Abstract

The propagation of electromagnetic pulses in a linear, dispersive Lorentz medium with two isolated resonance frequencies is presented within the context of the classical theory due to Sommerfeld and Brillouin. Particular attention is given to the modification of the dynamical field evolution when one proceeds from a single-resonance to a double-resonance medium. The appearance of a new field structure in the precursor field evolution is obtained, and the conditions for its evolution are described. The signal velocity for a pulsed field is obtained and compared with the energy-transport velocity in a double-resonance medium both when the conditions for the evolution of the modified precursor field structure are satisfied as well as when they are not satisfied and it is absent from the total propagated field structure.

© 1989 Optical Society of America

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References

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  1. A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
    [CrossRef]
  2. L. Brillouin, “Uber die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
    [CrossRef]
  3. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  4. K. E. Oughstun, “Propagation of optical pulses in dispersive media,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1978; University Microfilms, Ann Arbor, Mich.).
  5. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
    [CrossRef]
  6. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 5.18.
  7. G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
    [CrossRef]
  8. G. C. Sherman and K. E. Oughstun, “Physical model of pulse dynamics in a linear dispersive medium with absorption (the Lorentz medium),” to be submitted to J. Opt. Soc. Am. B.
  9. S. Shen, “Dispersive pulse propagation in a multiple resonance lorentz medium,” M.S. thesis (University of Wisconsin—Madison, Madison, Wisc., 1986).
  10. K. E. Oughstun and S. Shen, “The velocity of energy transport for a time-harmonic field in a multiple resonance Lorentz medium,” J. Opt. Soc. Am. B 5, 2395–2398 (1988).
    [CrossRef]

1988 (2)

1981 (1)

G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

1914 (2)

A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

L. Brillouin, “Uber die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

Brillouin, L.

L. Brillouin, “Uber die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

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

Oughstun, K. E.

K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
[CrossRef]

K. E. Oughstun and S. Shen, “The velocity of energy transport for a time-harmonic field in a multiple resonance Lorentz medium,” J. Opt. Soc. Am. B 5, 2395–2398 (1988).
[CrossRef]

G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

K. E. Oughstun, “Propagation of optical pulses in dispersive media,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1978; University Microfilms, Ann Arbor, Mich.).

G. C. Sherman and K. E. Oughstun, “Physical model of pulse dynamics in a linear dispersive medium with absorption (the Lorentz medium),” to be submitted to J. Opt. Soc. Am. B.

Shen, S.

K. E. Oughstun and S. Shen, “The velocity of energy transport for a time-harmonic field in a multiple resonance Lorentz medium,” J. Opt. Soc. Am. B 5, 2395–2398 (1988).
[CrossRef]

S. Shen, “Dispersive pulse propagation in a multiple resonance lorentz medium,” M.S. thesis (University of Wisconsin—Madison, Madison, Wisc., 1986).

Sherman, G. C.

K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
[CrossRef]

G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

G. C. Sherman and K. E. Oughstun, “Physical model of pulse dynamics in a linear dispersive medium with absorption (the Lorentz medium),” to be submitted to J. Opt. Soc. Am. B.

Sommerfeld, A.

A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 5.18.

Ann. Phys. (2)

A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

L. Brillouin, “Uber die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

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

Phys. Rev. Lett. (1)

G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

Other (5)

G. C. Sherman and K. E. Oughstun, “Physical model of pulse dynamics in a linear dispersive medium with absorption (the Lorentz medium),” to be submitted to J. Opt. Soc. Am. B.

S. Shen, “Dispersive pulse propagation in a multiple resonance lorentz medium,” M.S. thesis (University of Wisconsin—Madison, Madison, Wisc., 1986).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 5.18.

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

K. E. Oughstun, “Propagation of optical pulses in dispersive media,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1978; University Microfilms, Ann Arbor, Mich.).

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Figures (10)

Fig. 1
Fig. 1

Isotimic contours of both the real and imaginary parts of the complex phase function ϕ(ω, θ) in the right-hand half of the complex ω plane at θ = 1.6 for a double-resonance Lorentz medium. The isotimic contours of X(ω, θ) = Re{ϕ(ω, θ)}are parallel to the real axis at Re{ω} = 0 and at infinity, whereas the isotimic contours of Y(ω, θ) = Im{ϕ(ω, θ)} are parallel to the imaginary axis as |ω| goes to infinity.

Fig. 2
Fig. 2

Isotimic contours of X(ω, θ) = Re{ϕ(ω, θ)} at the saddle points of the complex phase function ϕ(ω, θ) in the right-hand half of the complex ω plane. (a) θ = 1.3, and the dominant saddle point is the distant saddle point SPd+. (b) θ = 1.6, and the upper middle saddle point SPm+ is dominant. (c) θ = 2.2, and the upper near saddle point SPn+ is dominant.

Fig. 3
Fig. 3

Dynamical evolution of the middle saddle points in the right-hand half of the complex ω plane as a function of θ. The medium parameters are the same as those used in Figs. 1 and 2.

Fig. 4
Fig. 4

The impulse response of a double-resonance Lorentz medium. (a) The middle saddle point is never the dominant saddle point. (b) The middle saddle point is the dominant saddle point for a finite θ interval. In both (a) and (b) the propagation distance is z = 48π × 10−6 cm.

Fig. 5
Fig. 5

Evolution of the precursor fields in a double-resonance Lorentz medium with propagation distance for an input delta-function pulse. The medium parameters are the same as those used in Fig. 4(b).

Fig. 6
Fig. 6

Approximate steepest-descent path segments at (a) θ = 1.6 and (b) θ = 2.2.

Fig. 7
Fig. 7

Propagated field structure due to an input unit-step-function-modulated signal with carrier frequency ωc = ωp at (a) z = 24π × 10−6 cm and (b) z = 48π × 10−6 cm. The horizontal dotted lines indicate the attenuated amplitude value for a monochromatic field of frequency ωc = ωp at each propagation distance.

Fig. 8
Fig. 8

Behavior of X[ωSP(θ), θ] = Re{ϕ[ωSP(θ), θ]} at the relevant saddle points as a function of θ. (a) The upper middle saddle point SPm is never dominant; (b) the upper middle saddle point is the dominant saddle point for θ ∈ (θsm, θmb).

Fig. 9
Fig. 9

Frequency dependence of the signal velocity (solid curves) and energy-transport velocity (dotted curves) for a double-resonance Lorentz medium. For the medium parameters chosen in (a) the upper middle saddle point is never dominant, whereas in (b) it is the dominant saddle point for a finite θ interval.

Fig. 10
Fig. 10

The frequency dependence of the energy-transport velocity ve for a double-resonance Lorentz medium and the predicted value of the frequency ωp at which ve attains its local maximum value in the frequency interval between the two resonance frequencies.

Equations (72)

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A ( z , t ) = ( 1 / 2 π ) C f ˜ ( ω ) exp [ ( z / c ) ϕ ( ω , θ ) ] d ω ,
f ˜ ( ω ) = - f ( t ) e i ω t d t
ϕ ( ω , θ ) = i ω [ n ( ω ) - θ ] ,
θ = c t / z
n ( ω ) = [ 1 - b 0 2 ω 2 - ω 0 2 + 2 i δ 0 ω - b 2 2 ω 2 - ω 2 2 + 2 i δ 2 ω ] 1 / 2 ,
ω ± ( 0 ) = ± ( ω 0 2 - δ 0 2 ) 1 / 2 - i δ 0 ,
ω ± ( 2 ) = ± ( ω 2 2 - δ 2 2 ) 1 / 2 - i δ 2
ω ± ( 1 ) ± ( ω 1 2 - b 0 2 b 2 2 ω 2 2 - ω 1 2 - δ 0 2 ) 1 / 2 - i δ 0 ,
ω ± ( 3 ) ± ( ω 3 2 + b 0 2 b 2 2 ω 3 2 - ω 0 2 - δ 2 2 ) 1 / 2 - i δ 2 ,
ω 1 2 = ω 0 2 + b 0 2 ,
ω 3 2 = ω 2 2 + b 2 2 ,
ω 0 = 1.0 × 10 16 / sec , ω 2 = 7.0 × 10 16 / sec , b 0 2 = 5.0 × 10 32 / sec 2 , b 2 2 = 20.0 × 10 32 / sec 2 , δ 0 = 0.1 × 10 16 / sec , δ 2 = 0.28 × 10 16 / sec .
ϕ ( - ω , θ ) = ϕ * ( ω * ) ,
n ( ω ) + ω n ( ω ) - θ = 0.
n ( ω ) 1 - b 0 2 2 ω ( ω + 2 i δ 0 ) - b 2 2 2 ω ( ω + 2 i δ 2 ) ,
ω ( ω + 2 i δ 0 ) ( ω + 2 i δ 2 ) - b 0 2 2 ( θ - 1 ) ( ω + 2 i δ 2 ) - b 2 2 2 ( θ - 1 ) ( ω + 2 i δ 0 ) 0 ,
ω SP d ± ( θ ) ± [ b 0 2 + b 2 2 2 ( θ - 1 ) ] 1 / 2 - i ( δ 0 + δ 2 ) ,
n ( ω ) θ 0 [ 1 + 1 2 ( 1 ω 0 2 - 1 ω A 2 ) ( ω 2 + 2 i δ 0 ω ) ] ,
θ 0 = n ( 0 ) = ( 1 + b 0 2 ω 0 2 + b 2 2 ω 2 2 ) 1 / 2 ,
ω A 2 = ω 0 2 + b 0 2 ω 2 2 ω 0 2 .
ω 2 + 4 3 i δ 0 ω + 2 3 1 - θ / θ 0 1 ω 0 2 - 1 ω A 2 0 ,
ω SP n ± ( θ ) ± ( 2 3 θ / θ 0 - 1 1 ω 0 2 - 1 ω A 2 - 4 9 δ 0 2 ) 1 / 2 - 2 3 i δ 0 .
θ 1 θ 0 [ 1 + 2 3 δ 0 2 ( 1 ω 0 2 - 1 ω A 2 ) ] ,
ω = ½ ( ω 1 + ω 2 ) + ξ ,
n ( ξ ) ( 1 - b 0 2 / ω s ξ + ω α - b 2 2 / ω s ξ + ω β ) 1 / 2 ,
ω s = ω 1 + ω 2 ,
ω α = ¼ ( ω 1 + ω 2 ) - ω 0 2 / ( ω 1 + ω 2 ) + i δ 0 ,
ω β = ¼ ( ω 1 + ω 2 ) - ω 2 2 / ( ω 1 + ω 2 ) + i δ 2 .
ξ ω 1 + ω 2
ξ ω α ,
ξ ω β
n ( ξ ) [ 1 ω s ( b 0 2 ω α 2 + b 2 2 ω β 2 ) ξ + 1 - 1 ω s ( b 0 2 ω α + b 2 2 ω β ) ] 1 / 2 ,
n ( ω ) ( α ω + β ) 1 / 2 ,
α = 1 ω s ( b 0 2 ω α 2 + b 2 2 ω β 2 ) ,
β = 1 - 1 ω s ( b 0 2 ω α + b 2 2 ω β ) - 1 2 ( b 0 2 ω α 2 + b 2 2 ω β 2 ) .
( 9 / 4 ) α 2 ω 2 + α ( 3 β - θ 2 ) ω + β ( β - θ 2 ) 0.
ω SP m ± ( θ ) ( 2 / 9 α ) [ θ 2 - 3 β ± θ ( θ 2 + 3 β ) 1 / 2 ] .
f ( t ) = δ ( t ) ,
ω 0 = 1.0 × 10 16 / sec , ω 2 = 4.0 × 10 16 / sec , b 0 2 = 5.0 × 10 32 / sec 2 , b 2 2 = 20.0 × 10 32 / sec 2 , δ 0 = 0.1 × 10 16 / sec , δ 2 = 0.28 × 10 16 / sec .
X ( ω SP ) - X ( ω ) 1.5 × 10 16 / sec ,
f ( t ) = u ( t ) sin ( ω c t ) ,
f ˜ ( ω ) = 1 / ( ω - ω c ) ,
θ 0 = 2.56 , θ 1 2.58.
α ( ω ) = ( ω / c ) n i ( ω )
( ω 1 2 - b 0 2 b 2 2 ω 2 2 - ω 1 2 - δ 0 2 ) 1 / 2 < ω c < ( ω 2 2 - δ 2 2 ) 1 / 2 .
X ( ω SP j , θ c j ) = X ( ω c ) ,
v c j = c / θ c j ,
X ( ω SP n , θ c ) = X ( ω c ) ,             θ c θ 0
v c = c / θ c .
X ( ω SP d + , θ c 1 ) = X ( ω c ) ,             1 < θ c 1 < θ SB .
v c 1 = c / θ c 1 ) ,             ω c > θ SB
X ( ω SP n + , θ c 2 ) = X ( ω c ) ,             θ SB < θ c 2 < θ 0 ,
v c 2 = c / θ c 2 ,             ω c > ω SB
c > v c 1 > c / θ SB > v c 2 > c / θ 0 v c .
v c m = c / θ c m ,
X ( ω SP m + , θ c m ) = X ( ω c ) ,             θ s m < θ c m < θ m b ,
c > v c 1 > c / θ s m > v c m > c / θ m b > v c 2 > c / θ 0 > v c .
v E ( ω ) = c / θ E ( ω ) ,
θ E ( ω ) = n r ( ω ) + 1 n r ( ω ) × [ b 0 2 ω 2 ( ω 2 - ω 0 2 ) 2 + 4 δ 0 2 ω 2 + b 2 2 ω 2 ( ω 2 - ω 2 2 ) 2 + 4 δ 2 2 ω 2 ] ,
θ p < θ 0 .
d θ E / d ω = 0.
θ E ( ω ) = n r ( ω ) + [ 1 / n r ( ω ) ] g ( ω ) ,
g ( ω ) = b 0 2 ω 2 ( ω 2 - ω 0 2 ) 2 + 4 δ 0 2 ω 2 + b 2 2 ω 2 ( ω 2 - ω 2 2 ) 2 + 4 δ 2 2 ω 2 .
n 2 ( ω ) = 1 - b 0 2 ( ω 2 - ω 0 2 - 2 i δ 0 ω ) ( ω 2 - ω 0 2 ) 2 + 4 δ 0 2 ω 2 - b 2 2 ( ω 2 - ω 2 2 - 2 i δ 2 ω ) ( ω 2 - ω 2 2 ) 2 + 4 δ 2 2 ω 2 ,
n r 2 ( ω ) 1 - 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 ,
[ n r 2 ( ω ) - g ( ω ) ] d n r 2 d ω + 2 n r 2 ( ω ) ( d g / d ω ) = 0.
[ 1 + b 0 2 ω 0 2 ( ω 2 - ω 0 2 ) 2 + 4 δ 0 2 ω 2 + b 2 2 ω 2 2 ( ω 2 - ω 2 2 ) 2 + 4 δ 2 2 ω 2 ] × d d ω [ b 0 2 ω 2 ( ω 2 - ω 0 2 ) 2 + 4 δ 0 2 ω 2 + b 2 2 ω 2 ( ω 2 - ω 2 2 ) 2 + 4 δ 2 2 ω 2 ] + [ 1 - b 0 2 ( 2 ω 2 - ω 0 2 ) ( ω 2 - ω 0 2 ) 2 + 4 δ 0 2 ω 2 - b 2 2 ( 2 ω 2 - ω 2 2 ) ( ω 2 - ω 2 2 ) 2 + 4 δ 2 2 ω 2 ] × d d ω [ b 0 2 ω 0 2 ( ω 2 - ω 0 2 ) 2 + 4 δ 0 2 ω 2 + b 2 2 ω 2 2 ( ω 2 - ω 2 2 ) 2 + 4 δ 2 2 ω 2 ] = 0.
4 [ b 0 2 ω 0 2 ( ω 2 - ω 0 2 ) 3 + b 2 2 ω 2 2 ( ω 2 - ω 2 2 ) 3 ] + ( ω 0 2 - ω 2 2 ) × [ b 0 2 ( ω 2 - ω 0 2 ) 3 + b 2 2 ( ω 2 - ω 0 2 ) 3 ] = 0 ,
ω p { [ b 0 2 ( 5 ω 0 2 + ω 2 2 ) ] 1 / 3 ω 2 2 + [ b 2 2 ( ω 0 2 + 5 ω 2 2 ) ] 1 / 3 ω 0 2 [ b 0 2 ( 5 ω 0 2 + ω 2 2 ) ] 1 / 3 + [ b 2 2 ( ω 0 2 + 5 ω 2 2 ) ] 1 / 3 } 1 / 2 .
A ( z , t ) f ˜ ( ω E ) 2 π [ - z ϕ ( 2 ) ( ω E ) ] 1 / 2 exp { i [ k ( ω E ) z - ω E t ] } ,
k ( ω ) = ( ω / c ) n ( ω )
k i ( ω ) = i ( ω / c ) n i ( ω )

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