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A closed form formula is found for the Mie scattering coefficents of incident complex focus beams (which are a nonparaxial generalization of Gaussian beams) with any numerical aperture. This formula takes the compact form of multipoles evaluated at a single complex point. Included are the cases of incident scalar fields as well as electromagnetic fields with many polarizations, such as linear, circular, azimuthal and radial. Examples of incident radially and azimuthally polarized beams are presented.
©2008 Optical Society of America
1. Introduction
Traditional Lorenz-Mie theory describes the scattering from spherical homogeneous dielectric particles of incident plane waves. However, the particles must be illuminated by focused beams in many applications, including optical trapping and manipulation [1, 2], and simultaneous measurements of particle size and velocity or chemical composition [3]. Any incident (focused or not) field can be expressed as a continuous superposition of plane waves, but the scattering computation for a focused field resulting from such a superpositionwould be very intensive. For this reason, Gouesbet’s group developed the so-called generalized Lorenz-Mie theory (GLMT) [4]–[5], with special attention paid to the case of Gaussian beams [6]. The GLMT has been formulated for particles that are not only axially shifted from the waist of the illuminating beam but also off-axis. It requires, however, the numerical computation of two sets of coefficients by using one of three different methods, which exhibit trade-offs between computational efficiency and accuracy. More recently, Kant [7] developed closed form solutions for on-axis scatterers in radially and azimuthally polarized fields. Also, a numerical method related to the GLMT for calculating the coefficients in a vectorial Laguerre-Gauss beam was proposed by van de Nes and Török [8].
We propose an alternate generalization of the Lorenz-Mie theory wherein the incident field is a complex focus field. The properties of such fields are described in Section 2. This generalization is advantageous as it does not require numerical integration. Further, it includes as special cases radially and azimuthally polarized fields.
2. Characteristics of complex focus fields
It was noted by Kravtsov [9] that a scalar monochromatic source placed at a complex point yields a field that, as the imaginary displacement is increased, tends towards a paraxial Gaussian beam. Cullen and Yu [10] generalized this concept to electromagnetic fields by placing a dipole source at the complex point and connected the resulting fields to nonparaxial laser cavity modes. A thorough history of complex source-point fields can be found in Refs. [11] and [12]. These fields, however, are not global solutions of the Helmholtz equation as they have a branch-point singularity at a ring in their waist plane. This problem was resolved by Sheppard and Saghafi [11], who instead considered displacing the focus of a spherical wave to a complex point. The resulting fields do not present the singularities, but have otherwise similar properties.
These fields, to which we refer herein as complex focus (CF) fields, are then given by the simple expression
where U 0 is a constant and ρ 0=r 0+iq. These fields do not include evanescent waves, but do include a significant counter-propagating component for small kq. For kq≫1, where q=|q| CF fields tend to paraxial Gaussian beams focused at r 0, traveling in the direction of q and with waist width proportional to . As such, CF fields are nonparaxial generalizations of Gaussian beams [11] and are therefore a good model to describe a tightly focused laser beam for kq≳3 (i.e., when the counter-propagating components are negligible).
Electromagnetic closed-form analogues of Gaussian beams can also be achieved by displacing an electric or magnetic dipole to a complex point ρ 0. The resulting CF fields are given by the expressions:
where p is a unit vector in the direction of the electric or magnetic dipole moment, respectively. Beams with several polarizations of interest can be obtained by using these fields. For example, a linearly polarized (in the sense of Richards and Wolf [13]) Gaussian-like beam focused at the origin, traveling in the z direction and whose polarization (before focusing) is in the y direction, can be written as E (E) CF(r;iq ẑ, ŷ)+E (B) CF (r;iq ẑ, x̂) [12]. On the other hand, beams with the so-called radial and azimuthal polarizations are given, respectively, by the expressions E (E) CF(r;iq ẑ, ẑ) and E (B) CF(r;iq ẑ, ẑ) [14]. These latter beams do not have a Gaussian profile, even in the paraxial limit; rather, they resemble combinations of first-order Hermite-Gaussian beams [15]. The azimuthally polarized fields have a null on-axis, while for the radially-polarized ones, the on-axis field is polarized along the axial direction, and its magnitude becomes more significant as the beam’s angular spread is increased.
3. Mie scattering of scalar CF fields
It is shown in Appendix A that the following relation holds for any real r a, r b and r c:
where the scalar multipoles Λlm(r) are defined as
Here, jl is the spherical Bessel function of the first kind of order l, (r, θr, ϕr) are the spherical coordinates of the position vector r and Ylm is a spherical harmonic, defined by
where l≥0, |m|≤l and Pml is the associated Legendre function [16].
It turns out that the analytic extension of Eq. (3) to cases where any of the points’ coordinates are complex is also valid. By using Eq. (3), a scalar CF field can then be written, in closed form, as a sum of multipoles. This result is obtained by letting r a=r (the spatial variable) and r b=ρ 0=r 0+iq. Let us now consider a scattering sphere of radius R and refractive index n. It is then convenient for r c to coincide with the center of the scatterer, which, for simplicity, is chosen as the origin (r c=0). The incident CF field is then given by the remarkably simple expression
where Λlm(ρ 0) is calculated via Eq. (4) with the complex values
where the branches must be chosen so that (ρ 0 sin θρ cos ϕρ, ρ 0 sin θρ sin ϕρ,ρ 0 cos θρ) indeed returns ρ 0. The compact form of the coefficients in the multipolar expansion in Eq. (6) is the key result of this manuscript; the coefficients are themselves multipoles evaluated at a single complex point.
Through the standard application of boundary conditions, the scattered field U (sc)(r) is then found to have the form
where ∏lm is the multipole field containing only outgoing components, i.e.,
Here, h (1) l is a spherical Hankel function of the first kind, so that ∏lm satisfies the Sommerfeld radiation condition. The coefficients cl are the standard scaling factors for the multipolar components, which result from imposing boundary conditions (i.e., continuity of the field and its derivative with respect to r) at the surface of the spherical scatterer. For a scalar field, these coefficients are given by [17]
where prime indicates differentiation with respect to the argument.
4. Mie scattering of electromagnetic fields
Equally simple results can be found in the electromagnetic case. Any incident electric field can be written as a linear combination of the electromagnetic multipolar fields, defined as
where Y lm(θ, ϕ) and Z lm(θ, ϕ) are the vector spherical harmonics [18, 19], which are defined here as
with the angular momentum operator L given by
[Note that the normalization in Eqs. (12) differs from that of the references.]
Incident electric and magnetic dipoles at the complex point ρ 0, defined in Eqs. (2), can then be written, respectively, as the linear combinations
where, as shown in Appendix B, the coefficients γ (I, II) lm are given in terms of multipole fields at a complex point:
As in the scalar case, the compact analytic form of the coefficients in Eq. (15) is the key to the convenience of this representation.
It is well known that no coupling occurs between Λ (I) and Λ (II) upon scattering. Thus, the scattered fields take the forms given below
where ∏ (I) lm and ∏ (II) lm are defined by replacing the spherical Bessel functions in Eqs. (11) with spherical Hankel functions of the first kind. The coefficients in Eqs. (16) may be calculated by applying the standard boundary conditions at a dielectric interface. These coefficients are found to be [17]
5. Examples: Radially and Azimuthally polarized beams
In this section, we will focus on the examples of radially and azimuthally polarized beams, resulting from aligning the electric and magnetic dipole moments, respectively, with, the z axis [14], which we have chosen as the direction of propagation. The far-field scattered radiant intensity can be calculated as the squared modulus of the sum of the vector spherical harmonics with the appropriate coefficients. For these figures, the sum is taken from l=1 to l max=12. Note that when the scatterer is centered on the beam’s axis, only the m=0 terms contribute. (For an arbitrarily polarized CF beam incident upon an on-axis scatterer, m must be summed only from -1 to 1.)
Figures 1 and 2 (and accompanying frames of the multimedia files) show the results, respectively, for radially- and azimuthally-polarized beams incident upon a spherical scatterer of radius kR=5 and refractive index n=2, where the focus of the beam is located at the upper edge of the sphere (i.e., k r 0=5x̂) and k q=4ẑ. Each figure includes (a) a depiction of the incident field’s magnitude over the x-z plane, so as to demonstrate the relative location of the scatterer to the beam, as well as (b) the magnitude of the total field (including scattered and inside the scattering sphere). Additionally, the coefficients for both Λ (I) lm(r) and Λ (II) lm(r) are depicted in (c) to demonstrate convergence for l max=12. Finally, the radiant intensity (i.e., irradiance in the far field) of the scattered field is shown in (d), configured to show back scattering in the center of the diagram, with forward scattering at the edges. Both figures have accompanying multimedia files which show the development of the fields, coefficients and radiant intensity as the focus moves laterally to the center of the scatterer, then axially to the front of the scatterer, and finally as the beam becomes more collimated (4≤kq≤9). The location of the focus and the value of the parameter q are indicated in (a).
Fig. 1. (a) Radially polarized beam with the specified focus and complex displacement, (b) total field (incident, scattered, and inside a scattering sphere of radius kR=5 and index n=2), (c) coefficients of the expansion and (d) radiant intensity. The animation shows the change of these plots as the focus of the beam moves laterally from the edge of the sphere to its center, then axially to the front of the sphere and finally as kq increases (i.e., the beam becomes more collimated). [Media 1]
6. Concluding remarks
A simple generalization to Mie scattering has been demonstrated, corresponding to incident CF fields, which are nonparaxial generalizations of (scalar as well as linearly, circularly or elliptically-polarized) Gaussian beams, or radially and azimuthally-polarized fields. This model leads to an easily calculable closed form for the coefficients in the multipolar expansion of these incident fields: the coefficients are themselves multipolar fields evaluated at a single complex point. The formulas apply regardless of the beam’s angular spread and relative position between the scatterer and the incident field’s focus, although the number of terms needed does depend on these parameters.
Fig. 2. (a) Azimuthally polarized beam with the specified focus and complex displacement, (b) total field (incident, scattered, and inside a scattering sphere of radius kR=5 and index n=2), (c) coefficients of the expansion and (d) radiant intensity. The animation shows the change of these plots as the focus of the beam moves laterally from the edge of the sphere to its center, then axially to the front of the sphere and finally as kq increases (i.e., the beam becomes more collimated). [Media 2]
While only uniform dielectric scatterers were considered here, these results can be applied to scatterers comprised of multiple spherical shells, each with different index of refraction. Moreover, in the scalar case, scattering of any rotationally symmetric beam can be calculated via this method by using the basis developed by Alonso, Borghi and Santarsiero [20], which relies on superpositions of CF beams. We are currently working on the expansion of that basis to non-rotationally symmetric scalar and electromagnetic fields, which will then allow these results to be expanded to model the scattering of any beam.
Acknowledgments
We acknowledge support from the National Science Foundation through the Career Award number PHY-0449708. We also wish to thank Zachary Smith and Andrew Berger for stimulating converstations that inspired this work.
A. Derivation of Eq. (3)
The left-hand side of Eq. (3) can be written as an integral of plane waves of the form
where the variables u and u′ are unit vectors integrated over the complete unit sphere, and δ(u, u′) is a Dirac delta function over the unit sphere. This function can be expressed in terms of the spherical harmonics, since they are a complete, orthonormal basis:
where θ and ϕ are the polar and azimuthal angles associated with u, and likewise for the prime variables. By inserting Eq. (19) into Eq. (18), and using the relation [19]
one obtains straightforwardly Eq. (3).
B. Derivation of Eq. (15)
The expressions in Eqs. (2) can be written as weighted superpositions of plane waves:
where δ⃡ ⊥ (u, u′) is a tensor Dirac delta function for transverse vector distributions over the sphere, which satisfies u·δ⃡ ⊥ (u, u′)=δ⃡ ⊥ (u, u′)·u=0. This function can be composed in terms of vector spherical harmonics as
where ⊗ denotes the outer product. By using the relations [19]
as well as Eq. (12b), it is easy to show that the substitution of Eq. (23) into Eqs. (21) and (22) gives precisely Eqs. (14) and (15).
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