Abstract

We study, numerically, the transitional behavior of precursors in the intermediate spectral regime between two contrasting regimes for which analytical solutions exist. The two opposite regimes are, respectively, defined as (1) off-resonance carrier with a spectral pulse width comparable to the medium’s absorption linewidth and (2) on-resonance carrier with the spectral width of the pulse much narrower than the absorption linewidth. Beyond these two regimes, there have been few studies of precursors. In this paper, we investigate precursor dynamics for the intermediate spectral regime using the finite-difference time domain method. The parameter that has the strongest influence on the precursor dynamics is found to be the plasma frequency, ωp. This research is applicable for controlling transients in fast communication systems.

© 2009 Optical Society of America

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References

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  1. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
  2. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).
  3. H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
    [CrossRef] [PubMed]
  4. W. R. LeFew, “Optical Precursor Behavior,” Duke UniversityPh.D. Dissertation, unpublished (2007).
  5. E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: Application to sound waves in superfluid 3B,” Phys. Rev. B 34, 7617-7640 (1986).
    [CrossRef]
  6. M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604-1611 (1970).
    [CrossRef]
  7. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
    [CrossRef]
  8. W. M. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antennas Propag. 45, 401-410 (1996).
    [CrossRef]
  9. S. C. Hagness, R. M. Joseph, and A. Taflove, “Direct time integration of Maxwell's equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses.” Opt. Lett. 16, 1412-1414 (1991).
    [CrossRef] [PubMed]
  10. R. Safian, C. D. Sarris, and M. Mojahedi, “Joint time-frequency and finite-difference time-domain analysis of precursor fields in dispersive media,” Phys. Rev. E 73, 066602-066610 (2006).
    [CrossRef]
  11. R. Safian, M. Mojahedi, and C. D. Sarris, “Asymptotic description of wave propagation in a Lorentzian medium,” Phys. Rev. E 75, 066611-066613 (2007).
    [CrossRef]
  12. A. Karlsson and S. Rikte, “Time-domain theory of forerunners,” J. Opt. Soc. Am. B 15, 487-502 (1998).
    [CrossRef]
  13. K. E. Oughstun, “Computational methods in ultrafast time-domain optics,” Comput. Sci. Eng. 5, 22-32 (2003).
    [CrossRef]
  14. K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210-2213 (1996).
    [CrossRef] [PubMed]
  15. W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
    [CrossRef]
  16. H. Jeong and U. L. Österberg, “Coherently transients: optical precursors and 0π pulses,” J. Opt. Soc. Am. B 25, B1-B5 (2008).
    [CrossRef]
  17. N. A. Cartwright and K. E. Oughstun, “Pulse centroid velocity of the Poynting vector,” J. Opt. Soc. Am. A 21, 439-450 (2004).
    [CrossRef]
  18. H. Jeong and U. L. Osterberg, “Steady-state pulse component in ultrafast pulse propagation in anomalously dispersive dielectric,” Phys. Rev. A 77, 021803(R) (2008).
    [CrossRef]
  19. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642-645 (1997).
    [CrossRef]
  20. J. D. Jackson, Electrodynamics, 3rd ed. (Wiley, 1999), pp. 322-339.

2009 (1)

W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
[CrossRef]

2008 (2)

H. Jeong and U. L. Osterberg, “Steady-state pulse component in ultrafast pulse propagation in anomalously dispersive dielectric,” Phys. Rev. A 77, 021803(R) (2008).
[CrossRef]

H. Jeong and U. L. Österberg, “Coherently transients: optical precursors and 0π pulses,” J. Opt. Soc. Am. B 25, B1-B5 (2008).
[CrossRef]

2007 (1)

R. Safian, M. Mojahedi, and C. D. Sarris, “Asymptotic description of wave propagation in a Lorentzian medium,” Phys. Rev. E 75, 066611-066613 (2007).
[CrossRef]

2006 (2)

R. Safian, C. D. Sarris, and M. Mojahedi, “Joint time-frequency and finite-difference time-domain analysis of precursor fields in dispersive media,” Phys. Rev. E 73, 066602-066610 (2006).
[CrossRef]

H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

K. E. Oughstun, “Computational methods in ultrafast time-domain optics,” Comput. Sci. Eng. 5, 22-32 (2003).
[CrossRef]

1998 (1)

A. Karlsson and S. Rikte, “Time-domain theory of forerunners,” J. Opt. Soc. Am. B 15, 487-502 (1998).
[CrossRef]

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, 642-645 (1997).
[CrossRef]

1996 (2)

K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210-2213 (1996).
[CrossRef] [PubMed]

W. M. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antennas Propag. 45, 401-410 (1996).
[CrossRef]

1991 (1)

1986 (1)

E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: Application to sound waves in superfluid 3B,” Phys. Rev. B 34, 7617-7640 (1986).
[CrossRef]

1970 (1)

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604-1611 (1970).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Avenel, O.

E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: Application to sound waves in superfluid 3B,” Phys. Rev. B 34, 7617-7640 (1986).
[CrossRef]

Balictsis, C. M.

K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210-2213 (1996).
[CrossRef] [PubMed]

Brillouin, L.

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

Cartwright, N. A.

Crisp, M. D.

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604-1611 (1970).
[CrossRef]

Dawes, A. M. C.

H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
[CrossRef] [PubMed]

Gauthier, D. J.

W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
[CrossRef]

H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
[CrossRef] [PubMed]

Hagness, S. C.

Jackson, J. D.

J. D. Jackson, Electrodynamics, 3rd ed. (Wiley, 1999), pp. 322-339.

Jeong, H.

H. Jeong and U. L. Österberg, “Coherently transients: optical precursors and 0π pulses,” J. Opt. Soc. Am. B 25, B1-B5 (2008).
[CrossRef]

H. Jeong and U. L. Osterberg, “Steady-state pulse component in ultrafast pulse propagation in anomalously dispersive dielectric,” Phys. Rev. A 77, 021803(R) (2008).
[CrossRef]

H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
[CrossRef] [PubMed]

Joseph, R. M.

Karlsson, A.

A. Karlsson and S. Rikte, “Time-domain theory of forerunners,” J. Opt. Soc. Am. B 15, 487-502 (1998).
[CrossRef]

LeFew, W. R.

W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
[CrossRef]

W. R. LeFew, “Optical Precursor Behavior,” Duke UniversityPh.D. Dissertation, unpublished (2007).

Mojahedi, M.

R. Safian, M. Mojahedi, and C. D. Sarris, “Asymptotic description of wave propagation in a Lorentzian medium,” Phys. Rev. E 75, 066611-066613 (2007).
[CrossRef]

R. Safian, C. D. Sarris, and M. Mojahedi, “Joint time-frequency and finite-difference time-domain analysis of precursor fields in dispersive media,” Phys. Rev. E 73, 066602-066610 (2006).
[CrossRef]

Osterberg, U. L.

H. Jeong and U. L. Osterberg, “Steady-state pulse component in ultrafast pulse propagation in anomalously dispersive dielectric,” Phys. Rev. A 77, 021803(R) (2008).
[CrossRef]

Österberg, U. L.

Oughstun, K. E.

N. A. Cartwright and K. E. Oughstun, “Pulse centroid velocity of the Poynting vector,” J. Opt. Soc. Am. A 21, 439-450 (2004).
[CrossRef]

K. E. Oughstun, “Computational methods in ultrafast time-domain optics,” Comput. Sci. Eng. 5, 22-32 (2003).
[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, 642-645 (1997).
[CrossRef]

K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210-2213 (1996).
[CrossRef] [PubMed]

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).

Rappaport, C. M.

W. M. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antennas Propag. 45, 401-410 (1996).
[CrossRef]

Rikte, S.

A. Karlsson and S. Rikte, “Time-domain theory of forerunners,” J. Opt. Soc. Am. B 15, 487-502 (1998).
[CrossRef]

Safian, R.

R. Safian, M. Mojahedi, and C. D. Sarris, “Asymptotic description of wave propagation in a Lorentzian medium,” Phys. Rev. E 75, 066611-066613 (2007).
[CrossRef]

R. Safian, C. D. Sarris, and M. Mojahedi, “Joint time-frequency and finite-difference time-domain analysis of precursor fields in dispersive media,” Phys. Rev. E 73, 066602-066610 (2006).
[CrossRef]

Sarris, C. D.

R. Safian, M. Mojahedi, and C. D. Sarris, “Asymptotic description of wave propagation in a Lorentzian medium,” Phys. Rev. E 75, 066611-066613 (2007).
[CrossRef]

R. Safian, C. D. Sarris, and M. Mojahedi, “Joint time-frequency and finite-difference time-domain analysis of precursor fields in dispersive media,” Phys. Rev. E 73, 066602-066610 (2006).
[CrossRef]

Sherman, G. C.

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).

Taflove, A.

Varoquaux, E.

E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: Application to sound waves in superfluid 3B,” Phys. Rev. B 34, 7617-7640 (1986).
[CrossRef]

Venakides, S.

W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
[CrossRef]

Weedon, W. M.

W. M. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antennas Propag. 45, 401-410 (1996).
[CrossRef]

Williams, G. A.

E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: Application to sound waves in superfluid 3B,” Phys. Rev. B 34, 7617-7640 (1986).
[CrossRef]

Xiao, H.

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642-645 (1997).
[CrossRef]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Comput. Sci. Eng. (1)

K. E. Oughstun, “Computational methods in ultrafast time-domain optics,” Comput. Sci. Eng. 5, 22-32 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

W. M. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antennas Propag. 45, 401-410 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

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

H. Jeong and U. L. Österberg, “Coherently transients: optical precursors and 0π pulses,” J. Opt. Soc. Am. B 25, B1-B5 (2008).
[CrossRef]

A. Karlsson and S. Rikte, “Time-domain theory of forerunners,” J. Opt. Soc. Am. B 15, 487-502 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (3)

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604-1611 (1970).
[CrossRef]

W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
[CrossRef]

H. Jeong and U. L. Osterberg, “Steady-state pulse component in ultrafast pulse propagation in anomalously dispersive dielectric,” Phys. Rev. A 77, 021803(R) (2008).
[CrossRef]

Phys. Rev. B (1)

E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: Application to sound waves in superfluid 3B,” Phys. Rev. B 34, 7617-7640 (1986).
[CrossRef]

Phys. Rev. E (2)

R. Safian, C. D. Sarris, and M. Mojahedi, “Joint time-frequency and finite-difference time-domain analysis of precursor fields in dispersive media,” Phys. Rev. E 73, 066602-066610 (2006).
[CrossRef]

R. Safian, M. Mojahedi, and C. D. Sarris, “Asymptotic description of wave propagation in a Lorentzian medium,” Phys. Rev. E 75, 066611-066613 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
[CrossRef] [PubMed]

K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210-2213 (1996).
[CrossRef] [PubMed]

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642-645 (1997).
[CrossRef]

Other (4)

J. D. Jackson, Electrodynamics, 3rd ed. (Wiley, 1999), pp. 322-339.

W. R. LeFew, “Optical Precursor Behavior,” Duke UniversityPh.D. Dissertation, unpublished (2007).

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

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).

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Figures (9)

Fig. 1
Fig. 1

Illustration of optical precursor behaviors in two different parameter regimes: (a) Brillouin’s parameter regime versus (b) resonant precursor regime.

Fig. 2
Fig. 2

Step-modulated pulse propagation for Brillouin’s parameter regime [blue solid lines]: ω c = 0.25 ω 0 , ω p = 1.1175 ω 0 , and δ = 0.07 ω 0 , and its transitional behaviors by changing each parameter [Table 1]. Numerically simulated using the FDTD method.

Fig. 3
Fig. 3

Sommerfeld precursors for different ω c by using (a) FDTD method, and (b) asymptotic saddle-points method in Brillouin’s regime ω 0 = 4 × 10 15 [ sec 1 ] , ω p = 1.1175 ω 0 , and δ = 0.07 ω 0 .

Fig. 4
Fig. 4

Transitional behaviors of frequency chirp in Brillouin regime: ω 0 = 4 × 10 15 [ sec 1 ] , ω p = 1.1175 ω 0 , and δ = 0.07 ω 0 . The left column shows field amplitude and the right column shows frequencies of each field components.

Fig. 5
Fig. 5

Validity test ( | β | test) of the asymptotic method near Brillouin’s parameter regime: ω c = 0.25 ω 0 , ω p = 1.1175 ω 0 , and δ = 0.07 ω 0 . (a) Changing δ from Brillouin’s parameter set (solid curve): δ = 0.035 ω 0 (dotted curve) and δ = 0.007 ω 0 (dashed curve). (b) Changing ω p from Brillouin’s parameter set (solid curve): ω p = 0.2 ω 0 (dotted curve) and ω p = 0.02 ω 0 (dashed curve).

Fig. 6
Fig. 6

Step-modulated pulse propagation for resonant parameter regime [dashed-dotted curve] [3]: ω c = ω 0 , ω p = 0.02 ω 0 , and δ = 0.007 ω 0 , and its transitional behaviors by changing each parameter [Table 1]. Numerically simulated using the FDTD method.

Fig. 7
Fig. 7

Transitional behaviors of frequency chirp in resonant regime: ω 0 = 4 × 10 15 [ sec 1 ] , ω p = 0.02 ω 0 , and δ = 0.007 ω 0 . The left column shows field amplitude and the right column shows frequencies of each field component.

Fig. 8
Fig. 8

Sommerfeld precursors for (a) ω p = 1.1175 ω 0 , and (b) ω p = 0.2 ω 0 by using asymptotic saddle-points method and FDTD method in resonant regime ω 0 = 4 × 10 15 [ sec 1 ] , ω c = 0.25 ω 0 , and δ = 0.007 ω 0 .

Fig. 9
Fig. 9

Validity test ( | β | test) of the asymptotic method near resonant parameter regime: ω c = ω 0 , ω p = 0.02 ω 0 , and δ = 0.007 ω 0 . (a) Changing δ from resonant parameter set (solid curve): δ = 0.035 ω 0 (dotted curve) and δ = 0.07 ω 0 (dashed curve). (b) Changing ω p from resonant parameter set (solid curve): ω p = 0.2 ω 0 (dotted curve) and ω p = 1.1175 ω 0 (dashed curve).

Tables (3)

Tables Icon

Table 1 Parameters Used in Different Regimes for Fixed ω 0 = 4.0 × 10 15 [ s 1 ] , and z = 10 3 [cm]

Tables Icon

Table 2 Arrival of Brillouin Precursor t B and Frequency Chirp t chirp in Fig. 2

Tables Icon

Table 3 Arrival of Brillouin Precursor t B and Frequency Chirp t chirp in Fig. 6

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

n ( ω ) = 1 ω p 2 ω 2 ω 0 2 + 2 i ω δ ,
E ( z , t ) = 1 2 π + i ϵ + i ϵ E ( 0 , ω ) e z c ϕ ( ω , t ) d ω ,
t B = z c ( θ 0 + 2 δ 2 ω p 2 θ 0 ω 0 2 ( 3 ϱ ω 0 2 4 ω p 2 ) ) ,
ϱ = 1 δ 2 ( 4 ω 0 2 + 5 ω p 2 ) 3 ω 0 2 ( ω 0 2 + ω p 2 ) .
ω S = ω 0 2 δ 2 + ω p 2 θ 2 θ 2 1 ,
ω B = ω 0 2 ( θ 2 θ 0 2 ) θ 2 θ 0 2 + 3 ϱ ω p 2 ω 0 2 δ 2 ( θ 2 θ 0 2 + 2 ω p 2 ω 0 2 θ 2 θ 0 2 + 3 ϱ ω p 2 ω 0 2 ) 2 ,
β = ω 0 z c ( ω + ω ) ( ω i δ ω 0 ) c τ z ,

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