Optimal pulse penetration in Lorentz-Model dielectrics using the Sommerfeld and Brillouin precursors

Kurt Edmund Oughstun

doi:10.1364/OE.23.026604

1. Introduction

The dynamical evolution of an ultrashort optical pulse propagating through a causally dispersive dielectric is a classic problem [1–6] in electromagnetic wave theory with application in imaging and remote sensing. The frequency dependent phase and attenuation in a causal medium are interrelated through a Hilbert transform pair [7] (an example of which are the Kramers-Kronig relations), resulting in fundamental change in the pulse structure with propagation. Because of phase dispersion, the phasal relationship between the spectral components of the pulse changes with propagation, and because of attenuation dispersion, the relative spectral amplitudes also change with propagation. These combined effects result in a complicated dynamical pulse evolution that is accurately described by the modern asymptotic theory [8–11] as the propagation distance exceeds a value set by the absorption depth at some characteristic frequency of the input pulse. For an ultrashort pulse, these effects manifest themselves through the formation of well-defined precursor fields that dominate the temporal field structure in the mature dispersion regime [5, 8–11]. Because the group velocity approximation, by its very nature [8], neglects frequency dispersion of the material attenuation, it is incapable of properly modeling precursor field formation in dispersive pulse dynamics [8, 12, 13].

The precursor fields are a characteristic of the material dispersion, the input pulse providing the requisite spectral energy in the appropriate frequency domain [8,14]. For a single-resonance Lorentz-model dielectric, the dynamical pulse evolution is dominated by an above resonance Sommerfeld precursor and a below resonance Brillouin precursor throughout the mature dispersion regime [8], whereas for a Debye-model dielectric, the dynamical pulse evolution is dominated by just a low-frequency Brillouin precursor [8, 14]. This is because the peak amplitude of the Brillouin precursor in either a Lorentz- or Debye-type dielectric decays only as the square root of the inverse of the propagation distance while the peak amplitude of the Sommerfeld precursor in a Lorentz-type dielectric possesses an exponential decay rate that is typically much smaller than that at the input pulse frequency. This unique property may then be used to advantage through the design of precursor-type pulses that possess optimal penetration into a given dispersive material. A detailed analysis of these properties for a Debye-model dielectric has been presented in [14] with application to the design of a Brillouin pulse that will optimally penetrate through a given Debye-model dielectric. Applications there include foliage and ground penetrating radar as well as bio-electromagnetic effects due to ultrawideband radar and related devices. Experimental design [15] and observation [16, 17] of Debye-model precursor decay in H₂O have been reported, confirming the original analysis [14] describing this phenomenon. The analysis presented here extends this research (in a nontrivial way) into the optical domain where the material dispersion is Lorentz-like. Applications include deep tissue imaging [18], tumor detection [19], and cellular therapy [20].

2. Asymptotic Description

Asymptotic description of optical pulse propagation in homogeneous, isotropic, locally linear, temporally dispersive media is based on the Fourier-Laplace integral representation [1–3, 6–8]

A (z, t) = \frac{1}{2 π} ℜ {i e^{- i ψ} \int_{i a - \infty}^{i a + \infty} \tilde{u} (ω - ω_{c}) e^{i} (\tilde{k} (ω) z - ω t) d ω}

for all z ≥ 0 and fixed a >γ_a set by the abscissa of absolute convergence [7] for the contour integral in Eq. (1), where A(0,t) = u(t)sin(ω_ct + ψ) describes the temporal behavior of the initial plane wave pulse at the z = 0 plane with fixed carrier frequency ω_c > 0, phase constant ψ, and real-valued envelope function u(t) with temporal frequency spectrum

\tilde{u} (ω) = \int_{- \infty}^{\infty} u (t) e^{i ω t} d t

. The temporal pulse spectrum

\tilde{A} (z, ω) = \int_{- \infty}^{\infty} A (z, t) e^{i ω t} d t

satisfies the Helmholtz equation

(\nabla^{2} + {\tilde{k}}^{2} (ω)) \tilde{A} (z, ω) = 0

for all z ≥ 0. Here

\tilde{k} (ω) = β (ω) + i α (ω) = (ω / c) n (ω)

is the complex wavenumber in a single resonance Lorentz model dielectric with resonance frequency ω₀ whose dispersive optical properties are described by the complex index of refraction

n (ω) = {(1 - \frac{ω_{p}^{2}}{ω^{2} - ω_{0}^{2} + 2 i δ ω})}^{1 / 2},

with damping constant δ > 0 and plasma frequency ω_p. For the numerical examples presented in this paper, ω₀ = 4.0 × 10¹⁶r/s,

ω_{p} = \sqrt{2} \times 10^{16} r / s

, and δ = 0.28 × 10¹⁶r/s.

In the saddle point asymptotic description [8–10], the integral representation (1) is rewritten as

A (z, t) = \frac{1}{2 π} ℜ {i e^{- i ψ} \int_{i a - \infty}^{i a + \infty} \tilde{u} (ω - ω_{c}) e^{(z / c) ϕ (ω, θ)} d ω}

for z > 0, where

ϕ (ω, θ) \equiv i \frac{c}{z} (\tilde{k} (ω) z - ω t) = i ω (n (ω) - θ)

is the complex phase function [2,3,6,8] with non dimensional space-time parameter θ ≡ ct/z. If the input pulse identically vanishes for all t < 0 [i.e., if A(0,t) = 0 ∀ t < 0], then Sommerfeld’s relativistic causality theorem [1, 6, 8] shows that A(z,t) = 0 for all space-time points θ < 1. The dynamical evolution of such a pulse is then completely determined by the integral representation over the luminal (θ = 1) to subluminal (θ > 1) space-time domain. The asymptotic approximation of the integral in Eq. (4) as z → ∞ for θ ≥ 1 is determined by the dynamical evolution of the saddle points of ϕ (ω,θ) with θ. The condition that ϕ (ω,θ) is stationary at a saddle point requires that ϕ′(ω,θ) = 0, the prime denoting differentiation with respect to ω, resulting in the saddle point equation

n (ω) + ω n^{'} (ω) = θ

For a single resonance Lorentz model dielectric, two sets of saddle points are found [2, 3, 6, 8, 9], each set symmetrically situated about the imaginary axis in the complex ω-plane. The distant saddle points

ω_{S P_{D}}^{\pm} (θ) = \pm ξ (θ) - i δ (1 + η (θ))

evolve with θ ≥ 1 in the high-frequency domain

| ω_{S P_{D}}^{\pm} (θ) | \geq ω_{1}

above the region of anomalous dispersion, where

ω_{1} \equiv \sqrt{ω_{0}^{2} + ω_{p}^{2}}

, while the near saddle points

ω_{S P_{N}}^{\pm} (θ) = {\begin{matrix} i [\pm | ψ (θ) | - \frac{2}{3} δ ζ (θ)], & 1 < θ \leq θ_{1} \\ \pm ψ (θ) - i \frac{2}{3} δ ζ (θ), & θ \geq θ_{1} \end{matrix}

evolve with θ > 1 in the low-frequency domain

| ω_{S P_{N}}^{\pm} (θ) | \leq ω_{0}

below the medium resonance frequency, where

θ_{0} \equiv n (0) = \sqrt{1 + ω_{p}^{2} / ω_{0}^{2}}

and

θ_{1} \approx θ_{0} + (2 δ^{2} ω_{p}^{2}) / (3 θ_{0} ω_{0}^{4})

. Approximate expressions for the functions ξ(θ), η(θ), ψ(θ), and ζ(θ) in terms of the Lorentz medium parameters ω₀, ω_p, and δ are given in Refs. [6, 8–11]. The asymptotic description of the propagated pulse may then be expressed either in the form [6, 8]

A (z, t) = A_{s} (z, t) + A_{b} (z, t) + A_{c} (z, t)

as for the Heaviside step-function signal, or as a linear combination of expressions of this form, where A_s(z,t) is the asymptotic contribution from the distant saddle points, A_b(z,t) from the near saddle points, and A_c(z,t) is the steady-state response or signal contribution (if any). For finite duration, sufficiently smooth envelope pulses (such as a gaussian pulse), the signal contribution is absent and the pulse evolves into Sommerfeld and Brillouin precursor field components [8, 12, 13] as illustrated in Fig. 1, an example of pulse-splitting due to dispersion.

Fig. 1 Pulse splitting into Sommerfeld and Brillouin precursor components due to an above-resonance (ω_c = 2.5ω₀) half-cycle gaussian envelope pulse at one absorption depth [z = z_d(ω_c) ≡ α⁻¹(ω_c)] in a single resonance Lorentz-model dielectric.

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3. The Brillouin Pulse

The Brillouin precursor A_b(z,t) describes the low-frequency response of the dispersive medium to the input pulse. Its uniform asymptotic approximation as z → ∞ is given by [8–11]

\begin{matrix} A_{b} (z, t) ~ e^{\frac{z}{c} α_{0}} {\frac{{(c / z)}^{2 / 3}}{2} ℜ {\tilde{u} (ω_{S P_{N}}^{+} - ω_{c}) | h^{+} | + \tilde{u} (ω_{S P_{N}}^{-} - ω_{c}) | h^{-} |} A_{i} (\pm | α_{1} | {(z / c)}^{2 / 3}) \\ - \frac{{(c / z)}^{2 / 3}}{2 {| α_{1} |}^{1 / 2}} ℜ {\tilde{u} (ω_{S P_{N}}^{+} - ω_{c}) | h^{+} | - \tilde{u} (ω_{S P_{N}}^{-} - ω_{c}) | h^{-} |} {A^{'}}_{1} (\pm | α_{1} | {(z / c)}^{2 / 3})} \end{matrix}

for all space-time points θ > 1. Here A_i(ζ) and

{A^{'}}_{i} (ζ)

denote the Airy function and its derivative with respect to its argument ζ, where ζ = +|α₁| (z/c)^2/3 for 1 < θ ≤ θ₁ and ζ = −|α₁| (z/c)^2/3 for θ ≥ θ₁. The Brillouin precursor is thus non-oscillatory over the initial space-time domain 1 < θ ≤ θ₁ and oscillatory for θ ≥ θ₁. The functions appearing in Eqs. (9)–(10) are given by [8]

α_{0} (θ) \equiv (1 / 2) [ϕ (ω_{S P_{N}}^{+}, θ) + ϕ (ω_{S P_{N}}^{-}, θ)]

,

α_{1}^{1 / 2} (θ) \equiv {(3 / 4) [ϕ (ω_{S P_{N}}^{+}, θ) - ϕ (ω_{S P_{N}}^{-}, θ)]}^{1 / 3}

, and

h^{\pm} (θ) \equiv {[\mp 2 α_{1}^{1 / 2} (θ) / ϕ^{″} (ω_{S P_{N}}^{\pm}, θ)]}^{1 / 2}

.

Because $ω_{S P_{N}}^{+} (θ_{0}) = 0$ , at which point the complex phase function ϕ (ω,θ) identically vanishes, the peak amplitude point of A_b(z,t) experiences zero exponential attenuation at the space-time point θ = θ₀ ≡ n(0). With $A_{i} (| α_{1} | {(z / c)}^{2 / 3}) ~ \frac{{(c / z)}^{1 / 6}}{2 \sqrt{π} {| α_{1} |}^{1 / 4}} e^{- (2 z / 3 c) {| α_{1} |}^{3 / 2}}$ and ${A^{'}}_{i} (| α_{1} | {(z / c)}^{2 / 3}) ~ - \frac{{(c / z)}^{1 / 6}}{2 \sqrt{π}} {| α_{1} |}^{1 / 4} e^{- (2 z / 3 c) {| α_{1} |}^{3 / 2}}$ , Eq. (9) becomes

A_{b} (z, t) ~ \frac{{(c / z)}^{1 / 2}}{4 \sqrt{π} {| α_{1} (θ) |}^{1 / 4}} ℜ {i [\tilde{u} (ω_{S P_{N}}^{+} - ω_{c}) | h^{+} | + \tilde{u} (ω_{S P_{N}}^{-} - ω_{c}) | h^{-} |]} e^{(z / c) ϕ (ω_{S P_{N}}^{+}, θ)}

as z → ∞ with 1 < θ < θ₁. Hence, at θ = θ₀ = ct₀/z,

A_{b} (z, θ_{0} z / c) ~ \frac{{(c / z)}^{1 / 2}}{4 \sqrt{π} {| α_{1} (θ_{0}) |}^{1 / 4}} ℜ {i [\tilde{u} (ω_{S P_{N}}^{+} - ω_{c}) | h^{+} (θ_{0}) | + \tilde{u} (ω_{S P_{N}}^{-} - ω_{c}) | h^{-} (θ_{0}) |]}

and the peak amplitude point, which propagates with velocity v_b = c/θ₀ = c/n(0), decays as z⁻¹^/² as z → ∞. An estimate of the effective oscillation frequency at this point is given by (see §13.3.3 in [8])

ω_{e f f} (θ_{0}) \approx \frac{3 π θ_{0} ω_{0}^{4}}{4 δ^{2} ω_{p}^{2} Δ_{f}} \frac{c}{z}

which approaches zero as z → ∞, where Δ_f is a numerically determined factor. The instantaneous frequency of the remaining Brillouin precursor evolution is found to be given by [6, 8]

ω_{b} (θ) \approx ℜ {ω_{S P_{N}}^{+} (θ)} = ψ (θ)

for θ > θ₁, so that the Brillouin precursor chirps up in frequency towards ω₀.

The dynamical evolution of the Brillouin precursor at ten absorption depths [z = 10z_d where z_d ≡ α⁻¹(ω_c)] due to a half-cycle rectangular envelope pulse u_r(t) = 1 for 0 < t < T and zero otherwise with spectrum ũ_r(ω) = (e^iωt− 1)/iω and a half-cycle gaussian envelope pulse $u_{g} (t) = e^{- {(t - τ_{0})}^{2} / T^{2}}$ centered at t = τ₀ with spectrum ${\tilde{u}}_{g} (ω) = \sqrt{π} T e^{- {(ω T / 2)}^{2}} e^{i ω τ_{0}}$ , each with initial pulse width 2T ≈ 1.571×10⁻¹⁶ s at the below resonance carrier frequency ω_c = ω₀/2, is illustrated in Fig. 2. Notice that the peak amplitude for each precursor pulse has been normalized to unity and shifted to the same instant of time t₀ = θ₀z/c. Any difference between the two precursor pulses is due to the difference in the input pulse spectra. This difference disappears as the initial pulse width decreases while the peak amplitude increases such that the pulse area remains constant and a delta-function pulse is approached.

Fig. 2 Brillouin precursor evolution due to a below-resonance (ω_c = ω₀/2) (a) half-cycle rectangular envelope pulse (blue curve) and (b) half-cycle gaussian envelope pulse (green curve) at ten absorption depths (z/z_d(ω_c) = 10) in a single resonance Lorentz dielectric.

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Because the Brillouin pulse spectrum is ultra-wideband and hence, relatively flat for frequencies below ω_c, accurate numerical determination of its effective peak frequency value becomes increasingly difficult as the propagation distance increases. As an illustration, consider the numerically determined peak frequency evolution of a below resonance (ω_c = ω₀/2) single-cycle gaussian envelope pulse illustrated in Fig. 3. The blue data points and dashed curve are obtained from numerical measurements of the half-period of the numerically determined Brillouin precursor at the given penetration depth, the solid blue curve depicts the behavior of the asymptotic estimate (12) with Δ_f = 11, and the green data points and dashed curve are obtained from the peak amplitude in the propagated pulse spectrum. Both dashed curves describe a cubic spline fit to the corresponding computed data points. Because the measured period T_eff of the Brillouin precursor over-estimates the actual period, the estimate of ω_eff from T_eff provides a lower bound to ω_eff (z). Notice that for z/z_d(ω_c) > 8, the numerical error in the spectral measure of ω_eff(z) increases as the propagated pulse spectrum becomes increasingly ultra-wideband and flattens out below ω_c, resulting in an erroneously rapid decrease of ω_eff(z) to zero instead of the correct asymptotic z⁻¹ behavior as z → ∞. Finally, notice that the Brillouin precursor is not a zero-frequency (or dc) event as istated elsewhere [21], as can be seen from the computed field structure illustrated in Fig. 2 and the effective frequency behavior depicted in Fig. 3; see §13.3.3 and §15.6.2 in [8] for a more detailed analysis of this point.

Fig. 3 Effective angular frequency of the peak Brillouin precursor amplitude as a function of the relative penetration depth z/z_d(ω_c) as given by (a) the asymptotic estimate in Eq. (13) with Δ_f = 11, (b) the numerically determined peak amplitude in the propagated pulse spectrum, and (c) the numerically measured period T_eff about the peak amplitude point.

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The numerically determined peak amplitude decay of the Brillouin precursor for (a) a Heaviside step-function signal u_H(t) (blue data points and dashed cubic spline fit curve) and (b) a single-cycle gaussian envelope pulse (green data points and dashed cubic spline fit curve) are illustrated in Fig. 4 for the below resonance case with ω_c = ω₀/2. The dashed black curve in the figure describes the pure exponential decay $e^{- z / z_{d} (ω_{c})}$ at the input pulse carrier frequency. Notice that the peak amplitude decay of the step-function signal Brillouin precursor initially follows this exponential decay for z < z_d(ω_c), referred to as the immature dispersion regime [5] wherein the Brillouin precursor is formed by the material dispersion. In the mature dispersion regime z > z_d, the Brillouin precursor is well-defined and increasingly satisfies the characteristic z⁻¹^/² peak amplitude decay described by Eq. (11), resulting in a significant departure from pure exponential decay. The peak amplitude decay of the single-cycle gaussian envelope pulse exhibits a similar dependence with a somewhat more rapid initial decay in the immature dispersion region z < z_d(ω_c) that is below the signal decay $e^{- z α (ω_{c})}$ followed by a transition to the characteristic z⁻¹^/² peak amplitude decay for z > z_d(ω_c).

Fig. 4 Peak amplitude decay of the Brillouin precursor for (a) a Heaviside step-function signal u_H(t)sin(ω_ct) and (b) a single-cycle gaussian envelope pulse u_g(t)cos(ω_ct). The lower dashed curve describes exponential signal decay $e^{- z α (ω_{c})}$ at the input carrier frequency ω_c = ω₀/2 and the upper dashed curve describes the frequency-chirped Lambert-Beer’s law limit given by $e^{- z α (ω_{e f f} (z))}$ from Eq. (14). In the immature dispersion regime, both pulses decay at or near to the signal rate $e^{- z α (ω_{c})}$ , but as the propagation distance enters the mature dispersion regime, the Brillouin precursor emerges with a decreased decay rate approaching the characteristic z⁻¹^/² asymptotic dependence. This transition between immature and mature dispersion regimes occurs at z/z_d(ω_c) ≃ 2.5 for the step-function signal and at z/z_d(ω_c) ≃ 1.5 for the single-cycle gaussian envelope pulse.

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Because the attenuation factor α(ω_eff(z)) varies with the propagation distance z through ω_eff(z), illustrated in Fig. 3, a proper comparison of the peak amplitude decay of the Brillouin precursor must be made with the decay factor

A (z / z_{d} (ω_{c})) / A_{0} = \exp {- \int_{0}^{z / z_{d} (ω_{c})} α (ω_{e f f} (ζ)) d ζ}

describing the optimal frequency-chirped Lambert-Beer’s law limit. A numerical evaluation of this exponential decay factor using the effective frequency behavior described by the green dashed curve in Fig. 3 for z/z_d(ω_c) < 8 and by Eq. (12) with Δ_f = 11 for z/z_d(ω_c) ≥ 8 (the value of Δ_f was chosen to provide a smooth transition between these two curves at z/z_d(ω_c) = 8) is illustrated by the upper black dashed curve in Fig. 4. The Brillouin precursor evolution is bounded above by this Lambert-Beer’s law limit because significant energy is lost from the input pulse in its creation. Such is not the case for the Brillouin pulse, as described below.

The peak amplitude decay of a Brillouin precursor pulse, depicted by the blue data points and cubic spline fit dashed curve in Fig. 5, decays as (z+z₀)⁻¹^/² for all z > 0 with fixed z₀ > 0. By its very nature, the Brillouin pulse is in the mature dispersion regime for all propagation distances z ≥ 0. The Brillouin precursor illustrated in Fig. 2 is an example of this Brillouin pulse and is used here as the initial pulse for that Lorentz-model dielectric. The optimal penetration of this Brillouin pulse presented in Fig. 5 dominates that for both Brillouin precursors presented in Fig. 4. Most importantly, to within the numerical accuracy of the numerical calculations presented here, it obeys the frequency-chirped Lambert-Beer’s law limit described by Eq. (14). Notice that $e^{- z α (ω_{e f f} (z))} \approx e^{- z α (ω_{e f f} (0))}$ when z ≈ 0 so that the initial Lambert-Beer’s law decay follows the simple exponential decay given by $e^{- z α (ω_{e f f} (0))}$ , departing from it as ω_eff(z) decreases with increasing penetration depth z > 0, whereas the Brillouin precursor pulse experiences zero exponential decay for all z > 0, attenuating algebraically as z⁻¹^/² as z → ∞. These results then show that the Brillouin pulse is precisely matched to the below-resonance dispersion properties of the dielectric material.

Fig. 5 Peak amplitude decay of the Brillouin precursor pulse (blue data points and dashed curve). The black dashed curve describes exponential decay $e^{- z α (ω_{e f f} (0))}$ at the initial (z = 0) effective oscillation frequency ω_eff(z) of the Brillouin pulse and the green dashed curve describes the optimal frequency-chirped Lambert-Beer’s law limit given by $e^{- z α (ω_{e f f} (z))}$ from Eq. (14).

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4. The Sommerfeld Pulse

The Sommerfeld precursor A_s(z,t) describes the high-frequency response of the dispersive medium to the input pulse. Its uniform asymptotic approximation as z → ∞ is given by [8–11]

A_{s} (z, t) ~ ℜ {{[2 \tilde{α} (θ) e^{- i \frac{π}{2}}]}^{ν} e^{- 1 \frac{z}{c} β (θ)} e^{- i ψ} [γ_{0} J_{ν} (\tilde{α} (θ) \frac{z}{c}) + 2 \tilde{α} (θ) e^{- i \frac{π}{2}} γ_{1} J_{ν + 1} (\tilde{α} (θ) \frac{z}{c})]}

for all θ ≥ 1. Here J_ν(ζ) is the Bessel function of the first kind of real order ν that is determined by the behavior of the spectral amplitude function ũ(ω) in the following manner [6,8,10,11]: let

\tilde{u} (ω) = ω^{- (1 + ν)} \tilde{q} (ω)

for large |ω| with ν > 0, where

\tilde{q} (ω)

possesses a Laurent series expansion convergent for |ω| ≥ R > 0 and is such that

\lim_{| ω | \to \infty} \tilde{q} (ω) \neq 0

; then

ν = \lim_{| ω | \to \infty} {\frac{\log | \tilde{q} (ω) | - \log | \tilde{u} (ω) |}{\log | ω |}} - 1

. If ν ≤ 0, then the uniform asymptotic expansion (15) remains valid for all θ ≥ 1 provided that its limiting value as θ → 1⁺ is finite. The functions appearing in Eq. (15) are

\tilde{α} (θ) \equiv \frac{i}{2} [ϕ (ω_{S P_{D}}^{+}, θ) - ϕ (ω_{S P_{D}}^{-}, θ)]

,

\tilde{β} (θ) \equiv \frac{i}{2} [ϕ (ω_{S P_{D}}^{+}, θ) + ϕ (ω_{S P_{D}}^{-}, θ)]

,

2 γ_{0} \equiv \tilde{u} (ω_{S P_{D}}^{+} - ω_{c}) {[2 \tilde{α}]}^{- (1 + ν)} {[\frac{4 {\tilde{α}}^{3}}{i ϕ^{″} (ω_{S P_{D}}^{+})}]}^{1 / 2} + \tilde{u} (ω_{S P_{D}}^{-} - ω_{c}) {[- 2 \tilde{α}]}^{- (1 + ν)} {[- \frac{4 {\tilde{α}}^{3}}{i ϕ^{″} (ω_{S P_{D}}^{-})}]}^{1 / 2}

, and

2 γ_{1} \equiv \tilde{u} (ω_{S P_{D}}^{+} - ω_{c}) {[2 \tilde{α}]}^{- (1 + ν)} {[\frac{4 {\tilde{α}}^{3}}{i ϕ^{″} (ω_{S P_{D}}^{+})}]}^{1 / 2} - \tilde{u} (ω_{S P_{D}}^{-} - ω_{c}) {[- 2 \tilde{α}]}^{- (1 + ν)} {[- \frac{4 {\tilde{α}}^{3}}{i ϕ^{″} (ω_{S P_{D}}^{-})}]}^{1 / 2}

. For θ > 1, each Bessel function in the uniform expansion (15) may be replaced by its asymptotic approximation

J_{ν} (ζ) ~ \sqrt{2 / π ζ} \cos (ζ - ν π / 2 - π / 4)

as ζ → ∞ with |arg(ζ)| < π, with the result [9]

A_{S} (z, t) ~ \sqrt{\frac{2 c}{π z}} ℜ {{(2 \tilde{α} (θ) e^{- i π / 2})}^{ν} e^{- i \tilde{β} (θ) z / c} [\frac{γ_{0}}{{\tilde{a}}^{1 / 2}} \cos (ζ (θ)) + 2 γ_{1} {\tilde{a}}^{1 / 2} \sin (ζ (θ))]}

as z → ∞ with θ > 1 + Δ, Δ > 0, where

ζ (θ) \equiv \tilde{a} (θ) z / c - ν π / 2 - π / 4

.

As illustrated in Fig. 6, the peak amplitude of the Heaviside step-function Sommerfeld precursor occurs just after the field arrival at the luminal space-time point θ ≡ ct/z = 1. Since $\tilde{β} (1) = 0$ for the Heaviside step-function signal, the Sommerfeld precursor front experiences zero attenuation with zero amplitude for integer ν ≥ 0. Unlike the Brillouin precursor, the peak amplitude of the Heaviside step-function Sommerfeld precursor experiences a small exponential decay with propagation distance in addition to the z⁻¹^/² asymptotic behavior appearing in Eq. (16). Notice that, because of the finite computation size imposed by computer memory limitations, the Sommerfeld precursor evolution displayed in Fig. 6 is a ω ∈ [0,ω_max] low-pass filtered version of the actual Sommerfeld precursor evolution where, for the numerical results presented here, sampling criteria sets ω_max ≃ 3.14 × 10²¹r/s. With this in mind, the solid blue curve in Fig. 7 displays a linear spline fit to the numerically determined peak amplitude decay of the Sommerfeld precursor appearing in the dynamical field evolution due to an above-resonance (ω_c = 2.5ω₀) Heaviside step-function signal. The initial peak amplitude decay of this precursor follows the exponential decay factor $e^{- α (ω_{c}) z}$ of the signal with fixed carrier frequency ω_c. Between z/z_d ≃ 1.5 and z/z_d ≃ 2.0, the Sommerfeld precursor emerges from the propagated field structure, as indicated by the change in behavior of the peak amplitude decay, decaying at a much slower rate for z/z_d > 2 and exceeding the signal amplitude $e^{- α (ω_{c}) z}$ when z/z_d > 3. A numerical determination of the effective oscillation frequency $ω_{e f f_{h}} (z)$ from the measured period of the field about the peak amplitude point is illustrated in Fig. 8 by the blue data points and dashed curve. Comparison of this numerical measurement of the effective oscillation frequency $ω_{e f f_{h}} (z)$ with that obtained from the asymptotic approximation (15) of the Sommerfeld precursor field for the Heaviside step-function signal, denoted by $ω_{e f f_{a s}} (z)$ and described by the black data points and dashed curve in Fig. 8, shows that $ω_{e f f_{h}} (z)$ has been decreased from its asymptotic behavior, this being due to the low-pass filter cut-off at ω_max inherent in the numerical field calculations. Computation of the resultant frequency-chirped Lambert-Beer’s law limit from Eq. (14) using $ω_{e f f_{h}} (z)$ for the numerically determined Sommerfeld precursor evolution is displayed by the blue dashed curve in Fig. 7. Notice that the Sommerfeld precursor decay (the blue data points and curve in Fig. 7@@) attains this optimal decay at seven absorption depths (z/z_d = 7) and then decays at a slightly slower rate for larger propagation distances.

Fig. 6 Sommerfeld precursor evolution due to an above-resonance (ω_c = 2.5ω₀) (a) Heaviside step-function signal (blue curve) and (b) single-cycle gaussian envelope pulse (green curve) at eight absorption depths (z/z_d(ω_c) = 8). Notice that the the peak amplitude for each precursor has been normalized to unity and shifted to the same instant of time.

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Fig. 7 Peak amplitude decay of the Sommerfeld precursor for (a) a Heaviside step-function signal u_H(t)sin(ω_ct) and (b) a single-cycle gaussian envelope pulse u_g(t)cos(ω_ct). The black dashed curve describes exponential decay $e^{- z α (ω_{c})}$ at the carrier frequency ω_c = 2.5ω₀ and the blue and green dashed curves describe the frequency-chirped Lambert-Beer’s law limit given by $e^{- z α (ω_{e f f_{j}} (z))}$ for the step-function (j = H) and gaussian (j = g) Sommerfeld precursors, respectively. In the immature dispersion regime, the step-function signal decays at the signal rate $e^{- z α (ω_{c})}$ , the transition to mature dispersion indicated by the abrupt change in behavior between z/z_d ≃ 1.5 and z/z_d ≃ 2.0 as the Sommerfeld precursor emerges, decaying at a much slower rate than the signal. The single-cycle gaussian pulse pulse decays faster than the signal rate in the immature dispersion regime, the transition to mature dispersion indicated by the change in behavior between z/z_d ≃ 0.8 and z/z_d ≃ 1.0 as the Sommerfeld precursor emerges, decaying at a slower rate than the signal.

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Fig. 8 Relative effective angular frequency ω_eff/ω_c of the peak Sommerfeld precursor amplitude as a function of the relative penetration depth z/z_d(ω_c) as derived from a numerical evaluation of the asymptotic approximation (15) of the Sommerfeld precursor for the Heaviside step-function signal $(ω_{e f f_{a s}} / ω_{c})$ , black data points and dashed curve), the numerically measured period $T_{e f f_{a s}}$ about the peak amplitude point in the step-function Sommerfeld precursor $(ω_{e f f_{a s}} / ω_{c})$ , blue data points and dashed curve), and the numerically determined peak amplitude in the propagated gaussian pulse spectrum $(ω_{e f f_{g}} / ω_{c})$ , green data points and dashed curve).

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The solid green curve in Fig. 7 displays a cubic spline fit to the numerically determined peak amplitude decay of the Sommerfeld precursor due to a single-cycle gaussian envelope pulse with the same above-resonance carrier frequency ω_c = 2.5ω₀ and initial pulse width 2T = 4π/ω_c ≃ 0.1257 f s. An illustration of this precursor evolution at eight absorption depths [z/z_d(ω_c) = 8] is depicted by the green curve in Fig. 6. Notice that, unlike the Sommerfeld precursor for the Heaviside step-function signal which has a sharply defined front at ct/z = 1, the gaussian Sommerfeld precursor does not as it possesses a continuously smooth turn-on. Although it decays at a faster initial rate than the exponential decay factor $e^{- α (ω_{c}) z}$ at the initial carrier frequency, it decays at a much slower rate for z/z_d > 1, exceeding the $e^{- α (ω_{c}) z}$ exponential decay factor for z/z_d > 2.0. A numerical determination of the effective oscillation frequency from the magnitude peak in the propagated pulse spectrum is described by the green data points and dashed curve in Fig. 8. The resultant computation of the corresponding frequency-chirped Lambert-Beer’s law limit from Eq. (14) for this gaussian Sommerfeld precursor evolution is displayed by the green dashed curve in Fig. 7. Because significant energy is lost from the input gaussian envelope pulse in its creation, the gaussian Sommerfeld precursor decay is bounded above by this Lambert-Beer’s law limit for all z > 0, paralleling it for z/z_d > 6.

5. Conclusion

The numerical results presented in this paper were performed with MATLAB 7.10.0 using a 2²⁴ point FFT-algorithm to compute the temporal structure of each propagated optical pulse based on a numerical synthesis of Eq. (1). The frequency structure of the dispersive Lorentz-model dielectric and pulse spectra were sampled from f_min = 2.98 × 10¹³Hz to f_max = 5.0 × 10²⁰Hz. This is adequate to properly sample both the material dispersion and the spectral content of the Brillouin and (f_max low-pass filtered) Sommerfeld precursors generated respectively by the below and above resonance input pulses considered here.

Nearly identical results to those presented here may be obtained through use of the weak dispersion equivalence relation given in §15.10.2 in [8] when the plasma frequency $ω_{p} \equiv \sqrt{N q_{e}^{2} / m e}$ is changed through the number density N. For example, when the plasma frequency is reduced from the value ω_p = 2×10¹⁶r/s used in the numerical examples presented here, corresponding to highly absorptive media with micron scale absorption depth z_d(ω_c) ≃ 2.1μm, to the value ω_p = 1×10¹²r/s, the dielectric becomes weakly dispersive with kilometer scale absorption depth z_d(ω_c) ≃ 389m, while the peak amplitude behavior presented here remains essentially unchanged.

The results presented here show that, because of their minimal decay in a given Lorentz model dielectric, the Brillouin and Sommerfeld pulses possess optimal penetration into that dispersive medium at the frequency-chirped Lambert-Beer’s law limit given in Eq. (14). The Brillouin pulse spans the below resonance normal dispersion region of the medium while the Sommerfeld pulse spans the above resonance normal dispersion region. Each is then ideally suited for penetrating its respective material frequency domain.

The experimental results and subsequent analysis of the Brillouin precursor behavior presented in Refs. [15–17, 21] have overlooked the fact that the Brillouin precursor evolution in water is determined by low-frequency dispersion (below infrared) of water due to Debye relaxation, as described in detail in Refs. [8, 14]; in particular, see §13.4 in [8]. It is established there that this Debye-model Brillouin precursor is not a dc field just as the Lorentz-model Brillouin precursor is not a dc field for finite propagation distances z, as is evident from Eq. (12).

The numerical generation of a Brillouin pulse for a given Lorentz-model dielectric may be obtained from Eq. (9) with ũ(ω −ω_c) = i/(ω −ω_c) and z = z_d(ω_c) ≡ α⁻¹(ω_c) where the fixed angular frequency ω_c is chosen to be near the resonance frequency ω₀ of the medium so as to eliminate the possible appearance of an artifact resonance peak in the Brillouin precursor when the near saddle point $ω_{S P_{N}}^{+} (θ)$ passes near the simple pole singularity at ω = ω_c when 0 < ω_c < ω₀ (see §9.2 in [6] and §15.5 in [8]). This complication can be avoided (as was done for the numerical results presented here) by numerically computing the propagated field due to either an input Heaviside step function pulse or a half-cycle gaussian envelope pulse with below resonance carrier frequency at a sufficiently large penetration distance such that (for the Heaviside step-function case) the signal contribution is exponentially negligible in comparison to the Brillouin precursor (typically z = 3z_d is sufficiently large). In addition, the Heaviside step-function signal was computed from the leading edge of a rectangular envelope pulse with sufficient temporal duration that the dispersive effects of the trailing edge were well-removed from the leading-edge precursors (see §15.6.1 in [8]). Similar remarks hold for the numerical generation of the Sommerfeld pulse. Laboratory generation of these optimal precursor pulses is left to my experimental colleagues.

Direct application of these results include deep tissue imaging [17,18], tumor detection [19], and cellular therapy [20], as well as to LIDAR for the detection of submerged bodies such as mines and submarines.

Acknowledgments

The more meaningful comparison of the precursor decay with the frequency-chirped Lambert-Beer’s law limit presented here was suggested by an unknown reviewer when I first submitted a much earlier version of this paper several years ago. I am indebted to that reviewer’s sage advice.

References and links

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Optimal pulse penetration in Lorentz-Model dielectrics using the Sommerfeld and Brillouin precursors

Abstract

1. Introduction

2. Asymptotic Description

3. The Brillouin Pulse

4. The Sommerfeld Pulse

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (16)

Optics Express