optimal pulse penetration in lorentz-model dielectrics ... · persive media (springer, 2006). 8. k....

Optimal pulse penetration inLorentz-Model dielectrics using the

Sommerfeld and Brillouin precursors

Kurt Edmund OughstunCollege of Engineering & Mathematical Sciences, University of Vermont, Burlington, VT

05405-0156 [email protected]

Abstract: Under proper initial conditions, the interrelated effects of phaseand attenuation dispersion in ultrawideband pulse propagation modify theinput pulse into precursor fields. Because of their minimal decay in a givendispersive medium, precursor-type pulses possess optimal penetration intothat material at the frequency-chirped Lambert-Beer’s law limit, makingthem 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 links1. A. Sommerfeld, “Uber die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. 44, 177-202 (1914).2. L. Brillouin, “Uber 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 Dis-

persive Media (Springer, 2006).8. K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive At-

tenuative 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. Osterberg, “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. Osterberg, “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 tumordetection,” 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, andK. 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. Osterberg, “Can precursors improve the transmission ofenergy at optical frequencies?,” J. Mod. Opt. 56 (9), 1083–1090 (2009).

1. Introduction

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

The precursor fields are a characteristic of the material dispersion, the input pulse providingthe requisite spectral energy in the appropriate frequency domain [8,14]. For a single-resonanceLorentz-model dielectric, the dynamical pulse evolution is dominated by an above resonanceSommerfeld precursor and a below resonance Brillouin precursor throughout the mature dis-persion regime [8], whereas for a Debye-model dielectric, the dynamical pulse evolution isdominated by just a low-frequency Brillouin precursor [8, 14]. This is because the peak am-plitude of the Brillouin precursor in either a Lorentz- or Debye-type dielectric decays onlyas the square root of the inverse of the propagation distance while the peak amplitude of theSommerfeld precursor in a Lorentz-type dielectric possesses an exponential decay rate that istypically much smaller than that at the input pulse frequency. This unique property may thenbe used to advantage through the design of precursor-type pulses that possess optimal penetra-tion into a given dispersive material. A detailed analysis of these properties for a Debye-modeldielectric has been presented in [14] with application to the design of a Brillouin pulse that willoptimally penetrate through a given Debye-model dielectric. Applications there include foliageand ground penetrating radar as well as bio-electromagnetic effects due to ultrawideband radarand related devices. Experimental design [15] and observation [16, 17] of Debye-model pre-cursor decay in H2O have been reported, confirming the original analysis [14] describing thisphenomenon. The analysis presented here extends this research (in a nontrivial way) into theoptical domain where the material dispersion is Lorentz-like. Applications include deep tissueimaging [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) =1

2πℜ{

ie−iψ∫ ia+∞

ia−∞u(ω −ωc)e

i(k(ω)z−ωt)dω}

(1)

for all z≥ 0 and fixed a> γa set by the abscissa of absolute convergence [7] for the contour inte-gral in Eq. (1), where A(0, t) = u(t)sin(ωct +ψ) describes the temporal behavior of the initialplane wave pulse at the z = 0 plane with fixed carrier frequency ωc > 0, phase constant ψ , andreal-valued envelope function u(t) with temporal frequency spectrum u(ω) =

∫ ∞−∞ u(t)eiωtdt.

The temporal pulse spectrum A(z,ω) =∫ ∞−∞ A(z, t)eiωtdt satisfies the Helmholtz equation(

∇2 + k2(ω))

A(z,ω) = 0 for all z ≥ 0. Here k(ω) ≡ β (ω)+ iα(ω) = (ω/c)n(ω) is the com-plex 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− ω2

p

ω2 −ω20 +2iδω

)1/2

, (2)

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

√2×1016r/s, and δ = 0.28×1016r/s.

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

A(z, t) =1

2πℜ{

ie−iψ∫ ia+∞

ia−∞u(ω −ωc)e

(z/c)φ(ω,θ)dω}

(3)

for z > 0, whereφ(ω,θ)≡ i

cz

(k(ω)z−ωt

)= iω(n(ω)−θ) (4)

is the complex phase function [2,3,6,8] with non dimensional space-time parameter θ ≡ ct/z. Ifthe input pulse identically vanishes for all t < 0 [i.e., if A(0, t) = 0 ∀ t < 0], then Sommerfeld’srelativistic 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 repre-sentation over the luminal (θ = 1) to subluminal (θ > 1) space-time domain. The asymptoticapproximation of the integral in Eq. (4) as z → ∞ for θ ≥ 1 is determined by the dynamicalevolution of the saddle points of φ(ω,θ) with θ . The condition that φ(ω,θ) is stationary at asaddle point requires that φ ′(ω,θ) = 0, the prime denoting differentiation with respect to ω ,resulting in the saddle point equation

n(ω)+ωn′(ω) = θ . (5)

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. Thedistant saddle points

ω±SPD

(θ) =±ξ (θ)− iδ (1+η(θ)) (6)

evolve with θ ≥ 1 in the high-frequency domain |ω±SPD

(θ)| ≥ ω1 above the region of anomalous

dispersion, where ω1 ≡√

ω20 +ω2

p , while the near saddle points

ω±SPN

(θ) =

{i[±|ψ(θ)|− 2

3 δζ (θ)], 1 < θ ≤ θ1

±ψ(θ)− i 23 δζ (θ), θ ≥ θ1

(7)


4 4.5 5 5.5 6

x 10-15

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-0.05

0

0.05

0.1

0.15

0.2

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t (s)

A(z

,t)

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

evolve with θ > 1 in the low-frequency domain |ω±SPN

(θ)| ≤ ω0 below the medium resonance

frequency, where θ0 ≡ n(0) =√

1+ω2p/ω2

0 and θ1 ≈ θ0 +(2δ 2ω2p)/(3θ0ω4

0 ). Approximate

expressions for the functions ξ (θ), η(θ), ψ(θ), and ζ (θ) in terms of the Lorentz mediumparameters ω0, ωp, and δ are given in Refs. [6, 8–11]. The asymptotic description of the prop-agated pulse may then be expressed either in the form [6, 8]

A(z, t) = As(z, t)+Ab(z, t)+Ac(z, t) (8)

as for the Heaviside step-function signal, or as a linear combination of expressions of thisform, where As(z, t) is the asymptotic contribution from the distant saddle points, Ab(z, t) fromthe near saddle points, and Ac(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 signalcontribution is absent and the pulse evolves into Sommerfeld and Brillouin precursor field com-ponents [8, 12, 13] as illustrated in Fig. 1, an example of pulse-splitting due to dispersion.

3. The Brillouin Pulse

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

Ab(z, t) ∼ ezc α0

{(c/z)2/3

2ℜ{

u(ω+SPN

−ωc)|h+|+ u(ω−SPN

−ωc)|h−|}

Ai

(±|α1|(z/c)2/3

)

− (c/z)2/3

2|α1|1/2ℜ{

u(ω+SPN

−ωc)|h+|− u(ω−SPN

−ωc)|h−|}

A′i

(±|α1|(z/c)2/3

)}(9)

for all space-time points θ > 1. Here Ai(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]

α0(θ)≡ (1/2)[φ(ω+

SPN,θ)+φ(ω−

SPN,θ)

], α1/2

1 (θ)≡{(3/4)

[φ(ω+

SPN,θ)−φ(ω−

SPN,θ)

]}1/3,

and h±(θ)≡[∓2α1/2

1 (θ)/

φ ′′(

ω±SPN

,θ)]1/2

.

9.12 9.125 9.13 9.135 9.14 9.145 9.15 9.155 9.16

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0

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0.4

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t (s)

Pro

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

Because ω+SPN

(θ0) = 0, at which point the complex phase function φ(ω,θ) identicallyvanishes, the peak amplitude point of Ab(z, t) experiences zero exponential attenuation at

the space-time point θ = θ0 ≡ n(0). With Ai(|α1|(z/c)2/3

) ∼ (c/z)1/6

2√

π|α1|1/4 e−(2z/3c)|α1|3/2and

A′i

(|α1|(z/c)2/3)∼− (c/z)1/6

2√

π |α1|1/4e−(2z/3c)|α1|3/2, Eq. (9) becomes

Ab(z, t)∼ (c/z)1/2

4√

π|α1(θ)|1/4ℜ{

i[u(ω+

SPN−ωc)|h+|+ u(ω−

SPN−ωc)|h−|

]}e(z/c)φ(ω+

SPN,θ)

(10)

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

Ab(z,θ0z/c)∼ (c/z)1/2

4√

π|α1(θ0)|1/4ℜ{

i[u(ω+

SPN−ωc)|h+(θ0)|+ u(ω−

SPN−ωc)|h−(θ0)|

]}(11)

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

ωe f f (θ0)≈ 3πθ0ω40

4δ 2ω2pΔ f

cz

(12)

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

ωb(θ)≈ ℜ{

ω+SPN

(θ)}= ψ(θ) (13)


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 = 10zd where

zd ≡ α−1(ωc)] due to a half-cycle rectangular envelope pulse ur(t) = 1 for 0 < t < T andzero otherwise with spectrum ur(ω) = (eiωt −1)/iω and a half-cycle gaussian envelope pulseug(t) = e−(t−τ0)

2/T 2centered at t = τ0 with spectrum ug(ω) =

√πTe−(ωT/2)2eiωτ0 , each with

initial pulse width 2T ≈ 1.571×10−16s at the below resonance carrier frequency ωc = ω0/2, isillustrated in Fig. 2. Notice that the peak amplitude for each precursor pulse has been normal-ized to unity and shifted to the same instant of time t0 = θ0z/c. Any difference between the twoprecursor pulses is due to the difference in the input pulse spectra. This difference disappearsas the initial pulse width decreases while the peak amplitude increases such that the pulse arearemains constant and a delta-function pulse is approached.

Because the Brillouin pulse spectrum is ultra-wideband and hence, relatively flat for frequen-cies below ωc, accurate numerical determination of its effective peak frequency value becomesincreasingly difficult as the propagation distance increases. As an illustration, consider the nu-merically determined peak frequency evolution of a below resonance (ωc = ω0/2) single-cyclegaussian envelope pulse illustrated in Fig. 3. The blue data points and dashed curve are obtained

0 2 4 6 8 10 12 14 16 18 2010

15

1016

1017

z/zd(ω

c)

ωef

f (r/

s)

(a)

(b)

(c)

Fig. 3. Effective angular frequency of the peak Brillouin precursor amplitude as a functionof the relative penetration depth z/zd(ωc) as given by (a) the asymptotic estimate in Eq.(13) with Δ f = 11, (b) the numerically determined peak amplitude in the propagated pulsespectrum, and (c) the numerically measured period Te f f about the peak amplitude point.

from numerical measurements of the half-period of the numerically determined Brillouin pre-cursor at the given penetration depth, the solid blue curve depicts the behavior of the asymptoticestimate (12) with Δ f = 11, and the green data points and dashed curve are obtained from thepeak amplitude in the propagated pulse spectrum. Both dashed curves describe a cubic spline fitto the corresponding computed data points. Because the measured period Te f f of the Brillouinprecursor over-estimates the actual period, the estimate of ωe f f from Te f f provides a lowerbound to ωe f f (z). Notice that for z/zd(ωc) > 8, the numerical error in the spectral measure ofωe f f (z) increases as the propagated pulse spectrum becomes increasingly ultra-wideband andflattens out below ωc, resulting in an erroneously rapid decrease of ωe f f (z) to zero instead ofthe correct asymptotic z−1 behavior as z → ∞. Finally, notice that the Brillouin precursor isnot 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.

0 2 4 6 8 10 12 14 16 18 2010

-2

10-1

100

z/zd(ω

c)

Pea

k A

mpl

itude

(a)

(b)

e-zα(ωc)

e-zα(ωeff(z))

Fig. 4. Peak amplitude decay of the Brillouin precursor for (a) a Heaviside step-functionsignal uH(t)sin(ωct) and (b) a single-cycle gaussian envelope pulse ug(t)cos(ωct). Thelower dashed curve describes exponential signal decay e−zα(ωc) at the input carrier fre-quency ωc = ω0/2 and the upper dashed curve describes the frequency-chirped Lambert-Beer’s law limit given by e−zα(ωe f f (z)) from Eq. (14). In the immature dispersion regime,both pulses decay at or near to the signal rate e−zα(ωc), but as the propagation distance en-ters the mature dispersion regime, the Brillouin precursor emerges with a decreased decayrate approaching the characteristic z−1/2 asymptotic dependence. This transition betweenimmature and mature dispersion regimes occurs at z/zd(ωc) � 2.5 for the step-functionsignal and at z/zd(ωc)� 1.5 for the single-cycle gaussian envelope pulse.

The numerically determined peak amplitude decay of the Brillouin precursor for (a) a Heav-iside step-function signal uH(t) (blue data points and dashed cubic spline fit curve) and (b) asingle-cycle gaussian envelope pulse (green data points and dashed cubic spline fit curve) areillustrated in Fig. 4 for the below resonance case with ωc = ω0/2. The dashed black curve inthe figure describes the pure exponential decay e−z/zd(ωc) at the input pulse carrier frequency.Notice that the peak amplitude decay of the step-function signal Brillouin precursor initiallyfollows this exponential decay for z < zd(ωc), referred to as the immature dispersion regime [5]wherein the Brillouin precursor is formed by the material dispersion. In the mature dispersionregime z > zd , the Brillouin precursor is well-defined and increasingly satisfies the characteris-tic z−1/2 peak amplitude decay described by Eq. (11), resulting in a significant departure frompure exponential decay. The peak amplitude decay of the single-cycle gaussian envelope pulseexhibits a similar dependence with a somewhat more rapid initial decay in the immature dis-persion region z < zd(ωc) that is below the signal decay e−zα(ωc) followed by a transition to thecharacteristic z−1/2 peak amplitude decay for z > zd(ωc).

Because the attenuation factor α(ωe f f (z)) varies with the propagation distance z throughωe f f (z), illustrated in Fig. 3, a proper comparison of the peak amplitude decay of the Brillouinprecursor must be made with the decay factor

A(z/zd(ωc))/A0 = exp

{−∫ z/zd(ωc)

0α(ωe f f (ζ ))dζ

}(14)


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

0 2 4 6 8 10 12 14 16 18 200.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

z/zd(ω

c)

Pea

k A

mpl

itude

e-zα(ωeff(z))

Brillouin Precursor Pulse

e-zα(ωeff(0))

Fig. 5. Peak amplitude decay of the Brillouin precursor pulse (blue data points and dashedcurve). The black dashed curve describes exponential decay e−zα(ωe f f (0)) at the initial (z =0) effective oscillation frequency ωe 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(ωe f f (z))

from Eq. (14).

The peak amplitude decay of a Brillouin precursor pulse, depicted by the blue data pointsand 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 dis-tances z ≥ 0. The Brillouin precursor illustrated in Fig. 2 is an example of this Brillouin pulseand is used here as the initial pulse for that Lorentz-model dielectric. The optimal penetrationof this Brillouin pulse presented in Fig. 5 dominates that for both Brillouin precursors presentedin Fig. 4. Most importantly, to within the numerical accuracy of the numerical calculations pre-sented here, it obeys the frequency-chirped Lambert-Beer’s law limit described by Eq. (14).Notice that e−zα(ωe f f (z)) ≈ e−zα(ωe f f (0)) when z ≈ 0 so that the initial Lambert-Beer’s law de-cay follows the simple exponential decay given by e−zα(ωe f f (0)), departing from it as ωe f f (z)decreases with increasing penetration depth z > 0, whereas the Brillouin precursor pulse ex-periences 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-resonancedispersion properties of the dielectric material.


4. The Sommerfeld Pulse

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

As(z, t)∼ ℜ{[

2α(θ)e−i π2

]νe−i z

c β (θ)e−iψ[γ0Jν

(α(θ)

zc

)+2α(θ)e−i π

2 γ1Jν+1

(α(θ)

zc

)]}(15)

for all θ ≥ 1. Here Jν(ζ ) is the Bessel function of the first kind of real order ν that isdetermined by the behavior of the spectral amplitude function u(ω) in the following man-ner [6,8,10,11]: let u(ω) =ω−(1+ν)q(ω) for large |ω| with ν > 0, where q(ω) possesses a Lau-rent series expansion convergent for |ω| ≥ R > 0 and is such that lim|ω|→∞ q(ω) = 0; then ν =

lim|ω|→∞

{log |q(ω)|−log |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 appear-

ing in Eq. (15) are α(θ)≡ i2

[φ(ω+

SPD,θ)−φ(ω−

SPD,θ)

], β (θ)≡ i

2

[φ(ω+

SPD,θ)+φ(ω−

SPD,θ)

],

2γ0 ≡ u(ω+SPD

−ωc)[2α]−(1+ν)[

4α3

iφ ′′(ω+SPD

)

]1/2

+ u(ω−SPD

−ωc)[−2α]−(1+ν)[− 4α3

iφ ′′(ω−SPD

)

]1/2

, and

2γ1 ≡ u(ω+SPD

−ωc)[2α]−(1+ν)[

4α3

iφ ′′(ω+SPD

)

]1/2

− u(ω−SPD

−ωc)[−2α]−(1+ν)[− 4α3

iφ ′′(ω−SPD

)

]1/2

. For

θ > 1, each Bessel function in the uniform expansion (15) may be replaced by its asymptoticapproximation Jν(ζ ) ∼

√2/πζ cos(ζ −νπ/2−π/4) as ζ → ∞ with |arg(ζ )| < π , with the

result [9]

AS(z, t)∼√

2cπz

ℜ{(

2α(θ)e−iπ/2)ν

e−iβ (θ)z/c[

γ0

α1/2cos(ζ (θ))+2γ1α1/2 sin(ζ (θ))

]}(16)

as z → ∞ with θ > 1+Δ, Δ > 0, where ζ (θ)≡ α(θ)z/c−νπ/2−π/4.

7.8 7.9 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

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(a)

(b)

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


0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

100

z/zd(ω

c)

Pea

k A

mpl

itude

(a)

(b)e-zα(ωc)

e-zα(ωeffg(z))

e-zα(ωeffh(z))

Fig. 7. Peak amplitude decay of the Sommerfeld precursor for (a) a Heaviside step-functionsignal uH(t)sin(ωct) and (b) a single-cycle gaussian envelope pulse ug(t)cos(ωct). Theblack dashed curve describes exponential decay e−zα(ωc) at the carrier frequency ωc =2.5ω0 and the blue and green dashed curves describe the frequency-chirped Lambert-Beer’s

law limit given by e−zα(ωe f f j (z)) for the step-function ( j = H) and gaussian ( j = g) Som-merfeld precursors, respectively. In the immature dispersion regime, the step-function sig-nal decays at the signal rate e−zα(ωc), the transition to mature dispersion indicated by theabrupt change in behavior between z/zd � 1.5 and z/zd � 2.0 as the Sommerfeld precursoremerges, decaying at a much slower rate than the signal. The single-cycle gaussian pulsepulse decays faster than the signal rate in the immature dispersion regime, the transition tomature dispersion indicated by the change in behavior between z/zd � 0.8 and z/zd � 1.0as 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 pre-cursor occurs just after the field arrival at the luminal space-time point θ ≡ ct/z = 1. Sinceβ (1) = 0 for the Heaviside step-function signal, the Sommerfeld precursor front experienceszero attenuation with zero amplitude for integer ν ≥ 0. Unlike the Brillouin precursor, the peakamplitude of the Heaviside step-function Sommerfeld precursor experiences a small exponen-tial decay with propagation distance in addition to the z−1/2 asymptotic behavior appearingin Eq. (16). Notice that, because of the finite computation size imposed by computer memorylimitations, the Sommerfeld precursor evolution displayed in Fig. 6 is a ω ∈ [0,ωmax] low-passfiltered version of the actual Sommerfeld precursor evolution where, for the numerical resultspresented here, sampling criteria sets ωmax � 3.14×1021r/s. With this in mind, the solid bluecurve in Fig. 7 displays a linear spline fit to the numerically determined peak amplitude decay ofthe Sommerfeld precursor appearing in the dynamical field evolution due to an above-resonance(ωc = 2.5ω0) Heaviside step-function signal. The initial peak amplitude decay of this precur-sor follows the exponential decay factor e−α(ωc)z of the signal with fixed carrier frequency ωc.Between z/zd � 1.5 and z/zd � 2.0, the Sommerfeld precursor emerges from the propagatedfield structure, as indicated by the change in behavior of the peak amplitude decay, decaying ata much slower rate for z/zd > 2 and exceeding the signal amplitude e−α(ωc)z when z/zd > 3.A numerical determination of the effective oscillation frequency ωe f fh(z) from the measuredperiod of the field about the peak amplitude point is illustrated in Fig. 8 by the blue data pointsand dashed curve. Comparison of this numerical measurement of the effective oscillation fre-


0 1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

3

3.5

4

4.5

5

5.5

z/zd(ω

c)

ωm

ax/ω

c

ωeffg

ωeffh

ωeffae/ωc

/ωc

/ωc

Fig. 8. Relative effective angular frequency ωe f f /ωc of the peak Sommerfeld precursoramplitude as a function of the relative penetration depth z/zd(ωc) as derived from a nu-merical evaluation of the asymptotic approximation (15) of the Sommerfeld precursor forthe Heaviside step-function signal (ωe f fas/ωc), black data points and dashed curve), thenumerically measured period Te f fas about the peak amplitude point in the step-functionSommerfeld precursor (ωe f fh/ωc), blue data points and dashed curve), and the numericallydetermined peak amplitude in the propagated gaussian pulse spectrum (ωe f fg/ωc), greendata points and dashed curve).

quency ωe f fh(z) with that obtained from the asymptotic approximation (15) of the Sommerfeldprecursor field for the Heaviside step-function signal, denoted by ωe f fas(z) and described by theblack data points and dashed curve in Fig. 8, shows that ωe f fh(z) has been decreased from itsasymptotic behavior, this being due to the low-pass filter cut-off at ωmax inherent in the numeri-cal field calculations. Computation of the resultant frequency-chirped Lambert-Beer’s law limitfrom Eq. (14) using ωe f fh(z) for the numerically determined Sommerfeld precursor evolutionis 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/zd = 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 determinedpeak amplitude decay of the Sommerfeld precursor due to a single-cycle gaussian envelopepulse with the same above-resonance carrier frequency ωc = 2.5ω0 and initial pulse width2T = 4π/ωc � 0.1257 f s. An illustration of this precursor evolution at eight absorption depths[z/zd(ωc) = 8] is depicted by the green curve in Fig. 6. Notice that, unlike the Sommerfeldprecursor 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(ωc)z at the initialcarrier frequency, it decays at a much slower rate for z/zd > 1, exceeding the e−α(ωc)z exponen-tial decay factor for z/zd > 2.0. A numerical determination of the effective oscillation frequencyfrom the magnitude peak in the propagated pulse spectrum is described by the green data pointsand dashed curve in Fig. 8. The resultant computation of the corresponding frequency-chirpedLambert-Beer’s law limit from Eq. (14) for this gaussian Sommerfeld precursor evolution isdisplayed by the green dashed curve in Fig. 7. Because significant energy is lost from the inputgaussian 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/zd > 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 basedon a numerical synthesis of Eq. (1). The frequency structure of the dispersive Lorentz-modeldielectric and pulse spectra were sampled from fmin = 2.98×1013Hz to fmax = 5.0×1020Hz.This is adequate to properly sample both the material dispersion and the spectral content ofthe Brillouin and ( fmax low-pass filtered) Sommerfeld precursors generated respectively by thebelow and above resonance input pulses considered here.

Nearly identical results to those presented here may be obtained through use of the weakdispersion equivalence relation given in §15.10.2 in [8] when the plasma frequency ωp ≡√

Nq2e/me is changed through the number density N. For example, when the plasma frequency

is reduced from the value ωp = 2x1016r/s used in the numerical examples presented here, cor-responding to highly absorptive media with micron scale absorption depth zd(ωc) � 2.1μm,to the value ωp = 1x1012r/s, the dielectric becomes weakly dispersive with kilometer scaleabsorption depth zd(ωc) � 389m, while the peak amplitude behavior presented here remainsessentially unchanged.

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

The experimental results and subsequent analysis of the Brillouin precursor behavior pre-sented in Refs. [15–17, 21] have overlooked the fact that the Brillouin precursor evolution inwater is determined by low-frequency dispersion (below infrared) of water due to Debye relax-ation, as described in detail in Refs. [8,14]; in particular, see §13.4 in [8]. It is established therethat this Debye-model Brillouin precursor is not a dc field just as the Lorentz-model Brillouinprecursor 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 beobtained from Eq. (9) with u(ω −ωc) = i/(ω −ωc) and z = zd(ωc)≡ α−1(ωc) where the fixedangular frequency ωc is chosen to be near the resonance frequency ω0 of the medium so asto eliminate the possible appearance of an artifact resonance peak in the Brillouin precursorwhen the near saddle point ω+

SPN(θ) passes near the simple pole singularity at ω = ωc when

0 < ωc < ω0 (see §9.2 in [6] and §15.5 in [8]). This complication can be avoided (as was donefor the numerical results presented here) by numerically computing the propagated field dueto either an input Heaviside step function pulse or a half-cycle gaussian envelope pulse withbelow resonance carrier frequency at a sufficiently large penetration distance such that (for theHeaviside step-function case) the signal contribution is exponentially negligible in comparisonto the Brillouin precursor (typically z = 3zd is sufficiently large). In addition, the Heavisidestep-function signal was computed from the leading edge of a rectangular envelope pulse withsufficient temporal duration that the dispersive effects of the trailing edge were well-removedfrom the leading-edge precursors (see §15.6.1 in [8]). Similar remarks hold for the numericalgeneration of the Sommerfeld pulse. Laboratory generation of these optimal precursor pulsesis 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 asmines and submarines.


Acknowledgment

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


optimal pulse penetration in lorentz-model dielectrics ... · persive media (springer, 2006). 8. k....

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