Conversion between PW and VSH

Now that both plane waves and vector spherical harmonics have been introduced, we want to show that they can be converted into each other. This will become important when experimental situations combine both planar and spherical elements. One can show ([47], p. 416) that

The unit vectors ˆep and ˆes correspond to so-called p- ands-waves, plane waves with a particular polarization that will be described in more detail in section 2.3.1. For now, it suffices to know the definitions

ˆ

The fomulas (2.50) and (2.51) are conveniently evaluated forr in cylindrical coordinates, r = (ρcosϕ,ρsinϕ, z), thenk⁰·r =k⁰sinθ⁰ρcos(ϕ⁰−ϕ) +k⁰cosθ⁰z. Theϕ⁰-integration can be carried out analytically with the help of the definition of the Bessel function [46],

J_`(ξ) = 1 Equations (2.45) and (2.46) can then be transformed to:

M<sub>`m</sub><sup>j</sup> (ρ, ϕ, z) = 1 Two exemplary plane-wave decompositions of vector spherical harmonics are shown in figure 2.4, namely M<sub>4,1</sub><sup>j</sup> and N<sub>1,−1</sub><sup>j</sup> . These particular exampls were chosen to illustrate again that the number of symmetry planes rises with ` and is different for each component. Since the functions are evaluated in the x-z-plane, ϕ = 0 in the right halves of the plots and ϕ=π in the left halves. Thus, the exponential functions exp(imϕ) are real, which means that ξ_m is real for even m and imaginary for odd m.

This results in purely real or purely imaginary components ofpm and sm, where a real p_m,x is always paired with an imaginary s_m,x and vice versa (and the same fory andz).

analytical

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-0.35 0.2 -0.35 0.35 -0.1 0.1

Im(M_4,1^j )_x Re(M_4,1^j )_y Im(M_4,1^j )_z

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-0.1 0.25 -0.25 0.1 -0.06 0.06

Re(N_1,−1^j )_x Im(N_1,−1^j )_y Re(N_1,−1^j )_z

Figure 2.4: Plane-wave decompositions of different VSH. Thex-,y- andz-component of M_{`m</sub><sup>j</sup> and N<sub>`m}^j are either purely real or purely imaginary, shown is always the nontrivial part in the x-z-plane. The top left corner depicts the analytical result according to equations (2.45,2.46), the bottom right corner the PW decomposition (2.56,2.57) (integral over 250 evenly spaced θ⁰-values between 0 and π, analytical ϕ⁰-integration, difference between analytical result and PW decomposition on the order of 10⁻⁶). Note the excellent agreement between analytical result and PW decomposition.

It follows from the formulas (2.56) and (2.57) that the x-, y- and z-components of each VSH are either purely real or purely imaginary. Figure 2.4 only shows these nontrivial components.

If the boundary conditions require the use of spherical Hankel functions instead of spherical Bessel functions, the PW decomposition becomes slightly more complicated.

Mathematically, it is possible to change an integral representation of j` to an integral representation of h<sub>`</sub> by changing the integration path in the complex plane, see e.g.

[45], §5.3. Physically, this can be justified as follows: While spherical Bessel functions are finite everywhere, spherical Hankel functions diverge at zero. This necessitates the inclusion of evanescent waves in the decomposition. Furthermore, as mentioned

in the previous section, spherical Hankel functions approach spherical waves for large arguments ([46], §19):

h_`(kr)^kr1−→ 1

kreikr−i(`+1)π/2

. (2.58)

Intuitively, this can be realized by only using plane waves moving away from the origin in the decomposition. In practice, this would mean only using k with θ⁰ ∈[0, π/2] for z >0 andθ⁰ ∈[π/2, π] for z <0. Similarly, evanescent waves with decreasing amplitude for longer distances from the origin are given by θ⁰ =π/2 +iθ⁰⁰, θ⁰⁰ ∈[−∞,0] for z >0 and θ⁰⁰ ∈ [0,∞] for z < 0. A comparison with the asymptotic form of the spherical Bessel function ([46], §19)

j_`(kr)^kr1−→ 1 shows that the functions do not only differ in the plane waves that are needed, but also in a prefactor of two. These thoughts result in the PW decompositions

M_`m^h (ρ, ϕ, z) = 1

These formulas, too, were implemented in Matlab. Since it is not possible to integrate to infinity numerically, a cutoff T has to be chosen for the imaginary part of θ⁰, i.e.

Im(θ⁰)∈[−T,0] (or [0, T] for z <0). The PW decompositions of the same functions M_4,1 and N1,−1 as before, but now with h instead of j, are presented for T = 20 in figure 2.5. As explained above, for M_{`m</sub><sup>j</sup> andN<sub>`m}^j , thex-,y- andz-components are each either purely real or purely imaginary in the x-z-plane. Since h` =j`+iy`, changing from spherical Bessel to Hankel functions only changes the previously zero real (or imaginary) part. Thus, only this new result is shown in figure 2.5. The most striking feature is the appearance of very large, rapidly oscillating values close to z = 0. We call them ringing-artefacts, because this effect is similar to the Gibbs phenomenon that causes ringing artefacts in signal processing (e.g. [48] §5.7). It is caused by the evanescent waves, which have large values at z = 0 and then drop off quickly. To get the correct result everywhere, one would have to include evanescent waves up to very large imaginary part of θ⁰, and with a dense sampling. This is further investigated in the chapter on the numerical implementation, 3.1.

The next step is to decompose a plane wave in vector spherical harmonics:

(E_peˆ_p+E_seˆ_s) e^ik0·r = 4π

The derivation of the formula can be found e.g. in [49]. As before, dashed quantities (θ⁰, ϕ⁰) describe the direction of the wavevector k⁰. Details on the convergence for a

finite approximation of the infinite sum are presented in section 3.1.

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-0.3 0.3 -0.5 0.5 -0.1 0.1

Re(M_4,1^h )_x Im(M_4,1^h )_y Re(M_4,1^h )_z

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-0.08 0.08 -0.08 0.08 -0.04 0.04

Im(N_1,−1^h )_x Re(N_1,−1^h )_y Im(N_1,−1^h )_z

Figure 2.5: Plane-wave decompositions of different VSH. Either the real or the imaginary part of the x-, y- andz-component ofM_{`m</sub><sup>h</sup> (N<sub>`m}^h ) is diverging at the origin. The other part is finite everywhere and identical to the same part of the same component ofM_{`m</sub><sup>j</sup> (N<sub>`m}^j ), and therefore not shown here. The color range was restricted in order to make the patterns better visible. The top left corner depicts the analytical result in the x-z-plane according to equations (2.45,2.46), the bottom right corner the PW decomposition (2.60,2.61) (integral over 250 evenly spaced θ⁰-values between 0 and π/2 and 200 evenly spaced θ⁰-values between π/2 and π/2 + 20ifor z >0; π minus these angles for z <0;

analyticalϕ⁰-integration). Note the appearance of atefacts near z= 0.