Electromagnetic field energy in dispersive materials

doi:10.1364/JOSAB.27.001215

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Electromagnetic field energy in dispersive materials

Kevin J. Webb and Shivanand »View Author Affiliations

JOSA B, Vol. 27, Issue 6, pp. 1215-1220 (2010)
http://dx.doi.org/10.1364/JOSAB.27.001215

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Abstract

A general expression for the electromagnetic energy density in a lossy dispersive medium, applicable for a field having a narrow temporal frequency bandwidth, is derived and compared with exact results for an example dielectric constant. Consequently, the possibility of negative time-averaged stored field energy is shown to have physical meaning. This observation is of interest in the study of dispersive metamaterials, such as those which can exhibit a negative refractive index.

1. INTRODUCTION

A negative refractive index [1

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ, and μ ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

] can be achieved when both the dielectric constant ϵ and the permeability μ are negative, which requires an operating frequency near a resonance in both constitutive parameters. In this frequency regime, the material properties can be highly dispersive for the candidate metamaterials, and the loss can be substantial [2

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

Poynting’s theorem, describing conservation of energy, can be written in a simple form when the material properties do not change with frequency. Brillouin suggested that in a causal system, one with frequency-dependent material properties, a different form of Poynting’s theorem was appropriate [3

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).

]. With this view, Veselago has presented requirements for positive energy density [1

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ, and μ ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

], based on the result given by Landau and Lifshitz [4

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

] and Brillouin [3

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).

] for transparent media. Loudon [5

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970). [CrossRef]

], Polevoi [6

V. G. Polevoi, “Maximum energy extractable from an electromagnetic field,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 33, 818–825 (1990).

], Ruppin [7

R. Ruppin, “Elecromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002). [CrossRef]

], Tretyakov [8

S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005). [CrossRef]

], Boardman and Marinov [9

A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006). [CrossRef]

], Cui and Kong [10

T. J. Cui and J. A. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B 70, 205106 (2004). [CrossRef]

], and Ziolkowski [11

R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic medium,” Phys. Rev. E 63, 046604 (2001). [CrossRef]

] have all studied energy density issues in dispersive media. This substantial body of work has concluded that all energies (total, stored, and dissipated) remain positive under all conditions, although Ziolkowski [11

R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic medium,” Phys. Rev. E 63, 046604 (2001). [CrossRef]

] has questioned whether this need be the case.

Here, we show that the stored energy near a material atomic resonance can be negative, through an exact decomposition into stored and dissipated energy. We derive general analytic energy expressions for fields having narrow temporal frequency bandwidth and compare these with exact numerical results for an example dispersive material response. We assume that there is no material spatial dispersion, and that the simple but physically meaningful set of physical constitutive parameters have frequency dispersion. These material parameters, the dielectric constant, for instance, can be uniquely determined by an appropriate set of measurements in a standard manner.

2. POYNTING’S THEOREM AND EXACT ENERGY DECOMPOSITION

With electric field E and magnetic field H, the vector identity

\nabla \cdot (E \times H) = H \cdot \nabla \times E - E \cdot \nabla \times H

, in conjunction with Maxwell’s curl equations, upon volume integration and application of the divergence theorem, leads to

\oint E \times H \cdot d s = - \int [E \cdot \frac{\partial D}{\partial t} + H \cdot \frac{\partial B}{\partial t}] d v,

(1)

where D and B are the electric and magnetic flux densities, respectively. Equation (1) is known as Poynting’s theorem, a statement of conservation of energy. A unique solution of Maxwell’s equations requires a model for the material constitutive parameters, which in turn can allow Poynting’s theorem to be expressed in terms of only E and H. Here it suffices to choose the linear and isotropic relationships

D = ϵ_{0} ϵ E

and

B = μ_{0} μ H

, in the frequency domain, where

ϵ_{0}

and

μ_{0}

are the free-space permittivity and permeability, respectively, ϵ is the dielectric constant, and μ is the relative permeability.

In the electric field case, with a Fourier superposition and dispensing with the spatial dependence,

\frac{\partial u_{E}}{\partial t} = E \cdot \frac{\partial D}{\partial t}

(2)

= ϵ_{0} E \frac{\partial}{\partial t} [\frac{1}{2 π} \int_{- \infty}^{\infty} ϵ (ω) E (ω) e^{- i ω t} d ω]

(3)

= \frac{ϵ_{0} E}{2 π} \int_{- \infty}^{\infty} - i ω [ϵ^{'} (ω) + i ϵ^{″} (ω)] [E^{'} (ω) + i E^{″} (ω)] [\cos (ω t) - i \sin (ω t)] d ω,

(4)

where

E (ω) = E^{'} (ω) + i E^{″} (ω)

and

ϵ (ω) = ϵ^{'} (ω) + i ϵ^{″} (ω)

. Imposing the necessary

E (ω)

and

ϵ (ω)

symmetry for

E (t)

and

ϵ (t)

to be real in Eq. (4) leads to

\frac{\partial u_{E}}{\partial t} = \frac{ϵ_{0} E}{2 π} \int_{- \infty}^{\infty} {ω ϵ^{'} (ω) [- E^{'} (ω) \sin (ω t) + E^{″} (ω) \cos (ω t)] + ω ϵ^{″} (ω) [E^{'} (ω) \cos (ω t) + E^{″} (ω) \sin (ω t)]} d ω = {\frac{\partial u_{E}}{\partial t} |}_{ϵ^{″} = 0} + {\frac{\partial u_{E}}{\partial t} |}_{ϵ^{'} = 0},

(5)

making it clear that exact separation into stored energy (which is returned to the field,

w_{E}

) and lost energy (converted to a nonelectromagnetic form, q) terms is possible. Note that we use time/frequency arguments only where appropriate for clarity. Evidently, this separation of energies has not been previously identified. The result in Eq. (5) is physical in the sense that the stored and lost energies at each frequency are dictated by the real and imaginary parts of the dielectric constant, respectively, and through spectral superposition the temporal energies are thus exact and physical. The real and imaginary parts of the dielectric constant are related by the Kramers–Kronig relations [12

J. D. Jackson, Classical Electrodynamics , 3rd ed. (Wiley, 1999).

], a consequence of a causal response. Causality therefore enforces a relationship between the two terms in Eq. (5). To further establish the significance of the separation in Eq. (5), we note that each of the two terms is causal.

3. ENERGY DENSITY FOR MODULATED LIGHT

The scalar electric displacement field, written as an exact Fourier decomposition, is

D = \frac{1}{2 π} \int_{- \infty}^{\infty} D (ω) e^{- i ω t} d ω

(6)

= \frac{ϵ_{0}}{2 π} \int_{- \infty}^{\infty} ϵ (ω) E (ω) e^{- i ω t} d ω,

(7)

where the complex dielectric constant is again

ϵ (ω) = ϵ^{'} (ω) + i ϵ^{″} (ω)

. We assume

E = e (t) \cos (ω_{0} t)

, with slowly varying modulation signal

e (t)

relative to

t_{0} = 2 π ∕ ω_{0}

, producing an effective bandwidth that is small relative to the features of

ϵ (ω)

. Therefore, in the spectral domain, the electric field becomes

E (ω) = \frac{1}{2} [e (ω - ω_{0}) + e (ω + ω_{0})],

(8)

where

e (ω)

is the Fourier transform of

e (t)

. Hence, substituting Eq. (8) into Eq. (7) gives

D = \frac{ϵ_{0}}{4 π} \int_{- \infty}^{\infty} ϵ (ω) e (ω - ω_{0}) e^{- i ω t} d ω + \frac{ϵ_{0}}{4 π} \int_{- \infty}^{\infty} ϵ (ω) e (ω + ω_{0}) e^{- i ω t} d ω .

(9)

Now, using the shift

ω \to ω + ω_{0}

, we can write the first term in Eq. (9) as

\frac{ϵ_{0}}{4 π} \int_{- \infty}^{\infty} ϵ (ω) e (ω - ω_{0}) e^{- i ω t} d ω = \frac{ϵ_{0}}{4 π} \int_{- \infty}^{\infty} ϵ (ω + ω_{0}) e (ω) e^{- i (ω + ω_{0}) t} d ω .

(10)

Also, using the shift

ω \to - (ω + ω_{0})

, we can write the second term in Eq. (9) as

\frac{ϵ_{0}}{4 π} \int_{- \infty}^{\infty} ϵ (ω) e (ω + ω_{0}) e^{- i ω t} d ω = \frac{ϵ_{0}}{4 π} \int_{- \infty}^{\infty} ϵ [- (ω + ω_{0})] e (- ω) e^{i (ω + ω_{0}) t} d ω .

(11)

Imposing the necessary

E (ω)

and

ϵ (ω)

symmetry for

E (t)

and

ϵ (t)

to be real leads to

ϵ [- (ω + ω_{0})] = ϵ^{*} (ω + ω_{0})

and

e (- ω) = e^{*} (ω)

. Therefore, the second term in Eq. (9) can be written as

\frac{ϵ_{0}}{4 π} \int_{- \infty}^{\infty} ϵ (ω) e (ω + ω_{0}) e^{- i ω t} d ω = \frac{ϵ_{0}}{4 π} \int_{- \infty}^{\infty} ϵ^{*} (ω + ω_{0}) e^{*} (ω) e^{i (ω + ω_{0}) t} d ω

(12)

= {[\frac{ϵ_{0}}{4 π} \int_{- \infty}^{\infty} ϵ (ω + ω_{0}) e (ω) e^{- i (ω + ω_{0}) t} d ω]}^{*} .

(13)

Consequently, Eq. (9) can be written as

D = \frac{ϵ_{0}}{4 π} \int_{- \infty}^{\infty} ϵ (ω + ω_{0}) e (ω) e^{- i (ω + ω_{0}) t} d ω + c.c.,

(14)

giving rise to

\frac{\partial D}{\partial t} = \frac{ϵ_{0}}{4 π} e^{- i ω_{0} t} \int [- i (ω + ω_{0}) ϵ (ω + ω_{0})] e (ω) e^{- i ω t} d ω + c.c.

(15)

The first two terms in a Taylor series expansion of

(ω + ω_{0}) ϵ (ω + ω_{0})

, appearing in Eq. (15), about

ω = 0

, are

(ω + ω_{0}) ϵ (ω + ω_{0}) = ω_{0} ϵ (ω_{0}) + ω {[\frac{\partial ((ω + ω_{0}) ϵ (ω + ω_{0}))}{\partial ω}] |}_{ω = 0} + \dots .

(16)

Thus, Eq. (15) can be written using Eq. (16) as

\frac{\partial D}{\partial t} \approx \frac{ϵ_{0}}{4 π} e^{- i ω_{0} t} \int [- i (ω_{0} ϵ (ω_{0}) + ω {[\frac{\partial ((ω + ω_{0}) ϵ (ω + ω_{0}))}{\partial ω}] |}_{ω = 0})] e (ω) e^{- i ω t} d ω + c.c.

(17)

Now, using

{\frac{\partial ((ω + ω_{0}) ϵ (ω + ω_{0}))}{\partial ω} |}_{ω = 0} = {\frac{\partial (ω ϵ (ω))}{\partial ω} |}_{ω = ω_{0}},

(18)

Eq. (17) can be written as

\frac{\partial D}{\partial t} \approx \frac{ϵ_{0}}{4 π} e^{- i ω_{0} t} \int [- i ω_{0} ϵ (ω_{0}) - i ω {\frac{\partial (ω ϵ (ω))}{\partial ω} |}_{ω = ω_{0}}] e (ω) e^{- i ω t} d ω + c.c. = \frac{ϵ_{0}}{2} e^{- i ω_{0} t} [\frac{- i ω_{0} ϵ (ω_{0})}{2 π} \int e (ω) e^{- i ω t} d ω + {\frac{\partial (ω ϵ (ω))}{\partial ω} |}_{ω = ω_{0}} \frac{1}{2 π} \int - i ω e (ω) e^{- i ω t} d ω] + c.c.

(19)

= \frac{ϵ_{0}}{2} [- i ω_{0} ϵ (ω_{0}) e (t) + {\frac{\partial (ω ϵ (ω))}{\partial ω} |}_{ω = ω_{0}} \frac{\partial e (t)}{\partial t}] e^{- i ω_{0} t} + c.c.

(20)

Evaluating Eq. (20) using the complex dielectric constant gives

\frac{\partial D}{\partial t} = ω_{0} ϵ_{0} e (t) [ϵ^{″} (ω_{0}) \cos (ω_{0} t) - ϵ^{'} (ω_{0}) \sin (ω_{0} t)] + ϵ_{0} \frac{\partial e (t)}{\partial t} [{\frac{\partial (ω ϵ^{'} (ω))}{\partial ω} |}_{ω = ω_{0}} \cos (ω_{0} t) + {\frac{\partial (ω ϵ^{″} (ω))}{\partial ω} |}_{ω = ω_{0}} \sin (ω_{0} t)] .

(21)

From Eqs. (2, 5, 21),

\frac{\partial w_{E}}{\partial t} \equiv {\frac{\partial u_{E}}{\partial t} |}_{ϵ^{″} = 0}

(22)

\approx - ω_{0} ϵ_{0} ϵ^{'} (ω_{0}) \frac{e^{2} (t)}{2} \sin (2 ω_{0} t) + ϵ_{0} {\frac{\partial (ω ϵ^{'} (ω))}{\partial ω} |}_{ω = ω_{0}} e (t) \frac{\partial e (t)}{\partial t} \cos^{2} (ω_{0} t),

(23)

and

\frac{\partial q}{\partial t} \equiv {\frac{\partial u_{E}}{\partial t} |}_{ϵ^{'} = 0}

(24)

\approx ω_{0} ϵ_{0} ϵ^{″} (ω_{0}) e^{2} (t) \cos^{2} (ω_{0} t) + ϵ_{0} \frac{e (t)}{2} \frac{\partial e (t)}{\partial t} {\frac{\partial (ω ϵ^{″} (ω))}{\partial ω} |}_{ω = ω_{0}} \sin (2 ω_{0} t) .

(25)

Equations (23, 25) apply regardless of the degree of loss, provided that two expansion terms in Eq. (16) are sufficiently accurate, and are a more general form than given previously for the lossless (transparency) case [4

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

]. Notice that the superposition of Eqs. (23, 25), when

e (t)

does not vary with time, giving

E = e_{0} \cos (ω_{0} t)

, results in

\frac{\partial u_{E}}{\partial t} = - ω_{0} ϵ_{0} e_{0}^{2} \cos (ω_{0} t) [ϵ^{'} (ω_{0}) \sin (ω_{0} t) - ϵ^{″} (ω_{0}) \cos (ω_{0} t)] .

(26)

Information transfer (or a causal system, where the field is zero at times prior to the propagation delay

l ∕ c

, with l the source-detector distance and c the speed of light in vacuum) implies modulation of a carrier signal. For the current purpose, this modulation signal can have arbitrarily small bandwidth relative to the carrier frequency,

ω_{0}

, or we assume the field has been applied for a sufficiently long time.

4. AVERAGE ENERGIES

We form energies (

u_{E}

w_{E}

, and q) from their temporal derivatives according to

f (t) = \int_{- \infty}^{t} \frac{\partial f (t^{'})}{\partial t^{'}} d t^{'} .

(27)

We use an average over

t_{0}

, giving for quantity

f (t)

⟨ f (t) ⟩ (t_{n}) = \frac{1}{t_{0}} \int_{t_{n} - t_{0} ∕ 2}^{t_{n} + t_{0} ∕ 2} f (t) d t,

(28)

with

t_{n}

the local reference time. Thus, the time average of

w_{E}

, obtained from Eq. (23), is

⟨ w_{E} ⟩ \approx \frac{1}{4} ϵ_{0} {\frac{\partial (ω ϵ^{'} (ω))}{\partial ω} |}_{ω = ω_{0}} e^{2} (t),

(29)

which is identical to an earlier result when loss is neglected [4

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

], and the average of Eq. (25) becomes

⟨ \frac{\partial q}{\partial t} ⟩ \approx \frac{1}{2} ω_{0} ϵ_{0} ϵ^{″} e^{2} (t) .

(30)

Conservation of energy for a passive medium dictates that

⟨ u_{E} ⟩ + ⟨ u_{H} ⟩ = ⟨ q ⟩ + ⟨ w_{E} ⟩ + ⟨ u_{H} ⟩ ⩾ 0,

(31)

where

⟨ q ⟩ ⩾ 0

is the lost energy (due to coupling to phonons, heating, and excited-state lifetime, which dictates the resonance linewidth), and

u_{H}

is the energy delivered in the magnetic field. Even if

⟨ w_{E} ⟩

in Eq. (29) is negative, net positive energy is delivered to the volume due to the loss in Eq. (31). With an appropriate set of measurements,

ϵ (ω)

and

μ (ω)

could be determined. These material responses must dictate the scatter and absorption at any frequency and for any set of frequencies, assuming linear susceptibilities. Such a measurement therefore could support the concept of negative field energy densities at particular frequencies near a resonance, as dictated by Eq. (29). However, a pure electromagnetic measurement would not provide the specific mechanisms involved in

ϵ^{″}

, for example.

5. IMPACT OF CAUSALITY

The Kramers–Kronig relations [12

J. D. Jackson, Classical Electrodynamics , 3rd ed. (Wiley, 1999).

], which result from causality, are given by the Hilbert transform pair

ζ^{'} (ω) - 1 = \frac{2}{π} \int_{0}^{\infty} - \frac{ω^{'} ζ^{″} (ω^{'})}{ω^{' 2} - ω^{2}} d ω^{'},

ζ^{″} (ω) = \frac{- 2 ω}{π} \int_{0}^{\infty} - \frac{ζ^{'} (ω^{'}) - 1}{ω^{' 2} - ω^{2}} d ω^{'},

(32)

where

ζ = ζ^{'} + i ζ^{″}

and

ζ = {μ, ϵ, n^{2}}

, with

n^{2}

the square of the refractive index, and the integral is interpreted as a principal value with the pole residue evaluated. As

ω \to \infty

ϵ \to 1

, from Eq. (32). As

ω \to 0

ϵ^{'}

must be positive (and

ϵ^{'} ⩾ 1

), and as is necessary for conservation of energy, for example, long after a capacitor containing the material is charged by a source. A bound

\partial (ω ϵ^{'}) ∕ \partial ω ⩾ 0

has been shown with negligible loss [4

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

], and this has been used in describing requirements for negative index materials [1

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ, and μ ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

, 13

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000). [CrossRef] [PubMed]

, 14

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006). [CrossRef]

]. In the neighborhood of a resonance, where

ϵ^{'} < 0

(and likewise

μ^{'} < 0

) and a negative refractive index is possible, there is no requirement on

ϵ^{'}

and hence no restriction on

\partial (ω ϵ^{'}) ∕ \partial ω

from Eq. (32). Without a constraint on the frequency dependence of the imaginary part of the refractive index, and based on the Kramers–Kronig relations, it has previously been noted that the frequency dependence of the derivative of the real part of the refractive index is unconstrained [14

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006). [CrossRef]

]. Therefore, at finite and nonzero frequencies, the sign of

⟨ w_{E} ⟩

in Eq. (29) is unconstrained, based on both energy and causality arguments.

6. ANALYTIC EXAMPLE

From a classical oscillator development [12

J. D. Jackson, Classical Electrodynamics , 3rd ed. (Wiley, 1999).

] or a quantum mechanical derivation [15

A. Yariv, Quantum Electronics , 2nd ed. (Wiley, 1975).

], the electric susceptibility can be written in the form

χ (ω) = \frac{e^{2}}{ϵ_{0} m} \sum_{j} \frac{Δ N_{j} f_{j}}{ω_{j}^{2} - ω^{2} - i γ_{j} ω},

(33)

where e is the electron charge, m is the mass,

Δ N_{j}

is the oscillator density for the

j th

resonance (

Δ N_{j} > 0

corresponds to loss and

Δ N_{j} < 0

to gain),

\sum_{j} f_{j} = 1

, and

γ_{j}

is the linewidth (

2 π γ_{j}^{- 1}

is the photon lifetime). It is assumed that any material operating in the regime where homogenization is meaningful can be represented by Eq. (33).

Writing the real part of the first term from Eq. (33),

ϵ^{'} (ω) = 1 + \frac{a_{1} (ω_{1}^{2} - ω^{2})}{{(ω_{1}^{2} - ω^{2})}^{2} + γ_{1}^{2} ω^{2}},

(34)

where

a_{1} = e^{2} Δ N_{1} ∕ (ϵ_{0} m)

. A perturbational expansion around

ω_{1}

, by setting

ω = ω_{1} + Δ ω

in Eq. (34), under the assumption that

Δ ω ⪡ γ_{1} ⪡ ω_{1}

, gives to order

(Δ ω ∕ ω_{1})

{\frac{\partial (ω ϵ^{'})}{\partial ω} |}_{ω = ω_{1} + Δ ω} = 1 - \frac{2 a_{1}}{γ_{1}^{2}} (1 - \frac{Δ ω}{ω_{1}}) .

(35)

With gain,

a_{1} < 0

, and

{\partial (ω ϵ^{'}) ∕ \partial ω |}_{ω = ω_{1} + Δ ω} > 0

. However, with loss, giving

a_{1} > 0

, it is possible to have

\partial {(ω ϵ^{'}) ∕ \partial ω |}_{ω = ω_{1} + Δ ω} < 0

. The stored energy density from Eqs. (23, 35) is

w_{E} = \frac{e^{2} (t) ϵ_{0}}{4} [1 - \frac{2 a_{1}}{γ_{1}^{2}} (1 - \frac{Δ ω}{ω_{1}})] {1 + \cos [2 (ω_{1} + Δ ω) t]},

(36)

and the time average with a narrowband modulation signal

e (t)

⟨ w_{E} ⟩ = \frac{e^{2} (t) ϵ_{0}}{4} [1 - \frac{2 a_{1}}{γ_{1}^{2}} (1 - \frac{Δ ω}{ω_{1}})] .

(37)

Notice that

{\partial (ω ϵ^{'}) ∕ \partial ω |}_{ω = ω_{1} + Δ ω} < 0

corresponds to

(2 a_{1} ∕ γ_{1}^{2}) [1 - (Δ ω ∕ ω_{1})] > 1

in Eq. (37).

7. NUMERICAL EXAMPLE

An example dispersive dielectric constant response having the form of one term in the susceptibility expansion in Eq. (33) is shown in Fig. 1 . We assume a Gaussian modulated field given by

E = {(σ \sqrt{2 π})}^{- 1} \exp [- {(t - t_{c})}^{2} {(2 σ^{2})}^{- 1}] \cos [ω_{0} (t - t_{c})],

(38)

with

ω_{0} = 8 ∕ 9

, 1 and

8 ∕ 7

t_{c} = 2^{10} (2 π)

, and

σ = 2^{7} (2 π)

. The expression for

\partial u_{E} ∕ \partial t

in Eq. (2) was evaluated numerically, with

D (t)

formed as in Eq. (3) by Fourier transforming (using an FFT) the temporal electric field, multiplying by

ϵ (ω)

, and then inverse Fourier transforming the result. The quantities

\partial w_{E} ∕ \partial t

and

\partial q ∕ \partial t

were formed using the exact decompositions in Eqs. (22, 24), respectively. Numerical integration over time using Eq. (27) resulted in

u_{E}

and

w_{E}

. Averages over each period were formed using Eq. (28). Subject to numerical precision, we consider these as exact results.

Figure 2 shows

⟨ w_{E} ⟩

obtained for the three different carrier frequencies and the approximate result of Eq. (29), the model. When the carrier frequency is sufficiently far from the resonant frequency,

⟨ w_{E} ⟩

is positive, just as it would have been in free space. However, when the carrier frequency is equal to the resonant frequency

(ω_{0} = 1)

⟨ w_{E} ⟩

becomes negative. Figure 3 shows

⟨ \partial q ∕ \partial t ⟩

obtained for the same three frequencies, along with the results from a model that is the approximate analytical expression in Eq. (30). Throughout, the total average energy

(⟨ u_{E} ⟩)

remains positive, as Fig. 4 shows. At times long after the pulse has gone,

⟨ q ⟩ = ⟨ u_{E} ⟩

, indicating that no additional energy beyond that delivered by the electromagnetic field is involved, despite

⟨ w_{E} ⟩

being negative for some time. Notice the excellent agreement between the exact results and the approximate analytical expressions in Figs. 2, 3.

When

⟨ \partial ω_{E} ∕ \partial t ⟩

is positive during the first half of the Gaussian and negative during the second half,

⟨ w_{E} ⟩

in Fig. 2 is positive throughout and zero for large time, after the pulse has left that point in space. In the case where we find negative stored energy near a resonance,

⟨ \partial w_{E} ∕ \partial t ⟩

is first negative and then positive. This opposite phase, relative to the cases where

⟨ w_{E} ⟩

is positive, results in negative stored energy. We interpret this to mean that the material first delivers some energy (that is dissipated), and then the field returns this energy. During the first half of the pulse, the field delivers less energy than is lost, and during the second half of the pulse the field delivers more energy than is lost. The negative field energy is energy borrowed from the material, and during the second half of the pulse the field repays this energy. This process is independent of carrier phase. The net energy delivered to this point in the material after the pulse has passed is equal to that dissipated. The net accumulation or declination is due to a small contribution over each period at the carrier frequency.

8. CIRCUIT ANALOG

There is a circuit analog of Eq. (29) that also supports the possibility of negative energy. The instantaneous power delivered to a capacitor can be written as

v d (v C) ∕ d t

, where v is the voltage across the capacitor. Assuming that

v = v_{0} (t) \cos (ω_{0} t)

, with

v_{0} (t)

a sufficiently slow modulation signal, the time average energy stored in the field in the capacitor is

(v_{0} {(t)}^{2} ∕ 4) {{\partial [ω C^{'} (ω)] ∕ \partial ω |}_{ω_{0}}}

, where the com plex capacitance is

C = C^{'} + i C^{″}

, with

C^{″}

describing loss, in direct analogy with Eq. (29). This is a resonant circuit, where

C^{'} < 0

corresponds to inductance. Note that in the case where the circuit values do not change with frequency, negative stored energy would not occur. In the case of low loss, Nedlin [16

G. Nedlin, “Energy in lossless and low-loss networks, and Foster’s reactance theorem,” IEEE Trans. Circuits Syst. 36, 561–567 (1989). [CrossRef]

] has derived a circuit equivalent to the result of Landau and Lifshitz [4

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

9. CONCLUSION

We have shown, based on Maxwell’s equations, and more specifically Poynting’s theorem, that the energy density stored in the electric field in a passive material can be negative in the neighborhood of a resonance. This frequency regime has become of interest in the study of metamaterials with negative constitutive parameters, including those with negative refractive index. While we focused on the electric field energy, that for the magnetic field follows from duality. We mathematically delineate between the stored and dissipated electric field energy in a physical system (the

ϵ^{'} = 0

and

ϵ^{″} = 0

terms). However, we note that physical materials have been proposed that have an operating frequency where

ϵ^{″} = 0

(and

ϵ^{'} ⩽ - 1

) [17

K. J. Webb and L. Thylén, “A perfect lens material condition from adjacent absorptive and gain resonances,” Opt. Lett. 33, 747–749 (2008). [CrossRef] [PubMed]

, 18

K. J. Webb and A. Ludwig, “Semiconductor quantum dot mixture as a lossless negative dielectric constant optical material,” Phys. Rev. B 78, 153303 (2008). [CrossRef]

] and hence satisfy one of these conditions at one frequency. Finally, we note that separability into stored and dissipated electric field energy was possible by defining the electric field as a basis. Consequently, this does not always allow the separation of the magnetic field energy in Eq. (31).

ACKNOWLEDGMENTS

We thank Alon Ludwig, Purdue University, for comments. This work was supported in part by the National Science Foundation (NSF) (ECCS-0524442 and ECCS-0901383), the Department of Energy (DOE) (DE-FG52-06NA27505), and the Army Research Office (W911NF-07-1-0019).

References and links

1.	V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ, and μ ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]
2.	R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]
3.	L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
4.	L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).
5.	R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970). [CrossRef]
6.	V. G. Polevoi, “Maximum energy extractable from an electromagnetic field,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 33, 818–825 (1990).
7.	R. Ruppin, “Elecromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002). [CrossRef]
8.	S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005). [CrossRef]
9.	A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006). [CrossRef]
10.	T. J. Cui and J. A. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B 70, 205106 (2004). [CrossRef]
11.	R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic medium,” Phys. Rev. E 63, 046604 (2001). [CrossRef]
12.	J. D. Jackson, Classical Electrodynamics , 3rd ed. (Wiley, 1999).
13.	D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000). [CrossRef] [PubMed]
14.	J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006). [CrossRef]
15.	A. Yariv, Quantum Electronics , 2nd ed. (Wiley, 1975).
16.	G. Nedlin, “Energy in lossless and low-loss networks, and Foster’s reactance theorem,” IEEE Trans. Circuits Syst. 36, 561–567 (1989). [CrossRef]
17.	K. J. Webb and L. Thylén, “A perfect lens material condition from adjacent absorptive and gain resonances,” Opt. Lett. 33, 747–749 (2008). [CrossRef] [PubMed]
18.	K. J. Webb and A. Ludwig, “Semiconductor quantum dot mixture as a lossless negative dielectric constant optical material,” Phys. Rev. B 78, 153303 (2008). [CrossRef]

Fig. 1 Real (solid curve) and imaginary (dotted curve) part of dielectric constant

ϵ = 1 + 10 {(ω_{1}^{2} - ω^{2} - i 0.1 ω)}^{- 1}

for

ω_{1} = 1

. The three different carrier frequencies

(ω_{0} = 8 ∕ 9, 1, 8 ∕ 7)

are marked by thin vertical solid lines.

Fig. 2

{⟨ u_{E} ⟩ |}_{ϵ^{″} = 0} = ⟨ w_{E} ⟩

obtained for

ω_{0} = 8 ∕ 9

(dotted curve),

ω_{0} = 1

(dashed-dotted curve), and

ω_{0} = 8 ∕ 7

(dashed curve).

ϵ^{'} = Real {1 + 10 {(ω_{1}^{2} - ω^{2} - i 0.1 ω)}^{- 1}}

and

ω_{1} = 1

. The model result (circles) plots Eq. (29), and the curves give exact results.

Fig. 3

{⟨ \partial u_{E} ∕ \partial t ⟩ |}_{ϵ^{'} = 0} = ⟨ \partial q ∕ \partial t ⟩

obtained for

ω_{0} = 8 ∕ 9

(dotted curve),

ω_{0} = 1

(dashed-dotted curve), and

ω_{0} = 8 ∕ 7

(dashed curve).

ϵ^{″} = Imag {1 + 10 {(ω_{1}^{2} - ω^{2} - i 0.1 ω)}^{- 1}}

and

ω_{1} = 1

. The model result is from Eq. (30), and the curves give exact results.

Fig. 4 Exact

⟨ u_{E} ⟩

obtained for

ω_{0} = 8 ∕ 9

(dotted curve),

ω_{0} = 1

(dashed-dotted curve), and

ω_{0} = 8 ∕ 7

(dashed curve).

ϵ = 1 + 10 {(ω_{1}^{2} - ω^{2} - i 0.1 ω)}^{- 1}

and

ω_{1} = 1

OCIS Codes
(260.2030) Physical optics : Dispersion
(260.2110) Physical optics : Electromagnetic optics
(160.3918) Materials : Metamaterials

ToC Category:
Physical Optics

History
Original Manuscript: February 18, 2010
Manuscript Accepted: March 4, 2010
Published: May 12, 2010

Citation
Kevin J. Webb and Shivanand, "Electromagnetic field energy in dispersive materials," J. Opt. Soc. Am. B 27, 1215-1220 (2010)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-27-6-1215

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References

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ, and μ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]
R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]
L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).
R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970). [CrossRef]
V. G. Polevoi, “Maximum energy extractable from an electromagnetic field,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 33, 818–825 (1990).
R. Ruppin, “Elecromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002). [CrossRef]
S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005). [CrossRef]
A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006). [CrossRef]
T. J. Cui and J. A. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B 70, 205106 (2004). [CrossRef]
R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic medium,” Phys. Rev. E 63, 046604 (2001). [CrossRef]
J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000). [CrossRef] [PubMed]
J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006). [CrossRef]
A. Yariv, Quantum Electronics, 2nd ed. (Wiley, 1975).
G. Nedlin, “Energy in lossless and low-loss networks, and Foster’s reactance theorem,” IEEE Trans. Circuits Syst. 36, 561–567 (1989). [CrossRef]
K. J. Webb and L. Thylén, “A perfect lens material condition from adjacent absorptive and gain resonances,” Opt. Lett. 33, 747–749 (2008). [CrossRef] [PubMed]
K. J. Webb and A. Ludwig, “Semiconductor quantum dot mixture as a lossless negative dielectric constant optical material,” Phys. Rev. B 78, 153303 (2008). [CrossRef]

Boardman, A. D.
Brillouin, L.
Cui, T. J.
Jackson, J. D.
Kong, J. A.
Kroll, N.
Landau, L. D.
Lifshitz, E. M.
Loudon, R.
Ludwig, A.
Marinov, K.
Nedlin, G.
Polevoi, V. G.
Ruppin, R.
Schultz, S.
Seip, K.
Shelby, R. A.
Skaar, J.
Smith, D. R.
Thylén, L.
Tretyakov, S. A.
Veselago, V. G.
Webb, K. J.
Yariv, A.
Ziolkowski, R. W.

IEEE Trans. Circuits Syst.

G. Nedlin, “Energy in lossless and low-loss networks, and Foster’s reactance theorem,” IEEE Trans. Circuits Syst. 36, 561–567 (1989). [CrossRef]

Izv. Vyssh. Uchebn. Zaved., Radiofiz.

V. G. Polevoi, “Maximum energy extractable from an electromagnetic field,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 33, 818–825 (1990).

J. Phys. A

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970). [CrossRef]

J. Phys. D

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006). [CrossRef]

Opt. Lett.

K. J. Webb and L. Thylén, “A perfect lens material condition from adjacent absorptive and gain resonances,” Opt. Lett. 33, 747–749 (2008). [CrossRef] [PubMed]

Phys. Lett. A

R. Ruppin, “Elecromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002). [CrossRef]
S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005). [CrossRef]

Phys. Rev. B

A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006). [CrossRef]
T. J. Cui and J. A. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B 70, 205106 (2004). [CrossRef]
K. J. Webb and A. Ludwig, “Semiconductor quantum dot mixture as a lossless negative dielectric constant optical material,” Phys. Rev. B 78, 153303 (2008). [CrossRef]

Phys. Rev. E

R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic medium,” Phys. Rev. E 63, 046604 (2001). [CrossRef]

Phys. Rev. Lett.

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000). [CrossRef] [PubMed]

Science

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

Sov. Phys. Usp.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ, and μ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

Other

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).
J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
A. Yariv, Quantum Electronics, 2nd ed. (Wiley, 1975).

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