Generalized Permeability Function and Field Energy Density in Artificial Magnetics Using the Equivalent Circuit Method

Pekka Ikonen; Sergei Tretyakov

REVIEW

Generalized Permeability Function and Field Energy Density in Artificial Magnetics Using the Equivalent Circuit Method

Pekka Ikonen, Sergei Tretyakov

Classical Physics · Submitted 27 Feb 2006 · 25 references

Pith's verdict: The generalized permeability with substrate losses is a genuine, validated contribution, but the energy-density result is not the stored energy because the mutual-inductance term is dropped.

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CitationPekka Ikonen, Sergei Tretyakov. "Generalized Permeability Function and Field Energy Density in Artificial Magnetics Using the Equivalent Circuit Method." arXiv:physics/0602182 (2006). https://pith.science/paper/SC6IH5WQ

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Editor's note

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Two things to know up front. The first half of the paper is a real contribution: adding a shunt resistor Rd across the split-ring capacitance and writing the permeability as (10) is the correct way to account for substrate losses, and the metasolenoid measurements in Sec. V support it. The old model is shown to overestimate Re{μeff} by roughly a factor of twenty in the example, which is a strong practical point. The second half is wrong. The energy density (24) is not the energy stored in the circuit of Fig. 3b.

Look at Eq. (20). The stored energy is written as 1/2(L0|I|^2 + L|IL|^2 + C|VC|^2). A coupled inductor pair also stores M Re{I IL*} (sign depending on winding sense), and Eq. (21) shows IL is not in quadrature with I once R or Rd or detuning is present. The mutual term is therefore real and cannot be dropped. I checked the algebra: in the lossless limit, adding the mutual term to (20) converts the bracket into the classical ∂(ωμ)/∂ω form, 1 + Aω^2(3ω0^2 − ω^2)/Δ^2. That exactly accounts for the difference between the EC and ED results that the paper blames on the quasi-static Lorentz model. So the claim that Eq. (24) is the microscopically correct stored energy, and the comparison with [21] in Sec. VI-A, are built on an incomplete accounting.

Are there other soft spots? Minor ones. The validation uses one sample and adjusts tan δ from 0.002 to 0.0025 to fit; that is sensitivity rather than independent confirmation, but the effect is large enough that the qualitative conclusion survives. The discussion of why black-box macroscopic definitions fail in lossy media is correct.

The audience is microwave and metamaterial researchers who use equivalent-circuit models. The permeability generalization is worth keeping and is likely citable as a model result. The energy-density result is wrong as stated and should not appear as a headline. For peer review: yes, send it to a competent referee. It is a serious paper with a serious flaw, and the fix is clear; as it stands I would not accept it, but a major revision should be given a chance.

Referee report

Significance

The permeability generalization with substrate losses is a practically useful contribution, and the experimental validation in Sec. V supports the proposed circuit modification; the polarizability comparison is a genuine strength of the paper, and the clarity with which the authors explain why microscopic information is needed for stored energy in lossy media is valuable. However, the central energy-density claim, Eq. (24), is built on an incomplete accounting of the stored energy in the coupled inductive circuit: the mutual-inductance energy is omitted. The omission is not small; in the lossless limit it exactly reconciles the circuit result with the classical formula ∂(ωμ)/∂ω, so the claimed discrepancy between the EC and ED methods is an artifact of the calculation rather than a physical property of the quasi-static permeability model. Because the headline result and the comparison in Sec. VI-A rest on this error, the energy-density contribution is not valid as presented.

Major objections (2)

[Sec. VI, Eq. (20)] The stored energy of the coupled circuit is incomplete. For the two inductively coupled loops of Fig. 3(b), the time-average magnetic energy contains the mutual term ½ M Re{I IL*} (in peak-amplitude convention), which is not included in W = ½(L0|I|² + L|IL|² + C|VC|²). This is not a small correction. In the lossless limit (R → 0, 1/Rd → 0), the loop equation gives IL = ω²M/(LΔ) I with Δ = ω0² − ω², so the omitted term equals ½(M²/L)(ω²/Δ)|I|². Adding it to the terms retained in Eq. (20) and dividing by ½ L0|I|² yields 1 + Aω²(3ω0² − ω²)/Δ², which is exactly ∂(ωμr)/∂ω for the permeability (1) with Γ = 0. Thus the difference between the EC result (24) and the classical expression (26) reported in Sec. VI-A is not a physical consequence of the quasi-static permeability model; it is an artifact of omitting the mutual-inductance energy.
[Sec. VI-A, Eqs. (24)-(26)] Because the mutual term is nonzero whenever IL is not in quadrature with I, the omission also invalidates the lossy expression (24) and the comparison with the ED-method result (25). With finite R and Rd, the loop current has a component in phase with I, so M Re{I IL*} contributes to W; Eq. (24) therefore does not represent the electromagnetic energy stored in the circuit of Fig. 3(b). Consequently, the claim in Sec. VI-A that the EC method gives a different and more physical result than the ED/classical expressions, and the discussion of the unphysical high-frequency behavior of (25), are not supported. A corrected calculation that includes the mutual term removes the discrepancy in the lossless limit and is likely to alter all of the paper's energy-density conclusions.

Minor objections (4)

[Eq. (2)] The notation is garbled: "µ0, l1 (2)" and later "restriction ˚1" should be a properly labeled inequality, presumably ∂(ωμ)/∂ω > μ0.
[Sec. VI, text after Eq. (24)] The sentence "then ~ω2 p → ωp and (24) reduces to (6)" contains symbols that are not defined in the paper; the intended statement is likely that ~ω0² → ω0² in the limit Γd → 0.
[Sec. V, Table I and Fig. 5] The validation uses an adjusted loss tangent (tan δ = 0.0025 instead of 0.002 from Ref. [8]) and the measured resonant frequency differs by about 10% from the calculated one; the authors mention this, but reporting the uncertainty in the fitted parameters would strengthen the claim.
[Throughout] The paper alternates between the symbols ω0 and ωp for the resonance frequency (e.g., Eq. (6) and the text near Eq. (24)); the notation should be unified.

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The paper

The paper claims

Substrate losses force a generalized permeability for split-ring magnetics, and the equivalent-circuit method gives a stored-energy density that stays valid under strong dispersion and loss.

The reading

Artificial magnetic materials — arrays of split rings or metasolenoids printed on a dielectric substrate — are usually modelled with a single-Lorentzian permeability that ignores the substrate. This paper claims that when the matrix is lossy, that standard form is insufficient: the permeability must be generalized, and the exact form depends on the substrate's loss mechanism (conducting versus Lorentzian-dispersive). Starting from an equivalent circuit with a resistor shunting the split-ring capacitance, the paper obtains a stored-energy density expression that remains valid under strong dispersion and loss at low frequencies and near resonance. The practical point is that matrix losses can dominate conductor losses in dense arrays, so designs using the old model can overestimate the achievable permeability by an order of magnitude or more.

What carries the argument

The load-bearing object is the equivalent circuit for one unit cell of an artificial magnetic medium: the probing solenoid inductance $L_0$ is magnetically coupled through $M$ to a loop with inductance $L$, capacitance $C$, conductor resistance $R$, and a resistor $R_d$ shunting $C$ that models dielectric loss in the substrate, with $Y=j\omega C(\epsilon'-j\epsilon'')$. The paper uses this circuit to derive the impedance seen by the source, identifies the generalized permeability function, and computes the reactive energy $W=\frac12(L_0|I|^2 + L|I_L|^2 + C|V_C|^2)$; converting $I$ to $H$ via $I=H/n$ turns $W$ into the energy-density expression (24).

Extended reading notes

Core claim

The paper's central claim is that the standard quasi-static permeability $\mu(\omega)=\mu_0(1 + A\omega^2/(\omega_0^2-\omega^2+j\omega\Gamma))$ omits the dominant loss channel of real split-ring structures — loss in the dielectric substrate or matrix — and that a circuit model with a resistor $R_d$ in parallel with the split-ring capacitance replaces it with $\mu(\omega)=\mu_0(1+(A\omega^2 - j\omega B)/( \tilde{\omega}_0^2-\omega^2+j\omega(\Gamma+\Gamma_d)))$. For a conducting matrix the loss resistor is frequency-independent; for a Lorentzian-dispersive matrix it scales as $1/\omega^2$ and produces a different permeability form. From this circuit the paper derives the stored-energy density $w_m = \frac{\mu_0}{2}\left(1 + \frac{A\omega^2(\omega_0^2 + \omega^2 + \Gamma_d^2)}{(\tilde{\omega}_0^2-\omega^2)^2+\omega^2(\Gamma+\Gamma_d)^2}\right)|H|^2$, stating that it is valid at low frequencies and near resonance even when dispersion and losses are strong, and that it avoids the unphysical high-frequency drop below the vacuum value exhibited by the electrodynamic-method result.

Load-bearing premise

The stored energy is computed as the sum of the energies in the coupling-free parts of the circuit — the solenoid inductance, the loop inductance, and the split-ring capacitance — with no contribution from the mutual inductance between the solenoid and the ring loop; the loop current is not exactly in quadrature with the solenoid current, so this omitted term is not automatically zero, and in the lossless limit it is large enough to shift the result toward the classical formula.

Editorial extensions

If this is right

In designs of artificial magneto-dielectric substrates with a lossy host matrix, the generalized permeability must replace the single-Lorentzian form; the measured metasolenoid example shows the old model overestimates polarizability and permeability by an order of magnitude or more.
At low frequencies and near resonance, the derived energy density stays positive and above the vacuum value under strong dispersion and loss, while the classical and electrodynamic-method results drop below the vacuum energy for $\omega>\sqrt{3}\omega_0$ in the lossless limit.
The correct permeability form depends on the substrate loss mechanism: a conducting matrix gives a frequency-independent $R_d$, whereas a Lorentzian-dispersive matrix gives a frequency-dependent loss term and a different permeability form.
When substrate losses vanish, the generalized energy density reduces to the earlier equivalent-circuit result, so the new expression extends rather than abandons the previous lossless energy formula.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not draw this consequence, but the energy sum in Eq. (20) omits the mutual-inductance contribution $M\,\mathrm{Re}\{I^{*}I_L\}$ between solenoid and loop; recomputing $W$ with the full two-inductor magnetic energy would include this term, and in the lossless limit that term moves the circuit result toward the classical formula.
A neighbouring problem: the same circuit method, applied to artificial dielectrics with substrate losses, should yield a permittivity analogue of the energy density, with the same caveat about mutual-energy terms.
Testable extension: because the circuit-theory energy density and the electrodynamic-method formula separate most strongly above resonance, a frequency-domain measurement of reactive stored power in a metasolenoid-loaded resonator above $\omega_0$ could distinguish the two predictions experimentally.

Original abstract

The equivalent circuit model for artificial magnetic materials based on various arrangements of split rings is generalized by taking into account losses in the substrate or matrix material. It is shown that a modification is needed to the known macroscopic permeability function in order to correctly describe these materials. Depending on the dominating loss mechanism (conductive losses in metal parts or dielectric losses in the substrate) the permeability function has different forms. The proposed circuit model and permeability function are experimentally validated. Furthermore, starting from the generalized circuit model we derive an explicit expression for the electromagnetic field energy density in artificial magnetic media. This expression is valid at low frequencies and in the vicinity of the resonance also when dispersion and losses in the material are strong. The presently obtained results for the energy density are compared with the results obtained using different methods.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,467 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.85 · human participants 0.02 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The equivalent circuit model for artificial magnetic materials based on various arrangements of split rings is generalized by taking into account losses in the substrate or matrix material. It is shown that a modification is needed to the known macroscopic permeability function in order to correctly describe these materials. Depending on the dominating loss mechanism (conductive losses in metal parts or dielectric losses in the substrate) the permeability function has different forms. The proposed circuit model and permeability function are experimentally validated. Furthermore, starting from the generalized circuit model we derive an explicit expression for the electromagnetic field energy density in artificial magnetic media. This expression is valid at low frequencies and in the vicinity of the resonance also when dispersion and losses in the material are strong. The presently obtained results for the energy density are compared with the results obtained using different methods.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The equivalent circuit model for artificial magnetic materials based on various arrangements of split rings is generalized by taking into account losses in the substrate or matrix material.	contribution 0.81	0.90	0.11	asserted 1.00
s1	It is shown that a modification is needed to the known macroscopic permeability function in order to correctly describe these materials.	result 0.59	0.84	0.32	asserted 1.00
s2	Depending on the dominating loss mechanism (conductive losses in metal parts or dielectric losses in the substrate) the permeability function has different forms.	result 0.90	0.80	0.16	asserted 1.00
s3	The proposed circuit model and permeability function are experimentally validated.	result 0.43	0.94	0.16	asserted 1.00
s4	Furthermore, starting from the generalized circuit model we derive an explicit expression for the electromagnetic field energy density in artificial magnetic media.	contribution 0.86	0.96	0.22	asserted 1.00
s5	This expression is valid at low frequencies and in the vicinity of the resonance also when dispersion and losses in the material are strong.	result 0.50	0.88	0.16	asserted 1.00
s6	The presently obtained results for the energy density are compared with the results obtained using different methods.	result 0.53	0.83	0.10	asserted 0.91

What kind of paper the classifier reports this is: theoretical result 0.87 · evidence from observation or experiment 0.69.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Split-ring resonator loaded with an infinitesimally

**Figure 3.** a) Magnetic material sample in the probe magnetic fiel

**Figure 4.** a) A schematic illustration of the metasolenoid. b) A

**Figure 5.** The analytically calculated (proposed model) and me

**Figure 6.** Magnetic field energy density given by different expr

Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:physics/0602182 by the authors.

References

Works cited (25)

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M. Gorkunov, M. Lapine, E. Shamonina, K. H. Ringhofer, “E ffective magnetic properties of a composite material with circular c onductive elements”, European Phys. Journal B , vol. 28, no. 3, pp. 263–269, July 2002

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A. N. Lagarkov, V . N. Semenenko, V . N. Kisel, V . A. Chistya ev, “Development and simulation of microwave artiﬁcial magnet ic com- posites utilizing nonmagnetic inclusions,” J. Magnetism and Magnetic Materials, vol. 258–259, pp. 161–166, March 2003

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B. Sauviac, C. R. Simovski, S. A. Tretyakov, “Double spli t-ring res- onators: Analytical modeling and numerical simulations,” Electromag- netics, vol. 24, no. 5, pp. 317–338, 2004

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J. D. Baena, R. Marqu´ es, F. Medina, J. Martel, “Artiﬁcia l magnetic metamaterial design by using spiral resonators,” Phys. Rev. B , 69, 014402, Jan. 2004

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[8]

S. I. Maslovski, P. Ikonen, I. A. Kolmakov, S. A. Tretyakov, M. Kaunisto, “Artiﬁcial magnetic materials based on the new magnetic par ticle: Metasolenoid,” Progress in Electromagnetics Research, vol. 54, pp. 61– 81, 2005

View on Pith 2005
[9]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeabil ity and permittivity,” Phys. Rev. Lett. , vol. 84, no. 18, pp. 4184–4187, May 2000

View on Pith 2000
[10]

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental ver iﬁcation of a negative index of refraction,” Science, vol. 292, pp. 77–79, April 2001

View on Pith 2001
[11]

R. C. Hansen and M. Burke, “Antenna with magneto-dielec trics,” Microwave Opt. Technol. Lett. , vol. 26, no. 2., pp. 75–78, July 2000

View on Pith 2000
[12]

S. Yoon and R. W. Ziolkowski, “Bandwidth of a microstrip patch antenna on a magneto-dielectric substrate,” IEEE Antennas Propa- gat. Soc. Int. Symposium , Columbus, Ohio, pp. 297–300, June 22-27, 2003

View on Pith 2003
[13]

H. Mossallaei and K. Sarabandi “Magneto-dielectrics i n electromagnet- ics: Concept and applications,”IEEE Trans. Antennas Propagat., vol. 52, no. 6, pp. 1558–1567, June 2004

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[14]

M. K. K¨ arkk¨ ainen, S. A. Tretyakov, P. Ikonen, “PIFA wi th dispersive material ﬁllings,” Microwave Opt. Technol. Lett., vol. 45, no. 1, pp. 5–8, April 2005

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[15]

M. E. Ermutlu, C. R. Simovski, M. K. K¨ arkk¨ ainen, P. Iko nen, S. A. Tretyakov, A. A. Sochava, “Miniaturization of patch antennas with new artiﬁcial magnetic layers,” 2005 IEEE Int. Workshop on Antenna Technology, Singapore, pp. 87–90, March 7–9, 2005

View on Pith 2005
[16]

K. Buell, H. Mosallaei, K. Sarabandi, “A substrate for s mall patch anten- nas providing tunable miniaturization factors,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 1., pp. 135–145, Jan. 2006

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[17]

P. Ikonen, S. Maslovski, C. Simovski, S. Tretyakov, “On artiﬁcial magneto-dielectric loading for improving the impedance ba ndwidth properties of microstrip antennas,” arXiv:physics/05091 39. (To appear in IEEE Trans. Antennas Propagat. , June 2006.) 7

View on Pith 2006
[18]

R. Ruppin, “Electromagnetic energy density in a disper sive and absorp- tive material,” Phys. Lett. A , vol. 299, pp. 309–312, July 2002

View on Pith 2002
[19]

T. J. Cui and J. A. Kong, “Time-domain electromagnetic e nergy in a frequency-dispersive left-handed medium,” Phys. Rev. B , vol. 70, 205106, Nov. 2004

View on Pith 2004
[20]

S. A. Tretyakov, “Electromagnetic ﬁeld energy density in artiﬁcial microwave materials with strong dispersion and loss,” Phys. Lett. A , vol. 343, pp. 231–237, June 2005

View on Pith 2005
[22]

L. D. Landau, E. M. Lifshits, Electrodynamics of continuous media, 2nd ed., Pergamon: Oxford, England, 1984

View on Pith 1984
[23]

L. A. Vainstein, Electromagnetic waves, 2nd ed., Radio i Sviaz: Moscow, 1988 (in Russian)

View on Pith 1988
[24]

J. D. Jackson, Classical electrodynamics, 3rd ed., John Wiley & Sons: New York, 1999

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[25]

A. Sihvola, Electromagnetic mixing rules and formulas , IEE Electro- magnetic Waves Series: London, 1999

View on Pith 1999

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [21]

A. D. Boardman and K. Marinov, “Electromagnetic energy in a disper- sive metamaterial,” Phys. Rev. B , vol. 73, 165110, April 2006

View on Pith 2006

[2] [1]

M. V . Kostin and V . V . Shevchenko, “Artiﬁcial magnetics b ased on double circular elements,” Proc. Bianisotropics’94, P´ erigueux, France, pp. 49–56, May 18–20, 1994

View on Pith 1994

[3] [2]

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “M agnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Mi- crowave Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999

View on Pith 1999

[4] [3]

R. Marqu´ es, F. Medina, R. Raﬁi-El-Idrissi, “Role of bia nisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B , vol. 65, 1444401(–6), April 2002

View on Pith 2002

[5] [4]

M. Gorkunov, M. Lapine, E. Shamonina, K. H. Ringhofer, “E ffective magnetic properties of a composite material with circular c onductive elements”, European Phys. Journal B , vol. 28, no. 3, pp. 263–269, July 2002

View on Pith 2002

[6] [5]

A. N. Lagarkov, V . N. Semenenko, V . N. Kisel, V . A. Chistya ev, “Development and simulation of microwave artiﬁcial magnet ic com- posites utilizing nonmagnetic inclusions,” J. Magnetism and Magnetic Materials, vol. 258–259, pp. 161–166, March 2003

View on Pith 2003

[7] [6]

B. Sauviac, C. R. Simovski, S. A. Tretyakov, “Double spli t-ring res- onators: Analytical modeling and numerical simulations,” Electromag- netics, vol. 24, no. 5, pp. 317–338, 2004

View on Pith 2004

[8] [7]

J. D. Baena, R. Marqu´ es, F. Medina, J. Martel, “Artiﬁcia l magnetic metamaterial design by using spiral resonators,” Phys. Rev. B , 69, 014402, Jan. 2004

View on Pith 2004

[9] [8]

S. I. Maslovski, P. Ikonen, I. A. Kolmakov, S. A. Tretyakov, M. Kaunisto, “Artiﬁcial magnetic materials based on the new magnetic par ticle: Metasolenoid,” Progress in Electromagnetics Research, vol. 54, pp. 61– 81, 2005

View on Pith 2005

[10] [9]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeabil ity and permittivity,” Phys. Rev. Lett. , vol. 84, no. 18, pp. 4184–4187, May 2000

View on Pith 2000

[11] [10]

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental ver iﬁcation of a negative index of refraction,” Science, vol. 292, pp. 77–79, April 2001

View on Pith 2001

[12] [11]

R. C. Hansen and M. Burke, “Antenna with magneto-dielec trics,” Microwave Opt. Technol. Lett. , vol. 26, no. 2., pp. 75–78, July 2000

View on Pith 2000

[13] [12]

S. Yoon and R. W. Ziolkowski, “Bandwidth of a microstrip patch antenna on a magneto-dielectric substrate,” IEEE Antennas Propa- gat. Soc. Int. Symposium , Columbus, Ohio, pp. 297–300, June 22-27, 2003

View on Pith 2003

[14] [13]

H. Mossallaei and K. Sarabandi “Magneto-dielectrics i n electromagnet- ics: Concept and applications,”IEEE Trans. Antennas Propagat., vol. 52, no. 6, pp. 1558–1567, June 2004

View on Pith 2004

[15] [14]

M. K. K¨ arkk¨ ainen, S. A. Tretyakov, P. Ikonen, “PIFA wi th dispersive material ﬁllings,” Microwave Opt. Technol. Lett., vol. 45, no. 1, pp. 5–8, April 2005

View on Pith 2005

[16] [15]

M. E. Ermutlu, C. R. Simovski, M. K. K¨ arkk¨ ainen, P. Iko nen, S. A. Tretyakov, A. A. Sochava, “Miniaturization of patch antennas with new artiﬁcial magnetic layers,” 2005 IEEE Int. Workshop on Antenna Technology, Singapore, pp. 87–90, March 7–9, 2005

View on Pith 2005

[17] [16]

K. Buell, H. Mosallaei, K. Sarabandi, “A substrate for s mall patch anten- nas providing tunable miniaturization factors,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 1., pp. 135–145, Jan. 2006

View on Pith 2006

[18] [17]

P. Ikonen, S. Maslovski, C. Simovski, S. Tretyakov, “On artiﬁcial magneto-dielectric loading for improving the impedance ba ndwidth properties of microstrip antennas,” arXiv:physics/05091 39. (To appear in IEEE Trans. Antennas Propagat. , June 2006.) 7

View on Pith 2006

[19] [18]

R. Ruppin, “Electromagnetic energy density in a disper sive and absorp- tive material,” Phys. Lett. A , vol. 299, pp. 309–312, July 2002

View on Pith 2002

[20] [19]

T. J. Cui and J. A. Kong, “Time-domain electromagnetic e nergy in a frequency-dispersive left-handed medium,” Phys. Rev. B , vol. 70, 205106, Nov. 2004

View on Pith 2004

[21] [20]

S. A. Tretyakov, “Electromagnetic ﬁeld energy density in artiﬁcial microwave materials with strong dispersion and loss,” Phys. Lett. A , vol. 343, pp. 231–237, June 2005

View on Pith 2005

[22] [22]

L. D. Landau, E. M. Lifshits, Electrodynamics of continuous media, 2nd ed., Pergamon: Oxford, England, 1984

View on Pith 1984

[23] [23]

L. A. Vainstein, Electromagnetic waves, 2nd ed., Radio i Sviaz: Moscow, 1988 (in Russian)

View on Pith 1988

[24] [24]

J. D. Jackson, Classical electrodynamics, 3rd ed., John Wiley & Sons: New York, 1999

View on Pith 1999

[25] [25]

A. Sihvola, Electromagnetic mixing rules and formulas , IEE Electro- magnetic Waves Series: London, 1999

View on Pith 1999