Abstract

We discuss the transient effects associated with short-pulse propagation in a linearly dispersive medium by comparing optical precursors with 0π pulses. The classical and quantum effects were, in the past, treated independently due to the different time scales. Using results from recent studies [Phys. Rev. Lett. 96, 143901 (2006) ] as well as the discussions by Varoquaux et al. [Phys. Rev. B. 34, 7617 (1989)] and Bürck [Hyperfine Interact. 123, 483 (1999)] , we show analytically that precursors and 0π pulses describe the same physical phenomenon. We also show that most published results of transient pulse propagation can be expressed using one overarching equation. Finally, we give a physical interpretation of the analogy between precursors and 0π pulses as a medium-impulse response using a Green’s function.

© 2008 Optical Society of America

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References

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  1. A. E. Fox and U. Österberg, “Observation of non-exponential absorption of ultra-fast pulses in water,” Opt. Express 14, 3688-3693 (2006).
    [CrossRef]
  2. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
  3. M. D. Crisp, “Propagation of small-area pulse of coherent light through a resonant medium,” Phys. Rev. A 1, 1604-1611 (1970).
    [CrossRef]
  4. S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
    [CrossRef]
  5. B. Segard, J. Zemmouri, and B. Macke, “Generation of electromagnetic pulses by stacking of coherent transients,” Europhys. Lett. 4, 47-52 (1987).
    [CrossRef]
  6. 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]
  7. H. Jeong, “Direct observation of optical precursors in a cold potassium gas,” Ph.D. dissertation (Duke University, Durham, N.C., USA, 2006).
  8. W. LeFew, “Optical precursor behavior,” Ph.D. dissertation (Duke University, Durham, N.C., USA, 2007).
  9. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).
  10. E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: application to sound waves in superfluid He3-B,” Phys. Rev. B 34, 7617-7640 (1986).
    [CrossRef]
  11. U. van Bürck, “Coherent pulse propagation through resonant media,” Hyperfine Interact. 123/124, 483-509 (1999).
    [CrossRef]
  12. T. W. Barrett, “Energy transfer and propagation and the dielectrics of materials: transient versus steady state effects,” in Ultra-Wide Band Radar Proceedings (CRC, 1991).
  13. H.-J. Hartmann and A. Laubereau, “Coherent pulse propagation in the infrared on the picosecond time scale,” Opt. Commun. 47, 117-122 (1983).
    [CrossRef]
  14. N. Dudovich, D. Oron, and Y. Silberberg, “Coherent transient enhancement of optically induced resonant transitions,” Phys. Rev. Lett. 88, 123004 (2002).
    [CrossRef]
  15. J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552-555 (1984).
    [CrossRef]
  16. O. Avenel, M. Rouff, E. Varoquaux, and G. A. Williams, “Resonant pulse propagation of sound in superfluid He3-B,” Phys. Rev. Lett. 50, 1591-1594 (1983).
    [CrossRef]
  17. J. Aaviksoo, J. Lippmaa, and J. Kuhl, “Observability of optical precursors,” J. Opt. Soc. Am. B 5, 1631-1635 (1988).
    [CrossRef]
  18. F. J. Lynch, R. E. Holland, and M. Hamermesh, “Time dependence of resonantly filtered gamma rays from Fe57,” Phys. Rev. 120, 513-520 (1960).
    [CrossRef]
  19. L. A. Vainshtein, “Propagation of pulses,” Sov. Phys. Usp. 19, 189-205 (1976).
    [CrossRef]
  20. A. L. Gutman, “Space-time green function and short pulse propagation in different media,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (Springer, 1999), pp. 301-311.

2006 (2)

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

2002 (1)

N. Dudovich, D. Oron, and Y. Silberberg, “Coherent transient enhancement of optically induced resonant transitions,” Phys. Rev. Lett. 88, 123004 (2002).
[CrossRef]

1999 (1)

U. van Bürck, “Coherent pulse propagation through resonant media,” Hyperfine Interact. 123/124, 483-509 (1999).
[CrossRef]

1988 (1)

1987 (1)

B. Segard, J. Zemmouri, and B. Macke, “Generation of electromagnetic pulses by stacking of coherent transients,” Europhys. Lett. 4, 47-52 (1987).
[CrossRef]

1986 (1)

E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: application to sound waves in superfluid He3-B,” Phys. Rev. B 34, 7617-7640 (1986).
[CrossRef]

1984 (1)

J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552-555 (1984).
[CrossRef]

1983 (2)

O. Avenel, M. Rouff, E. Varoquaux, and G. A. Williams, “Resonant pulse propagation of sound in superfluid He3-B,” Phys. Rev. Lett. 50, 1591-1594 (1983).
[CrossRef]

H.-J. Hartmann and A. Laubereau, “Coherent pulse propagation in the infrared on the picosecond time scale,” Opt. Commun. 47, 117-122 (1983).
[CrossRef]

1976 (1)

L. A. Vainshtein, “Propagation of pulses,” Sov. Phys. Usp. 19, 189-205 (1976).
[CrossRef]

1970 (1)

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

1969 (1)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
[CrossRef]

1960 (1)

F. J. Lynch, R. E. Holland, and M. Hamermesh, “Time dependence of resonantly filtered gamma rays from Fe57,” Phys. Rev. 120, 513-520 (1960).
[CrossRef]

Aaviksoo, J.

Avenel, O.

E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: application to sound waves in superfluid He3-B,” Phys. Rev. B 34, 7617-7640 (1986).
[CrossRef]

O. Avenel, M. Rouff, E. Varoquaux, and G. A. Williams, “Resonant pulse propagation of sound in superfluid He3-B,” Phys. Rev. Lett. 50, 1591-1594 (1983).
[CrossRef]

Balant, A. C.

J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552-555 (1984).
[CrossRef]

Barrett, T. W.

T. W. Barrett, “Energy transfer and propagation and the dielectrics of materials: transient versus steady state effects,” in Ultra-Wide Band Radar Proceedings (CRC, 1991).

Brillouin, L.

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

Crisp, M. D.

M. D. Crisp, “Propagation of small-area pulse 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]

Dudovich, N.

N. Dudovich, D. Oron, and Y. Silberberg, “Coherent transient enhancement of optically induced resonant transitions,” Phys. Rev. Lett. 88, 123004 (2002).
[CrossRef]

Fox, A. E.

Gauthier, D. J.

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]

Grischkowsky, D.

J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552-555 (1984).
[CrossRef]

Gutman, A. L.

A. L. Gutman, “Space-time green function and short pulse propagation in different media,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (Springer, 1999), pp. 301-311.

Hahn, E. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
[CrossRef]

Hamermesh, M.

F. J. Lynch, R. E. Holland, and M. Hamermesh, “Time dependence of resonantly filtered gamma rays from Fe57,” Phys. Rev. 120, 513-520 (1960).
[CrossRef]

Hartmann, H.-J.

H.-J. Hartmann and A. Laubereau, “Coherent pulse propagation in the infrared on the picosecond time scale,” Opt. Commun. 47, 117-122 (1983).
[CrossRef]

Holland, R. E.

F. J. Lynch, R. E. Holland, and M. Hamermesh, “Time dependence of resonantly filtered gamma rays from Fe57,” Phys. Rev. 120, 513-520 (1960).
[CrossRef]

Jeong, H.

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]

H. Jeong, “Direct observation of optical precursors in a cold potassium gas,” Ph.D. dissertation (Duke University, Durham, N.C., USA, 2006).

Kuhl, J.

Laubereau, A.

H.-J. Hartmann and A. Laubereau, “Coherent pulse propagation in the infrared on the picosecond time scale,” Opt. Commun. 47, 117-122 (1983).
[CrossRef]

LeFew, W.

W. LeFew, “Optical precursor behavior,” Ph.D. dissertation (Duke University, Durham, N.C., USA, 2007).

Lippmaa, J.

Lynch, F. J.

F. J. Lynch, R. E. Holland, and M. Hamermesh, “Time dependence of resonantly filtered gamma rays from Fe57,” Phys. Rev. 120, 513-520 (1960).
[CrossRef]

Macke, B.

B. Segard, J. Zemmouri, and B. Macke, “Generation of electromagnetic pulses by stacking of coherent transients,” Europhys. Lett. 4, 47-52 (1987).
[CrossRef]

McCall, S. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
[CrossRef]

Oron, D.

N. Dudovich, D. Oron, and Y. Silberberg, “Coherent transient enhancement of optically induced resonant transitions,” Phys. Rev. Lett. 88, 123004 (2002).
[CrossRef]

Österberg, U.

Oughstun, K. E.

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

Rothenberg, J. E.

J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552-555 (1984).
[CrossRef]

Rouff, M.

O. Avenel, M. Rouff, E. Varoquaux, and G. A. Williams, “Resonant pulse propagation of sound in superfluid He3-B,” Phys. Rev. Lett. 50, 1591-1594 (1983).
[CrossRef]

Segard, B.

B. Segard, J. Zemmouri, and B. Macke, “Generation of electromagnetic pulses by stacking of coherent transients,” Europhys. Lett. 4, 47-52 (1987).
[CrossRef]

Sherman, G. C.

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

Silberberg, Y.

N. Dudovich, D. Oron, and Y. Silberberg, “Coherent transient enhancement of optically induced resonant transitions,” Phys. Rev. Lett. 88, 123004 (2002).
[CrossRef]

Vainshtein, L. A.

L. A. Vainshtein, “Propagation of pulses,” Sov. Phys. Usp. 19, 189-205 (1976).
[CrossRef]

van Bürck, U.

U. van Bürck, “Coherent pulse propagation through resonant media,” Hyperfine Interact. 123/124, 483-509 (1999).
[CrossRef]

Varoquaux, E.

E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: application to sound waves in superfluid He3-B,” Phys. Rev. B 34, 7617-7640 (1986).
[CrossRef]

O. Avenel, M. Rouff, E. Varoquaux, and G. A. Williams, “Resonant pulse propagation of sound in superfluid He3-B,” Phys. Rev. Lett. 50, 1591-1594 (1983).
[CrossRef]

Williams, G. A.

E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: application to sound waves in superfluid He3-B,” Phys. Rev. B 34, 7617-7640 (1986).
[CrossRef]

O. Avenel, M. Rouff, E. Varoquaux, and G. A. Williams, “Resonant pulse propagation of sound in superfluid He3-B,” Phys. Rev. Lett. 50, 1591-1594 (1983).
[CrossRef]

Zemmouri, J.

B. Segard, J. Zemmouri, and B. Macke, “Generation of electromagnetic pulses by stacking of coherent transients,” Europhys. Lett. 4, 47-52 (1987).
[CrossRef]

Europhys. Lett. (1)

B. Segard, J. Zemmouri, and B. Macke, “Generation of electromagnetic pulses by stacking of coherent transients,” Europhys. Lett. 4, 47-52 (1987).
[CrossRef]

Hyperfine Interact. (1)

U. van Bürck, “Coherent pulse propagation through resonant media,” Hyperfine Interact. 123/124, 483-509 (1999).
[CrossRef]

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

Opt. Commun. (1)

H.-J. Hartmann and A. Laubereau, “Coherent pulse propagation in the infrared on the picosecond time scale,” Opt. Commun. 47, 117-122 (1983).
[CrossRef]

Opt. Express (1)

Phys. Rev. (2)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
[CrossRef]

F. J. Lynch, R. E. Holland, and M. Hamermesh, “Time dependence of resonantly filtered gamma rays from Fe57,” Phys. Rev. 120, 513-520 (1960).
[CrossRef]

Phys. Rev. A (1)

M. D. Crisp, “Propagation of small-area pulse of coherent light through a resonant medium,” Phys. Rev. A 1, 1604-1611 (1970).
[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 He3-B,” Phys. Rev. B 34, 7617-7640 (1986).
[CrossRef]

Phys. Rev. Lett. (4)

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]

N. Dudovich, D. Oron, and Y. Silberberg, “Coherent transient enhancement of optically induced resonant transitions,” Phys. Rev. Lett. 88, 123004 (2002).
[CrossRef]

J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552-555 (1984).
[CrossRef]

O. Avenel, M. Rouff, E. Varoquaux, and G. A. Williams, “Resonant pulse propagation of sound in superfluid He3-B,” Phys. Rev. Lett. 50, 1591-1594 (1983).
[CrossRef]

Sov. Phys. Usp. (1)

L. A. Vainshtein, “Propagation of pulses,” Sov. Phys. Usp. 19, 189-205 (1976).
[CrossRef]

Other (6)

A. L. Gutman, “Space-time green function and short pulse propagation in different media,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (Springer, 1999), pp. 301-311.

H. Jeong, “Direct observation of optical precursors in a cold potassium gas,” Ph.D. dissertation (Duke University, Durham, N.C., USA, 2006).

W. LeFew, “Optical precursor behavior,” Ph.D. dissertation (Duke University, Durham, N.C., USA, 2007).

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

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

T. W. Barrett, “Energy transfer and propagation and the dielectrics of materials: transient versus steady state effects,” in Ultra-Wide Band Radar Proceedings (CRC, 1991).

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Figures (3)

Fig. 1
Fig. 1

Illustration of transient pulse propagations in the temporal and spectral domains. (a) Traditional case of optical precursors (Brillouin’s case), (b) resonant precursors, and (c) small-area pulse ( 0 π pulse) propagation. In this paper, we compare the cases in (b) and (c) where the SVEA is valid.

Fig. 2
Fig. 2

Comparison of (a) the SVEA analysis [obtained by Eq. (5)] with (b) a numerical solution (FFT) [obtained by Eq. (4)]. Step-pulse propagation for on-resonance pulses ω c = ω 0 , based on the parameters in [6].

Fig. 3
Fig. 3

Step-pulse propagation for on-resonance pulses, ω c = ω 0 , based on the parameters in [6]. Note that the area is 0 π for large optical depths α 0 z = 10 and α 0 z = 100 .

Tables (5)

Tables Icon

Table 1 Comparison of Assumptions in [3] and [6]

Tables Icon

Table 2 Parameters Used in [2] and [6]

Tables Icon

Table 3 Notations in the Reference Papers

Tables Icon

Table 4 Parameters Used in Eqs. (9, 10)

Tables Icon

Table 5 Theories and Parameters Used in Each Reference

Equations (14)

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

θ ( z , t ) = μ A ( z , t ) d t ,
[ z i ω n 0 c + ( ω ) ] A ( z , ω ) = 0 ,
n ( ω ) = ϵ ( ω ) = j [ 1 ω p j 2 ( ω 2 ω j 2 + 2 i ω δ j ) ] ϵ ω p 2 ( ω 2 ω 0 2 + 2 i ω δ )
k ( ω ) = ω c n ( ω ) ω c [ n 0 ω p 2 n 0 4 ω ( ω ω 0 + i δ ) ] .
n ( ω ) = [ ϵ ω p 2 ( ω 2 ω 0 2 + 2 i ω δ ) ] 1 2
n 0 ω p 2 [ 4 n 0 ω ( ω ω 0 + i δ ) ] ,
E ( z , t ) = 1 2 π E ( 0 , ω ) e i [ k ( ω ) z ω t ] d ω
1 2 π E ( 0 , ω ) e i ω ( t n 0 z c ) p ω ω 0 + i δ d ω ,
E ( z , τ ) = A ( z , τ ) e i ω c τ ,
A ( z , τ ) = R { E 0 Θ ( τ ) [ e p Δ e Δ τ n = 1 ( p Δ ) n J n ( 2 p τ ) τ n 2 ] } ,
A ( z , τ ) = R [ E 0 Θ ( τ ) e Δ τ n = 0 ( Δ p ) n J n ( 2 p τ ) τ n 2 ] ,
G ( z , t ) = 1 2 π e i [ k ( ω ) z ω t ] d ω ,
E ( z , t ) = G ( z , t τ ) E ( 0 , τ ) d τ .
E ( z , t ) = E 0 [ Θ ( t ) α 0 z 0 δ τ J 1 ( 2 α 0 z v ) 2 α 0 z v e v d v ] ,

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