Abstract

We derive the instantaneous densities of electric energy and electric power dissipation in lossless and lossy dispersive media for a time-harmonic electric field. The instantaneous quantities are decomposed into DC and AC components, some of which are shown to be independent of the dispersion of dielectric constants. The AC component of the instantaneous energy density can be used to visualize propagation of electromagnetic waves through complex 3D structures.

© 2012 Optical Society of America

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References

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  1. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
  2. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A: Gen. Phys. 3, 233–245 (1970).
    [CrossRef]
  3. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth–Heinemann, 1984), 2nd ed. Section 80.
  4. V. Polevoi, “Maximum energy extractable from an electromanetic field,” Radiophys. Quantum Electron. 33, 603–609 (1990).
    [CrossRef]
  5. R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic metamaterial,” Phys. Rev. E 63, 046604 (2001).
    [CrossRef]
  6. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
    [CrossRef]
  7. T. Cui and J. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B 70, 205106 (2004).
  8. S. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005).
    [CrossRef]
  9. P.-G. Luan, “Power loss and electromagnetic energy density in a dispersive metamaterial medium,” Phys. Rev. E 80, 046601 (2009).
    [CrossRef]
  10. K. Webb and Shivanand, “Electromagnetic field energy in dispersive materials,” J. Opt. Soc. Am. B 27, 1215–1220(2010).
    [CrossRef]
  11. F. D. Nunes, T. C. Vasconcelos, M. Bezerra, and J. Weiner, “Electromagnetic energy density in dispersive and dissipative media,” J. Opt. Soc. Am. B 28, 1544–1552 (2011).
    [CrossRef]
  12. The derivation given in this paper for electrically dispersive media can be easily extended to magnetically dispersive media, which is described by a frequency-dependent magnetic permeability μ(ω)=μ′(ω)−iμ′′(ω).
  13. A. Raman and S. Fan, “Photonic band structure of dispersive metamaterials formulated as a Hermitian eigenvalue problem,” Phys. Rev. Lett.087401 (2010).
  14. J. D. Jackson, Classical Electrodynamics (Wiley, 1999), 3rd ed. Section 7.10.
  15. A. Sommerfeld, Mechanics (Academic, 1956). Section III.19.
  16. J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems (Saunders College, 1995), 4th ed. Section 3.6.
  17. A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices.” Appl. Opt. 37, 5271–5283 (1998).
    [CrossRef]
  18. W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mat. 22, 5120–5124 (2010).
    [CrossRef]
  19. G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” J. Lightwave Technol. 25, 2511–2521 (2007).
  20. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  21. E. D. Palik, ed., Handbook of Optical Constants of Solids(Academic, 1985).

2011 (1)

2010 (3)

K. Webb and Shivanand, “Electromagnetic field energy in dispersive materials,” J. Opt. Soc. Am. B 27, 1215–1220(2010).
[CrossRef]

A. Raman and S. Fan, “Photonic band structure of dispersive metamaterials formulated as a Hermitian eigenvalue problem,” Phys. Rev. Lett.087401 (2010).

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mat. 22, 5120–5124 (2010).
[CrossRef]

2009 (1)

P.-G. Luan, “Power loss and electromagnetic energy density in a dispersive metamaterial medium,” Phys. Rev. E 80, 046601 (2009).
[CrossRef]

2007 (1)

2005 (1)

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

2004 (1)

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

2002 (1)

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

2001 (1)

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

1998 (1)

1990 (1)

V. Polevoi, “Maximum energy extractable from an electromanetic field,” Radiophys. Quantum Electron. 33, 603–609 (1990).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1970 (1)

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

Bezerra, M.

Brillouin, L.

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

Brongersma, M. L.

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mat. 22, 5120–5124 (2010).
[CrossRef]

Cai, W.

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mat. 22, 5120–5124 (2010).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Cui, T.

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

Djurišic, A. B.

Elazar, J. M.

Fan, S.

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mat. 22, 5120–5124 (2010).
[CrossRef]

A. Raman and S. Fan, “Photonic band structure of dispersive metamaterials formulated as a Hermitian eigenvalue problem,” Phys. Rev. Lett.087401 (2010).

G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” J. Lightwave Technol. 25, 2511–2521 (2007).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999), 3rd ed. Section 7.10.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Kong, J.

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

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth–Heinemann, 1984), 2nd ed. Section 80.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth–Heinemann, 1984), 2nd ed. Section 80.

Loudon, R.

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

Luan, P.-G.

P.-G. Luan, “Power loss and electromagnetic energy density in a dispersive metamaterial medium,” Phys. Rev. E 80, 046601 (2009).
[CrossRef]

Majewski, M. L.

Marion, J. B.

J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems (Saunders College, 1995), 4th ed. Section 3.6.

Nunes, F. D.

Palik, E. D.

E. D. Palik, ed., Handbook of Optical Constants of Solids(Academic, 1985).

Polevoi, V.

V. Polevoi, “Maximum energy extractable from an electromanetic field,” Radiophys. Quantum Electron. 33, 603–609 (1990).
[CrossRef]

Rakic, A. D.

Raman, A.

A. Raman and S. Fan, “Photonic band structure of dispersive metamaterials formulated as a Hermitian eigenvalue problem,” Phys. Rev. Lett.087401 (2010).

Ruppin, R.

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

Shin, W.

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mat. 22, 5120–5124 (2010).
[CrossRef]

Shivanand,

Sommerfeld, A.

A. Sommerfeld, Mechanics (Academic, 1956). Section III.19.

Thornton, S. T.

J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems (Saunders College, 1995), 4th ed. Section 3.6.

Tretyakov, S.

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

Vasconcelos, T. C.

Veronis, G.

Webb, K.

Weiner, J.

Ziolkowski, R. W.

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

Adv. Mat. (1)

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mat. 22, 5120–5124 (2010).
[CrossRef]

Appl. Opt. (1)

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (2)

J. Phys. A: Gen. Phys. (1)

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

Phys. Lett. A (2)

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

S. 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 (2)

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

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. E (1)

P.-G. Luan, “Power loss and electromagnetic energy density in a dispersive metamaterial medium,” Phys. Rev. E 80, 046601 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

A. Raman and S. Fan, “Photonic band structure of dispersive metamaterials formulated as a Hermitian eigenvalue problem,” Phys. Rev. Lett.087401 (2010).

Radiophys. Quantum Electron. (1)

V. Polevoi, “Maximum energy extractable from an electromanetic field,” Radiophys. Quantum Electron. 33, 603–609 (1990).
[CrossRef]

Other (7)

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

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth–Heinemann, 1984), 2nd ed. Section 80.

The derivation given in this paper for electrically dispersive media can be easily extended to magnetically dispersive media, which is described by a frequency-dependent magnetic permeability μ(ω)=μ′(ω)−iμ′′(ω).

J. D. Jackson, Classical Electrodynamics (Wiley, 1999), 3rd ed. Section 7.10.

A. Sommerfeld, Mechanics (Academic, 1956). Section III.19.

J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems (Saunders College, 1995), 4th ed. Section 3.6.

E. D. Palik, ed., Handbook of Optical Constants of Solids(Academic, 1985).

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Figures (3)

Fig. 1.
Fig. 1.

Plots of u~e(t), q~e(t) and Re{E(t)2} for time-harmonic electric fields E(t)=x^E0eiωt in silver (Ag). The quantities are plotted for two vacuum wavelengths: (a) λ0=620nm and (b) λ0=1550nm. Note that u~e(t) is nearly 180° out-phase with Re{E(t)2} for both cases. Also notice that q~e(t) is out-of-phase with u~e(t) in (a) whereas it is nearly in-phase with u~e(t) in (b). The units of u~e(t), q~e(t), and Re{E(t)2} in the vertical axes are ε0E02, ωε0E02, and E02, respectively.

Fig 2.
Fig 2.

Metallic slot waveguide bend composed of two silver (Ag) films immersed in silica (SiO2). The two red arrows indicate the directions of the energy flow inside the slot region before and after the bend. The dashed blue arrow indicates the energy leakage into a mode bound to the metal film. The blue arcs indicate the radiation into the background silica. The dominant H-field component is shown for each energy flow channel. Note that the leakage channels have the dominant H-field polarized in different directions than the slot waveguide channel. The vacuum wavelength and relevant dimensions of the structure are indicated in the figure. At the specified vacuum wavelength, the dielectric constants of Ag [20] and SiO2 [21] are εAg=(129i3.28)ε0 and εSiO2=2.085ε0, respectively. The magnetic permeabilities of both materials are μ0.

Fig 3.
Fig 3.

Visualization of wave propagation through the metallic slot waveguide bend. The fields here are obtained by solving the frequency-domain Maxwell’s equations using the finite-difference frequency-domain method. The quantities (a) Hy, (b) time-averaged energy density u¯=u¯e+u¯m, and (c) AC component of instantaneous energy density u~=u~e+u~m of the solution of the frequency-domain Maxwell’s equations are plotted on two planes: a horizontal y=(const.) plane on top of the metal film and a vertical x=(const.) plane containing the central axis of the input port. Red, green, and blue indicate positive, zero, and negative, respectively. The area enclosed by the white dashed line in each figure is where the coupling of the EM energy into the surface plasmon mode is supposed to be observed. Note that (a) fails to capture such a coupling; (b) captures the coupling but loses the phase information; (c) displays both the coupling and phase information properly. Also notice the weak but discernible pattern of spherical wavefronts in the vertical plane in (c). The thin orange lines around the y=0 plane outline the two metal pieces.

Tables (1)

Tables Icon

Table 1. Dispersion Dependencies of the DC and AC Components of the Instantaneous Electric Energy Density ue(t) and Instantaneous Electric Power Dissipation Density qe(t) in Lossless and Lossy Dispersive Mediaa

Equations (41)

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u¯e=14d(ωε(ω))dω|E0|2.
q¯e=12ωε(ω)|E0|2,
uet=E·Dt=14(E·Dt+E*·Dt)+c.c.
Dt=12πi(ω+α)ε(ω+α)E0,αei(ω+α)tdα.
Dt[iωε(ω)E0(t)+d(ωε(ω))dωE0(t)t]eiωt.
uet=14[(iωεE0(t)2+d(ωε)dωE0(t)·E0(t)t)ei2ωt+(iωε|E0(t)|2+d(ωε)dωE0(t)*·E0(t)t)]+c.c.
uet=14[2ωεRe{iE0(t)2ei2ωt}+d(ωε)dωRe{2E0(t)·E0(t)tei2ωt}+d(ωε)dωt|E0(t)|2].
uet=12ω2dεdωRe{iE(t)2}+12d(ωε)dωtE(t)2.
ue(t)=12d(ωε)dωE(t)212ω2dεdωRe{iE(t)2dt}+C,
ue(t)=12d(ωε)dωE(t)214ωdεdωRe{E(t)2}+C=14d(ωε)dω|E0|2+14εRe{E(t)2}+C,
W2=14(W+W*)2=12|W|2+12Re{W2}
ue(t)=u¯e+u˜e(t),
u˜e(t)=14εRe{E(t)2}forε=0.
S(E×H)·da=V(E·Dt+H·Bt)dv
-S(E×H)·da=V(uet+umt)dv+V(qe+qm)dv,
ε(ω)=ε(1+i=1Nωp,i2ω0,i2ω2+iωΓi).
md2ridt2=mΓidridtmω0,i2rieE,
2Pit2+ΓiPit+ω0,i2Pi=ωp,i2εE,
-ω2Pi+iωΓiPi+ω0,i2Pi=ωp,i2εE,
Vit+ΓiVi+ω0,i2Pi=εωp,i2E.
E·Dt=i=1N1εωp,i2ΓiVi2+t[12εE2+i=1Nω0,i22εωp,i2Pi2+i=1N12εωp,i2Vi2].
qe(t)=i=1N1εωp,i2ΓiVi2,
ue(t)=12εE2+i=1Nω0,i22εωp,i2Pi2+i=1N12εωp,i2Vi2,
Pi(t)=εωp,i2ω0,i2ω2+iωΓiE(t),
Vi(t)=Pi(t)t=εiωωp,i2ω0,i2ω2+iωΓiE(t).
ue(t)=ue+u~e(t)
ue=14ε(1+i=1Nωp,i2(ω0,i2+ω2)(ω0,i2ω2)2+ω2Γi2)|E0|2
u~e(t)=14εRe{(1+i=1Nωp,i2(ω0,i2ω2)(ω0,i2ω2+iωΓi)2)E(t)2}.
-ueu~e(t)ue.
0ue(t)2ue,
qe(t)=qe+q~e(t)
qe=12εi=1Nωp,i2ω2Γi(ω0,i2ω2)2+ω2Γi2|E0|2
q~e(t)=12εRe{i=1Nωp,i2ω2Γi(ω0,i2ω2+iωΓi)2E(t)2}.
q~e(t+π/4ω)2ω=u~e(t)14Re{εE(t)2},
u~e(t)(Equation (13))14Re{εE(t)2}for|ε/ε|1,
ε(ω)=ε(1+ωp2ω02ω2+iωΓ).
u~e(t)=14Re{ε(1+ωp2(ω02ω2)(ω02ω2+iωΓ)2)E(t)2}.
ε(1+ωp2(ω02ω2)(ω02ω2+iωΓ)2)=ε(1+χe(ω)2Re{1χe(ω)}).
ε(1+ωp2(ω02ω2)(ω02ω2+iωΓ)2)=ε+(ε(ω)ε)2Re{1ε(ω)ε}.
u~e(t)=14εRe{E(t)2}+14Re{1ε(ω)ε}Re{(ε(ω)ε)2E(t)2}.
q~e(t)=12ωIm{(ε(ω)ε)E(t)2}+12ωRe{1ε(ω)ε}Im{(ε(ω)ε)2E(t)2}.

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