Generalizing speed-of-light limitations to arbitrary
passive linear media
Aaron Welters
Department of Mathematics1
Massachusetts Institute of Technology
Joint work with Steven G. Johnson1 and Yehuda Avniel1
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
1 / 20
Main Reference
A. Welters, S. G. Johnson, and Y. Avniel, Speed-of-light limitations in
passive linear media, In progress.
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
2 / 20
Motivation
(Relativistic Causality) Relativity implies information (signal) cannot
travel faster than c, the speed of light in vacuum.
Controversies over superliminal (i.e., > c) propagation of light in
dissipative and dispersive media especially related to the group
velocity and energy velocity.
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
3 / 20
Motivation
(Relativistic Causality) Relativity implies information (signal) cannot
travel faster than c, the speed of light in vacuum.
Controversies over superliminal (i.e., > c) propagation of light in
dissipative and dispersive media especially related to the group
velocity and energy velocity.
Obscurity of conditions which guarantee the speed-of-light restriction
on electromagnetic energy velocity ve .
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
3 / 20
Motivation
(Relativistic Causality) Relativity implies information (signal) cannot
travel faster than c, the speed of light in vacuum.
Controversies over superliminal (i.e., > c) propagation of light in
dissipative and dispersive media especially related to the group
velocity and energy velocity.
Obscurity of conditions which guarantee the speed-of-light restriction
on electromagnetic energy velocity ve .
The broad range of material properties including anisotropy,
bianisotropy, nonlocality, dispersion, periodicity and even delta
functions or similar generalized functions.
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
3 / 20
Motivation
(Relativistic Causality) Relativity implies information (signal) cannot
travel faster than c, the speed of light in vacuum.
Controversies over superliminal (i.e., > c) propagation of light in
dissipative and dispersive media especially related to the group
velocity and energy velocity.
Obscurity of conditions which guarantee the speed-of-light restriction
on electromagnetic energy velocity ve .
The broad range of material properties including anisotropy,
bianisotropy, nonlocality, dispersion, periodicity and even delta
functions or similar generalized functions.
Problem: What is the broadest range of linear materials and weakest
conditions which guarantee the speed-of-light restriction ||ve || ≤ c?
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
3 / 20
Why is the EM energy velocity ≤ c?
Why is the EM energy velocity ≤ c?
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
4 / 20
Why is the EM energy velocity ≤ c?
Why is the EM energy velocity ≤ c?
very long history
lots of results
still open questions
boil down to minimal mathematical conditions
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
4 / 20
Why is the EM energy velocity ≤ c?
Why is the EM energy velocity ≤ c?
very long history
lots of results
still open questions
boil down to minimal mathematical conditions
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
4 / 20
What is EM “information”/energy velocity?
Not generally phase velocity vph =
(speed of ripples)
Hamilton, 1839 & Rayleigh, 1881
group velocity (wave packet)
vg = dω
dk = energy velocity =
if loss is negligible
ω
k
energy flux
energy density
= ve
non-negligible loss/gain
”superluminal group velocity”6= information
velocity
(Sommerfeld, 1907...Milonni, 2005)
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
5 / 20
What is EM “information”/energy velocity?
Not generally phase velocity vph =
(speed of ripples)
Hamilton, 1839 & Rayleigh, 1881
group velocity (wave packet)
vg = dω
dk = energy velocity =
if loss is negligible
ω
k
energy flux
energy density
= ve
non-negligible loss/gain
”superluminal group velocity”6= information
velocity
(Sommerfeld, 1907...Milonni, 2005)
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
5 / 20
When is ||ve || ≤ c?
Brillouin (1914): true for homogeneous isotropic linear medium with
single Lorentzian resonance in (ω)
Landau & Lifshitz (1960): true for homogeneous isotropic linear
medium, assuming causality, positivity of dispersive energy, some
regularity of , µ
Yaghjian (2007): homogeneous isotropic passive linear medium,
assuming some regularity of , µ
Joannopoulos et al. (2008), (others?): periodic, isotropic,
dispersionless, linear medium, assuming real , µ ≥ 1
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
6 / 20
When is ||ve || ≤ c?
Brillouin (1914): true for homogeneous isotropic linear medium with
single Lorentzian resonance in (ω)
Landau & Lifshitz (1960): true for homogeneous isotropic linear
medium, assuming causality, positivity of dispersive energy, some
regularity of , µ
Yaghjian (2007): homogeneous isotropic passive linear medium,
assuming some regularity of , µ
Joannopoulos et al. (2008), (others?): periodic, isotropic,
dispersionless, linear medium, assuming real , µ ≥ 1
Today: passive linear medium including homogeneous or periodic
and anisotropic or bianisotropic.
passivity =⇒ causality, positivity, other analytic properties
with transparency window =⇒ vg = ve and ||ve || ≤ c
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
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Passive linear time-invariant media
Macroscopic Maxwell’s equations without sources:
1 ∂B
∇·B=0
∇×E+
=0
c ∂t
1 ∂D
=0
∇·D=0
∇×H−
c ∂t
Electric and magnetic polarizations P, M with linear constitutive
relations D = E + 4πP and B = H + 4πM via susceptibility χ:
Z ∞
P
E
E(t − t 0 )
0
=χ∗
=
χ(t )
dt 0
M
H
H(t − t 0 )
−∞
A homogeneous medium is passive if
Z t
∂M
∂P
E† 0 + H† 0 dt 0 ≥ 0
Re
∂t
∂t
−∞
(1)
(2)
for every t ∈ R and every [E, H]T ∈ D (H) – test functions from R
into Hilbert space H = C6 .
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
7 / 20
Passive linear time-invariant media
Macroscopic Maxwell’s equations without sources:
1 ∂B
∇·B=0
∇×E+
=0
c ∂t
1 ∂D
=0
∇·D=0
∇×H−
c ∂t
Electric and magnetic polarizations P, M with linear constitutive
relations D = E + 4πP and B = H + 4πM via susceptibility χ:
Z ∞
E
E(t − t 0 )
P
0
=χ∗
=
χ(t 0 )
0 ) dt
M
H
H(t
−
t
−∞
A periodic medium is passive in the unit cell V if
Z t Z
∂P
∂M
E† 0 + H† 0 drdt 0 ≥ 0
Re
∂t
∂t
−∞ V
(1)
(2)
for every t ∈ R and every [E, H]T ∈ D (H) – test functions from R
6
into Hilbert space H = L2 (V ) .
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
7 / 20
Consequences of passivity
Theorem 1 (Welters et al., 2013)
For a passive linear medium with susceptibility χ the following statements
are true:
1
2
3
(causality) Polarizations only depend on fields in the past, i.e.,
χ(t) = 0 for t < 0.
b(ω), is analytic for
(analyticity) Fourier transform of dχ
dt , i.e., −iω χ
Im ω > 0 in the norm topology on L(H) – bounded linear operators.
(positivity) The function h(ω) = ω χ
b(ω) satisfies Im h(ω) ≥ 0 for
Im ω > 0.
Proof (Sketch).
Linearity and passivity means dχ
dt ∗ is a “passive” convolution operator on
D (H). Deep results from linear response theory (Zemanian, 1972) on such
operators implies these results.
A. Welters (MIT)
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October 4, 2013
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Consequences of passivity
Theorem 1 (Welters et al., 2013)
For a passive linear medium with susceptibility χ the following statements
are true:
1
2
3
(causality) Polarizations only depend on fields in the past, i.e.,
χ(t) = 0 for t < 0.
(analyticity) Fourier transform of dχ
b(ω), is analytic for
dt , i.e., −iω χ
Im ω > 0 in the norm topology on L(H) – bounded linear operators.
(positivity) The function h(ω) = ω χ
b(ω) satisfies Im h(ω) ≥ 0 for
Im ω > 0.
Definition (Herglotz function)
Let H be a Hilbert space. A bounded-operator-valued Herglotz function is
analytic function from C+ (open complex upper half-plane) into L(H)
(bounded linear operators) whose values have positive semidefinite
imaginary part [e.g., h(ω) or (A − ωI)−1 with A self-adjoint].
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
8 / 20
Consequences of passivity
Example: Homogeneous isotropic media with integrable χ(t).
Time-harmonic fields E(t) = Ee −iωt , H(t) = He −iωt .
Linear constitutive relations are D(t) = De −iωt , B(t) = Be −iωt ,
where
D = (ω)E, B = µ(ω)H.
Elec. & magnetic polarizations are P(t) = Pe −iωt , M(t) = Me −iωt ,
where
(ω) − 1
µ(ω) − 1
P=
E, B =
H.
4π
4π
Scalar-valued permittivity (ω) and permeability µ(ω) related to
susceptibility via
χ
b(ω) = (4π)−1 diag{((ω) − 1)I, (µ(ω) − 1)I}.
The function h(ω) = ω χ
b(ω) and for ω ∈ R,
Im h(ω) = (4π)−1 ω diag{Im (ω)I, Im µ(ω)I}.
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Speed Limitations in Passive Media
October 4, 2013
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EM losses and transparency windows
Transparency window - frequency interval
(ω1 , ω2 ) ⊆ R where losses are negligible
(Landau & Lifshitz, 1960).
Electromagnetic losses (transfer of EM
energy into matter by absorption) ↔
boundary-values as Im ω ↓ 0 of Im h(ω).
Typically [e.g., integrable χ(t)], (ω1 , ω2 ) is
a transparency window if Im h(ω) = 0 for
all ω ∈ (ω1 , ω2 ).
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
10 / 20
EM losses and transparency windows
Example: Homogeneous isotropic media with integrable χ(t).
Scalar-valued permittivity (ω) and permeability µ(ω) related to
susceptibility via
χ
b(ω) = (4π)−1 diag{((ω) − 1)I, (µ(ω) − 1)I}.
Time-harmonic fields E(t) = Ee −iωt , H(t) = He −iωt : Time-average
of the power dissipated per unit time is
Z
0
1 T
d Re P(t 0 )
0 † d Re M(t )
+
Re
H(t
)
dt 0
Re E(t 0 )†
T 0
dt 0
dt 0
†
1 E
E
=
Im h(ω)
H
2 H
ω
=
[Im (ω)||E||2 + Im µ(ω)||H||2 ],
8π
where T = 2π/ω.
Well-known expression for EM losses in a dispersive isotropic medium
for monochromatic EM fields (Landau & Lifshitz, 1960).
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
11 / 20
EM losses and transparency windows
Example: Homogeneous isotropic media with integrable χ(t).
Scalar-valued permittivity (ω) and permeability µ(ω) related to
susceptibility via
χ
b(ω) = (4π)−1 diag{((ω) − 1)I, (µ(ω) − 1)I}.
Square-integrable fields E(t), H(t) with integrable dχ
dt (t): Total
energy dissipated by such fields is
Z ∞
0
dP(t 0 )
0 † dM(t )
Re
E(t 0 )†
+
H(t
)
dt 0
0
0
dt
dt
−∞
#†
"
#
Z ∞" b
b
dω
E(ω)
E(ω)
=
Im h(ω) b
b
H(ω)
H(ω) 2π
−∞
Z ∞ h
i
1
2
2 dω
b
b
ω Im (ω)||E(ω)||
+ Im µ(ω)||H(ω)||
=
.
4π −∞
2π
Well-known expression for EM losses in a dispersive isotropic medium
for non-monochromatic EM fields (Landau & Lifshitz, 1984).
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
11 / 20
EM losses and transparency windows
Theorem 2 (Welters et al., 2013)
For a passive linear medium with susceptibility χ, EM losses in any
bounded frequency interval (ω1 , ω2 ) ⊆ R can be quantified by a
nonnegative bounded-operator-valued measure Ω(·) on the bounded Borel
subsets of R via the limits (in the strong operator topology):
Z
ω2 −δ
Ω((ω1 , ω2 )) = lim lim
δ↓0 η↓0
ω1 +δ
1
Im h(ω + iη) dω.
π
(3)
Proof (Sketch).
The total energy dissipated by a field F (t) = [E(t), H(t)] ∈ D (H) is the
integral (2) with t = ∞. Welters et al. (2013) show using linear response
theory (Zemanian, 1972) that this time integral is given in terms of a
frequency integral involving only the measure Ω(·) and Fb (ω). The Stieltjes
inversion formula for Ω(·) from the theory of Herglotz functions (Gesztesy
et al., 2000, 2001) implies existence of the limits and equality (3).
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
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EM losses and transparency windows
Theorem 2 (Welters et al., 2013)
For a passive linear medium with susceptibility χ, EM losses in any
bounded frequency interval (ω1 , ω2 ) ⊆ R can be quantified by a
nonnegative bounded-operator-valued measure Ω(·) on the bounded Borel
subsets of R via the limits (in the strong operator topology):
Z
ω2 −δ
Ω((ω1 , ω2 )) = lim lim
δ↓0 η↓0
ω1 +δ
1
Im h(ω + iη) dω.
π
(3)
Definition (Transparency window)
For a passive linear medium with susceptibility χ, a transparency window is
a bounded frequency interval (ω1 , ω2 ) ⊆ R in which
Ω((ω1 , ω2 )) = 0.
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(4)
October 4, 2013
12 / 20
EM losses and transparency windows
Example: Homogeneous isotropic media with integrable χ(t).
Scalar-valued permittivity (ω) and permeability µ(ω) related to
susceptibility via
χ
b(ω) = (4π)−1 diag{((ω) − 1)I, (µ(ω) − 1)I}.
b(ω).
Measure is dΩω = π1 Im h(ω)dω, where h(ω) = ω χ
Transparency window is Ω((ω1 , ω2 )) = 0 ⇔ Im h(ω) = 0 for all
ω ∈ (ω1 , ω2 ).
Follows from positivity Im h(ω) ≥ 0 and
Z ω2 −δ
1
Ω((ω1 , ω2 )) = lim lim
Im h(ω + iη) dω
δ↓0 η↓0 ω1 +δ π
Z ω2 −δ
1
lim Im h(ω + iη) dω
= lim
δ↓0 ω1 +δ π η↓0
Z ω2
1
=
Im h(ω) dω.
ω1 π
A. Welters (MIT)
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October 4, 2013
13 / 20
Consequences of transparency windows
Theorem 3 (Welters et al., 2013)
For a passive linear medium with susceptibility χ and a transparency
window (ω1 , ω2 ) ⊆ R, i.e., Ω((ω1 , ω2 )) = 0, the following statements are
true:
1
(analytic continuation) The function h(ω) = ω χ
b(ω) can be analytic
continued through (ω1 , ω2 ).
2
(self-adjointness & monotonicity) For all ω ∈ (ω1 , ω2 ), Im h(ω) = 0
and h0 (ω) ≥ 0.
Proof (Sketch).
Theory of Herglotz functions (Gesztesy et al., 2000, 2001) implies
Z
1
λ
h (ω) = h0 + h1 ω +
dΩλ
−
, Im ω > 0.
λ − ω 1 + λ2
R
where Im h0 = 0, h1 ≥ 0 and (1 + λ2 )−1 is integrable w.r.t. Ω(·).
A. Welters (MIT)
Speed Limitations in Passive Media
October 4, 2013
14 / 20
Consequences of transparency windows
Theorem 3 (Welters et al., 2013)
For a passive linear medium with susceptibility χ and a transparency
window (ω1 , ω2 ) ⊆ R, i.e., Ω((ω1 , ω2 )) = 0, the following statements are
true:
1
(analytic continuation) The function h(ω) = ω χ
b(ω) can be analytic
continued through (ω1 , ω2 ).
2
(self-adjointness & monotonicity) For all ω ∈ (ω1 , ω2 ), Im h(ω) = 0
and h0 (ω) ≥ 0.
Proof (Sketch).
The hypothesis Ω((ω1 , ω2 )) = 0 and the integral representation implies
these results. Differentiating under the integral sign implies monotonicity:
Z
1
0 ≤ h0 (ω) = h1 +
dΩλ
, for all ω ∈ (ω1 , ω2 ).
(λ − ω)2
R\(ω1 ,ω2 )
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Speed Limitations in Passive Media
October 4, 2013
14 / 20
Homogeneous media: planewaves & energy velocity
Time-harmonic EM planewaves:
E(r, t) = Ee i(k·r−ωt) , H(r, t) = He i(k·r−ωt) ,
with nonzero [E, H]T ∈ C6 and real (k, ω).
Dispersion relation & group velocity: ω = ω(k), vg = ∇k ω(k).
Energy velocity: ve =
energy flux
energy density
=
Re S
U ,
S=
c
8π E
× H∗
Energy density (in transparency window):
1
†
†
Dispersionless, iso- or anisotropic: U = 16π E E + H µH .
1
† dωµ(ω)
Dispersive, iso- or anisotropic: U = 16π
E† dω(ω)
dω E + H
dω H
General form:
†
1
d
E
E
U=
ω [I + 4π χ
b(ω)]
H
16π H
dω
†
1
1
E
E
2
2
0
=
||E|| + ||H|| +
.
(5)
h (ω)
H
16π
4π H
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October 4, 2013
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Homogeneous media: speed-of-light limitation
Theorem 4 (Welters et al., 2013)
For any homogeneous passive linear medium and for any time-harmonic
EM planewave with frequency in a transparency window, the following
statements are true:
1
(energy positivity) The energy density is positive, i.e., U > 0.
2
(velocity equivalence) The group and energy velocities are equal, i.e.,
vg = ve .
3
(speed-of-light limitation) The energy velocity has the limitation
||ve || ≤ c.
Proof (Sketch).
Energy velocity: ve =
Re S
U ,S
=
c
8π E
× H∗ . Lagrange’s identity implies
||E × H∗ ||2 = ||E||2 ||H||2 − |ET H|2 ⇒ || Re S|| ≤
A. Welters (MIT)
Speed Limitations in Passive Media
c
||E||2 + ||H||2 .
16π
October 4, 2013
16 / 20
Homogeneous media: speed-of-light limitation
Theorem 4 (Welters et al., 2013)
For any homogeneous passive linear medium and for any time-harmonic
EM planewave with frequency in a transparency window, the following
statements are true:
1
(energy positivity) The energy density is positive, i.e., U > 0.
2
(velocity equivalence) The group and energy velocities are equal, i.e.,
vg = ve .
3
(speed-of-light limitation) The energy velocity has the limitation
||ve || ≤ c.
Proof (Sketch).
Transparency window implies monotonicity h0 (ω) ≥ 0 implies by (5),
cU ≥
A. Welters (MIT)
c
||E||2 + ||H||2 ≥ || Re S||.
16π
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October 4, 2013
16 / 20
Periodic media too!
Periodic media too!
periodicity ⇒ well-defined
Bloch waves, group velocity, &
energy velocity (transparency
window)
Our results works equally well
(previous work: only
dispersionless)
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October 4, 2013
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Periodic media: Bloch waves & energy velocity
Time-harmonic EM Bloch waves:
E(r, t) = E(r)e i(k·r−ωt) , H(r, t) = H(r)e i(k·r−ωt) ,
where F (r) = [E(r), H(r)]T is periodic function on the lattice with
6
unit cell V ⊆ R3 , F ∈ L2 (V ) , and real (k, ω).
Dispersion relation & group velocity: ω = ω(k), vg = ∇k ω(k).
R
Energy velocity: ve =
V
R
Re S(r)dr
,
U(r)dr
S(r) =
V
c
8π E(r)
× H(r)∗
Energy density (in transparency window):
local media,iso- or anisotropic:
1
† ∂ωµ(r,ω)
E(r)† ∂ω(r,ω)
E(r)
+
H(r)
H(r)
U(r) = 16π
∂ω
∂ω
local media, general form:
†
E(r)
E(r)
1
∂
U(r) = 16π
b(r, ω)]
∂ω ω [I + 4π χ
H(r)
H(r)
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Periodic media: Bloch waves & energy velocity
Time-harmonic EM Bloch waves:
E(r, t) = E(r)e i(k·r−ωt) , H(r, t) = H(r)e i(k·r−ωt) ,
where F (r) = [E(r), H(r)]T is periodic function on the lattice with
6
unit cell V ⊆ R3 , F ∈ L2 (V ) , and real (k, ω).
Dispersion relation & group velocity: ω = ω(k), vg = ∇k ω(k).
R
Energy velocity: ve =
V
R
Re S(r)dr
,
V U(r)dr
S(r) =
c
8π E(r)
× H(r)∗
Energy density (in transparency window):
6
General form: Unitary – (T (k)ψ)(r) = e ik·r ψ(r), ψ ∈ L2 (V ) ;
†
1
d
E
E
U=
T (−k)
ω [I + 4π χ
b(ω)] T (k)
H
16π H
dω
†
1
1
E
E
0
2
2
T (−k)h (ω)T (k)
.
=
||E|| + ||H|| +
H
16π
4π H
(6)
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Periodic media: speed-of-light limitation
Theorem 5 (Welters et al., 2013)
For any periodic passive (in a unit cell V ⊆ R3 ) linear medium and for any
time-harmonic EM Bloch wave with frequency in a transparency window,
the following statements are true:
1
(energy
R positivity) The spatially averaged energy density is positive,
i.e., V U(r)dr > 0.
2
(velocity equivalence) The group and energy velocities are equal
(under certain reservations), i.e., vg = ve .
3
(speed-of-light limitation) The energy velocity has the limitation
||ve || ≤ c.
Proof (Sketch).
An elementary inequality and Lagrange’s identity implies
Z
Z
Z
c
Re S(r)dr ≤
kRe
S(r)k
dr
≤
||E(r)||2 + ||H(r)||2 dr.
16π
V
V
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Periodic media: speed-of-light limitation
Theorem 5 (Welters et al., 2013)
For any periodic passive (in a unit cell V ⊆ R3 ) linear medium and for any
time-harmonic EM Bloch wave with frequency in a transparency window,
the following statements are true:
1
(energy
R positivity) The spatially averaged energy density is positive,
i.e., V U(r)dr > 0.
2
(velocity equivalence) The group and energy velocities are equal
(under certain reservations), i.e., vg = ve .
3
(speed-of-light limitation) The energy velocity has the limitation
||ve || ≤ c.
Proof (Sketch).
Transparency window implies monotonicity h0 (ω) ≥ 0 implies by (6),
Z
Z
Z
c
2
2
c
U(r)dr ≥
||E(r)|| + ||H(r)|| dr ≥ Re S(r)dr
.
16π V
V
V
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October 4, 2013
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What’s left?
Now:
arbitrary passive linear ⇒ causality, analyticity, positivity, ||ve || ≤ c
Open:
generalized assumptions for non-negligible loss
nonlocal media across unit-cell boundaries (e.g., P at r from E at r0 )
nonlinear, time-varying
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October 4, 2013
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Realizability theory for continuous linear systems,
Academic Press, New York, 1972.
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Electrodynamics of continuous media,
1st & 2nd Ed., Pergamon Press, Oxford, 1960 & 1984.
P. W. Milonni,
Fast light, slow light, and left-handed light,
Series in Optics and Optoelectronics, IOP, 2005.
J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade,
Photonic crystals: molding the flow of light,
2nd Ed., Princeton University Press, Princeton, 2008.
Sir W. R. Hamilton,
Researches respecting vibrations, connected with the theory of light,
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