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

Under proper initial conditions, the interrelated effects of phase and attenuation dispersion in ultrawideband pulse propagation modify the input pulse into precursor fields. Because of their minimal decay in a given dispersive medium, precursor-type pulses possess optimal penetration into that material at the frequency-chirped Lambert-Beer’s law limit, making them ideally suited for remote sensing and medical imaging. © 2015 Optical Society of America OCIS codes: (260.2030) Dispersion; (320.2250) Femtosecond phenomena; (320.5550) Pulses. References and links 1. A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. 44, 177-202 (1914). 2. L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. 44, 203-240 (1914). 3. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960). 4. M. Born and E. Wolf, Principals of Optics, 7th (expanded) ed. (Cambridge University Press, 1999), Ch. 1. 5. 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 (20), 1451–1454 (1981). 6. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, (Springer, 1994). 7. K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media (Springer, 2006). 8. K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive Attenuative Media (Springer, 2009). 9. 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 (4), 817-849 (1988). 10. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6(9), 1394-1420 (1988). 11. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49 (4), 628-648 (2007). 12. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78 (4), 642-645 (1997). 13. H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B 16 (10), 1773–1785 (1999). 14. K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop. 53, 1582–1590 (2005). 15. U. J. Gibson and U. L. Österberg, “Optical precursors and Beer’s law violations; non-exponential propagation losses in water,” Opt. Express 13 (6), 2105–2110 (2005). 16. A. E. Fox and U. Österberg, “Observation of non-exponential absorption of ultra-fast pulses in water,” Opt. Express 14 (8), 3688–3693 (2006). 17. Y. Okawachi, A. D. Slepkov, I. H. Agha, D. F. Geraghty, and A. L. Gaeta, “Absorption of ultrashort optical pulses in water,” J. Opt. Soc. Am. A 24 (10), 3343–3347 (2007). 18. J. Li, F. Jaillon, G. Dietsche, G. Maret, and T. Gisler, “Pulsation-resolved deep tissue dynamics measured with diffusing-wave spectroscopy,” Opt. Express 14, 7841-7851 (2006). #247567 Received 7 Aug 2015; revised 28 Sep 2015; accepted 29 Sep 2015; published 1 Oct 2015 (C) 2015 OSA 5 Oct 2015 | Vol. 23, No. 20 | DOI:10.1364/OE.23.026604 | OPTICS EXPRESS 26604 19. G. Pal, S. Basu, K. Mitra, and T. Vo-Dinh, “Time-resolved optical tomography using short-pulse laser for tumor detection,” Appl. Opt. 45, 6270-6282 (2006). 20. D. Stevenson, B. Agate, X. Tsampoula, P. Fischer, C. T. A. Brown, W. Sibbett, A. Riches, F. Gunn-Moore, and K. Dholakia, “Femtosecond optical transfection of cells: viability and efficiency,” Opt. Express 14, 7125-7133 (2006). 21. D. Lukofsky, J. Bessette, H. Jeong, E. Garmire, and U. Österberg, “Can precursors improve the transmission of energy at optical frequencies?,” J. Mod. Opt. 56 (9), 1083–1090 (2009).


Introduction
The dynamical evolution of an ultrashort optical pulse propagating through a causally dispersive dielectric is a classic problem [1][2][3][4][5][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][9][10][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][9][10][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 H 2 O 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].
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 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 evolve with θ ≥ 1 in the high-frequency domain |ω ± SP D (θ )| ≥ ω 1 above the region of anomalous dispersion, where ω 1 ≡ ω 2 0 + ω 2 p , while the near saddle points  evolve with θ > 1 in the low-frequency domain |ω ± SP N (θ )| ≤ ω 0 below the medium resonance frequency, where θ 0 ≡ n(0) = 1 + ω 2 p /ω 2 0 and θ 1 ≈ θ 0 + (2δ 2 ω 2 p )/(3θ 0 ω 4 0 ).Approximate expressions for the functions ξ (θ ), η(θ ), ψ(θ ), and ζ (θ ) in terms of the Lorentz medium parameters ω 0 , ω p , and δ are given in Refs.[6,[8][9][10][11].The asymptotic description of the propagated pulse may then be expressed either in the form [6,8] 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.
and the peak amplitude point, which propagates with velocity v b = c/θ 0 = 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]) 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] for θ > θ 1 , so that the Brillouin precursor chirps up in frequency towards ω 0 .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 ũr (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 = τ 0 with spectrum ũg (ω) = √ πTe −(ωT /2) 2 e iωτ 0 , each with initial pulse width 2T ≈ 1.571 × 10 −16 s at the below resonance carrier frequency ω c = ω 0 /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 t 0 = θ 0 z/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.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 = ω 0 /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 e f f of the Brillouin precursor over-estimates the actual period, the estimate of ω e f f from T e f f provides a lower bound to ω e f f (z).Notice that for z/z d (ω c ) > 8, the numerical error in the spectral measure of ω e f f (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 e f f (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.e -zα(ω eff (z)) Fig. 4. Peak amplitude decay of the Brillouin precursor for (a) a Heaviside step-function signal u H (t) sin (ω c t) and (b) a single-cycle gaussian envelope pulse u g (t) cos (ω c t).The lower dashed curve describes exponential signal decay e −zα(ω c ) at the input carrier quency ω c = ω 0 /2 and the upper dashed curve describes the frequency-chirped Lambert-Beer's law limit given by e −zα(ω e f f (z)) from Eq. ( 14).In the immature dispersion regime, both pulses decay at or near to the signal rate e −zα(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.
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 = ω 0 /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 −zα(ω c ) followed by a transition to the characteristic z −1/2 peak amplitude decay for z > z d (ω c ).
Because the attenuation factor α(ω e f f (z)) varies with the propagation distance z through ω e f f (z), illustrated in Fig. 3, a proper comparison of the peak amplitude decay of the Brillouin precursor must be made with the decay factor 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.Peak Amplitude e -zα(ω eff (z))

Brillouin Precursor
Pulse e -zα(ω eff (0)) Fig. 5. Peak amplitude decay of the Brillouin precursor pulse (blue data points and dashed curve).The black dashed curve describes exponential decay e −zα(ω e f f (0)) at the initial (z = 0) effective oscillation frequency ω e f f (z) of the Brillouin pulse and the green dashed curve describes the optimal frequency-chirped Lambert-Beer's law limit given by e −zα(ω e f f (z)) from Eq. ( 14).
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 + z 0 ) −1/2 for all z > 0 with fixed z 0 > 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 −zα(ω e f f (z)) ≈ e −zα(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 −zα(ω e f f (0)) , departing from it as ω e f f (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.

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][9][10][11] ) 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 ũ(ω) = ω −(1+ν) q(ω) for large |ω| with ν > 0, where q(ω) possesses a Laurent series expansion convergent for |ω| ≥ R > 0 and is such that lim |ω|→∞ q(ω) = 0; then ν = 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 α(θ , and . For θ > 1, each Bessel function in the uniform expansion (15) may be replaced by its asymptotic as  e -zα(ω effh (z)) Fig. 7. Peak amplitude decay of the Sommerfeld precursor for (a) a Heaviside step-function signal u H (t) sin (ω c t) and (b) a single-cycle gaussian envelope pulse u g (t) cos (ω c t).The black dashed curve describes exponential decay e −zα(ω c ) at the carrier frequency ω c = 2.5ω 0 and the and green dashed curves describe the frequency-chirped Lambert-Beer's law limit given by e −zα(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 −zα(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.
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 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 × 10 21 r/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 = 2.5ω 0 ) 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  quency 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, by ω e f f as (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.
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 = 2.5ω 0 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

Fig. 3 .
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 e f f about the peak amplitude point.

Fig. 6 .
Fig.6.Sommerfeld precursor evolution due to an above-resonance (ω c = 2.5ω 0 ) (a) Heaviside step-function signal (blue curve) and (b) 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.

Fig. 8 .
Fig.8.Relative effective angular frequency ω e f f /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 as /ω c ), black data points and dashed curve), the numerically measured period T e f f as about the peak amplitude point in the step-function Sommerfeld precursor (ω e f f h /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).