Spectral approach to scattering from spheres (scalar waves)

Transcription

1 Spectral approach to scattering from spheres (scalar waves) November 1, Introduction Consider the scalar wave equation [ 2 + ɛ(r)k 2 0]ψ(r) = S(r). (1) Here S(r) is the source which is usually zero in the region of interest, e.g., in the region where ɛ(r) 1 or where the measurements are being performed. I am interested in piece-wise constant potentials, but for now it is sufficient to consider the following shape of ɛ(r): ɛ(r) = 1 + 4πχΘ(r), (2) where Θ(r) = 1 if r V and Θ(r) = 0 otherwise. To simplify the problem even further, let V be a sphere of radius a and center at the origin. To make some connection to quantum mechanics: k0 2 = 2mE h 2, ɛ(r) = 1 U(r) E, (3) where E is the kinetic energy of the particle when it is far from the center of the scattering potential and U(r) is the scattering potential itself. Thus, in QM, k0 2 is a characteristic of the incident particles while ɛ(r) is a characteristic of the interaction potential. As we change the energy E, the potential does not change (apart from the trivial factor E in the denominator which can be easily removed), at least in the non-relativistic theory. 1

2 In electromagnetic scattering, the picture is different in the following respect: when we change k 0 (the frequency of the incident photons), the quantity ɛ(r) also changes in a complicated way. This is due to the effect of dispersion. Therefore, the usual spectral theory in which the potential is assumed to be frequency- or energy-independent does not work well here. 2 The Mie solution In the case when V is just a sphere, the scattering problem can be readily solved. Assuming that the incident field ψ inc (r) which satisfies [ 2 + k 2 0]ψ inc (r) = S(r) (4) can be approximated in the region of interest by a plane wave we have for the internal field ψ inc (r) = exp(ik inc r), (5) ψ in (r) = lm c l j l (mk 0 r)y lm(ˆk inc )Y lm (ˆr). (6) Here m is the solution to m 2 = 1 + 4πχ with positive imaginary part (the complex refractive index of the sphere) and c l is the Mie coefficient given below. The rest of the notations are self-explanatory. Outside of the spherical volume volume V, we have ψ(r) = ψ inc (r)+ψ s (r) where the scattered field is given by ψ s (r) = lm a l h (1) l (k 0 r)ylm(ˆk inc )Y lm (ˆr). (7) The Mie coefficients are given by c l = 4πil 1 i(k 0 a) 2 mh (1) l (k 0 a)j l (mk 0a) j l (mk 0 a)h (1) l (k 0 a), (8) a l = 4πi l j l (mk 0 a)j l(k 0 a) mj l (k 0 a)j l(mk 0 a) mh (1) l (k 0 a)j l (mk 0a) j l (mk 0 a)h (1) l (k 0 a). (9) The main problem with the Mie solution is the complex dependence of the above coefficients on the refractive index m and the size parameter x = k 0 a. 2

3 In other words, for each pair of m and x, the computationally-intensive part (computing all the coefficients) must be done anew. The reason why the Mie solution has this problem is that it is not a spectral solution. In other words, the individual terms in (6) and (7) are not eigenfunctions of any useful linear operator. These terms are simply the solutions to the original equation (1) in the regions r a and r > a, respectively, and the coefficients c l and a l are obtained by satisfying the boundary conditions at r = a (there is one more boundary condition at the infinity). 3 Spectral approach The spectral approach is based on the following eigenproblem: [ 2 + k 2 0]φ µ (r) = 4πk 2 0χ µ Θ(r)φ µ (r). (10) Of course, this eigenproblem can be formulated for a general shape of Θ(r), not just for a sphere. Now φ µ (r) are the eigenfunctions and χ µ are the eigenvalues. The eigenproblem (10) must be supplemented by the appropriate boundary conditions at the infinity (in our case, the usual Zommerfeld scattering condition). Equivalently, the eigenproblem can be re-formulated as φ µ (r) = χ µ G(r, r )Θ(r )φ µ (r )d 3 r, (11) where G(r, r ) is the usual retarded Green s function, or, in our case, as φ µ (r) = χ µ G(r, r )φ µ (r )d 3 r. (12) r a Now (10) is a generalized eigenproblem of the form ˆLφ µ = χ µ ˆDφµ. (13) It is supplemented by complex boundary conditions at the infinity. Therefore, it is not a Hermitian eigenproblem. The eigenvectors φ µ do not form an orthonormal basis. Instead, they obey a different orthogonality property (sometimes referred to as quasi-orthogonality), which is given below in (20). Similarly, the integral operator in the right-hand side of (11) is symmetric but not Hermitian. Note that the eigenproblem (11) is mathematically equivalent to (10) plus the boundary conditions at the infinity. 3

4 It is easy to see that the eigenfunctions have the following form: { jl [n lk (x)k 0 r]y lm (ˆr), r a φ lmk (r) = A lk (x)h (1) l (k 0 r)y lm (ˆr) r > a (14) Here n lk (x) is one of the complex roots (labeled by the index k) of the equation and j l [n lk (x)x]h (1) l (x) = n lk (x)j l[n lk (x)x]h (1) l (x). (15) A lk (x) = j l[n lk (x)x]. (16) (x) Not surprisingly, n lk (x) coincide with the values of the complex refractive index m at which the Mie coefficients (8),(9) have singularities (for a given value of x). The relationship between the eigenvalues χ lmk and the roots n lk is χ lmk (x) = n2 lk(x) 1. (17) 4π All quantities in (17) depend parametrically on the size parameter x = k 0 a. Thus, we have the obvious degeneracy with respect to the projection of the angular momentum, m (not to be confused with the refractive index). Below, I omit this index where appropriate. Further, it can be readily seen that a r 2 drj l [n lk (x)k 0 r]j l [n lk (x)k 0 r] = 0, if k k. (18) 0 Note that there is no complex conjugation in the above formula and n lk is complex. In fact, it can be stated as a theorem that (10) has no non-trivial solutions which satisfy the scattering boundary conditions for strictly real χ µ. The (quasi)-orthogonality rule (18) allows one to construct a dual bases h (1) l φ lmk (r) = j l [n lk (x)k 0 r]y lm(ˆr) (19) with the property that φ l m k (r)φ lmk(r)d 3 r = Z lk δ ll δ mm δ kk. (20) r a 4

5 The normalization factor Z lk can be readily obtained by integration; I omit here this result as it is somewhat lengthy. Using this orthogonality property, we can find the solution to the original equation (1) in the following form: ψ in = i l 1 k 3 0 lmk ψ s = i l 1 k 3 0 χ lmk A lk j l (n lk k 0 r)y lm(ˆk inc )Y lm (ˆr) Z lk (χ lk χ) [A 2 lk/χ lk ]h (1) l (k 0 r)ylm(ˆk inc )Y lm (ˆr) Z lk (χ lk χ), (21). (22) Here the dependence of χ lk on the size parameter x = k 0 a has been suppressed. I have also omitted some intermediate steps, such as calculation of some integrals. But now the solution depends on χ (or m) in a very simple way! For each new value of χ, all one has to do is simple summation (assuming all other quantities have been pre-computed). From these expressions, it is also easier to see where the resonances are and what are the resonance widths. One can define the resonance condition by Re(χ lk χ) = 0. By comparing with the Mie solution, we see that a l = i l 1 k 3 0 χ k [A 2 lk/χ lk ] Z lk (χ lk χ), (23) which is simply the pole expansion of a l. Another interesting formula is j l (k 0 r) = 1 A lk j l (n lk k 0 r). (24) 4πik0 3 k χ lk Z lk This equality must hold in the L 2 (V ) norm. The situation with the internal field coefficient c l is somewhat more complicated. We have c l j l (mk 0 r) = i l 1 k 3 0 k A lk j l (n lk k 0 r) Z lk (χ lk χ), (25) which must hold in the L 2 (V ) norm. The pole expansion of c l can then be obtained by taking the limit r 0. However, I have not yet verified this directly. Note that the complex refractive index m is related to the susceptibility χ by m 2 = 1 + 4πχ. The quantities χ lk in (21),(22) are the eigenvalues which are defined only by the geometry and do not depend in any way on χ. 5

6 4 What s good about the spectral solution The spectral solution contains more summations than the Mie solution. It can be viewed as a simple reformulation (pole expansion of the Mie coefficients). There are, however, three main advantages: 1. The dependence on the material properties is very simple. Averaging over the variable χ can be done quite easily, as long as all other quantities have been computed. 2. If we know the functions n lk (x), we can readily see where the resonances are, and what are their widths. 3. The formalism should be generalizable beyond the spherical shape. 5 Equation for the eigenmodes With some algebraic manipulation, equation (15) becomes j l (nx)xh (1) l+1 (x) = nj l+1(nx)h (1) l (x). (26) This must be viewed as an equation for n parameterized by x. The roots of this equation form a new set of special functions n lk (x). If we make the substitution z lk (x) = n lk (x)x, we also have an equation with respect to the variable z: j l (z)xh (1) l+1 (x) = zj l+1(z)h (1) l (x). (27) Further, if we use the explicit expressions for the spherical functions in terms of the elementary functions, namely, j l (ξ) = i(2ξ) (l+1) [e iξ P l ( 2iξ) e iξ P l (2iξ)], (28) h (1) l (ξ) = 2i(2ξ) (l+1) e iξ P l ( 2iξ), (29) where P l (x) is a polynomial of the form we can transform (27) to P l (x) = l j=0 (2l j)! j!(l j)! xj, (30) 6

7 x l+1 [e iz P l ( 2iz) e iz P l (2iz)] = z l+1 [e iz ( 1) l+1 e iz ]P l ( 2ix). (31) Numerically, this can be solved by the following method. Let us write z = ζ + 2πk and require that π < Re(ζ) π. Then in the above equation, the polynomials are shifted by 2πk, but exp(iz) = exp(iζ) can be expanded into power series up to some finite order. We will end up with an algebraic equation which has many roots. Algebraic equations are easily solvable numerically. Then we will only keep the roots ζ which satisfy the above inequality. Hopefully, there ll be only one such root. Then we repeat this for each integer k, which would generate a family of roots parameterized by k. A Solving integral equation by the spectral method [ 2 + ɛ(r)k 2 0]ψ(r) = S(r) (32) ɛ(r) = 1 + 4πχΘ(r) (33) [ 2 + k 2 0]ψ(r) = 4πχΘ(r)ψ(r) + S(r) (34) ψ(r) = ψ inc (r) + ψ s (r) (35) [ 2 + k 2 0]ψ inc (r) = S(r) (36) [ 2 + k 2 0]ψ s (r) = 4πχΘ(r)ψ(r) (37) [ 2 + k 2 0]G(r, r ) = 4πk 2 0δ(r r ) (38) In the above eq., one chooses G to satisfy the proper boundary conditions at infinity, i.e., the retreaded, rather than advanced, Green s function. ψ s (r) = χ G(r, r )Θ(r )ψ(r )d 3 r (39) 7

8 ψ(r) = ψ inc (r) + χ G(r, r )Θ(r )ψ(r )d 3 r (40) This is the integral equation. Write it in the operator form as ψ = ψ inc + χw ψ (41) Here W is a linear integral operator which is complex symmetric but not Hermitian (this is all due to the complex boundary conditions at infinity). Now let us expand the solution into the eigenfunctions φ µ : φ µ = χ µ W φ µ (42) ψ = µ c µ φ µ (43) Note that the basis φ µ is complete because W is non-defective. I demonstrate this explicitly by construction. The dual basis φ µ is defined in (19). Now substitute this expansion into the operator equation and use φ ν φ µ = Z ν δ ν,µ (note that for non-defective operators, Z ν 0), to obtain: c µ = φ µ ψ inc Z µ (1 χ/χ µ ) (44) This is the spectral solution. All that is left is to compute the integrals for Z µ and φ µ ψ inc, which is possible to do analytically (given the values of n µ ). 8