Abstract

The velocity of propagation of electromagnetic energy for a monochromatic plane-wave field in a causally dispersive dielectric, medium with absorption as described by the Lorentz model, is considered within the framework of Poynting’s theorem. A general, rigorous expression for the energy-transport velocity in a Lorentz medium with multiple-resonance frequencies is derived. From this rigorous result, an approximate expression for the energy velocity is obtained that is in a form that is independent of the medium model and so is likely to be applicable to general dispersive media.

© 1988 Optical Society of America

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References

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  1. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  2. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. 3, 233–245 (1970).
  3. S. Shen, Dispersive Pulse Propagation in a Multiple Resonance Lorentz Medium, M.S. thesis (University of Wisconsin–Madison, Madison Wisc., 1986).
  4. H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952).
  5. L. Rosenfeld, Theory of Electrons (Dover, New York, 1965).
  6. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 7.
  7. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sec. 1.1.4.

1970 (1)

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sec. 1.1.4.

Brillouin, L.

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 7.

Lorentz, H. A.

H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952).

Loudon, R.

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

Rosenfeld, L.

L. Rosenfeld, Theory of Electrons (Dover, New York, 1965).

Shen, S.

S. Shen, Dispersive Pulse Propagation in a Multiple Resonance Lorentz Medium, M.S. thesis (University of Wisconsin–Madison, Madison Wisc., 1986).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sec. 1.1.4.

J. Phys. (1)

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

Other (6)

S. Shen, Dispersive Pulse Propagation in a Multiple Resonance Lorentz Medium, M.S. thesis (University of Wisconsin–Madison, Madison Wisc., 1986).

H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952).

L. Rosenfeld, Theory of Electrons (Dover, New York, 1965).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 7.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Sec. 1.1.4.

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

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Figures (2)

Fig. 1
Fig. 1

Frequency dependence of the normalized energy-transport velocity vE/c (solid curve) and the normalized ratio WK/WE of the time-average kinetic energy density of the Lorentz oscillators with respect to the time-average electric field energy density (dotted curve) in a double-resonance Lorentz medium: ω0 = 1 × 1016/sec, b02 = 5 × 1032/sec2, δ0 = 0.1 × 1016/sec, ω2 =4 × 1016/sec, b22 = 20 ×1032/sec2, and δ2 = 0.28 × 1016/sec.

Fig. 2
Fig. 2

Frequency dependence of the normalized energy-transport velocity vE/c (solid curve) and the normalized ratio WK/WE of the time-average kinetic energy density of the Lorentz oscillators with respect to the time-average electric field energy density (dotted curve) in a double-resonance Lorentz medium. Parameters are the same as for Fig. 1 except that ω2 = 7 × 1016/sec.

Equations (36)

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m ( d 2 r j d t 2 + 2 δ j d r j d t + ω j 2 r j ) = - e E loc ,
r j = e / m ω 2 - ω j 2 + 2 i δ j ω E loc ,
p j = - e r j = - e 2 / m ω 2 - ω j 2 + 2 i δ j ω E loc .
P = j N j p j = E loc j N j α j ( ω ) ,
α j ( ω ) = - e 2 / m ω 2 - ω j 2 + 2 i δ j ω
N = j N j
P ( ω ) = χ e ( ω ) E ( ω ) ,
χ e ( ω ) = j N j α j ( ω ) = - j N j e 2 / m ω 2 - ω j 2 + 2 i δ j ω .
( ω ) = 1 + 4 π χ e ( ω ) = 1 - j b j 2 ω 2 - ω j 2 + 2 i δ j ω ,
b j 2 = 4 π N j e 2 m
n ( ω ) = [ ( ω ) ] 1 / 2 = ( 1 - j b j 2 ω 2 - ω j 2 + 2 i δ j ω ) 1 / 2 .
· S = - 1 4 π ( H · H t + E · E t + 4 π E · P t )
E = - m e ( d 2 r j d t 2 + 2 δ j d r j d t + ω j 2 r j )
P = - N 0 e r 0 - N 2 e r 2 .
E · P t = N 0 m ( d 2 r 0 d t 2 + 2 δ 0 d r 0 d t + ω 0 2 r 0 ) · d r 0 d t + N 2 m ( d 2 r 2 d t 2 + 2 δ 2 d r 2 d t + ω 2 2 r 2 ) · d r 2 d t = N 0 m [ 1 2 d d t ( d r 0 d t ) 2 + 2 δ 0 ( d r 0 d t ) 2 + ω 0 2 2 d d t ( r 0 ) 2 ] + N 2 m [ 1 2 d d t ( d r 2 d t ) 2 + 2 δ 2 ( d r 2 d t ) 2 + ω 2 2 2 d d t ( r 2 ) 2 ] .
W osc = 1 4 N 0 m [ ( d r 0 d t ) 2 + ω 0 2 ( r 0 ) 2 ] + 1 4 N 2 m [ ( d r 2 d t ) 2 + ω 2 2 ( r 2 ) 2 ] .
W osc = 1 4 N 0 e 2 m E 2 ω 2 + ω 0 2 ( ω 2 - ω 0 2 ) 2 + 4 δ 0 2 ω 2 + 1 4 N 2 e 2 m E 2 ω 2 + ω 2 2 ( ω 2 - ω 2 2 ) 2 + 4 δ 2 2 ω 2 .
W osc = 1 16 π × E 2 [ b 0 2 ( ω 2 + ω 0 2 ) ( ω 2 - ω 0 2 ) 2 + 4 δ 0 2 ω 2 + b 2 2 ( ω 2 + ω 2 2 ) ( ω 2 - ω 2 2 ) 2 + 4 δ 2 2 ω 2 ] .
W osc = 1 16 π E 2 j b j 2 ( ω 2 + ω j 2 ) ( ω 2 - ω j 2 ) 2 + 4 δ j 2 ω 2 .
W field = 1 16 π ( n r 2 + n i 2 + 1 ) E 2 .
n r 2 - n i 2 = r ( ω ) = 1 - j b j 2 ( ω 2 - ω j 2 ) ( ω 2 - ω j 2 ) 2 + 4 δ j 2 ω 2 ,
W total = W osc + W field = 1 8 π E 2 [ n r 2 + j b j 2 ω 2 ( ω 2 - ω j 2 ) 2 + 4 δ j 2 ω 2 ] ,
v E = S W total .
S = c 8 π n r E 2 ,
v E = c n r ( ω ) + 1 n r ( ω ) j b j 2 ω 2 ( ω 2 - ω j 2 ) 2 + 4 δ j 2 ω 2 .
W E = 1 16 π E 2
W H = 1 16 π n 2 E 2 = 1 16 π ( n r 2 + n i 2 ) E 2
W P = 1 16 π E 2 j b j 2 ω j 2 ( ω 2 - ω j 2 ) 2 + 4 δ j 2 ω 2 ,
W K = 1 16 π E 2 j b j 2 ω 2 ( ω 2 - ω j 2 ) 2 + 4 δ j 2 ω 2 .
v E = c n r ( ω ) + 16 π n r ( ω ) W K E 2 = c n r ( ω ) + 1 n r ( ω ) W K W E ,
W K W E = b 0 2 ω 2 ( ω 2 - ω 0 2 ) 2 + 4 δ 0 2 ω 2 ,
W K W E = ( r - 0 ) ( r - ) + i 2 0 - .
( ω ) = 1 - b 0 2 ω 2 - ω 0 2 + 2 i δ 0 ω ,
0 ( 0 ) ,
( )
v E = c n r ( ω ) + [ r ( ω ) - 0 ] [ r ( ω ) - ] + i 2 ( ω ) n r ( ω ) ( 0 - ) .

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