Abstract

The physical-realizability issues of scalar focus wave modes that are solutions of the homogeneous wave equation with the boundary data exp(−r2w0−2) given on a characteristic are addressed. The Gaussian beams are solutions of the paraxial equation with the same boundary data in a fixed plane. A comparison of these two situations suggests that the focus wave modes can be approximated by giving the exponential exp(−r2w0−2) on some moving plane. This suggestion is implemented.

© 1992 Optical Society of America

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References

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  1. A. Sezginer, “A general formulation of the focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
    [CrossRef]
  2. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. 2.
  3. P. Hillion, “The Goursat problem for the wave equation,”J. Math. Phys. 31, 1939–1942 (1990).
    [CrossRef]
  4. P. Hillion, “Focus wave modes: remarks,” J. Opt. Soc. Am. A 8, 695 (1991).
    [CrossRef]
  5. L. D. Landau, E. M. Lifschitz, Fluid Mechanics (Pergamon, Oxford, 1959).
  6. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
  7. P. Hillion, “Energy-momentum four-vector on a wavefront,” Int. J. Theor. Phys. 30, 197–204 (1991).
    [CrossRef]
  8. M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).
  9. E. Heyman, L. B. Felsen, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
    [CrossRef]
  10. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2023 (1989).
    [CrossRef] [PubMed]
  11. P. Hillion, “Splash wave modes in homogeneous Maxwell’s equations,”J. Electron. WaveAppl. 2, 725–739 (1988).
  12. R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
    [CrossRef] [PubMed]
  13. J. V. Candy, R. W. Ziolkowski, D. K. Lewis, “Transient waves: reconstruction and processing,”J. Acoust. Soc. Am. 88, 2248–2258 (1990).
    [CrossRef]
  14. A. Wünsche, “Embedding of focus wave modes into a wider class of approximate wave equation solutions,” J. Opt. Soc. Am. A 6, 1661–1668 (1989).
    [CrossRef]
  15. E. Heyman, “Focus wave modes: a dilemma with causality,”IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
    [CrossRef]
  16. A. Bélanger, “Lorentz transformation of packetlike solutions of the homogeneous-wave equation,”J. Opt. Soc. Am. 3, 541–542 (1986).
    [CrossRef]

1991 (2)

P. Hillion, “Energy-momentum four-vector on a wavefront,” Int. J. Theor. Phys. 30, 197–204 (1991).
[CrossRef]

P. Hillion, “Focus wave modes: remarks,” J. Opt. Soc. Am. A 8, 695 (1991).
[CrossRef]

1990 (2)

P. Hillion, “The Goursat problem for the wave equation,”J. Math. Phys. 31, 1939–1942 (1990).
[CrossRef]

J. V. Candy, R. W. Ziolkowski, D. K. Lewis, “Transient waves: reconstruction and processing,”J. Acoust. Soc. Am. 88, 2248–2258 (1990).
[CrossRef]

1989 (5)

E. Heyman, “Focus wave modes: a dilemma with causality,”IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[CrossRef]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

E. Heyman, L. B. Felsen, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
[CrossRef]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2023 (1989).
[CrossRef] [PubMed]

A. Wünsche, “Embedding of focus wave modes into a wider class of approximate wave equation solutions,” J. Opt. Soc. Am. A 6, 1661–1668 (1989).
[CrossRef]

1988 (1)

P. Hillion, “Splash wave modes in homogeneous Maxwell’s equations,”J. Electron. WaveAppl. 2, 725–739 (1988).

1986 (1)

A. Bélanger, “Lorentz transformation of packetlike solutions of the homogeneous-wave equation,”J. Opt. Soc. Am. 3, 541–542 (1986).
[CrossRef]

1985 (1)

A. Sezginer, “A general formulation of the focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

Bélanger, A.

A. Bélanger, “Lorentz transformation of packetlike solutions of the homogeneous-wave equation,”J. Opt. Soc. Am. 3, 541–542 (1986).
[CrossRef]

Candy, J. V.

J. V. Candy, R. W. Ziolkowski, D. K. Lewis, “Transient waves: reconstruction and processing,”J. Acoust. Soc. Am. 88, 2248–2258 (1990).
[CrossRef]

Cook, B. D.

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. 2.

Felsen, L. B.

Heyman, E.

E. Heyman, L. B. Felsen, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
[CrossRef]

E. Heyman, “Focus wave modes: a dilemma with causality,”IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[CrossRef]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. 2.

Hillion, P.

P. Hillion, “Energy-momentum four-vector on a wavefront,” Int. J. Theor. Phys. 30, 197–204 (1991).
[CrossRef]

P. Hillion, “Focus wave modes: remarks,” J. Opt. Soc. Am. A 8, 695 (1991).
[CrossRef]

P. Hillion, “The Goursat problem for the wave equation,”J. Math. Phys. 31, 1939–1942 (1990).
[CrossRef]

P. Hillion, “Splash wave modes in homogeneous Maxwell’s equations,”J. Electron. WaveAppl. 2, 725–739 (1988).

Kay, I. W.

M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

Kline, M.

M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

Landau, L. D.

L. D. Landau, E. M. Lifschitz, Fluid Mechanics (Pergamon, Oxford, 1959).

Lewis, D. K.

J. V. Candy, R. W. Ziolkowski, D. K. Lewis, “Transient waves: reconstruction and processing,”J. Acoust. Soc. Am. 88, 2248–2258 (1990).
[CrossRef]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Lifschitz, E. M.

L. D. Landau, E. M. Lifschitz, Fluid Mechanics (Pergamon, Oxford, 1959).

Sezginer, A.

A. Sezginer, “A general formulation of the focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

Wünsche, A.

Ziolkowski, R. W.

J. V. Candy, R. W. Ziolkowski, D. K. Lewis, “Transient waves: reconstruction and processing,”J. Acoust. Soc. Am. 88, 2248–2258 (1990).
[CrossRef]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2023 (1989).
[CrossRef] [PubMed]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. (1)

E. Heyman, “Focus wave modes: a dilemma with causality,”IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[CrossRef]

Int. J. Theor. Phys. (1)

P. Hillion, “Energy-momentum four-vector on a wavefront,” Int. J. Theor. Phys. 30, 197–204 (1991).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. V. Candy, R. W. Ziolkowski, D. K. Lewis, “Transient waves: reconstruction and processing,”J. Acoust. Soc. Am. 88, 2248–2258 (1990).
[CrossRef]

J. Appl. Phys. (1)

A. Sezginer, “A general formulation of the focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

J. Electron. WaveAppl. (1)

P. Hillion, “Splash wave modes in homogeneous Maxwell’s equations,”J. Electron. WaveAppl. 2, 725–739 (1988).

J. Math. Phys. (1)

P. Hillion, “The Goursat problem for the wave equation,”J. Math. Phys. 31, 1939–1942 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

A. Bélanger, “Lorentz transformation of packetlike solutions of the homogeneous-wave equation,”J. Opt. Soc. Am. 3, 541–542 (1986).
[CrossRef]

J. Opt. Soc. Am. A (3)

Phys. Rev. A (1)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2023 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Other (4)

L. D. Landau, E. M. Lifschitz, Fluid Mechanics (Pergamon, Oxford, 1959).

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. 2.

M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

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Equations (53)

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ϑ x 2 ψ + ϑ y 2 ψ + ϑ z 2 ψ - ϑ x 0 2 ψ = 0 ,             x 0 = c t .
ϑ x 2 ψ + ϑ y 2 ψ + 4 ϑ ξ ϑ η ψ = 0 ,
ψ = 1 a + i ξ exp ( i χ η - χ r 2 a + i ξ ) , r 2 = x 2 + y 2 ,             i = - 1 ,
D n = ϑ x j ϑ y l ϑ ξ m ϑ η p ,             n = j + l + m + p ,
( ϑ x p ) 2 + ( ϑ y p ) 2 + 4 ϑ ξ ϑ η p = 0.
g 00 = - g 11 = - g 22 = - g 33 = 1 ,             g μ ν = 0 ( μ ν ) ,
ψ ( r , ξ , η ) = 1 ξ exp ( i χ η + i χ r 2 ξ ) 0 ϕ ( s ) exp ( i χ s 2 ξ ) × J 0 ( 2 χ r s ξ ) s d s ,
θ ( r , ξ , η ) = χ ( η + ξ r 2 a 2 + ξ 2 ) ,
k 1 ( r , z , t ) = ϑ θ ϑ x = 2 χ ξ x a 2 + ξ 2 , k 2 ( r , z , t ) = ϑ θ ϑ y = 2 χ ξ y a 2 + ξ 2 , k 3 ( r , z , t ) = ϑ θ ϑ z = χ [ 1 + r 2 ( a 2 + ξ 2 ) ( a 2 + ξ 2 ) 2 ] ,
ω ( r , z , t ) = - ϑ θ ϑ t = - χ c [ 1 - r 2 ( a 2 - ξ 2 ) ( a 2 + ξ 2 ) 2 ] .
ϑ j ( r , z , t ) = ω ( k j / k 2 ) ,             k = ( k 1 2 + k 2 2 + k 3 2 ) 1 / 2 .
ω ( r , z , t ) = c k 3 ( r , z , t ) - 2 χ c ,
u 1 = u 2 = 0 ,             u 3 = ϑ ω ϑ k 3 = c .
P μ = μ η ν T μ ν d σ .
P 0 = - + ( T 00 - T 03 ) ξ = 0 d x d y d η .
T 00 - T 03 = ϑ r ψ ϑ r ψ ¯ + 4 ϑ η ψ ϑ η ψ ¯ ,
( T 00 - T 03 ) ξ = 0 = 4 χ 2 ( 1 + r 2 a 2 ) exp ( - 2 χ r 2 a ) ,
w 1 = w 2 = 0 ,             w 3 = c r 2 - a 2 r 2 + a 2 ,
ψ ^ = 1 ( 2 π ) 2 ψ exp ( i k ν x ν ) d x 0 d z d x d y .
ψ ^ = 2 π α i χ exp ( - a k 2 4 χ ) δ ( Ω c + χ + k 2 4 χ ) δ ( k z - χ + k 2 4 χ ) ,
k 0 = Ω / c ,             k 2 = k 1 2 + k 2 2 .
Ω c = - ( χ + k 2 4 χ ) ,             k z = χ - k 2 4 χ .
Ω 2 / c 2 = k z 2 + k 2 = k 2 ,
( 1 / c ) d Ω - d k z = 0.
ϕ = ψ ( χ ) F ( χ ) d χ ,
ϑ x 2 ψ o + ϑ y 2 ψ o + 4 ϑ z ϑ η ψ o = 0 ,
ψ o = 1 a + i z exp ( i χ η - χ r 2 a + i z ) ,
( ϑ χ p o ) 2 + ( ϑ y p o ) 2 + 4 ϑ z p o ϑ η p o = 0 ,
w 2 ( z ) = w 0 2 ( 1 + z 2 χ 2 w 0 4 ) .
ω o = - χ c ,             v j o = - χ c k j k - 2 .
ψ ^ o = 2 π a i χ exp ( - a k 2 4 χ ) δ ( χ + Ω c ) δ ( k z - χ + k z 2 4 χ ) .
Ω o c = χ ,             k z o = χ - k 2 4 χ .
( Ω o c ) 2 + Ω o c k z o - k 2 4 = 0.
ψ β = 1 a + i ξ β exp ( i χ η - χ r 2 a + i ξ β )
ϑ χ 2 ψ + ϑ y 2 ψ + 4 ϑ ξ β ϑ η ψ = 0 ,
ϑ x 2 ψ + ϑ y 2 ψ + 4 ( 1 + β ) 2 [ β ϑ z 2 ψ - ϑ x 0 2 ψ + ( 1 - β ) ϑ x 0 ϑ z ψ ] = 0.
β = 1 - μ 1 + μ .
ψ ^ β = 2 π a i χ exp ( - a k 2 4 χ ) δ ( Ω c + χ + β k 2 4 χ ) × δ ( k z - χ + k 2 4 χ ) .
Ω c = - ( χ + β k 2 4 χ ) ,             k z = χ - k 2 4 χ ,
Ω 2 c 2 + ( 1 - β ) Ω c k z - β k z 2 - ( 1 + β ) k 2 4 = 0 , 1 c d Ω - β d k z = 0.
V j = Ω ( k j / k 2 ) ,             Ω = ( c / 2 ) [ ( 1 + β ) k - ( 1 - β ) k j ] ,
U 1 = U 2 = 0 ,             U 3 = β c .
w 2 ( ξ β ) = w 0 2 ( 1 + ξ β 2 χ 2 w o 4 ) .
α β = [ 1 + ( 1 - β ) 2 z 2 χ 2 w 0 4 ] 1 / 2 .
z 2 χ 2 w 0 4 ( 1 - β ) 2 ,
d ξ = d z - β ( b x 0 ) d x 0 ,             d η = d z + d x 0
ϑ x 2 ψ + ϑ y 2 ψ + 4 ( 1 - β ) 2 [ β ϑ z 2 ψ - ϑ x 0 2 ψ + ( 1 - β ) ϑ x 0 ϑ z ψ - b β ( 1 + β ) 3 ( ϑ z ψ - ϑ x 0 ψ ) ] = 0 ,
lim b b 1 β ( b x 0 ) = 0 ,             lim b b 2 β ( b x 0 ) = 1 ,
lim b b 1 b β ( b x 0 ) = 0 ,             lim b b b β ( b x 0 ) = 0.
ψ = 1 a + i [ z - B ( b x 0 ) ] exp { i χ η - χ r 2 a + i [ z - B ( b x 0 ) ] } ,
ψ ^ = 2 π a i χ exp ( - a k 2 4 χ ) δ ( k z - χ + k 2 4 χ ) × - + exp [ i k 2 4 χ B ( b x 0 ) - i Ω c x 0 + i χ x 0 ] d x 0 ,
α = { 1 + [ z - B ( b x 0 ) ] 2 χ 2 w 0 4 } 1 / 2 ,
B ( b x 0 ) = sin [ b x 0 π 2 ( 1 + b 2 x 0 2 ) 1 / 2 ] ,

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