How to expand Debye potentials of a plane wave into spherical

[PuZHANG0702@gmail.com]

Dear all members,

In scattering problems involved with spherical geometry, plane wave
expansion into spherical harmonics is obtained in many textbooks. But if
the problem is posed in terms of Debye potentials, one has to expand the
Debye potential of the incident plane wave into spherical harmonics to
implement field boundary conditions. In a recent paper published in PRL
(http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRLTAO000099000006063903000001&idtype=cvips&gifs=yes), the expansion is given as Eq. (6). I can't
obtain this expression, however, from existing results for expansion of
a plane wave (like Eq. (1316) at
http://farside.ph.utexas.edu/teaching/jk1/lectures/node102.html). In the
PRL paper expression, associated Legendre functions P_n,1 are used,
while Legendre functions P_n,0 are used in a usual expansion. Could
anyone please tell me how this expression is obtained?

Thanks very much!

[Igor Khavkine]

On Jun 6, 10:34 am, "PuZHANG0...@gmail.com" <PuZHANG0...@gmail.com>
wrote:

> In scattering problems involved with spherical geometry, plane wave
> expansion into spherical harmonics is obtained in many textbooks. But
> if the problem is posed in terms of Debye potentials, one has to
> expand the Debye potential of the incident plane wave into spherical
> harmonics to implement field boundary conditions. In a recent paper
> published in PRL [1], the expansion is given as Eq. (6). I can't
> obtain this expression, however, from existing results for expansion
> of a plane wave (like Eq. (1316) at [2]). In the PRL paper
> expression, associated Legendre functions P_n,1 are used, while
> Legendre functions P_n,0 are used in a usual expansion. Could anyone
> please tell me how this expression is obtained?

Note that reference [2] expands a scalar harmonic plane wave in terms
of spherical Bessel functions and Legendre polynomials (or P_n,0
associated Legendre polynomials, as you've noted). In general such an
expansion is done in terms of spherical harmonics Y_n,m, where n and m
are respectively the orbital and magnetic quantum numbers. The
spherical harmonics are given by

Y_n,m(theta,phi) = N_l,m exp(i m phi) P_n,m(cos(theta)),

where P_n,m is the corresponding associated Legendre polynomial and
N_l,m is a normalization factor.

A scalar plane wave propagating along the z-axis possesses axial
symmetry. It is symmetric with respect to phi-rotations. Not
surprisingly, its expansion in terms of spherical harmonics will only
have Y_n,0 terms, which have trivial phi-dependence.

On the other hand, reference [1] states that it's is considering a
harmonic plane electromagnetic wave propagating along the z-axis with
E
field polarized along the x-axis (and consequently with the B field
polarized along the y-axis). The Debye potentials are closely related
to the radial components of the E and B fields [3]. In fact, the
expansion of the two Debye potentials in spherical harmonics will have
the exact same coefficients as the field radial components, up to
numerical factors.

However, it's quite obvious that the radial components of x- and y-
axis
polarized plane waves will behave respectively as cos(phi) and
sin(phi). Not surprisingly, the expansion of these potentials in
spherical harmonics will have only terms of the form Y_n,(+-1), which
involve P_n,1 associated Legendre polynomials. And this is precisely
the phi- and theta-dependence seen in equations (6)-(9).

[1] http://dx.doi.org/10.1103/PhysRevLett.99.063903
[2] http://farside.ph.utexas.edu/teaching/jk1/lectures/node102.html
[3] http://dx.doi.org/10.1119/1.11111

Hope this helps.

Igor