Analysis of numerical solutions to Sommerfeld integral relation of the half-space radiator problem

Transcription

1 Analysis of numerical solutions to Sommerfeld integral relation of the half-space radiator problem Arun I, Murugesan Venkatapathi Computational and Statistical Physics Laboratory, Department of Computational Science, Indian Institute of Science, Bangalore Abstract Sommerfeld integrals relate a spherical wave from a point source to a convolution set of plane and cylindrical waves. This relation does not have analytical solutions but it submits to a solution by numerical integration. Among others, it is significant for theoretical studies of many optical and radiation phenomena involving surfaces. This approach is preferred over discretized computational models of the surface because of the many orders of increased computations involved in the latter. One of the most widely used and accurate methods to compute these solutions is the numerical integration of the Sommerfeld integrand over a complex contour. We have analyzed the numerical advantages offered by this method, and have justified the optimality of the preferred contour of integration and the choice of two eigenfunctions used. In addition to this, we have also analyzed four other approximate methods to compute the Sommerfeld integral and have identified their regions of validity, and numerical advantages, if any. These include the high relative permittivity approximation, the short distance approximation, the exact image theory and Fourier expansion of the reflection coefficient. We also finally compare these five methods in terms of their computational cost. address: murugesh@serc.iisc.in (Murugesan Venkatapathi) Preprint submitted to Applied Numerical Mathematics December 11, 15

2 1. Introduction The problem of modelling radiating antennas near a plane boundary such as the earth-air interface is of considerable practical importance in the fields of nano optics, oceanography, geophysical exploration, and submarine communication and detection. The radiation characteristics of an antenna can be substantially affected by the presence of a lossy ground with finite conductivity. Applications in nano optics also require the computation of interactions between a large number of dipole sources and substrates, thus requiring a repetitive use of this method numerous times to obtain a final solution of the problem [7, 6, 8, 14,, 3, 1, 3, 9, 4]. Due to the vastly diminished scales involved in nano optics, the interaction between the dipole sources and the substrate is very significant [, 4, 5,, 19]. The classic formulation of this problem by Sommerfeld[18, 16, 17] assumes a homogeneous lossy half-space with finite conductivity (say, the earth), and an infinitesimal vertical point dipole embedded in the free space (say, air) above it. Maxwell s equations are applied subject to the half-space boundary conditions, and the solutions are obtained in the form of an inverse Fourier-Bessel integral. This integral modelling the interaction of the dipole with the surface is known as the Sommerfeld integral. Since the problem has cylindrical symmetry, it is convenient to express the solution in cylindrical coordinates as an integral of the eigenfunctions of the cylindrical Helmholtz operator. These eigenfunctions are in terms of the zeroth order Bessel function of the first kind for the ρ direction, and in terms of a complex exponential for the z direction. Because of the azimuthal symmetry in angle φ, the solutions are independent of φ. If cartesian coordinates had been used to express the solution, components along all three spatial directions x, y, and z would have been required and the resulting integral would have been a double integral. And, since handling higher dimensional integrals numerically is computationally more expensive and less stable, the cylindrical solutions of Sommerfeld are preferred.

3 The Sommerfeld integral has no closed form analytic solution, but there exist a few numerical methods to compute the solution. Since the integrand of the Sommerfeld integrand is oscillatory, traditional numerical integration schemes converge poorly if the path of integration is not optimally chosen. Approximate analytic expressions can also be obtained when certain constraints are applied on the parameters. The purpose of this work is to study the computational accuracy and efficiency of existing numerical methods for the evaluation of the Sommerfeld integral and also to suggest modifications in these methods where appropriate. One common form of the Sommerfeld integral is shown in (1). k zs J (k ρ ρ)e kz(z+h) k ρ dk ρ (1) εk z + k zs k z where ε is the relative permittivity of the surface, k ρ and k z are the ρ and z components respectively of the wave numbers in free space, k zs is the z component of the wave number under the surface, ρ and z are the coordinates of the observation point, and J is the zeroth order Bessel function of the first kind. Note the branch cuts and singularities of this integrand shown in Figure 1. We describe five methods to evaluate this integral; their regions of validity in terms of the permittivity of the half-space and the ρ / z ratios. Finally, in Section 7, we comment on the approximate computational costs involved in each of these methods.. Complex contour integration For analytical integration, as a consequence of Cauchy s residue theorem, all different complex contours are equivalent provided the closed loop formed by the different contours of integration do not enclose any poles or singularities of the integrand and the contours do not intersect any branch cut of the integrand. However, numerical quadrature schemes would encounter oscillatory or non-oscillatory integrands depending on the contour chosen, and this can significantly affect the rate of convergence and accuracy of the numerical result. 3

4 Branch cut Integration contour I{k ρ } k ε p O -k k R{k ρ } k ε A 1 i ρ p = k ε ε+1 Figure 1: Branch cuts and singularities of the Bessel function form of the Sommerfeld integrand in (1).1. Contour integration with Bessel functions We start by ascertaining the location of the poles, singularities and branch cuts of the Sommerfeld integrand. The integrand has a pole at ε k ρ = p = k 1 + ε corresponding to εk z + k zs = It also has branch point singularities at k ρ = ±k and k ρ = ±k ε due to kρ k and kρ εk respectively. The integrand has branch cuts due to kρ k and kρ εk. Choosing the conventional principal value of the square root function, the branch cuts are as follows. The detailed derivations which result in these branch cuts are shown 4

5 4 3 1 I{k ρ } R{k ρ } Figure : Domain coloring visualization of the Bessel function form of the Sommerfeld integrand on the complex k ρ plane for parameters k = 1, ε = 3+i, h = 1, ρ =.λ, z =.4λ (See Appendix A for a quantitative description of domain coloring of complex valued functions) in Appendix B. k ρ = t, t [ k, k] (a) k ρ = it, t R (b) k ρ = t + i ε yk t, t [ kr{ ε}, kr{ ε} ] (c) These branch cuts are shown in Figure 1 and may also be seen in the domain coloring visualization of the Sommerfeld integrand shown in Figure. preferred contour of integration due to Lytle and Lager [13], that avoids the branch cuts and singularities discussed above is also shown in Figure. Point A in the preferred contour of integration is chosen to be A 1 i ρ The Bessel function term in the integrand increases exponentially in the lower half of the complex plane. So, to avoid large cancellation errors, the contour The 5

6 No of integrand function calls Complex contour integration Real line integration z + h (in units of λ) Figure 3: Comparison of computational cost of integration over the real line and over the contour (for parameters k = 1, ε = 7 + 3i, ρ =.35λ) is chosen such that the argument of the Bessel function k ρ ρ remains roughly constant for different values of ρ. Hence, we have the 1 ρ term in A. The slope of OA given by the 1 i in the numerator is arbitrary. A comparison of the number of integrand evaluations required for convergence in the case of integration over the real line and integration over the complex contour is shown in Figure 3. Integration over the real line requires several orders of magnitude more number of integrand evaluations than the contour integration. Thus, it is clear from these numerical results, that contour integration has a significant computational advantage. Note that, to estimate the computational cost of real line integration, a contour arbitrarily close to the real line was chosen. This is because a contour over the real line will converge very poorly due to the singularities on the real line. Therefore, this is an optimistic estimate of the computational cost of real line integration. True real line integration would incur a higher computational cost. 6

7 Branch cut Integration contour I{k ρ } k ε p O -k k R{k ρ } k ε A B p = k ε ε+1 Figure 4: Branch cuts and singularities of the Hankel function form of the Sommerfeld integrand.. Contour integration with Hankel functions The Sommerfeld integral, usually expressed as a semi-infinite integral of Bessel functions, can also equivalently be expressed as an infinite integral of Hankel functions[18] as follows. k z J (k ρ ρ)e kzz k ρ dk ρ = 1 εk z + k zs k z k z H 1 (k ρ ρ)e kzz k ρ dk ρ εk z + k zs k z where H 1 is the zeroth order Hankel function of the first kind. In addition to the branch cuts of the Bessel function form of the Sommerfeld integrand, the Hankel function form of the Sommerfeld integrand has an additional branch cut along the negative real axis due to the Hankel function. This branch cut is given by k ρ = t (, ] 7

8 4 3 1 I{k ρ } R{k ρ } Figure 5: Domain coloring visualization of the Hankel function form of the Sommerfeld integrand on the complex k ρ plane for parameters k = 1, ε = 3+i, h = 1, ρ =.λ, z =.4λ (See Appendix A for a quantitative description of domain coloring of complex valued functions) The branch cuts of the Hankel function form of the Sommerfeld integrand along with the preferred contour of integration are shown graphically in Figure 4. Points A and B in the preferred contour of integration are given by A z iρ ρ B = kr{ ε} + I{A} The Hankel function term in the integrand increases exponentially in the lower half of the complex plane. So, to avoid large cancellation errors, the contour is chosen such that the argument of the Hankel function k ρ ρ remains roughly constant for different values of ρ. Hence, we have the 1 ρ term in A. The slope of OA given by the z iρ is chosen to approximate a contour corresponding to constant phase of the integrand. Smaller phase variations along the contour result in the numerical quadrature scheme encountering a less oscillatory integrand, and hence this is preferred. Point B is chosen such that when the contour 8

9 returns to the upper half of the complex plane, it does not intersect the branch cut due to k zs = kρ εk. The Bessel function form of the Sommerfeld integrand suffers from rapid oscillations for large ρ and slow decay for small z + h. Hence, for observation points along the planar interface and distant from the source, convergence is slow. In these cases, the Hankel function form of the integrand, which does not suffer from these issues, can be used. The infinite range of integration can be truncated to a relatively short range because of the fact that the Hankel function decays exponentially in the upper half of the complex plane. A domain coloring visualization of the Hankel function form of the Sommerfeld integrand is shown in Figure High relative permittivity approximation In his original work[18], Sommerfeld developed the following approximate image theory for surfaces of high permittivity. A z = k zs J (k ρ ρ)e kz(z+h) k ρ dk ρ εk z + k zs k z For infinitely large ε, the Sommerfeld integral vanishes. For large but not infinitely large ε, an useful limit case can be derived. This is as follows. The z component of the wave number under the surface k zs is given by k zs = k ρ εk k zs = k ρ εk εk 1 For large ε, that is, εk k ρ k zs = ik ε (3) Also, in the denominator, εk z + k zs εk z (4) 9

10 Applying approximations (3) and (4), A z = ik ε J (k ρ ρ)e kz(z+h) k ρ kz dk ρ This approximate integral can be interpreted as an image theory in the following way. For R = ρ + (z + h ), the Fourier-Bessel integral of the free-space Green s function can be expressed as e ikr R = J (k ρ ρ)e kz(z+h ) k ρ k z dk ρ Integrating with respect to h from h to, Thus, h h e ikr R dh = e ikr R dh = J (k ρ ρ)e kzz k ρ k z dk ρ J (k ρ ρ)e kzz k ρ kz dk ρ A z = ik ε h A z = ik ε where z = z + h and R = ρ + (z ) z+h e ikr R dh h e kzh dh e ikr R dz (5) This can be interpreted as the effect of an imaginary ray of dipoles starting at z = h and going to z =. The high permittivity integrand shown in (5) has branch cuts due to the square root function in R = ρ + (z ). Choosing the conventional principal value of the square root function, the branch cut is z = ±it, t [ρ, ) The detailed derivation which results in this branch cut is shown in Appendix C. The branch point corresponding to this branch cut is at z = ±iρ 1

11 Branch cut Integration contour I{z } A B iρ O iρ z + h C R{z } Figure 6: Branch cuts and singularities of the high relative permittivity integrand For the high permittivity integrand, contours parallel to the imaginary axis in the z plane encounter an exponentially decaying integrand. Contours parallel to the real axis in the z plane encounter an oscillatory integrand. Also, as the magnitude of z tends to infinity, the integrand approaches zero. Hence, for rapid convergence with numerical integration, we choose a contour as shown in Figure 6. Points A, B and C are chosen large enough such that the contribution of the contour segments AB and BC is negligibly small and need not be computed Error analysis The relative error in the integrand caused by approximation (3) is given by Relative error in integrand = 1 + i ε k ε = 1 + ik k zs kρ εk The relative error in the integrand caused by approximation (4) is given by Relative error in integrand = k zs = 1 kρ εk εk z ε kρ k 11

12 by Putting both approximations (3) and (4) together, the relative error is given Relative error in integrand = ik ( 1 + εk ) z εkz Relative error in integrand = Relative error in integrand = k i ε k ( 1 i + ε k z k zs ε k zs ) 1 ε + (6) k ρ k kρ εk This relative error in the integrand given by (6) shows maxima at k ρ = k and k ρ = k ε. Therefore, the high permittivity approximation will show lesser error for parameters whose corresponding integrands only have a small component at these points in the complex k ρ plane. Since all contours in the complex plane are equivalent, the error in the final result cannot be minimized by choosing a contour of integration that avoids these points. A comparison of the relative error for the high permittivity approximation vs complex contour integration for different values of relative permittivity is shown in Figure 7. As can be seen, the approximation becomes more accurate for larger permittivities. 4. Short distance approximation This approximation due to Panasyuk et al. [15] is applicable when the observation point is close to the reflection of the dipole source about the planar interface, that is for small kl where L = ρ + (z + h) is the distance between the observation point and the reflection of the dipole source about the planar interface. Detailed derivation of the results of the short distance approximation are shown in [15]. Here, we will consider only the approximations used and the region of validity of these approximations. 1

13 5.5 4 I{ε} R{ε}.15 Figure 7: High permittivity approximation vs complex contour integration - Relative error (for parameters k = 1, ρ = λ and z = λ) is One of the forms of the Sommerfeld integral dealt with in this approximation I = The integral is first split into three parts I = + + k k k J (k ρ ρ)e kz(z+h) k zs εk z k zs + εk z k 3 ρ k z dk ρ f(k ρ )J (k ρ ρ)e kz(z+h) dk ρ [f(k ρ ) g(k ρ )]J (k ρ ρ)e kz(z+h) dk ρ g(k ρ )J (k ρ ρ)e kz(z+h) dk ρ where f(k ρ ) = k3 ρ k zs εk z k z k zs + εk z ] g(k ρ ) = k3 ρ s(1 s)k [s + k z kρ s = 1 ε 1 + ε 13

14 Then, the variable of integration is transformed with ξ = 1 ( ) kρ k in the first integral, and ξ = integrals t 1, t and t 3. ( ) kρ k 1 in the last two integrals, resulting in I = k 3 [it 1 + t + t 3 ] where t 1 = 1 (1 ξ )J (kρ 1 ξ )e iξk(z+h) Q 1 (ξ)dξ (7) t = t 3 = (1 + ξ )J (kρ 1 + ξ )e ξk(z+h) [ ] s(1 s) Q (ξ) s (1 + ξ dξ (8) ) (1 + ξ )J (kρ 1 + ξ )e ξk(z+h) ξ + ε 1 εξ Q 1 (ξ) = ξ + ε 1 + εξ ξ ε + 1 εξ Q (ξ) = ξ ε εξ [ ] s(1 s) s + (1 + ξ dξ (9) ) Now, suitable short distance approximations are applied on integrals t 1, t and t 3. These approximations make the integrals analytically integrable and result in the expressions shown in (11), (1) and (14). The region of validity of these approximations is discussed here. In integrals t 1 and t, the Bessel function terms and the exponential terms are approximated by unity, which is the first term in their respective Taylor series. J (kρ 1 ξ ) 1 J (kρ 1 + ξ ) 1 e iξk(z+h) 1 e ξk(z+h) 1 (1a) (1b) (1c) (1d) 14

15 The approximations in (1) are valid only for very small arguments of the exponential and Bessel functions. Therefore, kρ and k(z + h) are required to be very small quantities. Thus, this approximation is valid only when ρ and z + h are much smaller than the free space wavelength λ. ρ λ z + h λ Substituting these approximations into (7) and (8), and evaluating the resulting integrals analytically results in (11) and (1) for integrals t 1 and t respectively. t 1 = t = ε 1 (ε 1)(ε + 1) 6ε 3 (ε 1)(ε + 1) [ ε (1 + ε) 1.5 {[.5 ln π + 1 ε 3 ε(1 + 3ε ε ) + ε 1 [ (ε + ε 1)(1 + ] ε + 1) ε + ε(ε + 1) ( ε 5 ( )]} ε + ) ε 1 ε π + iε ln[iε + 1 ε ] ε 1 ε ε 1 ] (11) (1) In integral t 3, the argument of the Bessel function grows linearly with the integration variable ξ at the upper limit of integration. Hence, direct approximation using the Taylor series is not possible. Hence, we make the argument of the Bessel function bounded as follows. J (kρ 1 + ξ ) = J [kρ(ξ + ξ)] where ξ = 1 + ξ ξ Then, using the addition theorem for Bessel functions, J (kρ 1 + ξ ) = J n (kρξ)j n (kρ ξ) n= Truncating the series due to the addition theorem at n = 1, J (kρ 1 + ξ ) = J 1 (kρξ)j 1 (kρ ξ) + J (kρξ)j (kρ ξ) J 1 (kρξ)j 1 (kρ ξ) 15

16 For Bessel functions of integer order n, J n (z) = ( 1) n J n (z) Therefore, J (kρ 1 + ξ ) = J (kρξ)j (kρ ξ) J 1 (kρξ)j 1 (kρ ξ) Now, that the argument kρ ξ is bounded, we can approximate it using the first term of the Taylor series as follows. J (kρ ξ) 1 J 1 (kρ ξ) kρ ξ Substituting these approximations, we obtain, J (kρ 1 + ξ ) = J (kρξ) kρ ξj 1 (kρξ) (13) Analogous to the approximations in (1), this approximation using the Taylor series is valid only for very small values of kρ, that is, when ρ is much smaller than the free space wavelength λ. ρ λ Substituting approximation (13) into (9), and evaluating the resulting integrals analytically results in (14) for integral t 3. [ t 3 = s ( ) z + h (kl) 3 3 1] 1 [ ] s(1 s) s + sr L kl kl Error analysis (14) Absolute error in t 1 = 1 (1 ξ )J (kρ 1 ξ )e iξk(z+h) Q 1 (ξ)dξ 1 (1 ξ )Q 1 (ξ)dξ 16

17 z + h z+h ρ ρ (a) Integral t1 (b) Integral t Figure 8: Short distance approximation vs complex contour integration - Relative error for a ρ z grid (for parameters k = 1, ε = 3 + i) Absolute error in t1 = Z 1 Absolute error in t = i h p (1 ξ ) J (kρ 1 ξ )eiξk(z+h) 1 Q1 (ξ)dξ Z i h p (1 + ξ ) J (kρ 1 + ξ )e ξk(z+h) 1 s(1 s) dξ Q (ξ) s (1 + ξ ) Since, in approximation (1), the oscillatory terms namely the Bessel and exponential functions where replaced by unity, the error introduced is also oscillatory. This is shown in the above expressions. 4.. Numerical results Numerical results computing the relative error of the short distance approximation for integrals t1 and t are shown in Figure 8. As can be seen from the figures, the error does not monotonically increase with increasing distance from the reflected source. Instead, there is an oscillating variation in the error. The source of this oscillating error is due to approximation by substitution of oscillatory terms in the integrand with constant terms, as explained in the previous section. 17

18 4.3. Observations and conclusions When the observation point is very close to the reflected source point, numerical integration of the Sommerfeld integral converges slowly. In such cases, this short distance approximation is useful. Also, when approximate analytical results are required, this short distance approximation provides a more accurate alternative to the well known electrostatic approximation. However, as an accurate general purpose numerical technique for all observation points, this approximation will not suffice. 5. Exact image theory An exact distributed image current function for the classical Sommerfeld half-space problem was obtained by Lindell and Alanen for excitation by a vertical magnetic dipole [1], a vertical electric dipole [11], and for arbitrarily oriented electric or magnetic dipoles [1]. We shall only consider excitation by a vertical electric dipole. The vector potential A(r) possesses only a z component and satisfies the Helmholtz wave equation, given by ( + k )A(r) = δ(r hẑ), z > ( + εk )A(r) =, z < The Fourier transform of the solution can be written as A z (k ρ ) = J (k ρ ρ) k z [ e kz(z h) + R e e kz(z+h)], z > (15) where R e is the reflection coefficient given by (15) can be split into two terms A 1 and A as R e (k z ) = εk z k zs εk z + k zs (16) A z = A 1 + A 18

19 where A 1 (k ρ ) = J (k ρ ρ) k z e kz(z+h) (17) A (k ρ ) = J (k ρ ρ) k z R e e kz(z+h) (18) A 1 corresponds to the field due to a vertical electric dipole at z = h radiating in free space. In the space domain, A 1 (r) = G(r hẑ) where G(r) is the free space Green s function given by G(r) = e ikr 4πr A corresponds to the field due to reflection from the surface. A (k ρ ) is the reflected field due to an incident wave with wave number k = k ρ ˆρ + k z ẑ Now, we try to express A as some superposition of point sources radiating in homogeneous space. For this, we first normalize the the propagation factor k z using q = kz ik ε 1 to get R e (q) = εq q + 1 εq + q + 1 If f is the Laplace transform of the reflection coefficient R e (q), then R e (q) = f(p)e qp dp Substituting this Laplace representation of R e (q) into (18), A (k ρ ) = J (k ρ ρ) e kz(z+h) k z Inverting the Fourier transform, A (k ρ ) = Swapping the order of the integrals, A (k ρ ) = J (k ρ ρ)e kz(z+h) k ρ k z f(p) f(p)e pq dp f(p)e pq dp dk ρ J (k ρ ρ)e kz(z+h) e pq k ρ k z dk ρ dp 19

20 Substituting q = A (k ρ ) = kz ik, ε 1 f(p) J (k ρ ρ)e kz [ ] z+h+ p ik kρ ε 1 k z dk ρ dp We find that this corresponds to a continuous line of point sources in complex z space, where f(p) is related to the image current function. ( ( ) ) p A (r) = f(p)g r + h + ik ẑ dp ε 1 Expressing this in more physically explicit quantities, where A (r) = e iθ h I(z )G(r z ẑ)dz (19) θ = arg(i ε 1) z p = h ik ε 1 I(z ) = ik ε 1f ( ik ε 1(z + h) ) The line of point sources exist along z, and I(z ) is the magnitude of the current along that line. Now, if we know f(p) we may simply substitute in (19), and find the net vector potential by superposition of the image point sources. where f(p) is simply the Laplace transform of R e (q). One of the forms of f(p) is f(p) = ε 1 ε 1 δ(p) εγ sinh(γp) + εγ ε + 1 ε + 1 γ = 1 ε 1 p sinh [γ(p s)] J (s) ds s More numerically suitable forms of f(p) are derived using various approximations in [11]. These approximations result in the series forms () and (1) for f(p). For p < 4, f(p) = ε 1 ε + 1 δ(p) + n= [ ] εγ ε 1 n 1 ε ε ( 1) m (m + 1)! (γp) n+1 (γ) m+1 m!(m + )! (n + 1)! m= ()

21 For p > 4, f(p) = ε γ 3 e γp + ε πp 3 ( sin p + π ) + 3(8 3ε ) ( 4 ε 3 cos p + π ) + (1) πp 5 4 These expressions can finally be used to compute the net effect of all the image sources using numerical integration of (19). Exact image theory provides an alternative way to understand the reflection as arising from a set of point sources. But note that these sources are positioned along a line in the complex plane and this may not have an apparent physical interpretation. Our analysis presented in Section 7 shows that the cost of computation using this approach may be higher than the contour integration, for ρ and z + h values less than λ. It is clear that as ρ increases, the computational cost of contour integration with the Bessel function form of the integrand increases. Therefore, exact image theory could potentially be exploited for mesoscale problems. 6. Fourier expansion of the reflection coefficient This technique approximates the reflection coefficient part of the Sommerfeld integral using a Fourier series. The resulting series of integrals is analytically integrable, thus converting the problem to a series summation. We start with a Sommerfeld integral of the form A z = J (k ρ ρ)e kz(z+h) [ εk z εk z + k zs ] kρ k z dk ρ The bracketed part is the reflection coefficient and is slowly varying with respect to the variable of integration and will be approximated using a Fourier series. Changing integration variables by substituting k ρ = k sin α and therefore k z = kρ k = ik cos α, we get A z = L + T 1

22 where π L = k T = k π +j π S L (cos α)j (kρ sin α)e k cos α[δ+i(z+h)] sin αdα () S T (i cos α)j (kρ sin α)e ik cos α(z+h) sin αdα (3) εψe kψδ S L (ψ) = εψ + ε 1 + ψ εψ S T (ψ) = εψ + i ε 1 + ψ Making a further substitution, ψ = cos α in () and ψ = i cos α in (3), we get L = ik T = k 1 S L (ψ)j (kρ 1 ψ )e kψ[δ+i(z+h)] dψ (4) S T (ψ)j (kρ 1 + ψ )e kψ(z+h) dψ (5) Now, we approximate the slowly varying parts of the integral namely S L (ψ) and S T (ψ) using a Fourier series as follows. Integral for L in (4) has finite limits ψ 1. So, S L can be approximated by extending it as an even function about ψ = 1, and then approximating it using an N L point Fourier sine transform. where ψ = N L s L (ν) = S L (ψ) = N L N L 1 ψ=1 N L 1 ν=1, ν odd ( S L (ψ) sin π ψ ψ ν N L ( s L (ν) sin π ψ ψ Integral for T in (5) has infinite limits. Therefore, the integral needs to be truncated at some ψ max, before S T can be approximated using a Fourier series. ψ max is chosen to be a point where the exponential function in the integral would have decayed to a very small value and therefore the contribution of the integral for ψ > ψ max can be neglected. ) ν N L )

23 ψ max is chosen to be ψ max = 1 ln (.5 1 (SD+)) kz min where SD is the number of significant digits accuracy required, and z min is the minimum z + h for which the Fourier series approximation will be used. Extending S T as an even function about ψ = ψ max, we get s T (ν) = S T (ψ) = N T N T 1 ψ=1 N T 1 ν=1, ν odd ( S T (ψ) sin π ψ ψ ν N T ( s T (ν) sin π ψ ψ Substituting the Fourier series approximations developed above back into (4) and (5), makes the integrals analytically integrable. Expressions for the results after analytical integration are available in [1]. The slowly varying parts of the integral S L and S T are parameterized only by the permittivity of the surface, and is independent of the positions of the source and observation points. Hence, for a given surface, the Fourier series approximations need to be computed only once and can be used repeatedly for various source and observation points, thus saving on computation. However, for observation points close to the surface, the convergence is slow, and a large number of sample points need to be used in the Fourier series approximation [1]. This is problematic and any computational advantage gained over numerical integration of the Sommerfeld integral may be lost here. ) ν N T ) 7. Comparison of computational cost In this section, we compare the computational cost of the various methods used to evaluate the Sommerfeld integral. In the case of contour integration, exact image theory and the high permittivity approximation, the number of integrand evaluations has been chosen as a measure of the computational cost. For exact image theory, since the series calculations for f(p) involve double summations, the number of terms in the series has also been added to the 3

24 No of integrand function calls Hankel Bessel High Index Exact Image Theory z + h (in units of λ) Figure 9: Comparison of computational costs involved in the various methods (for parameters k = 1, ε = 7 + 3i, ρ =.15λ) number of integrand evaluations performed during numerical integration. For the short distance approximation, only analytical expressions are involved, the cost of which is negligibly small compared to numerical integration. Hence we ignore it here. But note that its region of validity is limited as described in Section 4. The various methods of computation have different integrands each with their distinct computational costs. Therefore, the cost involved with a single integrand evaluation is not the same for different methods. However, this simplistic scheme assuming that all integrand and series evaluations are equivalent was chosen in order to decouple the measurement of computational cost from exact implementation details such as choice of numerical integration methods, choice of math libraries, compiler optimization, etc. Figures 9, 1 and 11 show a comparison of the number of integrand evaluations required for convergence in each method for different ρ and z + h. All numerical integration involved were computed using Romberg integration to a 4

25 No of integrand function calls Hankel Bessel High Index Exact Image Theory z + h (in units of λ) Figure 1: Comparison of computational costs involved in the various methods (for parameters k = 1, ε = 7 + 3i, ρ =.35λ) No of integrand function calls Hankel Bessel High Index Exact Image Theory z + h (in units of λ) Figure 11: Comparison of computational costs involved in the various methods (for parameters k = 1, ε = 7 + 3i, ρ = λ) 5

26 relative tolerance of 1 6. From the figures, we can see that the high permittivity approximation shows the lowest cost. But, as we saw earlier, its accuracy is too poor to be practically useful. Contour integration and exact image theory are closely matched, but for larger z + h, when the observation point tends to the far field, exact image theory has an order of magnitude lesser computational cost. Among the contour integrations, the Bessel function form of the integrand is slightly inferior to the Hankel function form for very low z + h, but at larger z + h, the increased damping effect of the exponential decay due to the large z + h gives the Bessel function form of the integrand an advantage. References [1] S. L. Dvorak. Application of the fast fourier transform to the computation of the sommerfeld integral for a vertical electric dipole above a half-space. Antennas and Propagation, IEEE Transactions on, 4(7):798 85, 199. [] A. M. Funston, C. Novo, T. J. Davis, and P. Mulvaney. Plasmon coupling of gold nanorods at short distances and in different geometries. Nano letters, 9(4): , 9. [3] V. V. Gozhenko, L. G. Grechko, and K. W. Whites. Electrodynamics of spatial clusters of spheres: substrate effects. Physical Review B, 68(1):154, 3. [4] A. Grigorenko, A. Geim, H. Gleeson, Y. Zhang, A. Firsov, I. Khrushchev, and J. Petrovic. Nanofabricated media with negative permeability at visible frequencies. Nature, 438(766): , 5. [5] T. R. Jensen, M. D. Malinsky, C. L. Haynes, and R. P. Van Duyne. Nanosphere lithography: tunable localized surface plasmon resonance spectra of silver nanoparticles. The Journal of Physical Chemistry B, 14(45): ,. 6

27 [6] M. W. Knight, Y. Wu, J. B. Lassiter, P. Nordlander, and N. J. Halas. Substrates matter: influence of an adjacent dielectric on an individual plasmonic nanoparticle. Nano letters, 9(5):188 19, 9. [7] C. Langhammer, B. Kasemo, and I. Zorić. Absorption and scattering of light by pt, pd, ag, and au nanodisks: Absolute cross sections and branching ratios. The Journal of chemical physics, 16(19):1947, 7. [8] J. Lerme, C. Bonnet, M. Broyer, E. Cottancin, D. Manchon, and M. Pellarin. Optical properties of a particle above a dielectric interface: cross sections, benchmark calculations, and analysis of the intrinsic substrate effects. The Journal of Physical Chemistry C, 117(1): , 13. [9] P. A. Letnes, I. Simonsen, and D. Mills. Substrate influence on the plasmonic response of clusters of spherical nanoparticles. Physical Review B, 83(7):7546, 11. [1] I. V. Lindell and E. Alanen. Exact image theory for the sommerfeld halfspace problem, part i: Vertical magnetic dipole. IEEE Transactions on Antennas and Propagation, AP-3(), February [11] I. V. Lindell and E. Alanen. Exact image theory for the sommerfeld halfspace problem, part ii: Vertical electric dipole. IEEE Transactions on Antennas and Propagation, AP-3(8), August [1] I. V. Lindell and E. Alanen. Exact image theory for the sommerfeld halfspace problem, part iii: General formulation. IEEE Transactions on Antennas and Propagation, AP-3(1), October [13] R. J. Lytle and D. L. Lager. Numerical evaluation of sommerfeld integrals. Technical report, Lawrence Livermore Laboratories, [14] S. Z. Malynych, N. L. Dmitruk, and I. E. Moroz. Influence of substrate on the optical properties of non-aggregated silver nanoparticles. The European Physical Journal Applied Physics, 64():4, 13. 7

28 [15] G. Y. Panasyuk, J. C. Schotland, and V. A. Markel. Short-distance expansion for the electromagnetic half-space green s tensor: general results and an application to radiative lifetime computations. Journal of Physics A: Mathematical and Theoretical, 4(7):753, 9. [16] A. Sommerfeld. Über die ausbreitung der wellen in der drahtlosen telegraphie. Annalen der Physik, 333(4): , 199. [17] A. Sommerfeld. Über die ausbreitung der wellen in der drahtlosen telegraphie. Annalen der Physik, 386(5): , 196. [18] A. Sommerfeld. Partial Differential Equations, chapter 6, pages Academic Press, [19] P. Valle, E. Ortiz, and J. Saiz. Near field by subwavelength particles on metallic substrates with cylindrical surface plasmon excitation. Optics communications, 137(4):334 34, [] M. Venkatapathi and A. K. Tiwari. Radiative and non-radiative effects of a substrate on localized plasmon resonance of particles. Journal of Applied Physics, 11(1):1359, 1. [1] K. C. Vernon, A. M. Funston, C. Novo, D. E. Gómez, P. Mulvaney, and T. J. Davis. Influence of particle- substrate interaction on localized plasmon resonances. Nano letters, 1(6):8 86, 1. [] Y. Wu and P. Nordlander. Finite-difference time-domain modeling of the optical properties of nanoparticles near dielectric substrates. The Journal of Physical Chemistry C, 114(16):73 737, 9. [3] G. Xu, Y. Chen, M. Tazawa, and P. Jin. Influence of dielectric properties of a substrate upon plasmon resonance spectrum of supported ag nanoparticles. Applied physics letters, 88(4):43114, 6. [4] J. Yan, K. S. Thygesen, and K. W. Jacobsen. Nonlocal screening of plasmons in graphene by semiconducting and metallic substrates: firstprinciples calculations. Physical review letters, 16(14):14683, 11. 8

29 1. Value(r) 1.8 Value r Figure A.1: Mapping of magnitude to Value (V) used in the domain coloring visualizations Appendix A. Domain coloring for visualizing complex functions Domain coloring is a technique for visualizing complex functions. Representing a complex function with separate magnitude and phase plots makes it difficult to put together and interpret. In domain coloring, the magnitude and phase of the complex function are mapped respectively to the value (V) and hue (H) components of that point in the HSV color space. The saturation (S) component of the color is set constant to unity. Thus, oscillations (that is, phase change) in the function are represented by changes in hue (color), and magnitude is represented by the brightness. Darker shades represent smaller magnitude, and brighter shades represent larger magnitude. For the domain coloring visualizations shown in Figures and 5, the chosen mapping of the complex number to the color was 9

30 where r and φ are the magnitude and phase of the complex number respectively. Hue(re iφ ) = φ Saturation(re iφ ) = 1 ( Value(re iφ ) = r ). The mapping of magnitude (r) to the Value (V) component of the HSV color is shown in Figure A.1. In the HSV color space, the maximum allowed value of the Value (V) component is 1. Hence the mapping is chosen such that it saturates for large magnitudes and does not exceed 1. Appendix B. Branch cuts of the Sommerfeld integrand Appendix B.1. Branch cuts due to k ρ k Letting k ρ = x + iy, where x and y are the real and imaginary parts of k ρ respectively, k ρ k = (x y k ) + ixy For the conventional principal value of the square root function, the branch cuts in the k ρ plane satisfy R{k ρ k } <, I{k ρ k } = Thus, x y k < xy = (B.1a) (B.1b) From (B.1b), x = or y =. If x =, from (B.1a), y > k 3

31 Since y and k are real numbers, y is always positive and k is always negative for all y, k, making the above inequality true for all y R. Thus, there exists an infinitely long branch cut along the imaginary axis in the k ρ plane. This can be mathematically expressed as k ρ = it, t R If y =, from (B.1a), x < k = x < k Since k is real and positive, x < k. Thus, the branch cut is Appendix B.. Branch cuts due to k ρ = t, t [ k, k] k ρ εk Letting k ρ = x + iy, where x and y are the real and imaginary parts of k ρ respectively, and ε = ε x + iε y, where ε x and ε y are the real and imaginary parts of ε respectively, k ρ εk = (x y ε x k ) + i(xy ε y k ) For the conventional principal value of the square root function, the branch cuts in the k ρ plane satisfy Thus, R{k ρ εk } <, I{k ρ εk } = x y ε x k < xy ε y k = (B.a) (B.b) From (B.b), Substituting in (B.a), y = ε yk x x 4 ε x k x ε yk < (B.3)

32 Completing squares, (x ε xk ) < k4 4 ( ε x + ε y) (x ε xk ) < k4 ε Thus, But, ε k (ε x ε )k < x ε xk ε = 4 < ε k < x < (ε x + ε )k ε x + ε y Therefore, ε ε x This means that ε x ε is negative. But, x is real, and so x is always positive. Therefore, x > (ε x ε ) k is always satisfied. This leaves us with the condition, x > (ε x + ε ) k (B.4) But, R{ ε} = ( ) 1 ε cos arg ε [R{ ( ) 1 ε}] = ε cos arg ε [R{ ε}] = ε [1 + cos(arg ε)] [R{ ε}] = ε [ 1 + ε ] x ε [R{ ε}] = ε x + ε Substituting in (B.4), x > k [R{ ε}] 3

33 Thus, the branch cut exists for kr{ ε} < x < kr{ ε} From (B.3), the branch cut is k ρ = t + i ε yk t, t [ kr{ ε}, kr{ ε} ] Appendix C. Branch cuts of the high permittivity integrand The high permittivity integrand shown in (5) has branch cuts due to the square root function in R = ρ + (z + h ) Letting z = x + y where x and y are the real and imaginary parts of z respectively, ρ + z = ρ + x y + ixy For the conventional principal value of the square root function, the branch cuts in the z plane satisfy R{ρ + z } <, I{ρ + z } = Thus, ρ + x y < xy = (C.1) (C.) From (C.), x = or y =. If x =, from (C.1), y > ρ = y > ρ If y =, from (C.1), x < ρ Since x is a real number, x is always positive. Since ρ is a real number, ρ is always negative. Therefore, the above condition is not possible for any real x. Therefore, the branch cut is given by z = ±it, t [ρ, ) 33