.C., et al. "Colloquium: Momentum of an electromagnetic wave in dielectric
media," Reviews of Modern Physics, Vol. 79, No. 4, 1197-1216, 2007.
16.
Veselago, V.G. "The electrodynamics of substances with simultaneously negative values of
permittivity and permeability," Sov. Phys. Usp., Vol. 10, 509-514, 1968.
17.
Shelby, R.A., D.R. Smith, and S. Schultz. "Experimental Verification of a Negative Index of
Refraction," Science, Vol. 292, No. 5514, 77-79, 2001.
18.
Pendry, J.B. "Negative Refraction Makes a Perfect Lens," Physical Review Letters, Vol. 85,
No. 18, 3966-3969, 2000.
19.
Kemp, B.A., J.A. Kong, and T.M. Grzegorczyk. "Reversal of wave momentum in isotropic
left-handed media," Physical Review A, Vol. 75, No. 5, 053810, 2007.
20.
Veselago, V.G. "Energy, linear momentum, and mass transfer by an electromagnetic wave in a
negative-refraction medium," Physics-Uspekhi, Vol. 52, No. 6, 649-654, 2009.
20