Abstract

A definition of the diffraction length characterizing the propagation of localized wave pulses launched from dynamic apertures is provided. Our inference of the diffraction range is based on an analysis of the spectral depletion of the spatial frequency components of the pulse as it propagates away from its source. We demonstrate the efficacy of our procedure by showing that it is capable of capturing some decay features of localized wave pulses that may be missed by other approaches.

© 1996 Optical Society of America

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References

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  1. R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
    [CrossRef]
  2. A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1988).
    [CrossRef]
  3. J. E. Hernandez, R. W. Ziolkowski, S. R. Parker, “Synthesis of the driving functions of an array for propagating localized wave energy,”J. Acoust. Soc. Am. 92, 550–562 (1992).
    [CrossRef]
  4. R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Experimental verification of the localized wave transmission effect,” Phys. Rev. Lett. 62, 147–150 (1989).
    [CrossRef] [PubMed]
  5. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [CrossRef] [PubMed]
  6. R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
    [CrossRef] [PubMed]
  7. R. Donnelly, R. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
    [CrossRef]
  8. R. W. Ziolkowski, D. K. Lewis, “Verification of the localized wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
    [CrossRef]
  9. R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
    [CrossRef]
  10. A. M. Vengsarkar, I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “Closed-form localized wave solutions in optical fiber waveguides,” J. Opt. Soc. Am. A 9, 937–949 (1992).
    [CrossRef]
  11. M. R. Palmer, R. Donnelly, “Focused wave modes and the scalar wave equation,”J. Math. Phys. 34, 4007–4013 (1993).
    [CrossRef]
  12. A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focused-wave-mode pulses from wide-band dynamic Gaussian apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
    [CrossRef]
  13. A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
    [CrossRef]
  14. B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am. A 8, 705–717 (1991).
    [CrossRef]
  15. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  16. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 11.3.

1995 (2)

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focused-wave-mode pulses from wide-band dynamic Gaussian apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
[CrossRef]

1993 (2)

1992 (3)

R. Donnelly, R. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

J. E. Hernandez, R. W. Ziolkowski, S. R. Parker, “Synthesis of the driving functions of an array for propagating localized wave energy,”J. Acoust. Soc. Am. 92, 550–562 (1992).
[CrossRef]

A. M. Vengsarkar, I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “Closed-form localized wave solutions in optical fiber waveguides,” J. Opt. Soc. Am. A 9, 937–949 (1992).
[CrossRef]

1991 (3)

B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am. A 8, 705–717 (1991).
[CrossRef]

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
[CrossRef]

1990 (1)

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[CrossRef]

1989 (2)

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

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

1988 (1)

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1988).
[CrossRef]

1987 (1)

Besieris, I. M.

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focused-wave-mode pulses from wide-band dynamic Gaussian apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
[CrossRef]

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[CrossRef]

A. M. Vengsarkar, I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “Closed-form localized wave solutions in optical fiber waveguides,” J. Opt. Soc. Am. A 9, 937–949 (1992).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1988).
[CrossRef]

Cook, B. D.

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

Donnelly, R.

M. R. Palmer, R. Donnelly, “Focused wave modes and the scalar wave equation,”J. Math. Phys. 34, 4007–4013 (1993).
[CrossRef]

R. Donnelly, R. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

Durnin, J.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 11.3.

Hafizi, B.

Hernandez, J. E.

J. E. Hernandez, R. W. Ziolkowski, S. R. Parker, “Synthesis of the driving functions of an array for propagating localized wave energy,”J. Acoust. Soc. Am. 92, 550–562 (1992).
[CrossRef]

Lewis, D. K.

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[CrossRef]

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

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 11.3.

Palmer, M. R.

M. R. Palmer, R. Donnelly, “Focused wave modes and the scalar wave equation,”J. Math. Phys. 34, 4007–4013 (1993).
[CrossRef]

Parker, S. R.

J. E. Hernandez, R. W. Ziolkowski, S. R. Parker, “Synthesis of the driving functions of an array for propagating localized wave energy,”J. Acoust. Soc. Am. 92, 550–562 (1992).
[CrossRef]

Sedky, S. M.

Shaarawi, A. M.

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focused-wave-mode pulses from wide-band dynamic Gaussian apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
[CrossRef]

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[CrossRef]

A. M. Vengsarkar, I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “Closed-form localized wave solutions in optical fiber waveguides,” J. Opt. Soc. Am. A 9, 937–949 (1992).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1988).
[CrossRef]

Sprangle, P.

Vengsarkar, A. M.

Ziolkowski, R.

R. Donnelly, R. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

Ziolkowski, R. W.

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focused-wave-mode pulses from wide-band dynamic Gaussian apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[CrossRef]

J. E. Hernandez, R. W. Ziolkowski, S. R. Parker, “Synthesis of the driving functions of an array for propagating localized wave energy,”J. Acoust. Soc. Am. 92, 550–562 (1992).
[CrossRef]

A. M. Vengsarkar, I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “Closed-form localized wave solutions in optical fiber waveguides,” J. Opt. Soc. Am. A 9, 937–949 (1992).
[CrossRef]

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
[CrossRef]

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[CrossRef]

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

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1988).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. E. Hernandez, R. W. Ziolkowski, S. R. Parker, “Synthesis of the driving functions of an array for propagating localized wave energy,”J. Acoust. Soc. Am. 92, 550–562 (1992).
[CrossRef]

J. Appl. Phys. (2)

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1988).
[CrossRef]

J. Math. Phys. (2)

M. R. Palmer, R. Donnelly, “Focused wave modes and the scalar wave equation,”J. Math. Phys. 34, 4007–4013 (1993).
[CrossRef]

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

J. Opt. Soc. Am. A (5)

Phys. Rev. A (2)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

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

Proc. IEEE (1)

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

R. Donnelly, R. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

Other (1)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 11.3.

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Figures (9)

Fig. 1
Fig. 1

Diffraction through a circular aperture of radius R.

Fig. 2
Fig. 2

Periodic and Gaussian time windows applied to the aperture.

Fig. 3
Fig. 3

Comparison of the decay of the centroid LW pulses and that of the centroid of quasi-monochromatic signals. In the upper plot, the solid curve represents the Gaussian FWM pulse, and the dotted curve represents the periodic FWM pulse.

Fig. 4
Fig. 4

Depletion of the spatial spectrum of the quasi-monochromatic signal with distance.

Fig. 5
Fig. 5

Oscillations that affect different sections of the Gaussian FWM spatial spectrum at z = ct = 525.27 m (dotted curve) and 0 (solid curve).

Fig. 6
Fig. 6

ω windows scanning the spatial spectrum of the Gaussian FWM pulse at z = ct = 0.

Fig. 7
Fig. 7

ω windows scanning the spatial spectrum of the periodic FWM pulse at z = ct = 0. The dashed curve represents the equivalent ω window corresponding to an effective cT value of 2.5 mm; the dotted curve represents ω windows equivalent to a Gaussian cT value of 9.375 mm.

Fig. 8
Fig. 8

Oscillations that affect different sections of the periodic FWM spatial spectrum at z = ct = 525.27 m (dotted curve) and 0 (solid curve).

Fig. 9
Fig. 9

Decay pattern of the periodic FWM pulse. Dashed curve, asymptotic behavior near the aperture; dotted curve, asymptotic behavior farther from the aperture.

Equations (30)

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Θ Δ κ n k l .
R d = R + Θ Z .
Z H S = R Θ .
Z H S = k l R Δ κ n .
Ψ i ( ρ , t ) = Re { Ψ ˆ i ( ρ , t ) }
Ψ ˆ i ( ρ , t ) = ( 1 2 π 0 + d χ χ J 0 ( χ ρ ) 0 + d ω exp ( i ω t ) × exp { i [ ( ω / c ) 2 χ 2 ] z } ϕ ( χ , ω ) ) z = 0 .
ϕ ( χ , ω ) = δ ˆ ( ω ω 0 ( χ ) ; T ) exp ( χ 2 w 2 / 16 ) ,
ω 0 ( χ ) = ( χ 2 4 β + β ) c .
δ ˆ [ ω ω 0 ( χ ) ; T ] = δ ˆ G [ ω ω 0 ( χ ) ; T ] = T π exp { T 2 [ ω ω 0 ( χ ) ] 2 } .
δ ˆ [ ω ω 0 ( χ ) ; T ] = δ ˆ P [ ω ω 0 ( χ ) ; T ] = 1 2 π sin { [ ω ω 0 ( χ ) ] 4 T } × { 1 [ ω ω 0 ( χ ) ] [ ω ω 0 ( χ ) ] [ ω ω 0 ( χ ) ] 2 ( 5 π / 4 T ) 2 } .
Ψ ˆ i G = β π ( a 1 i c t ) exp [ β ρ 2 / ( a 1 i c t ) ] × exp ( i β c t ) exp ( t 2 / 4 T 2 ) ,
Ψ ˆ i P = β π ( a 1 i c t ) exp [ β ρ 2 / a 1 i c t ] × exp ( i β c t ) cos 2 ( 5 π t / 8 T ) , for 4 T t + 4 T .
δ ˆ [ ω ω 0 ( χ ) ; T ] = δ ˆ q m [ ω ω 0 ( χ ) ; T ] = T π exp [ T 2 ( ω ω c ) 2 ] ,
Ψ ˆ i q m = 4 π w 2 exp ( 4 ρ 2 / w 2 ) exp ( t 2 / 4 T 2 ) exp ( i ω c t ) .
Ψ ( ρ , z , t ) = Re { Ψ ˆ ( ρ , z , t ) } ,
Ψ ˆ ( ρ , z , t ) = 1 2 π 0 + d χ χ J 0 ( χ ρ ) 0 + d ω exp ( i ω t ) × exp { i [ ( ω / c ) 2 χ 2 ] z } ϕ ( χ , ω ) .
Ψ ( ρ , z , t ) = 1 2 π 0 + d χ χ J 0 ( χ ρ ) 0 + d ω × δ ˆ ( [ ω ω 0 ( χ ) ; T ] exp ( χ 2 w 2 / 16 ) × cos { [ ( ω / c ) 2 χ 2 ] z ω t } .
Ψ ( ρ , z , t ) = 0 + d χ χ J 0 ( χ ρ ) ϕ s ( χ , z , t ) ,
ϕ s ( χ , z , t ) = 1 2 π 0 + d ω ϕ ( χ , ω ) × cos { [ ( ω / c ) 2 χ 2 ] z ω t } .
ϕ s ( χ , z , t ) = T exp ( χ 2 w 2 / 16 ) 2 π 3 / 2 × 0 + d ω exp [ T 2 ( ω ω c ) 2 ] × cos { [ ( ω / c ) 2 χ 2 ] z ω t } .
Θ max = [ ( ω c / c ) 2 χ max 2 ( ω c / c ) ] z .
Θ max = [ χ max 2 2 ( ω c / c ) ] z .
Z d = [ 4 π ( ω c / c ) χ max 2 ] .
ϕ s ( χ , z , t ) = T exp ( χ 2 w 2 / 16 ) 2 π 3 / 2 × 0 + d ω exp { T 2 [ ω ω 0 ( χ ) ] 2 } × cos { [ ( ω / c ) 2 χ 2 ] z ω t } .
Θ min = ( { [ ω 0 ( χ d ) / c ] 2 / c T } 2 χ d 2 { [ ω 0 ( χ d ) / c ] 2 / c T } ) z .
Θ min χ d 2 z 2 { [ ω 0 ( χ d ) / c ] 2 / c T } .
Θ max χ d 2 z 2 { [ ω 0 ( χ d ) / c ] + 2 / c T } .
Δ Θ d 2 c T χ d 2 z { [ ω 0 ( χ d ) / c ] 2 4 / ( c T ) 2 } .
Z d = π T { [ ω 0 ( χ d ) 2 ] 4 / T 2 } χ d 2 c π c T [ ω 0 ( χ d ) χ d c ] 2 ,
ϕ s ( χ , z , t ) = exp ( χ 2 w 2 / 16 ) 2 π × 0 + d ω δ ˆ P { [ ω ω 0 ( χ ) ] ; T } × cos { [ ( ω / c ) 2 χ 2 ] z ω t } .

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