Abstract

We point out inconsistencies in the recent paper by Oughstun et al. on Sommerfeld and Brillouin precursors [J. Opt. Soc. Am. B 27, 1664–1670 (2010)]. Their study is essentially numerical and, for the parameters used in their simulations, the difference between the two limits considered is not as clear cut as they state. The steep rise of the Brillouin precursor obtained in the singular limit and analyzed as a distinguishing feature of this limit simply results from an unsuitable time scale. In fact, the rise of the precursor is progressive and is perfectly described by an Airy function. In the weak dispersion limit, the equivalence relation, established at great length in Section 3 of that paper, appears as an immediate result in the retarded-time picture. Last but not least, we show that, contrary to the authors’ claim, the precursors are catastrophically affected by the rise time of the incident optical field, even when the latter is considerably faster than the medium relaxation time.

© 2011 Optical Society of America

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References

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  1. K. E. Oughstun, N. A. Cartwright, D. J. Gauthier, and H. Jeong, “Optical precursor in the singular and weak dispersion limits,” J. Opt. Soc. Am. B 27, 1664–1670 (2010).
    [CrossRef]
  2. Contrary to what is stated in , this (circular) frequency ω0 is not equal to the frequency of the resonance involved in the experiment on cold potassium atoms, but is 2π times smaller.
  3. L. Brillouin, “Propagation des ondes électromagnétiques dans les milieux matériels,” in Comptes Rendus du Congrès International d’Electricité (Gauthier-Villars, 1933), Vol.  2, pp 739–788. An English adaptation of this paper can be found in L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960), Ch. IV and Ch. V.
  4. Note that the Airy function A(x) used by Brillouin in differs from the standard one Ai(x) used in the present comment, with 2πAi(−x)=31/3A(31/3x). Peak amplitudes of A(x) and Ai(−x) are, respectively, 2.33 and 0.536.

2010 (1)

1933 (1)

L. Brillouin, “Propagation des ondes électromagnétiques dans les milieux matériels,” in Comptes Rendus du Congrès International d’Electricité (Gauthier-Villars, 1933), Vol.  2, pp 739–788. An English adaptation of this paper can be found in L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960), Ch. IV and Ch. V.

Brillouin, L.

L. Brillouin, “Propagation des ondes électromagnétiques dans les milieux matériels,” in Comptes Rendus du Congrès International d’Electricité (Gauthier-Villars, 1933), Vol.  2, pp 739–788. An English adaptation of this paper can be found in L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960), Ch. IV and Ch. V.

L. Brillouin, “Propagation des ondes électromagnétiques dans les milieux matériels,” in Comptes Rendus du Congrès International d’Electricité (Gauthier-Villars, 1933), Vol.  2, pp 739–788. An English adaptation of this paper can be found in L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960), Ch. IV and Ch. V.

Cartwright, N. A.

Gauthier, D. J.

Jeong, H.

Oughstun, K. E.

J. Opt. Soc. Am. B (1)

Other (3)

Contrary to what is stated in , this (circular) frequency ω0 is not equal to the frequency of the resonance involved in the experiment on cold potassium atoms, but is 2π times smaller.

L. Brillouin, “Propagation des ondes électromagnétiques dans les milieux matériels,” in Comptes Rendus du Congrès International d’Electricité (Gauthier-Villars, 1933), Vol.  2, pp 739–788. An English adaptation of this paper can be found in L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960), Ch. IV and Ch. V.

Note that the Airy function A(x) used by Brillouin in differs from the standard one Ai(x) used in the present comment, with 2πAi(−x)=31/3A(31/3x). Peak amplitudes of A(x) and Ai(−x) are, respectively, 2.33 and 0.536.

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Figures (3)

Fig. 1
Fig. 1

Brillouin precursor obtained for the parameters of Fig. 3 of [1], that is, for ω 0 = 3.9 × 10 14 rad / s , ω c = 3.0 × 10 14 rad / s , ω p = 3.05 × 10 14 rad / s , δ = 3.02 × 10 10 rad / s , and z = 7.232 × 10 2 m . The solid curve is the exact numerical solution obtained by fast Fourier transform and the dashed curve is the approximate analytical solution given by Eq. (7).

Fig. 2
Fig. 2

Same as Fig. 1 for the parameters of Fig. 5 of [1], that is, for ω 0 = 3.9 × 10 14 rad / s , ω c = 3.0 × 10 14 rad / s , ω p = 3.05 × 10 12 rad / s , δ = 3.02 × 10 12 rad / s , and z = 2.290 m . The rapid oscillations of small amplitude superposed upon the beginning of the Brillouin precursor are the end of the Sommerfeld precursor.

Fig. 3
Fig. 3

Effect of the rise time of the incident field on the transmi tted field. The parameters are those of Fig. 3 of [1], that is, ω 0 = 3.9 × 10 14 rad / s , ω c = 3.0 × 10 14 rad / s , ω p = 3.05 × 10 14 rad / s , δ = 3.02 × 10 10 rad / s , and z = 7.232 × 10 - 2 m . The vertical dashed line indicates the group delay τ g = d φ d ω | ω = ω c , where φ is the argument of H ( ω ) ( τ g = 471 ps ). τ g fixes the arrival of the “main field,” whose amplitude, equal to exp ( 10 ) 4.5 × 10 5 , obviously does not depend on the rise time. The different curves are obtained for (a)  T r = 0 , (b)  T r = τ / 500 , and (c)  T r = τ / 100 , where τ = 1 / δ is the relaxation or damping time of the medium.

Equations (8)

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n ( ω ) = ( 1 ω p 2 ω 2 ω 0 2 + 2 i δ ω ) 1 / 2 .
H ( ω ) = exp { i ω c z [ n ( ω ) 1 ] } ,
E ( z , t ) = 1 2 π + i a + + i a H ( ω ) E ˜ ( 0 , ω ) exp ( i ω t ) d ω .
E ( 0 , t ) = Θ ( t ) sin ( ω c t ) ,
n ( ω ) ( 1 + ω p 2 ω 0 2 ) 1 / 2 + ω 2 ω p 2 2 ω 0 3 ( ω 0 2 + ω p 2 ) 1 / 2
b = ω 0 [ 2 c ( ω 0 2 + ω p 2 ) 1 / 2 3 z ω p 2 ] 1 / 3 .
E ( z , t ) = 1 2 π ω c + exp ( i ω 3 3 b 3 i ω t ) d ω = b ω c Ai ( b t ) ,
H ( ω ) exp [ i ω 2 c ( ω p 2 z ω 2 ω 0 2 + 2 i δ ω ) ] .

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