Abstract

Starting from some general and plausible assumptions based on geometrical optics and on a common feature of the truncated Bessel beams, a heuristic derivation is presented of very simple analytical expressions capable of describing the longitudinal (on-axis) evolution of axially symmetric nondiffracting pulses truncated by finite apertures. The analytical formulation is applied to several situations involving subluminal, luminal, or superluminal localized pulses, and the results are compared with those obtained by numerical simulations of the Rayleigh–Sommerfeld diffraction integrals. The results are in excellent agreement. The present approach can be rather useful, because it yields, in general, closed-form expressions, avoiding the need for time-consuming numerical simulations, and also because such expressions provide a powerful tool for exploring several important properties of the truncated localized pulses, such as their depth of fields, the longitudinal pulse behavior, and the decaying rates.

© 2006 Optical Society of America

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  1. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bi-directional traveling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
    [CrossRef]
  2. J.-y. Lu and J. F. Greenleaf, "Nondiffracting X-waves: exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
    [CrossRef] [PubMed]
  3. J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
    [CrossRef]
  4. P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
    [CrossRef]
  5. E. Recami, "On localized X-shaped superluminal solutions to Maxwell equations," Physica A 252, 586-610 (1998).
    [CrossRef]
  6. M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, "New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies," Eur. J. Phys. 21, 217-228 (2002).
  7. M. A. Porras, S. Trillo, C. Conti, and P. Di Trapani, "Paraxial envelope X-waves," Opt. Lett. 28, 1090-1092 (2003).
    [CrossRef] [PubMed]
  8. G. Nyitray and S. V. Kukhlevsky, "Distortion-free tight confinement and step-like decay of fs pulses in free space," arXiv:physics/0310057vl, Oct. 13, 2003.
  9. M. Zamboni-Rached, H. E. Hernandez-Figueroa, and E. Recami, "Chirped optical X-type pulses," J. Opt. Soc. Am. A 21, 2455-2463 (2004).
    [CrossRef]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  11. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, "Aperture realizations of exact solutions to homogeneous wave-equations," J. Opt. Soc. Am. A 10, 75-87 (1993).
    [CrossRef]
  12. J.-y. Lu and J. F. Greenleaf, "Experimental verification of nondiffracting X-waves," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441-446 (1992).
    [CrossRef] [PubMed]
  13. J.-y. Lu, H.-h. Zou, and J. F. Greenleaf, "Biomedical ultrasound beam forming," Ultrasound Med. Biol. 20, 403-428 (1994).
    [CrossRef] [PubMed]
  14. M. Zamboni-Rached, A. Shaarawi, and E. Recami, "Focused X-shaped pulses," J. Opt. Soc. Am. A 21, 1564-1574 (2004).
    [CrossRef]
  15. This paper first appeared as e-print arXiv:physics/0512148vl, Dec. 16, 2005
  16. J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  17. M. Zamboni-Rached, "Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves," Opt. Lett. 12, 4001-4006 (2004).
  18. M. Zamboni-Rached, E. Recami, and H. Figueroa, "Theory of 'frozen waves:' modeling the shape of stationary wave fields," J. Opt. Soc. Am. A 22, 2465-2475 (2005).
    [CrossRef]
  19. M. Zamboni-Rached, "Diffraction-attenuation resistant beams in absorbing media," Opt. Express 14, 1804-1809 (2006). (Also available as e-print arXiv:physics/0506067v2 Jun. 15, 2005).
    [CrossRef] [PubMed]
  20. S. V. Kukhlevsky and M. Mechler, "Diffraction-free sub-wavelength beam optics at nanometer scale," Opt. Commun. 231, 35-43 (2004).
    [CrossRef]
  21. Interesting approximate analytical descriptions for special cases of truncated nondiffracting beams (not pulses) are discussed in Z. Jiang, "Truncation of a two-dimensional nondiffracting cos beam," J. Opt. Soc. Am. A 14, 1478-1481 (1997).
    [CrossRef]
  22. A. G. Sedukhin, "Marginal phase correction of truncated Bessel beams," J. Opt. Soc. Am. A 17, 1059-1066 (2000).
    [CrossRef]
  23. R. Donnelly and R. W. Ziolkowski, "Designing localized waves," Proc. R. Soc. London, Ser. A 440, 541-565 (1993).
    [CrossRef]

2006 (1)

2005 (2)

2004 (4)

M. Zamboni-Rached, "Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves," Opt. Lett. 12, 4001-4006 (2004).

S. V. Kukhlevsky and M. Mechler, "Diffraction-free sub-wavelength beam optics at nanometer scale," Opt. Commun. 231, 35-43 (2004).
[CrossRef]

M. Zamboni-Rached, A. Shaarawi, and E. Recami, "Focused X-shaped pulses," J. Opt. Soc. Am. A 21, 1564-1574 (2004).
[CrossRef]

M. Zamboni-Rached, H. E. Hernandez-Figueroa, and E. Recami, "Chirped optical X-type pulses," J. Opt. Soc. Am. A 21, 2455-2463 (2004).
[CrossRef]

2003 (2)

M. A. Porras, S. Trillo, C. Conti, and P. Di Trapani, "Paraxial envelope X-waves," Opt. Lett. 28, 1090-1092 (2003).
[CrossRef] [PubMed]

G. Nyitray and S. V. Kukhlevsky, "Distortion-free tight confinement and step-like decay of fs pulses in free space," arXiv:physics/0310057vl, Oct. 13, 2003.

2002 (1)

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, "New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies," Eur. J. Phys. 21, 217-228 (2002).

2000 (1)

1998 (1)

E. Recami, "On localized X-shaped superluminal solutions to Maxwell equations," Physica A 252, 586-610 (1998).
[CrossRef]

1997 (2)

1996 (1)

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

1994 (1)

J.-y. Lu, H.-h. Zou, and J. F. Greenleaf, "Biomedical ultrasound beam forming," Ultrasound Med. Biol. 20, 403-428 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (2)

J.-y. Lu and J. F. Greenleaf, "Experimental verification of nondiffracting X-waves," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441-446 (1992).
[CrossRef] [PubMed]

J.-y. Lu and J. F. Greenleaf, "Nondiffracting X-waves: exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
[CrossRef] [PubMed]

1989 (1)

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bi-directional traveling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

1968 (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Besieris, I. M.

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

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bi-directional traveling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

Conti, C.

Di Trapani, P.

Donnelly, R.

R. Donnelly and R. W. Ziolkowski, "Designing localized waves," Proc. R. Soc. London, Ser. A 440, 541-565 (1993).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Fagerholm, J.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Figueroa, H.

Friberg, A. T.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Greenleaf, J. F.

J.-y. Lu, H.-h. Zou, and J. F. Greenleaf, "Biomedical ultrasound beam forming," Ultrasound Med. Biol. 20, 403-428 (1994).
[CrossRef] [PubMed]

J.-y. Lu and J. F. Greenleaf, "Experimental verification of nondiffracting X-waves," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441-446 (1992).
[CrossRef] [PubMed]

J.-y. Lu and J. F. Greenleaf, "Nondiffracting X-waves: exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
[CrossRef] [PubMed]

Hernandez-Figueroa, H. E.

Hernández-Figueroa, H. E.

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, "New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies," Eur. J. Phys. 21, 217-228 (2002).

Huttunen, J.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Jiang, Z.

Kukhlevsky, S. V.

S. V. Kukhlevsky and M. Mechler, "Diffraction-free sub-wavelength beam optics at nanometer scale," Opt. Commun. 231, 35-43 (2004).
[CrossRef]

G. Nyitray and S. V. Kukhlevsky, "Distortion-free tight confinement and step-like decay of fs pulses in free space," arXiv:physics/0310057vl, Oct. 13, 2003.

Lu, J.-y.

J.-y. Lu, H.-h. Zou, and J. F. Greenleaf, "Biomedical ultrasound beam forming," Ultrasound Med. Biol. 20, 403-428 (1994).
[CrossRef] [PubMed]

J.-y. Lu and J. F. Greenleaf, "Experimental verification of nondiffracting X-waves," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441-446 (1992).
[CrossRef] [PubMed]

J.-y. Lu and J. F. Greenleaf, "Nondiffracting X-waves: exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
[CrossRef] [PubMed]

Mechler, M.

S. V. Kukhlevsky and M. Mechler, "Diffraction-free sub-wavelength beam optics at nanometer scale," Opt. Commun. 231, 35-43 (2004).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Morgan, D. P.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Nyitray, G.

G. Nyitray and S. V. Kukhlevsky, "Distortion-free tight confinement and step-like decay of fs pulses in free space," arXiv:physics/0310057vl, Oct. 13, 2003.

Porras, M. A.

Recami, E.

M. Zamboni-Rached, E. Recami, and H. Figueroa, "Theory of 'frozen waves:' modeling the shape of stationary wave fields," J. Opt. Soc. Am. A 22, 2465-2475 (2005).
[CrossRef]

M. Zamboni-Rached, H. E. Hernandez-Figueroa, and E. Recami, "Chirped optical X-type pulses," J. Opt. Soc. Am. A 21, 2455-2463 (2004).
[CrossRef]

M. Zamboni-Rached, A. Shaarawi, and E. Recami, "Focused X-shaped pulses," J. Opt. Soc. Am. A 21, 1564-1574 (2004).
[CrossRef]

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, "New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies," Eur. J. Phys. 21, 217-228 (2002).

E. Recami, "On localized X-shaped superluminal solutions to Maxwell equations," Physica A 252, 586-610 (1998).
[CrossRef]

Reivelt, K.

P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

Saari, P.

P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

Salomaa, M. M.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Sedukhin, A. G.

Shaarawi, A.

Shaarawi, A. M.

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

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bi-directional traveling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

Trillo, S.

Zamboni-Rached, M.

Ziolkowski, R. W.

R. Donnelly and R. W. Ziolkowski, "Designing localized waves," Proc. R. Soc. London, Ser. A 440, 541-565 (1993).
[CrossRef]

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

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bi-directional traveling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

Zou, H.-h.

J.-y. Lu, H.-h. Zou, and J. F. Greenleaf, "Biomedical ultrasound beam forming," Ultrasound Med. Biol. 20, 403-428 (1994).
[CrossRef] [PubMed]

Eur. J. Phys. (1)

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, "New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies," Eur. J. Phys. 21, 217-228 (2002).

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (2)

J.-y. Lu and J. F. Greenleaf, "Nondiffracting X-waves: exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
[CrossRef] [PubMed]

J.-y. Lu and J. F. Greenleaf, "Experimental verification of nondiffracting X-waves," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441-446 (1992).
[CrossRef] [PubMed]

J. Math. Phys. (1)

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bi-directional traveling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Commun. (1)

S. V. Kukhlevsky and M. Mechler, "Diffraction-free sub-wavelength beam optics at nanometer scale," Opt. Commun. 231, 35-43 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

M. Zamboni-Rached, "Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves," Opt. Lett. 12, 4001-4006 (2004).

M. A. Porras, S. Trillo, C. Conti, and P. Di Trapani, "Paraxial envelope X-waves," Opt. Lett. 28, 1090-1092 (2003).
[CrossRef] [PubMed]

Phys. Rev. E (1)

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Phys. Rev. Lett. (2)

P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Physica A (1)

E. Recami, "On localized X-shaped superluminal solutions to Maxwell equations," Physica A 252, 586-610 (1998).
[CrossRef]

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

R. Donnelly and R. W. Ziolkowski, "Designing localized waves," Proc. R. Soc. London, Ser. A 440, 541-565 (1993).
[CrossRef]

Ultrasound Med. Biol. (1)

J.-y. Lu, H.-h. Zou, and J. F. Greenleaf, "Biomedical ultrasound beam forming," Ultrasound Med. Biol. 20, 403-428 (1994).
[CrossRef] [PubMed]

Other (3)

G. Nyitray and S. V. Kukhlevsky, "Distortion-free tight confinement and step-like decay of fs pulses in free space," arXiv:physics/0310057vl, Oct. 13, 2003.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

This paper first appeared as e-print arXiv:physics/0512148vl, Dec. 16, 2005

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Figures (9)

Fig. 1
Fig. 1

Typical Bessel beam truncated by a finite aperture.

Fig. 2
Fig. 2

Peak intensity evolution of the subluminal TNP for the three cases: (1) V = 0.995 c and b = 1.5 × 10 15 Hz , (2) V = 0.998 c and b = 6 × 10 14 Hz , (3) V = 0.9992 c and b = 2.4 × 10 14 Hz . In all cases R = 4 mm . The continuous curves represent results obtained from our closed-form analytical expression (23), and dotted curves represent results from the numerical simulation of the Rayleigh–Sommerfeld formula (8).

Fig. 3
Fig. 3

On-axis evolution of the subluminal TNP, at the times t = 0.11 ns , t = 0.22 ns , and t = 0.33 ns , for each case cited in Fig. 2; (a), (b), and (c), represent cases (1), (2), and (3), respectively. The continuous curves are the results obtained from our closed-form analytical expression (23) and dotted curves represent results from the numerical simulation of the Rayleigh–Sommerfeld formula (8).

Fig. 4
Fig. 4

Peak-intensity evolution of the truncated luminal FWM pulse for the three cases: (1) a = 1.6 × 10 16 and b = 5 × 10 11 Hz , (2) a = 1.25 × 10 16 s and b = 3 × 10 11 Hz , (3) a = 1 × 10 16 and b = 2 × 10 11 Hz . In all cases R = 2 mm . The continuous curves are the results obtained from our closed-form analytical expression (25) and dotted curves represent results from the numerical simulation of the Rayleigh–Sommerfeld formula (8).

Fig. 5
Fig. 5

On-axis evolution of the truncated luminal FWM pulse, at the times t = 0.22 ns , t = 0.44 ns , and t = 0.66 ns , for each of the cases cited in Fig. 4; (a), (b), and (c) represent cases (1), (2), and (3), respectively. The continuous curves are the results obtained from our closed-form analytical expression (25), and dotted curves represent results from the numerical simulation of the Rayleigh–Sommerfeld formula (8).

Fig. 6
Fig. 6

Peak intensity evolution of the truncated superluminal FWM pulse for the three cases: (1) V = 1.0002 c , b = 3 × 10 12 Hz , and a = 2.5 × 10 17 s ; (2) V = 1.0001 c , b = 1 × 10 12 Hz , and a = 5 × 10 17 s ; and (3) V = 1.00008 c , b = 2 × 10 12 Hz , and a = 1.1 × 10 17 s . In all cases R = 3 mm . The continuous curves are results obtained from our closed-form, analytical expression (28), and dotted curves represent results from the numerical simulation of the Rayleigh–Sommerfeld formula (8).

Fig. 7
Fig. 7

On-axis evolution of the truncated SFWM pulse, at the times t = 0.14 ns , t = 0.29 ns , and t = 0.43 ns , for each of the cases cited in Fig. 6; (a), (b), and (c) represent cases (1), (2), and (3), respectively. The continuous curves are the results obtained from our closed-form, analytical expression (28), and results represented by dotted curves come from the numerical simulation of the Rayleigh–Sommerfeld formula (8).

Fig. 8
Fig. 8

Peak-intensity evolution of the truncated luminal MPS pulse for the three cases: (1) a = 1.6 × 10 16 , b 0 = 5 × 10 11 Hz , and q = 2 × 10 11 s ; (2) a = 1.25 × 10 16 s , b 0 = 3 × 10 11 Hz , and q = 10 × 10 11 s ; and (3) a = 1 × 10 16 , b 0 = 2 × 10 11 Hz , and q = 20 × 10 11 s . In all cases R = 2 mm . The continuous curves are results obtained from our closed-form analytical expression (32), and dotted curves represent results from the numerical simulation of the Rayleigh–Sommerfeld formula (8).

Fig. 9
Fig. 9

On-axis evolution of the truncated luminal MPS pulse, at the times t = 0.22 ns , t = 0.44 ns , and t = 0.66 ns , for each of the three cases considered in Fig. 8; (a), (b), and (c) represent cases (1), (2), and (3), respectively. The continuous curves are the results obtained from our closed-form analytical expression (32), dotted curves represent results from the numerical simulation of the Rayleigh–Sommerfeld formula (8).

Equations (34)

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

Ψ IBb ( ρ , z , t ) = J 0 ( k ρ ρ ) e i β z e i ω t ,
Ψ TBb ( ρ = 0 , z , t ) { e i β z e i ω t for z R tan θ 0 for z > R tan θ } ,
Ψ TBb ( ρ = 0 , z , t ) e i β z e i ω t [ H ( z ) H ( z R tan θ ) ] ,
Ψ ( ρ , z , t ) = 0 d ω ω c ω c d β S ¯ ( ω , β ) J 0 ( ρ ω 2 c 2 β 2 ) e i β z e i ω t
ω = V β + b
Ψ INP ( ρ , z , t ) = e i b z V ω min ω max d ω S ( ω ) × J 0 ( ρ ( 1 c 2 1 V 2 ) ω 2 + 2 b ω V 2 b 2 V 2 ) e i ω ζ V ,
k ρ 2 = ( 1 c 2 1 V 2 ) ω 2 + 2 b ω V 2 b 2 V 2 0 , β 0 ,
Ψ TNP ( ρ , z , t ) = 0 2 π d ϕ 0 R d ρ ρ 1 2 π D { [ Ψ INP ] ( z z ) D 2 + [ c t Ψ INP ] ( z z ) D } ,
R 1 k ρ = c ω sin θ ( ω ) ,
Ψ TNP ( ρ = 0 , z , t ) e i b z V ω min ω max d ω S ( ω ) × e i ω ζ V [ H ( z ) H ( z R tan θ ( ω ) ) ] ,
Ψ TNP ( ρ = 0 , z > 0 , t ) e i b z V [ ω min ω max S ( ω ) e i ω ζ V d ω ω min ω max S ( ω ) e i ω ζ V H ( z R tan θ ( ω ) ) d ω ] ,
R tan θ ( ω ) = R 1 ( c V b c V ω ) 2 ( c V b c V ω )
H ( z R tan θ ( ω ) ) = H ( z R ( c V b c V ω ) 1 ( c V b c V ω ) 2 ) = { 1 for z R ( c V b c V ω ) 1 ( c V b c V ω ) 2 0 for z < R ( c V b c V ω ) 1 ( c V b c V ω ) 2 } .
Ψ TNP ( ρ = 0 , z , t ) e i b z V [ ω min ω max S ( ω ) e i ω ζ V d ω ω min ω max S ( ω ) e i ω ζ V H ( z R ( c V b c V ω ) 1 ( c V b c V ω ) 2 ) d ω ] .
H ( z R ( c V b c V ω ) 1 ( c V b c V ω ) 2 )
= { 1 for ω b c ( c z V z 2 + R 2 ) 0 for ω > b c ( c z V z 2 + R 2 ) } .
Ψ TNP ( ρ = 0 , z > 0 , t ) e i b z V [ b b c ( c V ) S ( ω ) e i ω ζ V d ω b b c ( c z V z 2 + R 2 ) S ( ω ) e i ω ζ V d ω ] = e i b z V [ b c ( c z V z 2 + R 2 ) b c ( c V ) S ( ω ) e i ω ζ V d ω ] ,
H ( z R ( c V b c V ω ) 1 ( c V b c V ω ) 2 )
= { 1 for ω b ( 1 z z 2 + R 2 ) 0 for ω > b ( 1 z z 2 + R 2 ) } ,
Ψ TNP ( ρ = 0 , z > 0 , t ) e i b z c [ b S ( ω ) e i ω ζ c d ω b b ( 1 z z 2 + R 2 ) S ( ω ) e i ω ζ c d ω ] = e i b z c [ b ( 1 z z 2 + R 2 ) S ( ω ) e i ω ζ c d ω ] .
H ( z R ( c V b c V ω ) 1 ( c V b c V ω ) 2 ) = { 1 for ω b c ( c z V z 2 + R 2 ) and z R V 2 c 2 1 1 for ω b c ( c z V z 2 + R 2 ) and z R V 2 c 2 1 0 otherwise } .
Ψ TNP ( ρ = 0 , z > 0 , t ) { e i b z V b c ( c V ) S ( ω ) e i ω ζ V d ω for z R V 2 c 2 1 e i b z V b c ( c V ) b c ( c z V z 2 + R 2 ) S ( ω ) e i ω ζ V d ω for z R V 2 c 2 1 } .
Ψ TNP ( ρ = 0 , z > 0 , t ) { e i b z V b c ( c z V z 2 + R 2 ) S ( ω ) e i ω ζ V d ω for z R V 2 c 2 1 0 for z R V 2 c 2 1 } .
Ψ INP ( ρ , z , t ) = exp ( i b V γ 2 η c 2 ) sinc ( b γ c ρ 2 + γ 2 ζ 2 ) ,
Ψ TNP ( ρ = 0 , z > 0 , t ) c 2 b V γ 2 e i b z V [ b c ( c z V z 2 + R 2 ) b c ( c V ) e i ω ζ V d ω ] = c V 2 b V γ 2 i ζ e i b z V { exp [ i b c V ( c V ) ζ ] exp [ i b c z 2 + R 2 V ( c z 2 + R 2 z V ) ζ ] } ,
Ψ INP ( ρ , z , t ) = a c a c i ζ exp ( i b 2 c η ) exp [ b ρ 2 2 c ( a c i ζ ) ] ,
Ψ TNP ( ρ = 0 , z > 0 , t ) a e a b 2 e i b z c b ( 1 z z 2 + R 2 ) e a ω e i ω ζ c d ω = a c a c i ζ e a b 2 e i b z c exp [ b z 2 + R 2 ( a c i ζ ) c ( z 2 + R 2 z ) ] ,
Ψ INP ( ρ , z , t ) = a V exp ( i b 2 V η ) X exp { b ( V 2 + c 2 ) 2 V ( V 2 c 2 ) × [ ( a V i ζ ) X 1 ] } ,
X = [ ( a V i ζ ) 2 + ( V 2 c 2 1 ) ρ 2 ] 1 2 .
Ψ TNP ( ρ = 0 , z > 0 , t ) { a V e a b 2 e i b z V a V i ζ exp [ b c z 2 + R 2 ( a V i ζ ) V ( c z 2 + R 2 z V ) ] for z R V 2 c 2 1 0 for z R V 2 c 2 1 } ,
Ψ ANP ( ρ , z , t ) = b min b max d b ω min ω max d ω S ( ω , b ) × J 0 ( ρ ( 1 c 2 1 V 2 ) ω 2 + 2 b ω V 2 b 2 V 2 ) e i ω ζ V e i b z V ,
S ( b ) = H ( b b 0 ) q exp [ q ( b b 0 ) ] ,
Ψ ANP ( ρ , z , t ) = b 0 a q c a c i ζ exp ( i b 2 c η ) exp [ b ρ 2 2 c ( a c i ζ ) ] exp [ q ( b b 0 ) ] d b = 2 a c 2 q ( 2 c q + i η ) ( a c i ζ ) + ρ 2 exp ( i b 0 2 c η ) exp [ b 0 ρ 2 2 c ( a c i ζ ) ] ,
Ψ TNP ( ρ = 0 , z > 0 , t ) b 0 a q c a c i ζ e a b 2 e i b z c × exp [ q ( b b 0 ) ] exp [ b z 2 + R 2 ( a c i ζ ) c ( z 2 + R 2 z ) ] d b = a q c e a b 0 2 e i b 0 z c ( a c i ζ ) [ q a 2 + i z c + z 2 + R 2 ( a c i ζ ) c ( z 2 + R 2 z ) ] exp [ b 0 z 2 + R 2 ( a c i ζ ) c ( z 2 + R 2 z ) ] ,

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