Abstract

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

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References

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  1. A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. 44, 177–202 (1914).
    [Crossref]
  2. L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. 44, 203–240 (1914).
    [Crossref]
  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.
    [Crossref]
  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).
    [Crossref]
  6. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, (Springer, 1994).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  11. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49(4), 628–648 (2007).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  14. K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop. 53, 1582–1590 (2005).
    [Crossref]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]

2009 (1)

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).
[Crossref] [PubMed]

2007 (2)

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).
[Crossref]

N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49(4), 628–648 (2007).
[Crossref]

2006 (4)

2005 (2)

K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop. 53, 1582–1590 (2005).
[Crossref]

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).
[Crossref] [PubMed]

1999 (1)

1997 (1)

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).
[Crossref]

1988 (2)

1981 (1)

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).
[Crossref]

1914 (2)

A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. 44, 177–202 (1914).
[Crossref]

L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

Agate, B.

Agha, I. H.

Basu, S.

Bessette, J.

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).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principals of Optics, 7th (expanded) ed. (Cambridge University Press, 1999), Ch. 1.
[Crossref]

Brillouin, L.

L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).

Brown, C. T. A.

Cartwright, N. A.

N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49(4), 628–648 (2007).
[Crossref]

Dholakia, K.

Dietsche, G.

Fischer, P.

Fox, A. E.

Gaeta, A. L.

Garmire, E.

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).
[Crossref] [PubMed]

Geraghty, D. F.

Gibson, U. J.

Gisler, T.

Gunn-Moore, F.

Jaillon, F.

Jeong, H.

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).
[Crossref] [PubMed]

Li, J.

Lukofsky, D.

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).
[Crossref] [PubMed]

Maret, G.

Mitra, K.

Okawachi, Y.

Österberg, U.

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).
[Crossref] [PubMed]

A. E. Fox and U. Österberg, “Observation of non-exponential absorption of ultra-fast pulses in water,” Opt. Express 14 (8), 3688–3693 (2006).
[Crossref] [PubMed]

Österberg, U. L.

Oughstun, K. E.

N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49(4), 628–648 (2007).
[Crossref]

K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop. 53, 1582–1590 (2005).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, (Springer, 1994).
[Crossref]

K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media (Springer, 2006).

K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive Attenuative Media (Springer, 2009).

Pal, G.

Riches, A.

Sherman, G. C.

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, (Springer, 1994).
[Crossref]

Sibbett, W.

Slepkov, A. D.

Sommerfeld, A.

A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. 44, 177–202 (1914).
[Crossref]

Stevenson, D.

Tsampoula, X.

Vo-Dinh, T.

Wolf, E.

M. Born and E. Wolf, Principals of Optics, 7th (expanded) ed. (Cambridge University Press, 1999), Ch. 1.
[Crossref]

Xiao, H.

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).
[Crossref]

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).
[Crossref]

Ann. Phys. (2)

A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. 44, 177–202 (1914).
[Crossref]

L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Ant. Prop. (1)

K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop. 53, 1582–1590 (2005).
[Crossref]

J. Mod. Opt. (1)

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).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

Opt. Express (4)

Phys. Rev. Lett. (2)

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).
[Crossref]

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).
[Crossref]

SIAM Rev. (1)

N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49(4), 628–648 (2007).
[Crossref]

Other (5)

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, (Springer, 1994).
[Crossref]

K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media (Springer, 2006).

K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive Attenuative Media (Springer, 2009).

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).

M. Born and E. Wolf, Principals of Optics, 7th (expanded) ed. (Cambridge University Press, 1999), Ch. 1.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Pulse splitting into Sommerfeld and Brillouin precursor components due to an above-resonance (ωc = 2.5ω0) half-cycle gaussian envelope pulse at one absorption depth [z = zd(ωc) ≡ α1(ωc)] in a single resonance Lorentz-model dielectric.
Fig. 2
Fig. 2 Brillouin precursor evolution due to a below-resonance (ωc = ω0/2) (a) half-cycle rectangular envelope pulse (blue curve) and (b) half-cycle gaussian envelope pulse (green curve) at ten absorption depths (z/zd(ωc) = 10) in a single resonance Lorentz dielectric.
Fig. 3
Fig. 3 Effective angular frequency of the peak Brillouin precursor amplitude as a function of the relative penetration depth z/zd(ωc) as given by (a) the asymptotic estimate in Eq. (13) with Δf = 11, (b) the numerically determined peak amplitude in the propagated pulse spectrum, and (c) the numerically measured period Teff about the peak amplitude point.
Fig. 4
Fig. 4 Peak amplitude decay of the Brillouin precursor for (a) a Heaviside step-function signal uH(t)sin(ωct) and (b) a single-cycle gaussian envelope pulse ug(t)cos(ωct). The lower dashed curve describes exponential signal decay e z α ( ω c ) at the input carrier frequency ωc = 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 z1/2 asymptotic dependence. This transition between immature and mature dispersion regimes occurs at z/zd(ωc) ≃ 2.5 for the step-function signal and at z/zd(ωc) ≃ 1.5 for the single-cycle gaussian envelope pulse.
Fig. 5
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 ωeff(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).
Fig. 6
Fig. 6 Sommerfeld precursor evolution due to an above-resonance (ωc = 2.5ω0) (a) Heaviside step-function signal (blue curve) and (b) single-cycle gaussian envelope pulse (green curve) at eight absorption depths (z/zd(ωc) = 8). Notice that the the peak amplitude for each precursor has been normalized to unity and shifted to the same instant of time.
Fig. 7
Fig. 7 Peak amplitude decay of the Sommerfeld precursor for (a) a Heaviside step-function signal uH(t)sin(ωct) and (b) a single-cycle gaussian envelope pulse ug(t)cos(ωct). The black dashed curve describes exponential decay e z α ( ω c ) 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 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/zd ≃ 1.5 and z/zd ≃ 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/zd ≃ 0.8 and z/zd ≃ 1.0 as the Sommerfeld precursor emerges, decaying at a slower rate than the signal.
Fig. 8
Fig. 8 Relative effective angular frequency ωeffc of the peak Sommerfeld precursor amplitude as a function of the relative penetration depth z/zd(ωc) 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).

Equations (16)

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A ( z , t ) = 1 2 π { i e i ψ i a i a + u ˜ ( ω ω c ) e i ( k ˜ ( ω ) z ω t ) d ω }
n ( ω ) = ( 1 ω p 2 ω 2 ω 0 2 + 2 i δ ω ) 1 / 2 ,
A ( z , t ) = 1 2 π { i e i ψ i a i a + u ˜ ( ω ω c ) e ( z / c ) ϕ ( ω , θ ) d ω }
ϕ ( ω , θ ) i c z ( k ˜ ( ω ) z ω t ) = i ω ( n ( ω ) θ )
n ( ω ) + ω n ( ω ) = θ
ω S P D ± ( θ ) = ± ξ ( θ ) i δ ( 1 + η ( θ ) )
ω S P N ± ( θ ) = { i [ ± | ψ ( θ ) | 2 3 δ ζ ( θ ) ] , 1 < θ θ 1 ± ψ ( θ ) i 2 3 δ ζ ( θ ) , θ θ 1
A ( z , t ) = A s ( z , t ) + A b ( z , t ) + A c ( z , t )
A b ( z , t ) ~ e z c α 0 { ( c / z ) 2 / 3 2 { u ˜ ( ω S P N + ω c ) | h + | + u ˜ ( ω S P N ω c ) | h | } A i ( ± | α 1 | ( z / c ) 2 / 3 ) ( c / z ) 2 / 3 2 | α 1 | 1 / 2 { u ˜ ( ω S P N + ω c ) | h + | u ˜ ( ω S P N ω c ) | h | } A 1 ( ± | α 1 | ( z / c ) 2 / 3 ) }
A b ( z , t ) ~ ( c / z ) 1 / 2 4 π | α 1 ( θ ) | 1 / 4 { i [ u ˜ ( ω S P N + ω c ) | h + | + u ˜ ( ω S P N ω c ) | h | ] } e ( z / c ) ϕ ( ω S P N + , θ )
A b ( z , θ 0 z / c ) ~ ( c / z ) 1 / 2 4 π | α 1 ( θ 0 ) | 1 / 4 { i [ u ˜ ( ω S P N + ω c ) | h + ( θ 0 ) | + u ˜ ( ω S P N ω c ) | h ( θ 0 ) | ] }
ω e f f ( θ 0 ) 3 π θ 0 ω 0 4 4 δ 2 ω p 2 Δ f c z
ω b ( θ ) { ω S P N + ( θ ) } = ψ ( θ )
A ( z / z d ( ω c ) ) / A 0 = exp { 0 z / z d ( ω c ) α ( ω e f f ( ζ ) ) d ζ }
A s ( z , t ) ~ { [ 2 α ˜ ( θ ) e i π 2 ] ν e 1 z c β ( θ ) e i ψ [ γ 0 J ν ( α ˜ ( θ ) z c ) + 2 α ˜ ( θ ) e i π 2 γ 1 J ν + 1 ( α ˜ ( θ ) z c ) ] }
A S ( z , t ) ~ 2 c π z { ( 2 α ˜ ( θ ) e i π / 2 ) ν e i β ˜ ( θ ) z / c [ γ 0 a ˜ 1 / 2 cos ( ζ ( θ ) ) + 2 γ 1 a ˜ 1 / 2 sin ( ζ ( θ ) ) ] }

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