Optical precursors in the singular and weak dispersion limits

2015

Phys. Rev. A

We theoretically study the formation of Brillouin precursors in Debye media. We point out that the precursors are visible only at propagation distances such that the impulse response of the medium is essentially determined by the frequency dependence of its absorption and is practically Gaussian. By simple convolution, we then obtain explicit analytical expressions of the transmitted waves generated by reference incident waves, distinguishing precursor and main signal by a simple examination of the long-time behavior of the overall signal. These expressions are in good agreement with the signals obtained in numerical or real experiments performed on water in the radio-frequency domain and explain in particular some observed shapes of the precursor. Results are obtained for other remarkable incident waves. In addition, we show quite generally that the shape of the Brillouin precursor appearing alone at sufficiently large propagation distance and the law giving its amplitude as a function of this distance do not depend on the precise form of the incident wave but only on its integral properties. The incidence of a static conductivity of the medium is also examined and explicit analytical results are again given in the limit of weak and strong conductivities.

Section: Introductionmentioning

confidence: 99%

Brillouin precursors in Debye media

2015

Phys. Rev. A

“…An abundant bibliography on forerunners can be found in [16]. For most recent studies related to the forerunners in the sense of Sommerfeld and Brillouin, see [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

From Sommerfeld and Brillouin forerunners to optical precursors

2013

Phys. Rev. A

The Sommerfeld and Brillouin forerunners generated in a single-resonance absorbing medium by an incident step-modulated pulse are theoretically considered in the double limit where the susceptibility of the medium is weak and the resonance is narrow. Combining direct LaplaceFourier integration and calculations by the saddle-point method, we establish an explicit analytical expression of the transmitted field valid at any time, even when the two forerunners significantly overlap. We examine how their complete overlapping, occurring for shorter propagation distances, originates the formation of the unique transient currently named resonant precursor or dynamical beat. We obtain an expression of this transient identical to that usually derived within the slowly varying envelope in spite of the initial discontinuity of the incident field envelope. The dynamical beats and 0π pulses generated by ultra-short incident pulses are also briefly examined.

“…OCIS codes: 260.2030, 320.5550, 320.2250. In a recent paper [1], Oughstun et al revisit the classical problem of the propagation of a step modulated pulse in a Lorentz model medium. They specifically consider the case where the absorption line is narrow (singular limit) and the one where the refractive index of the medium keeps very close to unity at every frequency (weak dispersion limit).…”

mentioning

confidence: 99%

“…Although it abundantly refers to the theoretical results obtained by the asymptotic method, the study reported in [1] is mainly numerical. All the simulations are made for ω 0 = 3.9 × 10 14 rad/s [2] and ω c = 3.0 × 10 14 rad/s in a normal dispersion region.…”

mentioning

confidence: 99%

“…In their study of the weak dispersion limit, Oughstun et al [1] mention a "curious difficulty in the numerical FFT simulation of pulse propagation" and, in order to overcome it, they develop at great length an equivalence relation. In fact the difficulty is completely avoided in the retarded-time picture and their equivalence relation then appears as an immediate result.…”

mentioning

confidence: 99%

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Comment on “Optical precursors in the singular and weak dispersion limits”

2011

J. Opt. Soc. Am. B

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 a Airy function. In the weak dispersion limit, the equivalence relation, established at great length in Section 3 of the 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.OCIS codes: 260.2030, 320.5550, 320.2250.PACS numbers: 42.25. Bs, 42.50.Md, 41.20.Jb In a recent paper [1], Oughstun et al. revisit the classical problem of the propagation of a step modulated pulse in a Lorentz model medium. They specifically consider the case where the absorption line is narrow (singular limit) and the one where the refractive index of the medium keeps very close to unity at every frequency (weak dispersion limit). The medium is characterized by its complex refractive indexHere ω, ω p , ω 0 and δ respectively designate the current optical frequency, the plasma frequency, the resonance frequency and the damping or relaxation rate. The wave propagates in the z-direction. In the following we use the retarded time t, equal to the real time minus z/c where c is the velocity of light in vacuum (retarded time picture). The medium is then characterized by the transfer functionand the field transmitted E(z, t) at the abscissa z reads asHere a is a positive constant and E(0, ω) is the Fourier transform of the incident field E(0, t). In [1], the latter is assumed to have the idealized formwhere Θ(t) is the unit step function and ω c is the frequency of the optical carrier.Although it abundantly refers to the theoretical results obtained by the asymptotic method, the study reported in [1] is mainly numerical. All the simulations are made for ω 0 = 3.9 × 10 14 rad/s [2] and ω c = 3.0 × 10 14 rad/s in a normal dispersion region. The singular and weak dispersion limits are respectively attained when δ ≪ ω 0 and ω 2 p ≪ δω 0 [see Eq. (1)]. Oughstun et al. emphasize that these two limiting cases "are fundamentally different in their effects upon propagation" but, surprisingly enough, they take for their simulations in the weak dispersion limit a value of δ for which the singular limit nearly holds (δ < ω 0 /100). Consequently, mutatis mutandis, the results obtained in the two limits appears qualitatively similar. The steep rise of the Brillouin precursor obtained in the singular limit (their Fig.3) and analyzed as a distinguishi...