Ellipsoids in the electrostatic approximation

Appendix G

Ellipsoids in the electrostatic approximation In this appendix, we provide a number of analytic expressions in relation with the problem of the ellipsoid in the electrostatic approximation (see Section 5.1.4). Particular attention is given to the special cases of spheroids (oblate and prolate) for which expressions in closed form can be obtained. These can be used in a fairly straightforward manner to implement numerically most calculations related to this problem. Expansions in terms of the aspect ratio are also given to study the limiting cases of a ‘flat’ oblate spheroid and a needle-like prolate spheroid. Finally, we also give a brief description of a possible Matlab implementation of these results in the context of SERS modeling. A discussion of some of these aspects, including additional references, can be found in electromagnetic textbooks, e.g. [95,149]. For a direct connection with SERS, see review articles like Refs. [4,218]. G.1. GENERAL CASE We do not give here the details of the derivation, which can for example be found in Ref. [149], but only the most important results, which are then used to discuss some of the key EM indicators of interest for SERS. G.1.1. Some definitions A general ellipsoid can be described, in the appropriate coordinate frame, by the equation: x2 y2 z2 + 2 + 2 = 1, 2 a b c

(G.1) 573

574

G. ELLIPSOIDS IN THE ELECTROSTATIC APPROXIMATION

Figure G.1. Schematic representation of special cases of ellipsoids: a sphere, an oblate spheroid and a prolate spheroid (both with an aspect ratio of a/c = 3).

where a ≥ b ≥ c (by convention) are the semi-axis lengths. Several special cases can be identified (see Fig. G.1): • If a = b = c, the ellipsoid has maximum symmetry and is a sphere of radius a. • If a = b > c, we have an ellipsoid of revolution (around the z-axis). It can be viewed as a sphere squashed along one direction (pumpkin-like), and it is called an oblate spheroid. • If a > b = c, we have again an ellipsoid of revolution (around the x-axis). It can be viewed as a sphere elongated along one direction (rugby-ball-like), and it is called a prolate spheroid 1 . • If a > b > c, the ellipsoid may be called a scalene ellipsoid. The two special cases of oblate and prolate spheroids are usually sufficient to approximate many particles of interest. They also yield simpler analytical expressions. They can moreover be conveniently characterized (up to a global scaling of dimensions) by a single quantity: their aspect ratio defined here as h = a/c. G.1.2. Ellipsoidal coordinates Unfortunately, in order to solve the Laplace equation for an ellipsoid using separation of variables, it is necessary to use ellipsoidal coordinates, which are not as ‘nice’ as spherical coordinates, and with which most people are not very familiar. There is an infinite family of ellipsoidal coordinates, each being the natural coordinate system of an ellipsoid with given aspect ratios (semi-axis ratios). The expressions are in fact different for oblate and prolate spheroids, and we therefore exclude this case for the moment and consider 1

It seems more appropriate from New Zealand to call this shape ‘rugby ball’ instead of ‘football’, as it would be surely called in the US.

G.1 GENERAL CASE

575

in the following: an ellipsoid with a > b > c (the spheroid cases can then be obtained a posteriori by taking the appropriate limit). The corresponding ellipsoidal coordinates are denoted (ξ, η, ζ) and are defined as the three solutions of the following equation (where u is the unknown): x2 y2 z2 + 2 + 2 = 1, +u b +u b +u

(G.2)

a2 with

−a2 < ζ < −b2 ,

−b2 < η < −c2 ,

−c2 < ξ < ∞.

(G.3)

The ellipsoid boundary equation (Eq. (G.1)) therefore corresponds simply to the surface ξ = 0. One can show that to a given point with Cartesian coordinates (x, y, z) corresponds one set of ellipsoidal coordinates (ξ, η, ζ). The opposite is not exactly true: to a given (ξ, η, ζ), corresponds a unique triplet (x2 , y 2 , z 2 ), and therefore 8 points related by symmetries around the axes. More specifically, we have:  2 2 2  x2 = (a + ξ)(a + η)(a + ζ)   2 2 2  (b − a )(c − a2 )   2 (b + ξ)(b2 + η)(b2 + ζ) y2 =  (a2 − b2 )(c2 − b2 )   2  (c + ξ)(c2 + η)(c2 + ζ)   z 2 = . (a2 − c2 )(b2 − c2 )

(G.4)

G.1.3. The electrostatic solution Solution for the potential

We consider an ellipsoid, with dielectric constant (ω), embedded in a dielectric medium with dielectric constant M , and placed in a constant and uniform external electric field E0 = E0 ez = −∇φ0 . Here the field polarization is chosen along one of the main axes; the general case can be obtained by superposition and will be discussed later. The electric potential solutions inside and outside are given in ellipsoidal coordinates as [149]: φin =

φ0 3M φ0 , −M = 3L3  + M (3 − 3L3 ) 1 + L3 M

(G.5)

576

G. ELLIPSOIDS IN THE ELECTROSTATIC APPROXIMATION

and φout = φ0 − φ0 3β3 F (c, ξ).

(G.6)

As we shall see, β3 is the non-dimensional polarizability (for excitation along the z-axis) and is analogous to βS for the sphere (Eq. (6.15)):

β3 (ω) =

(ω) − M . 3L3 (ω) + M (3 − 3L3 )

(G.7)

F (d, ξ) is an auxiliary function given by: Z F (d, ξ) = abc ξ

∞

dq , 2(d2 + q)f (q)

(G.8)

where f (q) =

p

(a2 + q)(b2 + q)(c2 + q).

(G.9)

L3 is a geometrical factor (discussed later) given as: Z L3 = F (c, 0) = abc 0

∞

dq . 2(c2 + q)f (q)

(G.10)

Field solution and polarizability The electric field can then be derived from the electric potential. The internal field is constant and aligned with the incident field, as was the case for a sphere: Ein =

3M E0 = (1 − 3β3 L3 )E0 , 3L3  + M (3 − 3L3 )

(G.11)

from which we derive a similar relation for the electric polarization inside the ellipsoid (see Section 6.2.1): PM = 30 M β3 E0 .

(G.12)

As for the sphere, the polarization is uniform across the volume (VE = (4/3)πabc)) of the ellipsoid, and equivalent to a dipole moment pM = α3 E0 ,

G.1 GENERAL CASE

577

where the dipolar polarizability (for excitation along the z-axis) is:

α3 = 30 M VE β3 = 4π0 M abc

 − M . 3L3  + M (3 − 3L3 )

(G.13)

The outside field is the sum of the external field E0 and the scattered field. After some manipulation, it can be expressed as: Eout = E0 + Esca = E0 (1 − 3β3 F (c, ξ)) ez + 3β3 E0

abc (eξ · ez ) eξ , f (ξ) (G.14)

where a mixture of ellipsoidal and Cartesian coordinates was used for simplicity. Expressed in this form, the similarities with the case of a perfect sphere (Section 6.2.3) are easily noted. The unit vector eξ is the normal to the ellipsoid surface for points on this surface (i.e. when ξ = 0), and can otherwise be expressed as the general expression: eξ = q



1 x2 (a2 +ξ)2

+

y2 (b2 +ξ)2

+

z2 (c2 +ξ)2

 x y z ex + 2 ey + 2 ez . a2 + ξ b +ξ c +ξ (G.15)

Effect of incident polarization

The previous expressions were obtained for an exciting field polarized along z. Because the ellipsoid is not fully symmetric like the sphere, the results should depend on the incident polarization. The previous treatment in fact remains valid for polarization along x or y, only replacing z by x or y, L3 by L1 = F (a, 0) or L2 = F (b, 0), and β3 (and α3 ) by the corresponding β1 (and α1 ) or β2 (and α2 ). The Li ’s are in general called geometrical factors or depolarization factors and are amongst the most important parameters for the optical properties of the ellipsoids. They can, in fact, be interpreted as depolarization factors [149], but only when M = 1 or when one considers the polarization PM with respect to the embedding medium. For further discussion of this aspect, see Ref. [149]. We have in addition the important properties: L1 + L2 + L3 = 1

and

0 ≤ L1 ≤ L2 ≤ L3 ≤ 1,

(G.16)

578

G. ELLIPSOIDS IN THE ELECTROSTATIC APPROXIMATION

where the latter inequalities arise from the convention a ≥ b ≥ c. Note that it is possible to obtain ‘simpler’ analytical expressions for the Li ’s in the special case of oblate or prolate spheroids as given in Sections G.2 and G.3. Since the Li ’s are in general distinct (except for the special case of the sphere where L1 = L2 = L3 = 1/3), the dipolar polarizabilities are also different depending on the incident field polarization. For a general incident polarization, the solution is simply the sum of the solutions for each of the three components along the main ellipsoid axes. If E0 = E0x ex + E0y ey + E0z ez , the induced dipole, for example, is: pM = α1 E0x ex + α2 E0y ey + α3 E0z ez .

(G.17)

The induced dipole is therefore not necessarily aligned with the incident polarization (except when it is aligned with one of the main axes of the ellipsoid). In the following discussion, we will focus again on the special case of incident polarization along one of the main axes, z here, but the results can easily be extended to the other two axes, or to a general polarization (by superposition). G.1.4. Some important EM indicators for ellipsoids Far-field properties Following the treatment of the sphere and using again the dipolar approximation, it is straightforward to obtain the absorption (approximately equal to extinction in this approximation) and scattering cross-sections of the ellipsoid in the ES approximation for incident polarization along one of the main axes: σExt ≈ σAbs = 4πkM abcIm(βi (ω)),

(G.18)

and σSca =

8π (kM )4 (abc)2 |βi (ω)|2 , 3

(G.19)

where kM = nM ω/c and i = 1, 2, 3 depending on the axis of the incident polarization. Local fields in the ES approximation Of particular interest to us is the electric field just outside the ellipsoid at the surface, i.e. at ξ = 0. Using F (c, 0) = L3 and f (0) = abc, we have: Eout (ξ = 0) = E0 [(1 − 3L3 β3 )ez + 3β3 (eξ · ez ) eξ ] .

(G.20)

G.1 GENERAL CASE

579

The similarity with the expression for the sphere (Eq. (6.18)) is again noted, and it can easily be recovered here by taking L3 = 1/3. We can now write, as for the sphere, a local field intensity enhancement factor (LFIEF), normal (⊥) and parallel (//) to the ellipsoid surface, at any point r on the surface (ξ = 0): |Eξ |2 2 = A⊥ 3 (ω) |eξ · ez | E02 2 A⊥ 3 (ω) = |1 + (3 − 3L3 )β3 (ω)| ;

(G.21)

 2 2 M k (r, ω) = |Eη | + |Eζ | = Ak (ω)[1 − |e · e |2 ] ξ z 3 Loc 2 E0  k where A3 (ω) = |1 − 3L3 β3 (ω)|2 .

(G.22)

 

⊥ MLoc (r, ω) =



where

and

We can deduce the LFIEF on the surface as: k

k

2

MLoc (r, ω) = A3 (ω) + (A⊥ 3 (ω) − A3 (ω)) |eξ · ez | ,

(G.23)

and the SERS EF derives from: 0 = (MLoc )2 . FE4

(G.24)

Note that the scalar product can for example be obtained as a function of the Cartesian coordinates of a point on the surface (i.e. for ξ = 0 only) as: 2

|eξ · ez | =

z2 . c4 (x2 /a4 + y 2 /b4 + z 2 /c4 )

(G.25)

Finally, we have the same useful expression as for the sphere: k

(M )2 A3 (ω) = . |(ω)|2 A⊥ 3 (ω) See Section 6.2.3 for a discussion of this relation.

(G.26)

580

G. ELLIPSOIDS IN THE ELECTROSTATIC APPROXIMATION

Average enhancement factors in the ES approximation The average enhancement factors cannot be written here in closed form, but 2 4 can be expressed in terms of the surface averages h|eξ · ez | i and h|eξ · ez | i. We have for example: 2

⊥ hMLoc (ω)i = A⊥ 3 (ω)h|eξ · ez | i, k

(G.27) k

2

hMLoc (ω)i = A3 (ω) + (A⊥ 3 (ω) − A3 (ω))h|eξ · ez | i,

(G.28)

and, h i2 h i k k k 2 0 hFE4 (ωL )i = A3 (ωL ) + 2A3 (ωL ) A⊥ 3 (ωL ) − A3 (ωL ) h|eξ · ez | i h i2 k 4 + A⊥ (ω ) − A (ω ) h|eξ · ez | i. (G.29) L L 3 3 2

4

The actual calculation of the surface averages h|eξ · ez | i and h|eξ · ez | i is not an easy task. In fact, in the case of a general ellipsoid, even the surface area cannot be expressed analytically in a simple form. It is however possible to calculate these averages analytically for the special cases of spheroids. The results are given for reference in Sections G.2 and G.3 along with a number of other analytic expressions relevant to spheroids. Most of these expressions are usually sufficiently complicated by themselves, and their utility could be questioned. They are, however, very useful for numerical calculations of the spheroid optical properties in the ES approximation and are provided mostly to this end.

Depolarization and radiative corrections for spheroids It was proposed [218,219] that the corrections to the polarizability of the sphere discussed in Section 6.2.1 could be generalized to the case of the ellipsoid and it was argued that such corrections agree well with exact results. However, as already pointed out in Section 6.2.1, these corrections are already inadequate for the sphere, and they are therefore unlikely to perform any better for the ellipsoid. These corrections are therefore no better a priori than the ESA itself and are in fact identical to it in the limit of small kM a. If they appear to agree with exact results, it is most likely because the ESA would also agree, and the additional complications of these corrections are then unnecessary. As for the sphere, more investigations are needed in this area to clarify the situation.

G.1 GENERAL CASE

581

G.1.5. Some aspects of the numerical implementation Geometrical factors The geometrical (or depolarization) factors can, by definition, be obtained from the integrals: L1 = F (a, 0) = L2 = F (b, 0) = L3 = F (c, 0) =

abc 2 abc 2 abc 2

∞

Z

0 Z ∞

∞

0

(G.30)

dq , + q)3/2 (c2 + q)1/2

(G.31)

dq , (a2 + q)1/2 (b2 + q)1/2 (c2 + q)3/2

(G.32)

(a2

0

Z

dq , (a2 + q)3/2 (b2 + q)1/2 (c2 + q)1/2 +

q)1/2 (b2

and from a ≥ b ≥ c, we recall that: L1 ≤ L2 ≤ L3

and L1 + L2 + L3 = 1.

(G.33)

One can also show that these factors can be expressed as surface integrals on the surface of the ellipsoid (S) as: L3 =

1 4π

Z Z S

z (eξ · ez )dS, r3

(G.34)

where r = |r|, and with equivalent relations for L1 and L2 .

Other important surface integrals Also of interest are the integrals defining the surface area S of the ellipsoid 2 4 and the surface averages h|eξ · ez | i and h|eξ · ez | i, and their counterparts for the x and y axes. We have: Z Z S=

dS,

(G.35)

S 2

h|eξ · ez | i =

1 S

Z Z

2

|eξ · ez | dS,

(G.36)

S

and similar expressions for the other averages. All these surface integrals (including the ones to calculate the Li ’s) can in principle be written as integrals on 2 angles, for example, by carrying out the

582

G. ELLIPSOIDS IN THE ELECTROSTATIC APPROXIMATION

following change of variables:  x = a sin θ cos φ y = b sin θ sin φ  z = c cos θ

0≤θ≤π 0 ≤ φ < 2π.

with

(G.37)

Note that in this context, θ and φ are not the usual angles of the spherical coordinates. The infinitesimal surface element on the surface of the ellipsoid then takes the form: √ dS = sin θ sdθdφ,

(G.38)

where s = (abc)

2



x2 y2 z2 + + a4 b4 c4



= b2 c2 sin2 θ cos2 φ + a2 c2 sin2 θ sin2 φ + a2 b2 cos2 θ.

(G.39)

Note that for a point on the surface (ξ = 0), we moreover have the simple relation: abc h x y z i eξ = √ e + e + ez . x y b2 c2 s a2

(G.40)

Example of numerical implementation in Matlab

We give below examples of some of these integrals using this change of variables. 2π

Z

π

Z

S=

sin θ 0

p

s(θ, φ)dθdφ,

(G.41)

0

Z

1 h|eξ · ez | i = S 2

2π

Z

0

π

a2 b2 sin θ cos2 θ p dθdφ, s(θ, φ)

0

(G.42)

and L3 =

1 4π

Z 0

2π

Z 0

π

(a2

2

sin

θ cos2

abc cos2 θ sin θ dθdφ. φ + b2 sin2 θ sin2 φ + c2 cos2 θ)3/2 (G.43)

G.2 OBLATE SPHEROID (PUMPKIN)

583

These (horrible) expressions can be rewritten using special functions called incomplete elliptic integrals, but are probably best left for numerical estimation. As an example, the surface area of the ellipsoid can be simply computed in Matlab with the following (assuming, a, b, and c have been defined): Sdiff = @(t,f) sin(t).*(b^2*c^2*sin(t).^2.*cos(f).^2+ ... a^2*c^2*sin(t).^2.*sin(f).^2+a^2*b^2*cos(t).^2).^(1/2); S = dblquad(Sdiff,0,pi,0,2*pi) The first command defines the function of θ (t) and φ (f ) to integrate and the second simply computes the integral. All the important properties of the ellipsoid (including the Li ’s) can be computed as surface integrals in this way. Many key EM indicators then derive easily. These are implemented in a few ready-to-use Matlab scripts that can be found on the book website (www.victoria.ac.nz/raman/book). Finally, in the special case of spheroids, these surface integrals can in fact be computed and closed-form analytical expressions can be obtained. These may be more convenient for some studies and can also be used to understand the limiting cases. Matlab scripts using these direct expressions are also provided. The relevant formulas for the two possible cases of spheroids, oblate and prolate, are listed for reference without further justification in the following two sections. G.2. OBLATE SPHEROID (PUMPKIN) We will go rapidly in this section through the most important expressions that can be derived for oblate spheroids (see Fig. G.1). G.2.1. Geometrical factors For an oblate spheroid (a = b > c), the ellipsoid has symmetry of revolution around the z-axis, its aspect ratio is h = a/c, and the eccentricity is by definition: eo = 1 − c2 /a2

(0 ≤ eo < 1).

(G.44)

We moreover have L1 = L2 < 1/3 < L3 and: # 1 − e2o arcsin(eo ) − (1 − e2o ) , eo " # p 1 − e2o 1 L3 = 1 − 2L1 = 2 1 − arcsin(eo ) . eo eo

1 L1 = L2 = 2 2eo

"p

This reduces to L1 = L2 = L3 = 1/3 for eo → 0 (sphere), as expected.

(G.45)

584

G. ELLIPSOIDS IN THE ELECTROSTATIC APPROXIMATION

G.2.2. Surface averages

The surface averages can be written conveniently using the auxiliary function fo defined as: fo =

1 − e2o ln 2eo



1 + eo 1 − eo

 .

(G.46)

The surface area is then: S = 2πa2 [1 + fo ] .

(G.47)

The surface averages involved in the calculations of local field enhancement factors are: 2

h|eξ · ez | i =

1 1 − fo · , e2o 1 + fo

(G.48)

from which we derive by symmetry: 1 2 2 2 h|eξ · ex | i = h|eξ · ey | i = (1 − h|eξ · ez | i) 2 1 (1 + e2o )fo − (1 − e2o ) = 2· . 2eo 1 + fo

(G.49)

Moreover, 4

h|eξ · ez | i =

1 3 − 2e2o − 3fo · , e4o 1 + fo

(G.50)

from which we derive by symmetry: 3 4 4 2 4 h|eξ · ex | i = h|eξ · ey | i = (1 − 2h|eξ · ez | i + h|eξ · ez | i) 8 3(1 − e2o ) 3 − e2o − (3 + e2o )fo · . = 8e4o 1 + fo

(G.51)

Finally, in the limit of eo → 0 (sphere), we have: 2 2 fo ≈ 1 − e2o − e4o + O(e6o ), 3 15

(G.52)

G.3 PROLATE SPHEROID (RUGBY BALL)

585 2

from which we recover the sphere results: h|eξ · ex,y,z | i = 1/3 and 4 h|eξ · ex,y,z | i = 1/5, as expected. G.2.3. Limit of large aspect ratio Also of interest is the case where eo → 1, i.e. a very flat, disk-like, pumpkin, with a large aspect ratio (h = a/c p  1). We can then expand the previous results in terms of 1/h = c/a = 1 − e2o → 0, and we obtain: c2 πc − 2 +O 4a a

L1 = L2 ≈



c3 a3

 and L3 ≈ 1.

(G.53)

Moreover fo → 0 and, more precisely: fo ≈

c2 ln a2



2a c



 +O

c4 ln a4



2a c

 ,

(G.54)

which implies: S ≈ 2πa2 ,

(G.55)

2

h|eξ · ez | i ≈ 1,

(G.56)        c2 2a 2a 1 c4 2 2 h|eξ · ex | i = h|eξ · ey | i ≈ 2 ln ln − +O ,(G.57) 4 a c 2 a c 4

h|eξ · ez | i ≈ 1,

(G.58)

and 3c2 h|eξ · ex | i = h|eξ · ey | i ≈ 2 + O 4a 4

4



c4 ln a4



2a c

 .

(G.59)

G.3. PROLATE SPHEROID (RUGBY BALL) We again go rapidly in this section through the most important expressions that can be derived for prolate spheroids (see Fig. G.1). G.3.1. Geometrical factors For a prolate spheroid (b = c < a), the ellipsoid has symmetry of revolution around the x-axis, its aspect ratio is h = a/b, and the eccentricity is by definition: ep = 1 − b2 /a2

(0 ≤ ep < 1).

(G.60)

586

G. ELLIPSOIDS IN THE ELECTROSTATIC APPROXIMATION

In addition, we have L1 < 1/3 < L2 = L3 and:    1 − e2p 1 1 + ep L1 = −1 + ln , e2p 2ep 1 − ep "  # 1 − e2p 1 1 + ep L3 = L2 = (1 − L1 )/2 = 2 1 − ln . 2ep 2ep 1 − ep

(G.61)

This reduces to L1 = L2 = L3 = 1/3 for ep → 0 (limit of a sphere), as expected.

G.3.2. Surface averages

The surface averages can be written conveniently using the auxiliary function fp defined as: with

1

fp = ep

q

1 − e2p

arcsin(ep ).

(G.62)

The surface area is then S = 2πb2 [1 + fp ] .

(G.63)

The surface averages involved in the calculations of local field enhancement factors are: 2

h|eξ · ex | i =

1 − e2p fp − 1 · , e2p 1 + fp

(G.64)

and 2

2

h|eξ · ey | i = h|eξ · ez | i =

1 1 − (1 − 2e2p )fp . · 2e2p 1 + fp

(G.65)

3 3 − 2e2p − (3 − 4e2 )fp · , 8e4p 1 + fp

(G.66)

Furthermore: 4

4

h|eξ · ey | i = h|eξ · ez | i =

G.3 PROLATE SPHEROID (RUGBY BALL)

587

and 8 4 2 4 h|eξ · ex | i = 1 − 4h|eξ · ez | i + h|eξ · ez | i 3 1 − e2p 3 − e2p − 3(1 − e2p )fp · = . e4p 1 + fp

(G.67)

Finally, in the limit of ep → 0 (sphere), we have: 2 8 fp ≈ 1 + e2p + e4p + O(e6p ), 3 15

(G.68)

from which we recover again the sphere results, as expected. G.3.3. Limit of large aspect ratio Also of interest is the case where ep → 1, i.e. a ‘very elongated rugby ball’, more like a Cuban cigar, with a large aspect ratio (h = a/b q  1). We can then expand the previous results in terms of 1/h = b/a = we obtain: b2 L1 ≈ 2 a



 ln

2a b



1 − e2p → 0, and

  4   b 2a −1 +O ln and L2 = L3 ≈ 1/2. (G.69) 4 a b

Moreover fp → ∞ and, more precisely: fp ≈

πa πb 2 b2 +O −1+ − 2b 4 a 3 a2



b3 a3

 ,

(G.70)

which implies: S ≈ π 2 ab,

(G.71)

1 2 2 h|eξ · ey | i = h|eξ · ez | i ≈ , 2  2 b2 b 2 h|eξ · ex | i ≈ 2 + O , a a2 4

4

h|eξ · ey | i = h|eξ · ez | i ≈

(G.72) (G.73)

3 , 8

(G.74)

and 4 b3 b4 h|eξ · ex | i ≈ − 3 +O π a3 a4 4



b4 a4

 .

(G.75)