Vector spherical harmonics at a spherical interface

After having found a description for planar interfaces, we now turn our attention to spherical interfaces. We encounter these e.g. when measuring dye molecules embedded in polymer beads, fluorescent proteins in solution close to metal nanospheres, or when scattering a laser beam at a glass bead. Historically, the term reflection is used when light interacts with a planar boundary, and when the previously found law “angle of incidence equals angle of reflected light” is valid. On the other hand, if light changes its propagation direction due to encountering spherical particles, the term scattering is used¹⁰. The most obvious difference between these situations is that upon reflection of an incident beam of light, all light travels in the same direction, while scattering generally leads to a broad distribution of propagation directions. However, the physical mechanism behind both reflection and scattering is the same, namely the interaction of electromagnetic fields with the interface between two media with different refractive indices. The general boundary conditions derived at the beginning of this chapter are still valid: The tangential component of the electric field, as well as the product of n² and the perpendicular component of the electric field have to be constant across the interface. In a spherical geometry where the origin of the coordinate system coincides with the center of the sphere, the tangential field components are given by Eθ and Eφ, while the perpendicular component is E_r. Conveniently, these are exactly the same components which are used in the definition of vector spherical harmonics. Thus, in the following, the origin of the coordinate system is always chosen to coincide with the center of the spherical interface. We will now set out to derive how an incident field made up of VSH creates a scattered and a transmitted field that are also decomposed into VSH.

Before doing any calculations, we have to choose which VSH to include in the model. As stated in section 2.1.3, VSH with different radial functions exist. Since the differential equation (2.31) describing the r-dependence¹¹ is of second order, a linear combination of two linearly independent solutions is needed to fulfill general boundary conditions.

It is convenient to use j_{`</sub>(kr) and h<sup>1</sup><sub>`}(kr): The former is finite at the origin, while the latter has the form of outgoing spherical waves in the far field. Furthermore, both are needed in the VSH decomposition of the field of an oscillating dipole, as seen in section 2.2.2. Since we ultimately want to describe the interaction of a dipole emitter with spherical particles, it is crucial that we can use a dipole field as the incident field. Thus, the general structure of the solution can be summarised as one linear combination of {M_{`m</sub><sup>j</sup> ,N<sub>`m}^j ,M_{`m</sub><sup>h</sup> ,N<sub>`m}^h | `∈N, m∈Z,|m| ≤`} on the inside of the spherical interface, and another such linear combination on the outside of the interface. However, not all types of VSH are always needed to fulfill the boundary conditions. This can be illustrated on the example of a sphere embedded in a medium and illuminated by different types of light sources:

10It is assumed here that the particle is smaller than the diameter of the beam of light, such that the surface cannot be approximated as one plane across the whole cross section of the beam. A counterexample where the term reflection is more appropriate is the interaction of the beam of light of a laser pointer hitting a metal sphere the size of a watermelon. There, if the surface is sufficiently smooth, all reflected light will travel in roughly the same direction.

11 1 r²

∂

∂r r^2∂R_∂r +

k²−^`(`+1)_r2

R= 0 (2.31).

light source VSH inside sphere VSH outside of sphere M_{`m</sub><sup>j</sup> ,N<sub>`m}^j M_{`m</sub><sup>h</sup> ,N<sub>`m}^h M_{`m</sub><sup>j</sup> ,N<sub>`m}^j M_{`m</sub><sup>h</sup> ,N<sub>`m}^h plane wave transmitted

- incident scattered

from outside field field field

dipole emitter scattered incident

- transmitted

within sphere field field field

dipole emitter transmitted

- incident scattered

outside of sphere field field field

The list of needed VSH always refers to the field directly at the interface¹². It becomes apparent that out of the four possible types of function sets, only three are needed in a particular situation. For example, if the light source is situated outside the sphere, there is no reason why the field should become infinitely large at the center of the sphere. Thus, VSH with spherical Hankel functions are not needed inside the sphere.

Contrarily, if the light source is situated inside the sphere, then the electric field far away from the sphere has to have the form of outgoing spherical waves. Thus, VSH with spherical Bessel functions are not needed outside the sphere. This observation is similar to the reflection of plane waves at a planar boundary, where generally four types of waves are defined: In each medium, there could be waves traveling towards or away from the interface. However, if we assume an incoming wave in one of the media, then outgoing waves in both media are generated (the reflected and transmitted waves), but no incoming wave in the second medium occurs. Thus, for plane waves at a planar interface, always one of the waves traveling towards the interface is missing, while the other one is the incident field. Analogously, in the VSH decomposition of the field in a spherical geometry, eitherM_{`m</sub><sup>h</sup> ,N<sub>`m}^h inside the sphere orM_{`m</sub><sup>j</sup> ,N<sub>`m}^j outside the sphere are unneeded, while the other set defines the incident field. We use the former case to now quantify the relation between incident, scattered and transmitted fields.

Imagine a sphere with radius R and refractive index nsph embedded in a homogeneous medium with refractive index n_med. Let us assume that the incident field in the environment is given by N_`m^j . Then, the conditions for the transmitted fieldE_tr inside the sphere and the scattered field Esc in the environment are:

E_tr ·eˆ_θ = N_`m^j +E_sc

·eˆ_θ, E_tr·eˆ_φ = N_`m^j +E_sc

·eˆ_φ, n²_sphE_tr ·eˆ_r =n²_med N_`m^j +E_sc

·eˆ_r. (2.114)

According to our previous considerations, we now define E_tr =

∞

X

`⁰=1

`⁰

X

m⁰=−`⁰

a^tr_{`</sub>0m<sup>0</sup>M<sub>`}^j0m⁰ +b^tr_{`</sub>0m<sup>0</sup>N<sub>`}^j0m⁰, (2.115)

12Recall the piecewise definition of the VSH decomposition of a dipole field: When there is a dipole inside the sphere at a distancer0 from the origin,M_{`m</sub><sup>j</sup> andN<sub>`m}^j are needed for the field atr < r0. Conversely, if the dipole is outside the sphere at distancer0from the origin, the scattered field and the original dipole field are both made up ofM_{`m</sub><sup>h</sup> andN<sub>`m}^h atr > r0.

E_sc = Let us now look at the first condition of equation (2.114), using the shorthandf_`⁰(kR) :=

1/(kR) d[rf_`(kr)]/dr|_r=R: This seems rather daunting. However, we know that the equation has to be fulfilled at every set (θ, φ) – which means that we can make use of the orthogonality and thus the linear independence of the angle-dependent functions. For example, since

Z 2π 0

e^imφe^−im0φ= 2πδ_m,m⁰ , (2.118) the function e^imφ cannot be expressed as a sum over exponential functions with all other m⁰ 6=m,

e^imφ 6= X

m⁰6=m

c_m⁰e^im0φ. (2.119)

Therefore, equality at all φ ∈ [0,2π] can only be reached if the decomposition only uses terms with m⁰ =m=m⁰⁰. For the θ-dependent functions, recall that we showed orthogonality of (M_{`,m</sub>,N<sub>`,m}), (M_{`m</sub>,M<sub>`}⁰_m⁰) and (N_{`m</sub>,N<sub>`}⁰_m⁰) in section 2.1.3 (eq.

2.47-2.49). Thus, the complicated equation (2.117) reduces to:

b^tr_{`m</sub>j<sub>`}⁰(k_sphR) =j_{`</sub><sup>0</sup>(k<sub>med</sub>R) +b<sup>sc</sup><sub>`m}h⁰_`(k_medR). (2.120) We now have one equation with two unknowns, so a second equation is needed to determine the expansion coefficients. For that, we turn to the radial component of the electric field and find: Combined, this leads to a matrix equation similar to eq. (2.97) and (2.100):

j_{`</sub><sup>0</sup>(k<sub>sph</sub>R) −h<sup>0</sup><sub>`}(k_medR) where we now allowed an arbitrary prefactorb^i,med_{`m</sub> for the initial wave in the environment, and also added an initial wave b<sup>i,sph</sup><sub>`m} N_`m^h inside the sphere to account for the other

possible illumination scenarios mentioned above. The same procedure leads to an equation for VSH of type M_{`m</sub><sup>j,h</sup>, with the one difference that there the condition B<sub>θ</sub><sup>sph</sup> =B<sub>θ</sub><sup>med</sup>, B<sub>φ</sub><sup>sph</sup> =B<sub>φ</sub><sup>med</sup> (with B=∇ ×E/(ik0) and ∇ ×M<sub>`m}^f =kN_{`m</sub><sup>f</sup> ) has to be used as second equation because M<sub>`m}^f does not have anr-component:

j_{`</sub>(k<sub>sph</sub>R) −h<sub>`}(k_medR) Thus, the expansion coefficients for the scattered and transmitted fields can be found by simple matrix operations. Since these equations are independent for different sets (`, m) and for VSH typeM andN, one can calculate a “lookup table” for given refractive indices n_sph, n_med and given sphere radius R. Then, the interaction of any initial field that has been decomposed into VSH with the sphere can easily be determined.

As a side note, we want to mention that this method scales readily to the case of several concentric shells with different refractive indices n₁, n₂, n₃, . . . and radii r₁, r₂, r₃, . . . and the environment n_med. Then, the conditions treated above have to be valid at each single interface, and one simply gets larger matrices.