Abstract

In this Letter, we analyze the reflection of cylindrical waves (CWs) at planar interfaces. We consider the reflected CW proposed in the literature as a spectral integral. We present a Laurent series expansion of the Fresnel coefficient convergent on the whole real axis and we use it to solve analytically the reflected-wave integral. We found a solution that involves both Bessel functions and Anger–Weber functions, i.e., solutions of both the homogeneous and inhomogeneous Bessel differential equations. We compare the analytical solution with the numerical results obtained with a quadrature formula presented in the literature. Moreover, we present a physical interpretation that connects our solution to the image principle.

© 2014 Optical Society of America

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  1. R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, J. Opt. Soc. Am. A 13, 483 (1996).
    [CrossRef]
  2. H. Liu and P. Lalanne, Nature 452, 728 (2008).
    [CrossRef]
  3. G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, Opt. Commun. 95, 192 (1993).
    [CrossRef]
  4. F. Frezza, G. Schettini, and N. Tedeschi, Opt. Commun. 284, 3867 (2011).
    [CrossRef]
  5. R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, J. Electromagn. Waves Appl. 14, 1353 (2000).
    [CrossRef]
  6. F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, IEEE Geosci. Remote Sens. Lett. 10, 179 (2013).
    [CrossRef]
  7. A. Coatanhay and J. M. Conoir, J. Comput. Acoust. 12, 233 (2004).
    [CrossRef]
  8. P. Pawliuk and M. Yedlin, IEEE Trans. Antennas Propag. 60, 5296 (2012).
    [CrossRef]
  9. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, 1944).
  10. F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge, 2010).
  11. F. J. W. Whipple, Proc. Lond. Math. Soc. 16, 94 (1917).

2013

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, IEEE Geosci. Remote Sens. Lett. 10, 179 (2013).
[CrossRef]

2012

P. Pawliuk and M. Yedlin, IEEE Trans. Antennas Propag. 60, 5296 (2012).
[CrossRef]

2011

F. Frezza, G. Schettini, and N. Tedeschi, Opt. Commun. 284, 3867 (2011).
[CrossRef]

2008

H. Liu and P. Lalanne, Nature 452, 728 (2008).
[CrossRef]

2004

A. Coatanhay and J. M. Conoir, J. Comput. Acoust. 12, 233 (2004).
[CrossRef]

2000

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, J. Electromagn. Waves Appl. 14, 1353 (2000).
[CrossRef]

1996

1993

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, Opt. Commun. 95, 192 (1993).
[CrossRef]

1917

F. J. W. Whipple, Proc. Lond. Math. Soc. 16, 94 (1917).

Boisvert, R. F.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge, 2010).

Borghi, R.

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, J. Electromagn. Waves Appl. 14, 1353 (2000).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, J. Opt. Soc. Am. A 13, 483 (1996).
[CrossRef]

Cincotti, G.

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, Opt. Commun. 95, 192 (1993).
[CrossRef]

Clark, C. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge, 2010).

Coatanhay, A.

A. Coatanhay and J. M. Conoir, J. Comput. Acoust. 12, 233 (2004).
[CrossRef]

Conoir, J. M.

A. Coatanhay and J. M. Conoir, J. Comput. Acoust. 12, 233 (2004).
[CrossRef]

Frezza, F.

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, IEEE Geosci. Remote Sens. Lett. 10, 179 (2013).
[CrossRef]

F. Frezza, G. Schettini, and N. Tedeschi, Opt. Commun. 284, 3867 (2011).
[CrossRef]

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, J. Electromagn. Waves Appl. 14, 1353 (2000).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, J. Opt. Soc. Am. A 13, 483 (1996).
[CrossRef]

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, Opt. Commun. 95, 192 (1993).
[CrossRef]

Furnò, F.

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, Opt. Commun. 95, 192 (1993).
[CrossRef]

Gori, F.

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, J. Opt. Soc. Am. A 13, 483 (1996).
[CrossRef]

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, Opt. Commun. 95, 192 (1993).
[CrossRef]

Lalanne, P.

H. Liu and P. Lalanne, Nature 452, 728 (2008).
[CrossRef]

Liu, H.

H. Liu and P. Lalanne, Nature 452, 728 (2008).
[CrossRef]

Lozier, D. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge, 2010).

Olver, F. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge, 2010).

Pajewski, L.

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, IEEE Geosci. Remote Sens. Lett. 10, 179 (2013).
[CrossRef]

Pawliuk, P.

P. Pawliuk and M. Yedlin, IEEE Trans. Antennas Propag. 60, 5296 (2012).
[CrossRef]

Ponti, C.

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, IEEE Geosci. Remote Sens. Lett. 10, 179 (2013).
[CrossRef]

Santarsiero, M.

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, J. Electromagn. Waves Appl. 14, 1353 (2000).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, J. Opt. Soc. Am. A 13, 483 (1996).
[CrossRef]

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, Opt. Commun. 95, 192 (1993).
[CrossRef]

Santini, C.

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, J. Electromagn. Waves Appl. 14, 1353 (2000).
[CrossRef]

Schettini, G.

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, IEEE Geosci. Remote Sens. Lett. 10, 179 (2013).
[CrossRef]

F. Frezza, G. Schettini, and N. Tedeschi, Opt. Commun. 284, 3867 (2011).
[CrossRef]

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, J. Electromagn. Waves Appl. 14, 1353 (2000).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, J. Opt. Soc. Am. A 13, 483 (1996).
[CrossRef]

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, Opt. Commun. 95, 192 (1993).
[CrossRef]

Tedeschi, N.

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, IEEE Geosci. Remote Sens. Lett. 10, 179 (2013).
[CrossRef]

F. Frezza, G. Schettini, and N. Tedeschi, Opt. Commun. 284, 3867 (2011).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, 1944).

Whipple, F. J. W.

F. J. W. Whipple, Proc. Lond. Math. Soc. 16, 94 (1917).

Yedlin, M.

P. Pawliuk and M. Yedlin, IEEE Trans. Antennas Propag. 60, 5296 (2012).
[CrossRef]

IEEE Geosci. Remote Sens. Lett.

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, IEEE Geosci. Remote Sens. Lett. 10, 179 (2013).
[CrossRef]

IEEE Trans. Antennas Propag.

P. Pawliuk and M. Yedlin, IEEE Trans. Antennas Propag. 60, 5296 (2012).
[CrossRef]

J. Comput. Acoust.

A. Coatanhay and J. M. Conoir, J. Comput. Acoust. 12, 233 (2004).
[CrossRef]

J. Electromagn. Waves Appl.

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, J. Electromagn. Waves Appl. 14, 1353 (2000).
[CrossRef]

J. Opt. Soc. Am. A

Nature

H. Liu and P. Lalanne, Nature 452, 728 (2008).
[CrossRef]

Opt. Commun.

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, Opt. Commun. 95, 192 (1993).
[CrossRef]

F. Frezza, G. Schettini, and N. Tedeschi, Opt. Commun. 284, 3867 (2011).
[CrossRef]

Proc. Lond. Math. Soc.

F. J. W. Whipple, Proc. Lond. Math. Soc. 16, 94 (1917).

Other

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, 1944).

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge, 2010).

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Figures (6)

Fig. 1.
Fig. 1.

Geometry of the problem.

Fig. 2.
Fig. 2.

Conformal mapping of the real path n(,+), on the plane of w=u+ιv.

Fig. 3.
Fig. 3.

Amplitude of the RW0(ρ,θ), with μ1=μ2=1, and ε1=2, ε2=8, for ρ=0.5, and θ[π/2;π/2], obtained with the quadrature method proposed in [5] (dashed line) and with the Eq. (7) (solid line).

Fig. 4.
Fig. 4.

Amplitude of the RW0(ρ,θ), in the same scenario of Fig. 3, with ρ=20, obtained with the quadrature method proposed in [5] (circles) and with Eq. (7) (solid line).

Fig. 5.
Fig. 5.

Amplitude of the RW1(ρ,θ), in the same scenario of Fig. 3, in H polarization with ρ=4, obtained with the quadrature method proposed in [5] (circles) and with Eq. (7) (solid line).

Fig. 6.
Fig. 6.

Amplitude of the RWm(ρ,θ), in the same scenario of Fig. 3, with ρ=4 and θ=0°, as a function of the order m, obtained with the quadrature method proposed in [5] (circles) and with Eq. (7) (solid line).

Equations (14)

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

RWm(ξ,ζ)=12π+RE/HCW^meιζndn,
w=n±n21n=w2+12w,
RE/H(w)=k=0RkE/Hwk,
RkE/H=12πιCRE/H(w)wk+1dw
RE/H(n)=k=0RkE/H(n±n21)k.
RE/H(n)=k=0R2kE/He2kιarccos|n|.
RWm(ξ,ζ)=k=0+RkE/H[Fm2k(ξ,ζ)+(1)mFm2k(ξ,ζ)],
Fp(ξ,ζ)=12π0+CW^p(ξ,n)eιζndn.
Fp(ρ,θ)=(ι)pπ0π2ιeιρcos(tθ)+ιptdt
(ι)pπθπ2eιρcost+ιptdt=l=+ap,l(θ)Jl(ρ),
ap,l(θ)=2πιlpeι(l+p)(π4θ2)(π4+θ2)×sinc[(l+p)(π4+θ2)].
Ap(ρ)=1π0+eρsinhvpvdv,
Fp(ρ,θ)=eιpθ[l=+[ap,l(θ)Jl(ρ)]ιAp(ρ)].
Vr(ρ,θ)=RWm(ξ2χ,ζ)=RWm(ρχ,θχ).

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