May 1, 2014 / Vol. 39, No. 9 / OPTICS LETTERS
2727
Generalized image principle for cylindrical waves Fabrizio Frezza and Nicola Tedeschi* Department of Information Engineering, Electronics, and Telecommunications, “La Sapienza” University of Rome, Roma 00184, Italy *Corresponding author: [email protected] Received February 24, 2014; revised March 31, 2014; accepted April 3, 2014; posted April 3, 2014 (Doc. ID 206962); published April 28, 2014 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 OCIS codes: (050.1755) Computational electromagnetic methods; (050.1960) Diffraction theory; (260.2110) Electromagnetic optics; (290.2558) Forward scattering. http://dx.doi.org/10.1364/OL.39.002727
The interaction of cylindrical waves (CWs) with a planar interface is of great importance in optic research, e.g., in the scattering problem by cylindrical structures near plane interfaces [1] or in microscopy [2]. Let us consider a plane interface between two media: medium 1 with relative permittivity and permeability ε1 and μ1 , respectively, and medium 2 with ε2 and μ2 . We define a coordinate system on the plane x; z containing the perpendicular vector to the interface, where x is the axis perpendicular to the interface and z is the axis parallel to it. We normalize the coordinates with respect to the wave number of medium 1: ξ k1 x and ζ k1 z, see Fig. 1. The reflection of a CW by a plane interface can be expressed by the so-called cylindrical reflected wave (RW) [1] 1 RWm ξ; ζ 2π
Z
∞
−∞
d m eιζn∥ dn∥ ; RE∕H CW
(1)
where n∥ is the component parallel to the interface of the propagation vector normalized with respect to k1 , and d m ξ; n∥ is the plane-wave spectrum of the cylindrical CW function of order m [3,4]. The function in Eq. (1) is obtained by the plane wave spectrum of the incident cylindrical function where each elementary wave is multiplied by the reflection coefficient, RE∕H n∥ , in E or H polarization, respectively, depending on the direction of the electric field of the impinging wave. Being n∥ the normalized component parallel to the interface of the propagation vector of the elementary plane wave of the spectrum, the reflection at the interface depends on it and, in particular, for jn∥ j > arcsink2 ∕k1 each elementary plane wave is totally reflected by the interface. The numerical evaluation of the integral in Eq. (1) has been widely discussed in the literature [5] and it is the main difficulty in the solution of the scattering problem by cylindrical structures placed near plane interfaces between either lossless or lossy media [6]. Recently, a new approach, based on the Fourier expansion of the reflection coefficient inside the integral, has been attempted for the evaluation of the RW [7,8]. The main difficulty of this approach is that the Fourier expansion is possible only for n∥ ∈ −1; 1. By adding and subtracting two integrals without the reflection coefficient in the intervals 0146-9592/14/092727-04$15.00/0
jn∥ j ∈ 1; ∞, the authors of such papers obtained the expression of the CW. However, they must calculate the remaining integrals, in the intervals jn∥ j ∈ 1; ∞, numerically. This procedure has been connected to the image method, but, in our opinion, in the formulation presented it cannot be easily related to this method due to the unsolved integrals. The main obstacle faced in [7,8], for the solution of the integral Eq. (1), was on the series expansion of the reflection coefficient RE∕H n∥ . The authors assert that the Fourier expansion in n∥ ∈ −1; 1 is the only possible expansion, being that the Laurent expansion is not applicable because of a polar singularity of the reflection coefficient on the complex plane. We present a power expansion of the reflection coefficient, in both polarizations, that converges on the whole real axis. The reflection coefficients, in E or H polarization, cannot be expanded in a Laurent series in the whole complex plane because of a polar singularity away from the real axis. On the other hand, we are interested in the values of the function only on the real axis, where the function does not have singularities. Our idea is to perform a conformal mapping of the function to transform the real axis in a circumference away from the singularities. Let us consider the following conformal mapping:
Fig. 1. Geometry of the problem. © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 9 / May 1, 2014
w n∥
q w2 1 ; n2∥ − 1 → n∥ 2w
(2)
where the is to indicate that the function w has two branch points in n∥ 1. The presence of these branch points is extremely important as we will see in the following. Now, we study how the real axis on the plane of n∥ transforms on the plane of w: when w → 0− , then n∥ → −∞; when w → −1, then n∥ → −1; when w → ι, then n∥ → 0; when w → 1, then n∥ → 1; and when w → 0 , then n∥ → ∞. The path n∥ ∈ −∞; ∞ can be mapped on the plane of w as a semi-circumference of radius 1, centered in the origin, see Fig. 2. It can be seen that the reflection coefficient as a function of w does not have any singularities inside a circle of unitary radius centered in the origin. Therefore, the reflection coefficient can be expanded in a Laurent series inside this circle ∞ X
RE∕H w
k0
RkE∕H wk ;
(3)
where the coefficients RkE∕H are the Laurent coefficients RkE∕H
1 2πι
Z
RE∕H w dw k1 C w
(4)
and the path C can be chosen as the circumference of unitary radius centered in the origin. We can write the series as a function of n∥ RE∕H n∥
∞ X k0
qk RkE∕H n∥ n2∥ − 1 :
(5)
Now, the importance of the branch points of the square roots appears clear. To write the expression in a more understandable way, we can note that the reflection coefficient is an even function of w, so the odd coeffiE∕H cients R2n1 are all equal to zero. We can multiply into the brackets for 1, and, if we choose the sign − in the path n∥ ∈ −∞; 1, and the sign in the path n∥ ∈ 1; ∞, the Laurent series can be written as follows: RE∕H n∥
∞ X k0
E∕H 2kι arccos jn∥ j R2k e :
At this point, we can substitute the Laurent series for the reflection coefficient inside the RW expression in Eq. (1), and, by dividing the integral in the positive and negative integral paths, we can write the RW as follows: RWm ξ; ζ
∞ X k0
RkE∕H F m−2k ξ; ζ −1m F −m−2k ξ; −ζ; (7)
where the function F p ξ; ζ is defined as follows: F p ξ; ζ
1 2π
Z
∞ 0
d p ξ; n∥ eιζn∥ dn∥ : CW
(8)
It is the plane-wave expansion of a cylindrical function integrated only for positive values of n∥ . The series in Eq. (7) converges quickly: the relative error is less than 10−3 , if it is truncated at k 10, and less than 10−4 at k 20. The convergence speed slows down weakly as ρ increases. Now, our problem becomes the computation of the integral in Eq. (8). We make the following change of variable: n∥ sin t, i.e., we write the plane-wave spectrum as in the Sommerfeld representation [3,9] Z ι−p π2−ι∞ ιρ cost−θιpt e dt (9) F p ρ; θ π 0 the asymptote in π∕2 − ι∞ can be translated of a quantity θ, as well explained in [9]. The integral in Eq. (9) can be written as the sum of two integrals. The first integral ι−p π
Z
π 2
−θ
eιρ cos tιpt dt
∞ X l−∞
ap;l θJ l ρ;
(10)
where we made use of the well-known Jacobi expansion of the exponential expιρ cos t and with π θ 2 π θ ap;l θ ιl−p eιlp 4−2 π 4 2 π θ : × sinc l p 4 2
(11)
The second integral (6) Ap ρ
This representation is convergent on the whole real axis.
1 π
Z
∞ 0
e−ρ sinh v−pv dv;
(12)
where v π∕2 − ιt. It is the associate Anger–Weber function that is a solution of the inhomogeneous Bessel differential equation [10]. This kind of function has been employed in the scattering problem by a wedge [9,11]. We can write the function in Eq. (8) as follows: F p ρ; θ Fig. 2. Conformal mapping of the real path n∥ ∈ −∞; ∞, on the plane of w u ιv.
eιpθ
X ∞
ap;l θJ l ρ − ιAp ρ :
(13)
l−∞
This function is a solution of the inhomogeneous Bessel differential equation, i.e., the solution of the wave
May 1, 2014 / Vol. 39, No. 9 / OPTICS LETTERS
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).
equation in cylindrical coordinates with cylindrical sources. By inserting Eq. (13) in Eq. (7), we find an analytical expression for the RW of order m. Before providing a physical interpretation of the results, we want to validate them by comparisons with the numerical evaluation of the integral Eq. (1) obtained by the adaptive quadrature method proposed in the literature [5,6]. Let us consider the case of two nonmagnetic media, μ1 μ2 1. We consider the following relative permittivities: ε1 2 and ε2 8. We calculate the RW for a fixed radius as a function of the angle θ. In Fig. 3, the amplitude of the RW of order zero, for a radius ρ 0.5, in E polarization is shown. We can see that the solution obtained with the quadrature method is oscillating around the analytical result. Actually, up to now the quadrature formula cannot be tested directly but only by the comparisons of the electric field in such a scenario of scattering from cylinders near an interface. Now we can see that the analytical procedure returns the true value of the reflected CW, while the quadrature method presents some numerical errors due to the high oscillation of the integrand function [1]. Being the RW presented in Fig. 3 in the near field, we see that the errors of the quadrature formula are not negligible. However, if we consider the far field, i.e., high values of the radius, the oscillations of the numerical solution disappear, and the two results perfectly coincide. In Fig. 4, the amplitude of the RW in the same
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).
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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).
situation of Fig. 3 is shown when the radius is ρ 20. Moreover, in order to show a result for the other polarization, the RW1 for H polarization, in the same situation of Fig. 3, is shown in Fig. 5 with a radius ρ 4. Again, we see that the agreement is very good, but the quadrature method shows some small oscillations near θ 0. Finally, to validate the procedure for high orders, in Fig. 6 the amplitude of the RW in the same scenario of Fig. 3, with ρ 4 and θ 0° , as a function of the order is shown. We recognize the exponential growth with the order typical of the Hankel function. In Eq. (13) this behavior is due to the associate Anger–Weber function, strongly connected both to the Weber function and to the Bessel function of the second kind. Now, we attempt a physical interpretation of the results found. Let us consider the incidence of a CW of order m on a planar interface between the two media 1 and 2. We suppose that the impinging wave originates at a distance h from the interface, and we normalize this distance with respect to k1 : χ k1 h, see Fig. 1. From a physical point of view, the problem can always be considered as the incidence of a CW with either the electric or the magnetic field parallel to the interface (directed along y in our reference frame); the general solution would always be a superposition of these two cases. We can use the scalar function V i ρ; θ CWm ρ; θ to represent either the electric or the magnetic incident field. The reflected CW can be expressed with the
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).
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OPTICS LETTERS / Vol. 39, No. 9 / May 1, 2014
RW, as explained in [1]. We have to consider the planewave spectrum of the CW. Each elementary wave propagates for a distance χ toward the interface, it is multiplied by the reflection coefficient and it propagates again toward the origin of the reference frame and, after it, continues to propagate in the positive ξ direction. Therefore, we have to consider a shift of −2χ in the argument of the RW. Defining ρχ ξ − 2χ2 ζ2 1∕2 and θχ arctanζ∕ξ − 2χ, as in Fig. 1, we can write the reflected field as follows: V r ρ; θ RWm ξ − 2χ; ζ RWm ρχ ; θχ :
(14)
The question is: how we can interpret this result? Looking at Eq. (13), we recognize that the RW is written as a superposition of Bessel functions of the first kind and of the associated Anger–Weber functions. These last functions can be written as the superposition of Bessel functions of the first and second kind and by the Weber functions [9,10]. The Weber functions are solutions of the inhomogeneous Bessel differential equation with an inhomogeneous term that is composed by algebraic functions: f n ρ −ρ n −1n ρ − n∕πρ2 [10]. Therefore, the RW is a solution of an inhomogeneous Bessel differential equation, being the superposition of solutions of the homogeneous differential equation and a particular solution of the inhomogeneous one. Moreover, the inhomogeneous term has its origin at the same point that we take as origin of the RW. From the previous considerations, the solution can be interpreted as if the interface were substituted by a cylindrical source placed symmetrically at the origin of the incident CW. We can call the method a generalized image principle for CWs. In fact, in the classical image principle, when an electric charge is in front of a perfect electric plane, the plane itself can be substituted with an electric charge, of opposite sign, placed symmetrically with respect to the original one. In the same way, we can elimi-
nate the dielectric interface by placing an equivalent cylindrical source symmetrically with respect to the origin of the incident wave. We can, in general, say that: when a CW impinges on a plane interface, it can be substituted by a superposition of cylindrical sources. In conclusion, we presented an analysis of the reflection of CWs by planar interfaces and we solved analytically the reflected-wave integral Eq. (1). We found an expansion of the Fresnel reflection coefficient, for both polarizations, convergent on the whole real axis and we solved the RW integral, finding a solution that involves solutions of both the homogeneous and inhomogeneous Bessel differential equations. We validated the expression found by comparisons with the results obtained with the quadrature method presented in [5] and we gave a physical interpretation that connects the found function to the image principle. References 1. R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, J. Opt. Soc. Am. A 13, 483 (1996). 2. H. Liu and P. Lalanne, Nature 452, 728 (2008). 3. G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, Opt. Commun. 95, 192 (1993). 4. F. Frezza, G. Schettini, and N. Tedeschi, Opt. Commun. 284, 3867 (2011). 5. R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, J. Electromagn. Waves Appl. 14, 1353 (2000). 6. F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, IEEE Geosci. Remote Sens. Lett. 10, 179 (2013). 7. A. Coatanhay and J. M. Conoir, J. Comput. Acoust. 12, 233 (2004). 8. P. Pawliuk and M. Yedlin, IEEE Trans. Antennas Propag. 60, 5296 (2012). 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).
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Generalized image principle for cylindrical waves Fabrizio Frezza and Nicola Tedeschi* Department of Information Engineering, Electronics, and Telecommunications, “La Sapienza” University of Rome, Roma 00184, Italy *Corresponding author: [email protected] Received February 24, 2014; revised March 31, 2014; accepted April 3, 2014; posted April 3, 2014 (Doc. ID 206962); published April 28, 2014 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 OCIS codes: (050.1755) Computational electromagnetic methods; (050.1960) Diffraction theory; (260.2110) Electromagnetic optics; (290.2558) Forward scattering. http://dx.doi.org/10.1364/OL.39.002727
The interaction of cylindrical waves (CWs) with a planar interface is of great importance in optic research, e.g., in the scattering problem by cylindrical structures near plane interfaces [1] or in microscopy [2]. Let us consider a plane interface between two media: medium 1 with relative permittivity and permeability ε1 and μ1 , respectively, and medium 2 with ε2 and μ2 . We define a coordinate system on the plane x; z containing the perpendicular vector to the interface, where x is the axis perpendicular to the interface and z is the axis parallel to it. We normalize the coordinates with respect to the wave number of medium 1: ξ k1 x and ζ k1 z, see Fig. 1. The reflection of a CW by a plane interface can be expressed by the so-called cylindrical reflected wave (RW) [1] 1 RWm ξ; ζ 2π
Z
∞
−∞
d m eιζn∥ dn∥ ; RE∕H CW
(1)
where n∥ is the component parallel to the interface of the propagation vector normalized with respect to k1 , and d m ξ; n∥ is the plane-wave spectrum of the cylindrical CW function of order m [3,4]. The function in Eq. (1) is obtained by the plane wave spectrum of the incident cylindrical function where each elementary wave is multiplied by the reflection coefficient, RE∕H n∥ , in E or H polarization, respectively, depending on the direction of the electric field of the impinging wave. Being n∥ the normalized component parallel to the interface of the propagation vector of the elementary plane wave of the spectrum, the reflection at the interface depends on it and, in particular, for jn∥ j > arcsink2 ∕k1 each elementary plane wave is totally reflected by the interface. The numerical evaluation of the integral in Eq. (1) has been widely discussed in the literature [5] and it is the main difficulty in the solution of the scattering problem by cylindrical structures placed near plane interfaces between either lossless or lossy media [6]. Recently, a new approach, based on the Fourier expansion of the reflection coefficient inside the integral, has been attempted for the evaluation of the RW [7,8]. The main difficulty of this approach is that the Fourier expansion is possible only for n∥ ∈ −1; 1. By adding and subtracting two integrals without the reflection coefficient in the intervals 0146-9592/14/092727-04$15.00/0
jn∥ j ∈ 1; ∞, the authors of such papers obtained the expression of the CW. However, they must calculate the remaining integrals, in the intervals jn∥ j ∈ 1; ∞, numerically. This procedure has been connected to the image method, but, in our opinion, in the formulation presented it cannot be easily related to this method due to the unsolved integrals. The main obstacle faced in [7,8], for the solution of the integral Eq. (1), was on the series expansion of the reflection coefficient RE∕H n∥ . The authors assert that the Fourier expansion in n∥ ∈ −1; 1 is the only possible expansion, being that the Laurent expansion is not applicable because of a polar singularity of the reflection coefficient on the complex plane. We present a power expansion of the reflection coefficient, in both polarizations, that converges on the whole real axis. The reflection coefficients, in E or H polarization, cannot be expanded in a Laurent series in the whole complex plane because of a polar singularity away from the real axis. On the other hand, we are interested in the values of the function only on the real axis, where the function does not have singularities. Our idea is to perform a conformal mapping of the function to transform the real axis in a circumference away from the singularities. Let us consider the following conformal mapping:
Fig. 1. Geometry of the problem. © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 9 / May 1, 2014
w n∥
q w2 1 ; n2∥ − 1 → n∥ 2w
(2)
where the is to indicate that the function w has two branch points in n∥ 1. The presence of these branch points is extremely important as we will see in the following. Now, we study how the real axis on the plane of n∥ transforms on the plane of w: when w → 0− , then n∥ → −∞; when w → −1, then n∥ → −1; when w → ι, then n∥ → 0; when w → 1, then n∥ → 1; and when w → 0 , then n∥ → ∞. The path n∥ ∈ −∞; ∞ can be mapped on the plane of w as a semi-circumference of radius 1, centered in the origin, see Fig. 2. It can be seen that the reflection coefficient as a function of w does not have any singularities inside a circle of unitary radius centered in the origin. Therefore, the reflection coefficient can be expanded in a Laurent series inside this circle ∞ X
RE∕H w
k0
RkE∕H wk ;
(3)
where the coefficients RkE∕H are the Laurent coefficients RkE∕H
1 2πι
Z
RE∕H w dw k1 C w
(4)
and the path C can be chosen as the circumference of unitary radius centered in the origin. We can write the series as a function of n∥ RE∕H n∥
∞ X k0
qk RkE∕H n∥ n2∥ − 1 :
(5)
Now, the importance of the branch points of the square roots appears clear. To write the expression in a more understandable way, we can note that the reflection coefficient is an even function of w, so the odd coeffiE∕H cients R2n1 are all equal to zero. We can multiply into the brackets for 1, and, if we choose the sign − in the path n∥ ∈ −∞; 1, and the sign in the path n∥ ∈ 1; ∞, the Laurent series can be written as follows: RE∕H n∥
∞ X k0
E∕H 2kι arccos jn∥ j R2k e :
At this point, we can substitute the Laurent series for the reflection coefficient inside the RW expression in Eq. (1), and, by dividing the integral in the positive and negative integral paths, we can write the RW as follows: RWm ξ; ζ
∞ X k0
RkE∕H F m−2k ξ; ζ −1m F −m−2k ξ; −ζ; (7)
where the function F p ξ; ζ is defined as follows: F p ξ; ζ
1 2π
Z
∞ 0
d p ξ; n∥ eιζn∥ dn∥ : CW
(8)
It is the plane-wave expansion of a cylindrical function integrated only for positive values of n∥ . The series in Eq. (7) converges quickly: the relative error is less than 10−3 , if it is truncated at k 10, and less than 10−4 at k 20. The convergence speed slows down weakly as ρ increases. Now, our problem becomes the computation of the integral in Eq. (8). We make the following change of variable: n∥ sin t, i.e., we write the plane-wave spectrum as in the Sommerfeld representation [3,9] Z ι−p π2−ι∞ ιρ cost−θιpt e dt (9) F p ρ; θ π 0 the asymptote in π∕2 − ι∞ can be translated of a quantity θ, as well explained in [9]. The integral in Eq. (9) can be written as the sum of two integrals. The first integral ι−p π
Z
π 2
−θ
eιρ cos tιpt dt
∞ X l−∞
ap;l θJ l ρ;
(10)
where we made use of the well-known Jacobi expansion of the exponential expιρ cos t and with π θ 2 π θ ap;l θ ιl−p eιlp 4−2 π 4 2 π θ : × sinc l p 4 2
(11)
The second integral (6) Ap ρ
This representation is convergent on the whole real axis.
1 π
Z
∞ 0
e−ρ sinh v−pv dv;
(12)
where v π∕2 − ιt. It is the associate Anger–Weber function that is a solution of the inhomogeneous Bessel differential equation [10]. This kind of function has been employed in the scattering problem by a wedge [9,11]. We can write the function in Eq. (8) as follows: F p ρ; θ Fig. 2. Conformal mapping of the real path n∥ ∈ −∞; ∞, on the plane of w u ιv.
eιpθ
X ∞
ap;l θJ l ρ − ιAp ρ :
(13)
l−∞
This function is a solution of the inhomogeneous Bessel differential equation, i.e., the solution of the wave
May 1, 2014 / Vol. 39, No. 9 / OPTICS LETTERS
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).
equation in cylindrical coordinates with cylindrical sources. By inserting Eq. (13) in Eq. (7), we find an analytical expression for the RW of order m. Before providing a physical interpretation of the results, we want to validate them by comparisons with the numerical evaluation of the integral Eq. (1) obtained by the adaptive quadrature method proposed in the literature [5,6]. Let us consider the case of two nonmagnetic media, μ1 μ2 1. We consider the following relative permittivities: ε1 2 and ε2 8. We calculate the RW for a fixed radius as a function of the angle θ. In Fig. 3, the amplitude of the RW of order zero, for a radius ρ 0.5, in E polarization is shown. We can see that the solution obtained with the quadrature method is oscillating around the analytical result. Actually, up to now the quadrature formula cannot be tested directly but only by the comparisons of the electric field in such a scenario of scattering from cylinders near an interface. Now we can see that the analytical procedure returns the true value of the reflected CW, while the quadrature method presents some numerical errors due to the high oscillation of the integrand function [1]. Being the RW presented in Fig. 3 in the near field, we see that the errors of the quadrature formula are not negligible. However, if we consider the far field, i.e., high values of the radius, the oscillations of the numerical solution disappear, and the two results perfectly coincide. In Fig. 4, the amplitude of the RW in the same
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).
2729
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).
situation of Fig. 3 is shown when the radius is ρ 20. Moreover, in order to show a result for the other polarization, the RW1 for H polarization, in the same situation of Fig. 3, is shown in Fig. 5 with a radius ρ 4. Again, we see that the agreement is very good, but the quadrature method shows some small oscillations near θ 0. Finally, to validate the procedure for high orders, in Fig. 6 the amplitude of the RW in the same scenario of Fig. 3, with ρ 4 and θ 0° , as a function of the order is shown. We recognize the exponential growth with the order typical of the Hankel function. In Eq. (13) this behavior is due to the associate Anger–Weber function, strongly connected both to the Weber function and to the Bessel function of the second kind. Now, we attempt a physical interpretation of the results found. Let us consider the incidence of a CW of order m on a planar interface between the two media 1 and 2. We suppose that the impinging wave originates at a distance h from the interface, and we normalize this distance with respect to k1 : χ k1 h, see Fig. 1. From a physical point of view, the problem can always be considered as the incidence of a CW with either the electric or the magnetic field parallel to the interface (directed along y in our reference frame); the general solution would always be a superposition of these two cases. We can use the scalar function V i ρ; θ CWm ρ; θ to represent either the electric or the magnetic incident field. The reflected CW can be expressed with the
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).
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OPTICS LETTERS / Vol. 39, No. 9 / May 1, 2014
RW, as explained in [1]. We have to consider the planewave spectrum of the CW. Each elementary wave propagates for a distance χ toward the interface, it is multiplied by the reflection coefficient and it propagates again toward the origin of the reference frame and, after it, continues to propagate in the positive ξ direction. Therefore, we have to consider a shift of −2χ in the argument of the RW. Defining ρχ ξ − 2χ2 ζ2 1∕2 and θχ arctanζ∕ξ − 2χ, as in Fig. 1, we can write the reflected field as follows: V r ρ; θ RWm ξ − 2χ; ζ RWm ρχ ; θχ :
(14)
The question is: how we can interpret this result? Looking at Eq. (13), we recognize that the RW is written as a superposition of Bessel functions of the first kind and of the associated Anger–Weber functions. These last functions can be written as the superposition of Bessel functions of the first and second kind and by the Weber functions [9,10]. The Weber functions are solutions of the inhomogeneous Bessel differential equation with an inhomogeneous term that is composed by algebraic functions: f n ρ −ρ n −1n ρ − n∕πρ2 [10]. Therefore, the RW is a solution of an inhomogeneous Bessel differential equation, being the superposition of solutions of the homogeneous differential equation and a particular solution of the inhomogeneous one. Moreover, the inhomogeneous term has its origin at the same point that we take as origin of the RW. From the previous considerations, the solution can be interpreted as if the interface were substituted by a cylindrical source placed symmetrically at the origin of the incident CW. We can call the method a generalized image principle for CWs. In fact, in the classical image principle, when an electric charge is in front of a perfect electric plane, the plane itself can be substituted with an electric charge, of opposite sign, placed symmetrically with respect to the original one. In the same way, we can elimi-
nate the dielectric interface by placing an equivalent cylindrical source symmetrically with respect to the origin of the incident wave. We can, in general, say that: when a CW impinges on a plane interface, it can be substituted by a superposition of cylindrical sources. In conclusion, we presented an analysis of the reflection of CWs by planar interfaces and we solved analytically the reflected-wave integral Eq. (1). We found an expansion of the Fresnel reflection coefficient, for both polarizations, convergent on the whole real axis and we solved the RW integral, finding a solution that involves solutions of both the homogeneous and inhomogeneous Bessel differential equations. We validated the expression found by comparisons with the results obtained with the quadrature method presented in [5] and we gave a physical interpretation that connects the found function to the image principle. References 1. R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, J. Opt. Soc. Am. A 13, 483 (1996). 2. H. Liu and P. Lalanne, Nature 452, 728 (2008). 3. G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, Opt. Commun. 95, 192 (1993). 4. F. Frezza, G. Schettini, and N. Tedeschi, Opt. Commun. 284, 3867 (2011). 5. R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, J. Electromagn. Waves Appl. 14, 1353 (2000). 6. F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, IEEE Geosci. Remote Sens. Lett. 10, 179 (2013). 7. A. Coatanhay and J. M. Conoir, J. Comput. Acoust. 12, 233 (2004). 8. P. Pawliuk and M. Yedlin, IEEE Trans. Antennas Propag. 60, 5296 (2012). 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).
