Abstract

We predict the existence of a specific wave object that arises when a spatially limited electromagnetic wave with an abrupt leading edge propagates in a vacuum. This object demonstrates a precursorlike behavior, since the diffraction spreading of the field proceeds behind it. The transverse structure and the field strength at the leading edge of the object remain unchanged during propagation, whereas the object’s energy decreases as z−1 (z is the distance of propagation) because of diffractive erosion of its trailing edge. We show that this phenomenon resembles to a certain extent the Sommerfeld precursor formation in a temporally dispersive medium.

© 1994 Optical Society of America

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References

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  1. W. Kaiser, ed., Ultrashort Optical Pulses, 3rd ed. (Springer-Verlag, New York, 1987).
  2. IEEE J. Quantum Electron. 12, 2415–2682 (1989) (special issue on ultrafast phenomena).
  3. E. M. Belenov, A. V. Nazarkin, “Solutions of nonlinear-optics equations found outside the approximation of slowly varying amplitudes and phases,” JETP Lett. 51, 288–292 (1990); “Dynamics of the propagation and interaction of electromagnetic pulses in two-level media,” Sov. Phys. JETP 73, 422–428 (1991).
  4. I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985); “Transmission of radiation with rapid temporal modulation through optical instruments,” Opt. Quantum Electron. 17, 353–357 (1985).
    [CrossRef]
  5. T. T. Wu, “Electromagnetic missiles,” J. Appl. Phys. 57, 2370–2373 (1985); J. M. Myers, H.-M. Shen, T. T. Wu, H. E. Brandt, “Curved electromagnetic missiles,” J. Appl. Phys. 65, 2604–2610 (1989).
    [CrossRef]
  6. P. Wolfe, “Diffraction of a pulse by a strip,”J. Math. Anal. Appl. 48, 170–182 (1974).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.
  8. M. A. Leontovitch, “On the problem of electromagnetic wave propagation along a well-conducting surface,” Izv. Acad. Nauk SSSR Ser. Fiz. 8, 16–22 (1944). (In Russian).
  9. S. A. Akhmanov, A. P. Sukhorukov, A. S. Chirkin, “Non-stationary phenomena and space–time analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–756 (1969).
  10. A. Sommerfeld, “About the propagation of light in dispersive media,” in Wave Propagation and Group Velocity, L. Brillouin, ed. (Academic, New York, 1960), pp. 17–42.
  11. L. Brillouin, “Propagation of electromagnetic waves in material media,” in Wave Propagation and Group Velocity, L. Brillouin, ed. (Academic, New York, 1960), pp. 85–111.
  12. M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
    [CrossRef]
  13. E. Varoquaux, G. A. Williams, O. Avenel, “Pulse propagation in a resonant medium: application to sound waves in superfluid 3He–B,” Phys. Rev. B 34, 7617–7640 (1986).
    [CrossRef]
  14. K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
    [CrossRef]
  15. D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electrooptic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
    [CrossRef]

1990 (1)

E. M. Belenov, A. V. Nazarkin, “Solutions of nonlinear-optics equations found outside the approximation of slowly varying amplitudes and phases,” JETP Lett. 51, 288–292 (1990); “Dynamics of the propagation and interaction of electromagnetic pulses in two-level media,” Sov. Phys. JETP 73, 422–428 (1991).

1989 (1)

IEEE J. Quantum Electron. 12, 2415–2682 (1989) (special issue on ultrafast phenomena).

1988 (1)

1986 (1)

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

1985 (2)

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985); “Transmission of radiation with rapid temporal modulation through optical instruments,” Opt. Quantum Electron. 17, 353–357 (1985).
[CrossRef]

T. T. Wu, “Electromagnetic missiles,” J. Appl. Phys. 57, 2370–2373 (1985); J. M. Myers, H.-M. Shen, T. T. Wu, H. E. Brandt, “Curved electromagnetic missiles,” J. Appl. Phys. 65, 2604–2610 (1989).
[CrossRef]

1984 (1)

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electrooptic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

1974 (1)

P. Wolfe, “Diffraction of a pulse by a strip,”J. Math. Anal. Appl. 48, 170–182 (1974).
[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]

1969 (1)

S. A. Akhmanov, A. P. Sukhorukov, A. S. Chirkin, “Non-stationary phenomena and space–time analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–756 (1969).

1944 (1)

M. A. Leontovitch, “On the problem of electromagnetic wave propagation along a well-conducting surface,” Izv. Acad. Nauk SSSR Ser. Fiz. 8, 16–22 (1944). (In Russian).

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, A. S. Chirkin, “Non-stationary phenomena and space–time analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–756 (1969).

Auston, D. H.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electrooptic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Avenel, O.

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

Belenov, E. M.

E. M. Belenov, A. V. Nazarkin, “Solutions of nonlinear-optics equations found outside the approximation of slowly varying amplitudes and phases,” JETP Lett. 51, 288–292 (1990); “Dynamics of the propagation and interaction of electromagnetic pulses in two-level media,” Sov. Phys. JETP 73, 422–428 (1991).

Brillouin, L.

L. Brillouin, “Propagation of electromagnetic waves in material media,” in Wave Propagation and Group Velocity, L. Brillouin, ed. (Academic, New York, 1960), pp. 85–111.

Cheung, K. P.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electrooptic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Chirkin, A. S.

S. A. Akhmanov, A. P. Sukhorukov, A. S. Chirkin, “Non-stationary phenomena and space–time analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–756 (1969).

Christov, I. P.

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985); “Transmission of radiation with rapid temporal modulation through optical instruments,” Opt. Quantum Electron. 17, 353–357 (1985).
[CrossRef]

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]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.

Kleinman, D. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electrooptic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Leontovitch, M. A.

M. A. Leontovitch, “On the problem of electromagnetic wave propagation along a well-conducting surface,” Izv. Acad. Nauk SSSR Ser. Fiz. 8, 16–22 (1944). (In Russian).

Nazarkin, A. V.

E. M. Belenov, A. V. Nazarkin, “Solutions of nonlinear-optics equations found outside the approximation of slowly varying amplitudes and phases,” JETP Lett. 51, 288–292 (1990); “Dynamics of the propagation and interaction of electromagnetic pulses in two-level media,” Sov. Phys. JETP 73, 422–428 (1991).

Oughstun, K. E.

Sherman, G. C.

Sommerfeld, A.

A. Sommerfeld, “About the propagation of light in dispersive media,” in Wave Propagation and Group Velocity, L. Brillouin, ed. (Academic, New York, 1960), pp. 17–42.

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, A. S. Chirkin, “Non-stationary phenomena and space–time analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–756 (1969).

Valdmanis, J. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electrooptic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Varoquaux, E.

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

Williams, G. A.

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

Wolfe, P.

P. Wolfe, “Diffraction of a pulse by a strip,”J. Math. Anal. Appl. 48, 170–182 (1974).
[CrossRef]

Wu, T. T.

T. T. Wu, “Electromagnetic missiles,” J. Appl. Phys. 57, 2370–2373 (1985); J. M. Myers, H.-M. Shen, T. T. Wu, H. E. Brandt, “Curved electromagnetic missiles,” J. Appl. Phys. 65, 2604–2610 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

IEEE J. Quantum Electron. 12, 2415–2682 (1989) (special issue on ultrafast phenomena).

Izv. Acad. Nauk SSSR Ser. Fiz. (1)

M. A. Leontovitch, “On the problem of electromagnetic wave propagation along a well-conducting surface,” Izv. Acad. Nauk SSSR Ser. Fiz. 8, 16–22 (1944). (In Russian).

J. Appl. Phys. (1)

T. T. Wu, “Electromagnetic missiles,” J. Appl. Phys. 57, 2370–2373 (1985); J. M. Myers, H.-M. Shen, T. T. Wu, H. E. Brandt, “Curved electromagnetic missiles,” J. Appl. Phys. 65, 2604–2610 (1989).
[CrossRef]

J. Math. Anal. Appl. (1)

P. Wolfe, “Diffraction of a pulse by a strip,”J. Math. Anal. Appl. 48, 170–182 (1974).
[CrossRef]

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

JETP Lett. (1)

E. M. Belenov, A. V. Nazarkin, “Solutions of nonlinear-optics equations found outside the approximation of slowly varying amplitudes and phases,” JETP Lett. 51, 288–292 (1990); “Dynamics of the propagation and interaction of electromagnetic pulses in two-level media,” Sov. Phys. JETP 73, 422–428 (1991).

Opt. Commun. (1)

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985); “Transmission of radiation with rapid temporal modulation through optical instruments,” Opt. Quantum Electron. 17, 353–357 (1985).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. B (1)

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

Phys. Rev. Lett. (1)

D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electrooptic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Sov. Phys. JETP (1)

S. A. Akhmanov, A. P. Sukhorukov, A. S. Chirkin, “Non-stationary phenomena and space–time analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–756 (1969).

Other (4)

A. Sommerfeld, “About the propagation of light in dispersive media,” in Wave Propagation and Group Velocity, L. Brillouin, ed. (Academic, New York, 1960), pp. 17–42.

L. Brillouin, “Propagation of electromagnetic waves in material media,” in Wave Propagation and Group Velocity, L. Brillouin, ed. (Academic, New York, 1960), pp. 85–111.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.

W. Kaiser, ed., Ultrashort Optical Pulses, 3rd ed. (Springer-Verlag, New York, 1987).

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Figures (1)

Fig. 1
Fig. 1

Evolution of the pulse’s stepwise leading edge resulting from diffraction. a, The pulse’s leading edge at some initial moment; b, the diffractive precursor formation after some distance of passage.

Equations (35)

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2 E z 2 + Δ E - 1 c 2 2 E t 2 = 0 ,
Δ = 2 x 2 + 2 y 2 .
E ( r , t ) = E ( z , r , τ ) ,
r = e x x + e y y .
2 E z τ - 1 / 2 Δ E = 0.
E ( z , r , τ ) = E ˜ ( z , r , τ ) exp ( i ω τ ) + c . c .
E z - 1 2 c Δ - τ E ( τ ) d τ = 0.
W = c 4 π - d x d y - d τ E 2
d w d z = c 2 4 π - d τ - d x d y E Δ - τ d τ E ( τ ) .
d w d z = - c 2 4 π - d τ - d x d y { ( E x ) [ x - τ d τ E ( τ ) ] + ( E y ) [ y - τ d τ E ( τ ) ] } .
d w d z = - c 2 8 π - d x d y { [ x Ψ ( r , ) ] 2 + [ y Ψ ( r , ) ] 2 } ,
Ψ ( r , τ ) = - τ E ( z , r , τ ) d τ .
E z = c x Ψ ( r , τ ) ,             H z = - c y Ψ ( r , τ ) .
- E ( z , r , τ ) d τ = 0.
E ( r ) = ( 2 π ) - 2 - d k x d k y exp ( i k r ) E ( k ) .
E k z = - 1 2 c k 2 - τ E k ( τ ) d τ ,
E ( 0 , r , t ) = E ( r ) f ( t ) ,
f ( t ) = { cos ω o t t > 0 0 t < 0 .
2 E k z 2 - 1 c 2 2 E k t 2 = c 2 k 2 E k ,
E k ( o , t ) = E ( k ) f ( t ) .
2 E k z 2 - 1 c 2 2 E k t 2 = 4 π c 2 2 P k t 2 ,
P k ( z , t ) = c 2 k 2 4 π - t d t - t d t E k ( z , t ) .
k 11 2 ( ω ) = ω 2 c 2 ɛ ( ω ) = ω 2 c 2 ( 1 - ω p 2 ω 2 ) ,
ω p 2 = c 2 k 2 .
E k ( z , t ) = ( 2 π ) - 1 E ( k ) Re - exp [ i ( ω t - k 11 ( ω ) z ) ] i ( ω - ω o ) d ω
k k o = ω o / c .
E k ( z , τ ) = ( 2 π ) - 1 E ( k ) × Re - i ω - 1 exp [ i ( ω τ + ω p 2 z 2 c ω ) ] d ω .
E k ( z , τ ) = E ( k ) J o ( 2 k 2 c z τ ) ,
k 11 ( ξ ) k 11 ( ω o ) + k 11 ( ω o ) ξ - 1 / 2 k 11 ( ω o ) ξ 2 ,
k 11 = k 11 ω , k 11 = 2 k 11 ω 2 ,
E k ( z , t ) = E ( k ) Re { exp [ i ( ω o t - k 11 ( ω o ) z ) ] × Φ [ t - k 11 z ( 2 k 11 z ) 1 / 2 ] } ,
Φ ( ω ) = ( 2 π ) - 1 / 2 ( 1 - i ) - w exp ( i η 2 ) d η
E ( z , r , τ ) = ( 2 π ) - 2 exp ( i ω o τ ) - d k x d k y E ( k ) × exp ( i k r + i z k 2 / 2 k o ) × Φ [ ω o τ - z k 2 / 2 k o ( 2 z k 2 / k o ) 1 / 2 ] ,
E ( z , r , τ ) = ( 2 π ) - 2 - d k x d k y E ( k ) × exp ( i k r ) J o ( 2 k 2 c z τ ) 1 / 2 .
E ( z , y , τ ) = E o 2 { 1 - ξ 2 + ζ θ + [ ( 1 - ξ 2 + ζ θ ) 2 + 4 ξ 2 ] 1 / 2 ( 1 - ξ 2 + ζ θ ) 2 + 4 ξ 2 } 1 / 2 ,

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